[PPT] - 0-0 Well-posedness and asymptotic analysis for a Penrose-Fife type PowerPoint 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`

, 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

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NOTATION

Ω ⊂ RN a bounded, connected domain in RN, N ≤ 3,

with smooth boundary ∂Ω,

T > 0 final time.

Variables

ϑ the absolute temperature of the system, ϑc the phase change temperature, χ order parameter (e.g., local proportion

f solid/liquid phase in melting,

fraction of pointing up spins in Ising ferromagnets)

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The Penrose-Fife phase field system (I)

εϑt + λχt − ∆(− 1 ϑ) = f in Ω × (0, T), δχt − ∆χ + β(χ) + σ′(χ) + λ ϑ − λ ϑc ∋ 0 in Ω × (0, T), β : R → 2R a maximal monotone graph, β = ∂ ˆ β for convex ˆ β, σ′ : R → R a Lipschitz function, ε, δ relaxation parameters.

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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 ∂te + 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) = χ3 − χ, derivative of the double well potential W(χ) = (χ2−1)2

4

.

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Singular limit as ε ↓ 0 (formal)

       εϑt + χt − ∆(− 1

ϑ) = f,

δχt − ∆χ + β(χ) + σ′(χ) ∋ − 1

ϑ,

∂nχ = ∂n(− 1

ϑ) = 0

n ∂Ω × (0, T),

↓ ε = 0    χt − ∆(δχt − ∆χ + β(χ) + σ′(χ)) ∋ f, ∂nχ = ∂n(−∆χ + ξ + σ′(χ)) = 0 ξ ∈ β(χ),

n ∂Ω × (0, T).

In the limit ε ↓ 0, we formally obtain the viscous Cahn Hilliard equation with source term and nonlinearities.

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Singular limit as ε, δ ↓ 0 (formal)

       εϑt + χt − ∆(− 1

ϑ) = f,

δχt − ∆χ + β(χ) + σ′(χ) ∋ − 1

ϑ,

∂nχ = ∂n(− 1

ϑ) = 0

n ∂Ω × (0, T),

↓ ε = δ = 0    χt − ∆(−∆χ + β(χ) + σ′(χ)) ∋ f, ∂nχ = ∂n(−∆χ + ξ + σ′(χ)) = 0 ξ ∈ β(χ),

n ∂Ω × (0, T).

In the limit ε, δ ↓ 0, we formally obtain the Cahn Hilliard equation with source term and nonlinearities.

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The Cahn Hilliard equation

χt − ∆(−∆χ + χ3 − χ) = f a.e. in Ω × (0, T), χ(·, 0) = χ0

(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 |Ω|

Ω

f(x, t)dx = 0 for a.e. t ∈ (0, T), = ⇒ χ is a conserved parameter , i.e. m(χ(t)) := 1 |Ω|

Ω

χ(x, t) dx = m(χ0) ∀t ∈ [0, T].

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The viscous Cahn Hilliard equation

χt − ∆(χt − ∆χ + χ3 − χ) = f a.e. in Ω × (0, T), χ(·, 0) = χ0

(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´

dka ’95] for (CH)) accounts for, e.g., a

constraint on the values of χ.

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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].

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A bad approximation

Reformulate (VCH) in terms of the chemical potential u . See that the asymptotic analysis    εϑεt + χεt − ∆(− 1

ϑε ) = f,

δχεt − ∆χε + ξε + σ′(χε) ∋ − 1

ϑε ,

ξε ∈ β(χε) ↓ ε ↓ 0    χt − ∆u = f, δχt − ∆χ + ξ + σ′(χ) = u, ξ ∈ β(χ), is ill-posed (& same considerations for the (CH)):

poor estimates, no bounds on ϑε!
if u = limε↓0 − 1

ϑε in a “reasonable” topology, then u ≤ 0 a.e. in:

a sign constraint does not pertain to the (VCH)!!

