Introductioncvgmt.sns.it/media/doc/paper/3726/20191029.pdf · 2019. 10. 29. · HEAT AND ENTROPY...

HEAT AND ENTROPY FLOWS IN CARNOT GROUPS

LUIGI AMBROSIO AND GIORGIO STEFANI

Abstract We prove the correspondence between the solutions of the sub-elliptic heatequation in a Carnot group G and the gradient flows of the relative entropy functionalin the Wasserstein space of probability measures on G Our result completely answers aquestion left open in a previous paper by N Juillet where the same correspondence wasproved for G = Hn the n-dimensional Heisenberg group

1 Introduction

Since the pioneering works [2335] and the monograph [4] in the last twenty years therehas been an increasing interest in the study of the relation between evolution equationsand gradient flows of energy functionals in a large variety of different frameworks see [57 13ndash16192122263033343840]

The prominent case in the literature is represented by the connection between the heatequation and the relative entropy functional It is well-known that the heat equation

983099983103

983101parttut = ∆ut in (0 +infin) times Rn

u0 = u isin L2(Rn) on 0 times Rn(HE)

can be seen as the gradient flow in L2(Rn) of the Dirichlet energy D(u) =983125Rn |nablau|2 dx

accordingly to the general approach introduced in [11] If the initial datum u isin L2(Rn)is such that micro0 = u L n isin P2(Rn) where

P2(Rn) =983069

micro isin P(Rn) 983133

Rn|x|2 dmicro(x) lt +infin

983070

then the solution (ut)tge0 of (HE) induces a curve (microt)tge0 sub P2(Rn) microt = ut L n If weendow the set P2(Rn) with its usual Wasserstein distance W then the curve (microt)tge0 islocally absolutely continuous with locally integrable squared W-derivative in [0 +infin) Onthe one hand since (ut)tge0 satisfies (HE) the curve (microt)tge0 naturally solves in the weaksense the continuity equation

983099983103

983101parttmicrot + div(vtmicrot) = 0 in (0 +infin) times Rn

micro0 = u L n on 0 times Rn(CE)

where the velocity vector field (vt)tge0 is given by vt = minusnablautut On the other hand therelative entropy

Ent(micro) =983133

Rnu log u dx for micro = u L n isin P2(Rn)

Date October 29 20192010 Mathematics Subject Classification 53C17 28A33 35K05Key words and phrases Carnot group entropy gradient flow sub-elliptic heat equation

1

HEAT AND ENTROPY FLOWS IN CARNOT GROUPS 2

computed along a curve (microt)tge0 sub P2(Rn) satisfying (CE) for a given velocity field (vt)tge0is such that

d

dtEnt(microt) = minus

983133

Rn(log ut + 1) div(vtut) dx =

983133

Rn

983071vt

nablaut

ut

983072dmicrot

In analogy with the Hilbertian case but using Ottorsquos calculus [35] in the interpretation ofthe right hand side one says that (microt)tge0 is a gradient flow of the entropy in (P2(Rn) W)if and only if the curve t 983041rarr Ent(microt) has maximal dissipation rate This happens if andonly if vt = minusnablautut ie when (ut)tge0 satisfies (HE)

Although not fully rigorous the argument presented above contains all the key toolsneeded to establish the correspondence between the heat flow and the entropy flow in ageneral metric measure space (X dm) Both the heat equation (HE) and the continuityequation (CE) have been adequately understood in this general context For (HE) onerelaxes the Dirichlet energy to the so-called Cheeger energy

Ch(u) = inf983069

lim infn

983133

X|Dun|2 dm un rarr u in L2(X dm) un isin Lip(X)

983070

where the local Lipschitz constant |Du|(x) = lim supyrarrx

|u(y)minusu(x)|d(xy) of u isin Lip(X) plays the

same role of the absolute value of the gradient in Rn It can be shown that the naturallyassociated Sobolev space W 12(X dm) is a Banach space (not Hilbertian in general) andthat the functional Ch is convex so that (HE) can still be interpreted as its gradientflow in the Hilbert space L2(X dm) see [5] For (CE) one introduces an appropriatespace S2(X) of test functions in W 12(X dm) and says that (microt)tgt0 satisfies the continuityequation with respect to a family of maps (Lt)tgt0 S2(X) rarr R if t 983041rarr

983125X f dmicrot is absolutely

continuous for every f isin S2(X) with ddt

983125X f dmicrot = Lt(f) for ae t gt 0 see [20]

The notion of gradient flow of the entropy functional

Entm(micro) =983133

X983172 log 983172 dm for micro = 983172m isin P2(X)

in the Wasserstein space (P2(X) W) can be rigorously defined by requiring the validityof the following sharp energy dissipation inequality

Entm(microt) + 12

983133 t

s|micror|2 dr + 1

2

983133 t

s|DminusEntm|2(micror) dr le Entm(micros)

for all 0 le s le t where t 983041rarr |microt| is the W-derivative of the curve (microt)tgt0 sub P2(X) and

|DminusEntm|(micro) = lim supνrarrmicro

max983083

Entm(micro) minus Entm(ν)W(micro ν) 0

983084

is the so-called descending slope of the entropy Note that this definition is consistentwith the standard one in Hilbert spaces since uprime(t) = minusnablaE(u(t)) is equivalent to

12 |uprime|2(t) + 1

2 |nablaE(u(t))|2 le minus d

dtE(u(t))

by combining the chain rule with CauchyndashSchwarz and Youngrsquos inequalitiesAs pointed out in [519] this abstract approach provides a complete equivalence between

the two gradient flows if the entropy is K-convex along geodesics in (P2(X) W) for some


K isin R that is if

Entm(microt) le (1 minus t)Entm(micro0) + t Entm(micro1) minus K

2 t(1 minus t)W(micro0 micro1)2 t isin [0 1] (K)

holds for a class of constant speed geodesics (microt)tisin[01] sub P2(X) sufficiently large to joinany pair of points in P2(X) The K-convexity (also known as displacement convexity) ofthe entropy heavily depends on the structure of (X dm) and encodes a precise informationabout the ambient space if X is a Riemannian manifold then (K) is valid if and onlyif the Ricci curvature satisfies Ric ge K see [37] For this reason if property (K) holdsthen (X dm) is called a space with generalized Ricci curvature bounded from below orsimply a CD(K infin) space

According to this general framework the correspondence between heat flow and en-tropy flow has been proved on Riemannian manifolds with Ricci curvature bounded frombelow see [14] and on compact Alexandrov spaces see [212233] Alexandrov spaces areconsidered as metric measure spaces with generalized sectional curvature bounded frombelow (a condition stronger than (K) see [36])

If (X dm) is not a CD(K infin) space then the picture is less clear As stated in [5 The-orem 85] the correspondence between heat flow and entropy flow still holds if the de-scending slope |DminusEntm| of the entropy is an upper gradient of Entm and satisfies a preciselower semicontinuity property basically equivalent to the equality between |DminusEntm| andthe so-called Fisher information These assumptions are weaker than (K) but not easyto check for a given non-CD(K infin) space

In [24 25] it was proved that the Heisenberg group Hn is a non-CD(K infin) space inwhich nevertheless the correspondence between heat flow and gradient flow holds TheHeisenberg group is the simplest non-commutative Carnot group Carnot groups are oneof the most studied examples of CarnotndashCaratheacuteodory spaces see [10 27 32] and thereferences therein for an account on this subject The proof of the correspondence ofthe two flows in Hn presented in [25] essentially splits into two parts The first partshows that a solution of the sub-elliptic heat equation parttut + ∆Hnut = 0 corresponds toa gradient flow of the entropy in the Wasserstein space (P2(Hn) WHn) induced by theCarnot-Caratheacuteodory distance dcc The direct computations needed are justified by someprecise estimates on the sub-elliptic heat kernel in Hn given in [28 29] The second partproves that a gradient flow of the entropy in (P2(Hn) WHn) induces a sub-elliptic heatdiffusion in Hn The argument is based on a clever regularization of the gradient flow(microt)tge0 based on the particular structure of the Lie algebra of Hn

An open question arisen in [25 Remark 53] was to extend the correspondence of thetwo flows to any Carnot group The aim of the present work is to give a positive answerto this problem To prove that a solution of the sub-elliptic heat equation correspondsto a gradient flow of the entropy we essentially follow the same strategy of [25] Sincethe results of [28 29] are not known for a general Carnot group we instead rely on theweaker estimates given in [39] valid in any nilpotent Lie group To show that a gradientflow of the entropy induces a sub-elliptic heat diffusion we regularize the gradient flow(microt)tge0 both in time and space via convolution with smooth kernels This regularizationdoes not depend on the structure of the Lie algebra of the group but nevertheless allowsus to preserve the key quantities involved such as the continuity equation and the Fisherinformation In the presentation of the proofs we also take advantage of a few results


taken from the general setting of metric measures spaces developed in [5] and in thereferences therein

The paper is organized as follows In Section 2 we collect the standard definitionsand well-known facts that are used throughout the work The precise statement of ourmain result is given in Theorem 24 at the end of this part In Section 3 we extendthe technical results presented in [25 Sections 3 and 4] to any Carnot group with minormodifications and we prove that Carnot groups are non-CD(K infin) spaces (see Propo-sition 36) generalizing the analogous result obtained in [24] Finally in Section 4 weprove the correspondence of the two flows

2 Preliminaries

21 AC curves entropy and gradient flows Let (X d) be a metric space let I sub Rbe a closed interval and let p isin [1 +infin] We say that a curve γ I rarr X belongs toACp(I (X d)) if

d(γs γt) le983133 t

sg(r) dr s t isin I s lt t (21)

for some g isin Lp(I) The space ACploc(I (X d)) is defined analogously The case p = 1

corresponds to absolutely continuous curves and is simply denoted by AC(I (X d)) Itturns out that if γ isin ACp(I (X d)) there is a minimal function g isin Lp(I) satisfying (21)called metric derivative of the curve γ which is given by

|γt| = limsrarrt

d(γs γt)|s minus t| for ae t isin I

See [4 Theorem 112] for the simple proof We call (X d) a geodesic metric space if forevery x y isin X there exists a curve γ [0 1] rarr X such that γ(0) = x γ(1) = y and

d(γs γt) = |s minus t| d(γ0 γ1) foralls t isin [0 1]Let Rlowast = R cup minusinfin +infin and let f X rarr Rlowast be a function We define the effective

domain of f asDom(f) = x isin X f(x) isin R

Given x isin Dom(f) we define the local Lipschitz constant of f at x by

|Df |(x) = lim supyrarrx

|f(y) minus f(x)|d(x y)

The descending slope and the ascending slope of f at x are respectively given by

|Dminusf |(x) = lim supyrarrx

[f(y) minus f(x)]minusd(x y) |D+f |(x) = lim sup

yrarrx

[f(y) minus f(x)]+d(x y)

Here a+ and aminus denote the positive and negative part of a isin R respectively Whenx isin Dom(f) is an isolated point of X we set |Df |(x) = |Dminusf |(x) = |D+f |(x) = 0 Byconvention we set |Df |(x) = |Dminusf |(x) = |D+f |(x) = +infin for all x isin X Dom(f)

Definition 21 (Gradient flow) Let E X rarr R cup +infin be a function We say thata curve γ isin ACloc([0 +infin) (X d)) is a (metric) gradient flow of E starting from γ0 isinDom(E) if the energy dissipation inequality (EDI)

E(γt) + 12

983133 t

s|γr|2 dr + 1

2

983133 t

s|DminusE|2(γr) dr le E(γs) (22)


holds for all s t ge 0 with s le t

Note that if (γt)tge0 is a gradient flow of E then γt isin Dom(E) for all t ge 0 andγ isin AC2

loc([0 +infin) (X d)) with t 983041rarr |DminusE|(γt) isin L2loc([0 +infin)) Moreover the function

t 983041rarr E(γt) is non-increasing on [0 +infin) and thus ae differentiable and locally integrable

Remark 22 As observed in [5 Section 25] if the function t 983041rarr E(γt) is locally absolutelycontinuous on (0 +infin) then (22) holds as an equality by the chain rule and Youngrsquosinequality In this case (22) is also equivalent to

d

dtE(γt) = minus|γt|2 = minus|DminusE|2(γt) for ae t gt 0

22 Wasserstein space We now briefly recall some properties of the Wasserstein spaceneeded for our purposes For a more detailed introduction to this topic we refer theinterested reader to [2 Section 3]

