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ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

ENTROPY FORMULATION FOR FORWARD-BACKWARD PARABOLIC EQUATION

A.Terracina University La Sapienza, Roma, Italy 25/06/2012

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Collaboration

Corrado Mascia Flavia Smarrazzo Alberto Tesei

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Introduction

Forward-backward parabolic equation ut = ∆φ(u) (1) where the function φ ∈ Liploc(R) is decreasing in some interval.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Introduction

Forward-backward parabolic equation ut = ∆φ(u) (1) where the function φ ∈ Liploc(R) is decreasing in some interval. Example 1 Model of phase separation φ′(u) > 0 if u ∈ (−∞, b) ∪ (a, ∞), φ′(u) < 0 if u ∈ (b, a);

u v φ(u) B A c b a d

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Example 2 Model of image processing (1D), Perona–Malik equation φ(u) =

u 1+u2 .

In this case the instability region is unbounded.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Example 2 Model of image processing (1D), Perona–Malik equation φ(u) =

u 1+u2 .

In this case the instability region is unbounded. Example 3 Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ(u) = ue−u u ≥ 0.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Example 2 Model of image processing (1D), Perona–Malik equation φ(u) =

u 1+u2 .

In this case the instability region is unbounded. Example 3 Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ(u) = ue−u u ≥ 0. Problems are ill–posed.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Example 2 Model of image processing (1D), Perona–Malik equation φ(u) =

u 1+u2 .

In this case the instability region is unbounded. Example 3 Model of population dynamic, Padron (Comm. Partial Differential Equations 1998) φ(u) = ue−u u ≥ 0. Problems are ill–posed. Hollig (Trans. Amer. Math. Soc. 83) φ piecewise linear, there is an infinite number of solutions of the Neumann boundary problem.

a ! u v B A b d !

" +

! c

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

IDEA : Introducing a viscous regularization that gives a good formulation in analogy with first order conservation laws

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

IDEA : Introducing a viscous regularization that gives a good formulation in analogy with first order conservation laws Problem is ill-posed since some relevant physical terms are neglected

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Phase transition, Cahn–Hilliard equation ut = ∆(φ(u) − δ∆u).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Phase transition, Cahn–Hilliard equation ut = ∆(φ(u) − δ∆u). An analogous approximation for the Perona–Malik equation (1D)

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Phase transition, Cahn–Hilliard equation ut = ∆(φ(u) − δ∆u). An analogous approximation for the Perona–Malik equation (1D) Model of population dynamic, Padron ut = (φ(u) + ǫut)xx.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Phase transition

Viscous Cahn–Hilliard Novick Cohen (1988) Gurtin (1996) based on microforce balance ut = ∆(φ(u) − δ∆u + ǫut)

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Phase transition

Viscous Cahn–Hilliard Novick Cohen (1988) Gurtin (1996) based on microforce balance ut = ∆(φ(u) − δ∆u + ǫut) In the following δ = 0, φ is of cubic type.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Novick Cohen-Pego (Trans. Amer. Math. Soc. 1991) study the viscosity problem            ut = ∆v in Ω × (0, T] =: QT

∂v ∂ν = 0

in ∂Ω × (0, T] u = u0 in Ω × {0}, (2) where v := φ(u) + ǫut (ǫ > 0) , (3) is the chemical potential, Ω ⊆ I Rn is bounded , ∂Ω regular, T > 0.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Partial differential equation in (2) can be rewritten ut = −1 ǫ (I − (I − ǫ∆)−1)φ(u) that corresponds to the Yosida approximation of the operator ∆.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Partial differential equation in (2) can be rewritten ut = −1 ǫ (I − (I − ǫ∆)−1)φ(u) that corresponds to the Yosida approximation of the operator ∆. Moreover v = (I − ǫ∆)−1φ(u).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Partial differential equation in (2) can be rewritten ut = −1 ǫ (I − (I − ǫ∆)−1)φ(u) that corresponds to the Yosida approximation of the operator ∆. Moreover v = (I − ǫ∆)−1φ(u). Using the standard theory of ODE in the Banach spaces we have

