two phase solutions for a forward-backward equation

Two phase solutions for

a forward-backward equation

Corrado Mascia

Joint works with P.Lafitte (Lille),

A.Terracina (Roma 1), A.Tesei (Roma 1)

Laboratoire Jacques-Louis Lions

Universite Pierre et Marie Curie

Seminar on Compressible Fluids – March 30th, 2011

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1 Forward-backward diffusionAn ill-posed problemTransitions dynamics

2 Origin of the modelPhase transitionsMemory effects

3 The pseudo-parabolic regularizationPiecewise smooth solutionsNumerical experiments

4 Links to hyperbolic equationsThe van der Waals fluidsWhich synthesis?

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Forward-backward diffusion

Plan





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Forward-backward diffusion

Model equation

Nonlinear diffusion equation:

∂u

∂t= ∆φ(u)

If φ′ ≥ ν > 0, the equation is parabolic;if φ′ ≥ 0, the equation is degenerate parabolic,

if φ′ changes sign, the equation is forward-backward.

Typical examples: J given bounded interval

Cahn–Hilliard: φ (phase transitions)

φ′(u) < 0 ⇐⇒ u ∈ J;

Perona–Malik: φ (image processing).

φ′(u) > 0 ⇐⇒ u ∈ J.

See also Barenblatt, Bertsch, Dal Passo, Ughi (1993), turbulent flows

Padron (1993), aggregating populations.

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Forward-backward diffusion An ill-posed problem

An ill-posed problem

Here: diffusion function φ withan “increasing” cubic-like shape

∂u

∂t= ∆φ(u)

spinodal region: interval J = (b, a)

the Cauchy problem is ill-posed.

Passing to weak formulation, solutions to

the Cauchy problem may become infinite.

Hollig (1983): φ piecewise linear, x ∈ R1:for (smooth) initial data u0 such that

u0(±∞) < b < supx∈R

u0(x)

the problem has infinite solutions.

Generalization in: Zhang (2006)general φ, x ∈ R1

(via differential inclusions).

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Forward-backward diffusion An ill-posed problem

Infinitely many evolutions

Starting from an initial datum with values in the spinodal region

it is possible to build (infinitely many) solutions assuming the initial datum in L1 as

t → 0+ given by (infinitely many) transitions from one stable phase to the other

(Precisely, Hollig considers v such that ∂v/∂x = u.)

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Forward-backward diffusion Transitions dynamics

Transitions dynamics

— 1. Transitions formation —how is the initial transition pattern determined?

→ 2. Transitions evolution←which are the rules governing the dynamics of the transitionbetween stable phases?

3 unknownsu± (right/left limits)

ξ (jump position)

2 conditions[φ(u)] = 0ξ′ [u] + [∂xφ(u)] = 0 (R–H)

[f ] jump of function f

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Forward-backward diffusion Transitions dynamics

Two possible versions

Two possible versions to assignthe missing condition:

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5u

Á(u)

1. Steady transition (given boundary)[∂φ(u)

∂x

]= 0

From R–H, it follows ξ′ ≡ 0;

φ(u±) varies in time.

2. Moving transition (free boundary)

φ(u−) = φ(u+) = C

From R–H, it follows ξ′ = − [∂xφ(u)]

[u]

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Origin of the model

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Origin of the model Phase transitions

Phase transitions

Phase: a distinct and homogeneousform of matter separated by its surfacefrom other forms.

Transition: the process of changingfrom one state to another.

Here, spinodal decomposition:mechanism generating a separation in a compoundin regions with different chemical and physical properties.

Paradigm: Emergence of oscillations starting from an (almost) homogeneousconfiguration may be generated by the presence of negative diffusion:

– Region φ′ < 0: spinodal region or unstable phase;– Region φ′ > 0: stable phase.

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Origin of the model Phase transitions

Cahn–Hilliard equation

Generalized diffusion equation for the concentration u

∂u

∂t= −D div (−grad v) = D ∆v (D > 0)

(div : divergence theorem, grad : Fick’s law),where v denotes the (chemical) potential.

Cahn, Hilliard (1958): the potential v coincides with the static potential

vs :=δF

δuwhere F [u] :=

∫ Φ(u) +

1

2γ |grad u|2

dx .

Thus∂u

∂t= D ∆

(φ(u)− γ∆u

)con φ := Φ′.

Usually, the function Φ has a double-well form.

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Origin of the model Memory effects

Memory effects

Jackle, Frisch (1985): the potential v is given by

v = vs + vm

where vm denotes the memory potential

vm := Ψ(u)(x , t) +

∫ t

−∞θ′(t − s) Ψ(u)(x , s) ds

=

∫ t

−∞θ′(t − s) (Ψ(u)(x , s)−Ψ(u)(x , t)) ds

with θ (decreasing) such that θ(0) = 1 and θ(+∞) = 0.

The term vm vanishes:– at the equilibrium, u = u(x);

– or, in absence of memory, θ′ = −δ0.

