Sinteza rezultatelor obtinute in anii 2007-2010in cadrul grantului PN-II-IDEI 8/28.09.2007, cod ID 404

Metode functionale, deterministe sistochastice in dinamica fluidelor

Director proiect: prof.dr. Catalin Lefter

ArticoleAnul 2008:1. V.Barbu, G. Da Prato, M. Roeckner, Stochastic nonlinear diffusion equations with singular diffusivity, SIAM J. Math.Anal., 41(3), 2009, 1106-1120.2. V.Barbu, G.Da Prato, M. Roeckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions, Ann. Probab., 37(2), 2009, 428-452.3. G. Marinoschi, Periodic solutions to fast diffusion equations with non Lipschitz convective terms, 2007, Nonlinear Anal., Real World Appl. 10(2), 2009, 1048-1067.4. C. Ciutureanu, G. Marinoschi, Convergence of the finite difference scheme for a fast diffusion equation in porous media, Numer. Funct. Anal. Optim., 29(9-10), 2008, 1034-1063.5. Mimmo Iannelli, Gabriela Marinoschi, Well-posedness for a hyperbolic-parabolic Cauchy problem arising in population dynamics, 2008, Differential and Integral Equations, 21(9-10), 2008, 917-934.6. Adriana Ioana Lefter, Nonlinear feedback controllers for the Navier-Stokes equations, Nonlinear Anal., Theory Methods Appl., 71(1-2), 2009, 301-316.7. C.Lefter, On a unique continuation property related to the boundary stabilization of magnetohydrodynamic equations, An. Stiint. Univ. Al.I.Cuza Iaşi, Ser. Noua. Mat., 56(1), 2010, 1-15.

Anul 2009:1. Viorel Barbu, Giuseppe Da Prato, Luciano Tubaro, Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space, Ann. Probab., 37(4), 2009, 1427–1458.2. V.Barbu, G.Da Prato, M. Roeckner, Finite time extinction for solutions to fast diffusion stochastic porous media equations, C. R. , Math., Acad. Sci. Paris, 347(1-2), 2009, 81-84.3. Angelo Favini, Gabriela Marinoschi, Periodic behavior for a degenerate fast diffusion equation, J. Math. Anal. Appl. 351 (2), 2009, 509—521.4. Jean Michel Coron, Catalin Lefter, Andreea Grigoriu, Gabriel Turinici, Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling, New Journal of Physics, 11(10), 2009, 22 pp.5. Angelo Favini, Gabriela Marinoschi, Identification of the time derivative coefficient in a fast diffusion degenerate equation, J. Optim. Theory Appl., 145(2), 2010, 249–269 .6. C. Ciutureanu, COMSOL modelling for a water infiltration model in an unsaturated medium, An. Stiint. Univ. Ovidius Constanta , Ser. Mat. 17(3), 2009, 87-98.

Anul 2010:1. Viorel Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations, Annual Reviews in Control, 34 (1), 2010, 52-61.; doi:10.1016/ j.arcontrol. 2009. 12. 002;2. M. Fabrizio, A.Favini, Gabriela Marinoschi, An optimal control problem for a singular system of solid-liquid phase transition, Numer. Funct. Anal. Optim., 31(9), 2010, 989-1022.3. Catalin-George Lefter, Alfredo Lorenzi, Approximate controllability for an integro-differential control problem, Applicable Analysis, trimis spre publicare;4. Adriana-Ioana Lefter, On the feedback stabilization of Boussinesq equations, An. Stiint. Univ. Al.I.Cuza Iaşi, Ser. Noua. Mat., acceptat pentru publicare (va apare in 2011).

Directii de cercetare abordate ın anul 2007

Ecuatia Kolmogorov pentru sistemul Navier-Stokes 2-D ıntr-un canal(Viorel Barbu)

Se considera o ecuatie Navier-Stokes 2-D ıntr-un canal, cu conditii periodice pe axa,conditii Navier pe perete si perturbata cu o forta stochastica

√QW :

dX = (ν∆X − (X · ∇X) +∇p)dt +√

QdW (t),

∇ ·X = 0, ın (0,∞)×O,

∂X1

∂ξ2(t, ξ1, 0) =

∂X1

∂ξ2(t, ξ1, 1) = 0, ın (0,∞)× IR,

X2(t, ξ1, 0) = X2(t, ξ1, 1) = 0, ın (0,∞)× IR,

X(t, ξ1 + L, ξ2) = X(t, ξ1, ξ2), ın (0,∞)×O,

X(0, ξ) = x(ξ)0, ın O,

unde ξ = (ξ1, ξ2) este un punct generic din O = IR × (0, 1), X = (X1, X2) estecampul vitezelor si W este un proces Wiener cilindric ın spatiul Hilbert H,

H =x ∈ [L2((0, L)× (0, 1))]2 : ∇ · x = 0, x(ξ1 + L, ξ2) = x(ξ1, ξ2),

∫ L

0

∫ 1

0x1(ξ1, ξ2)dξ1dξ2 = 0, x2(ξ1, 0) = x2(ξ1, 1) = 0

.

Q este un operator ın H, autoadjunct, nenegativ, cu urma.S-a urmarit constructia operatorului Kolmogorov asociat cu procesul stochas-

tic corespunzator exprimat ın vorticitate. Rezultatul principal este ca operatorulKolmogorov definit pe un spatiu de functii C2 netede este esential m− disipativın L2(H0, µ), unde H0 =

u ∈ [L2((0, L)× (0, 1))]2 : u(ξ1 + L, ξ2) = u(ξ1, ξ2)

este

spatiul starilor pentru ecuatia ın vorticitate iar µ este o masura invarianta.

Solutii periodice pentru ecuatii de difuzie rapida cu termen convectivnon Lipschitz (Gabriela Marinoschi)

In aceasta etapa a fost pregatita pentru publicare lucrarea Periodic solutions to fastdiffusion equations with non Lipschitz convective terms, autor Gabriela Marinoschi.

Lucrarea studiaza existenta solutiilor periodice pentru ecuatia de difuzie rapida,singulara cu termen convectiv puternic neliniar, ın cazul nedegerat. Modelul constaın ecuatia

∂θ

∂t−∆β∗(θ) +∇ ·K(θ) 3 f ın Ω×R, (1)

unde Ω este un deschis marginit din RN , cu frontiera neteda pe portiuni, cu conditiila limita si conditii de periodicitate.

1

In aceasta problema β∗ : (−∞, θs] → R este o functie multivoca definita astfel:

β∗(r) = ∫ r

0 β(ξ)dξ, daca r < θs,[K∗

s ,+∞), daca r = θs,(2)

unde β : (−∞, θs) → R, reprezentand coeficientul de difuzie, este o functie continua,pozitiva (ın cazul nedegenerat), monoton crescatoare, care explodeaza ın θ = θs,adica

β(r) ≥ ρ > 0, pentru r < θs, β(r) = ρ pentru r ≤ 0, (3)

limrθs

β(r) = +∞. (4)

In consecinta, β∗ are proprietatile:(i) (β∗(r)− β∗(r′)) (r − r′) ≥ ρ(r − r′)2, pentru orice r, r′ ∈ (−∞, θs],(ii) limrθs β∗(r) = K∗

s ,(iii) limr→−∞ β∗(r) = −∞,

unde ρ, θs si K∗s sunt constante pozitive.

In aceasta lucrare facem urmatoarele ipoteze asupra lui K = Kjj=1,...,N :

Kj : [0, θs] → [0,Kjs ], Kj(0) = 0, Kj(θs) = Kj

s > 0,

si presupunem ca fiecare componenta a vectorului K este continua pe [0, θs].In plus consideram ca fiecare Kj este diferentiabil pe [0, θs), aceasta implicand

ca este local Lipschitz, adica exista Ml > 0 astfel ıncat(iK) |Kj(r)−Kj(r)| ≤ Ml |r − r|, pentru orice r, r ∈ [0, θl] cu θl < θs, j =

1, ..., N.Aceasta ipoteza include si situatia ın care unii Kj au derivata explodand la

stanga lui θs,lim

rθs

K ′j(r) = +∞. (5)

Extindem pe Kj la stanga lui 0 si la dreapta lui θs prin continuitate, astfel:

Kj(r) = 0 pentru r ≤ 0, (6)

Kj(r) = Kjs pentru r ≥ θs. (7)

Consideram frontiera compusa din doua parti Γu si Γα, Γu ∩ Γα = ∅ si atasamconditii la limita de tip Neumann neomogen si Robin,

(K(θ)−∇β∗(θ)) · ν 3 u pe Γu ×R, (8)

(K(θ)−∇β∗(θ)) · ν − αβ∗(θ) 3 f0 pe Γα ×R, (9)

unde α : Γα → [αm, αM ] este o functie continua, pozitiva αm > 0, reprezentandpermeabilitatea frontierei Γα.

Scopul final este de a studia existenta solutiilor periodice

θ(x, T ) = θ(x, t + T ) a.p.t. x ∈ Ω, a.p.t. t ∈ R, T > 0, (10)

ın ipoteza T− periodicitatii datelor problemei, f, u si f0.

2

Problema intermediara Fie T un numar pozitiv arbitrar, fixat. In primul rand sestudiaza o problema intermediara pe intervalul de timp (0, T ), urmand ca rezultatulobtinut sa fie extins prin periodicitate. Consideram problema

∂θ

∂t−∆β∗(θ) +∇ ·K(θ) 3 f ın Q = Ω× (0, T ), (11)

(K(θ)−∇β∗(θ)) · ν 3 u pe Σu = Γu × (0, T ), (12)

(K(θ)−∇β∗(θ)) · ν − αβ∗(θ) 3 f0 pe Σα = Γα × (0, T ), (13)

θ(x, 0) = θ(x, T ) ın Ω. (14)

si introducem spatiul V = H1(Ω) cu norma

‖ψ‖V =(∫

Ω|∇ψ|2 dx +

∫

Γα

α(x) |ψ|2 dσ

)1/2

(15)

si dualul sau V ′. Introducem operatorul A : D(A) ⊂ V ′ → V ′,

〈Aθ, ψ〉V ′,V =∫

Ω(∇ζ · ∇ψ −K(θ) · ∇ψ) dx +

∫

Γα

αζψdσ, (16)

∀ψ ∈ V, unde ζ(x) ∈ β∗(θ(x)) a.p.t. pe Ω.

D(A) = θ ∈ L2(Ω); ∃ζ ∈ V, ζ(x) ∈ β∗(θ(x)) a.p.t. x ∈ Ω (17)

si definim B ∈ L(L2(Γu);V ′) si fΓ ∈ L2(0, T ; V ′) prin

Bu(ψ) = −∫

Γu

uψdσ, fΓ(t)(ψ) = −∫

Γα

f0ψdσ, ∀ψ ∈ V. (18)

Cu aceste notatii introducem problema Cauchy

dθ

dt+ Aθ 3 f + Bu + fΓ, a.p.t. t ∈ (0, T ), (19)

θ(0) = θ(T ). (20)

Teorema 1. Fie

f ∈ L∞(0, T ; V ′), u ∈ L∞(0, T ; L2(Σu)), f0 ∈ L∞(0, T ; L2(Σu)). (21)

Exista cel putin o solutie θ a problemei (11)-(14) cu proprietatile

θ ∈ C([0, T ]; L2(Ω)) ∩W 1,2(0, T ; V ′) ∩ L2(0, T ;V ), (22)θ(x, t) ≤ θs a.p.t. (x, t) ∈ Q, η ∈ L2(0, T ; V ),

unde η(x, t) ∈ β∗(θ(x, t)) a.e. (x, t) ∈ Q. Solutia satisface estimarea∫ T

0

∥∥∥∥dθ

dτ(τ)

∥∥∥∥2

V ′dτ +

∫ T

0‖η(τ)‖2

V dτ (23)

≤ γ0(αm)∫ T

0

(‖f(τ)‖2

V ′ + ‖u(τ)‖2L2(Γu) + ‖f0(τ)‖2

L2(Γα)

)dτ + 1

3

unde γ0 (αm) depinde doar de constantele problemei. In plus, daca

f ≥ 0 a.p.t. (x, t) ∈ Q, (24)

u(x, t) ≤ 0 a.p.t. (x, t) ∈ Σu, (25)

f0(x, t) ≤ 0 a.p.t. (x, t) ∈ Σα, (26)

atunci θ(x, t) ≥ 0 a.p.t. x ∈ Ω, pentru orice t ∈ [0, T ].

Demonstratie. Pentru fiecare ε > 0 fixat introducem problema aproximativa

∂θε

∂t−∆β∗ε (θε) +∇ ·Kε(θ) = f ın Q, (27)

(Kε(θε)−∇β∗ε (θε)) · ν = u pe Σu, (28)

(Kε(θε)−∇β∗ε (θε)) · ν = αβ∗ε (θε) + f0 pe Σα, (29)

θε(x, 0) = θε(x, T ) ın Ω, (30)

ın care β∗ε si Kε sunt aproximatii regulate ale lui β∗ si K. Demonstratia include treipasi:

1. Existenta unei solutii pentru problema Cauchy aproximativa (27)-(29) cuconditia θε(x, 0) = θ0 fixat ın L2(Ω).

2. Demonstrarea existentei unei solutii pentru problema (27)-(30) utilizand unrezultat de punct fix.

3. Trecerea la limita pentru ε → 0 ın (27)-(30).

Existenta unei solutii globale Corolar 2. Fie

f ∈ L∞(R;V ′), u ∈ L∞(R;L2(Γu)), f0 ∈ L∞(R; L2(Γα)),

f(x, t) = f(x, t + T ), u(x, t) = u(x, t + T ), f0(x, t) = f0(x, t + T ) a.p.t. ın Ω×R,

si presupunem ipotezele (24)-(26) din Teorema 1. Atunci problema (1)-(8)-(10) arecel putin o solutie

θ ∈ C(R; L2(Ω)) ∩W 1,2(R; V ′) ∩ L2loc(R; V ),

0 ≤ θ(x, t) ≤ θs a.p.t. (x, t) ∈ Ω×R,

η ∈ L2loc(R; V ), unde η(x, t) ∈ β∗(θ(x, t)) a.p.t. (x, t) ∈ Ω×R.

