Long time behaviour of periodic solutions of uniformly elliptic integro-differential equations

Transcription

1 1/ 15 Long time behaviour of periodic solutions of uniformly elliptic integro-differential equations joint with Barles, Chasseigne (Tours) and Ciomaga (Chicago) C. Imbert CNRS, Université Paris-Est Créteil Valparaíso, January 7-10, 2013

2 2/ 15 Outline of the talk 1 The problem 2 The corrector 3 The large time behaviour

3 3/ 15 Outline of the talk 1 The problem 2 The corrector 3 The large time behaviour

4 4/ 15 A typical example The model equation. t u+ ( 2 x 1,x 1 )u }{{} Diffusion in x ( x 2,x 2 ) α/2 u }{{} Anomalous diffusion in x 2 } {{ } mixed diffusion + u m }{{} Superlinear first order term = f (x) with u(0, x) = u 0 (x) u 0 and f periodic m 1 and α > 1 Fractional Laplacian. ( x 2 2,x 2 ) α/2 u = dz (u(x 1, x 2 + z) u(x 1, x 2 )) z d 1+α.

5 5/ 15 Steady state solutions. If u(t, x) = λt + v(x), then λ + ( 2 x 1,x 1 )v + ( 2 x 2,x 2 ) α/2 v + v m = f (x). Expected. There should exists (λ, v) and λ unique. Problem. Can we prove u(t, x) λt v(x) 0 as t for some (λ, v) solving the cell problem?

6 5/ 15 Steady state solutions. If u(t, x) = λt + v(x), then λ + ( 2 x 1,x 1 )v + ( 2 x 2,x 2 ) α/2 v + v m = f (x). Expected. There should exists (λ, v) and λ unique. Problem. Can we prove u(t, x) λt v(x) 0 as t for some (λ, v) solving the cell problem? Techniques. Maximum principle.

7 6/ 15 Large time behaviour of Hamilton-Jacobi equations Lions (1982) Fathi, Namah-Roquejoffre (HJ1-per) Capuzzo-Dolcetta-Lions, Arisawa, Lions (ergodic control) Barles-Souganidis (HJ1-per and quasilinear eq ns) Barles, Ishii, Mitake (HJ1, non-periodic, BVP etc) Tchambat, Barles-Porretta-Tchambat (viscous HJ)... Solving the ergodic problem Lions-Papanicolaou-Varadhan (1987) Schwab (integro-differential equations)...

8 7/ 15 Outline of the talk 1 The problem 2 The corrector 3 The large time behaviour

9 8/ 15 Theorem There exists a solution (λ, v) of the cell problem. Proof. Consider the following approximated cell problem: δv δ + ( x 2 1,x 1 )v δ + ( x 2 2,x 2 ) α/2 v δ + v δ m = f (x) Fact. a solution for such a problem (Perron s method) with δv δ f Hope. (λ δ, ṽ δ ) = (δv δ, v δ v δ (0)) (λ, v) solving the cell problem Key point. Estimates on the modulus of continuity of v δ Reduction. Enough to prove that (ṽ δ ) δ is unif. bounded [BCCI 2012]

10 9/ 15 Proof (continued). Argue by contradiction (Arisawa-Lions, Schwab): c δ = ṽ δ along a subsequence and rescale ṽ δ w δ = ṽ δ Assume equi-continuity of (w δ ) δ (hard part) Pass to the limit and get c δ w m 0 i.e. w 1. But this contradicts w(0) = 0.

11 10/ 15 Proof (continued). Equi-continuity. Prove that for any µ (0, 1), L(µ) s.t. w δ (x) w δ (y) L(µ) x y α + 2(1 µ) Reformulation. the maximum of µw δ (x) w δ (y) L x y α is nonpositive Argue by contradiction: in particular, x y (2/L) 1/α Write down the equation satisfied by µw δ Write viscosity inequalities and combine them to get c m 1 (1 µ 1 m ) P m O(1) (1 µ)l m/α O(1) For large L s, this yields a contradiction

12 11/ 15 Proof (continued). Prove next that λ is unique and v is up to an additive constant The comparison principle implies that λ 1 t + v 1 (x) λ 2 t + v 2 (x) + C with C = max(v 1 v 2 ). Divide by t and t. Then λ 1 λ 2. Now λt + v 1 (x) and λt + v 2 (x) are two solutions of the Cauchy problem, Lipschitz in x [BCCI 2012]. Then the strong maximum principle [Ciomaga 2012] implies that v 1 v 2 is constant.

13 12/ 15 Outline of the talk 1 The problem 2 The corrector 3 The large time behaviour

14 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R

15 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m

16 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m w(t, x) bounded hence equi-lipschitz in x [BCCI 2012]

17 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m w(t, x) bounded hence equi-lipschitz in x [BCCI 2012] extract a converging subsequence w(φ(n), x) v

18 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m w(t, x) bounded hence equi-lipschitz in x [BCCI 2012] extract a converging subsequence w(φ(n), x) v hence w(t + φ(n), x) w(t, x) with w(0, x) = v

19 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m w(t, x) bounded hence equi-lipschitz in x [BCCI 2012] extract a converging subsequence w(φ(n), x) v hence w(t + φ(n), x) w(t, x) with w(0, x) = v pass to the limit to get: m = max( w(t, ) v) and w = v = v + m [Ciomaga 2012]

20 13/ 15 Theorem For any periodic Lipschitz u 0 and any solution u of the Cauchy problem, there exists a solution of the cell problem (λ, v) such that u(t, x) λt v(x) 0 as t. Proof. Follows closely [Barles-Souganidis-2001] Wanted: w(t, x) := u(t, x) λt v(x) + C for some C R m(t) = max(w(t, ) v) is nonincreasing and m(t) m w(t, x) bounded hence equi-lipschitz in x [BCCI 2012] extract a converging subsequence w(φ(n), x) v hence w(t + φ(n), x) w(t, x) with w(0, x) = v pass to the limit to get: m = max( w(t, ) v) and w = v = v + m [Ciomaga 2012] hence w(t, x) v(x) + m

21 14/ 15 More general equations Example 2 t u + ( ) α/2 u + b u = f b τ-hölder, f continuous, α 1 τ Example 3 t u (u(x + j(x, z)) u(x))dµ(z) + u m = f j(x, z) Lipschitz in x (... ) and f Hölder General form t u+f 1 (x 1, x1 u, Dx 2 1 u, I x1 [x 1, u])+f 2 (x 2,..., I x2 [x 2, u])+h(du) = f with a very long list of Assumptions...

22 15/ 15 References Known results. Arisawa, Lions. On ergodic stochastic control, CPDE Schwab. Periodic homogenization of nonlinear integro-diff equations, SIAM Companion papers. With Barles, Chasseigne, Ciomaga. Lipschitz regularity of solutions of mixed integro-differential equations, JDE Ciomaga. On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Advances in Differential Equations, The paper. With Barles, Chasseigne, Ciomaga. Large time behaviour of periodic viscosity solutions of uniformly elliptic integro-differential equations, arxiv: v1.