Theorem

(Novick Cohen-Pego) Given u0 ∈ L∞(Ω), ǫ > 0 there exists a unique solution (uǫ, vǫ) defined in (0, Tǫ), uǫ ∈ C 1([0, Tǫ), L∞(Ω)).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

A priori estimates

For every g ∈ C 1(R) such that g ′ ≥ 0 G(u) = u g(φ(s)) ds + c.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

A priori estimates

For every g ∈ C 1(R) such that g ′ ≥ 0 G(u) = u g(φ(s)) ds + c. Then [G(uǫ)]t = div

• g(vǫ)∇vǫ
• − g ′(vǫ)|∇vǫ|2+

−1 ǫ

• g(φ(uǫ)) − g(vǫ)
• (φ(uǫ) − vǫ) .

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

A priori estimates

For every g ∈ C 1(R) such that g ′ ≥ 0 G(u) = u g(φ(s)) ds + c. Then [G(uǫ)]t = div

• g(vǫ)∇vǫ
• − g ′(vǫ)|∇vǫ|2+

−1 ǫ

• g(φ(uǫ)) − g(vǫ)
• (φ(uǫ) − vǫ) .

Integrating in Ω and using boundary condition we have d dt

• Ω

G(uǫ(x, t)) dx ≤ 0 that gives a priori estimates in L∞. Moreover choosing g(u) ≡ u we have

• QT
• |∇v ǫ|2 + ǫ|∂tuǫ|2

dxdt ≤ C2 .

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy formulation

In analogy with conservation laws we characterize an entropy solution

• f problem

           ut = ∆φ(u) in Ω × (0, T] = QT

∂φ(u) ∂ν

= 0 in ∂Ω × (0, T] u = u0 in Ω × {0}, (4) as that obtained as limit of the solutions of problem (2) when ǫ → 0+.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy formulation

In analogy with conservation laws we characterize an entropy solution

• f problem

           ut = ∆φ(u) in Ω × (0, T] = QT

∂φ(u) ∂ν

= 0 in ∂Ω × (0, T] u = u0 in Ω × {0}, (4) as that obtained as limit of the solutions of problem (2) when ǫ → 0+. For every ǫ > 0 and g ∈ C 1(R), g ′ ≥ 0 we have

• QT
• G(uǫ)ψt − g(v ǫ)∇v ǫ · ∇ψ − g ′(v ǫ)|∇v ǫ|2ψ
• ≥ 0

(5) for every ψ ∈ C ∞

0 (QT), ψ ≥ 0.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy formulation

In analogy with conservation laws we characterize an entropy solution

• f problem

           ut = ∆φ(u) in Ω × (0, T] = QT

∂φ(u) ∂ν

= 0 in ∂Ω × (0, T] u = u0 in Ω × {0}, (4) as that obtained as limit of the solutions of problem (2) when ǫ → 0+. For every ǫ > 0 and g ∈ C 1(R), g ′ ≥ 0 we have

• QT
• G(uǫ)ψt − g(v ǫ)∇v ǫ · ∇ψ − g ′(v ǫ)|∇v ǫ|2ψ
• ≥ 0

(5) for every ψ ∈ C ∞

0 (QT), ψ ≥ 0.

The idea is to pass in the limit in (5) to characterize an entropy solution of (4).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Plotnikov’s results

Study of the singular limit, Plotnikov (J. Math. Sci. 1993).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Plotnikov’s results

Study of the singular limit, Plotnikov (J. Math. Sci. 1993). Using previous a priori estimate we deduce that there exist two subsequence {uǫn}, {v ǫn} and a couple (u, v) u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T); H1(Ω)) such that for every T > 0: uǫn ∗ ⇀ u in L∞(QT) , v ǫn ∗ ⇀ v in L∞(QT) , v ǫn ⇀ v in L2((0, T), H1(Ω)) .