Asupra proprietatilor de unica continuare aplicate la stabilizareafeedback a ecuatiilor magnetohidrodinamicii (Catalin Lefter)

Fie Ω ⊂ IR2 multime deschisa, marginita, conexa cu frontiera regulata ∂Ω ∈ C2 siω ⊂⊂ Ω o submultime deschisa a lui Ω. Fie Q = Ω × (0,∞), Σ = ∂Ω × (0,∞), n

4

este normala exterioara la ∂Ω. Consideram urmatorul sistem MHD controlat:

∂y

∂t− ν∆y + (y · ∇)y − (B · ∇)B +∇(

12B2 + p) = f + B1u ın Q,

∂B

∂t+ η curl curl B + (y · ∇)B − (B · ∇)y = g + B2v ın Q,

∇ · y = 0, ∇ ·B = 0 ın Q,

y = 0, B · n = 0, rot B = 0 pe Σ

y(·, 0) = y0, B(·, 0) = B0 in Ω.

(31)

Functiile care apar ın sistem au urmatoarele semnificatii fizice: y = (y1, y2, y3)T :Ω×(0, T ) → IR3 este campul vitezelor, p : Ω×(0, T ) → IR reprezinta presiunea, B =(B1, B2, B3)T : Ω×(0, T ) → IR3 este campul magnetic iar f = (f1, f2, f3)T : Ω → IR3

reprezinta densitatea fortelor exterioare ((· · ·)T semnifica matricea transpusa) iardiv g = 0. Coeficientii ν, η sunt vascozitatea cinematica si respectiv rezistivitateamagnetica. Notam cu

rot B =∂B2

∂x1− ∂B1

∂x2

versiunea scalara a operatorului curl .Functiile u, v : ω × (0, T ) → IR ,u, v ∈ U := L2(0, T ; (L2(ω))) sunt controale iar

operatorii B1,B2 sunt operatorii de control.Ne propunem sa studiem stabilizarea feedback ın jurul unei stari stationare. Daca

vrem ca si functiile B1u,B2v sa fie cu suportul ın ω, atunci stabilizarea sistemuluiliniarizat (care este etapa principala a metodei folosite) conduce la o problema deunica continuare pentru sistemul liniarizat adjunct.Vom alege ın acest sens operatorii B1,B2 ın felul urmator:Fie ϕ solutia urmatoarei probleme la limita:

∆ϕ = u− 1|ω|

∫

ωudx ın ω,

∂ϕ

∂n= 0 pe ∂ω

(32)

Definim B1 : L2(ω) → L2(Ω) prin formula

B1u = χω

− ∂ϕ

∂x2

∂ϕ

∂x1

, (33)

unde χω este operatorul de prelungire cu 0 la Ω a functiilor definite pe ω.Fie ψ solutia urmatoarei probleme la limita

∆ψ = v ın ω,

ψ = 0 pe ∂ω(34)

5

Definim B2 : L2(ω) → L2(Ω) prin formula

B2v = χω

− ∂ψ

∂x2

∂ψ

∂x1

. (35)

Sistemul liniarizat adjunct, ın jurul solutiei stationare (y, B), este:

−ζt −∆ζ + (∇C)B − (∇ζ)y + (∇B)T C + (∇y)T ζ +∇π = 0 ın Q,

−Ct −∆C + (∇ζ)B − (∇C)y − (∇B)T ζ − (∇y)T C +∇ρ = 0 ın Q,

∇ · ζ = 0, ∇ · C = 0 ın Q

ζ = 0, C · n = 0, rot C = 0 pe Σ.

(36)

Proprietatea de unica continuare care trebuie demonstrata este:

ζ = 0, rot C = 0 ın ω × (0, T ) ⇒ ζ = 0, C = 0 ın Ω× (0, T ).

Pentru a demonstra aceasta proprietate se reduce sistemul la un sistem de ecuatiiparabolice ın rot ζ, rot C pentru care ne propunem obtinerea de inegalitati de tipCarleman locale (pentru stabilizare) sau globale (daca vrem sa obtinem rezultate decontrolabilitate). S-au obtinut rezultate partiale ın aceasta directie.

Stabilizarea feedback a ecuatiilor Navier-Stokes ın conditii de invarianta.Ecuatii Navier-Stokes cu potential (Adriana-Ioana Lefter)

Se considera ecuatiile Navier-Stokes controlate, cu conditii Dirichlet omogene lafrontiera:

∂y

∂t− ν∆y + (y · ∇)y +∇p + ∂IK(y) = f + u ın Ω× (0, T ),

∇ · y = 0 ın Ω× (0, T ),

y = 0 pe ∂Ω× (0, T ),

y(·, 0) = y0 ın Ω,

(37)

unde Ω este un domeniu marginit din IRd, d = 2, 3, cu frontiera ∂Ω. ∂IK estesubdiferentiala functiei indicatoare a unei multimi convexe si ınchise date K, iaru este un control feedback. Controalele ın forma feedback sunt mai avantajoase ınaplicatii practice, deoarece sunt mai robuste (mai putin sensibile la perturbatii)decat cele cu bucla deschisa.

Se urmareste obtinerea unor rezultate de stabilizare exponentiala cu conservareaanumitor proprietati ale solutiei (ceea ce s-a tradus matematic prin mentinerea

6

solutiei ın multimea K). Avem ın vedere un control feedback distribuit pe ıntregdomeniul , caz ın care s-a demonstrat un rezultat de stabilizare globala. De aseme-nea, vom studia posibilitatea utilizarii unui control feedback dintr-un spatiu liniarfinit dimensional, distribuit ıntr-un subdomeniu al lui Ω; ın acest caz s-ar puteadeduce un rezultat de stabilizare locala.

Pentru a obtine stabilitatea este nevoie sa demonstram existenta globala pentrusistemul controlat rezultat, care este un sistem Navier-Stokes perturbat cu un opera-tor maximal monoton de tip subdiferentiala. De aceea, o prima etapa este enuntareaunor teoreme de existenta si, unde este posibil, unicitate pentru solutiile ecuatiilorNavier-Stokes cu potential. Rezultate ın acest sens au fost deja obtinute de V. Barbu(Journal of Mathemathical Analysis and Applications, 2001), A.I. Lefter (Abstract& Applied Analysis, 2007).

7

Stochastic nonlinear diffusion equations withsingular diffusivity

Viorel Barbu ∗,University Al. I. Cuza

andInstitute of Mathematics “Octav Mayer”, Iasi, Romania ,

Giuseppe Da Prato †,Scuola Normale Superiore di Pisa, Italy

andMichael Rockner ‡

Fakultat fur Mathematik,Universitat Bielefeld, D-33501 Bielefeld, Germany

andDepartment of Mathematics and Statistics,

Purdue University, W. Lafayette, IN 47907, USA.

March 20, 2008

Abstract

This paper is concerned with the stochastic diffusion equationdX(t) = div[sgn(∇(X(t))]dt +

√Q dW (t) in (0,∞) × O where O

is a bounded open subset of Rd, d = 1, 2, W (t) is a cylindrical Wienerprocess on L2(O) and sgn(∇X) = ∇X/|∇X|d if ∇X 6= 0 and sgn(0) = v ∈ Rd : |v|d ≤ 1. The multivalued and highly singular dif-fusivity term sgn(∇X) introduces nonlocal diffusion effects and thesolution X = X(t) might be viewed as the stochastic flow generatedby the gradient of the total variation ‖DX‖. Our main result says

∗Supported by the Grant PN-II ID-404(2007-2010) of Romanian Minister of Research.†Supported by the research program “Equazioni di Kolmogorov” from the Italian “Min-

istero della Ricerca Scientifica e Tecnologica”‡Supported by the SFB-701 and the BIBOS-Research Center.

1

that this problem is well posed in the space of processes with boundedvariation in the spatial variable ξ. The above equation is relevant formodeling crystal growth as well as for total variation based techniquesin image restoration.

2000 Mathematics Subject Classification AMS: 60H15, 35K55Key words: Stochastic diffusion equation, bounded variation, Wiener pro-cess.

1 Introduction

We are concerned here with the following stochastic diffusion equation onH = L2(O)

dX(t) = div[sgn (∇(X(t))]dt+√Q dW (t) in (0,∞)× O

X(t) = 0 on ∂O × (0, T )

X(0) = x in O,

(1.1)

where O is a bounded open subset of Rd, d = 1, 2, W (t) is a cylindricalWiener process on L2(O) of the form

W (t) =∞∑k=1

βk(t)ek, t ≥ 0,

where βk is a sequence of mutually independent real Brownian motionson a filtered probability spaces (Ω,F , Ftt≥0,P) (see [8]) and ek is anorthonormal basis in H = L2(O). The operator Q ∈ L(H) is symmetric,self-adjoint and of trace class. To be more specific we shall take ek, k ∈ N,to be the eigenfunction of Q, i.e.,

Qek = λkek, k ∈ N.

For example, we can take Q = A−1−δ where δ = 0 if d = 1, δ > 0 if d = 2and

A = −∆, D(A) = H2(O) ∩H10 (O). (1.2)

The multi-valued function u→ sgn u from Rd into Rd is defined by

sgn u =

u|u|d, if u 6= 0,

v ∈ Rd : |v|d ≤ 1, if u = 0.

2

(Here | · |d is the Euclidean norm and 〈·, ·〉d is the Euclidean inner product.)Equation (1.1) is relevant in material science to describe the motion of

grain boundaries and in image processing. The first model is concerned withfaced growth of cristals derived from spatially homogeneous energy

E(X) =

∫O

|∇X|d dξ,

which formally leads to the gradient system

dX(t) = −div

(∇X(t)

|∇X(t)|d

)dt, t ≥ 0, (1.3)

or to (1.1) in presence of the Gaussian perturbation√Q dW . (We refer

to [11], [12], [17] for the presentation and treatment of the correspondingdeterministic models.)

The total variation based image restoration model based on E(X) hasbeen proposed in [19] (see also [7], [8], [16], [17]), i.e., as the solution to theminimization problem,

min

∫O

(|∇X|d +

1

2|X − f |2

)dξ, (1.4)

where f is the given image and X is the restored image. The minimizationproblem (1.4) leads to a flow X = X(t) generated by the evolution equation

dX(t) = −div

(∇X(t)

|∇X(t)|d

)dt− (X(t)− f(t))dt, t ≥ 0, (1.5)

which perturbed by a Gaussian process leads to equation (1.1). This restora-tion model was designed with the explicit aim to preserve edges and sharpdiscontinuities of the image.

In both equations (1.4) and (1.5) the discontinuous map u→ u|u|d

shouldbe replaced of course by its multi-valued maximal monotone graph u→ signu obtained by filling the jumps. It should also be mentioned that equation(1.1) (as well as the deterministic version (1.4) or (1.5)) is highly nonlinearand so the diffusion effect is non local. For instance in 1-D equation (1.1)has the form

dX(t) = −δ(∇X(t))∆X(t)dt =√Q dW (t), t ≥ 0, (1.6)

where δ is the Dirac measure at zero on O. Of course this is only a formalrepresentation because the multiplier δ(∇X(t)) is not well defined and so(1.6) does not make sense.

3

The main result established here (see Theorem 3.2 below) is concerned,however, with existence and uniqueness of a variational solution for d = 1, 2in the space of functions with bounded variation in the spatial variable ξ ∈O. A similar result is proved in Section 6 for equation (1.1) with linearmultiplicative noise along with positivity of solutions.

It should be noted that, though equation (1.1) arises in a variational set-ting, its existence theory is not covered by the classical results of E. Pardoux[14] or N. Krylov and B. Rozovskii [13] (see also [15], [18]). Indeed a generalstochastic equation of the form

dX(t) = div (a(∇(X(t)))dt+√Q dW (t) in (0,∞)× O

X(t) = 0 on ∂O × (0, T )X(0) = x in O,

(1.7)

where a : Rd → Rd is a monotonically increasing, continuous and coercivevector field with polynomial growth can be solved in the abstract variationalsetting

dX(t) + AX(t)dt =√Q dW (t)

X(0) = x(1.8)

where A : V → V ′ is a nonlinear monotone and demi-continuous operator(see [2]) such that

(Ax, x) ≥ ω‖x‖pV − ω1|x|2H ,

‖Ax‖V ′ ≤ C1‖x‖p′

V + C2,1

p+

1

p′= 1.

(Here V ⊂ H ⊂ V ′ is a classical variational Gelfand triple.)This is exactly the variational stochastic framework developed in [14],

[13], which however does not apply in this situation. As a matter of fact, thesituation considered here is a limit case of (1.7)-(1.8) and this fact will beexploited later to obtain existence of solutions for (1.1).

Notations. Everywhere in the following H is the Hilbert space L2(O)with the scalar product (·, ·) and the norm | · |). Lp(O), p ≥ 1, and V =H1

0 (O), V ′ = H−1(O) are the standard spaces of integrable functions andSobolev spaces on O. By CW ([0, T ];H) we shall denote the space of allcontinuous functions from [0, T ] to L2(Ω;H) which are adapted to W . Thespaces L2

W (0, T ;V ) and L2W (0, T ;V ′) are similarly defined. We shall denote

by ‖ · ‖ the norm of V , and spatial variables in O are denoted by ξ.

4

Existence of strong solutions for stochasticporous media equation under general

monotonicity conditions

Viorel Barbu ∗,University Al. I. Cuza

andInstitute of Mathematics “Octav Mayer”, Iasi, Romania ,

Giuseppe Da Prato †,Scuola Normale Superiore di Pisa, Italy

andMichael Rockner ‡

Faculty of Mathematics, University of Bielefeld, Germanyand

Department of Mathematics and Statistics, Purdue University,U. S. A.

February 5, 2008

Abstract

This paper addresses existence and uniqueness of strong solutionsto stochastic porous media equations dX −∆Ψ(X)dt = B(X)dW (t)in bounded domains of Rd with Dirichlet boundary conditions. HereΨ is a maximal monotone graph in R×R (possibly multivalued) with

∗Supported by PN-II ID.404 (2007-2010).†Supported by the research program “Equazioni di Kolmogorov” from the Italian “Min-

istero della Ricerca Scientifica e Tecnologica”‡Supported by the SFB-701 and the BIBOS-Research Center.

1

the domain and range all of R. Compared with the existing literatureon stochastic porous media equations no growth condition on Ψ isassumed and the diffusion coefficient Ψ might be multivalued anddiscontinuous. The latter case is encountered in stochastic models forself-organized criticality or phase transition.

AMS subject Classification 2000: 76S05, 60H15.Key words: stochastic porous media equation, Wiener process, convex

functions, Ito’s formula.

