Phase-field systems with nonlinear coupling and dynamic boundary conditions

Transcription

1 1 / 46 Phase-field systems with nonlinear coupling and dynamic boundary conditions Cecilia Cavaterra Dipartimento di Matematica F. Enriques Università degli Studi di Milano cecilia.cavaterra@unimi.it VIII Workshop on Partial Differential Equations Rio de Janeiro, August 25-28, 2009 Joint work with C.G.Gal, M.Grasselli, A.Miranville

2 Phase-field system 2 / 46 A well-known system of partial differential equations which describes the behavior of a two-phase material in presence of temperature variations, and neglecting mechanical stresses, is the so-called Caginalp phase-field system { δψt D ψ + f (ψ) λ (ψ)θ = 0, Ω (0, ) (εθ + λ(ψ)) t k θ = 0, Ω (0, ) Ω R 3, bounded domain with smooth boundary Γ ψ order parameter (or phase-field) θ (relative) temperature δ, D, ε, k positive coefficients λ function related to the latent heat f = F, F nonconvex potential (e.g., double well potential)

3 A naive derivation of the system 3 / 46 bulk free energy functional E Ω (ψ, θ) = Ω [ D 2 ψ 2 + F(ψ) λ(ψ)θ ε ] 2 θ2 dx θ E Ω (ψ, θ), ψ E Ω (ψ, θ): Frechét derivatives of E Ω (ψ, θ) equation for the temperature ( θ E Ω (ψ, θ)) t + q = 0, q = k θ equation for the order parameter ψ t = ψ E Ω (ψ, θ)

4 Motivation for studying the case λ nonlinear 4 / 46 Assuming λ linear is satisfactory for solid-liquid phase transitions However, when one deals, for instance, with phase transitions in ferromagnetic materials, where ψ represents the fraction of lattice sites at which the spins are pointing up, then a quadratic λ is a more appropriate choice (see M.Brokate & J.Sprekels (1996))

5 Standard boundary conditions for ψ and θ 5 / 46 ψ: Neumann boundary condition n: outward normal to Γ n ψ = 0 θ: Dirichlet, Neumann, Robin boundary conditions b n θ + cθ = 0 (D): b = 0, c > 0 (N): b > 0, c = 0 (R): b > 0, c > 0

6 Dynamic boundary condition for ψ 6 / 46 In order to account for possible interaction with the boundary Γ, we can consider dynamic boundary conditions for ψ surface free energy functional [ α E Γ (ψ) = 2 Γψ 2 + β ] 2 ψ2 + G(ψ) ds Γ tangential gradient operator Γ α, β > 0, G nonconvex boundary potential ψ E Γ (ψ): Frechét derivative of E Γ (ψ) ψ t = ψ E Γ (ψ, θ) n ψ ψ t α Γ ψ + n ψ + βψ + g(ψ) = 0 Γ Laplace-Beltrami operator, g = G

7 (IBVP) for the phase-field system 7 / 46 Without loss of generality we take D = k = δ = ε = 1 evolution equations in Ω (0, ) ψ t ψ + f (ψ) λ (ψ)θ = 0 (θ + λ(ψ)) t θ = 0 boundary conditions on Γ (0, ) ψ t α Γ ψ + n ψ + βψ + g(ψ) = 0 b n θ + cθ = 0 initial conditions on Ω θ(0) = θ 0, ψ(0) = ψ 0

8 Main goals 8 / 46 Well posedness Existence of the global attractor Existence of an exponential attractor Convergence of solutions to single equilibria

9 Definitions of global attractor and exponential attractor Dynamical system: X : complete metric sp. (X, S(t)) S(t) : X X semigroup of op. Hausdorff semidistance: dist X (W, Z) = sup inf w W z Z The global attractor A X is a compact set in X : A is fully invariant (S(t)A = A, t 0) A is an attracting set w.r.t. H-semidistance: bdd B lim t dist X (S(t)B, A) = 0 d(w, z) An exponential attractor E X is a compact set in X : E is invariant (S(t)E E, t 0) E has finite fractal dimension E is an exponential attracting set w.r.t. H-semidistance: bdd B C B > 0, ω > 0 s.t. dist X (S(t)B, E) C B e ωt, t 0 9 / 46