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Plotnikov’s results

Study of the singular limit, Plotnikov (J. Math. Sci. 1993). Using previous a priori estimate we deduce that there exist two subsequence {uǫn}, {v ǫn} and a couple (u, v) u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T); H1(Ω)) such that for every T > 0: uǫn ∗ ⇀ u in L∞(QT) , v ǫn ∗ ⇀ v in L∞(QT) , v ǫn ⇀ v in L2((0, T), H1(Ω)) . Unfortunately this is not enough to pass to the limit in (5).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Plotnikov’s results

Study of the singular limit, Plotnikov (J. Math. Sci. 1993). Using previous a priori estimate we deduce that there exist two subsequence {uǫn}, {v ǫn} and a couple (u, v) u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T); H1(Ω)) such that for every T > 0: uǫn ∗ ⇀ u in L∞(QT) , v ǫn ∗ ⇀ v in L∞(QT) , v ǫn ⇀ v in L2((0, T), H1(Ω)) . Unfortunately this is not enough to pass to the limit in (5). Plotnikov gives a characterization of the Young measure ν(x, t) associate to the sequence {uǫn} proving that this is given by superposition of Dirac measures, ν(x,t)(τ) =

2

• i=0

λi(x, t)δ(τ − βi(v(x, t))) a.e in QT

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

where βi(v), i = 0, 1, 2 are the three branches of the graph v = φ(u)

u v φ(u) B A c b a d β₁(v) β₂(v) β₀(v)

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

where βi(v), i = 0, 1, 2 are the three branches of the graph v = φ(u)

u v φ(u) B A c b a d β₁(v) β₂(v) β₀(v)

Moreover, 0 ≤ λi ≤ 1 e 2

i=0 λi(x, t) = 1.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

where βi(v), i = 0, 1, 2 are the three branches of the graph v = φ(u)

u v φ(u) B A c b a d β₁(v) β₂(v) β₀(v)

Moreover, 0 ≤ λi ≤ 1 e 2

i=0 λi(x, t) = 1.

It could be proved that a subsequence of {v ǫn} converges strongly to the function v and (u, v) satisfies equation ut = ∆v in the weak sense,

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

where βi(v), i = 0, 1, 2 are the three branches of the graph v = φ(u)

u v φ(u) B A c b a d β₁(v) β₂(v) β₀(v)

Moreover, 0 ≤ λi ≤ 1 e 2

i=0 λi(x, t) = 1.

It could be proved that a subsequence of {v ǫn} converges strongly to the function v and (u, v) satisfies equation ut = ∆v in the weak sense, meanwhile u satisfies ut = ∆(φ(u)) in the sense of measured valued solutions.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Letting ǫn → 0+ in the viscous entropy inequality

• QT
• G(uǫn)ψt − g(v ǫn)∇v ǫn · ∇ψ − g ′(v ǫn)|∇v ǫn|2ψ
• dxdt+
• Ω

G(u0(x))ψ(x, 0)dx ≥ 0

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Letting ǫn → 0+ in the viscous entropy inequality

• QT
• G(uǫn)ψt − g(v ǫn)∇v ǫn · ∇ψ − g ′(v ǫn)|∇v ǫn|2ψ
• dxdt+
• Ω

G(u0(x))ψ(x, 0)dx ≥ 0 we have

• QT
• G(u)ψt − g(v)∇v · ∇ψ − g ′(v)|∇v 2|ψ
• dxdt+
• Ω

G(u0(x))ψ(x, 0) dx ≥ 0 (6) where G(u) = 2

i=0 λiG(βi(v)).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy solution

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy solution

Definition

Given u0 ∈ L∞(Ω) an entropy solution of problem forward–backward (4) is given by the functions λi ∈ L∞(QT), i = 0, 1, 2, u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T), H1(Ω)). Such that (i) 2

i=0 λi = 1, λi ≥ 0, u = 2 i=0 λiβi(v);