1 Introduction

This work is concerned with existence and uniqueness of solutions to stochas-tic porous media equations

dX(t)−∆Ψ(X(t))dt = B(X(t))dW (t) in (0, T )× O := QT ,Ψ(X(t)) = 0 on (0, T )× ∂O := ΣT ,X(0) = x in O,

(1.1)

where O is an open, bounded domain of Rd, d ≥ 1, with smooth boundary∂O, W (t) is a cylindrical Wiener process on L2(O), while B is a Lipschitzcontinuous operator from H := H−1(O) to the space of Hilbert–Schmidtoperators on L2(O). (See Hypothesis H2 below).

The function Ψ : R → R (or more generally the multivalued functionΨ : R → 2R) is a maximal monotone graph in R× R. (See the definition inSection 1.1 below).

Existence results for equation (1.1) were obtained in [8] (see also [3],[4])in the special case B =

√Q, with Q linear nonnegative, Tr Q < +∞ and

Ψ ∈ C1(R) satifying the growth condition

k3 + k1|s|r−1 ≤ Ψ′(s) ≤ k2(1 + |s|r−1), s ∈ R, (1.2)

where k1, k2 > 0, k3 ∈ R, r > 1.Under these growth conditions on Ψ, equation (1.1) covers many impor-

tant models describing the dynamics of an ideal gas in a porous medium(see e.g. [1]) but excludes, however, other significant physical models such asplasma fast diffusion ([5]) which arises for Ψ(s) =

√s and phase transitions

or dynamics of saturated underground water flows (Richard’s equation). In

2

the later case multivalued monotone graphs Ψ might appear as in [12]. Re-cently in [15] (see also [14]) the existence results of [8] were extended to thecase of monotone nonlinearities Ψ such that s 7→ sΨ(s) is (comparable to) a∆2-regular Young function (cf. assumption (A1) in [15]) thus including thefast diffusion model. As a matter of fact, in the line of the classical work ofN. Krylov and B. Rozovskii [10] the approach used in [15] is a variational onei.e. one considers the stochastic equation (1.1) in a duality setting inducedby a functional triplet V ⊂ H ⊂ V ′ and this requires to find appropriatespaces V and H. This was done in [15] in an elaborate way even with Ounbounded and with ∆ replaced by very general (not necessarily differential)operators L.

The method we use here is quite different and essentially an L1-approachrelying on weak compacteness techniques in L1(QT ) via the Dunford-Pettistheorem which involve minimal growth assumptions on Ψ. Restricted to sin-gle valued continuous functions Ψ the main result, Theorem 2.2 below, givesexistence and uniqueness of solutions only assuming that lims→+∞Ψ(s) =+∞, lims→−∞Ψ(s) = −∞, Ψ monotonically increasing and

lim sup|s|→+∞

∫ −s0

Ψ(t)dt∫ s0

Ψ(t)dt< +∞. (1.3)

We note that the assumptions on Ψ in [15]) imply our assumptions. In thissense, under assumption (H2) below on the noise, the results on this paperextend those in [15] in case L = ∆ if O is bounded and if the coefficients donot depend on (t, ω). The latter two were not assumed in [15]. On the otherhand a growth condition on Ψ is imposed in [15] (cf. [15, Lemma 3.2]) whichis not done here. Another main progress of this paper is that Ψ is no longerassumed to be continuous, it might be multivalued and with exponentialgrowth to ±∞ (for instance of the form exp (a|x|p)). We note that (1.3) isnot a growth condition at +∞ but a kind of symmetry condition about thebehaviour of Ψ at ±∞. If Ψ is a maximal monotone graph with potential j(i.e. Ψ = ∂j) then (1.3) takes the form (see Hypothesis (H3) below)

lim sup|s|→+∞

j(−s)j(s)

< +∞.

Anyway this condition is automatically satisfied for even monotonically in-creasing functions Ψ or e.g. if a condition of the form (1.2) is satisfied. We

3

note, however, that because of our very general conditions on Ψ the solu-tion of (1.1) will be pathwise only weakly continuous in H. The question ofpathwise strong continuity of solutions, however, remains open. The mainreason is the absence of a variational setting for problem (1.1) (see [10],[14])in the present situation. Other major technical difficulties encountered herein the proofs are that e.g. the integration by parts formula or Ito’s formula(see Lemmas 3.1 and 3.2 below) cannot be applied directly because of thesame reason. It should be said, however, that the L1 approach used here,which allows to treat very general nonlinearities, is applicable to determin-istic equations as well and seems to be new also in that context. On theother hand, the existence for the deterministic part of equation (1.1) is animmediate consequence of the Crandall–Liggett generation theorem for non-linear semigroups of contractions (see [2]) which is, however, not applicableto stochastic equations.

1.1 Notations

O is a bounded open subset of Rd, d ≥ 1, with smooth boundary ∂O. Weset

QT = (0, T )× O, ΣT = (0, T )× ∂O.

Lp(O), Lp(QT ), p ≥ 1, are standard Lp- function spaces and H10 (O), Hk(O)

are Sobolev spaces on O. By H := H−1(O) we denote the dual of H10 (O)

with the norm and the scalar product given by

|u|−1 := (A−1u, u)1/2, 〈u, v〉−1 = (A−1u, v),

respectively, where (·, ·) is the pairing between H10 (O) and H−1

0 and thescalar product of L2(O). Here A denotes the Laplace operator with Dirichlethomogeneous boundary conditions, i.e.

Au = −∆u, u ∈ D(A) = H2(O) ∩H10 (O). (1.4)

Given a Hilbert space U , the norm of U will be denoted by | · |U and thescalar product by (·, ·)U . By C([0, T ];U) we shall denote the space of U -valued continuous functions on [0, T ] and by Cw([0, T ];U) the space of weaklycontinuous functions from [0, T ] to U .

Given two Hilbert spaces U and V we shall denote the space of linearcontinuous operators from U to V by L(U, V ) and the space of Hilbert-

4

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a

CONVERGENCE OF THE FINITE DIFFERENCE SCHEME FORA FASTDIFFUSION EQUATION IN POROUS MEDIA

Cornelia Ciutureanu, Institute of Mathematical Statistics and Applied Math-ematics, Bucharest, Romania

Gabriela Marinoschi1 , Institute of Mathematical Statistics and Applied Math-ematics, Bucharest, Romania

Keywords Boundary value problems for nonlinear parabolic PDE; PDEwith multivalued right-hand sides; Free boundary problems for PDE; Stabilityand convergence of numerical methods; Flows in porous media.

AMS Subject Classication 35K60; 35R70; 35R35; 65M12; 76S05.

Abstract.We are concerned with an implicit scheme for the nite di¤erence solution

to a nonlinear parabolic equation with a multivalued coe¢ cient which describesthe fast di¤usion in a porous medium. The boundary conditions contain themultivalued function as well. We prove the stability and the convergence of thescheme, emphasizing the precise nature of convergence in this specic case andcompute the error level of the approximating solution. The method is aimed tosimplify the numerical computations for the solutions to equations of this type,without performing an approximation of the multivalued function. The theoryis illustrated by numerical results.

1 Introduction

Apart from other di¤usion processes, the di¤usion of a uid in a porous medium,consisting in a solid matrix and a void part, has a specic behavior in relationwith the medium structure, specically with the pore geometry and the volumeof voids. Under certain conditions depending on the soil structure, the rate atwhich the uid may be supplied on a part of the domain boundary, the initialdistribution of the uid in the soil, the presence of underground sources andthe boundary permeability, a part or more of the ow domain may saturateand the pores in these regions become completely lled with the uid. In thesaturated part the uid concentration in the pores attains the maximum value,or the saturation value, which will be denoted in this paper by s: This is apositive number equal to the value of the medium porosity (assuming e.g., that

1Address correspondence to Gabriela Marinoschi, Institute of Mathematical Statisticsand Applied Mathematics, Calea 13 Septembrie 13, Bucharest, Romania; E-mail: gabi-marinoschi@yahoo.com, gmarino@acad.ro

1

it is constant) and it is specic to each material. At the interface between thesaturated and unsaturated parts a free boundary is formed and it may beginto advance towards the unsaturated part. In the model below its presence ischaracterized by a multivalued function.We consider an open bounded subset of RN (N 2 N = f1; 2; :::g); with

the boundary := @ piecewise smooth, composed by two disjoint parts uand : We denote the space variable by x := (x1; :::; xN ) 2 and the time byt 2 (0; T ); with T nite. We are concerned with the boundary value problemconsisting of a di¤usion equation with a transport term

@

@t() +r K() 3 f in Q := (0; T ) ; (1.1)

with the initial datum(0; x) = 0 in : (1.2)

Various boundary conditions on (0; T ) can be attached but we shall illus-trate the theory by considering nonhomogeneous Neumann and Robin boundaryconditions of the form

(K()r()) 3 u on u := (0; T ) u; (1.3)

(K()r()) () 3 f0 on := (0; T ) ; (1.4)

where is the normal vector to : The second condition expresses the fact thatthe boundary is semipermeable (with characterizing its variable permeabil-ity). The meaning of the equations (1.1)-(1.4) will be explained a little furtherin the denition of the solution.In this problem : R! R is a graph dened as

(r) :=

8<:R r0()d; if r < s;

[Ks ;+1); if r = s;

?; if r > s;(1.5)

where : (1; s) ! R is a positive, di¤erentiable, strictly monotonicallyincreasing function, which blows up at r = s, but having the integral nite atthis point. Namely we set

(r) > 0; for each r < s; (r) := for r 0; (1.6)

limr%s

(r) = +1, (1.7)

limr%s

Z r

0

()d = Ks : (1.8)

Consequently, has the properties

((r) (r)) (r r) (r r)2; for every r; r 2 (1; s]; (1.9)

limr!1

(r) = 1; (1.10)

2

limr%s

(r) = Ks : (1.11)

A particular case of this model was set up in [1] for describing the saturated-unsaturated water inltration in soils. The hypotheses (1.7)-(1.8) reveal thecharacter of fast di¤usion (see [2], [1]).We notice that ()1 : R!(1; s] ! is single-valued, monotonically

increasing on (1;Ks ) and constant for r 2 [K

s ;+1); i.e., ()1(r) = s:Also, by (1.9) it follows that ()1 is Lipschitz continuous on R with theconstant 1

:

The vector K(r) = (K1(r); :::;KN (r)) has each component nonnegative andbounded,

0 Ki(r) Ksi for any r 2 R: (1.12)

Moreover we consider that

Ki(r) = 0 for r 0 and Ki(r) = Ksi for r s: (1.13)

In the above relationships ; s;Ks and K

si are positive known constants

and we shall denoteKs = max

i=1;:::;NfKs

i g: (1.14)

We shall study the case in which Ki is Lipschitz, satisfying

jKi(r)Ki(r)j Mi jr rj for all i = 1; :::; N: (1.15)

Finally, we assume that : ! [m; M ] is a positive continuous function,i.e., m > 0:

1.1 Functional framework and preliminary results

For simplicity, we shall denote the scalar product and norm in L2() with nosubscript, i.e., by (; ) and kk ; respectively, and consider V = H1() with thenorm

k kV =Z

jr (x)j2 dx+Z

(x) j (x)j2 d1=2

; (1.16)

which is equivalent with the standard norm on the Hilbert space H1(): Indeed,there exist the positive constants denoted cH ; c ; cu depending on

1=2m ; the

domain and the dimension N; such that

k kH1() cH k kV ; k kL2() c k kV ; k kL2(u) cu k kV ;(1.17)

for any 2 V: These relationships are easily deduced via the trace theorem andPoincaré inequality (see [1], page 137).We endow the dual V 0 of V with the scalar product

(; )V 0 = h; iV 0;V ; for any ; 2 V 0; (1.18)

3

WELL-POSEDNESS FOR A HYPERBOLIC-PARABOLICCAUCHY PROBLEM ARISING IN

POPULATION DYNAMICS1

MIMMO IANNELLI

Mathematics Department, University of Trento, via Sommarive 14, 38050Povo (Trento), Italy, e-mail: iannelli@science.unitn.it

GABRIELA MARINOSCHI

Institute of Mathematical Statistics and Applied Mathematics, Calea 13Septembrie 13, 050711 Bucharest, Romania, e-mail: gmarino@acad.ro

Abstract. In some previous work [1]-[3], the authors have considered the dif-fusion of a population in a multilayered habitat, taking into account both thedemographic structure, due to the age distribution of the individuals, and thespatial distribution related to population spread and diffusion. The developmentof the mathematical framework for this kind of problems leads the attention ona linear problem which incorporates all the features that make this kind of prob-lems unusual. This model is represented by a system of PDE with discontinuouscoefficients and data and sources at the boundaries between layers with differentstructure. In this paper we provide well-posedness to such a problem together withregularity conditions, using m-accretiveness and fixed points techniques.

1 Introduction

The modelling of age-structured populations, spreading in a geographical region,leads to analyze non linear P.D.E. with non-local terms that are strictly related tohereditary effects such as those represented in the framework of integral equationsof Volterra type. In particular, in some previous work [1]-[3], the authors have con-sidered the diffusion of a population in a multilayered habitat, taking into accountboth the demographic structure, due to the age distribution of the individuals,and the spatial distribution related to population spread and diffusion.

1AMS Subject classification: 35K90, 35M10, 35R05, 92D25

1

However, the same problems studied in [1]-[3] lead to consider several techni-cal problems mainly focused on well-posedness of a certain linear system whichincorporates all the features that make this kind of problems unusual. Namelythe hyperbolic character with respect to the age variable (denoted by a ∈ [0, a+]),interacts with the parabolic features due to the spatial one (denoted by y ∈ [0, L]).The model is represented by a system of PDE with discontinuous coefficients anddata and sources at the boundaries between layers with different structure. Thus,in the present paper we provide a systematic approach to the problem, stating somebasic results in the framework that allows to treat various problems in populationdynamics such as the control problems approached in [3].

The same results may be of interest for other modelling problems with hyperbolic-parabolic behaviour and discontinuous coefficients and data.

We consider the domain Ω = (0, a+) × (y0, yn) composed of n parallel layers

Ωj = (0, a+) × (yj−1, yj), j = 1, 2, ...n,

and denoteΓyj

= (a, yj); a ∈ (0, a+), j = 0, ..., n,

where Γyjwith j = 0 and j = n are the exterior boundaries, while Γyj

withj = 1, . . . , n − 1 represent the interior ones. The time t runs within the finiteinterval (0, T ).

