0-0
Well-posedness and asymptotic analysis for a Penrose-Fife type phase-field system Buona positura e analisi asintotica per un sistema di phase field di tipo Penrose-Fife Salò, 3-5 Luglio 2003 Riccarda Rossi Dipartimento di Matematica F.Casorati, Università di Pavia. Dipartimento di Matematica F.Enriques, Università di Milano. riccarda@dimat.unipv.it
NOTATION Ω R N a bounded, connected domain in R N, N 3, with smooth boundary Ω, T > 0 final time. Variables ϑ the absolute temperature of the system, ϑ c the phase change temperature, χ order parameter (e.g., local proportion of solid/liquid phase in melting, fraction of pointing up spins in Ising ferromagnets) Salò, 3 Luglio 2003 2
The Penrose-Fife phase field system (I) εϑ t + λχ t ( 1 ) = f in Ω (0, T ), ϑ δχ t χ + β(χ) + σ (χ) + λ ϑ λ ϑ c 0 in Ω (0, T ), β : R 2 R a maximal monotone graph, β = ˆβ for convex ˆβ, σ : R R a Lipschitz function, ε, δ relaxation parameters. Salò, 3 Luglio 2003 3
The Penrose-Fife phase field system (II) Let e := ϑ + χ be the internal energy, and let q q = (l(ϑ)), l(ϑ) = 1 ϑ be the heat flux: = the first equation is indeed t e + div q = f, the balance law for the internal energy. The second equation is a Cahn-Allen type dynamics for χ, in which e.g., β(χ) + σ (χ) = χ 3 χ, derivative of the double well potential W(χ) = ( χ 2 1) 2 4. Salò, 3 Luglio 2003 4
Singular limit as ε 0 (formal) εϑ t + χ t ( 1 ϑ ) = f, δχ t χ + β(χ) + σ (χ) 1 ϑ, n χ = n ( 1 ϑ ) = 0 on Ω (0, T ), ε = 0 χ t (δχ t χ + β(χ) + σ (χ)) f, n χ = n ( χ + ξ + σ (χ)) = 0 ξ β(χ), on Ω (0, T ). In the limit ε 0, we formally obtain the viscous Cahn Hilliard equation with source term and nonlinearities. Salò, 3 Luglio 2003 5
Singular limit as ε, δ 0 (formal) εϑ t + χ t ( 1 ϑ ) = f, δχ t χ + β(χ) + σ (χ) 1 ϑ, n χ = n ( 1 ϑ ) = 0 on Ω (0, T ), ε = δ = 0 χ t ( χ + β(χ) + σ (χ)) f, n χ = n ( χ + ξ + σ (χ)) = 0 ξ β(χ), on Ω (0, T ). In the limit ε, δ 0, we formally obtain the Cahn Hilliard equation with source term and nonlinearities. Salò, 3 Luglio 2003 6
The Cahn Hilliard equation χ t ( χ + χ 3 χ) = f a.e. in Ω (0, T ), χ(, 0) = χ 0 (CH) models phase separation : χ is the concentration of one of the two components in a binary alloy. Homogeneous Neumann boundary conditions on χ and χ, source term f spatially homogeneous, i.e. 1 f(x, t)dx = 0 for a.e. t (0, T ), Ω Ω = χ is a conserved parameter, i.e. m(χ(t)) := 1 χ(x, t) dx = m(χ 0 ) t [0, T ]. Ω Ω Salò, 3 Luglio 2003 7
The viscous Cahn Hilliard equation χ t (χ t χ + χ 3 χ) = f a.e. in Ω (0, T ), χ(, 0) = χ 0 (VCH) was introduced by [Novick-Cohen 88] to model viscosity effects in the phase separation of polymeric systems; derived by [Gurtin 96] in a model accounting for working of internal microforces, see also [Miranville 00, 02]... Homogeneous Neumann b.c. and spatial homogeneity of f, = χ is a conserved parameter. The maximal monotone graph β ([Blowey-Elliott 91], [Kenmochi-Niezgódka 95] for (CH)) accounts for, e.g., a constraint on the values of χ. Salò, 3 Luglio 2003 8