(ii) ut = ∆v in the weak sense; (iii) u and v satisfy (6) for every g ∈ C 1(R), g ′ ≥ 0 (entropy condition).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Entropy solution

Definition

Given u0 ∈ L∞(Ω) an entropy solution of problem forward–backward (4) is given by the functions λi ∈ L∞(QT), i = 0, 1, 2, u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T), H1(Ω)). Such that (i) 2

i=0 λi = 1, λi ≥ 0, u = 2 i=0 λiβi(v);

(ii) ut = ∆v in the weak sense; (iii) u and v satisfy (6) for every g ∈ C 1(R), g ′ ≥ 0 (entropy condition). Problems: Existence in a stronger sense? Uniqueness? Study of the evolution of the different phases.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Two phase entropy solution

Case n = 1. Let Ω = (−L, L), u0 ≤ b in (−L, 0), u0 ≥ a in (0, L), initial data in the two stable phases.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Two phase entropy solution

Case n = 1. Let Ω = (−L, L), u0 ≤ b in (−L, 0), u0 ≥ a in (0, L), initial data in the two stable phases. φ(u0) ∈ BC([−L, L]), φ(u0) ∈ C 1([−L, 0]), φ(u0) ∈ C 1([0, L])

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Two phase entropy solution

Case n = 1. Let Ω = (−L, L), u0 ≤ b in (−L, 0), u0 ≥ a in (0, L), initial data in the two stable phases. φ(u0) ∈ BC([−L, L]), φ(u0) ∈ C 1([−L, 0]), φ(u0) ∈ C 1([0, L]) We search a solution in which the two stable phases are separated by an interface ξ, ξ(0) = 0,

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Two phase entropy solution

Case n = 1. Let Ω = (−L, L), u0 ≤ b in (−L, 0), u0 ≥ a in (0, L), initial data in the two stable phases. φ(u0) ∈ BC([−L, L]), φ(u0) ∈ C 1([−L, 0]), φ(u0) ∈ C 1([0, L]) We search a solution in which the two stable phases are separated by an interface ξ, ξ(0) = 0, V1 := {(x, t) ∈ QT | − L ≤ x < ξ(t), t ∈ (0, T)}, V2 := QT \ V 1

x t

• L

L ξ(t) V2 V1

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

An entropy solution is a triple of functions (ξ, u, v) such that : (a) ξ ∈ C 1((0, T)), ξ(0) = 0, u ∈ L∞(QT), v ∈ L∞(QT) ∩ L2((0, T) ; H1((−L, L)); (b) u, v satisfy u = βi(v) in Vi (i = 1, 2) (v = φ(u)); (c) ut = vxx in the weak sense, entropy condition, boundary and initial condition.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Determinate conditions for the interface.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Determinate conditions for the interface.

Theorem

(Evans–Portilheiro Math. Models Methods Appl. Sci. (2004)) Let u, v, ξ a two phase entropy “piecewise regular” solution then (i) Rankine-Hugoniot condition: ξ′ = −[vx] [u] a.e on γ := {(ξ(t), t) : t ∈ (0, T))}. (ii) entropy condition: ξ′ [G(u)] ≥ −g(v)[vx] a.e. su γ .

where [h] := h(ξ(t)+, t) − h(ξ(t)−, t).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Admissibility condition for the interface

Choosing properly the function g we can select admissibility conditions for the interface γ = (ξ(t), t)

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Admissibility condition for the interface

Choosing properly the function g we can select admissibility conditions for the interface γ = (ξ(t), t)

Theorem

(a) ξ′(t) > 0 = ⇒ φ(u(ξ(t), t)) = A ; (b) ξ′(t) < 0 = ⇒ φ(u(ξ(t), t)) = B.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Admissibility condition for the interface