Let (X d) be a Polish space ie a complete and separable metric space We denote byP(X) the set of probability Borel measures on X The Wasserstein distance W betweenmicro ν isin P(X) is given by

W(micro ν)2 = inf983069983133

XtimesXd(x y)2 dπ π isin Γ(micro ν)

983070 (23)

whereΓ(micro ν) = π isin P(X times X) (p1)π = micro (p2)π = ν (24)

Here pi X times X rarr X i = 1 2 are the the canonical projections on the components Asusual if micro isin P(X) and T X rarr Y is a micro-measurable map with values in the topologicalspace Y the push-forward measure T(micro) isin P(Y ) is defined by T(micro)(B) = micro(T minus1(B))for every Borel set B sub Y The set Γ(micro ν) introduced in (24) is call the set of admissibleplans or couplings for the pair (micro ν) For any Polish space (X d) there exist optimalcouplings where the infimum in (23) is achieved

The function W is a distance on the so-called Wasserstein space (P2(X) W) where

P2(X) =983069

micro isin P(X) 983133

Xd(x x0)2 dmicro(x) lt +infin for some and thus any x0 isin X

983070

The space (P2(X) W) is Polish If (X d) is geodesic then (P2(X) W) is geodesic aswell Moreover micron

Wminusrarr micro if and only if micron micro and983133

Xd(x x0)2 dmicron(x) rarr

983133

Xd(x x0)2 dmicro(x) for some x0 isin X

As usual we write micron micro if983125

X ϕ dmicron rarr983125

X ϕ dmicro for all ϕ isin Cb(X)

23 Relative entropy Let (X dm) be a metric measure space where (X d) is a Polishmetric space and m is a non-negative Borel and σ-finite measure We assume that thespace (X dm) satisfies the following structural assumption there exist a point x0 isin Xand two constants c1 c2 gt 0 such that

m (x isin X d(x x0) lt r) le c1ec2r2

(25)


The relative entropy Entm P2(X) rarr (minusinfin +infin] is defined as

Entm(micro) =

983099983105983103

983105983101

983133

X983172 log 983172 dm if micro = 983172m isin P2(X)

+infin otherwise(26)

According to our definition micro isin Dom(Entm) implies that micro isin P2(X) and that theeffective domain Dom(Entm) is convex As pointed out in [5 Section 71] the structuralassumption (25) guarantees that in fact Ent(micro) gt minusinfin for all micro isin P2(X)

When m isin P(X) the entropy functional Entm naturally extends to P(X) is lowersemicontinuous with respect to the weak convergence in P(X) and positive by Jensenrsquosinequality In addition if F X rarr Y is a Borel map then

EntFm(Fmicro) le Entm(micro) for all micro isin P(X) (27)with equality if F is injective see [4 Lemma 945]

When m(X) = +infin if we set n = eminusc d(middotx0)2m for some x0 isin X where c gt 0 is chosen

so that n(X) lt +infin (note that the existence of such c gt 0 is guaranteed by (25)) thenwe obtain the useful formula

Entm(micro) = Entn(micro) minus c983133

Xd(x x0)2 dmicro for all micro isin P2(X) (28)

This shows that Entm is lower semicontinuous in (P2(X) W)

24 Carnot groups Let G be a Carnot group ie a connected simply connected andnilpotent Lie group whose Lie algebra g of left-invariant vector fields has dimension n andadmits a stratification of step κ

g = V1 oplus V2 oplus middot middot middot oplus Vκ

withVi = [V1 Viminus1] for i = 1 κ [V1 Vκ] = 0

We set mi = dim(Vi) and hi = m1 + middot middot middot + mi for i = 1 κ with h0 = 0 and hκ = nWe fix an adapted basis of g ie a basis X1 Xn such that

Xhiminus1+1 Xhiis a basis of Vi i = 1 κ

Using exponential coordinates we can identify G with Rn endowed with the group lawdetermined by the CampbellndashHausdorff formula (in particular the identity e isin G corre-sponds to 0 isin Rn and xminus1 = minusx for x isin G) It is not restrictive to assume that Xi(0) = ei

for any i = 1 n therefore by left-invariance for any x isin G we getXi(x) = dlxei i = 1 n (29)

where lx G rarr G is the left-translation by x isin G ie lx(y) = xy for any y isin G Weendow g with the left-invariant Riemannian metric 〈middot middot〉G that makes the basis X1 Xn

orthonormal For any i = 1 n we define the gradient with respect to the layer Vi as

nablaVif =

hi983131

j=himinus1+1(Xjf) Xj isin Vi

We let HG sub TG be the horizontal tangent bundle of the group G ie the left-invariantsub-bundle of the tangent bundle TG such that HeG = X(0) X isin V1 We use thedistinguished notation nablaG = nablaV1 for the horizontal gradient


For any i = 1 n we define the degree d(i) isin 1 κ of the basis vector field Xi

as d(i) = j if and only if Xi isin Vj With this notion the one-parameter family of groupdilations (δλ)λge0 G rarr G is given by

δλ(x) = δλ(x1 xn) = (λx1 λd(i)xi λκxn) for all x isin G (210)The Haar measure of the group G coincides with the n-dimensional Lebesgue mea-sure L n and has the homogeneity property L n(δλ(E)) = λQL n(E) where the integerQ = 983123κ

i=1 i dim(Vi) is the homogeneous dimension of the groupWe endow the group G with the canonical CarnotndashCaratheacuteodory structure induced

by HG We say that a Lipschitz curve γ [0 1] rarr G is a horizontal curve if γ(t) isin Hγ(t)Gfor ae t isin [0 1] The CarnotndashCaratheacuteodory distance between x y isin G is then defined as

dcc(x y) = inf983069983133 1

0983042γ(t)983042G dt γ is horizontal γ(0) = x γ(1) = y

983070

By ChowndashRashevskiirsquos Theorem the function dcc is in fact a distance which is alsoleft-invariant and homogeneous with respect to the dilations defined in (210) preciselydcc(zx zy) = dcc(x y) and dcc(δλ(x) δλ(y)) = λdcc(x y) for all x y z isin G and λ ge 0The resulting metric space (G dcc) is a Polish geodesic space We let BG(x r) be thedcc-ball centred at x isin G of radius r gt 0 Note that L n(BG(x r)) = cnrQ wherecn = L n(BG(0 1)) In particular the metric measure space (G dcc L n) satisfies thestructural assumption (25)

Let us write x = (x1 xκ) where xi = (xhiminus1+1 xhi) for i = 1 κ As proved

in [18 Theorem 51] there exist suitable constants c1 = 1 c2 ck isin (0 1) dependingonly on the group structure of G such that

dinfin(x 0) = max983153ci|xi|1i

Rmi i = 1 κ

983154 x isin G (211)

induces a left-invariant and homogeneous distance dinfin(x y) = dinfin(yminus1x 0) x y isin Gwhich is equivalent to dcc

Let 1 le p lt +infin and let Ω sub Rn be an open set The horizontal Sobolev spaceW 1p

G (Ω) = u isin Lp(Ω) Xiu isin Lp(Ω) i = 1 m1 (212)endowed with the norm

983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)

is a reflexive Banach space see [17 Proposition 112] By [17 Theorem 123] the setCinfin(Ω) cap W 1p

G (Ω) is dense in W 1pG (Ω) By a standard cut-off argument we get that

Cinfinc (Rn) is dense in W 1p

G (Rn)

25 Riemannian approximation The metric space (G dcc) can be seen as the limitin the pointed GromovndashHausdorff sense as ε rarr 0 of a family of Riemannian manifolds(Gε dε)εgt0 defined as follows see [12 Theorem 212] For any ε gt 0 we define theRiemannian approximation (Gε dε) of the Carnot group (G dcc) as the manifold Rn en-dowed with the Riemannian metric gε(middot middot) equiv 〈middot middot〉ε that makes orthonormal the vectorfields εd(i)minus1Xi i = 1 n ie such that

〈Xi Xj〉ε = ε2minusd(i)minusd(j)δij i j = 1 n


We let dε be the Riemannian distance induced by the metric gε Note that dε is left-invariant and satisfies dε le dcc for all ε gt 0 For any ε gt 0 the ε-Riemannian gradient isdefined as

nablaεf =n983131

i=1ε2(d(i)minus1)(Xif) Xi =

κ983131

i=1ε2(iminus1)nablaVi

f

By (29) we get thatgε(x) = (dlx)T Dε (dlx) x isin G

where Dε is the diagonal block matrix given by

Dε = diag(1m1 εminus21m2 εminus2(iminus1)1mi εminus2(κminus1)1mκ)

As a consequence the Riemannian volume element is given by

volε =983156

det gε dx1 and middot middot middot and dxn = εnminusQL n

We remark that for each ε gt 0 the n-dimensional Riemannian manifold (Gε dε) has Riccicurvature bounded from below More precisely there exists a constant K gt 0 dependingonly on the Carnot group G such that

Ricε ge minusKεminus2 for all ε gt 0 (213)

By scaling invariance the proof of inequality (213) can be reduced to the case ε = 1which in turn is a direct consequence of [31 Lemma 11]

In the sequel we will consider the metric measure space (Gε dε L n) ie the Riemann-ian manifold (Gε dε volε) with a rescaled volume measure Both these two spaces satisfythe structural assumption (25) Moreover we have

Entvolε(micro) = Ent(micro) + log(εQminusn) for all ε gt 0 (214)

Here and in the following Ent denotes the entropy with respect to the reference mea-sure L n

26 Sub-elliptic heat equation We let ∆G = 983123m1i=1 X2

i be the so-called sub-Laplacianoperator Since the horizontal vector fields X1 Xh1 satisfy Houmlrmanderrsquos conditionby Houmlrmanderrsquos theorem the sub-elliptic heat operator partt minus ∆G is hypoelliptic meaningthat its fundamental solution h (0 +infin) times G rarr (0 +infin) ht(x) = h(t x) the so-calledheat kernel is smooth In the following result we collect some properties of the heatkernel that will be used in the sequel We refer the reader to [39 Chapter IV] and to thereferences therein for the proof

Theorem 23 (Properties of the heat kernel) The heat kernel h (0 +infin)timesG rarr (0 +infin)satisfies the following properties

(i) ht(xminus1) = ht(x) for any (t x) isin (0 +infin) times G(ii) hλ2t(δλ(x)) = λminusQht(x) for any λ gt 0 and (t x) isin (0 +infin) times G

(iii)983125G ht dx = 1 for any t gt 0

(iv) there exists C gt 0 depending only on G such that

ht(x) le CtminusQ2 exp983075

minusdcc(x 0)2

4t

983076

forall(t x) isin (0 +infin) times G (215)


(v) for any ε gt 0 there exists Cε gt 0 such that

ht(x) ge CεtminusQ2 exp

983075

minusdcc(x 0)2

4(1 minus ε)t

983076


(vi) for every j l isin N and ε gt 0 there exists Cε(j l) gt 0 such that

|(partt)lXi1 middot middot middot Xijht(x)| le Cε(j l)tminus Q+j+2l

2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076

forall(t x) isin (0 +infin) times G

(217)where Xi1 middot middot middot Xij

isin V1

Given 983172 isin L1(G) the function

983172t(x) = (983172 983183 ht)(x) =983133

Ght(yminus1x) 983172(y) dy (t x) isin (0 +infin) times G (218)

is smooth and is a solution of the heat diffusion problem983099983103

983101partt983172t = ∆G983172t in (0 +infin) times G

9831720 = 983172 on 0 times G(219)

The initial datum is assumed in the L1-sense ie limtrarr0

983172t = 983172 in L1(G) As a consequence ofthe properties of the heat kernel if 983172 ge 0 then the solution (983172t)tge0 in (218) is everywherepositive and satisfies 983133

G983172t(x) dx = 983042983172983042L1(G) forallt gt 0

In addition if 983172L n isin P2(G) then (983172tL n)tge0 sub P2(X) Indeed by (215) we have

Ct =983133

Gdcc(x 0)2 ht(x) dx lt +infin forallt gt 0

Thus by triangular inequality we have

(dcc(middot 0)2 983183 ht)(x) =983133

Gdcc(xyminus1 0)2 ht(y) dy le 2 dcc(x 0)2 + 2Ct

so that for all t gt 0 we get983133

Gdcc(x 0)2 983172t(x) dx =

983133

G(dcc(middot 0)2983183ht)(x) 983172(x) dx le 2

983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)