Motivations for the asymptotic analyses Taking the limits ε 0 (physically: small specific heat density) and ε, δ 0: passage from a non-conserved dynamics to a conserved dynamics; Proving convergence results for ε 0 : obtain existence results for the viscous Cahn-Hilliard equation with nonlinearities, never obtained so far. Analogy with a similar asymptotic analysis for the Caginalp phase field model [Caginalp 90, Stoth 95, R. 03], [Laurençot et al.: attractors]. Salò, 3 Luglio 2003 9
A bad approximation Reformulate (VCH) in terms of the chemical potential u. See that the asymptotic analysis εϑ εt + χ εt ( 1 ϑ ε ) = f, δχ εt χ ε + ξ ε + σ (χ ε ) 1 ϑ ε, ξ ε β(χ ε ) ε 0 χ t u = f, δχ t χ + ξ + σ (χ) = u, ξ β(χ), is ill-posed (& same considerations for the (CH)): poor estimates, no bounds on ϑ ε! if u = lim ε 0 1 ϑ ε in a reasonable topology, then u 0 a.e. in: a sign constraint does not pertain to the (VCH)!! Salò, 3 Luglio 2003 10
A new approximating system (I) Colli & Laurençot 98: an alternative heat flux law, better for large temperatures : and α increasing. l(ϑ) = 1 ϑ α(ϑ) ϑ 1 ϑ, The new approximating system for ε 0: replace l(ϑ) α ε (ϑ) = ε 1/2 ϑ 1 ϑ in each equation: you obtain Problem P ε : εϑ t + χ t (ε 1/2 ϑ 1 ϑ ) = f, δχ t χ + ξ + σ (χ) = ε 1/2 ϑ 1 ϑ, ξ β( χ). + hom. N.B.C. on χ and α ε (ϑ) and I.C. on χ and ϑ. Salò, 3 Luglio 2003 11
A new approximating system (II) The previous difficulties are overcome: the term ε 1/2 ϑ allows for estimates on the approximate sequence ϑ ε ; No more sign constraints on u = lim ε 0 α ε (ϑ ε ) = ε 1/2 ϑ ε 1 ϑ ε : α ε ranges over the whole of R! Salò, 3 Luglio 2003 12
A phase field model with double nonlinearity (I) In general, we investigate the phase field system ϑ t + χ t u = f, u α(ϑ), in Ω (0, T ), χ t χ + ξ + σ (χ) = u, ξ β(χ), in Ω (0, T ), α & β maximal monotone graphs on R 2, with the initial conditions ϑ(, 0) = ϑ 0, χ(, 0) = χ 0 a.e. in Ω, and the boundary conditions n χ = n u = 0 in Ω (0, T ). Problem: well-posedness? Salò, 3 Luglio 2003 13
Analytical difficulties The double nonlinearity of α & β; The homogeneous Neumann boundary conditions on both χ and u. Usually, third type boundary conditions for u n u + γu = γh, γ > 0, h L 2 ( Ω (0, T )) are given: they allow to recover a H 1 (Ω)-bound on u from the first equation. How to deal with homogeneous N.B.C.? Salò, 3 Luglio 2003 14
Previous contributions Kenmochi-Kubo 99: OK double nonlinearity; third type b.c. on u; Zheng 92: α(ϑ) = 1 ϑ, OK for N.B.C. on u in 1D. Ito-Kenmochi-Kubo 02: α(ϑ) = 1 ϑ, OK for N.B.C. on u under additional constraints. Fill the gap? Salò, 3 Luglio 2003 15
General setting H := L 2 (Ω), V := H 1 (Ω), W := { v H 2 (Ω) : n v = 0 }, with dense and compact embeddings W V H = H V W. Consider the realization of the Laplace operator with homog. N.B.C., i.e. the operator A : V V defined by Au, v := u v dx u, v V. Ω The inverse operator N is defined for the elements v V of zero mean value m(v). Take on V and V the equivalent norms: u 2 V := Au, u + (u, m(u)) u V v 2 V := v, N (v m(v)) + (v, m(v)) v V. Salò, 3 Luglio 2003 16