10 Literature: λ linear + standard b.c. + smooth potentials 10 / 46 C.M.Elliott & S.Zheng (1990); A.Damlamian, N.Kenmochi & N. Sato (1994); G.Schimperna (2000) well-posedness V.K.Kalantarov (1991); P.W.Bates & S.Zheng (1992); D.Brochet, X.Chen & D.Hilhorst (1993); O.V.Kapustyan (2000); A.Jiménez-Casas & A.Rodríguez-Bernal (2002) longtime behavior of solutions existence and smoothness of global attractors existence of exponential attractors Z.Zhang (2005) asymptotic behavior of single solutions

11 Literature: λ nonlinear + standard b.c. + smooth potentials 11 / 46 Ph.Laurençot (1996); M.Grasselli & V.Pata (2004) well-posedness, global and exponential attractors M.Grasselli, H.Wu & S.Zheng (2008) global and exponential attractors, asymptotic behavior of single solutions (nonhomogenous b.c.)

12 Literature: λ nonlinear + standard b.c. + singular potentials 12 / 46 M.Grasselli, H.Petzeltová & G.Schimperna (2006) well-posedness and asymptotic behavior

13 Literature: λ linear + dynamic b.c. + smooth potentials 13 / 46 R.Chill, E.Fašangová & J.Prüss (2006) F with polynomial controlled growth of degree 6, G 0 well-posedness convergence to single equilibria via Łojasiewicz-Simon inequality (when F is also real analytic) S.Gatti & A.Miranville (2006) construction of a s-continuous dissipative semigroup global attractor A ε upper semicontinuous at ε = 0 exponential attractors E ε

14 Literature: λ linear + dynamic b.c. + smooth potentials 14 / 46 C.G.Gal, M.Grasselli & A.Miranville (2007) family of exponential attractors {E ε } stable as ε 0 when n θ = 0 C.G.Gal & M.Grasselli (2008, 2009) F and G smooth potentials (more general than S.Gatti & A.Miranville) (possibly) dynamic boundary condition for θ (a, b, c 0) aθ t + b n θ + cθ = 0 construction of a dissipative semigroup (larger phase spaces w.r.t. S.Gatti & A.Miranville) global attractor, exponential attractors

15 Literature: λ linear + dynamic b.c. + singular potential 15 / 46 L.Cherfils & A.Miranville (2007) F singular potential defined on ( 1, 1) G smooth potential (sign restrictions) construction of a s-continuous dissipative semigroup global attractor of finite fractal dimension convergence to single equilibria via Ł-S method (when F is real analytic and G 0) S.Gatti, L.Cherfils & A.Miranville (2007, 2008) F (strongly) singular potential defined on ( 1, 1) G smooth potential (sign restrictions are removed) separation property and existence of global solutions existence of global and exponential attractors

16 Notations 16 / 46 p p,γ norm on L p (Ω) norm on L p (Γ), 2 usual scalar product inducing the norm on L 2 (Ω) (even for vector-valued functions), 2,Γ usual scalar product inducing the norm on L 2 (Γ) (even for vector-valued functions) H s (Ω) norm on H s (Ω), for any s > 0 H s (Γ) norm on H s (Γ), for any s > 0

17 17 / 46 The operators A K In order to account different cases of boundary conditions, we introduce the linear operators A K = : D(A K ) L 2 (Ω) D(A K ) = H 1 0 (Ω) H2 (Ω), if K = D D(A K ) = {θ H 2 (Ω) : b n θ + cθ = 0}, if K = N, R D, N, R stand for Dirichlet, Neumann, or Robin bdry conds A K generates an analytic semigroup e A K t on L 2 (Ω) A K is nonnegative and self-adjoint on L 2 (Ω)