27 Main result We are now ready to state the main result of the paper The proof isgiven in Section 4 and deals with the two parts of the statement separately

Theorem 24 Let (G dcc L n) be a Carnot group and let 9831720 isin L1(G) be such thatmicro0 = 9831720L n isin Dom(Ent) If (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t

with initial datum 9831720 then microt = 983172tL n is a gradient flow of Ent in (P2(G) WG) startingfrom micro0

Conversely if (microt)tge0 is a gradient flow of Ent in (P2(G) WG) starting from micro0 thenmicrot = 983172tL n for all t ge 0 and (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t withinitial datum 9831720


3 Continuity equation and slope of the entropy

31 The Wasserstein space on the approximating Riemannian manifold Let(P2(Gε) Wε) be the Wasserstein space introduced in Section 22 relative to the metricmeasure space (Gε dε L n) As observed in Section 25 (Gε dε L n) is an n-dimensionalRiemannian manifold (with rescaled volume measure) whose Ricci curvature is boundedfrom below Here we collect some known results taken from [1440] concerning the space(P2(Gε) Wε) In the original statements the canonical reference measure is the Rie-mannian volume Keeping in mind that volε = εnminusQL n and the relation (214) in ourstatements each quantity is rescaled accordingly All time-dependent vector fields appear-ing in the sequel are tacitly understood to be Borel measurable

Let micro isin P2(Gε) be given We define the space

L2ε(micro) =

983069ξ isin S (TGε)

983133

Gε

983042ξ9830422ε dmicro lt +infin

983070

Here S (TGε) denotes the set of sections of the tangent bundle TGε Moreover we definethe lsquotangent spacersquo of (P2(Gε) Wε) at micro as

Tanε(micro) = nablaεϕ ϕ isin Cinfinc (Rn)L2

ε(micro)

The lsquotangent spacersquo Tanε(micro) was first introduced in [35] We refer the reader to [4Chapter 12] and to [40 Chapters 13 and 15] for a detailed discussion on this space

Let ε gt 0 be fixed Given I sub R an open interval and a time-dependent vector fieldvε I times Gε rarr TGε (t x) 983041rarr vε

t (x) isin TxGε we say that a curve (microt)tisinI sub P2(Gε) satisfiesthe continuity equation

parttmicrot + div(vεt microt) = 0 in I times Gε (31)

in the sense of distributions if983133

I

983133

Gε

983042vεt (x)983042ε dmicrot(x) dt lt +infin

and 983133

I

983133

Gε

parttϕ(t x) + 〈vεt (x) nablaεϕ(t x)〉ε dmicrot(x) dt = 0 forallϕ isin Cinfin

c (I times Rn)

We can thus state the following result see [14 Proposition 25] for the proof Here andin the sequel the metric derivative in the Wasserstein space (P2(Gε) Wε) of a curve(microt)tisinI sub P2(Gε) is denoted by |microt|ε

Proposition 31 (Continuity equation in (P2(Gε) Wε)) Let ε gt 0 be fixed and let I sub Rbe an open interval If (microt)t isin AC2

loc(I P2(Gε)) then there exists a time-dependent vectorfield vε I times Gε rarr TGε with t 983041rarr 983042vε

t 983042L2ε(microt) isin L2

loc(I) such thatvε

t isin Tanε(microt) for ae t isin I (32)and the continuity equation (31) holds in the sense of distributions The vector field vε

t

is uniquely determined in L2ε(microt) by (31) and (32) for ae t isin I and we have

983042vεt 983042L2

ε(microt) = |microt|ε for ae t isin I

Conversely if (microt)tisinI sub P2(Gε) is a curve satisfying (31) for some (vεt )tisinI such that

t 983041rarr 983042vεt 983042L2

ε(microt) isin L2loc(I) then (microt)t isin AC2

loc(I (P2(Gε) Wε)) with|microt|ε le 983042vε

t 983042L2ε(microt) for ae t isin I


We can interpret the time-dependent vector field (vεt )tisinI given by Proposition 31 as the

lsquotangent vectorrsquo of the curve (microt)tisinI in (P2(Gε) Wε) As remarked in [14 Section 2] forae t isin I the vector field vε

t has minimal L2ε(microt)-norm among all time-dependent vector

fields satisfying (31) Moreover this minimality is equivalent to (32)In the following result and in the sequel |Dminus

ε Ent|(micro) denotes the descending slope ofthe entropy Ent at the point micro isin P2(Gε) in the Wasserstein space (P2(Gε) Wε)

Proposition 32 Let ε gt 0 be fixed and let micro = 983172L n isin P2(Gε) The following state-ments are equivalent

(i) |Dminusε Ent|(micro) lt +infin

(ii) 983172 isin W 11loc (Gε) and nablaε983172 = wε983172 for some wε isin L2

ε(micro)In this case wε isin Tanε(micro) and |Dminus

ε Ent|(micro) = 983042wε983042L2ε(micro) Moreover for any ν isin P2(Gε)

we haveEnt(ν) ge Ent(micro) minus 983042wε983042L2

ε(micro) Wε(ν micro) minus K2ε2 Wε(ν micro)2 (33)

where K gt 0 is the constant appearing in (213)

The equivalence part in Proposition 32 is proved in [14 Proposition 43] Inequal-ity (33) is the so-called HWI inequality and follows from [40 Theorem 2314] see [40 Re-mark 2316]

The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n

appearing in Proposition 32 is the so-called Fisher information of micro = 983172L n isin P2(Gε)The inequality Fε(983172) le |Dminus

ε Ent|(micro) holds in the context of metric measure spaces see [5Theorem 74] The converse inequality does not hold in such a generality and heavilydepends on the lower semicontinuity of the descending slope |Dminus

ε Ent| see [5 Theorem 76]

32 The Wasserstein space on the Carnot group Let (P2(G) WG) be the Wasser-stein space introduced in Section 22 relative to the metric measure space (G dcc L n)In this section we discuss the counterparts of Proposition 31 and Proposition 32 in thespace (P2(G) WG) All time-dependent vector fields appearing in the sequel are tacitlyunderstood to be Borel measurable

Let micro isin P2(G) be given We define the space

L2G(micro) =

983069ξ isin S (HG)

983133

G983042ξ9830422

G dmicro lt +infin983070

Here S (HG) denotes the set of sections of the horizontal tangent bundle HG Moreoverwe define the lsquotangent spacersquo of (P2(G) WG) at micro as

TanG(micro) = nablaGϕ ϕ isin Cinfinc (Rn)L2

G(micro)

Given I sub R an open interval and a horizontal time-dependent vector field vG I timesG rarrHG (t x) 983041rarr vG

t (x) isin HxG we say that a curve (microt)tisinI sub P2(G) satisfies the continuityequation

parttmicrot + div(vGt microt) = 0 in I times Gε (34)


I

983133

G983042vG

t (x)983042G dmicrot(x) dt lt +infin


and 983133

I

983133

Gparttϕ(t x) +

983111vG

t (x) nablaGϕ(t x)983112

Gdmicrot(x) dt = 0 forallϕ isin Cinfin

c (I times Rn)

The following result is the exact analogue of Proposition 31 Here and in the sequel themetric derivative in the Wasserstein space (P2(G) WG) of a curve (microt)tisinI sub P2(G) isdenoted by |microt|G

Proposition 33 (Continuity equation in (P2(G) WG)) Let I sub R be an open intervalIf (microt)t isin AC2

loc(I (P2(G) WG)) then there exists a horizontal time-dependent vectorfield vG I times G rarr HG with t 983041rarr 983042vG

t 983042L2G(microt) isin L2

loc(I) such that

vGt isin TanG(microt) for ae t isin I (35)

and the continuity equation (34) holds in the sense of distributions The vector field vGt

is uniquely determined in L2G(microt) by (34) and (35) for ae t isin I and we have

983042vGt 983042L2

G(microt) = |microt|G for ae t isin I

Conversely if (microt)tisinI sub P2(G) is a curve satisfying (34) for some (vGt )tisinI such that

t 983041rarr 983042vGt 983042L2

G(microt) isin L2loc(I) then (microt)t isin AC2

loc(I (P2(G) WG)) with

|microt|G le 983042vGt 983042L2

G(microt) for ae t isin I

As for Proposition 31 we can interpret the horizontal time-dependent vector field (vGt )tisinI

given by Proposition 33 as the lsquotangent vectorrsquo of the curve (microt)tisinI in (P2(G) WG) Aneasy adaptation of [14 Lemma 24] to the sub-Riemannian manifold (G dcc L n) againshows that for ae t isin I the vector field vG

t has minimal L2G(microt)-norm among all time-

dependent vector fields satisfying (34) and moreover that this minimality is equivalentto (35)

Proposition 33 can be obtained applying the general results obtained in [20] to themetric measure space (G dcc L n) Below we give a direct proof exploiting Proposition 31The argument is very similar to the one of [25 Proposition 31] and we only sketch itProof If (microt)t isin AC2

loc(I (P2(G) WG)) then also (microt)t isin AC2loc(I (P2(Gε) Wε)) for

every ε gt 0 since dε le dcc Let vε I times Gε rarr TGε be the time-dependent vector fieldgiven by Proposition 31 Note that

983133

G983042vε

t 9830422ε dmicrot = |microt|2ε le |microt|2G for ae t isin I (36)

Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131

i=2ε2(1minusi)983042vεVi

t 98304221 for all ε gt 0 (37)

where vεVit denotes the projection of vε

t on Vi Combining (36) and (37) we find asequence (εk)kisinN with εk rarr 0 and a horizontal time-dependent vector field vG I timesG rarrHG such that vεkV1 vG and vεkVi rarr 0 for all i = 2 κ as k rarr +infin locally in timein the L2-norm on I times G naturally induced by the norm 983042 middot 9830421 and the measure dmicrotdt Inparticular t 983041rarr 983042vG


loc(I) and 983042vGt 983042L2

G(microt) le |microt|G for ae t isin I To prove (34)fix a test function ϕ isin Cinfin

c (I times Rn) and pass to the limit as ε rarr 0+ in (31)Conversely if (microt)tisinI sub P2(G) satisfies (34) for some horizontal time-dependent vector

field (vGt )tisinI such that t 983041rarr 983042vG


loc(I) then we can apply Proposition 31 for


ε = 1 By the superposition principle stated in [9 Theorem 58] applied to the Riemannianmanifold (G1 d1 L n) we find a probability measure ν isin P(C(I (G1 d1))) concentratedon AC2

loc(I (G1 d1)) such that microt = (et)ν for all t isin I and with the property that ν-aecurve γ isin C(I (G1 d1)) is an absolutely continuous integral curve of the vector field vGHere et C(I (G1 d1)) rarr G denotes the evaluation map at time t isin I Since vG ishorizontal ν-ae curve γ isin C(I (G1 d1)) is horizontal Therefore for all s t isin I s lt twe have

dcc(γ(t) γ(s)) le983133 t

s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr

and we can thus estimate

W2G(microt micros) le

983133

GtimesGd2

cc(x y) d(et es)ν(x y) =983133

AC2loc

d2cc(γ(t) γ(s)) dν(γ)

le (t minus s)983133

AC2loc

983133 t

s983042vG

r (γ(r))9830422G drdν(γ) = (t minus s)

983133 t

s

983133

G983042vG

r 9830422G dmicrordr

This immediately gives |microt|G le 983042vGt 983042L2

G(microt) for ae t isin I which in turn proves (35)

To establish an analogue of Proposition 32 we need to prove the two inequalitiesseparately For micro = 983172L n isin P2(G) the inequality FG(983172) le |Dminus

GEnt|(micro) is stated inProposition 34 below Here and in the sequel |Dminus

GEnt|(micro) denotes the descending slopeof the entropy Ent at the point micro isin P2(G) in the Wasserstein space (P2(G) WG)

Proposition 34 Let micro = 983172L n isin P2(G) If |DminusGEnt|(micro) lt +infin then 983172 isin W 11

G loc(G) andnablaG983172 = wG983172 for some horizontal vector field wG isin L2(micro) with 983042wG983042L2

G(micro) le |DminusGEnt|(micro)

Proposition 34 can be obtained by applying [5 Theorem 74] to the metric measure space(G dcc L n) Below we give a direct proof of this result which is closer in the spirit to theone in the Riemannian setting see [14 Lemma 42] See also [25 Proposition 31]

