On a stochastic mass conserved Allen-Cahn equation with nonlinear - PowerPoint PPT Presentation

On a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion Perla El Kettani 1 , Danielle Hilhorst 2 , Kai Lee 3 1 University of Paris-Sud 2 CNRS and University of Paris-Sud 3 University of Tokyo, Japan March 25th, 2019 Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 1 / 35

Two related topics I - A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion; • nonlinear diffusion, mass conservation, additive noise II - Stochastic nonlocal Allen-Cahn equation with a multiplicative noise. • linear diffusion, no mass conservation, multplicative noise Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 2 / 35

The mathematical problem Prove the existence and uniqueness of the solution of the nonlocal stochastic reaction-diffusion equation with nonlinear diffusion  ∂ϕ ∂ t = div ( A ( ∇ ϕ )) + f ( ϕ ) − 1 � f ( ϕ ) dx + ∂ W ∂ t , in D × ( 0 , T )   | D |  D   ( P ) A ( ∇ ϕ ) .ν = 0 , on ∂ D × ( 0 , T )    x ∈ D ϕ ( x , 0 ) = ϕ 0 ( x ) ,   Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 3 / 35

Motivation [Rubinstein and Sternberg, 1992] Deterministic mass conserved 1 Allen-Cahn equation with linear diffusion. Binary mixture undergoing phase separation. [Boussaïd, Hilhorst and Nguyen, 2015] proved the 2 well-posedness and the stabilization of the solution for large times for the corresponding Neumann problem. [Antonopoulou, Bates, Blömker and Karali, 2016] 3 • use the stochastic mass conserved equation with linear diffusion to describe the motion of a droplet; • They study the singular limit of the solution of this equation, letting a small parameter tend to zero; • They consider the colored noise, namely a white noise in time which also depends on space. [Funaki and Yokoyama, 2016] study the singular limit of the 4 solution of the stochastic mass conserved Allen-Cahn equation with a smoothened one dimensional white noise in time. Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 4 / 35

Hypotheses on the domain D and on the function A • D open bounded set of R n , with a smooth boundary ∂ D ; • A is such that | A ( a ) − A ( b ) | ≤ C | a − b | ; • A = ∇ Ψ : R n → R n for some convex C 1 , 1 -function Ψ and A ( 0 ) = 0; • A is monotone ( A ( a ) − A ( b )) . ( a − b ) ≥ C 0 ( a − b ) 2 , C 0 > 0 ; for all a , b ∈ R n . [Funaki, Spohn,1997] . Remark: If A = I ⇒ div ( A ( ∇ u )) = ∆ u . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 5 / 35

Usual diffusion Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 6 / 35

Anisotropic diffusion Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 7 / 35

The nonlinear function f The nonlinear function f is given by 2 p − 1 b j s j with b 2 p − 1 < 0 , p ≥ 2 , � f ( s ) = j = 0 which also includes the Allen-Cahn equation with f ( s ) = s ( 1 − s 2 ) . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 8 / 35

The Q-Brownian motion • The function W ( x , t ) is a Q-Brownian motion in L 2 ( D ) . ∞ ∞ 1 � � � 2 e k ( x ) = W ( x , t ) = β k ( t ) Q λ k β k ( t ) e k ( x ) , k = 1 k = 1 where { e k } k ≥ 1 is an orthonormal basis in L 2 ( D ) diagonalizing Q ; 1 { λ k } k ≥ 1 are the corresponding eigenvalues for all k ≥ 1; 2 Q is a nonnegative definite symmetric operator on L 2 ( D ) with 3 Tr Q < + ∞ . ∞ ∞ � � Tr Q = � Qe k , e k � L 2 ( D ) = λ k ≤ Λ 0 ; k = 1 k = 1 ∞ � λ k � e k � 2 L ∞ ( D ) ≤ Λ 1 ; k = 1 { β k ( t ) } k ≥ 1 is a sequence of independent ( F t ) -Brownian motions 4 defined on the probability space (Ω , F , P) . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 9 / 35

The nonlinear stochastic heat equation We consider the nonlinear stochastic heat equation  ∂ W A = div ( A ( ∇ W A )) + ∂ W ∂ t , in D × ( 0 , T )    ∂ t   ( P 1 ) A ( ∇ ( W A )) .ν = 0 , on ∂ D × ( 0 , T )    W A ( x , 0 ) = 0 , x ∈ D   [Krylov, Rozovskii, 1981] Definition W A ∈ L ∞ ( 0 , T ; L 2 (Ω × D )) ∩ L 2 (Ω × ( 0 , T ); H 1 ( D )) ; 1 div ( A ( ∇ W A )) ∈ L 2 (Ω × ( 0 , T ); ( H 1 ( D )) ′ ); 2 W A satisfies a.s. for a.e. t ∈ ( 0 , T ) the problem 3 � t  in ( H 1 ( D )) ′ , W A ( t ) = div ( A ( ∇ W A ( s ))) ds + W ( t )  0 A ( ∇ W A ( t )) .ν = 0 ,  in the sense of distributions on ∂ D . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 10 / 35

