On a generalisation of compactness by compensation in theLp–Lq setting
Marin Misur
email: [email protected] of Mathematics, Faculty of Science
University of Zagreb
joint work with Darko Mitrovic, University of Montenegro and University of Bergen
October 11, 2013
Motivation - Maxwell’s equationsLet ⌦ ✓ R
3. Denote by E and H the electric and magnetic field, and by D andB the electric and magnetic induction. Let ⇢ denote the charge, and j thecurrent density. Maxwell’s system of equations reads:
@t
B+ rot E = G,
div B = 0,
@t
D+ j� rot H = F,
div D = ⇢.
Assume that properties of the material can be expressed by following linearconstitutive equations:
D = ✏E, B = µH.
The energy of electromagnetic field at time t is given by:
T (t) =1
2
Z
⌦
(D · E+ B · H)dx.
It’s natural to consider
D,B 2 L1([0, T ]; L2
div (⌦;R3)),
E,H 2 L1([0, T ]; L2
rot (⌦;R3)),
J 2 L1([0, T ]; L2(⌦;R3)), F,G 2 L2([0, T ]; L2(⌦;R3)).
Let us consider a family of problems:
@t
Bn + rot En = Gn,
@t
Dn + Jn � rot Hn = Fn,
with constitution equations:
Dn = ✏nEn, Bn = µnHn, Jn = �nEn.
What can we say about energy T (t) if we know
Tn(t) =1
2
Z
⌦
(Dn · En + Bn · Hn)dx.
Div-rot lemma in L2
Theorem 1. Assume that ⌦ is open and bounded subset of R3, and that itholds:
u
n
* u in L2(⌦;R3),
v
n
* v in L2(⌦;R3),
rotun
bounded in L2(⌦;R3), divvn
bounded in L2(⌦).
Thenu
n
· vn
* u · v
in the sense of distributions.
Quadratic theorem
Theorem 2. (Quadratic theorem) Assume that ⌦ ✓ R
d is open and that⇤ ✓ R
r is defined by
⇤ :=n
� 2 R
r : (9 ⇠ 2 R
d \ {0})d
X
k=1
⇠k
A
k� = 0o
,
where Q is a real quadratic form on R
r, which is nonnegative on ⇤, i.e.
(8� 2 ⇤) Q(�) > 0 .
Furthermore, assume that the sequence of functions (un
) satisfies
u
n
�* u weakly in L2
loc
(⌦;Rr) ,
⇣
X
k
A
k@k
u
n
⌘
relatively compact in H�1
loc
(⌦;Rq) .
Then every subsequence of (Q � un
) which converges in distributions to it’slimit L, satisfies
L > Q � u
in the sense of distributions.
IntroductionClassical results
L2 theoryResult by Panov
H-distributionsDefinitionLocalisation principle
Application to the parabolic type equation
The most general version of the classical L2 results has recently been proved byE. Yu. Panov (2011):
Assume that the sequence (un
) is bounded in Lp(Rd;Rr), 2 p < 1, andconverges weakly in D0(Rd) to a vector function u.Let q = p0 if p < 1, and q > 1 if p = 1. Assume that the sequence
⌫
X
k=1
@k
(Ak
u
n
) +
d
X
k,l=⌫+1
@kl
(Bkl
u
n
)
is precompact in the anisotropic Sobolev space W�1,�2;q
loc
(Rd;Rm), wherem⇥ r matrices Ak and B
kl have variable coe�cents belonging to L2q(Rd),q = p
p�2
if p > 2, and to the space C(Rd) if p = 2.
We introduce the set ⇤(x)
⇤(x) =
(
� 2 C
r
�
� (9⇠ 2 R
d \ {0}) : (1)
i
⌫
X
k=1
⇠k
A
k(x)� 2⇡
d
X
k,l=⌫+1
⇠k
⇠l
B
kl(x)
!
� = 0
m
)
,
and consider the bilinear form on C
r
q(x,�,⌘) = Q(x)� · ⌘, (2)
where Q 2 Lq
loc
(Rd; Symr
) if p > 2 and Q 2 C(Rd; Symr
) if p = 2.Finally, let q(x,u
n
,un
)* ! weakly in the space of distributions.
Result by Panov
The following theorem holds
Theorem 3. [P, 2011] Assume that (8� 2 ⇤(x)) q(x,�,�) � 0 (a.e. x 2 R
d)and u
n
* u, then q(x,u(x),u(x)) !.
The connection between q and ⇤ given in the previous theorem, we shall callthe consistency condition.