Proof Let V isin Cinfinc (G HG) be a smooth horizontal vector field with compact support

Then there exists δ gt 0 such that for any t isin (minusδ δ) the flow map of the vector field Vat time t namely

Ft(x) = expx(tV ) x isin G

is a diffeomorphism and Jt = det(DFt) is such that cminus1 le Jt le c for some c ge 1 Bythe change of variable formula the measure microt = (Ft)micro is such that microt = 983172tL n withJt983172t = 983172 F minus1

t for t isin (minusδ δ) Let us set H(r) = r log r for r ge 0 Then for t isin (minusδ δ)

Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt

983062Jt dx = Ent(micro) minus

983133

G983172 log(Jt) dx lt +infin

Note that J0 = 1 J0 = div V and that t 983041rarr JtJminus1t is uniformly bounded for t isin (minusδ δ)

Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx

On the other hand we have

W2G(microt micro) = W2

G((Ft)micro micro) le983133

Gd2

cc(Ft(x) x) dmicro(x)


and solim sup

trarr0

W2G(microt micro)|t|2 le

983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0

[Ent(microt) minus Ent(micro)]minusWG(microt micro) middot WG(microt micro)

|t| le |DminusGEnt|(micro)

983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx

983055983055983055983055 le |DminusGEnt|(micro)

983061983133

G983042V 9830422

G dmicro983062 1

2

By Riesz representation theorem we conclude that there exists a horizontal vector fieldwG isin L2

G(micro) such that 983042wG983042L2G(micro) le |Dminus

GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112

Gdmicro for all V isin Cinfin

c (G HG)

This implies that nablaG983172 = wG983172 and the proof is complete We call the quantity

FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n

appearing in Proposition 34 the horizontal Fisher information of micro = 983172L n isin P2(G)On its effective domain FG is convex and sequentially lower semicontinuous with respectto the weak topology of L1(G) see [5 Lemma 410]

Given micro = 983172L n isin P2(G) it is not clear how to prove the inequality |DminusGEnt|2(micro) le

FG(983172) under the mere condition |DminusGEnt|(micro) lt +infin Following [25 Proposition 34] in

Proposition 35 below we show that the condition |Dminusε Ent|(micro) lt +infin for some ε gt 0 (and

thus any) implies that |DminusGEnt|2(micro) le FG(983172)

Proposition 35 Let micro = 983172L n isin P2(G) If |Dminusε Ent|(micro) lt +infin for some ε gt 0 then

also |DminusGEnt|(micro) lt +infin and moreover FG(983172) = |Dminus

GEnt|2(micro)

Proof Since |Dminusε Ent|(micro) lt +infin we have Ent(micro) lt +infin Since dε le dcc and so Wε le

WG we also have |DminusGEnt|(micro) le |Dminus

ε Ent|(micro) By Proposition 34 we conclude that983172 isin W 11

G loc(G) and nablaG983172 = wG983172 for some horizontal vector field wG isin L2G(micro) with

983042wG983042L2G(micro) le |Dminus

GEnt|(micro) We now prove the converse inequality Since |Dminusε Ent|(micro) lt +infin

by Proposition 32 we have Fε(983172) = |Dminusε Ent|2(micro) and

Ent(ν) ge Ent(micro) minus F12ε (983172) Wε(ν micro) minus K

2ε2 W2ε(ν micro) (38)

for any ν isin P2(G) Take ε = WG(ν micro)14 and assume ε lt 1 Since Wε le WG from (38)we get

Ent(ν) ge Ent(micro) minus F12ε (983172) WG(ν micro) minus K

2ε2 W2G(ν micro)

= Ent(micro) minus F12ε (983172) WG(ν micro) minus K

2 W32G (ν micro) (39)

We need to bound Fε(983172) from above in terms of FG(983172) To do so observe that

nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131

i=2ε2(iminus1)983042nablaVi

9831729830422G


In particular nablaVi983172

983172isin L2

G(micro) for all i = 2 k Recalling the inequality (1+r) le9830591 + r

2

9830602

for r ge 0 we can estimate

Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)

Inserting (310) into (39) we finally get

Ent(ν) ge Ent(micro) minus F12G (983172) WG(ν micro) minus C W32

G (ν micro)

for some C gt 0 independent of ε This immediately leads to |DminusGEnt|(micro) le F12

G (983172)

33 Carnot groups are non-CD(K infin) spaces As stated in [5 Theorem 76] if themetric measure space (X dm) is Polish and satisfies (25) then the properties

(i) |DminusEnt|2(micro) = F(983172) for all micro = 983172m isin Dom(Ent)(ii) |DminusEnt| is sequentially lower semicontinuous with respect to convergence with mo-

ments in P(X) on sublevels of Entare equivalent We do not know if property (ii) is true for the space (G dcc L n) and thisis why in Proposition 35 we needed the additional assumption |Dminus

ε Ent|(micro) lt +infinBy [5 Theorem 93] property (ii) holds true if (X dm) is CD(K infin) for some K isin R

As the following result shows (non-commutative) Carnot groups are not CD(K infin) sothat the validity of property (ii) in these metric measure spaces is an open problem Notethat Proposition 36 below was already known for the Heisenberg groups see [24]

Proposition 36 If (G dcc L n) is a non-commutative Carnot group then the metricmeasure space (G dcc L n) is not CD(K infin) for any K isin R

Proof By contradiction assume that (G dcc L n) is a CD(K infin) space for some K isin RSince the DirichletndashCheeger energy associate to the horizontal gradient is quadratic onL2(G L n) (see [6 Section 43] for a definition) by [3 Theorem 61] we deduce that(G dcc L n) is a (σ-finite) RCD(K infin) space By [3 Theorem 72] we deduce that(G dcc L n) satisfies the BE(K infin) property that is

983042nablaG(Ptf)9830422G le eminus2KtPt(983042nablaf9830422

G) for all t ge 0 f isin Cinfinc (Rn) (311)

Here and in the rest of the proof we set Ptf = f 983183 ht for short Arguing similarly asin the proof of [41 Theorem 11] it is possible to prove that (311) is equivalent to thefollowing reverse Poincareacute inequality

Pt(f 2) minus (Ptf)2 ge 2I2K(t) 983042nabla(Ptf)9830422G for all t ge 0 f isin Cinfin

c (Rn) (312)

where IK(t) = eKtminus1K

if K ∕= 0 and I0(t) = t Now by [8 Propositions 25 and 26] thereexists a constant Λ isin

983147Q

2m1 Q

m1

983148(where Q and m1 are as in Section 24) such that the


inequality

Pt(f 2) minus (Ptf)2 ge t

Λ 983042nabla(Ptf)9830422G for all t ge 0 f isin Cinfin

c (Rn) (313)

holds true and moreover is sharp Comparing (312) and (313) we thus must have thatΛ le t

2I2K(t) for all t gt 0 Passing to the limit as t rarr 0+ we get that Λ le 12 so that

Q le m1 This immediately implies that G is commutative a contradiction

4 Proof of the main result

41 Heat diffusions are gradient flows of the entropy In this section we prove thefirst part of Theorem 24 The argument follows the strategy outlined in [25 Section 41]

The following technical lemma will be applied to horizontal vector fields in the proof ofProposition 42 below The proof is exactly the same of [25 Lemma 41] and we omit it

Lemma 41 Let V Rn rarr Rn be a vector field with locally Lipschitz coefficients suchthat |V |Rn isin L1(Rn) and div V isin L1(Rn) Then

983125Rn div V dx = 0

Proposition 42 below states that the function t 983041rarr Ent(983172tL n) is locally absolutelycontinuous if (983172t)tge0 solves the sub-elliptic heat equation (219) with initial datum 9831720 isinL1(G) such that micro0 = 9831720L n isin P2(G) By [5 Proposition 422] this result is true underthe stronger assumption that 9831720 isin L1(G) cap L2(G) Here the point is to remove the L2-integrability condition on the initial datum exploiting the estimates on the heat kernelcollected in Theorem 23 see also [25 Section 411]

Proposition 42 (Entropy dissipation) Let 9831720 isin L1(G) be such that micro0 = 9831720L n isinDom(Ent) If (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t with initial da-tum 9831720 then the map t 983041rarr Ent(microt) microt = 983172tL n is locally absolutely continuous on (0 +infin)and it holds

d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dx for ae t gt 0 (41)

Proof Note that microt isin P2(G) for all t gt 0 by (220) Hence Ent(microt) gt minusinfin for all t gt 0Since Ct = supxisinG ht(x) lt +infin for each fixed t gt 0 by (215) we get that 983172t le Ct for allt gt 0 Thus Ent(microt) lt +infin for all t gt 0

For each m isin N define

zm(r) = minm max1 + log r minusm Hm(r) =983133 r

0zm(s) ds r ge 0 (42)

Note that Hm is of class C1 on [0 +infin) with H primem is globally Lipschitz and bounded We

claim thatd

dt

983133

GHm(983172t) dx =

983133

Gzm(983172t)∆G983172t dx forallt gt 0 forallm isin N (43)

Indeed we have |Hm(983172t)| le m983172t isin L1(G) and given [a b] sub (0 +infin) by (217) thefunction x 983041rarr suptisin[ab] |∆Ght(x)| is bounded Thus

suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]

(9831720 983183 |∆Ght|) le m 9831720 983183 suptisin[ab]

|∆Ght| isin L1(G)


Therefore (43) follows by differentiation under integral sign We now claim that983133

Gzm(983172t)∆G983172t dx = minus

983133

eminusmminus1lt983172tltemminus1

983042nablaG983172t9830422G

983172t

dx forallt gt 0 forallm isin N (44)

Indeed by CauchyndashSchwarz inequality we have

983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G

9831569831720(xyminus1) 983042nablaGht(y)983042G983156

ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133

G9831720(xyminus1) 983042nablaGht(y)9830422

Ght(y) dy

983076 983061983133

G9831720(xyminus1) ht(y) dy

983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076

(x) for all x isin G

(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183

983042nablaGht9830422G

ht

dx =983133

G


ht

dx lt +infin (46)

This together with (43) proves that

div(zm(983172t)nablaG983172t) = zprimem(983172t)983042nablaG983172t9830422

G minus zm(983172t)∆G983172t isin L1(G) (47)

Thus (44) follows by integration by parts provided that983133

Gdiv(zm(983172t)nablaG983172t) dx = 0 (48)

To prove (48) we apply Lemma 41 to the vector field V = zm(983172t)nablaG983172t By (47) wealready know that div V isin L1(G) so we just need to prove that |V |Rn isin L1(G) Notethat 983133

G|V |Rn dx le m

983133

G9831720 983183 |nablaGht|Rn dx = m

983133

G|nablaGht|Rn dx

so it is enough to prove that |nablaGht|Rn isin L1(G) But we have

|nablaGht(x)|Rn le p(x1 xn)983042nablaGht(x)983042G x isin G

where p Rn rarr [0 +infin) is a function with polynomial growth because the horizontalvector fields X1 Xh1 have polynomial coefficients Since dcc is equivalent to dinfinwhere dinfin was introduced in (211) by (217) we conclude that |nablaGht|Rn isin L1(G) Thiscompletes the proof of (48)

Combining (43) and (44) we thus getd

dt

983133

GHm(983172t) dx = minus

983133


983042nablaG983172t9830422G

983172t

dx forallt gt 0 forallm isin N

Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t

dx isin L1loc(0 +infin) (49)


Indeed from (216) and (217) we deduce that

t 983041rarr983133

G


ht

dx isin L1loc(0 +infin)

Recalling (45) and (46) this immediately implies (49) Therefore983133

GHm(983172t1) dx minus

983133

GHm(983172t0) dx = minus

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt

for any t0 t1 isin (0 +infin) with t0 lt t1 and m isin N We now pass to the limit as m rarr +infinNote that Hm(r) rarr r log r as m rarr +infin and that for all m isin N

r log r le Hm+1(r) le Hm(r) for r isin [0 1] (410)and

0 le Hm(r) le 1 + r log r for r isin [1 +infin) (411)Thus

limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx

by the monotone convergence theorem on 983172t le 1 and by the dominated convergencetheorem on 983172t gt 1 Moreover

limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt

by the monotone convergence theorem This concludes the proof We are now ready to prove the first part of Theorem 24 The argument follows the

strategy outlined in [25 Section 41] See also the first part of the proof of [5 Theorem 85]