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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) ❀ αε(ϑ) = ε1/2ϑ − 1

ϑ in each equation: you obtain

Problem Pε: εϑt + χt − ∆(ε1/2ϑ − 1 ϑ) = f, δχt − ∆χ + ξ + σ′(χ) = ε1/2ϑ − 1 ϑ, ξ ∈ β(χ). + hom. N.B.C. on χ and αε(ϑ) and I.C. on χ and ϑ.

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A new approximating system (II)

The previous difficulties are overcome:

the term ε1/2ϑ allows for estimates
n the approximate sequence ϑε;
No more sign constraints on

u = limε↓0 αε(ϑε) = ε1/2ϑε −

1 ϑε :

αε ranges over the whole of R!

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A phase field model with double nonlinearity (I)

In general , we investigate the phase field system ϑt + χt − ∆u = f, u ∈ α(ϑ), in Ω × (0, T), χt − ∆χ + ξ + σ′(χ) = u, ξ ∈ β(χ), in Ω × (0, T),

α & β maximal monotone graphs on R2,
with the initial conditions

ϑ(·, 0) = ϑ0, χ(·, 0) = χ0 a.e. in Ω,

and the boundary conditions

∂nχ = ∂nu = 0 in ∂Ω × (0, T).

Problem: well-posedness?

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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

∂nu + γu = γh, γ > 0, h ∈ L2(∂Ω × (0, T)) are given: they allow to recover a H1(Ω)-bound on u from the first equation.

How to deal with homogeneous N.B.C. ?

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Previous contributions

Kenmochi-Kubo ’99: OK double nonlinearity; third type b.c.
n 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?

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General setting

H := L2(Ω), V := H1(Ω), W :=

v ∈ H2(Ω) : ∂nv = 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 :=

Ω

∇u∇v dx ∀u, v ∈ V.

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: u2

V := Au, u + (u, m(u))

∀u ∈ V v2

V ′ := v, N(v − m(v)) + (v, m(v))

∀v ∈ V ′.

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A phase field model with double nonlinearity (II)

Variational formulation Problem P Given χ0 ∈ V ϑ0 ∈ H satisfying suitable conditions, find ϑ ∈ H1(0, T; V ′) ∩ L∞(0, T; H) and χ ∈ L∞(0, T; V ) ∩ H1(0, T; H) ∩ L2(0, T; W) such that ϑ ∈ D(α), χ ∈ D(β) a.e. in Q, ∂tϑ + ∂tχ + Au = f in V ′ for a.e. t ∈ (0, T), for u ∈ L2(0, T; V ) with u ∈ α(ϑ) a.e. in Q, ∂tχ + Aχ + ξ + σ′(χ) = u in H for a.e. t ∈ (0, T), for ξ ∈ L2(0, T; H) with ξ ∈ β(χ) a.e. in Q, χ(·, 0) = χ0, ϑ(·, 0) = ϑ0.

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A double approximation procedure (I)

As in [Ito-Kenmochi-Kubo’02], a first approximate problem: Problem Pν. Find ϑ and χ such that the initial conditions hold and ∂tϑ + ∂tχ + νu + Au = f, u ∈ α(ϑ) in V ′ for a.e. t ∈ (0, T), ∂tχ + Aχ + ξ + σ′(χ) = u, ξ ∈ β(χ) in H for a.e. t ∈ (0, T),

Pν is coercive: consider the equivalent scalar product (

( · , · ) ) on V ( (v, w) ) := ν

Ω

vw dx +

Ω

∇v∇w dx ∀v, w ∈ V. Then you recover the full V -norm of u from the first equation. = ⇒

OK for the boundary conditions!

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A double approximation procedure (II)

Existence for Pν

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 ′: ∂tϑ + ∂tχ + νu + Au = f ❀ ∂tϑ + ∂tχ + ∂V ′ϕ(ϑ) ∋ f

A further regularization: Approximate the maximal

monotone graph α with {αn}, αn increasing, bi-Lipschitz continuous, αn ❀ ϕn; approximate β with its Yosida regularization βn.