A phase field model with double nonlinearity (II) Variational formulation Problem P Given χ 0 V ϑ 0 H satisfying suitable conditions, find ϑ H 1 (0, T ; V ) L (0, T ; H) and χ L (0, T ; V ) H 1 (0, T ; H) L 2 (0, T ; W ) such that ϑ D(α), χ D(β) a.e. in Q, t ϑ + t χ + Au = f in V for a.e. t (0, T ), for u L 2 (0, T ; V ) with u α(ϑ) a.e. in Q, t χ + Aχ + ξ + σ (χ) = u in H for a.e. t (0, T ), for ξ L 2 (0, T ; H) with ξ β(χ) a.e. in Q, χ(, 0) = χ 0, ϑ(, 0) = ϑ 0. Salò, 3 Luglio 2003 17
A double approximation procedure (I) As in [Ito-Kenmochi-Kubo 02], a first approximate problem: Problem P ν. Find ϑ and χ such that the initial conditions hold and t ϑ + t χ + νu + Au = f, u α(ϑ) in V for a.e. t (0, T ), t χ + Aχ + ξ + σ (χ) = u, ξ β(χ) in H for a.e. t (0, T ), P ν is coercive: consider the equivalent scalar product ((, )) on V ((v, w)) := ν vw dx + v w dx v, w V. Ω Then you recover the full V -norm of u from the first equation. = OK for the boundary conditions! Ω Salò, 3 Luglio 2003 18
A double approximation procedure (II) Existence for P ν A subdifferential approach [Kenmochi-Kubo 99]: Reformulate the first equation as a subdifferential inclusion in V by means of a proper, l.s.c, convex functional ϕ on V : t ϑ + t χ + νu + Au = f t ϑ + t χ + V ϕ(ϑ) f A further regularization: Approximate the maximal monotone graph α with {α n }, α n increasing, bi-lipschitz continuous, α n ϕ n ; approximate β with its Yosida regularization β n. Solve the approximate system (subdifferential inclusion for ϕ n ) + (Cahn-Allen eq. with β n ) Passage to the limit as n (standard): = Well-posedness for P ν!! Salò, 3 Luglio 2003 19
A double approximation procedure (III) Passage to the limit for ν 0 t ϑ + t χ + νu + Au = f, t χ + Aχ + ξ + σ (χ) = u u α(ϑ), ξ β(χ), Problem: lack of a V -bound for u (now ν tends to 0!!) Additional assumption on β [Colli-Gilardi-Rocca-Schimperna 03]: M β 0 s. t. ξ M β (1 + ˆβ(r)) ξ β(r), r R. Then you can pass to the limit: the solutions of P ν converge as ν 0 to the unique solution of P! Salò, 3 Luglio 2003 20
Main Existence Result Theorem 1. [R., 03] Problem P admits a unique solution (χ, ϑ). In particular, for every ε > 0 there exists a unique solution (χ ε, ϑ ε ) to the Problem P ε : ε t ϑ ε + t χ ε (ε 1/2 ϑ ε 1 ϑ ε ) = f δ t χ ε χ ε + ξ ε + σ (χ ε ) = ε 1/2 ϑ ε 1 ϑ ε, ξ ε β(χ ε ) + hom. N.B.C. on χ ε and α ε (ϑ ε ) + I.C. on χ ε and ϑ ε. Set u ε := α ε (ϑ ε ) = ε 1/2 ϑ ε 1 ϑ ε. Passage to the limit for ε 0 (VCH)? Salò, 3 Luglio 2003 21
Variational formulation of the Neumann problem for the viscous Cahn-Hilliard equation Problem P δ. Given the data χ 0 V, β( χ 0 ) L 1 (Ω), f L 2 (0, T ; V 1 ), Ω Ω f(x, t)dx = 0 for a.e. t (0, T ), find χ H 1 (0, T ; H) L (0, T ; V ) L 2 (0, T ; W ), u L 2 (0, T ; V ) s.t. t χ + Au = f in V, a.e. in (0, T ), δ t χ + Aχ + ξ + σ (χ) = u in H, a.e. in (0, T ) for ξ L 2 (0, T ; H), ξ β(χ) a.e. in Q. χ(, 0) = χ 0. Continuous dependence on the data holds for P δ : uniqueness of the solution χ; existence via approximation. Salò, 3 Luglio 2003 22
Approximation Approximating data: Given the data χ 0 and f of Problem P δ, consider the approximating data {χ ε 0 }, {ϑ ε 0 }, and {f ε } fulfilling f ε L 2 (0, T ; H), f ε f in L 2 (0, T ; V ) as ε 0 χ ε 0 χ 0 in H as ε 0 and suitable boundedness conditions. Approximate solutions: Let {(χ ε, ϑ ε )} be the sequence of solutions to P ε supplemented with the data {χ ε 0 }, {ϑ ε 0 } and {f ε }. Salò, 3 Luglio 2003 23