The model we are going to analyze reads

∂qj

∂t+∂qj

∂a−

∂

∂y

(Kj(a)

∂qj

∂y

)+ Tj(t)qj = fj

in (0, T ) × Ωj, j = 1, ..., n

qj(0, a, y) = q0j (a, y) in Ωj, j = 1, ..., n,

qj(t, 0, y) = Fj(t, y) in (0, T ) × (yj−1, yj), j = 1, ..., n,qj = qj+1 on (0, T ) × Γyj

, j = 1, ...n − 1,

Kj(a)∂qj

∂y= Kj+1(a)

∂qj+1

∂y+ kj(t, a)

on (0, T ) × Γyj, j = 1, ...n − 1,

−K1(a)∂q1

∂y= k0(t, a) on (0, T ) × Γy0

,

Kn(a)∂qn

∂y= kn(t, a) on (0, T ) × Γyn

,

(1.1)

assuming that

q0j ∈ L2(Ωj), fj ∈ L2(0, T ;L2(Ωj)), Fj ∈ L2(0, T ;L2(yj−1, yj)),

Tj ∈ C([0, T ];L(L2(Ωj), L2(Ωj)), kj ∈ L2(0, T ;L2(0, a+)),

Kj ∈ L∞(0, a+),Kj(a) ≥ K0 > 0 a.e. in (0, a+).

(1.2)

2

We introduce the generic notation defining the function Φ(t, a, y) in the set(0, T ) × Ω, by

Φ(t, a, y) = Φj(t, a, y), for y ∈ (yj−1, yj) (1.3)

where Φj stands for any function defined on (0, T ) × Ωj (see also the notations(25)-(32) from [1]). Vice-versa, if Φ is a function defined on (0, T ) × Ω, we defineΦj in (0, T ) × Ωj by setting

Φj(t, a, y) = Φ(a, t, y), y ∈ (yj−1, yj). (1.4)

Then, denoting

HΩ = L2(Ω), H = L2(0, L), V = H1(0, L),

with the standard norms, and using assumptions (1.2) we have

q0 ∈ HΩ, f ∈ L2(0, T ;HΩ), F ∈ L2(0, T ;H),

k ∈ L2(0, T ;L2(0, a+)), K ∈ L∞(0, a+).(1.5)

Moreover, we may define the operator T (t) : HΩ → HΩ setting

(T (t)θ)j = (Tj(t)θj),

where θ ∈ HΩ and we have used both (1.3) and (1.4). Then T (t) is continuous onHΩ, for each t ∈ [0, T ], and we have that

‖T (t)θ‖HΩ≤M ‖θ‖HΩ

,∀θ ∈ HΩ, M > 0. (1.6)

Concerning the previous problem we are lead to adopt the following definition:

Definition 1.1. A weak solution to problem (1.1) is a function

q ∈ C([0, T ];HΩ) ∩ L2(0, T ;L2(0, a+, V )) ∩ C([0, a+];L2(0, T ;H)) (1.7)

satisfying

−

∫ T

0

∫

Ωq∂ψ

∂tdydadt −

∫ T

0

⟨∂ψ

∂a(t), q(t)

⟩dt

+

∫

Ωq(T, a, y)ψ(T, a, y)dady +

∫ T

0

∫ L

0q(t, a+, y)ψ(t, a+, y)dydt

−

∫

Ωq0(a, y)ψ(0, a, y)dyda −

∫ T

0

∫ L

0F (t, y)ψ(t, 0, y)dydt

+

∫ T

0

∫

Ω

(K(a, y)

∂q

∂y

∂ψ

∂y+ (T (t)q(t)) (a, y)ψ(t, a, y)

)dydadt

=

∫ T

0

∫

Ωfψ dydadt +

∫ T

0

∫ a+

0

n∑

j=0

kj(t, a)ψ(t, a, yj)dadt,

(1.8)

3

for any function ψ such that

ψ ∈ L2(0, T ;L2(0, a+;V )) ∩W 1,2(0, T ;HΩ),

ψa ∈ L2(0, T ;L2(0, a+;V ′)).

(1.9)

We specify that 〈·, ·〉 denotes the duality between L2(0, a+;V ) and L2(0, a+;V ′)defined as

〈h, g〉 =

∫ a+

0〈h(a), g(a)〉V ′,V da, ∀h ∈ L2(0, a+;V ′), g ∈ L2(0, a+;V ),

where 〈·, ·〉V ′,V is the pairing between V ′ and V .The aim of this paper is to prove existence and uniqueness to (1.1). To this end

we will proceed by steps, considering different particular cases of the full problem.

2 The basic problem

To approach our problem we first introduce the linear operator

B0 : D(B0) ⊂ L2(0, a+;V ) → L2(0, a+;V ′)

on the domain

D(B0) = v ∈ L2(0, a+;V ); va ∈ L2(0, a+;V ′), v(0, y) = 0, (2.1)

where we note that the condition v(0, y) = 0 is meaningful because v ∈ D(B0)implies that v ∈ C([0, a+];H).

We define B0 by setting, for v ∈ D(B0)

〈(B0v)(a), ψ〉V ′,V = 〈va(a, ·), ψ〉V ′,V +

∫ L

0K(a, y)vy(a, y)ψy(y)dy, (2.2)

for a.e. a ∈ [0, a+] and any ψ ∈ V .Then we define the operator B : D(B) ⊂ HΩ → HΩ, by

Bv = B0v,

on the domainD(B) = v ∈ D(B0); Bv ∈ HΩ.

4

1

NONLINEAR FEEDBACK CONTROLLERS FOR THE NAVIER-STOKESEQUATIONS

Adriana-Ioana LefterFaculty of Mathematics, “Al.I.Cuza” University, Iasi, Romania;

“O.Mayer” Institute of Mathematics, Iasi, Romania.1

Abstract. In this paper we prove the feedback stabilization of the Navier-Stokesequations preserving the invariance of a given convex set. To this aim we first de-duce an existence theorem concerning weak solutions for the Navier-Stokes systemperturbed with a subdifferential.2000 Mathematics Subject Classification: 76D05, 76D03, 47H05, 47J05, 35B35.Keywords and phrases: Navier-Stokes equations, strong and weak solution, mono-tone operator, feedback controller, exponential stability.

Let T > 0 and let Ω ⊂ IRd, d = 2, 3 be an open and bounded domain, witha smooth boundary ∂Ω (of class C2, e.g.). Consider the controlled Navier-Stokesequations

∂y

∂t(x, t)− ν∆y(x, t) + (y(x, t) · ∇)y(x, t)

+∇p(x, t) = f0(x, t) + u0(x, t), (x, t) ∈ Q = Ω× (0, T ),

div y(x, t) = 0, (x, t) ∈ Q,

y(x, t) = 0, (x, t) ∈ Σ = ∂Ω× (0, T ),

y(x, 0) = y0(x), x ∈ Ω,

(1)

where y = (y1, y2, . . . , yd) is the velocity field, p is the scalar pressure, the densityof external forces is f0 = (f01, f02, . . . , f0d), the constant ν > 0 is the kinematicviscosity coefficient and u0 = (u01, u02, . . . , u0d) is a distributed control on Ω.

We intend to find feedback controllers which would insure the exponential sta-bilization of the Navier-Stokes equations in invariance conditions. To this aim, letΦ be an operator satisfying the following hypotheses:(h1) Φ = ∂ϕ, where ϕ : H → IR is a lower semicontinuous proper convex

function (hence Φ is a maximal monotone operator in H ×H);(h2) 0 ∈ D(Φ);(h3) there exist two constants γ ≥ 0, α ∈ (0, (1/ν)) such that

(Aw,Φλ(w)) ≥ −γ(1 + |w|2)− α|Φλ(w)|2, ∀λ > 0,∀w ∈ D(A), (2)

where A is the Stokes operator and Φλ : H → H is the Yosida approximation of Φ.In order to prove stability we need a global solution for the controlled problem.

The results we have previously proved for strong solutions say that in dimensiond = 3 we may get only local strong solutions. That is why a stability resultin the three dimensional case requires an existence theorem for weak solutions.We give such a result in Section 2. In Section 3 we state and prove exponentialstabilization theorems. The first example uses a feedback controller distributed onthe entire domain (§3.1) and the stability result is global. In the second examplethe feedback controller belongs to a finite dimensional space and it is distributedon a subdomain(§3.2); the stability result is local.

1MAILING ADRESS: Faculty of Mathematics, “Al.I.Cuza” University, Bd.Carol I nr.11,

700506, Iasi, Romania. FAX: +40 232 201160 E-MAIL: ilefter@uaic.ro

This work was supported by CNCSIS Grant PN II ID 404/2008.

1

On a unique continuation property related to the

boundary stabilization of magnetohydrodynamic

equations

Catalin-George Lefter∗

Abstract

We prove that the MHD system in space dimension 2 is exponentiallystabilizable with boundary controllers. This result relies on a unique con-tinuation property for the adjoint linearized MHD system,

2000 Mathematics Subject Classification: 93D15, 35Q35, 76W05, 35Q30,35Q60, 93B07.

Keywords and phrases: Magnetohydrodynamic equations, feedback sta-bilization, stream function, unique continuation

1 Introduction

This paper is concerned with a unique continuation result which is appliedto the study of the local exponential stabilization for the magnetohydro-dynamic (MHD) equations in space dimension 2, with boundary feedbackcontrollers. We reduce the study, by a usual procedure, to the case ofinternally distributed controllers with compact support. It is howevernecessary to have divergence free controllers in the second equation.

The method we use for the stabilization is to linearize the systemaround the stationary state and then construct a feedback controller stabi-lizing the linear system. Then we show that the same controller stabilizes,locally in a specified space, the nonlinear system.

The stabilization of the linearized system is obtained via a spectral de-composition of the elliptic part. Thus, we project the system on the stableand unstable subspaces corresponding to this decomposition. The unsta-ble subspace is finite dimensional and the corresponding projected systemis exactly controllable, as a consequence of the approximate controllabil-ity of the original linearized system; one may thus construct a feedbackstabilizing this finite dimensional linear system. The projected system onthe stable subspace is asymptotically stable and the feedback for the fi-nite dimensional system is stabilizing the initial linearized equations. Theapproximate controllability of the linearized system is a consequence ofthe unique continuation result we describe below:

Let Ω ⊂ R2 be a bounded, simply connected open set with C2 bound-ary ∂Ω and ω ⊂⊂ Ω an open subset of Ω. Let Q = Ω × (0,∞), Σ =

∗This work was supported by the Minister of Education research grant PN II ID 404/2007-2010

1

∂Ω × (0,∞), n is the unit exterior normal to ∂Ω. We consider in thepaper the following MHD controlled system:

∂y

∂t− ν∆y + (y · ∇)y − (B · ∇)B +∇(

1

2B2 + p) = f + χωu in Q,

∂B

∂t+ η curl curl B + (y · ∇)B − (B · ∇)y = g + χωv in Q,

∇ · y = 0, ∇ ·B = 0 in Q,

y = 0, B · n = 0, rot B = 0 on Σ

y(·, 0) = y0, B(·, 0) = B0 in Ω.(1)

The functions that appear in the system have the following physical mean-ing: y = (y1, y2)T : Ω×(0, T )→ R2 is the velocity field, p : Ω×(0, T )→ Ris the pressure, B = (B1, B2)T : Ω × (0, T ) → R2 is the magnetic fieldand f = (f1, f2)T : Ω → R2 represents the density of the exterior forces((· · ·)T means the matrix transpose). The coefficients ν, η are the positivekinematic viscosity and the magnetic resistivity coefficients. We denoteby

rot B =∂B2

∂x1− ∂B1

∂x2

the scalar version of the curl operator. We also note the formula curl curl B =−∆B+∇div B. So, knowing the fact that the solution B will remain di-vergence free, we may write −∆B instead of curl curl B but we will keepthe notation to keep in mind that the system models in fact phenomenain a 3 dimensional cylindrical body and the data depend only on x1, x2

variables.The functions u, v : ω × (0, T ) → R2 ,u, v ∈ U := L2(0, T ; (L2(ω))2)

are the controllers and χω : L2(ω)→ L2(Ω) is the operator extending thefunctions in L2(ω) with 0 to the whole Ω. Moreover, div v = 0 in ω andv · n = 0 on ∂ω, where n is the normal to the boundary of ω.

Let (y, B) be a stationary solution. Let ξ =

(ζC

)with ζ, C written

as colon vectors. Then, the dual of the linearized system is:

−ζt −∆ζ + (∇C)B − (∇ζ)y + (∇B)TC + (∇y)T ζ +∇π = 0 in Q,

−Ct −∆C + (∇ζ)B − (∇C)y − (∇B)T ζ − (∇y)TC +∇ρ = 0 in Q,

∇ · ζ = 0, ∇ · C = 0 in Q

ζ = 0, C · n = 0, rot C = 0 on Σ.(2)

The central result is the following unique continuation property:

ζ = 0, rot C = 0 in ω × (0, T )⇒ ζ = 0, C = 0 in Ω× (0, T ). (3)

2

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 81–84http://france.elsevier.com/direct/CRASS1/

Probability Theory

Finite time extinction for solutions to fast diffusion stochastic porousmedia equations

Viorel Barbu a, Giuseppe Da Prato b, Michael Röckner c,d

a Institute of Mathematics “Octav Mayer”, 700506 Iasi, Romaniab Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

c Faculty of Mathematics, University of Bielefeld, Germanyd Department of Mathematics and Statistics, Purdue University, USA

Received 26 August 2008; accepted 27 November 2008

Available online 18 December 2008

Presented by Paul Malliavin

Abstract

We prove that the solutions to fast diffusion stochastic porous media equations have finite time extinction with strictly positiveprobability. To cite this article: V. Barbu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Extinction en temps fini pour les solutions des équations des milieu poreux avec diffusion rapide. Nous prouvons l’extinc-tion avec une probabilité strictement positive pour les solutions des équations des milieux poreux avec diffusion rapide. Pour citercet article : V. Barbu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

1. Introduction

Consider the stochastic porous media equationdX(t) − ρ(|X|α(t) signX(t))dt − (Ψ (X(t))dt = σ(X(t))dW(t), in (0,∞) × O,

X = 0 on (0,∞) × ∂O, X(0, x) = x on O,(1)

where ρ > 0, α ∈ (0,1), Ψ is a continuous monotonically nondecreasing function of linear growth and σ(X)dW =∑∞k=1 μkXek dβk , t 0, where βk is a sequence of independent real Brownian motions on a filtered proba-

bility space (Ω, F , Ft ,P) and ek is an orthonormal basis in L2(O) which for convenience will be taken asthe eigenfunction system for the Laplace operator with Dirichlet boundary conditions, i.e., −ek = λkek in O,ek = 0 on ∂O, where O is an open and bounded subset of R

d , with smooth boundary ∂O. We shall assume that∑∞k=1 μ2

kλ2k < ∞. Eq. (1) for 0 < α < 1 is relevant in the mathematical modelling of the dynamics of an ideal gas in

E-mail addresses: vb41@uaic.ro (V. Barbu), daprato@sns.it (G. Da Prato), roeckner@Mathematik.Uni-Bielefeld.de (M. Röckner).