Solve the approximate system

(subdifferential inclusion for ϕn) + (Cahn-Allen eq. with βn)

Passage to the limit as n ↑ ∞ (standard):

= ⇒ Well-posedness for Pν!!

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A double approximation procedure (III)

Passage to the limit for ν ↓ 0 ∂tϑ + ∂tχ + νu + Au = f, u ∈ α(ϑ), ∂tχ + Aχ + ξ + σ′(χ) = u 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]: ∃ Mβ ≥ 0 s. t. ξ ≤ Mβ(1 + ˆ β(r)) ∀ξ ∈ β(r), ∀r ∈ R.

⇒ Then you can pass to the limit:

the solutions of Pν converge as ν ↓ 0 to the unique solution of P!

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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

(χε, ϑε) to the Problem Pε: ε∂tϑε + ∂tχε − ∆(ε1/2ϑε − 1 ϑε ) = f δ∂tχε − ∆χε + ξε + σ′(χε) = ε1/2ϑε − 1 ϑε , ξε ∈ β(χε) + hom. N.B.C. on χε and αε(ϑε) + I.C. on χε and ϑε. Set uε := αε(ϑε) = ε1/2ϑε −

1 ϑε .

Passage to the limit for ε ↓ 0 ❀ (VCH)?

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Variational formulation of the Neumann problem for the viscous Cahn-Hilliard equation

Problem Pδ. Given the data χ0 ∈ V,

β(χ0) ∈ L1(Ω),

f ∈ L2(0, T; V ′), 1 |Ω|

Ω

f(x, t)dx = 0 for a.e. t ∈ (0, T), find χ ∈ H1(0, T; H) ∩ L∞(0, T; V ) ∩ L2(0, T; W), u ∈ L2(0, T; V ) s.t.              ∂tχ + Au = f in V ′, a.e. in (0, T), δ∂tχ + Aχ + ξ + σ′(χ) = u in H, a.e. in (0, T) for ξ ∈ L2(0, T; H), ξ ∈ β(χ) a.e. in Q. χ( · , 0) = χ0. Continuous dependence on the data holds for Pδ: ⇒ uniqueness of the solution χ; existence via approximation.

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Approximation

Approximating data: Given the data χ0 and f of Problem Pδ,

consider the approximating data {χε0}, {ϑε0}, and {fε} fulfilling fε ∈ L2(0, T; H), fε → f in L2(0, T; V ′) as ε ↓ 0 χε

0 → χ0 in H as ε ↓ 0

and suitable boundedness conditions .

Approximate solutions: Let {(χε, ϑε)} be the sequence of

solutions to Pε supplemented with the data {χε0}, {ϑε0} and {fε}.

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Asymptotic behaviour of Pε as ε ↓ 0 and existence for Problem Pδ.

Theorem 2. [R.,’03] There exists a triplet (χ, u, ξ) such that the following convergences hold as ε ↓ 0, and along a subsequence {εk :} χε⇀∗χ in H1(0, T; H) ∩ L∞(0, T; V ) ∩ L2(0, T; W), χε → χ in C0([0, T]; H) ∩ L2(0, T; V ), εϑε → 0 in L∞(0, T; H), εϑε ⇀ 0 in H1(0, T; V ′), uεk ⇀ u as k ↑ ∞, in L2(0, T; V ), ξεk ⇀ ξ as k ↑ ∞, in L2(0, T; V ), and ξ ∈ β(χ) a.e. in Ω × (0, T). Moreover, the triplet (χ, u, ξ) solves Problem Pδ .