Theorem 43 Let 9831720 isin L1(G) be such that micro0 = 9831720L n isin Dom(Ent) If (983172t)tge0 solves thesub-elliptic heat equation partt983172t = ∆G983172t with initial datum 9831720 then microt = 983172tL n is a gradientflow of Ent in (P2(G) WG) starting from micro0

Proof Note that (microt)tgt0 sub P2(G) see the proof of Proposition 42 Moreover (microt)tgt0satisfies (34) with vG

t = nablaG983172t983172t for t gt 0 Note that t 983041rarr 983042vGt 983042L2

G(microt) isin L2loc(0 +infin)

by (216) (217) (45) and (46) By Proposition 33 we conclude that

|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t


By Proposition 42 the map t 983041rarr Ent(microt) is locally absolutely continuous on (0 +infin) andso by the chain rule we get

minus d

dtEnt(microt) le |Dminus

GEnt|(microt) middot |microt|G for ae t gt 0 (413)

Thus if we prove that

|DminusGEnt|2(microt) =

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t


then combining this equality with (41) (412) and (413) we find that

|microt|G = |DminusGEnt|(microt)

d

dtEnt(microt) = minus|Dminus

GEnt|(microt) middot |microt|G for ae t gt 0


so that (microt)tge0 is a gradient flow of Ent starting from micro0 as observed in Remark 22We now prove (414) To do so we apply Proposition 35 We need to check that

|Dminusε Ent|(microt) lt +infin for some ε gt 0 To prove this we apply Proposition 32 Since

dε le dcc we have microt isin P2(Gε) for all ε gt 0 Moreover

Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx

Since FG(983172t) lt +infin we just need to prove that983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin

for all i = 2 κ Indeed arguing as in (45) by CauchyndashSchwarz inequality we have

983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht

Therefore by (216) and (217) we get983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183

983042nablaViht9830422

Ght

dx =983133

G


Ght

dx lt +infin

This concludes the proof

42 Gradient flows of the entropy are heat diffusions In this section we provethe second part of Theorem 24 Our argument is different from the one presented in [25Section 42] However as observed in [25 Remark 53] the techniques developed in [25Section 42] can be adapted in order to obtain a proof of Theorem 48 below for anyCarnot group G of step 2

Let us start with the following remark If (microt)tge0 is a gradient flow of Ent in (P2(G) WG)then recalling Definition 21 we have that microt isin Dom(Ent) for all t ge 0 By (26) thismeans that microt = 983172tL n for some probability density 983172t isin L1(G) for all t ge 0 In additiont 983041rarr |Dminus

GEnt|(microt) isin L2loc([0 +infin)) and the function t 983041rarr Ent(microt) is non-increasing therefore

ae differentiable and locally integrable on [0 +infin)Lemma 44 below shows that it is enough to establish (415) in order to prove the second

part of Theorem 24 For the proof see also the last paragraph of [25 Section 42]

Lemma 44 Let 9831720 isin L1(G) be such that micro0 = 9831720L n isin Dom(Ent) Assume (microt)tge0 is agradient flow of Ent in (P2(G) WG) starting from micro0 with microt = 983172tL n for all t ge 0 Let(vG

t )tgt0 and (wGt )tgt0 wG

t = nablaG983172t983172t be the horizontal time-dependent vector fields givenby Proposition 33 and Proposition 34 respectively If it holds

minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112

Gdmicrot for ae t gt 0 (415)

then (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t with initial datum 9831720

Proof From Definition 21 we get that

Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t

s|DminusEnt|2G(micror) dr le Ent(micros)


for all s t ge 0 with s le t Therefore

minus d

dtEnt(microt) ge 1

2 |microt|2G + 12 |DminusEnt|2G(microt) for ae t gt 0

By Youngrsquos inequality Proposition 33 and Proposition 34 we thus get

minus d

dtEnt(microt) ge |microt|G middot |Dminus

GEnt|(microt) ge 983042vGt 983042L2

G(microt) middot 983042wGt 983042L2

G(microt) for ae t gt 0 (416)

Combining (415) and (416) by CauchyndashSchwarz inequality we conclude that vGt =

minuswGt = minusnablaG983172t983172t in L2

G(microt) for ae t gt 0 This immediately implies that (983172t)tge0 solves thesub-elliptic heat equation partt983172t = ∆G983172t with initial datum 9831720 in the sense of distributionsie

983133 +infin

0

983133

Gparttϕt(x) + ∆Gϕt(x) dmicrot dt +

983133

Gϕ0(x) dmicro0(x) = 0 forallϕ isin Cinfin

c ([0 +infin) times Rn)

By well-known results on hypoelliptic operators this implies that (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t with initial datum 9831720

To prove Theorem 48 below we need some preliminaries The following two lemmasare natural adaptations of [7 Lemma 214] to our setting

Lemma 45 Let micro isin P(G) and σ isin L1(G) with σ ge 0 Let ν isin M (GRm) be aRm-valued Borel measure with finite total variation and such that |ν| ≪ micro Then

983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)

In addition if (σk)kisinN sub L1(G) σk ge 0 weakly converges to the Dirac mass δ0 andνmicro

isin L2(G micro) then

limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)

Proof Inequality (417) follows from Jensen inequality and is proved in [7 Lemma 214]We briefly recall the argument for the readerrsquos convenience Consider the map Φ Rm timesR rarr [0 +infin] given by

Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0

0 if (z t) = (0 0)+infin if either t lt 0 or t = 0 z ∕= 0

Then Φ is convex lower semicontinuous and positively 1-homogeneous By Jensenrsquosinequality we have

Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)

for any Borel function ψ G rarr Rm+1 and any positive and finite measure ϑ on G Fixx isin G and apply (419) with ψ(y) =

983059νmicro(y) 1

983060and dϑ(y) = σ(xyminus1)dmicro(y) to obtain

983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν

micro(y) σ(xyminus1) dmicro(y)

983133

Gσ(xyminus1) dmicro(y)

983076


le983133

GΦ

983075ν

micro(y) 1

983076

σ(xyminus1) dmicro(y) =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

(y) σ(xyminus1) dmicro(y)

which immediately gives (417) The limit in (418) follows by the joint lower semiconti-nuity of the functional (ν micro) 983041rarr

983125G

983055983055983055 νmicro

9830559830559830552

dmicro see Theorem 234 and Example 236 in [1] In Lemma 46 below and in the rest of the paper we let f lowast g be the convolution of the

two functions f g with respect to the time variable We keep the notation f 983183 g for theconvolution of f g with respect to the space variable

Lemma 46 Let microt = 983172tL n isin P(G) for all t isin R and let ϑ isin L1(R) ϑ ge 0 If thehorizontal time-dependent vector field v RtimesG rarr HG satisfies vt isin L2

G(microt) for ae t isin Rthen

983133

G

983056983056983056983056983056ϑ lowast (983172middotvmiddot)(t)

ϑ lowast 983172middot(t)

983056983056983056983056983056

2

Gϑ lowast 983172middot(t) dx le ϑ lowast

983061983133

G983042vmiddot9830422

G dmicromiddot

983062(t) for all t isin R (420)

In addition if (ϑj)jisinN sub L1(G) ϑj ge 0 weakly converges to the Dirac mass δ0 then

limjrarr+infin

983133

G

983056983056983056983056983056ϑj lowast (983172middotvmiddot)(t)

ϑj lowast 983172middot(t)

983056983056983056983056983056

2

Gϑj lowast 983172middot(t) dx =

983133

G983042vt9830422

G dmicrot for ae t isin R (421)

Proof Inequality (420) follows from (419) in the same way of (417) so we omit thedetails For (421) set microj

t = ϑj lowast micromiddot(t) and νjt = ϑj lowast (vmiddotmicromiddot)(t) for all t isin R and j isin N

Then 983042νjt 983042G ≪ microj

t and νjt νt = vtmicrot for ae t isin R so that

lim infjrarr+infin

983133

G



983056983056983056983056983056

2

Gϑj lowast 983172middot(t) dx = lim inf

jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot

for ae t isin R by Theorem 234 and Example 236 in [1] The following lemma is an elementary result relating weak convergence and convergence

of scalar products of vector fields We prove it here for the readerrsquos convenience

Lemma 47 For k isin N let microk micro isin P(G) and let vk wk v w G rarr TG be Borel vectorfields Assume that microk micro vkmicrok vmicro and wkmicrok wmicro as k rarr +infin If

lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422

G dmicro lt +infin and lim supkrarr+infin

983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133

G〈vk wk〉G dmicrok =

983133

G〈v w〉G dmicro (422)

Proof By lower semicontinuity we know that limkrarr+infin

983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422

G dmicro for all t isin R

Expanding the squares we get

lim infkrarr+infin

9830612t

983133

G〈vk wk〉G dmicrok +

983133

G983042wk9830422

G dmicrok

983062ge 2t

983133

G〈v w〉G dmicro for all t isin R

Choosing t gt 0 dividing both sides by t and letting t rarr +infin gives the lim inf inequalityin (422) Choosing t lt 0 a similar argument gives the lim sup inequality in (422)


We are now ready to prove the second part of Theorem 24

Theorem 48 Let 9831720 isin L1(G) be such that micro0 = 9831720L n isin Dom(Ent) If (microt)tge0 is agradient flow of Ent in (P2(G) WG) starting from micro0 then microt = 983172tL n for all t ge 0and (983172t)tge0 solves the sub-elliptic heat equation partt983172t = ∆G983172t with initial datum 9831720 Inparticular t 983041rarr Ent(microt) is locally absolutely continuous on (0 +infin)

Proof By Lemma 44 we just need to show that the map t 983041rarr Ent(microt) satisfies (415) Itis not restrictive to extend (microt)tge0 in time to the whole R by setting microt = micro0 for all t le 0So from now on we assume microt isin AC2

loc(R (P2(G) WG)) The time-dependent vector field(vG)tgt0 given by Proposition 33 extends to the whole R accordingly Note that (microt)tisinR isa gradient flow of Ent in the following sense for each h isin R (microt+h)tge0 is a gradient flowon Ent starting from microh By Definition 21 we get t 983041rarr |Dminus

GEnt|(microt) isin L2loc(R) so that

t 983041rarr FG(983172t) isin L1loc(R) by Proposition 34

We divide the proof in three main stepsStep 1 smoothing in the time variable Let ϑ R rarr R be a symmetric smooth mollifier

in R ieϑ isin Cinfin

c (R) supp ϑ sub [minus1 1] 0 le ϑ le 1983133

Rϑ(t) dt = 1

We set ϑj(t) = j ϑ(jt) for all t isin R and j isin N We define

microjt = 983172j

tLn 983172j

t = (ϑj lowast 983172middot)(t) =983133

Rϑj(t minus s) 983172s ds forallt isin R forallj isin N

For any s t isin R let πst isin Γ0(micros microt) be an optimal coupling between micros and microt Aneasy computation shows that πj

t isin P(G times G) given by983133

GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133

GtimesGϕ(x y) dπst(x y) ds

for any ϕ G times G rarr [0 +infin) Borel is a coupling between microjt and microt Hence we get

WG(microjt microt)2 le

983133

Rϑj(t minus s) WG(micros microt)2 ds forallt isin R forallj isin N

Therefore limjrarr+infin WG(microjt microt) = 0 for all t isin R This implies that microj

t microt as j rarr +infinand

limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133

Gdcc(x 0)2 dmicrot(x) forallt isin R (423)

In particular (microjt)tisinR sub P2(G) Ent(microj

t) gt minusinfin for all j isin N and

lim infjrarr+infin

Ent(microjt) ge Ent(microt) forallt isin R (424)

We claim thatlim supjrarr+infin

Ent(microjt) le Ent(microt) for ae t isin R (425)

Indeed define the new reference measure ν = eminusc d2cc(middot0)L n where c gt 0 is chosen so

that ν isin P(G) Since the function H(r) = r log r + (1 minus r) for r ge 0 is convex andnon-negative by Jensenrsquos inequality we have

