W I A S Uniqueness in nonlinearly coupled PDE systems DICOP 08, Cortona, September 26, 2008

Transcription

1 W I A S Weierstrass Institute for Applied Analysis and Stochastics in Forschungsverbund B erlin e.v. Pavel Krejčí Uniqueness in nonlinearly coupled PDE systems joint work with Lucia Panizzi DICOP 8, Cor tona, S eptember 26, 28 Mohrenstrasse 39, 1117 B erlin, Germany krejci krejci@wias-berlin.de

2 Model problem: sur face hardening of steel In a domain Q T =, T, R 3 bounded and smooth, consider the system θ t θ = rθ, c c t div Dθ, c c = with boundar y conditions on Σ T =, T θ ν + hx, θ, θ Γx, t = Dθ, c c = bx, t ν and initial conditions θx, = θ x, cx, = c x. θ θ >, θ Γ θ, b L 2 Σ T, c L 2... given data r, D [d, d 1 ], hx,,... smooth constitutive func tions DICOP 8, Cor tona S eptember 26, 28 2/17 W I A S

3 Existence and regularit y I. Assumption on h : a > : θ, θ Γ θ = hx, θ, θ Γ θ θ Γ aθ θ Γ 2. Existence can be proved by Schauder argument. Maximal expec ted regularit y for c : c L 2 Q T c t L 2, T ; H 1 c L, T ; L 2 Regularit y for θ : θ t, θ L 2 Q T θ θ θ L Q T maximum principle Alikakos-Moser This is not enough for uniqueness! DICOP 8, Cor tona S eptember 26, 28 3/17 W I A S

4 Par tial Kirchhoff transform We introduce new func tions u = F θ, c := c Dθ, c dc c = Gθ, u Rθ, u := rθ, Gθ, u, Hθ, u := θ F θ, Gθ, u, Ux, t := We assume that F, G, R, H are Lipschitz, and H is bounded. t ux, τdτ In terms of θ, U, the system of equations has the form θ t ϕ + θ ϕ Rθ, U t ϕ dx + hx, θ, θ Γ x, t ϕ ds = Gθ, U t t ψ + U t ψ Hθ, U t θ ψ dx + bx, tψ ds = for ever y test func tions ϕ, ψ H 1. DICOP 8, Cor tona S eptember 26, 28 4/17 W I A S

5 Uniqueness I. Consider t wo solutions θ 1, U 1, θ 2, U 2, and set θ = θ1 θ 2, Ū = U 1 U 2. Test the difference of the equations for θ 1, θ 2 by ϕ = θ ; Integrate the equations for U 1, U 2 in time, and test the difference by ψ = Ūt. On the left hand side, we have d dt θt 2 L 2 + Ūt 2 L 2 + θt 2 L 2 + Ūtt 2 L 2 We omit the good terms on the right hand side that can be estimated by Gronwall s argument. The only bad term is the integral t Hθ 1, U 1 t θ 1 Hθ 2, U 2 t θ 2 dτ We have to integrate by par ts in time! Ūtx, t dx DICOP 8, Cor tona S eptember 26, 28 5/17 W I A S

6 Uniqueness II. Two bad terms remain: 1 d t Hθ 1, U 1 t θ 1 Hθ 2, U 2 t θ 2 dτ Ūx, t dx dt 2 Hθ 1, U 1 t θ 1 Hθ 2, U 2 t θ 2 Ūx, t dx We integrate in time. Omitting again the terms that can be handled via Gronwall, we obtain for the left hand side t θt 2 L 2 + Ūt 2 L 2 + θτ 2 L 2 + Ūtτ 2 L 2 dτ the upper bounds t 1 Ūx, t t 2 θ + Ū t θ 1 x, τ dτ dx Ū θ + Ūt θ 1 x, τ dx dτ DICOP 8, Cor tona S eptember 26, 28 6/17 W I A S

7 Regularit y II. A necessar y and sufficient condition to continue the above uniqueness estimates reads θ L 2, T ; L It is proved by differentiating θ in tangential direc tions and using the boundar y condition to check the regularit y of the normal component. B oundar y of must be of class C 2,1 Different regularit y is obtained for the normal and for the tangential derivatives Anisotropic Sobolev embedding theorem For a vec tor p = p 1,..., p N we define the space L p R N as the subspace of L 1 R N of func tions u with finite norm p2 /p pn /p 1/p N N 1 1 u p =... ux p 1 dx 1 dx 2... dx N R R R DICOP 8, Cor tona S eptember 26, 28 7/17 W I A S

