INVERSE VISCOSITY BOUNDARY VALUE PROBLEM FOR THE STOKES EVOLUTIONARY EQUATION

Transcription

1 INVERSE VISCOSITY BOUNDARY VALUE PROBLEM FOR THE STOKES EVOLUTIONARY EQUATION Sebastián Zamorano Aliaga Departamento de Ingeniería Matemática Universidad de Chile Workshop Chile-Euskadi 9-10 December 2014

2 Joint Work with Jaime Ortega Departamento de Ingeniería Matemática, Universidad de Chile, Chile. Rodrigo Lecaros Centro de Modelación Matemática (CMM), Universidad de Chille, Chile. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

3 Introduction Contenido 1 Introduction 2 Main Result 3 Proof Main Result 4 References (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

4 Introduction Isakov Result Consider the parabolic problem t u div (a u) + cu = 0, in Ω (0, T ), u = g 0, on Ω (0, T ), u(x, 0) = 0, in Ω, We define the Dirichlet-to-Neumann map as Γ l (g 0 ) = u n, on Ω (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

5 Introduction Isakov Result Theorem 9.4.1, Isakov [3] Let a be a scalar matrix. Then the lateral Dirichlet-to-Neumann map Γ l determines a and b. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

6 Introduction Isakov Result Theorem 9.4.1, Isakov [3] Let a be a scalar matrix. Then the lateral Dirichlet-to-Neumann map Γ l determines a and b. The proof is based in make use of stabilization of solutions of parabolic problem when t, reducing the inverse parabolic problem to inverse elliptic problem with parameter. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

7 Introduction Stokes Equations (1) u t div (σ µ (u, p)) = 0, in Ω (0, T ), div u = 0, in Ω (0, T ), u(x, 0) = u 0 (x), in Ω, where u = (u 1, u 2, u 3 ) is the velocity vector field and p is the pressure and σ µ (u, p) = 2µe(u) pi 3 is the stress tensor, where e(u) = (( u) + ( u) T )/2, and µ > 0 is the viscosity function. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

8 Introduction We are interested in the following inverse problem: (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

9 Introduction We are interested in the following inverse problem: Identifiability of µ by overdetermined data. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

10 Introduction We note that when µ is constant, the problem (1) can be reduced to the following familiar form (2) u t µ u + p = 0, in Ω (0, T ), div u = 0, in Ω (0, T ), u(x, 0) = u 0 (x), in Ω, If u is independent of time, then problem (1) can be formulated as { div (σµ (u, p)) = 0, in Ω, (3) div u = 0, in Ω (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

11 Introduction Heck, Li, and Wang (2007): Identifiability for system (3) (Stationary Stokes Equation.) (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

12 Introduction Heck, Li, and Wang (2007): Identifiability for system (3) (Stationary Stokes Equation.) Lai, Uhlmann, and Wang (2014) : Identifiability for Stokes and Navier-Stokes equations in the plane. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

13 Introduction Heck, Li, and Wang (2007): Identifiability for system (3) (Stationary Stokes Equation.) Lai, Uhlmann, and Wang (2014) : Identifiability for Stokes and Navier-Stokes equations in the plane. Isakov (1999) : Some inverse problem for the diffusion equation. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

14 Main Result Contenido 1 Introduction 2 Main Result 3 Proof Main Result 4 References (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

15 Main Result Mathematical Setup Let Ω R 3 be an open bounded connected domain with boundary Ω C 2. Consider the following boundary value problem u t div (σ µ (u, p)) = 0, in Ω (0, T ), div u = 0, in Ω (0, T ), (4) u = g, on Ω (0, T ), u(x, 0) = u 0 (x), in Ω, where g satisfies the compatibility condition g nds = 0, Ω where n is the unit outer normal of Ω, (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

16 Main Result Assume that the solution of (4) exists and the trace is well defined. σ µ (u, p) n Ω (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

17 Main Result Assume that the solution of (4) exists and the trace is well defined. σ µ (u, p) n Ω Physically, σ(u, p) n Ω is the Cauchy forces acting on the boundary Ω. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

18 Main Result Assume that the solution of (4) exists and the trace is well defined. σ µ (u, p) n Ω Physically, σ(u, p) n Ω is the Cauchy forces acting on the boundary Ω. Define the set of Cauchy data for (4) S µ = {(u Ω, σ µ (u, p) n Ω ) : (u, p) solution to (4)}. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

19 Main Result Inverse Problem : Determine µ from S µ. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

20 Main Result Inverse Problem : Determine µ from S µ. Identifiablity : if S µ1 = S µ2 then µ 1 = µ 2?. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

21 Main Result Inverse Problem : Determine µ from S µ. Identifiablity : if S µ1 = S µ2 then µ 1 = µ 2?. Theorem 1 Assume that µ 1 and µ 2 are two viscosity functions satisfying µ 1, µ 2 C k (Ω) for k 8 and (5) (6) µ i 1, i = 1, 2. µ 1 (x) = µ 2 (x), x Ω. Let S µ1 and S µ2 be the Cauchy data associated with µ 1 and µ 2, respectively. If S µ1 = S µ2, then µ 1 = µ 2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

22 Proof Main Result Contenido 1 Introduction 2 Main Result 3 Proof Main Result 4 References (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