18 18 / 46 The functional spaces Z 1 K Z 1 D = H1 0 (Ω) Z 1 K = H1 (Ω), if K {N, R} θ 2 2, if K = D, θ 2 Z 1 = θ c K b θ Γ 2, if K = R, 2,Γ θ θ 2 Ω, if K = N, where we have set v Ω := Ω 1 Ω v (x) dx the norm in Z 1 K is equivalent to the standard H1 -norm

19 19 / 46 The function spaces V s V s = C ( s Ω ) Vs, s > 0 ( ψ Vs = ψ 2 H s (Ω) + ψ Γ 2 H s (Γ) V s = H s (Ω) H s (Γ) V 0 = L 2 (Ω) L 2 (Γ) ) 1/2 V s is compactly embedded in V s 1, s 1 H 1 (Ω) L 6 (Ω) H 1/2 (Γ) L 4 (Γ) H 1 (Γ) L s (Γ), s 1

20 Enthalpy conservation 20 / 46 In the case K = N we define the enthalpy I N (ψ(t), θ(t)) := λ (ψ(t)) + θ(t) Ω and this quantity is conserved in time for any given solution

21 The weak formulation Problem P w K (ψ 0, θ 0 ) V 1 L 2 (Ω) find (ψ, θ) C ( [0, + ); V 1 L 2 (Ω) ) : ψ t L 2 ([0, + ); V 0 ), θ L 2 ( [0, + ); (L 2 (Ω)) 3) ψ t, u 2 + ψ, u 2 + f (ψ) λ (ψ) θ, u 2 + ψ t, u 2,Γ +α Γ ψ, Γ u 2,Γ + βψ + g (ψ), u 2,Γ = 0, u V 1, a.e. in (0, ) (θ + λ (ψ)) t, v 2 + θ, v 2 + d θ, v 2,Γ = 0, v Z 1 K, a.e. in (0, ) θ(0) = θ 0, ψ(0) = ψ 0 and, if K = N, I N (ψ(t), θ(t)) = I N (ψ 0, θ 0 ), t 0 Here d = c b if K = R, d = 0 otherwise 21 / 46

22 The strong formulation 22 / 46 Problem P s K (ψ 0, θ 0 ) V 2 Z 1 K find (ψ, θ) C ( [0, + ); V 2 Z 1 ) K : (ψ t, θ t ) L 2 ( [0, + ); V 1 L 2 (Ω) ) A K θ L 2 ([0, + ) Ω) ψ t ψ + f (χ) λ (ψ)θ = 0, a.e. in Ω (0, + ) (θ + λ(ψ)) t θ = 0, a.e. in Ω (0, + ) ψ t α Γ ψ + n ψ + βψ + g(ψ) = 0, a.e. in Γ (0, + ) b n θ + cθ = 0, a.e. in Γ (0, + ) θ(0) = θ 0, ψ(0) = ψ 0 and, if K = N, I N (ψ(t), θ(t)) = I N (ψ 0, θ 0 ), t 0

23 Assumptions on the nonlinearities f, g and λ (H1) f, g C 1 (R) satisfy lim inf f (y) > 0, y + lim inf y + g (y) > 0 (H2) c f, c g > 0, q [1, + ) : f (y) c f (1 + y 2), g (y) c g ( 1 + y q ), y R (H3) λ C 2 (R) : λ (y) c λ, y R, where c λ > 0 (H4) a > 0 and γ C 2 (R) W 1, (R) : λ (y) = γ (y) ay 2, y R (H5) η 1 > 0 and η 2 0 : f (y) y η 1 y 4 η 2, y R 23 / 46

24 Remarks Remark The growth restriction (H2) is only needed to analyze P w K. Remark Assumption (H4) is only needed to handle the case K = N. Note that (H4) is justified from a physical viewpoint (see, e.g., M.Brokate & J.Sprekels (1996)). 24 / 46

25 25 / 46 The global existence result for P w K Theorem Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Then, for each K {D, N, R}, problem P w K weak solution. admits a global