8 Regularit y III. For a matrix P = P ij N i,j=1, P ij = 1/p ij, we define the anisotropic Sobolev space { } W 1,P R N = u L 1 R N u : L p i R N, i = 1,..., N, p i = p i1,..., p in x i We denote by ϱp the spec tral radius of P, by I the identit y N N matrix, and by 1 the vec tor 1 = 1, 1,..., 1. Theorem Let ϱp < 1, and let I P 1 1 = b = b 1,..., b N. Then W 1,P R N is compac tly embedded in L R N, and there exists a constant C > such that each u W 1,P R N has for all x, z R N the Hölder proper t y uz ux C u W 1,P R N N z i x i 1/b i. i=1 DICOP 8, Cor tona S eptember 26, 28 8/17 W I A S

9 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 Ūx, t t 2 θτ 2 L 2 + Ūtτ 2 L 2 dτ θ + Ū t θ 1 x, τ dτ dx Ū θ + Ūt θ 1 x, τ dx dτ we first use the Cauchy-Schwar z inequalit y t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 9/17 W I A S

10 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side 1 Ūt L 2 t 2 t θτ 2 L 2 + Ūtτ 2 L 2 dτ θ + Ū t 2 θ 1 x, τ dτ dx Ū θ + Ūt θ 1 x, τ dx dτ, we first use the Cauchy-Schwar z inequalit y 1/2, t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 1/17 W I A S

11 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 θ + Ū t 2 θ 1 x, τ dτ dx, t 2 then the Young inequalit y Ū θ + Ūt θ 1 x, τ dx dτ, θτ 2 L 2 + Ūtτ 2 L 2 dτ t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 11/17 W I A S

12 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 θ + Ū t 2 θ 1 x, τ dτ dx, t 2 Ū θ + Ūt θ 1 x, τ dx dτ, θτ 2 L 2 + Ūtτ 2 L 2 dτ we finally eliminate again the terms subjec t to Gronwall s argument. t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 12/17 W I A S

13 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 θ + Ū t 2 θ 1 x, τ dτ dx, t 2 Ū θ + Ūt θ 1 x, τ dx dτ, θτ 2 L 2 + Ūtτ 2 L 2 dτ we finally eliminate again the terms subjec t to Gronwall s argument. t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 13/17 W I A S

14 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 θ + Ū t 2 θ 1 x, τ dτ dx, t 2 Ū θ + Ūt θ 1 x, τ dx dτ, θτ 2 L 2 + Ūtτ 2 L 2 dτ we finally eliminate again the terms subjec t to Gronwall s argument. t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 14/17 W I A S

15 Uniqueness III. In the uniqueness estimates with left hand side θt 2 L 2 + Ūt 2 L 2 + t and right hand side t 1 θ + Ū t 2 θ 1 x, τ dτ dx, t 2 Ū θ + Ūt θ 1 x, τ dx dτ, θτ 2 L 2 + Ūtτ 2 L 2 dτ the remaining term is estimated using Minkowski s inequalit y t Ūt θ 1 dτ 2 dx t Ūt 2 θ 1 2 dx 1/2 dτ 2 DICOP 8, Cor tona S eptember 26, 28 15/17 W I A S

16 Uniqueness IV. The critical inequalit y has the form t Ūtτ 2 L 2 dτ t 2 γτ Ūtτ L 2 dτ with a func tion γ L 2, T : γt = C sup ess x θ 1 x, t. Lemma L p Gronwall inequalit y Let s L p, T, γ L p, T, α L, T be nonnegative func tions such that t 1/p s p τ dτ αt + t γτ sτ dτ for ever y t [, T ], with a fixed p > 1. Then there exists M γ > such that T s p τ dτ 1/p M γ sup ess [,T ] α. DICOP 8, Cor tona S eptember 26, 28 16/17 W I A S

17 Conclusions In coupled parabolic systems, lack of regularit y in one equation can be compensated by a higher regularit y in the other equation; Anisotropic Sobolev embeddings might be useful; Gronwall s inequalit y is not the only argument for uniqueness; These obser vations might be of interest also in other situations Penrose-Fife with state dependent heat conduc tivit y, say; In this case, the phase variable is the more regular one. DICOP 8, Cor tona S eptember 26, 28 17/17 W I A S