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 1 / 46
Phase-field system 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 � δψ t − 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) 2 / 46
A naive derivation of the system bulk free energy functional � � D 2 |∇ ψ | 2 + F ( ψ ) − λ ( ψ ) θ − ε � 2 θ 2 E Ω ( ψ, θ ) = dx Ω ∂ 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 ) 3 / 46
Motivation for studying the case λ nonlinear 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)) 4 / 46
Standard boundary conditions for ψ and θ ψ : Neumann boundary condition ∂ n ψ = 0 n : outward normal to Γ θ : Dirichlet, Neumann, Robin boundary conditions b ∂ n θ + c θ = 0 (D): b = 0 , c > 0 (N): b > 0 , c = 0 (R): b > 0 , c > 0 5 / 46
Dynamic boundary condition for ψ In order to account for possible interaction with the boundary Γ , we can consider dynamic boundary conditions for ψ surface free energy functional � α 2 |∇ Γ ψ | 2 + β � � 2 ψ 2 + G ( ψ ) E Γ ( ψ ) = dS Γ ∇ Γ tangential gradient operator 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 g = G ′ ∆ Γ Laplace-Beltrami operator , 6 / 46
(IBVP) for the phase-field system 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 + c θ = 0 initial conditions on Ω θ ( 0 ) = θ 0 , ψ ( 0 ) = ψ 0 7 / 46
Main goals Well posedness Existence of the global attractor Existence of an exponential attractor Convergence of solutions to single equilibria 8 / 46
Definitions of global attractor and exponential attractor Dynamical system: ( X , S ( t )) X : complete metric sp. S ( t ) : X → X semigroup of op. Hausdorff semidistance: dist X ( W , Z ) = sup z ∈Z d ( w , z ) inf w ∈W 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 ⇒ t →∞ dist X ( S ( t ) B , A ) = 0 lim 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 − ω t , ∀ t ≥ 0 9 / 46
Literature: λ linear + standard b.c. + smooth potentials 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 10 / 46
Literature: λ nonlinear + standard b.c. + smooth potentials 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.) 11 / 46
Literature: λ nonlinear + standard b.c. + singular potentials M.Grasselli, H.Petzeltová & G.Schimperna (2006) well-posedness and asymptotic behavior 12 / 46
Literature: λ linear + dynamic b.c. + smooth potentials 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 13 / 46
Literature: λ linear + dynamic b.c. + smooth potentials C.G.Gal, M.Grasselli & A.Miranville (2007) ∃ family of exponential attractors {E ε } stable as ε ց 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 ) a θ t + b ∂ n θ + c θ = 0 construction of a dissipative semigroup (larger phase spaces w.r.t. S.Gatti & A.Miranville ) ∃ global attractor, ∃ exponential attractors 14 / 46
Literature: λ linear + dynamic b.c. + singular potential 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 15 / 46
Notations norm on L p (Ω) �·� p norm on L p (Γ) �·� p , Γ usual scalar product inducing the norm on L 2 (Ω) �· , ·� 2 (even for vector-valued functions) usual scalar product inducing the norm on L 2 (Γ) �· , ·� 2 , Γ (even for vector-valued functions) norm on H s (Ω) , for any s > 0 �·� H s (Ω) norm on H s (Γ) , for any s > 0 �·� H s (Γ) 16 / 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) ∩ H 2 (Ω) , if K = D D ( A K ) = { θ ∈ H 2 (Ω) : b ∂ n θ + c θ = 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) 17 / 46
The functional spaces Z 1 K Z 1 D = H 1 0 (Ω) Z 1 K = H 1 (Ω) , if K ∈ { N , R } �∇ θ � 2 2 , if K = D , � 2 � θ � 2 �∇ θ � 2 2 + c � � K = � θ | Γ 2 , Γ , if K = R , Z 1 b �∇ θ � 2 2 + � θ � 2 Ω , if K = N , where we have set � � v � Ω := | Ω | − 1 v ( x ) dx Ω the norm in Z 1 K is equivalent to the standard H 1 -norm 18 / 46
The function spaces V s � �·� V s , V s = C s � Ω s > 0 � 1 / 2 � � 2 � ψ � 2 � � � ψ � V s = H s (Ω) + � ψ | Γ H s (Γ) V s = H s (Ω) ⊕ H s (Γ) V 0 = L 2 (Ω) ⊕ L 2 (Γ) 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 19 / 46
Enthalpy conservation 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 20 / 46
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
Recommend
More recommend