26 Sketch of the proof Proof. suitable Faedo-Galerkin approximation scheme based on the characterization of a suitable self-adjoint positive operator B on V 0 satisfying Wentzell b.c. (cf. C.G.Gal, G.Goldstein, J.Goldstein, S.Romanelli & M.Warma (2009)) system of nonlinear ODEs for the approximating solutions a priori estimates for the approximating solutions global existence (in time) + strong convergence passage to the limit in the nonlinear terms convergence of the approximating solutions to the solution of P w K 26 / 46

27 27 / 46 The global existence result for P s K Theorem Let f, g, λ satisfy (H1) and (H3). Then, for each K {D, N, R}, problem P s K admits a global strong solution. Proof. global existence result for P w K higher order estimates

28 P w K : continuous dependence estimate 28 / 46 Lemma Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Let (ψ wi, θ wi ) be two global solutions to P w K corresponding to the initial data (ψ 0i, θ 0i ) V 1 L 2 (Ω), i = 1, 2. Then, for any t 0, the following estimate holds: (ψ w1 ψ w2 ) (t) 2 V 1 + (θ w1 θ w2 ) (t) 2 2 t + 0 [ ] (ψ w1 ψ w2 ) t (s) 2 V 0 + (θ w1 θ w2 ) (s) 2 Z 1 ds K C w e Lw t ( ψ 01 ψ 02 2 V 1 + θ 01 θ Here C w and L w are positive constants depending on the norms of the initial data in V 1 L 2 (Ω), on Ω and on the parameters of the problem, but are both independent of time. )

29 P s K : continuous dependence estimate 29 / 46 Lemma Let f, g, λ satisfy (H1) and (H3). Let (ψ si, θ si ) two global solutions to P s K corresponding to the initial data (ψ 0i, θ 0i ) V 2 Z 1 K, i = 1, 2. Then, for any t 0, the following estimate holds: (ψ s1 ψ s2 ) (t) 2 V 1 + (θ s1 θ s2 ) (t) 2 2 t + 0 [ ] (ψ s1 ψ s2 ) t (s) 2 V 0 + (θ s1 θ s2 ) (s) 2 Z 1 ds K C s e Lst ( ψ 01 ψ 02 2 V 1 + θ 01 θ Here C s and L s are positive constants depending on the norms of the initial data in V 2 Z 1 K, on Ω and on the parameters of the problem, but are both independent of time. )

30 30 / 46 The dynamical system generated by P w K Corollary Let f, g, λ satisfy assumptions (H1), (H2) and (H3). Then, for each K {D, N, R}, we can define a strongly continuous semigroup by setting, for all t 0, S w K (t) : V 1 L 2 (Ω) V 1 L 2 (Ω) S w K (t) (ψ 0, θ 0 ) = (ψ w (t), θ w (t)) where (ψ w, θ w ) is the unique solution to P w K.

31 31 / 46 The dynamical system generated by P s K Corollary Let f, g, λ satisfy (H1) and (H3). Then, for each K {D, N, R}, we can define a semigroup SK s (t) : V 2 Z 1 K V 2 Z 1 K by setting, for all t 0, SK s (t) (ψ 0, θ 0 ) = (ψ s (t), θ s (t)) where (ψ s, θ s ) is the unique solution to P s K.

32 32 / 46 The phase spaces for S w K and Ss K Due to the enthalpy conservation, in the case K = N we define the complete metric spaces w.r.t. the metrics induced by the norms where the constraint M 0 is fixed ( ) M { } V 1 L 2 (Ω) := (u, v) V 1 L 2 (Ω) : I N (u, v) M ( ) M { } V 2 Z 1 N := (u, v) V 2 Z 1 N : I N (u, v) M Phase-space for SK w(t) { V1 L 2 (Ω), if K {D, R} Y 0,K = ( V1 L 2 (Ω) ) M, if K = N Phase-space for SK s (t) { V2 Z 1 K, if K {D, R} Y 1,K = ( V2 Z 1 ) M N, if K = N