H-distributions
H-distributions were introduced by N. Antonic and D. Mitrovic (2011) as anextension of H-measures to the Lp � Lq context.
M. Lazar and D. Mitrovic (2012) extended and applied them on a velocityaveraging problem.
We need Fourier multiplier operators with symbols defined on a manifold Pdetermined by d-tuple ↵ 2 (R+)d:
P =n
⇠ 2 R
d :d
X
k=1
|⇠k
|l↵k = 1o
,
where l is the smallest number such that l↵k
> d for each k. In order toassociate an Lp Fourier multiplier to a function defined on P, we extend it toR
d\{0} by means of the projection
(⇡P
(⇠))j
= ⇠j
⇣
|⇠1
|l↵1 + · · ·+ |⇠d
|l↵d
⌘�1/l↵j, j = 1, . . . , d.
Theorem 4. [LM, 2012] Let (un
) be a bounded sequence in Ls(Rd), s > 1,and let (v
n
) be a bounded sequence of uniformly compactly supportedfunctions in L1(Rd). Then, after passing to a subsequence (not relabelled),for any s 2 (1, s) there exists a continuous bilinear functional B on
Ls
0(Rd)⌦ Cd(P) such that for every ' 2 Ls
0(Rd) and 2 Cd(P) it holds
B(', ) = limn!1
Z
Rd'(x)u
n
(x)�
A Pvn
�
(x)dx ,
where A P is the Fourier multiplier operator on R
d associated to � ⇡P
.
Corollary 1. [LM, 2012] The bilinear functional B defined in the previoustheorem can be extended by continuity to a continuous functional onLs
0(Rd;Cd(P)).
We need the following extension of the results given above.
Theorem 5. Let (un
) be a bounded sequence in Lp(Rd), p > 1, and let (vn
)be a bounded sequence of uniformly compactly supported functions in Lq(Rd),1/q + 1/p < 1. Then, after passing to a subsequence (not relabelled), for anys 2 (1, pq
p+q
) there exists a continuous bilinear functional B on
Ls
0(Rd)⌦ Cd(P) such that for every ' 2 Ls
0(Rd) and 2 Cd(P), it holds
B(', ) = limn!1
Z
Rd'(x)u
n
(x)�
A Pvn
�
(x)dx ,
where A P is the Fourier multiplier operator on R
d associated to � ⇡P
.The bilinear functional B can be continuously extended as a linear functionalon Ls
0(Rd;Cd(P)).
Localisation principle
Lemma
Assume that sequences (un
) and (vn
) are bounded in Lp(Rd;Rr) andLq(Rd;Rr), respectively, and converge toward u and 0 in the sense ofdistributions.Furthermore, assume that sequence (u
n
) satisfies:
G
n
:=d
X
k=1
@↵kk
(Ak
u
n
) ! 0 in W�1,p(⌦;Rm), (3)
where either ↵k
2 N, k = 1, . . . , d or ↵k
> d, k = 1, . . . , d, and elements ofmatrices Ak belong to Ls
0(Rd), s 2 (1, pq
p+q
).Finally, by µ denote a matrix H-distribution corresponding to subsequences of(u
n
) and (vn
� v). Then the following relation holds
⇣
d
X
k=1
(2⇡i⇠k
)↵kA
k
⌘
µ = 0.
Proof of the lemma
Assume, without loosing any generality, that v = 0. Denote by B
the Fouriermultiplier operator with the symbol
( � ⇡P
)(⇠)(1� ✓(⇠))
(|⇠1
|l↵1 + · · ·+ |⇠d
|l↵d)1/l.
According to [LM 2012, Lemma 5], for any 2 Cd(P) and any s > 1, themultiplier operator B
: L2(Rd) \ Ls(Rd) ! W↵;s(Rd), is bounded (with Ls
norm considered on the domain of operator B
and ↵ = (↵1
, . . . ,↵d
))Take a test function g
n
given by:
g
n
(x) = B
�
�vn
�
(x),
where 2 Cd(P) and � 2 C1c
(Rd) and multiply with G
n
.
Proof of the lemma - continued
We get
Z
RdG
n
· gn
dx =
Z
Rd
d
X
k=1
A
k
u
n
· A( �⇡P)(⇠)
(1�✓(⇠))(2⇡i⇠k)↵k
(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v
n
)dx
=
Z
Rd
d
X
k=1
A
k
u
n
· A( �⇡P)(⇠)
(2⇡i⇠k)↵k
(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v
n
)dx +
+
Z
Rd
d
X
k=1
A
k
u
n
· A( �⇡P)(⇠)
✓(⇠)(2⇡i⇠k)↵k
(|⇠1|l↵1+···+|⇠d|l↵d)1/l(�v
n
)dx.