Entν(microjt) =

983133

GH

983059ec d2

cc(middot0) ϑj lowast 983172middot(t)983060

dν =983133

GH

983059ϑj lowast

983059ec d2

cc(middot0)983172middot983060

(t)983060

dν


le983133

Gϑj lowast H

983059ec d2


(t) dν = ϑj lowast Entν(micromiddot)(t) forallt isin R forallj isin N

Therefore lim supjrarr+infin Entν(microjt) le Entν(microt) for ae t isin R Thus (425) follows by (28)

and (423) Combining (424) and (425) we getlim

jrarr+infinEnt(microj

t) = Ent(microt) for ae t isin R (426)

Let (vGt )tisinR be the horizontal time-dependent vector field relative to (microt)tisinR given by

Proposition 33 Let (vjt )tisinR be the horizontal time-dependent vector field given by

vjt = ϑj lowast (983172middotvmiddot)(t)

983172jt

forallt isin R (427)

We claim that vjt isin L2

G(microjt) for all t isin R Indeed applying Lemma 46 we get

983133

G983042vj

t 9830422G dmicroj

t le ϑj lowast983061983133

G983042vmiddot9830422

G dmicromiddot

983062(t) = ϑj lowast |micromiddot|2G(t) forallt isin R

We also claim that (microjt)tisinR solves parttmicro

jt + div(vj

t microjt) = 0 in the sense of distributions for all

j isin N Indeed if ϕ isin Cinfinc (R times Rn) then also ϕj = ϑj lowast ϕ isin Cinfin

c (R times Rn) so that983133

R

983133

Gparttϕ

j(t x) +983111nablaGϕj(t x) vj

t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133

Gparttϕ(middot x) + 〈nablaGϕ(middot x) vmiddot(x)〉G dmicromiddot(x)

983062(t) dt

=983133

R

983133

Gparttϕ(t x) + 〈nablaGϕ(t x) vt(x)〉G dmicrot(x) dt = 0 forallj isin N

By Proposition 33 we conclude that (microjt)t isin AC2

loc(R (P2(G) WG)) with |microjt |2G le ϑj lowast

|micromiddot|2G(t) for all t isin R and j isin NFinally we claim that FG(983172j

t) le ϑj lowast FG(983172middot)(t) for all t isin R and j isin N Indeed arguingas in (45) by CauchyndashSchwarz inequality we have

983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic

983172middot983042nablaG983172middot983042Gradic

983172middot

9830769830782

(t) le 983172jt ϑj lowast

983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j

t) le ϑj lowast983075983133

983172middotgt0

983042nablaG983172middot9830422G

983172middotdx

983076

(t) = ϑj lowast FG(983172middot)(t)

Step 2 smoothing in the space variable Let j isin N be fixed Let η G rarr R be asymmetric smooth mollifier in G ie a function η isin Cinfin

c (Rn) such that

supp η sub B1 0 le η le 1 η(xminus1) = η(x) forallx isin G983133

Gη(x) dx = 1

We set ηk(x) = kQ η(δk(x)) for all x isin G and k isin N We define

microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133

Gηk(xyminus1)983172j

t(y) dy x isin G

for all t isin R and k isin N Note thatmicrojk

t =983133

G(ly)microj

t ηk(y)dy forallt isin R forallk isin N (428)

where ly(x) = yx x y isin G denotes the left-translation


Note that (microjkt )tisinR sub P2(G) for all k isin N Indeed arguing as in (220) we have

983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133

Gdcc(x 0)2 ηk(x) dx

(429)

for all t isin R In particular Ent(microjkt ) gt minusinfin for all t isin R and k isin N Clearly microjk

t microjt as

k rarr +infin for each fixed t isin R so that

lim infkrarr+infin

Ent(microjkt ) ge Ent(microj

t) forallt isin R (430)

Moreover we claim that

lim supkrarr+infin

Ent(microjkt ) le Ent(microj


Indeed let ν isin P(G) and H as in Step 1 Recalling (428) by Jensenrsquos inequality we get

Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)

Define νy = (ly)ν and note that νy = eminusc dcc(middoty)L n for all y isin G Thus by (27) we have

Entν((ly)microjt) = Entνyminus1 (microj

t) = Ent(microjt) + c

983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G

By the dominated convergence theorem we get that y 983041rarr Entν((ly)microjt) is continuous and

therefore (431) follows by passing to the limit as k rarr +infin in (432) Combining (430)and (431) we get

limkrarr+infin

Ent(microjkt ) = Ent(microj


Let (vjt )tisinR be as in (427) and let (vjk

t )tisinR be the horizontal time-dependent vectorfield given by

vjkt = ηk 983183 (983172j

tvjt )

983172jkt

forallt isin R forallk isin N (434)

We claim that vjkt isin L2

G(microjkt ) for all t isin R Indeed applying Lemma 45 we get

983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj

t le |microjt |2G forallt isin R forallk isin N (435)

We also claim that (microjkt )tisinR solves parttmicro

jkt + div(vjk

t microjkt ) = 0 in the sense of distributions

for all k isin N Indeed if ϕ isin Cinfinc (R times Rn) then also ϕk = ηk 983183 ϕ isin Cinfin


R

983133

Gparttϕ

k(t x) +983111nablaGϕk(t x) vjk

t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133

Rparttϕ(t middot) +

983111nablaGϕ(t middot) vj

t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +

983111nablaGϕ(t x) vj

t (x)983112

Gdmicroj

t(x) dt = 0 forallk isin N

Here we have exploited a key property of the space (G dcc L n) which cannot be expectedin a general metric measure space that is the continuity equation in (34) is preservedunder regularization in the space variable By Proposition 33 and (435) we conclude


that (microjkt )t isin AC2

loc(R (P2(G) WG)) with |microjkt |G le 983042vjk

t 983042L2G(microjk

t ) le 983042vjt 983042L2

G(microjt ) le |microj

t |G forall t isin R and k isin N

Finally we claim that FG(983172jkt ) le FG(983172j

t) for all t isin R and k isin N Indeed arguing asin (45) by CauchyndashSchwarz inequality we have

983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)

Step 3 truncated entropy Let j k isin N be fixed For any m isin N consider the mapszm Hm [0 +infin) rarr R defined in (42) We set zm(r) = zm(r) + m for all r ge 0 andm isin N Since 983172jk

t isin P(G) for all t isin R differentiating under the integral sign we getd

dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk

t ) partt983172jkt dx

for all t isin R Fix t0 t1 isin R with t0 lt t1 Then983133

GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk

t ) partt983172jkt dxdt (436)

Let (αi)iisinN sub Cinfinc (t0 t1) such that 0 le αi le 1 and αi rarr χ(t0t1) in L1(R) as i rarr +infin

Let i isin N be fixed and consider the function ut(x) = zm(983172jkt (x)) αi(t) for all (t x) isin RtimesRn

We claim that there exists (ψh)hisinN sub Cinfinc (Rn+1) such that

limhrarr+infin

983133

R

983133

G|ut(x) minus ψh

t (x)|2 + 983042nablaGut(x) minus nablaGψht (x)9830422

G dx dt = 0 (437)

Indeed consider the direct product Glowast = R times G and note that Glowast is a Carnot groupRecalling (212) we know that Cinfin

c (Rn+1) is dense in W 12Glowast (Rn+1) Thus to get (437) we

just need to prove that u isin W 12Glowast (Rn+1) (in fact the L2-integrability of parttu is not strictly

necessary in order to achieve (437)) We have 983172jk partt983172jk isin Linfin(Rn+1) because

983042983172jk983042Linfin(Rn+1) le 983042ηk983042Linfin(Rn) 983042partt983172jk983042Linfin(Rn+1) le 983042ϑprime

j983042L1(R)983042ηk983042Linfin(Rn)

by Youngrsquos inequality Moreover 983172jkαi partt983172jkαi isin L1(Rn+1) because

983042983172jkαi983042L1(Rn+1) = 983042αi983042L1(R) 983042partt983172jkαi983042L1(Rn+1) le 983042ϑprime

j983042L1(R)983042αi983042L1(R)

Therefore 983172jkαi partt983172jkαi isin L2(Rn+1) which immediately gives u isin L2(Rn+1) Now by

Step 2 we have that983133

G983042nablaG983172jk

t 9830422G dx le 983042983172jk

t 983042Linfin(Rn)FG(983172jkt ) le 983042983172jk

t 983042Linfin(Rn)FG(983172jt) forallt isin R

so that by Step 1 we get983042nablaG983172jkαi9830422

L2(Rn+1) le 983042ηk983042Linfin(Rn)983042α2i middot ϑj lowast FG(983172middot)983042L1(R)

This prove that 983042nablaGu983042G isin L2(Rn+1) The previous estimates easily imply that alsoparttu isin L2(Rn+1) This concludes the proof of (437)


Since 983172jk partt983172jk isin Linfin(Rn+1) by (437) we get that

limhrarr+infin

983133

R

983133

Gψh

t partt983172jkt dxdt =

983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )

983111nablaG983172jk

t vjkt

983112

Gdmicrojk

t dt (439)

for each fixed i isin N Since parttmicrojkt +div(vjk

t microjkt ) = 0 in the sense of distributions by Step 2

for each h isin N we have983133

R

983133

Gψh

t partt983172jkt dxdt = minus

983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt

We can thus combine (438) and (439) to get983133

Rαi(t)

983133

Gzm(983172jk

t ) partt983172jkt dxdt =

983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)

Passing to the limit as i rarr +infin in (440) we finally get that983133 t1

t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)

by the dominated convergence theorem Combining (436) and (441) we get983133

GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

eminusmminus1lt983172jkt ltemminus1

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)

for all t0 t1 isin R with t0 lt t1 where wjkt = nablaG983172jk

t 983172jkt in L2

G(microjkt ) for all t isin R

We can now conclude the proof We pass to the limit as m rarr +infin in (436) and we get

Ent(microjkt1 ) minus Ent(microjk

t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)

for all t0 t1 isin R with t0 lt t1 For the left-hand side of (436) recall (410) and (411) andapply the monotone convergence theorem on

983153983172jk

t le 1983154

and the dominated convergencetheorem on

983153983172jk

t gt 1983154 For the right-hand side of (436) recall that t 983041rarr 983042vjk

t 983042L2G(microjk

t ) isinL2

loc(R) and that t 983041rarr 983042wjkt 983042L2

G(microjkt ) = F12

G (983172jkt ) isin L2

loc(R) by Step 2 and apply CauchyndashSchwarz inequality and the dominated convergence theorem

We pass to the limit as k rarr +infin in (443) and we get

Ent(microjt1) minus Ent(microj

t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)

for all t0 t1 isin R with t0 lt t1 where wjt = nablaG983172j

t983172jt in L2

G(microjt) for all t isin R For

the left-hand side of (443) recall (433) For the right-hand side of (443) recall thatt 983041rarr 983042vj

t 983042L2G(microj

t ) isin L2loc(R) and that t 983041rarr 983042wj

t 983042L2G(microjk

t ) = F12G (983172j

t) isin L2loc(R) by Step 1 so

that the conclusion follows applying Lemma 45 Lemma 47 CauchyndashSchwarz inequalityand the dominated convergence theorem

We finally pass to the limit as j rarr +infin in (444) and we get

Ent(microt1) minus Ent(microt0) =983133 t1

t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)


for all t0 t1 isin R N with t0 lt t1 where N sub R is the set of discontinuity pointsof t 983041rarr Ent(microt) and wG

t = nablaG983172t983172t in L2G(microt) for ae t isin R by Proposition 34 For

the left-hand side of (444) recall (426) For the right-hand side of (444) recall thatt 983041rarr 983042vG


loc(R) and that t 983041rarr 983042wGt 983042L2

G(microjkt ) = F12

G (983172t) isin L2loc(R) so that the

conclusion follows applying Lemma 46 Lemma 47 CauchyndashSchwarz inequality and thedominated convergence theorem From (445) we immediately deduce (415) and we canconclude the proof by Lemma 44 In particular by Proposition 42 the map t 983041rarr Ent(microt)is locally absolutely continuous on (0 +infin)

References[1] L Ambrosio N Fusco and D Pallara Functions of bounded variation and free discontinuity prob-

lems Oxford Mathematical Monographs The Clarendon Press Oxford University Press New York2000

[2] L Ambrosio and N Gigli A userrsquos guide to optimal transport Modelling and optimisation of flowson networks Lecture Notes in Math vol 2062 Springer Heidelberg 2013 pp 1ndash155

[3] L Ambrosio N Gigli A Mondino and T Rajala Riemannian Ricci curvature lower bounds inmetric measure spaces with σ-finite measure Trans Amer Math Soc 367 (2015) no 7 4661ndash4701