1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.11.018

82 V. Barbu et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 81–84

a porous medium and, in particular, in a plasma fast diffusion model (for α = 1/2) (see e.g. [4]). The existence anduniqueness of a strong solution in the sense to be defined below was studied in [1–3,5] for more general nonlinearstochastic equations of the form (1). In [3] (see also [1]) it was also proven that for α = 0 and d = 1 the solutionX = X(t, x) to (1) has the finite extinction property: P(τ n) 1 − |x|−1

ργ(∫ n

0 e−CNs ds)−1 for |x|−1 < C−1N ργ where

τ = inft 0: |X(t, x)|−1 = 0 = supt 0: |X(t, x)|−1 > 0 and CN,γ are constants related to the Wiener processW and respectively to the domain O ⊂ R

1.The following notations will be used in the sequel. H = L2(O), p 1, with the norm denoted by | · |2 and scalar

product 〈·, ·〉. H−1(O) is the dual of the Sobolev space H 10 (O) and is endowed with the scalar product 〈u,v〉−1 =

〈u, (−)−1v〉, where is the Laplace operator with domain H 2(O) ∩ H 10 (O). All processes X = X(t) arising here

are adapted with respect to the filtration Ft . For a Banach space E, LpW(0, T ;E) denotes the space of all adapted

processes in Lp(0, T ;E). We shall use standard notation for Sobolev spaces and spaces of integrable functions on O.

2. The main result

Definition 2.1. Let x ∈ H . An H -valued continuous (Ft )-adapted process X = X(t, x) is called a solution to (1) on[0, T ] if X ∈ Lp(Ω × (0, T ) × O) ∩ L2(0, T ;L2(Ω,H)), p 2, such that P-a.s. ∀j ∈ N, t ∈ [0, T ],

⟨X(t, x), ej

⟩ = 〈x, ej 〉 +t∫

0

∫O

(ρ∣∣X(s, x)(ξ)

∣∣α signX(s, x)(ξ) + Ψ(X(s, x)(ξ)

))ej (ξ)dξ ds

+∞∑

k=1

μk

t∫0

⟨X(s, x)ek, ej

⟩dβk(s). (2)

For x ∈ Lp(O), p 4 and d = 1,2,3 there is a unique solution X ∈ L∞W (0, T ;Lp(Ω,H)) to (1) in the sense of

Definition 2.1. Moreover, if x 0 a.e. in O then X 0 a.e. in Ω × [0, T ] × O.By the proof of [3, Theorem 2.2] and [3, Proposition 3.4] we also know that for λ → 0,

Xλ → X strongly both in L2(0, T ;L2(Ω,L2(O))) and in L2(Ω;C([0, T ];H)),

weakly in Lp(Ω × (0, T ) × O), and weak∗ in L∞(0, T ;Lp(Ω;Lp(O))),(3)

where Xλ, λ > 0, is the solution to approximating equation⎧⎪⎨⎪⎩

dXλ(t) − (Ψλ(Xλ(t)) + λXλ(t) + Ψ (Xλ(t)))dt = σ(Xλ(t))dW(t),

Ψλ(Xλ) + λXλ + Ψ (Xλ) = 0 on ∂O, Xλ(0, x) = x,

Ψλ(x) = 1λ(x − (1 + λΨ0)

−1(x)) = Ψ0((1 + λΨ0)−1(x)), Ψ0(x) = ρ|x|α signx.

(4)

Everywhere in the sequel X = X(t, x) is the solution to (1) in the sense of Definition 2.1 where x ∈ L4(O). Belowγ shall denote the minimal constant arising in the Sobolev embedding Lα+1(O) ⊂ H−1(O) (see (7) below) andC∗ = ∑∞

k=1 μ2k|ek|2

H 10 (O)

= ∑∞k=1 μ2

kλ2k. Theorem 2.2 is the main result of the paper.

Theorem 2.2. Assume that d = 1,2,3 and that 0 < α < 1 if d = 1,2, 15 α < 1 if d = 3. Let τ := inft 0:

|X(t, x)|−1 = 0. Then we have |X(t, x)|−1 = 0, for t τ , P-a.s. Furthermore

P(τ t) 1 − |x|1−α−1

(1 − α)ργ 1+α

( t∫0

e−C∗(1−α)s ds

)−1

.

In particular, if |x|1−α−1 < ργ 1+α/C∗, then P(τ < ∞) > 0, and if C∗ = 0, then τ |x|1−α

−1 /((1 − α)ργ 1+α).

Remark 1. This result extends to O ⊂ Rd with d 4, if α ∈ [ d−2

d+2 ,1). However, we have to strengthen the assumptionon μk , k ∈ N, see [1, Section 4] and in particular [6, Remark 2.9(iii)] for a detailed discussion.

V. Barbu et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 81–84 83

3. Proof of Theorem 2.2

We shall proceed as in the proof of [3, Theorem 4.2]. Consider the solution Xλ ∈ L2W(0, T ;L2(Ω;H 1

0 (O))) toEq. (4). Then by applying the classical Itô formula to the real valued semi-martingale |Xλ(t)|2−1, t ∈ [0, T ], and to thefunction ϕε(r) = (r + ε2)(1−α)/2, r ∈ R, we find that

dϕε

(∣∣Xλ(t)∣∣2−1

) + (1 − α)(∣∣Xλ(t)

∣∣2−1 + ε2)−(1+α)/2⟨

Xλ(t),Ψλ

(Xλ(t)

) + λXλ(t) + Ψλ

(Xλ(t)

)⟩dt

= 1

2

∞∑k=1

μ2k(1 − α)

|Xλ(t)ek|2−1(|Xλ(t)|2−1 + ε2) − (1 − α)2|〈Xλ(t)ek,Xλ(t)〉−1|2)(|Xλ(t)|2−1 + ε2)(3+α)/2

dt

+ ⟨σ(Xλ(t)

)dW(t),ϕ′

ε

(∣∣Xλ(t)∣∣2−1

)Xλ(t)

⟩−1

1

2

∞∑k=1

μ2k

(1 − α)|Xλ(t)ek|2−1

(|Xλ(t)|2−1 + ε2)(1+α)/2dt + ⟨

σ(Xλ(t)

)dW(t),ϕ′

ε

(∣∣Xλ(t)∣∣2−1

)Xλ(t)

⟩−1

C∗ (1 − α)|Xλ(t)ek|2−1

(|Xλ(t)|2−1 + ε2)(1+α)/2dt + ⟨

σ(Xλ(t)

)dW(t),ϕ′

ε

(∣∣Xλ(t)∣∣2−1

)Xλ(t)

⟩−1. (5)

Then letting λ → 0, by (3) we get that lim infλ→0∫ T

0 〈Ψλ(Xλ(t)),Xλ(t)〉dt ρ∫ T

0 |X(t)|1+α

L1+α(O)dt , P-a.s. and hence

ϕε

(∣∣X(t)∣∣2−1

) + (1 − α)ρ

t∫r

|X(s)|α+1Lα+1(O)

(|X(s)|2−1 + ε2)(1+α)/2ds ϕε

(∣∣X(r)∣∣2−1

)

+ C∗t∫

r

(1 − α)|X(s)|2−1

(|X(s)|2−1 + ε2)(1+α)/2ds + 2

t∫r

⟨σ(X(s)

)dW(s),ϕ′

ε

(∣∣X(s)∣∣2−1

)X(s)

⟩−1, P-a.s., r < t. (6)

Next by the Sobolev embedding theorem we have|u|−1 γ |u|Lα+1(O), ∀u ∈ Lα+1(O), if d > 2 and α d−2

d+2 , and ∀α > 0, if d = 1,2. (7)

Then substituting (7) into (6) we get

ϕε

(∣∣X(t)∣∣2−1

) + (1 − α)ργ 1+α

t∫r

|X(s)|α+1−1

(|X(s)|2−1 + ε2)(1+α)/2ds ϕε

(∣∣X(r)∣∣2−1

)

+ C∗t∫

r

(1 − α)|X(s)|2−1

(|X(s)|2−1 + ε2)(1+α)/2ds +

t∫r

⟨σ(X(s)

)dW(s),ϕ′

ε

(∣∣X(s)∣∣2−1

)X(s)

⟩−1, P-a.s., r < t. (8)

Now for ε → 0 we have

∣∣X(t)∣∣1−α

−1 + (1 − α)ργ 1+α

t∫r

1|X(s)|−1>0 ds ∣∣X(r)

∣∣1−α

−1 + C∗(1 − α)

t∫r

∣∣X(s)∣∣1−α

−1 ds

+ (1 − α)

t∫r

⟨σ(X(s)

)dW(s),

∣∣X(s)∣∣−(α+1)

−1 X(s)⟩−1, P-a.s., r < t.

Hence by Itô’s product rule

e−C∗(1−α)t∣∣X(t)

∣∣1−α

−1 + (1 − α)ργ 1+α

t∫r

e−C∗(1−α)s1|X(s)|−1>0 ds

e−C∗(1−α)r∣∣X(r)

∣∣1−α

−1 + (1 − α)

t∫e−C∗(1−α)s

⟨σ(X(s)

)dW(s),

∣∣X(s)∣∣−(α+1)

−1 X(s)⟩−1, P-a.s., r < t. (9)

r

84 V. Barbu et al. / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 81–84

From this it immediately follows that e−C∗(1−α)t |X(t)|1−α−1 , t 0, is an (Ft )-supermartingale, hence |X(t)|−1 = 0 for

all t τ . So, (9) with r = 0 after taking expectation implies that∫ t

0 e−C∗(1−α)sP(τ > s)ds |x|1−α

−1 /((1 − α)ργ 1+α),

t 0. This implies that P(τ > t) |x|1−α−1 /((1 − α)ργ 1+α)(

∫ t

0 e−C∗(1−α)s ds)−1, t 0, and the assertion follows.

Acknowledgements

This work has been supported in part by the PIN-II ID-404 (2007–2010) project of Romanian Minister of Research,the DFG-International Graduate School “Stochastics and Real World Models”, the SFB-701 and the BiBoS-ResearchCenter.’, the research programme “Equazioni di Kolmogorov” from the Italian “Ministero della Ricerca Scientifica eTecnologica” and “FCT, POCTI-219, FEDER”.

References

[1] V. Barbu, G. Da Prato, M. Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equation, Indiana Univ.Math. J. 57 (2008) 187–212.

[2] V. Barbu, G. Da Prato, M. Röckner, Existence of strong solutions for stochastic porous media equation under general monotonicity conditions,Ann. Probab., in press.

[3] V. Barbu, G. Da Prato, M. Röckner, Stochastic porous media equations and self-organized criticality, Comm. Math. Phys., in press.[4] J. Berryman, C. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (4) (1980) 379–388.[5] J. Ren, M. Röckner, F.Y. Wang, Stochastic generalized porous media and fast diffusion equations, J. Differential Equations 238 (1) (2007)

118–152.[6] M. Röckner, F.Y. Wang, Non-monotone stochastic generalized porous media equations, J. Differential Equations, in press.

Author's personal copy

J. Math. Anal. Appl. 351 (2009) 509–521

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Periodic behavior for a degenerate fast diffusion equation

Angelo Favini a, Gabriela Marinoschi b,∗

a Università di Bologna, Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italyb Institute of Mathematical Statistics and Applied Mathematics, Calea 13, Septembrie 13, 050711 Bucharest, Romania

a r t i c l e i n f o a b s t r a c t

Article history:

Received 4 April 2008

Available online 29 October 2008

Submitted by M. Iannelli

Keywords:

Degenerate parabolic PDE

Periodic solutions

Fixed point theorem

Flows in porous media

This work deals with the study of periodic solutions to a degenerate fast diffusion equation.

The existence of the periodic solution to an intermediate problem restraint to a period T is

proved first and then the result is extended by the data periodicity to all time real space.

The approach involves an appropriate approximating problem whose periodic solution is

proved via a fixed point theorem. Next, a passing to the limit procedure leads to the

existence of the solution to the original problem on a time period. Finally, the behavior

at large time of the solution to a Cauchy problem with periodic data is characterized.

2008 Elsevier Inc. All rights reserved.

1. Statement of the problem

Periodic problems for possibly degenerate equations of the type

d

dt

(My(t)

)+ Ly(t) = f (t), 0 t 1, (1.1)

with the periodic condition

(My)(0) = (My)(1) (1.2)

have been studied in the paper [2], for L and M two closed linear operators from a complex Banach space into itself, under

the assumptions that the domain D(L) of L is continuously embedded in D(M) and L has a bounded inverse. Assuming

suitable hypotheses on the modified resolvent (λM+ L)−1 , it has been proved that problem (1.1)–(1.2) admits one 1-periodic

solution. Some examples of applications to partial differential equations and ordinary differential equations have been given.

The latter case has been studied in the paper [3], too.

In this paper we shall approach a concrete PDE problem (1.1)–(1.2) where L is a nonlinear multivalued operator.

We consider Ω an open bounded subset of RN (N ∈ N∗ = 1,2, . . .), with the boundary Γ := ∂Ω of class C1 and denote

the space variable by x := (x1, . . . , xN ) ∈ Ω and the time by t ∈ R. We are concerned with the study of periodic solutions to

a nonlinear model consisting in a degenerate diffusion equation with homogeneous Dirichlet boundary conditions

∂(m(x)u)

∂t− β∗(u) ∋ f in Ω × R,

u(x, t) = 0 on Γ × R,

This work was partially supported by RFO of the University of Bologna, under the auspices of INDAM, Italy and the projects CEEX-05-D11-36/2005 and

PN II IDEI ID_404, financed by the Romanian Ministry of Education and Research.

* Corresponding author.

E-mail addresses: gabimarinoschi@yahoo.com, gmarino@acad.ro (G. Marinoschi).