Choosing properly the function g we can select admissibility conditions for the interface γ = (ξ(t), t)

Theorem

(a) ξ′(t) > 0 = ⇒ φ(u(ξ(t), t)) = A ; (b) ξ′(t) < 0 = ⇒ φ(u(ξ(t), t)) = B. Condition for the phase change.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Admissibility condition for the interface

Choosing properly the function g we can select admissibility conditions for the interface γ = (ξ(t), t)

Theorem

(a) ξ′(t) > 0 = ⇒ φ(u(ξ(t), t)) = A ; (b) ξ′(t) < 0 = ⇒ φ(u(ξ(t), t)) = B. Condition for the phase change. We can pass from phase 1 to phase 2 only if v = B

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

If v ∈ (A, B) phase does not change

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Piecewise linear case

φ(u) =      φ−(u) if u ≤ b φ0(u) if b < u < a φ+(u) if u ≥ a , where φ±(u) := α± u + β± , φ0(u) := A(u − b) − B(u − a) a − b .

a ! u v B A b d !

" +

! c

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Theorem

(Mascia, T., Tesei, Arch. Rat. Mech 2009) There exists at most a unique two phase entropy solution

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Theorem

(Mascia, T., Tesei, Arch. Rat. Mech 2009) There exists at most a unique two phase entropy solution

Theorem

(Mascia, T., Tesei) Suppose that one of the following conditions is satisfies i) α−u0(0−) + β− = α+u0(0+) + β+ ∈ (A, B); ii) α−u′

0(0−) = α+u′ 0(0+).

Then there exists τ > 0 such that the two phase problem has solution in R × (0, τ).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Theorem

(Mascia, T., Tesei, Arch. Rat. Mech 2009) There exists at most a unique two phase entropy solution

Theorem

(Mascia, T., Tesei) Suppose that one of the following conditions is satisfies i) α−u0(0−) + β− = α+u0(0+) + β+ ∈ (A, B); ii) α−u′

0(0−) = α+u′ 0(0+).

Then there exists τ > 0 such that the two phase problem has solution in R × (0, τ). PROOF (idea) Local existence using two auxiliary problems: moving boundary problem and steady boundary problem.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

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Extension in time of the solution

We can extend in time the solution until a first time τ such that ξ′(τ) = 0 and φ(u(ξ(τ), τ)) = A or B (critical case).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Extension in time of the solution

We can extend in time the solution until a first time τ such that ξ′(τ) = 0 and φ(u(ξ(τ), τ)) = A or B (critical case). Is it possible to obtain the solution with a sequence of solutions of moving boundary problems that alternate in time with solutions of steady boundary problems?

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Extension in time of the solution

We can extend in time the solution until a first time τ such that ξ′(τ) = 0 and φ(u(ξ(τ), τ)) = A or B (critical case). Is it possible to obtain the solution with a sequence of solutions of moving boundary problems that alternate in time with solutions of steady boundary problems? It is necessary to study the critical case

Theorem

(T. Siam J. Mat. Anal. 2011) Let (ξ, u) be a solution of the two phase problem in Qτ. Let t1 < τ such that in (t1, τ) the solution is given by the solution either of the moving boundary problem or of the steady boundary problem. Then there exists t2 > τ such that the solution of the two phase problem can be extended in (0, t2).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

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Nolinear φ

Smarrazzo, T. (Discrete and Continuous dynamical systems A.) Rankine–Hugoniot has to be understood in a weak sense, the same for the entropy condition (theory of vector filelds with divergence–measure). We obtain the same compatibility condition for the two phase problem.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Nolinear φ

Smarrazzo, T. (Discrete and Continuous dynamical systems A.) Rankine–Hugoniot has to be understood in a weak sense, the same for the entropy condition (theory of vector filelds with divergence–measure). We obtain the same compatibility condition for the two phase problem. Uniqueness