[4] L Ambrosio N Gigli and G Savareacute Gradient flows in metric spaces and in the space of probabilitymeasures 2nd ed Lectures in Mathematics ETH Zuumlrich Birkhaumluser Verlag Basel 2008

[5] L Ambrosio N Gigli and G Savareacute Calculus and heat flow in metric measure spaces and applica-tions to spaces with Ricci bounds from below Invent Math 195 (2014) no 2 289ndash391

[6] L Ambrosio N Gigli and G Savareacute Metric measure spaces with Riemannian Ricci curvaturebounded from below Duke Math J 163 (2014) no 7 1405ndash1490

[7] L Ambrosio and G Savareacute Gradient flows of probability measures Handbook of differential equa-tions evolutionary equations Vol III Handb Differ Equ ElsevierNorth-Holland Amsterdam2007 pp 1ndash136

[8] F Baudoin and M Bonnefont Reverse Poincareacute inequalities isoperimetry and Riesz transforms inCarnot groups Nonlinear Anal 131 (2016) 48ndash59

[9] P Bernard Young measures superposition and transport Indiana Univ Math J 57 (2008) no 1247ndash275

[10] A Bonfiglioli E Lanconelli and F Uguzzoni Stratified Lie groups and potential theory for theirsub-Laplacians Springer Monographs in Mathematics Springer Berlin 2007

[11] H Breacutezis Opeacuterateurs maximaux monotones et semi-groupes de contractions dans les espaces deHilbert North-Holland Publishing Co Amsterdam-London American Elsevier Publishing Co IncNew York 1973 (French) North-Holland Mathematics Studies No 5 Notas de Matemaacutetica (50)

[12] L Capogna D Danielli S D Pauls and J T Tyson An introduction to the Heisenberg group andthe sub-Riemannian isoperimetric problem Progress in Mathematics vol 259 Birkhaumluser VerlagBasel 2007

[13] E A Carlen and J Maas An analog of the 2-Wasserstein metric in non-commutative probabilityunder which the fermionic Fokker-Planck equation is gradient flow for the entropy Comm MathPhys 331 (2014) no 3 887ndash926

[14] M Erbar The heat equation on manifolds as a gradient flow in the Wasserstein space Ann InstHenri Poincareacute Probab Stat 46 (2010) no 1 1ndash23

[15] M Erbar and J Maas Gradient flow structures for discrete porous medium equations DiscreteContin Dyn Syst 34 (2014) no 4 1355ndash1374

[16] S Fang J Shao and K-T Sturm Wasserstein space over the Wiener space Probab Theory RelatedFields 146 (2010) no 3-4 535ndash565

[17] B Franchi R Serapioni and F Serra Cassano Meyers-Serrin type theorems and relaxation ofvariational integrals depending on vector fields Houston J Math 22 (1996) no 4 859ndash890

[18] On the structure of finite perimeter sets in step 2 Carnot groups J Geom Anal 13 (2003)no 3 421ndash466


[19] N Gigli On the heat flow on metric measure spaces existence uniqueness and stability Calc VarPartial Differential Equations 39 (2010) no 1-2 101ndash120

[20] N Gigli and B-X Han The continuity equation on metric measure spaces Calc Var Partial Dif-ferential Equations 53 (2015) no 1-2 149ndash177

[21] N Gigli K Kuwada and S-I Ohta Heat flow on Alexandrov spaces Comm Pure Appl Math 66(2013) no 3 307ndash331

[22] N Gigli and S-I Ohta First variation formula in Wasserstein spaces over compact Alexandrovspaces Canad Math Bull 55 (2012) no 4 723ndash735

[23] R Jordan D Kinderlehrer and F Otto The variational formulation of the Fokker-Planck equationSIAM J Math Anal 29 (1998) no 1 1ndash17

[24] N Juillet Geometric inequalities and generalized Ricci bounds in the Heisenberg group Int MathRes Not IMRN 13 (2009) 2347ndash2373

[25] Diffusion by optimal transport in Heisenberg groups Calc Var Partial Differential Equations50 (2014) no 3-4 693ndash721

[26] B Khesin and P Lee A nonholonomic Moser theorem and optimal transport J Symplectic Geom7 (2009) no 4 381ndash414

[27] E Le Donne A primer on Carnot groups homogenous groups Carnot-Caratheacuteodory spaces andregularity of their isometries Anal Geom Metr Spaces 5 (2017) 116ndash137

[28] H-Q Li Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de HeisenbergJ Funct Anal 236 (2006) no 2 369ndash394

[29] Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg C RMath Acad Sci Paris 344 (2007) no 8 497ndash502

[30] J Maas Gradient flows of the entropy for finite Markov chains J Funct Anal 261 (2011) no 82250ndash2292

[31] J Milnor Curvatures of left invariant metrics on Lie groups Advances in Math 21 (1976) no 3293ndash329

[32] R Montgomery A tour of subriemannian geometries their geodesics and applications MathematicalSurveys and Monographs vol 91 American Mathematical Society Providence RI 2002

[33] S-i Ohta Gradient flows on Wasserstein spaces over compact Alexandrov spaces Amer J Math131 (2009) no 2 475ndash516

[34] S-i Ohta and K-T Sturm Heat flow on Finsler manifolds Comm Pure Appl Math 62 (2009)no 10 1386ndash1433

[35] F Otto The geometry of dissipative evolution equations the porous medium equation Comm PartialDifferential Equations 26 (2001) no 1-2 101ndash174

[36] A Petrunin Alexandrov meets Lott-Villani-Sturm Muumlnster J Math 4 (2011) 53ndash64[37] M-K von Renesse and K-T Sturm Transport inequalities gradient estimates entropy and Ricci

curvature Comm Pure Appl Math 58 (2005) no 7 923ndash940[38] G Savareacute Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds

C R Math Acad Sci Paris 345 (2007) no 3 151ndash154[39] N Th Varopoulos L Saloff-Coste and T Coulhon Analysis and geometry on groups Cambridge

Tracts in Mathematics vol 100 Cambridge University Press Cambridge 1992[40] C Villani Optimal transport old and new Fundamental Principles of Mathematical Sciences

vol 338 Springer-Verlag Berlin 2009[41] F-Y Wang Equivalent semigroup properties for the curvature-dimension condition Bull Sci Math

135 (2011) no 6-7 803ndash815

Scuola Normale Superiore Piazza Cavalieri 7 56126 Pisa ItalyEmail address luigiambrosiosnsit

Scuola Normale Superiore Piazza Cavalieri 7 56126 Pisa ItalyEmail address giorgiostefanisnsit


computed along a curve (microt)tge0 sub P2(Rn) satisfying (CE) for a given velocity field (vt)tge0is such that

d


983133

Rn(log ut + 1) div(vtut) dx =

983133

Rn

983071vt

nablaut

ut

983072dmicrot

In analogy with the Hilbertian case but using Ottorsquos calculus [35] in the interpretation ofthe right hand side one says that (microt)tge0 is a gradient flow of the entropy in (P2(Rn) W)if and only if the curve t 983041rarr Ent(microt) has maximal dissipation rate This happens if andonly if vt = minusnablautut ie when (ut)tge0 satisfies (HE)

Although not fully rigorous the argument presented above contains all the key toolsneeded to establish the correspondence between the heat flow and the entropy flow in ageneral metric measure space (X dm) Both the heat equation (HE) and the continuityequation (CE) have been adequately understood in this general context For (HE) onerelaxes the Dirichlet energy to the so-called Cheeger energy

Ch(u) = inf983069

lim infn

983133

X|Dun|2 dm un rarr u in L2(X dm) un isin Lip(X)

983070

where the local Lipschitz constant |Du|(x) = lim supyrarrx

|u(y)minusu(x)|d(xy) of u isin Lip(X) plays the

same role of the absolute value of the gradient in Rn It can be shown that the naturallyassociated Sobolev space W 12(X dm) is a Banach space (not Hilbertian in general) andthat the functional Ch is convex so that (HE) can still be interpreted as its gradientflow in the Hilbert space L2(X dm) see [5] For (CE) one introduces an appropriatespace S2(X) of test functions in W 12(X dm) and says that (microt)tgt0 satisfies the continuityequation with respect to a family of maps (Lt)tgt0 S2(X) rarr R if t 983041rarr

983125X f dmicrot is absolutely

continuous for every f isin S2(X) with ddt

983125X f dmicrot = Lt(f) for ae t gt 0 see [20]

The notion of gradient flow of the entropy functional

Entm(micro) =983133

X983172 log 983172 dm for micro = 983172m isin P2(X)

in the Wasserstein space (P2(X) W) can be rigorously defined by requiring the validityof the following sharp energy dissipation inequality

Entm(microt) + 12

983133 t

s|micror|2 dr + 1

2

983133 t

s|DminusEntm|2(micror) dr le Entm(micros)

for all 0 le s le t where t 983041rarr |microt| is the W-derivative of the curve (microt)tgt0 sub P2(X) and

|DminusEntm|(micro) = lim supνrarrmicro

max983083

Entm(micro) minus Entm(ν)W(micro ν) 0

983084

is the so-called descending slope of the entropy Note that this definition is consistentwith the standard one in Hilbert spaces since uprime(t) = minusnablaE(u(t)) is equivalent to

12 |uprime|2(t) + 1

2 |nablaE(u(t))|2 le minus d

dtE(u(t))

by combining the chain rule with CauchyndashSchwarz and Youngrsquos inequalitiesAs pointed out in [519] this abstract approach provides a complete equivalence between

the two gradient flows if the entropy is K-convex along geodesics in (P2(X) W) for some


K isin R that is if











2 Preliminaries






|γt| = limsrarrt











yrarrx




E(γt) + 12

983133 t

s|γr|2 dr + 1

2

983133 t








d




W(micro ν)2 = inf983069983133


983070 (23)




P2(X) =983069



983070




983133







(25)



Entm(micro) =

983099983105983103

983105983101

983133




















nablaVif =

hi983131









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




K isin R that is if











2 Preliminaries






|γt| = limsrarrt











yrarrx




E(γt) + 12

983133 t

s|γr|2 dr + 1

2

983133 t








d




W(micro ν)2 = inf983069983133


983070 (23)




P2(X) =983069



983070




983133







(25)



Entm(micro) =

983099983105983103

983105983101

983133




















nablaVif =

hi983131









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815






2 Preliminaries






|γt| = limsrarrt











yrarrx




E(γt) + 12

983133 t

s|γr|2 dr + 1

2

983133 t








d




W(micro ν)2 = inf983069983133


983070 (23)




P2(X) =983069



983070




983133







(25)



Entm(micro) =

983099983105983103

983105983101

983133




















nablaVif =

hi983131









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815









d




W(micro ν)2 = inf983069983133


983070 (23)




P2(X) =983069



983070




983133







(25)



Entm(micro) =

983099983105983103

983105983101

983133




















nablaVif =

hi983131









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





Entm(micro) =

983099983105983103

983105983101

983133




















nablaVif =

hi983131









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815









dcc(x y) = inf983069983133 1


983070





Rmi i = 1 κ

983154 x isin G (211)




983042u983042W 1pG (Ω) = 983042u983042Lp(Ω) +

m1983131

i=1983042Xiu983042Lp(Ω)




G (Rn)





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





nablaεf =n983131


κ983131


f





volε =983156















minusdcc(x 0)2

4t

983076





983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815






983075

minusdcc(x 0)2

4(1 minus ε)t

983076




2 exp983075

minusdcc(x 0)2

4(1 + ε)t

983076



isin V1


983172t(x) = (983172 983183 ht)(x) =983133




9831720 = 983172 on 0 times G(219)





Ct =983133



(dcc(middot 0)2 983183 ht)(x) =983133



Gdcc(x 0)2 983172t(x) dx =

983133


983133

Gdcc(x 0)2 983172(x) dx+2Ct (220)









L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815







L2ε(micro) =


983133

Gε


983070



ε(micro)






I

983133

Gε


and 983133

I

983133

Gε


c (I times Rn)





loc(I) such thatvε


t


983042vεt 983042L2



t 983041rarr 983042vεt 983042L2



















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815


















The quantity

Fε(983172) = 983042wε9830422L2

ε(micro) =983133

Gεcap983172gt0

983042nablaε9831729830422ε

983172dL n






L2G(micro) =


983133

G983042ξ9830422




G(micro)