Taking into account the preceeding theorem and strong precompactess ofsequence (G
n
), we get
⇣
d
X
k=1
(2⇡i⇠k
)↵kA
k
⌘
µ = 0.
Strong consistency condition
Introduce the set
⇤D =n
µ 2 Ls(Rd; (Cd(P))0)r :⇣
n
X
k=1
(2⇡i⇠k
)↵kA
k
⌘
µ = 0
m
o
,
where the given equality is understood in the sense of Ls(Rd; (Cd(P))0)m.
Definition
We say that set ⇤D, bilinear form q from (2) and matrixµ = [µ
1
, . . . ,µr
],µj
2 Ls(Rd; (Cd(P))0)r satisfy the strong consistencycondition if (8j 2 {1, . . . , r}) µ
j
2 ⇤D, and it holds
h�Q⌦ 1,µi � 0, � 2 Ls(Rd;R+
0
).
Compactness by compensation
Let us assume that coe�cients of the bilinear form q from (2) belong to spaceLt
loc
(Rd), where 1/t+ 1/p+ 1/q < 1.
Theorem 6. Assume that sequences (un
) and (vn
) are bounded inLp(Rd;Rr) and Lq(Rd;Rr), respectively, and converge toward u and v in thesense of distributions.Assume that (3) holds and that
q(x;un
,vn
)* ! in D0(Rd).
If the set ⇤D, the bilinear form (2), and matrix H-distribution µ,corresponding to subsequences of (u
n
� u) and (vn
� v), satisfy the strongconsistency condition, then
q(x;u,v) ! in D0(Rd).
Proof of the theorem
Let us abuse the notation by denoting u
n
= u
n
� u* 0 andv
n
= v
n
� v * 0.According to the theorem on existence of H-distributions, for any non-negative� 2 D(Rd) it holds
limn!1
Z
RdQu
n
· vn
� dx = h�Q⌦ 1,µi,
where µ is matrix H-distribution corresponding to (subsequences of)u
n
,vn
* 0. Since, according to the localisation principle, for every fixedk 2 {1, . . . , r}, the r-tuple µ
k
belongs to ⇤D, we conclude from the strongconsistency condition that
h�Q⌦ 1,µi � 0.
From the fact that
q(x;un
,vn
)* ! � q(x;u,v) � 0 in D0(Rd),
the statement of the theorem follows.
Application to the parabolic type equation
Now, let us consider the non-linear parabolic type equation
L(u) = @t
u� div div (g(t,x, u)A(t,x)),
on (0,1)⇥ ⌦, where ⌦ is an open subset of Rd. We assume that
u 2 Lp((0,1)⇥ ⌦), g(t,x, u(t,x)) 2 Lq((0,1)⇥ ⌦), 1 < p, q,
A 2 Ls
loc
((0,1)⇥ ⌦)d⇥d, where 1/p+ 1/q + 1/s < 1,
and that the matrix A is strictly positive definite, i.e.
A⇠ · ⇠ > 0, ⇠ 2 R
d \ {0}, (a.e.(t,x) 2 (0,1)⇥ ⌦).
Furthermore, assume that g is a Caratheodory function and non-decreasingwith respect to the third variable.
Then we have the following theorem.
Theorem 7. Assume that sequences (ur
) and g(·, ur
) are such thatur
, g(ur
) 2 L2(R+ ⇥R
d) for every r 2 N; assume that they are bounded inLp(R+ ⇥R
d), p 2 (1, 2], and Lq(R+ ⇥R
d), q > 2, respectively, where1/p+ 1/q < 1; furthermore, assume u
r
* u and, for some,f 2 W�1,�2;p(R+ ⇥R
d), the sequence
L(ur
) = fr
! f strongly in W�1,�2;p(R+ ⇥R
d).
Under the assumptions given above, it holds
L(u) = f in D0(R+ ⇥R
d).
References
[AM] N. Antonic, D. Mitrovic, H-distributions: An Extension of H-Measures to anLp � Lq Setting, Abstr. Appl. Anal. Volume 2011, Article ID 901084, 12pages.
[LM] M. Lazar, D. Mitrovic, Velocity averaging – a general framework, Dynamics ofPDEs 9 (2012) 239–260.
[P] E. J. Panov, Ultraparabolic H-measures and compensated compactness,Annales Inst. H.Poincare 28 (2011) 47–62.