A preliminary change of functions We remark that W A ∈ L ∞ ( 0 , T ; L q (Ω × D )) for all q ≥ 2, and define: u ( t ) := ϕ ( t ) − W A ( t ) , ∂ u  ∂ t = div ( A ( ∇ ( u + W A )) − A ( ∇ W A )) + f ( u + W A )      − 1 �   in D × ( 0 , T ) f ( u + W A ) dx ,  ( P 2 ) | D | D  A ( ∇ ( u + W A )) .ν = 0 , on ∂ D × ( 0 , T )      u ( x , 0 ) = ϕ 0 ( x ) , x ∈ D   Remark: The conservation of mass property holds, namely � � u ( x , t ) dx = ϕ 0 ( x ) dx , for a.e. t ∈ ( 0 , T ) . D D Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 11 / 35

Definition of the solution Definition u ∈ L ∞ ( 0 , T ; L 2 (Ω × D )) ∩ L 2 (Ω × ( 0 , T ); H 1 ( D )) ∩ L 2 p (Ω × ( 0 , T ) × D ) , 1 div [ A ( ∇ ( u + W A ))] ∈ L 2 (Ω × ( 0 , T ); ( H 1 ( D )) ′ ) ; u satisfies the integral equation a.s. for a.e. t ∈ ( 0 , T ) in the sense 2 of distributions in D � t u ( t ) = ϕ 0 + div [ A ( ∇ ( u + W A )) − A ( ∇ W A )] ds 0 � t � t 1 � + f ( u + W A ) − f ( u + W A ) dxds | D | 0 0 D u satisfies the natural boundary condition in the sense of 3 distributions on ∂ D. Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 12 / 35

Existence of a solution of Problem ( P 2 ) Theorem There exists a unique solution of Problem ( P 2 ) . We work with the following spaces: � � � v ∈ L 2 ( D ) , , V = H 1 ( D ) ∩ H and Z = V ∩ L 2 p H = v = 0 D Proof: We apply the Galerkin method, and use the following notations • 0 < γ 1 < γ 2 ≤ ... ≤ γ k ≤ ... eigenvalues of − ∆ with homogeneous Neumann boundary conditions. • w k , k = 0 , ... smooth unit eigenfunctions in L 2 ( D ) . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 13 / 35

Existence of a solution of Problem ( P 2 ) We look for an approximate solution of the form m u im ( t ) w i , M = 1 � � u m ( x , t ) − M = ϕ 0 ( x ) dx | D | D i = 1 which satisfies the equation: � ∂ ∂ t ( u m ( x , t ) − M ) w j D � � = − [ A ( ∇ ( u m − M + W A )) − A ( ∇ ( W A ))] ∇ w j + f ( u m + W A ) w j D D − 1 � � � � f ( u m + W A ) dx w j dx , | D | D D for all w j , j = 1 , ..., m . m � ( ϕ 0 , w i ) w i converges strongly in L 2 ( D ) to ϕ 0 as u m ( x , 0 ) = M + i = 1 m → ∞ . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 14 / 35

Existence of a solution of Problem ( P 2 ) Remark: The contribution of the nonlocal term vanishes !! � w j ( x ) dx = 0 , for all j � = 0 D ⇓ − 1 � � � � f ( u m + W A ) dx w j = 0 | D | D D � ∂ ∂ t ( u m ( x , t ) − M ) w j D � � = − [ A ( ∇ ( u m − M + W A )) − A ( ∇ ( W A ))] ∇ w j + f ( u m + W A ) w j D D for all w j , j = 1 , ..., m . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 15 / 35

A priori estimates Lemma There exists a positive constant C such that � ( u m ( t ) − M ) 2 dx ≤ C , for all t ∈ [ 0 , T ] , E D � T � |∇ ( u m − M ) | 2 dxdt ≤ C , E 0 D � T � ( u m − M ) 2 p dxdt ≤ C , E 0 D � T � 2 p ( f ( u m + W A )) ≤ C , E 2 p − 1 0 D � T � divA ( ∇ ( u m + W A )) � 2 ≤ C . E ( H 1 ( D )) ′ 0 Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 16 / 35

Weak convergence properties Hence there exist a subsequence which we denote again by { u m − M } and functions u − M ∈ L 2 (Ω × ( 0 , T ); V ) ∩ L 2 p (Ω × ( 0 , T ) × D ) ∩ L ∞ ( 0 , T ; L 2 (Ω × D )) , χ and Φ such that: weakly in L 2 (Ω × ( 0 , T ); V ) u m − M ⇀ u − M and L 2 p (Ω × ( 0 , T ) × D ) weakly star in L ∞ ( 0 , T ; L 2 (Ω × D )) u m − M ⇀ u − M 2 p 2 p − 1 (Ω × ( 0 , T ) × D ) f ( u m + W A ) weakly in L ⇀ χ L 2 (Ω × ( 0 , T ); ( H 1 ) ′ ) div ( A ( ∇ ( u m + W A ))) ⇀ Φ weakly in as m → ∞ . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 17 / 35

Passing to the limit It remains to prove that : � Φ + χ, w � = � div ( A ( ∇ ( u + W A ))) + f ( u + W A ) , w � for all w ∈ V ∩ L 2 p ( D ) . Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 18 / 35