I

983133

G983042vG



and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




and 983133

I

983133

Gparttϕ(t x) +

983111vG



c (I times Rn)





loc(I) such that




983042vGt 983042L2



t 983041rarr 983042vGt 983042L2












983133

G983042vε


Moreover983042vε

t 9830422ε = 983042vεV1

t 98304221 +

κ983131


t 98304221 for all ε gt 0 (37)




loc(I) and 983042vGt 983042L2










s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815







s983042γ(r)983042G dr =

983133 t

s983042vG

r (γ(r))983042G dr



983133

GtimesGd2


AC2loc



AC2loc

983133 t

s983042vG


983133 t

s

983133

G983042vG
















Ent(microt) =983133

GH(983172t) dx =

983133

GH

983061983172

Jt


983133



Thus we haved

dtEnt(microt)

983055983055983055983055983055t=0

= minus d

dt

983133

G983172 log(Jt) dx

983055983055983055983055983055t=0

dx =983133

Gminus983172

Jt

Jt

983055983055983055983055983055t=0

dx = minus983133

G983172 div V dx




Gd2



and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




and solim sup

trarr0


983133

Glim sup

trarr0

d2cc(Ft(x) x)

|t|2 dmicro(x) =983133

G983042V 9830422

G dmicro

Hence

minus d

dtEnt(microt)

983055983055983055983055983055t=0

le lim suptrarr0



983061983133

G983042V 9830422

G dmicro983062 1

2

and thus 983055983055983055983055983133

G983172 div V dx


983061983133

G983042V 9830422

G dmicro983062 1

2



GEnt|(micro) and

minus983133

G983172 div V dx =

983133

G

983111wG V

983112


c (G HG)


FG(983172) = 983042wG9830422L2G(micro) =

983133

Gcap983172gt0

983042nablaG9831729830422G

983172dL n








GEnt|2(micro)












2ε2 W2G(ν micro)


2 W32G (ν micro) (39)


nablaε983172 = nablaG983172 +k983131


983172 983042nablaε9831729830422ε = 983042nablaG9831729830422

G +k983131


9831729830422G



983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





983172isin L2


2

9830602


Fε(983172) = FG(983172) +k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

= FG(983172)983075

1 + 1FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

le FG(983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

9830762

and thus

F12ε (983172) le F12

G (983172)983075

1 + 12FG(983172)

k983131

i=2ε2(iminus1)

983056983056983056nablaVi983172

983172

9830569830569830562

L2G(micro)

983076

(310)



G (ν micro)


G (983172)












c (Rn) (312)



983147Q

2m1 Q

m1



inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




inequality



c (Rn) (313)








983125Rn div V dx = 0



d


983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t





0zm(s) ds r ge 0 (42)


claim thatd

dt

983133

GHm(983172t) dx =

983133



suptisin[ab]

983055983055983055983055983055d

dtHm(983172t)

983055983055983055983055983055 le m suptisin[ab]


|∆Ght| isin L1(G)




983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815






983133


983042nablaG983172t9830422G

983172t



983042nablaG983172t(x)9830422G

983172t(x) le [(9831720 983183 983042nablaGht983042G)(x)]2

(9831720 983183 ht)(x)

= 1(9831720 983183 ht)(x)

983093

983095983133

G


ht(y)middot

9831569831720(xyminus1)

983156ht(y) dy

983094

9830962

le 1(9831720 983183 ht)(x)

983075983133


Ght(y) dy

983076 983061983133


983062

le983075

9831720 983183983042nablaGht9830422

Ght

983076


(45)

Thus by (216) and (217) we get983133


983042nablaG983172t9830422G

983172t

dx le983133

G9831720 983183


ht

dx =983133

G


ht

dx lt +infin (46)







G|V |Rn dx le m

983133


983133

G|nablaGht|Rn dx





dt

983133


983133


983042nablaG983172t9830422G

983172t


Note that

t 983041rarr983133

G

983042nablaG983172t9830422G

983172t




t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





t 983041rarr983133

G


ht




983133


983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt




limmrarr+infin

983133

GHm(983172t) dx =

983133

G983172t log 983172t dx


limmrarr+infin

983133 t1

t0

983133


983042nablaG983172t9830422G

983172t

dxdt =983133 t1

t0

983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t

dxdt








|microt|2 le983133

Gcap983172tgt0

983042nabla983172t9830422G

983172t



minus d





983133

Gcap983172tgt0

983042nablaG983172t9830422G

983172t




d







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815







Fε(983172t) = FG(983172t) +k983131

i=2ε2(iminus1)

983133

Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx lt +infin


983042nablaVi983172t9830422

G983172t

le (983172 983183 983042nablaViht983042G)2

983172 983183 ht

= 1983172 983183 ht

983077

983172 983183

983075983042nablaVi

ht983042Gradicht

983156ht

9830769830782

le 983172 983183983042nablaVi

ht9830422G

ht


Gcap983172tgt0

983042nablaVi983172t9830422

G983172t

dx le983133

G983172 983183


Ght

dx =983133

G


Ght

dx lt +infin










minus d

dtEnt(microt) le

983133

G

983111minuswG

t vGt

983112




Ent(microt) + 12

983133 t

s|micror|2G dr + 1

2

983133 t




minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





minus d

dtEnt(microt) ge 1



minus d








983133 +infin

0

983133


983133






983133

G

983055983055983055983055983055σ 983183 ν

σ 983183 micro

983055983055983055983055983055

2

σ 983183 micro dx le983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (417)



limkrarr+infin

983133

G

983055983055983055983055983055σk 983183 ν

σk 983183 micro

983055983055983055983055983055

2

σk 983183 micro dx =983133

G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2

dmicro (418)


Φ(z t) =

983099983105983105983105983105983103

983105983105983105983105983101

|z|2t

if t gt 0



Φ983061983133

Gψ(x) dϑ(x)

983062le

983133

GΦ(ψ(x)) dϑ(x) (419)


983059νmicro(y) 1


983055983055983055983055983055(σ 983183 ν)(x)(σ 983183 micro)(x)

983055983055983055983055983055

2

(σ 983183 micro)(x) = Φ983075983133

G

ν


983133


983076


le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




le983133

GΦ

983075ν

micro(y) 1

983076


G

983055983055983055983055983055ν

micro

983055983055983055983055983055

2



983125G

983055983055983055 νmicro

9830559830559830552





983133

G



983056983056983056983056983056

2


983061983133

G983042vmiddot9830422

G dmicromiddot



limjrarr+infin

983133

G



983056983056983056983056983056

2


983133

G983042vt9830422






lim infjrarr+infin

983133

G



983056983056983056983056983056

2


jrarr+infin

983133

G

983056983056983056983056983056νj

t

microjt

983056983056983056983056983056

2

Gdmicroj

t ge983133

G

983056983056983056983056983056νt

microt

983056983056983056983056983056

2

Gdmicrot




lim supkrarr+infin

983133

G983042vk9830422

G dmicrok le983133

G983042v9830422


983133

G983042wk9830422

G dmicrok lt +infin

thenlim

krarr+infin

983133


983133



983125G 983042vk9830422

G dmicrok =983125G 983042v9830422

G dmicro and

lim infkrarr+infin

983133

G983042tvk + wk9830422

G dmicrok ge983133

G983042tv + w9830422



lim infkrarr+infin

9830612t

983133


983133

G983042wk9830422

G dmicrok

983062ge 2t

983133













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815













Rϑ(t) dt = 1


microjt = 983172j

tLn 983172j





GtimesGϕ(x y) dπj

t (x y) =983133

Rϑj(t minus s)

983133




983133




limjrarr+infin

983133

Gdcc(x 0)2 dmicroj

t(x) =983133




lim infjrarr+infin






Entν(microjt) =

983133

GH

983059ec d2


dν =983133

GH

983059ϑj lowast

983059ec d2


(t)983060

dν


le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815




le983133

Gϑj lowast H

983059ec d2










983172jt




983133

G983042vj

t 9830422G dmicroj


G983042vmiddot9830422

G dmicromiddot



jt + div(vj




R

983133

Gparttϕ


t (x)983112

Gdmicroj

t(x) dt =

=983133

Rϑj lowast

983061983133


983062(t) dt

=983133

R

983133






983042nablaG983172jt9830422

G le983077

ϑj lowast983075

χ983172gt0radic


983172middot

9830769830782


983075983042nablaG983172middot9830422

G983172middot

χ983172middotgt0

983076

(t)

so thatFG(983172j


983172middotgt0


983172middotdx

983076



c (Rn) such that


Gη(x) dx = 1


microjkt = 983172jk

t L n 983172jkt (x) = ηk 983183 983172j

t(x) =983133


t(y) dy x isin G


t =983133

G(ly)microj





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





983133

Gdcc(x 0)2 dmicrojk

t (x) =983133

G

983059dcc(middot 0)2 983183 ηk

983060(x) dmicroj

t(x)

le 2983133

Gdcc(x 0)2 dmicroj

t(x) + 2983133


(429)


t microjt as


lim infkrarr+infin




lim supkrarr+infin




Entν(microjkt ) le

983133

GEntν((ly)microj

t) ηk(y)dy (432)




983133

Gdcc(yx 0)2 dmicroj

t(x) forally isin G



limkrarr+infin





vjkt = ηk 983183 (983172j

tvjt )

983172jkt




983133

G983042vjk

t 9830422G dmicrojk

t le983133

G983042vj

t 9830422G dmicroj



jkt + div(vjk




R

983133

Gparttϕ


t (x)983112

Gdmicrojk

t (x) dt =

=983133

Gηk 983183

983061983133



t (middot)983112

G983172j

t(middot) dt983062

(x) dx

=983133

R

983133

Gparttϕ(t x) +


t (x)983112

Gdmicroj






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815






t 983042L2G(microjk

t ) le 983042vjt 983042L2





983042nablaG983172jkt 9830422

G le983093

983095ηk 983183

983091

983107χ983172jt gt0

983156983172j

t

983042nablaG983172jt983042G983156

983172jt

983092

983108

983094

9830962

le 983172jkt ηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

so that

FG(983172jkt ) le

983133

Gηk 983183

983075983042nablaG983172j

t9830422G

983172jt

χ983172jt gt0

983076

dx =983133

983172jt gt0

983042nablaG983172jt9830422

G

983172jt

dx = FG(983172jt)



dt

983133

GHm(983172jk

t ) dx =983133

Gzm(983172jk



GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133

Gzm(983172jk





limhrarr+infin

983133

R

983133

G|ut(x) minus ψh


G dx dt = 0 (437)






j983042L1(R)983042ηk983042Linfin(Rn)



j983042L1(R)983042αi983042L1(R)




t 9830422G dx le 983042983172jk








limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815





limhrarr+infin

983133

R

983133

Gψh


983133

Rαi(t)

983133

Gzm(983172jk


andlim

hrarr+infin

983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt =983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (439)




R

983133

Gψh


983133

R

983133

Gparttψ

ht dmicrojk

t dt =983133

R

983133

G

983111nablaGψh

t vjkt

983112

Gdmicrojk

t dt


Rαi(t)

983133

Gzm(983172jk


983133

Rαi(t)

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (440)


t0

983133

Gzm(983172jk


983133 t1

t0

983133

Gzprime

m(983172jkt )


t vjkt

983112

Gdmicrojk

t dt (441)


GHm(983172jk

t1 ) dx minus983133

GHm(983172jk

t0 ) dx =983133 t1

t0

983133


983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (442)


t 983172jkt in L2




t0 ) =983133 t1

t0

983133

G

983111minuswjk

t vjkt

983112

Gdmicrojk

t dt (443)


983153983172jk

t le 1983154


983153983172jk


t 983042L2G(microjk

t ) isinL2


G(microjkt ) = F12





t0) =983133 t1

t0

983133

G

983111minuswj

t vjt

983112

Gdmicroj

t dt (444)


t983172jt in L2



t 983042L2G(microj


t 983042L2G(microjk

t ) = F12G (983172j





t0

983133

G

983111minuswG

t vGt

983112

Gdmicrot dt (445)







G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815









G(microjkt ) = F12













































135 (2011) no 6-7 803ndash815


























135 (2011) no 6-7 803ndash815



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Introductioncvgmt.sns.it/media/doc/paper/3726/20191029.pdf · 2019. 10. 29. · HEAT AND ENTROPY...

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