0022-247X/$ – see front matter 2008 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2008.10.048

Author's personal copy

510 A. Favini, G. Marinoschi / J. Math. Anal. Appl. 351 (2009) 509–521

u(x, t) = u(x, t + T ) in Ω × R, (1.3)

under the assumption of the T -periodicity of the function f ,

f (x, t) = f (x, t + T ) for (x, t) ∈ Ω × R, 0 < T < ∞. (1.4)

In this problem β∗ : (−∞,us] → R is a multivalued function defined as

β∗(r) :=

∫ r0 β(ξ)dξ, if r < us,

[K ∗s ,+∞), if r = us,

(1.5)

where β : (−∞,us) → R is a positive differentiable, monotonically increasing function, which blows up at r = us , but having

the integral finite at this point. Namely we set

β(r) ρ > 0, for each r < us, β(r) := ρ for r 0, (1.6)

limrրus

β(r) = +∞ and limrրus

r∫0

β(ξ)dξ = K ∗s . (1.7)

Consequently, β∗ has the properties(β∗(r) − β∗(r)

)(r − r) ρ(r − r)2, for every r, r ∈ (−∞,us], (1.8)

limr→−∞

β∗(r) = −∞, (1.9)

limrրus

β∗(r) = K ∗s . (1.10)

In the above relationships ρ , us and K ∗s are positive known constants and the hypotheses (1.7) reveal the character of fast

diffusion (see [1,4]).

We also notice that (β∗)−1 : R →(−∞,us] is single-valued, monotonically increasing on (−∞, K ∗s ) and constant for

r ∈ [K ∗s ,+∞), i.e., (β∗)−1(r) = us . Also, it follows that (β∗)−1 is Lipschitz with the constant 1

ρ .

We still assume that

m ∈ C1(Ω), 0 m(x) 1, x ∈ Ω. (1.11)

More exactly, we consider that the degeneration of the equation may occur on Ω0 , where Ω0 is an open bounded subset

of Ω , strictly contained in Ω . The upper bound of m can be taken any positive constant, but by rescaling, we may consider

it equal to 1, without any loss of generality.

The model (1.3) with initial data (m(x)u(x,0) = v0(x) instead of the periodic condition) was studied in [4] where it was

proved that it has a unique weak solution in appropriate functional spaces. In fact, the model was introduced in [7] and it

describes for example the water infiltration in a unsaturated porous medium in which saturation can occur. This event is

mathematically modeled by both the blow-up of the function β at us and the multivalued function β∗ . The function m(x)

characterizes the space variable porosity of the nonhomogeneous medium, while the vanishing of m indicates the existence

of impermeable intrusions in the soil.

A study of the periodic solutions to fast diffusion equations with m(x) = 1 was done in [8] for the case with a nonlinear

convection, in connection with some results given in [6].

The paper is organized as follows: first we shall prove that the problem

∂(m(x)u)

∂t− β∗(u) ∋ f in Ω × R,

u(x, t) = 0 on Γ × R,

m(x)(u(x, t) − u(x, t + T )

)= 0 in Ω × R (1.12)

has a unique solution.

In order to prove the existence for problem (1.12) we shall establish the existence for the solution to the problem on a

time period

∂(m(x)u)

∂t− β∗(u) ∋ f in Q := Ω × (0, T ),

u(x, t) = 0 on Σ := Γ × (0, T ),

m(x)(u(x,0) − u(x, T )

)= 0 in Ω. (1.13)

This will be done by a fixed point argument in Section 2. The result obtained for (1.13) will be extended by periodicity to

all t ∈ R and the longtime behavior of the solution corresponding to a periodic f and a certain initial datum v0 will be

established in connection with the periodic solution to (1.12).

Author's personal copy

A. Favini, G. Marinoschi / J. Math. Anal. Appl. 351 (2009) 509–521 511

Finally, we shall show that the existence of the unique periodic solution to (1.12) implies the existence of the unique

periodic solution to (1.3).

Functional framework and preliminaries. For approaching the problems previously specified we shall consider the Hilbert

space V = H10(Ω) with the usual Hilbertian norm and its dual V ′ = H−1(Ω), endowed with the scalar product (u,u)V ′ :=

〈u,ψ〉V ′,V , where ψ ∈ V satisfies −ψ = u, ψ = 0 on Γ , and 〈u,ψ〉V ′,V is the pairing between V ′ and V .

For simplicity, we shall denote by (·,·) and ‖ · ‖ the scalar product and the norm in L2(Ω), respectively.

Definition 1.1. Let

m ∈ C1(Ω), f ∈ L∞(0, T ; V ′). (1.14)

We call a solution to (1.13) a function u which satisfies

u ∈ L2(0, T ; V ), u us, a.e. (x, t) ∈ Q ,

mu ∈ C([0, T ]; L2(Ω)

)∩ W 1,2(0, T ; V ′),

ζ ∈ L2(0, T ; V ), ζ(x, t) ∈ β∗(u(x, t)

), a.e. (x, t) ∈ Q , (1.15)

the condition m(x)(u(x,0) − u(x, T )) = 0 in Ω and the equation

T∫0

⟨d(m(x)u)

dt(t),φ(t)

⟩V ′,V

dt +

∫Q

∇ζ(x, t) · ∇φ(x, t)dxdt =

T∫0

⟨f (t),φ(t)

⟩V ′,V

dt, a.e. t ∈ (0, T ), (1.16)

for each φ ∈ L2(0, T ; V ), where ζ(x, t) ∈ β∗(u(x, t)), a.e. (x, t) ∈ Q .

On the domain

D(A) :=u ∈ L2(Ω); there exists η ∈ V , such that η(x) ∈ β∗

(u(x)

), a.e. x ∈ Ω

we define the multivalued operator A : D(A) ⊂ V ′ → V ′ by

〈Au,ψ〉V ′,V :=

∫Ω

∇η · ∇ψ dx, for each ψ ∈ V , where η(x) ∈ β∗(u(x)

), a.e. x ∈ Ω.

We remark that u ∈ D(A) implies u ∈ V , due to the Lipschitz property of the inverse of β∗ .

Next, we introduce the multiplication operator M : D(A) → L2(Ω), Mu := mu, whose inverse is multivalued. Thus, we

can write the abstract problem

d

dt

(Mu(t)

)+ Au(t) ∋ f (t), a.e. t ∈ (0, T ), (1.17)

M(u(0) − u(T )

)= 0 (1.18)

and notice that the solution to (1.17)–(1.18) is a solution to (1.13) in the sense of Definition 1.1.

Denoting v(x, t) :=m(x)u(x, t) we can rewrite (1.17)–(1.18) in terms of v as,

dv

dt+ AM v ∋ f , a.e. t ∈ (0, T ),

v(0) = v(T ), (1.19)

where AM = AM−1 and

D(AM) :=

v ∈ L2(Ω);

v

m∈ L2(Ω), ∃ζ ∈ V , ζ(x) ∈ β∗

(v

m(x)

), a.e. x ∈ Ω

.

We easily see that v ∈ D(AM) if and only if u = vm

∈ D(A).

For a later use we define j : R → (−∞,+∞] by

j(r) :=

∫ r0 β∗(ξ)dξ, r us,

+∞, r > us,(1.20)

where the left limit of β∗ at us was specified in (1.10). The function j is proper, convex, lower semicontinuous and

∂ j(r) =

β∗(r), r < us,

[K ∗s ,+∞), r = us,

∅, r > us

(1.21)

(see [7, p. 166]).

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512 A. Favini, G. Marinoschi / J. Math. Anal. Appl. 351 (2009) 509–521

Also, we recall a result proved in [4] (see Theorem 3.2) related to the problem

∂(m(x)u)

∂t− β∗(u) ∋ f in Q ,

u(x, t) = 0 on Σ,

m(x)u(x,0) = v0 in Ω. (1.22)

Theorem 1.2. Let

m ∈ C1(Ω), f ∈ L2(0, T ; V ′),v0

m∈ L2(Ω),

v0

m us, a.e. x ∈ Ω.

Then, the Cauchy problem (1.22) has a unique solution u, such that

mu ∈ C([0, T ]; L2(Ω)

)∩ W 1,2(0, T ; V ′),

β∗(u) ∈ L2(0, T ; V ),

u ∈ L2(0, T ; V ), u us, a.e. (x, t) ∈ Q .

2. Existence on the time period (0, T )

In this section we shall study the existence of the solution to the problem (1.13) defined on the time period (0, T ). To

this end we shall establish first an existence result for the approximate problem obtained by replacing m by

mε(x) :=m(x) + ε, where ε mε(x) 1+ ε

and β∗ by the single-valued function β∗ε : R → R,

β∗ε (r) :=

β∗(r), if r < us − ε,

β∗(us − ε) +K ∗s −β∗(us−ε)

ε [r − (us − ε)], if r us − ε,(2.1)

for each positive ε. The function β∗ε is continuous and monotonically increasing on R, differentiable on R \ us −ε, but with

lateral finite derivatives at u = us − ε, satisfies (1.8) for any r, r ∈ R and

limr→−∞

β∗ε (r) = −∞, lim

r→+∞β∗

ε (r) = +∞.

We denote by βε the derivative of β∗ε defined as

βε(r) :=

β(r), if r < us − ε,

K ∗s −β∗(us−ε)

ε , if r us − ε(2.2)

and remark that βε(r) ρ for any r ∈ R.

Then we introduce Aε : D(Aε) ⊂ V ′ → V ′ by

〈Aεu,ψ〉V ′,V :=

∫Ω

∇β∗ε (u) · ∇ψ dx, for every ψ ∈ V ,

D(Aε) :=u ∈ L2(Ω); β∗

ε (u) ∈ V

and consider the periodic approximating problem

d(mεuε)

dt+ Aεuε = f , a.e. t ∈ (0, T ), (2.3)

mε

(uε(0) − uε(T )

)= 0 (2.4)

which is equivalent with

dvε

dt+ Bεvε = f , a.e. t ∈ (0, T ),

vε(0) = vε(T ), (2.5)

by the function replacement

vε =mεuε. (2.6)

Here, Bεvε = Aε(vεmε

). We are going to prove the following existence result.

Quantum control design by Lyapunov trajectory

tracking for dipole and polarizability coupling

Jean Michel Coron1†, Andreea Grigoriu2§, Catalin Lefter3,4‖

and Gabriel Turinici2§1Laboratoire Jacques-Louis Lions, Universite Pierre et Marie Curie, 175 rue du

Chevaleret 75013 Paris, France et Institut Universitaire de France.2CEREMADE, Universite Paris Dauphine, Place du Marechal De Lattre De

Tassigny, 75016, Paris, France.3 Faculty of Mathematics, University “Al. I. Cuza”, Bd. Carol I nr. 11, 700506, Iasi,

Romania.4”Octav Mayer” Institute of Mathematics, Romanian Academy, Bd. Carol I, nr. 8,

700505 Iasi, Romania.

E-mail: coron@ann.jussieu.fr, grigoriu@ceremade.dauphine.fr,

catalin.lefter@uaic.ro and gabriel.turinici@dauphine.fr

Abstract. We analyse in this paper the Lyapunov trajectory tracking of the

Schrodinger equation for a coupling control operator containing both a linear (dipole)

and a quadratic (polarizability) term. We show numerically that the contribution

of the quadratic part cannot be exploited by standard trajectory tracking tools and

propose two improvements: discontinuous feedback and periodic (time-dependent)

feedback. For both cases we present theoretical results and support them by numerical

illustrations.

PACS numbers: 32.80.Qk,03.65.Yz,78.20.Bh.

Submitted to: New Journal of Physics

† Financial support from “Agence Nationale de la Recherche” (ANR), Projet Blanc C-QUID number

BLAN-3-139579.§ Financial support from INRIA Rocquencourt projet MicMac and OMQP, from “Agence Nationale

de la Recherche” (ANR), Projet Blanc C-QUID number BLAN-3-139579 and from PICS NSF-CNRS.‖ Financial support by Universite Paris Dauphine and CNCSIS grant PNII ID-404/2007-2010

acknowledged

Quantum trajectory control for polarizability 2

1. Introduction

We consider in this work the evolution of a quantum system with wavefunction Ψ(t)

under the external influence of a laser field; the system satisfies the Time Dependent

Schrodinger equation (TDSE)

id

dtΨ(t) = H(t)Ψ(t), (1)

with H(t) a Hermitian operator; the control is realized by selecting a convenient laser

intensity u(t). When the laser is shut off H(t) is the internal Hamiltonian of the system,

denoted H0; when the laser is present H(t) is the sum of H0 and additional terms that

describe the interaction of the system with the laser field. The first order term is the

dipole coupling [30] of the form u(t)H1; in the limit of small laser intensities this term

may be enough to adequately describe the interaction.

However, situations exist where the dipole coupling does not have enough influence

on the system to reach the control goal; the goal may become accessible only after

adding a polarizability term u2(t)H2 in the expansion of H(t) (see e.g. [13, 14] and

related works); to make effective use of this term one needs higher laser intensities u(t).

The focus of the paper is on practical procedures to find suitable control fields u(t)

for the Hamiltonian H(t) = H0+u(t)H1+u2(t)H2 by adapting feedback tracking control

procedures to this setting. Here and in the following H0, H1 and H2 are n×n Hermitian

matrices with complex coefficients and the control is the laser intensity u(t) ∈ R.

In what concerns the mere possibility to find a control, we recall that the

controllability of the finite dimensional quantum system evolving with equation

id

dtΨ(t) = (H0 + u(t)H1 + u2(t)H2)Ψ(t), (2) eq_gen

can be studied via the general accessibility criteria [4, 32] based on Lie brackets; more

specific results can be found in [34].

Let us consider for a moment the system with Hamiltonian H0 + u(t)H1 + v(t)H2,

v(t) being a second control independent of u(t). It can be shown [34] that this system

is controllable under the same circumstances as H0 + u(t)H1 + u2(t)H2 i.e. all target

states that can be reached with Hamiltonian H0 + u(t)H1 + v(t)H2 can also be reached

by H0 + u(t)H1 + u2(t)H2 (although obviously the second Hamiltonian is a particular

case of the first for v(t) = u2(t)). This rather counter-intuitive result suggests that u2(t)

can be considered, for the purpose of theoretical controllability, as independent of u(t);

however, u2(t) having a particular functional dependence on u(t) will play a role at the

level of the numerical procedure to find the control: in general finding the control for

H0 + u(t)H1 + u2(t)H2 is more difficult than for H0 + u(t)H1 + v(t)H2.