Asymptotic behaviour of P ε as ε 0 and existence for Problem P δ. Theorem 2. [R., 03] There exists a triplet (χ, u, ξ) such that the following convergences hold as ε 0, and along a subsequence {ε k :} χ ε χ in H 1 (0, T ; H) L (0, T ; V ) L 2 (0, T ; W ), χ ε χ in C 0 ([0, T ]; H) L 2 (0, T ; V ), εϑ ε 0 in L (0, T ; H), εϑ ε 0 in H 1 (0, T ; V ), u εk u as k, in L 2 (0, T ; V ), ξ εk ξ as k, in L 2 (0, T ; V ), and ξ β(χ) a.e. in Ω (0, T ). Moreover, the triplet (χ, u, ξ) solves Problem P δ. Salò, 3 Luglio 2003 24
Error estimates for ε 0 There exists a constant C err 0, depending on T, Ω and L only, such that the error estimates χ ε χ C0 ([0,T ];H) L 2 (0,T ;V ) ( ) C err ε 1/8 + χ 0 ε χ 0 1/2 V + χ 0 χ 0 ε H + f f ε 1/2 L 2 (0,T ;V ), εϑ ε L (0,T ;H) Cε 1/4 hold for every ε (0, 1). Salò, 3 Luglio 2003 25
Asymptotic analysis for ε and δ 0 We investigate the asymptotic behaviour as ε and δ 0 of the solutions χ εδ, ϑ εδ to ε t ϑ εδ + t χ εδ (ε 1/2 ϑ εδ 1 ϑ εδ ) = f, δ t χ εδ χ εδ + χ εδ 3 χ εδ = ε 1/2 ϑ εδ 1 ϑ εδ, + hom. N.B.C. on χ εδ and α ε (ϑ εδ ) + I.C. on χ εδ and ϑ εδ. Let u εδ := α ε (ϑ εδ ) = ε 1/2 ϑ εδ 1 ϑ εδ. We refer to this problem as Problem P εδ. The limiting problem is the standard Cahn Hilliard equation with source term f. Salò, 3 Luglio 2003 26
Approximation of the Cahn Hilliard equation Variational formulation of the limit problem. Given χ 0 V and f L 2 (0, T ; V ) with null mean value, find χ H 1 (0, T ; V ) L (0, T ; V ) L 2 (0, T ; W ), u L 2 (0, T ; V ) s.t. t χ + Au = f in V, for a.e. t (0, T ), Aχ + χ 3 χ = u in H, for a.e. t (0, T ). χ(, 0) = χ 0. Approximation of the initial data χ 0 and f: consider the sequences {χ 0 εδ } V, {ϑ0 εδ } H and {f εδ} L 2 (0, T ; H) χ 0 εδ χ 0 in H, f εδ f in L 2 (0, T ; V ) Approximate solutions: for every ε, δ > 0 consider the pair (χ εδ, ϑ εδ ) solving P εδ with the data χ 0 εδ, ϑ0 εδ and f εδ. Salò, 3 Luglio 2003 27
Asymptotic behaviour of P εδ as ε and δ 0 Theorem 3. [R., 03] Under analogous assumptions on the approximating initial data {χ 0 εδ }, {ϑ0 εδ }, and {f εδ}, there exists a pair (χ, u) such that the following convergences hold as ε, δ 0: χ εδ χ in L (0, T ; V ) L 2 (0, T ; W ), χ εδ χ in C 0 ([0, T ]; V ) L 2 (0, T ; V ), u εδ u in L 2 (0, T ; V ), εϑ εδ 0 δ t χ εδ 0 in L 2 (0, T ; H), in L 2 (0, T ; H), Furthermore, χ C 0 ([0, T ]; H) and it is the unique solution for the Neumann problem for the Cahn-Hilliard equation. Salò, 3 Luglio 2003 28
Error estimates for ε, δ 0 There exists a constant M err 0, only depending on T and Ω, such that the error estimates hold for every ε, δ (0, 1). χ χ εδ C 0 ([0,T ];V ) L 2 (0,T ;V ) ( ) M err χ 0 χ 0 εδ V + δ 1/2 χ 0 χ 0 εδ H ) +M err ( f f εδ L 2 (0,T ;V ) + ε 1/8 + δ εϑ ε L (0,T ;H) Cε 1/4 Salò, 3 Luglio 2003 29
Open problem Asymptotic analysis as ε 0 for the Penrose-Fife phase field system with special heat flux law: εϑ t + χ t (ϑ 1 ) = f in Ω (0, T ), ϑ χ t χ + β(χ) + σ (χ) 1 ϑ in Ω (0, T ). Difficult! Existence for the limit problem: Work in progress... χ t (ϑ 1 ) = f in Ω (0, T ), ϑ χ t χ + β(χ) + σ (χ) 1 ϑ in Ω (0, T ). Salò, 3 Luglio 2003 30