Theorem

There exists at most one two–phase solution of problem . Let (u1, v 1, ξ1) and (u2, v 2, ξ2) be two two–phase solutions then we have

• QT
• (u1 − u2)ψt − (v 1

x − v 2 x )ψx

• dxdt = 0

(7) for every test function ψ ∈ H1(QT) ∩ C(QT) such that ψ(·, T) ≡ 0.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

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Choosing ψ(x, t) := t

T

{v 1(x, s) − v 2(x, s)} ds, (8) equation reads

• QT

(u1−u2)(v 1−v 2)dxdt=

• QT

(v 1

x −v 2 x )

• t

T

(v 1

x (x, s) − v 2 x (x, s)ds

• dxdt.

(9) Concerning the right-hand side of (9) we have

• QT

(v 1

x − v 2 x )

t

T

(v 1

x (x, s) − v 2 x (x, s) ds

• dxdt

(10) = 1 2 ω2

ω1

T d dt t

T

v 1

x (x, s) − v 2 x (x, s) ds

2 dtdx ≤ 0.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Existence

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Existence Solutions of the approximation problems ut = (φ(u) + ǫut)xx converge to the solution of the two phase problem ?

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

Existence Solutions of the approximation problems ut = (φ(u) + ǫut)xx converge to the solution of the two phase problem ? Suppose that one of the following conditions is satisfied

• there exists δ > 0 such that φ(u0) < B in (0, δ);
• there exists δ > 0 such that φ(u0) > A in (−δ, 0));

then there exists τ > 0 such that uǫn converges strongly to the solution of the two phase problem with fixed boundary (ξ ≡ 0) in (−L, L) × (0, τ).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

1) Maximum principle for the viscous problem

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

1) Maximum principle for the viscous problem If φ(u0(·)) ≤ B in (−L, x0) and v ǫ(x0, ·) ≤ B in (0, t0) then φ(uǫ), v ǫ ≤ B in (−L, x0) × (0, t0). Moreover if u0(·) ≤ b (−L, x0) then uǫ ≤ b in (−L, x0) × (0, t0).

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

1) Maximum principle for the viscous problem If φ(u0(·)) ≤ B in (−L, x0) and v ǫ(x0, ·) ≤ B in (0, t0) then φ(uǫ), v ǫ ≤ B in (−L, x0) × (0, t0). Moreover if u0(·) ≤ b (−L, x0) then uǫ ≤ b in (−L, x0) × (0, t0). 2) A priori estimates

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

1) Maximum principle for the viscous problem If φ(u0(·)) ≤ B in (−L, x0) and v ǫ(x0, ·) ≤ B in (0, t0) then φ(uǫ), v ǫ ≤ B in (−L, x0) × (0, t0). Moreover if u0(·) ≤ b (−L, x0) then uǫ ≤ b in (−L, x0) × (0, t0). 2) A priori estimates v ǫk

t L2(Qτǫk ) + v ǫk x L∞(0,τǫk ;L2((−L,L)) ≤ C

(11) where τǫk is the maximum time in which the two phases are separated by the stationary interface.

ENTROPY FORMULATION FOR FORWARD- BACKWARD PARABOLIC EQUATION

• A. Terracina

1) Maximum principle for the viscous problem If φ(u0(·)) ≤ B in (−L, x0) and v ǫ(x0, ·) ≤ B in (0, t0) then φ(uǫ), v ǫ ≤ B in (−L, x0) × (0, t0). Moreover if u0(·) ≤ b (−L, x0) then uǫ ≤ b in (−L, x0) × (0, t0). 2) A priori estimates v ǫk

t L2(Qτǫk ) + v ǫk x L∞(0,τǫk ;L2((−L,L)) ≤ C

(11) where τǫk is the maximum time in which the two phases are separated by the stationary interface. Using (11) we can pass to the limit strongly and prove that there exists τ > 0 such that τǫk ≥ τ for k large enough.