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Error estimates for ε ↓ 0

There exists a constant Cerr ≥ 0, depending on T, |Ω| and L only, such that the error estimates χε − χC0([0,T ];H)∩L2(0,T ;V ) ≤ Cerr

ε1/8 + χε

0 − χ01/2 V ′ + |χ0 − χε 0|H + f − fε1/2 L2(0,T ;V ′)

,

εϑεL∞(0,T ;H) ≤ Cε1/4 hold for every ε ∈ (0, 1).

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Asymptotic analysis for ε and δ ↓ 0

We investigate the asymptotic behaviour as ε and δ ↓ 0 of the

solutions χεδ, ϑεδ to ε∂tϑεδ + ∂tχεδ − ∆(ε1/2ϑεδ − 1 ϑεδ ) = f, δ∂tχεδ − ∆χεδ + χεδ

3 − χεδ = ε1/2ϑεδ − 1

ϑεδ , + hom. N.B.C. on χεδ and αε(ϑεδ) + I.C. on χεδ and ϑεδ. Let uεδ := αε(ϑεδ) = ε1/2ϑεδ −

1 ϑεδ .

We refer to this problem as Problem Pεδ.

The limiting problem is the standard Cahn Hilliard equation

with source term f.

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Approximation of the Cahn Hilliard equation

Variational formulation of the limit problem. Given χ0 ∈ V and f ∈ L2(0, T; V ′) with null mean value, find χ ∈ H1(0, T; V ′) ∩ L∞(0, T; V ) ∩ L2(0, T; W) , u ∈ L2(0, T; V ) s.t. ∂tχ + Au = f in V ′, for a.e. t ∈ (0, T), Aχ + χ3 − χ = u in H, for a.e. t ∈ (0, T). χ( · , 0) = χ0.

Approximation of the initial data χ0 and f: consider the

sequences {χ0

εδ} ⊂ V , {ϑ0 εδ} ⊂ H and {fεδ} ⊂ L2(0, T; H)

χ0

εδ → χ0 in H,

fεδ → f in L2(0, T; V ′)

Approximate solutions: for every ε, δ > 0 consider the pair

(χεδ, ϑεδ) solving Pεδ with the data χ0

εδ, ϑ0 εδ and fεδ. Sal`

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Asymptotic behaviour of Pεδ as ε and δ ↓ 0

Theorem 3. [R.,’03] Under analogous assumptions on the approximating initial data {χ0

εδ}, {ϑ0 εδ}, and {fεδ}, there exists a pair

(χ, u) such that the following convergences hold as ε, δ ↓ 0: χεδ⇀∗χ in L∞(0, T; V ) ∩ L2(0, T; W), χεδ → χ in C0([0, T]; V ′) ∩ L2(0, T; V ), uεδ ⇀ u in L2(0, T; V ), εϑεδ → 0 in L2(0, T; H), δ∂tχεδ → 0 in L2(0, T; H), Furthermore, χ ∈ C0([0, T]; H) and it is the unique solution for the Neumann problem for the Cahn-Hilliard equation.

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Error estimates for ε, δ ↓ 0

There exists a constant Merr ≥ 0, only depending on T and |Ω|, such that the error estimates χ − χεδC0([0,T ];V ′)∩L2(0,T ;V ) ≤ Merr

χ0 − χ0

εδV ′ + δ1/2|χ0 − χ0 εδ|H

+Merr
f − fεδL2(0,T ;V ′) + ε1/8 + δ
εϑεL∞(0,T ;H) ≤ Cε1/4

hold for every ε, δ ∈ (0, 1).

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Open problem

Asymptotic analysis as ε ↓ 0 for the Penrose-Fife phase field

system with special heat flux law: εϑt + χt − ∆(ϑ − 1 ϑ) = f in Ω × (0, T), χt − ∆χ + β(χ) + σ′(χ) ∋ − 1 ϑ in Ω × (0, T). Difficult! Work in progress... Existence for the limit problem: χt − ∆(ϑ − 1 ϑ) = f in Ω × (0, T), χt − ∆χ + β(χ) + σ′(χ) ∋ − 1 ϑ in Ω × (0, T).

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