33 33 / 46 A dissipative estimate for S w K and Ss K Lemma Let f, g satisfy assumptions (H1) and (H5). Let λ satisfies either (H3), if K {D, R}, or (H4), if K = N. Then, (ψ 0, θ 0 ) Y 0,K, the following estimate holds (ψ (t), θ (t)) 2 Y 0,K t+1 ( ) + ψ t (s) 2 V 0 + θ (s) 2 Z 1 + ψ (s) 4 L K 4 (Ω) ds t ) C K ( (ψ 0, θ 0 ) 2 Y 0,K + F (ψ 0 ), G (ψ 0 ), 1 2,Γ C K, t 0 where F (y) = y 0 f (r) dr, G (y) = y 0 g (r) dr, y R. Here ρ, C K, C K are independent of t and of the initial data. e ρt

34 Existence of a compact absorbing set Lemma Let f, g, λ satisfy (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, R 0 > 0, a positive nondecreasing monotone function Q and t 0 = t 0 (R 0 ) > 0 : S w K (t) (ψ 0, θ 0 ) V2 H 2 (Ω) Q(R 0), t t 0 (ψ 0, θ 0 ) B(R 0 ) Y 0,K, where B(R 0 ) is a ball of radius R 0. Lemma Let f, g, λ satisfy (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, R 1 > 0, a positive nondecreasing monotone function Q and t 1 = t 1 (R 1 ) > 0 : S s K (t) (ψ 0, θ 0 ) V3 H 3 (Ω) Q(R 1), t t 1 (ψ 0, θ 0 ) B(R 1 ) Y 1,K, where B(R 1 ) is a ball of radius R / 46

35 Existence of the global attractor 35 / 46 Theorem Let f, g, λ satisfy assumptions (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, SK w (t) possesses the connected global attractor A w K Y 0,K, which is bounded in V 2 H 2 (Ω). Theorem Let f, g, λ satisfy assumptions (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, SK s (t) possesses the connected global attractor A s K Y 1,K, which is bounded in V 3 H 3 (Ω).

36 Existence of exponential attractors for S w K (t) 36 / 46 Theorem Let f, g C 2 (R) and λ C 3 (R) satisfy (H1), (H2), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, SK w (t) has an exponential attractor M w K, bounded in V 2 H 2 (Ω), namely, (I) M w K is compact and positively invariant w.r.t Sw K (t), i.e., S w K (t) (Mw K ) Mw K, t 0. (II) The fractal dimension of M w K w.r.t. Y 0,K -metric is finite. (III) There exist a positive nondecreasing monotone function Q w and a constant ρ w > 0 such that dist Y0,K (S w K (t) B, Mw K ) Q w( B Y0,K )e ρw t, t 0, where B is any bounded set of initial data in Y 0,K. Here dist Y0,K denotes the non-symmetric Hausdorff distance in Y 0,K and B Y0,K stands for the size of B in Y 0,K.

37 Existence of exponential attractors for S s K (t) 37 / 46 Theorem Let f, g C 2 (R) and λ C 3 (R) satisfy (H1), (H3) and (H5). If K = N, assume also λ fulfilling (H4). Then, SK s (t) has an exponential attractor M s K, bounded in V 3 H 3 (Ω), namely, (I) M s K is compact and positively invariant w.r.t. Ss K (t), i.e., S s K (t) (Ms K ) Ms K, t 0. (II) The fractal dimension of M s K w.r.t. Y 1,K -metric is finite. (III) There exist a positive nondecreasing monotone function Q s and a constant ρ s > 0 such that dist Y1,K (S s K (t) B, Ms K ) Q s( B Y1,K )e ρst, t 0, where B is any bounded set of initial data in Y 1,K. Here dist Y1,K denotes the non-symmetric Hausdorff distance in Y 1,K and B Y1,K stands for the size of B in Y 1,K.

