0-0 Well-posedness and asymptotic analysis for a Penrose-Fife type - PowerPoint PPT Presentation

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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 Sal` o, 3-5 Luglio 2003 Riccarda Rossi Dipartimento di Matematica “F.Casorati”, Universit` a di Pavia. Dipartimento di Matematica “F.Enriques”, Universit` a 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) Sal` o, 3 Luglio 2003 2

The Penrose-Fife phase field system (I) εϑ t + λχ t − ∆( − 1 in Ω × (0 , T ), ϑ ) = f δχ t − ∆ χ + β ( χ ) + σ ′ ( χ ) + λ ϑ − λ ∋ 0 in Ω × (0 , T ) , ϑ c β : R → 2 R a maximal monotone graph, β = ∂ ˆ β for convex ˆ β , σ ′ : R → R a Lipschitz function, ε , δ relaxation parameters . Sal` o, 3 Luglio 2003 3

The Penrose-Fife phase field system (II) • Let e := ϑ + χ be the internal energy , and let q ℓ ( ϑ ) = − 1 q = −∇ ( ℓ ( ϑ )) , ϑ ⇒ the first equation is indeed be the heat flux: = ∂ 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 Sal` o, 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 on ∂ Ω × (0 , T ) , ϑ ) = 0 ↓ 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. Sal` o, 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 on ∂ Ω × (0 , T ) , ϑ ) = 0 ↓ 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. Sal` o, 3 Luglio 2003 6

The Cahn Hilliard equation χ t − ∆( − ∆ χ + χ 3 − χ ) = f χ ( · , 0) = χ 0 a.e. in Ω × (0 , T ) , • (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 Sal` o, 3 Luglio 2003 7

The viscous Cahn Hilliard equation χ t − ∆( χ t − ∆ χ + χ 3 − χ ) = f χ ( · , 0) = χ 0 a.e. in Ω × (0 , T ) , • (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-Niezg´ odka ’95] for (CH)) accounts for, e.g., a constraint on the values of χ . Sal` o, 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] , [Lauren¸ cot et al. : attractors] . Sal` o, 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 ξ ε ∈ β ( χ ε ) ϑ 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)!! Sal` o, 3 Luglio 2003 10

A new approximating system (I) • Colli & Lauren¸ cot ’98 : an alternative heat flux law , better for large temperatures : ℓ ( ϑ ) = − 1 ϑ ❀ α ( ϑ ) ∼ ϑ − 1 ϑ, and α increasing. • 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 ξ ∈ β ( χ ) . ϑ, + hom. N.B.C. on χ and α ε ( ϑ ) and I.C. on χ and ϑ . Sal` o, 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 1 u = lim ε ↓ 0 α ε ( ϑ ε ) = ε 1 / 2 ϑ ε − ϑ ε : α ε ranges over the whole of R ! Sal` o, 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? Sal` o, 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 γ > 0, h ∈ L 2 ( ∂ Ω × (0 , T )) ∂ n u + γu = γh, 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. ? Sal` o, 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 1 D . • Ito-Kenmochi-Kubo ’02 : α ( ϑ ) = − 1 ϑ , OK for N.B.C. on u under additional constraints. Fill the gap? Sal` o, 3 Luglio 2003 15

General setting � � H := L 2 (Ω) , V := H 1 (Ω) , v ∈ H 2 (Ω) : ∂ n v = 0 W := , = H ′ ⊂ V ′ ⊂ W ′ . with dense and compact embeddings W ⊂ V ⊂ H ∼ • 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 ′ . V ′ := � v, N ( v − m ( v )) � + ( v, m ( v )) Sal` o, 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 , in V ′ for a.e. t ∈ (0 , T ) , ∂ t ϑ + ∂ t χ + Au = f 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 . Sal` o, 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 in V ′ for a.e. t ∈ (0 , T ) , ∂ t ϑ + ∂ t χ + νu + Au = f, u ∈ α ( ϑ ) ∂ 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 Ω Then you recover the full V -norm of u from the first equation. OK for the boundary conditions! = ⇒ Sal` o, 3 Luglio 2003 18