The characterization of the controllability does not provide in general a simple

and efficient way for open-loop trajectory generation. Optimal control techniques (cf.,

[23] and [30] and the references herein) provide a first set of methods. A different

approach consists in using feedback to generate trajectories and open-loop steering

control [5, 19, 22]. More recent results can be found in [27] for decoupling techniques,

Quantum trajectory control for polarizability 3

in [3, 15, 17, 23, 31, 35, 36] for Lyapunov-based techniques and in [1, 7, 28] for

factorizations techniques of the unitary group.

In order to study feedback control of systems with Hamiltonian H(t) = H0 +

u(t)H1 +u2(t)H2 we adapt the analysis [20, 24], initially proposed for bilinear quantum

systems H0 + u(t)H1. In the previous work it has been shown that the success of the

feedback control depends on whether there exists (non-zero) direct coupling, through H1,

between the target state and all other eigenstates. When H1 has the same property for

H(t) = H0 + u(t)H1 + u2(t)H2 we show that same feedback formulas hold. However we

argued that the polarizability term u2(t)H2 is added when dipole u(t)H1 is not enough

to control the system; consequently the most interesting question is what happens

when some of the (direct) coupling is realized by H2 instead of H1. We show that

the previous feedback formulas do not hold any more and we propose two alternatives.

Our method is valid to track any eigenstate trajectory of a Schrodinger equation (2)

when the Hamiltonian includes a second order coupling operator.

The balance of the paper is as follows: in Section 2 we introduce the main

notations and the Lyapunov tracking feedback for a particular case. Section 3

contains the presentation of two types of feedback: discontinuous and time-dependent

(periodic) forcing, that can be applied for all types of second order coupling operators.

Both sections present theoretical results on the convergence illustrated by numerical

simulations. Concluding remarks are presented in Section 4.

2. Tracking feedback design

eedback:sec

2.1. Dynamics and global phasegauge:ssec

We consider a n-level quantum system evolving under the equation (2). The wave

function Ψ = (Ψj)nj=1 is a vector in Cn, verifying

∑n

j=1 |Ψj|2 = 1, thus it lives on the

unit sphere S2n−1 of C

n. Physically, Ψ and eiθ(t)Ψ describe the same physical state for

any global phase θ(t) ∈ R, i.e. Ψ1 and Ψ2 are identified when exists θ(t) ∈ R such that

Ψ1 = exp(iθ(t))Ψ2. To take into account such non trivial geometry we add a second

control ω corresponding to θ (see also [24]). Thus we consider the following control

system

id

dtΨ(t) = (H0 + u(t)H1 + u2(t)H2 + ω(t))Ψ(t), (3) dyne:eq

where ω ∈ R is a new control playing the role of a gauge degree of freedom. We can

choose it arbitrarily without changing the physical quantities attached to Ψ. With such

additional fictitious control ω, we will assume in the sequel that the state space is S2n−1

and the dynamics given by (3) admits two independent controls u and ω.

2.2. Lyapunov control designdesign:ssec

Take a reference trajectory t 7→ (Ψr(t), ur(t), ωr(t)), i.e., a smooth solution of (3):

Comsol modelling for a water infiltration problem

in an unsaturated medium

Cornelia-Andreea Ciutureanu

Keywords Boundary value problems for nonlinear parabolic PDE; Stabilityand convergence of numerical methods; Flows in porous media.

AMS Subject Classification 35K60; 65M12; 76S05.

Abstract

The paper deals with the COMSOL modelling of fluid diffusion in

unsaturated porous media. A representative phenomenon in this class of

problems is water infiltration in soils.

The model we are concerned of describes the water infiltration into anisotropic, nonhomogeneous, unsaturated porous medium with a variable poros-ity. It consists of a diffusion equation with a transport term in addition with ainitial data and a Dirichlet boundary condition

m(x)∂u

∂t− ∆β∗(u) +

∂K(u)

∂x3

= F in Q := Ω × (0, T ), (1)

m(x)u(x, 0) = θ0(x) in Ω, (2)

u(x, t) = g(x) < us on Σ := Γ × (0, T ). (3)

The domain Ω is an open bounded subset of R3, with the boundary Γ := ∂Ω

piecewise smooth. We denote the space variable by x := (x1, x2, x3) ∈ Ω and thetime by t ∈ (0, T ), with T finite. The model is written in dimensionless form.The porosity is denoted by m, the function u stands for the water saturation,while by us we shall denote its maximum value.

The volumetric water content is given by mu and θ0 is the initial volumetricwater content.

Hypothesis

In the unsaturated case the diffusivity β : (−∞, us) → [ρ, +∞) is a continu-ous and monotonically increasing function that satisfies the following hypothe-ses:

1

Annual Reviews in Control xxx (2010) xxx–xxx

G Model

JARAP-340; No. of Pages 10

Self-organized criticality and convergence to equilibrium of solutions to nonlineardiffusion equations§

Viorel Barbu

Department of Mathematics, ‘‘Al.I. Cuza’’ University and ‘‘Octav Mayer’’ Institute of Mathematics of the Romanian Academy, 700506 Iasi, Romania

A R T I C L E I N F O

Article history:

Received 5 November 2009

Accepted 26 December 2009

Keywords:

Self-organized criticality

Nonlinear diffusion equation

Sand-pile model

Stochastic differential equation

A B S T R A C T

This article is a survey on some recent results on longtime behaviour of solutions to nonlinear singular

diffusion equations with main emphasis on fast diffusion porous media equations and the sand-pile

model of self-organized criticality. Deterministic as well stochastic equations are presented. One of the

key problem considered here is the extinction in finite time of solutions.

2010 Elsevier Ltd. All rights reserved.

Contents lists available at ScienceDirect

Annual Reviews in Control

journa l homepage: www.e lsev ier .com/ locate /arcontro l

1. Introduction

A critical point (critical state) specifies the conditions (tem-perature, pressure) at which the phase boundary ceases to exist(for instance, liquid–vapor system). Such a situation is usually metin phase-transition and in equilibrium systems, where the criticalpoint is reached only by tuning a control parameter precisely. Insome non-equilibrium system, however, the critical point is anattractor of the dynamics and s elf-organized criticality (SOC) is aproperty of dynamical systems which have a critical point as anattractor and emerges spontaneously to this attractor. This is oneof the mechanisms by which the emergence of complexity fromsimple local interactions could be spontaneous and therefore asource of natural complexity.

The Bak–Tang–Wiesenfeld (BTW) model (so-called sand-pilemodel) (Bak, Tang, & Wiesenfeld, 1988) is the standard model ofself-organized criticality and it can briefly be described asfollows. The sand is slowly dropped onto a plane surface forminga pile. As the pile grows, avalanches occur which carry sandfrom the top to the bottom of the pile. It turns out that aftersome critical height value the shape of the pile becomesindependent (in time) of the rate at which the system is drivenby dropping sand; this is the self-organized critical shape. In thiscase, the distribution of avalanches as function of size is given bya power law, which is the signature of the criticality. Because thesand-pile tends to adjust the shape of its sides it arrives in a

§ An earlier version of this article was presented at the 8th IFAC Workshop on

Time Delay Systems, Sinaia, Romania, September 1–3, 2009.

E-mail address: vb41@uaic.ro.

Please cite this article in press as: Barbu, V. Self-organized criticality aequations. Annual Reviews in Control (2010), doi:10.1016/j.arcontrol

1367-5788/$ – see front matter 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.arcontrol.2009.12.002

certain critical equilibrium state the criticality is called ‘‘self-

organized’’.The precise mathematical description of this model will be

presented in Section 2 below; here, we confine to give a generalpresentation of self-organized criticality. If X ¼ Xðt; jÞ is the stateof system at time t, where Xðt; jÞ, t0, j2O, is a numerical variabledistributed in a spatial domain O, then it can be in one of thefollowing three spatial regions: critical region Ot

0, whereXðt; jÞ ¼ Xc , subcritical regionOt

¼ fj2O; Xðt; jÞ<Xcg and super-critical region Ot

þ ¼ fj2O; Xðt; jÞ>Xcg. The essential feature ofself-organized criticality is that subcritical and supercritical zonesare unstable and are absorbed in time by the critical zone. In fact, inoff-critical zones the diffusion is very slow if not zero and thesystem moves towards the critical zone, where the evolutionprocess is very fast. Here, the fluctuations and changes eveninfinitesimal ones have a great effect in configuration of thesystem, which moves apparently spontaneously to a newequilibrium. Arrived here a nonlinear instability behaviour some-what similar to that encountered in the ‘‘chaos’’ theory, i.e., the‘‘butterfly’’ effect occurs. The principal difference betweenunstable system with chaotic behaviour and that with self-organized critically is that the latter develops an autonomousmechanism for evolution to a stable equilibrium which is of coursethe critical state. A problem of great interest, which will bediscussed in detail below, is whether and when such a systemindeed reaches the critical value Xc , i.e., system is ‘‘absorbed’’ intocritical zone. Most self-organized criticality examples are fromphysical and applied sciences, but several models of this type havebeen proposed in economics (see, e.g., Bak & Chen, 1991; Bak, Chen,Schneinkman, & Wooldford, 1992; Dıaz-Guilera, 1994, pp. 177–182; Arenas et al., 2002) to explain the instabilities which occur in

nd convergence to equilibrium of solutions to nonlinear diffusion.2009.12.002

V. Barbu / Annual Reviews in Control xxx (2010) xxx–xxx2

G Model

JARAP-340; No. of Pages 10

macroeconomics. It is generally accepted now that large econom-ics have intrinsically unstable dynamics which have non smoothattractors, i.e., develop ‘‘chaos’’. In particular, in Krugman (1996) isdiscussed the general significance of self-organized criticality inthe dynamic of the economy viewed as a dynamic process, inwhich ‘‘order spontaneously emerges from instability’’ andconnects it with the Keynes ideas of the ‘‘nonlinear businesscycle’’ theory. Most of these models refer to an economy involvinga large number of firms which interact by selling or buyingproducts and goods. Such a model is studied by Bak et al. (1992)and we shall briefly refer to it below. In this case, the inventorydynamic X ¼ XðtÞ is described by the equation:

Xðt þ 1Þ ¼ XðtÞ þ YðtÞ ZðtÞ;

where YðtÞ (the outproduct) and ZðtÞ (the sales) are related by anonlinear feedback law YðtÞ ¼ FðXðsÞ; SðtÞÞ. The self-organizedcriticality in this situation is a consequence of specific form ofnonlinear function F. For other literature on SOC, we cite Jensen,1988; Carlson, Chayes, Grannan, and Swindle (1990a), Carlson,Chayes, Grannan, and Swindle (1990b), Carlson et al. (1995),Giacometti and Dıaz-Guilera (1998), Hentschel and Family (1991)and Janosi and Kertesz (1993). We do not discuss here in detailsthese models and confine to presentation of mathematicaldescription of sand-pile model and longtime behaviour ofsolutions to nonlinear diffusion equation which describes thismodel. We consider both deterministic and stochastic model.

The plan of this survey paper is the following. In Section 2, themathematical formulation of the sand-pile model is presented, viacellular automaton device. In Section 3, a few basic results onnonlinear diffusion equations associated with this model aresurveyed. In Section 4, few asymptotic results for the solutions toequations modeling the sand-pile model are briefly discussed. InSection 5, the self-organized criticality model perturbed byGaussian noise is considered and existence, longtime-behaviourand absorption in finite time are discussed. One must emphasizethat only sketches of proofs are given and the reader is referred tothe works Barbu, Da Prato, and Roeckner (2008) and Barbu, DaPrato, and Roeckner (2009c) for details and complete results.

2. Mathematical description of the sand-pile model

The sand-pile model heuristically presented above can beformalized via cellular automaton by the following argument (seeBantay & Janosi, 1992). Consider an N N square matrixrepresenting a spatial discrete region O ¼ fXi jgN

i; j¼1. To each siteði; jÞ is assigned at moment t a nonnegative (integer) variable Xi jðtÞ.The dynamic of the N2-valued variable XðtÞ ¼ fXi jðtÞgN

i; j¼1 isdescribed by the equation:

Xi‘ðt þ 1Þ!Xi jðtÞ Zk‘i j for ðk; ‘Þ 2G i j; (2.1)

where G i j ¼ fðiþ 1; jÞ; ði; jþ 1Þ; ði 1; jÞ; ði; j 1Þg is the set of all4 nearest neighbors of ði; jÞ and

Zk‘i j ¼

4 if i ¼ k; j ¼ ‘;1 if ðk; ‘Þ 2G i j;0 if ðk; ‘Þ 2G i j:

8<: (2.2)

The algebraic law (2.1) describes rigorously what happens with the‘‘activated’’ site ði; jÞ (i.e., a site which has attained or is over thecritical height Xc): it looses four grains of sands which move tonearest neighbors in the interval of time ðt; t þ 1Þ. This is a small‘‘avalanche’’ which leads to a new configuration of sand-pile.

This transition from XðtÞ to Xðt þ 1Þ can be written as

Xi jðt þ 1Þ Xi jðtÞ ¼ Zi jHðXi jðtÞ XcÞ; i; j ¼ 1; . . . ;N; (2.3)

Please cite this article in press as: Barbu, V. Self-organized criticality aequations. Annual Reviews in Control (2010), doi:10.1016/j.arcontrol

where H is the Heaviside function

HðrÞ ¼ 1 if r>00 if r<0

(2.4)

and Zi j ¼ fZk‘i j gk;‘2G i j

.