23 Proof Main Result Sketch Proof Stabilization of solutions of following Stokes problem u t div (σ µ (u, p)) + u = 0, en Ω (0, T ), div u = 0, en Ω (0, T ), (7) u = g, sobre Ω (0, T ), u(x, 0) = u 0 (x), en Ω. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

24 Proof Main Result Sketch Proof Stabilization of solutions of following Stokes problem u t div (σ µ (u, p)) + u = 0, en Ω (0, T ), div u = 0, en Ω (0, T ), (7) u = g, sobre Ω (0, T ), u(x, 0) = u 0 (x), en Ω. Identifiability result for the stationary Stokes problem div (σ µ (u, p )) + u = 0, en Ω, div u = 0, en Ω, u = g(t ), sobre Ω. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

25 Proof Main Result Sketch Proof Stabilization of solutions of following Stokes problem u t div (σ µ (u, p)) + u = 0, en Ω (0, T ), div u = 0, en Ω (0, T ), (7) u = g, sobre Ω (0, T ), u(x, 0) = u 0 (x), en Ω. Identifiability result for the stationary Stokes problem div (σ µ (u, p )) + u = 0, en Ω, div u = 0, en Ω, u = g(t ), sobre Ω. Conclusion of result. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

26 Proof Main Result Asymptotic Stability Evolutionary Stokes Solution Consider the following evolutionary Stokes system u t div (σ µ (u, p)) + u = 0, in Ω (0, T ), div u = 0, in Ω (0, T ), (8) u = g, on Ω (0, T ), u(x, 0) = u 0 (x), in Ω. Then, we can prove the following result for solutions of this systems (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

27 Proof Main Result Theorem 2 Let (u, p) be a solution of (8). Assume that g do not depend on t, for all t (T /2, T ) for some T > 0. Then, there exists constants C 1, C 2 > 0 such that u(t) u H1 (Ω) u(t /2) u 2 L 2 (Ω) e 2C1t + u(t /2) u 2 L 2 (Ω) e 2C 2 t µ, t > T 2, where (u, p ) H 2 (Ω) H 1 (Ω) is a solution to the (stationary) Stokes problem div (σ µ (u, p )) + u = 0, in Ω, (9) div u = 0, in Ω, u = g(t ), on Ω. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

28 Proof Main Result Identifiability of Viscosity: Stationary Case Let (u, p) be the solution to the stationary Stokes problem div (σ µ (v, q )) + v = 0, in Ω, (10) div v = 0, in Ω, v = g 0, on Ω. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

29 Proof Main Result Theorem 3 Let (u, p) be the solution to the stationary Stokes problem (10). Assume that µ 1 (x) and µ 2 (x) are two viscosity function satisfying µ 1, µ 2 C k (Ω), k 8. µ i 1, i = 1, 2. µ 1 (x) = µ 2 (x), x Ω. Let S E µ 1 and S E µ 2 be the Cauchy data associated with µ 1 and µ 2, respectively. If S E µ 1 = S E µ 2, then µ 1 = µ 2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

30 Proof Main Result Proof Theorem 1 Consider the substitution u = ve λt, p = qe λt, with λ > 0. Then (11) { vt div (σ µ (v, q)) + λv = 0, en Ω (0, T ), div v = 0, en Ω (0, T ). (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

31 Proof Main Result Proof Theorem 1 Consider the substitution u = ve λt, p = qe λt, with λ > 0. Then (11) { vt div (σ µ (v, q)) + λv = 0, en Ω (0, T ), div v = 0, en Ω (0, T ). Let g = g 0 φ, where g 0 any function in C 2 (Ω) and φ(t) C (R) satisfies the conditions φ(t) = 0 on (, T /4) and φ(t) = e λt on (T /2, ). (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

32 Proof Main Result Proof Theorem 1 Consider the substitution u = ve λt, p = qe λt, with λ > 0. Then (11) { vt div (σ µ (v, q)) + λv = 0, en Ω (0, T ), div v = 0, en Ω (0, T ). Let g = g 0 φ, where g 0 any function in C 2 (Ω) and φ(t) C (R) satisfies the conditions φ(t) = 0 on (, T /4) and φ(t) = e λt on (T /2, ). The boundary data of the equation (11) is independent of t > T /2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

33 Proof Main Result We can define the Cauchy data, S E µ, associated with the stationary problem (10), using Theorem 2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

34 Proof Main Result We can define the Cauchy data, S E µ, associated with the stationary problem (10), using Theorem 2. Then S µ1 = S µ2 then S E µ 1 = S E µ 2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

35 Proof Main Result We can define the Cauchy data, S E µ, associated with the stationary problem (10), using Theorem 2. Then S µ1 = S µ2 then S E µ 1 = S E µ 2. Theorem 3 implies that µ 1 = µ 2. (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

36 References Contenido 1 Introduction 2 Main Result 3 Proof Main Result 4 References (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

37 References References G. Eskin and J. Ralston. On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), H. Heck, R. Li, and J.N. Wang, Identification of Viscosity in an Incompressible Fluid, Indiana Univ. Math. J. 56 (2007), no. 5, 2489?2510. V. Isakov, Some inverse problems for the diffusion equation, Inverse Problems 15 (1999) R. Lai, G.Uhlmann, and J-N. Wang, Inverse Boundary Value Problem for the Stokes and the Navier-Stokes equations in the plane, arxiv febrero (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24

38 References Thank you for your attention (BCAM - Bilbao) Workshop Chile-Euskadi 9-10 December / 24