38 Finite fractal dimension of the global attractor Remark Thanks to the existence of exponential attractors for SK w (t) and SK s (t) we deduce that the global attractors Aw K and As K have finite fractal dimension. 38 / 46

39 Lyapunov functional (strong solutions) 39 / 46 Proposition Let f, g, λ satisfy (H1) and (H3). Then SK s (t) has a (strict) Lyapunov functional defined by the free energy, namely, L K (ψ 0, θ 0 ) := 1 [ ] ψ α Γψ 0 2 2,Γ + β ψ 0 2 2,Γ + θ F (ψ 0 (x)) dx + G (ψ 0 (S)) ds, K {D, N, R} Ω where F (y) = y 0 f (r) dr, G (y) = y 0 g (r) dr, y R. In particular, for all t > 0, we have d dt L K (S s K (t)(ψ 0, θ 0 )) = ψ t (t) 2 2 ψ t (t) 2 2,Γ θ (t) 2 2 d θ Γ (t) 2 2,Γ Γ

40 Equilibrium points (strong solutions) 40 / 46 (ψ, θ ) Y 1,K is an equilibrium for P s K if and only if it is a solution to the boundary value problem ψ + f (ψ ) λ (ψ )θ = 0, in Ω α Γ ψ + n ψ + βψ + g (ψ ) = 0, θ = 0, in Ω, b n θ + cθ = 0, on Γ on Γ if K {D, R}, then θ 0 if K = N, then θ = I N (ψ 0, θ 0 ) λ(ψ ) Ω If K = N, then ψ is solution of a nonlocal boundary value problem ( a special version of the Łojasiewicz-Simon inequality is needed)

41 K = N: convergence to equilibrium (strong solutions) 41 / 46 Theorem Let f, g, λ satisfy assumptions (H1), (H3), (H4) and (H5). Suppose, in addition, that the nonlinearities F, G, λ are analytic. Then, (ψ 0, θ 0 ) Y 1,N, the solution (ψ (t), θ (t)) = SN s (t)(ψ 0, θ 0 ) converges to a single equilibrium (ψ, θ ) in the topology of V 2 Z 1 N, that is, ) lim ( ψ(t) ψ t + V + θ(t) θ 22 Z = 0 1N Moreover, C > 0 and ξ (0, 1/2) depending on (ψ, θ ) : ψ(t) ψ V θ(t) θ Z 1 N + ψ t (t) V 0 2 C(1 + t) ξ/(1 2ξ) for all t 0.

42 Remarks Remark A s K coincides with the unstable manifold of the set of equilibria. Remark Since we have gradient systems with a set of equilibria bounded in the phase-space, we could avoid to prove the existence of a bounded absorbing set and we could directly show the existence of the global attractor. However, the existence of a uniform dissipative estimate can be easily adapted to some nonautonomous systems. 42 / 46

43 Further issue: coupled dynamic boundary conditions 43 / 46 surface free energy functional [ α E Γ (ψ, θ) = 2 Γψ 2 + β 2 ψ2 + G(ψ) l(ψ)θ a ] 2 θ2 ds Γ ψ E Γ (ψ, θ), θ E Γ (ψ, θ): Frechét derivatives of E Γ (ψ, θ) ψ t = ψ E Γ (ψ, θ) n ψ ( θ E Γ (ψ, θ)) t = b n θ cθ ψ t α Γ ψ + n ψ + βψ + g(ψ) l (ψ)θ = 0 (aθ + l(ψ)) t + b n θ + cθ = 0 a > 0, g = G l C 2 (R) : l (y) cl, y R, where c l > 0

44 Open issue: singular potentials 44 / 46 bdry coupling with singular F (and, possibly, G) of the form F(s) = γ 1 [(1 + s) ln(1 + s) + (1 s) ln(1 s)] γ 2 s 2

45 Open issue: memory effects 45 / 46 ψ t + h 1 (s)( ψ + f (ψ) λ (ψ)θ)(t s)ds = 0 0 (θ + λ(ψ)) t h 2 (s) θ(t s)ds = 0 0 ψ t + h 3 (s)( Γ ψ + ψ + g(ψ) + n ψ)(t s)ds = 0 0 b n θ + cθ = 0 θ(s) = θ 0 ( s), ψ(s) = ψ 0 ( s) in Ω, s 0 h 1, h 2, h 3 0 smooth exp. decreasing relaxation kernels

46 Thanks for your attention! 46 / 46