The exact meaning of (2.3) is that the transfer dynamic (2.1)works in critical or supercritical region only, i.e., in an activated siteði; jÞ, where Xi j >Xc . Otherwise, one can consider that Xi j remainsunchanged. Of course, since (2.3) is valid for all i; j, we canrepresent it as

Xðt þ 1Þ XðtÞ ¼ ZHðXðtÞ XcÞ; (2.5)

where Z ¼ fZi jgNi; j¼1. It can be seen that Zi j is the second order

difference operator in the spatial domain O, i.e.,

Zi jðYi jÞ ¼ Yiþ1; j þ Yi1; j þ Yi; jþ1 4Yi j þ Yi; j1

for alli; j ¼ 1; . . . ;N;(2.6)

and so Eq. (2.5) is the discrete version of the partial differentialequation of parabolic type

@X

@tðtÞ ¼ DHðXðtÞ XcÞ inO; (2.7)

where D is the discrete version of the 2-D Laplace operator:

@2

@x21

þ @2

@x22

Therefore, if replace the finite dimensional region O by acontinuous domain in 2-dimensional space (for instanceO ¼ ð0;1Þ ð0;1Þ) and replace the site location ði; jÞ by a point jin O, the above model reduces to nonlinear diffusion Eq. (2.7) onthe spatial domain OR R and on the continuous time interval½0; T. More generally, one can consider an equation in a domain Oof the Euclidean space Rd, d ¼ 1;2;3, and replace H by a continuousfunction with jump at 0. Of course, to the partial differentialEq. (2.7) one must associate an initial value condition:

Xð0; jÞ ¼ xðjÞ; j2O; (2.8)

representing the initial configuration x ¼ xðjÞ and also conditionson the boundary @O of domain O. The most common is theDirichlet condition from mathematical physics

Xðt; jÞ ¼ 0; 8 ðt; jÞ 2 ð0;1Þ @O: (2.9)

Nonlinear diffusion equations of the form (2.7) with conditions(2.8) and (2.9), known also as the porous media equation, wereintensively studied in the literature in recent years (see, e.g., Barbu,1976, 2010 and references given there) and one knows that underquite general assumptions it is well posed in the space of integrablefunctions on O and its solution X can be obtained as limit of finitedifference scheme:

Xiþ1 ¼ Xi þ hDHðXiþ1 XcÞ; i ¼ 0;1; . . . ;N; (2.10)

where h!0. Moreover, if X0Xc , then XðtÞXc for all t and so thesystem remains in supercritical region. (More is said about this inSection 3 below.)

It should be said that compared with the standard porous mediaequation governing the dynamic of gas in porous media, which isusually described by (2.7) when H is replaced by a polynomialfunction, Eq. (2.7) is highly singular in the neighborhood of criticalvalue Xc and this is the main reason of anomalies and instabilitiesexhibited by the self-organized criticality modeled by thisequation.

Perhaps a few words about the difference between the abovecontinuous and discrete model of SOC are in order. The continuousmodel described by nonlinear parabolic Eq. (2.7) takes into account

nd convergence to equilibrium of solutions to nonlinear diffusion.2009.12.002

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Numerical Functional Analysis and Optimization, 31(9):1–34, 2010Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630563.2010.512691

AN OPTIMAL CONTROL PROBLEM FOR A SINGULAR SYSTEMOF SOLID–LIQUID PHASE TRANSITION

Mauro Fabrizio1, Angelo Favini1, and Gabriela Marinoschi2

1Dipartimento di Matematica, Università di Bologna, Bologna, Italy2Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania

In this article, we deal with a control problem for a singular system regarding a phase-fieldmodel which describes a solid–liquid transition by the Ginzburg–Landau theory. The purposeis to control the system by the means of the heat supply r able to guide it into a certainstate with a solid (or liquid) part in a prescribed subset 0 of the space domain , andmaintain it in this state during a period of time. The transition is described by a nonlineardifferential system of two equations for the phase field and temperature. The control problemis set for some expressions of the cost functional which might reveal cases of physical interest.An approximating control problem is introduced and the existence of at least an optimal pair isproved. The first-order optimality conditions for the approximating problem are determined anda convergence result is given.

Keywords Optimal control for singular systems; Phase transitions.

AMS Subject Classification 80A22; 49J20.

1. INTRODUCTION

The model that describes the solid–liquid phase transformation isgiven by a Ginzburg–Landau equation [9], fitted to a first order transition,as proved in [8]. There it is was suggested that this equation canbe obtained by a balance law on the order structure. In this article,the ice–water phase transition is described by an order parameter ,such that 0 ≤ ≤ 1 When = 0, we are in water phase, while theice phase is represented by = 1. Finally, if ∈ (0, 1) we are in the

Received 11 May 2010; Revised 21 July 2010; Accepted 23 July 2010.Address correspondence to Gabriela Marinoschi, Institute of Mathematical Statistics and

Applied Mathematics, POB 1-24, Calea 13, Septembrie 13, Bucharest 050711, Romania; E-mail:gabimarinoschi@yahoo.com

1

44454647484950515253545556575859606162636465666768697071727374757677787980818283848586

2 M. Fabrizio et al.

melt phase. Moreover, the model makes use of a density decomposition,such that

1

= 11

+ 12

(1)

where, usually the component 1, of the density , is related with thepressure. Namely, the pressure depends on 1,1 but is independent of2, while the variation of the component 2 is connected with thephase transition. Therefore, 2 will be related with the order parameter by a function obtained as a restriction coming from the laws ofthermodynamics.

Of course, besides the order parameter , the decomposition (1)introduces a new unknown variable 2. Anyway, we need only one newequation which will be the Ginzburg–Landau equation, because we can usethe relation between the phase and the order density 2

In this article, we study the ice–water phase transition under thehypothesis of a steady motion for the ice and water. Then, thephenomenon is described by a nonlinear differential system of twoequations only for the phase field and temperature. The heat equation isobtained by using the first law of thermodynamics by means of the internalbalance powers of the order structure and heat.

The control problem we propose refers to the determination of theheat supply able to bring and maintain the system into a certain (solid orliquid) state in a subset 0 of the space domain , during a period oftime. Various expressions of the cost functional which might reflect somecases of physical interest are proposed.

Since we do not prove the well-posedness of the state system, this beingbeyond the scope of the article, we deal with a control problem for asingular system, in the sense of Lions (see [13]).

A certain approximating control problem depending on new controlsis introduced and the existence of at least an optimal pair for it is proved.The necessary optimality conditions for the approximating problem aredetermined and a convergence result is proved if it is a priori knownthat the solution to the original control problem exists and has certainproperties.

2. MATHEMATICAL MODEL FOR ICE–WATER PHASETRANSITIONS

Following Landau [9–11], we describe the solid–liquid phase transitionby the means of an order parameter ∈ [0, 1] able to represent the

1Of course, when the phases are given by incompressible materials, the density 1 will beconstant and the pressure turns out to be an undetermined quantity.

APPROXIMATE CONTROLLABILITY FOR AN

INTEGRO-DIFFERENTIAL CONTROL PROBLEM

CATALIN-GEORGE LEFTER AND ALFREDO LORENZI

1. Introduction

This paper is devoted to the study of a controlled linear parabolic equationwith a memory term represented by a generalized convolution. The controlacts on a subdomain of the domain where the system lives.

The study of the controllability properties for parabolic equations withinternal controllers acting on subdomains raises delicate problems relatedto unique continuation properties and to global Carleman estimates for theadjoint problem. In fact, the exact null controllability in this case wasproved by O.Yu. Imanuvilov and a sequence of papers followed, for thecase of nonlinear parabolic equations, Navier-Stokes equations, etc. (see[6, 5, 3]) Another issue of the Carlelman inequalities is an estimate for thecost of approximate controllability and this was achieved by E. Fernandez-Cara and E. Zuazua in [4], where a very refined analysis of the Carlemanestimate is performed and precise estimates for the constants appearing inthe Carleman inequalities are deduced.

For the linear parabolic equation with memory we are going to studythe same attempt for treating the approximate controllability produces anadjoint equation, also with integral term. The corresponding property ofunique continuation turns out to be equivalent to the approximate control-lability of the initial problem. However, at this point, one is no more able toderive Carleman estimates because of the integral term which is not at allclear how to handle in the classical computations. However, we are able toreduce our problem to a problem of unique continuation at initial time forparabolic equations. A result in this direction was obtained by P. Albanoand D. Tataru in [1]. This is a very precise result but has a local character.In order to obtain a global unique continuation result at initial time we makeuse of the paper [4] and the result we obtain is interesting in itself, statingthat if the solution to a homogenous parabolic equation, with homogeneousboundary conditions, vanishes at initial time faster than exp(−f(t)/t), wherelimt→0 f(t) = +∞ and f is convex, then it should vanish everywhere. Thisresult is the key for proving the approximate controllability for our problem

2000 Mathematics Subject Classification. Primary: 35K10, 47G20 , 93B05, 93B07,93C20.

Key words and phrases. Approximate controllability, integro-differential parabolicequations, unique continuation, Carleman estimates.

1

2 CATALIN-GEORGE LEFTER AND ALFREDO LORENZI

under appropriate hypotheses for the memory kernel, regarding its shapenear 0.

Another approach of the problem, under different hypotheses imposed onthe kernel (more precisely on the Laplace transform of the kernel) was donein [2]. Concerning this subject, one may find in the literature also the paper[7] which, unfortunately, does not seem to be correct.

2. Preliminaries and main results

Let Ω ⊂ Rn be an open set with a C2-boundary. Let ω ⊂⊂ Ω be an opensubset. Consider an elliptic operator of the form

Ay =

n∑i,j=1

αi,j∂2y

∂xi∂xj+

n∑j=1

βj∂y

∂xj+ γy (2.1)

wherei) αi,j ∈ C1(Ω), αi,j = αj,i, i, j = 1, . . . , n, and define a uniformly positivedefinite matrix:

n∑i,j=1

αi,j(x)ξiξj ≥ C|ξ|2,

for some positive constant C and all x ∈ Ω, ξ ∈ RN .

ii) βi, γ ∈ L∞(Ω).

Let

E(T ) = (t, s) ∈ R2 : 0 ≤ s ≤ t ≤ T. (2.2)

and for a ∈ Lp(E(T )) we define the following operator, denoted also by aand acting on a subspace of L1(0, T ) to be defined, given by the generalizedconvolution “ ?” as

(a ? y)(t) =

∫ t

0a(t, s)y(s) ds. (2.3)

The generalized convolution coincides with the usual convolution when a(t, s) =a0(t− s) for some a0 ∈ Lp(0, T ):

(a ? y)(t) = (a0 ∗ y)(t) :=

∫ t

0a0(t− s)y(s)ds.

Now, given a pair (a, b) ∈ Lp1(E(T )) × Lp2(E(T )), we define the integralcomposition of the two functions, denoted again by a ? b, by the formula

(a ? b)(t, s) =

∫ t

sa(t, τ)b(τ, s)dτ,

whenever it makes sense.If a = a(t) depends on only one variable and b = b(t, s), then it is easy to

see that

[a ∗ (b ? y)](t) = [(a ? b) ? y](t), t ∈ [0, T ].

where we have defined by a(t, s) = a(t− s).

APPROXIMATE CONTROLLABILITY 3

Consider now the controlled integro-differential equationy′ = A(y(t) + (a ? y)(t)) +Bu(t), t ∈ (0, T ),

y(0) = y0,(2.4)

We assume that the memory kernel a ∈ Lp(E(T )), with Dta ∈ Lp(E(T ))and p ∈ (1,+∞], satisfies

a(t, t) = c = const, t ∈ (0, T ), (2.5)

|Djta(t, s)| ≤ aj(t− s), 0 ≤ s ≤ t ≤ T, j = 0, 1, (2.6)

wherea0, a1 ∈ Lp(0, T ). (2.7)

The control operator B : L2(ω)→ L2(Ω) is the extension by zero of functionsdefined on ω.

For a given u ∈ U := L2(0, T, L2(ω)) denote by yu(t) the solution toproblem (2.4).

In this paper we study the following problem:

P Prove that under suitable conditions on the behaviour of a near t = s, forgiven T > 0, the closure in L2(Ω) of the set yu(T ) : u ∈ U is the wholespace. This means that system (2.4) is approximately controllable.

Our main result is the following:

Theorem 2.1. Assume that the original kernel a admits further the follow-ing representation

a(t, s) = cec(t−s) + ϕγ(t− s)a(t, s), (t, s) ∈ E(T ), (2.8)

where c ∈ R, γ ∈ (1,+∞), a ∈ W 1,p(E(T );R), p ∈ (1,+∞], and ϕγ is thenondecreasing function defined by

ϕγ(t) =

e−γt

−1f(t), t ∈ (0, t0)

e−γt−10 f(t0), t ∈ [t0,+∞),

(2.9)

f ∈ C1((0, t0); (0,+∞)) being a convex function such that f(t) → +∞ ast → 0+. Then problem P is solvable, that is the system (2.4) is approxi-mately controllable in time T > 0.

ON THE FEEDBACK STABILIZATION OF BOUSSINESQEQUATIONS

ADRIANA-IOANA LEFTER

Abstract. This paper provides feedback stabilization results, preserving theinvariance of a given convex set, for the Boussinesq system, in 2D and 3D. Theproofs use an existence theorem for weak solutions.

2000 Mathematics Subject Classification: 76D05, 47H05, 35B35.Keywords and phrases: Navier-Stokes equations, Boussinesq equations, weaksolutions, monotone operators, stabilization.

1. Introduction

Let T > 0 and Ω ⊂ IRd, d = 2, 3, be an open and bounded domain, with a smoothboundary ∂Ω (of class C2 for instance). Let Q = Ω× (0, T ), Σ = ∂Ω× (0, T ) andconsider the controlled Boussinesq system,

(1)

∂y

∂t− ν∆y + (y · ∇)y − γ(θ − θ0)ed +∇p = f0 + u0, in Q,

∂θ

∂t− k∆θ + y · ∇θ = f0 + u0, in Q,

div y = 0, in Q,

y = 0, θ = 0, on Σ,

y(·, 0) = y0, θ(·, 0) = θ0, in Ω,

where y = (y1, y2, . . . , yd) denotes the velocity field, while p and θ are scalar func-tions, representing the scalar pressure and the temperature of the fluid. The densi-ties of external forces are f0 = (f01, f02, . . . , f0d) and f0. The constants ν, k > 0 arethe kinematic viscosity coefficient and the thermic diffusivity, respectively. Also,γ = g/θ0 > 0, where g is the gravitational constant and θ0 > 0 is a constant ref-erence temperature, while ed = (0, . . . , 0, 1) is the dth unit vector of IRd. Finally,u0 = (u01, u02, . . . , u0d) and u0 are controls distributed on Ω.

In this section we recall the functional framework and put the Boussinesq systemin an abstract form. In order to give a stability result we have to prove first theglobal existence for the solution of the involved problem. Because in the case d = 3the strong solutions are in general only local, in Section 2 we will give an existenceresult for weak solutions. Section 3 concerns the exponential feedback stabilizationof the Boussinesq equations; the solution remains in a closed convex set. The firststability result is global and the control used is distributed on the entire domain.The second result is a local stability one; the control belongs to a finite dimensionalspace and it is distributed on a subdomain.

This work was supported by CNCSIS Grant PN II ID 404/2007-2010.

1