The form of the term vm is inspired by the Boltzmann

superposition principle used in viscoelasticity.13 / 34


An exponential decaying memory

Given τ > 0, let us choose

θ(t) := exp(− t

τ

)⇒ ∂

∂t

(vm −Ψ(u)

)= −1

τvm

Coupling with the diffusion equation

∂u

∂t= D ∆

(vs + vm

)we get

τ∂2u

∂t2+∂u

∂t= D ∆

(vs + τ

∂

∂t

(vs + Ψ(u)

))that gives, setting ψ := τ(φ+ Ψ), the fifth order equation

τ∂2u

∂t2+∂u

∂t= D ∆

(φ(u) +

∂ψ(u)

∂t− γ∆u − γ τ ∆

∂u

∂t

)14 / 34


An intricated equation

The complete equation is rather complicate

τ∂2u

∂t2+∂u

∂t= D ∆

(φ(u) +

∂ψ(u)

∂t− γ∆u − γ τ ∆

∂u

∂t

)A natural approach is to consider some simplifications

τ = 0 :∂u

∂t= D ∆

(φ(u)− γ∆u

)(Cahn–Hilliard)

γ = 0 : τ∂2u

∂t2+∂u

∂t= D ∆

(φ(u) +

∂ψ(u)

∂t

)...

Which are the differences (if any) in the dynamicsdictated by the different reductions?

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The pseudo-parabolic regularization

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The regularized equation (ε > 0)

Following Novick-Cohen, Pego (1991), let us consider

ψ(u) = ε u, γ = 0, τ = 0.

so that we end up with a pseudoparabolic (or Sobolev type) equation

∂u

∂t= D ∆

(φ(u) + ε

∂u

∂t

)The same third order term appears also in Benjamin, Bona, Mahony (1973).

1. Well posed-ness. the Cauchy problem (zero-flux boundary conditions) withinitial datum u0 ∈ L∞ has a unique global solution in C 1

([0,∞), L∞

).

2. Equilibrium stability. If U is such that φ(U) = C ∈ R, C ∈ (A,B),∫Ω

(u0 − U) dx = 0 =⇒ |u(·, t)− U|L∞ ≤ c e−ν t |u0 − U|

L∞

for |u0 − U|L∞ small, ν > 0.

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The singular limit ε→ 0+

Plotnikov (1994) analyze the limit ε→ 0+ by means of Young measures(see Slemrod (1991), Demoulini (1996), . . . , Horstmann, Schweizer (2008))

A fundamental role is played by the entropy inequality:given vε := φ(uε) + ε ∂tu

ε, for g ∈ C 1, g ′ > 0, setting

G (u) :=

∫ u

u0

g(φ(s))ds;

there holds

∂G (uε)

∂t− div

(g(v ε)∇v ε

)+ g ′(v ε)|∇v ε|2

= −1

ε

[g(φ(uε))− g(v ε)

](φ(uε)− v ε) ≤ 0

giving a corresponding inequality in the limit ε→ 0+.

The term entropy is chosen for the analogy

with the case of conservation laws.18 / 34

The pseudo-parabolic regularization Piecewise smooth solutions

Piecewise smooth solution for ε = 0

Evans, Portilheiro (2004): solutions with pure stable phases, i.e.the space Ω× [0,T ] is divided into two regions V1, V2 such that

u ≤ b in V1, u ≥ a in V2,

If u is a weak entropy solution, smooth in the interior of V1,V2,with smooth boundary Γ := V1 ∩ V2, (normal vector (νx , νt)),

1. u is a classical solution at the interior of V1 and of V2;

2. at any point of Γ, the function φ(u) is continuous and

νt [u] = νx · [∇φ(u)] (Rankine–Hugoniot);

3. finally, along Γ, there holdsνt = 0 φ(u) 6= A,B,

νt ≥ 0 φ(u) = A,

νt ≤ 0 φ(u) = B.

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Transition rules, 1/2

Steady transition: ifφ(u) ∈ (A,B), then ξ′ ≡ 0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T= 0.005

x

u

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x

phi(u

)

The profiles of u and φ(u): intial time (blue), evolution (red).

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Transition rules, 2/2

Moving transition: ifφ(u) = B, then ξ′ ≤ 0

-3

-2

-1

0

1

2

3

4

5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T= 0.1

x

u

-2

-1

0

1

2

3

4

5

6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0x

phi(u

)

The profiles of u and φ(u): intial time (blue), evolution (red).

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Combined motion

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-3 -2 -1 0 1 2 3u

Á(u)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5u

Á(u)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-3 -2 -1 0 1 2 3u

Á(u)

Such rules guarantee uniqueness of the solution to the Cauchy problem

(at least in the case of a single transition layer).

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Uniqueness of two phase solutions

Two phase solution: a triple (ξ, u, v) such that

∂u

∂t=∂2v

∂x2

for x 6= ξ(t);

u = β±(v) for ± x > ξ(t)

β± : inverses of φ in (−∞, b) and (a,+∞).

Mascia, Terracina, Tesei (2009): The jump conditions guaranteeuniqueness of the two phase solution for the Cauchy problem

∂u

∂t=∂2φ(u)

∂x2u(x , 0) = u0(x)

with φ(u0) ∈ BC (R) such that

u0 ≤ b in (−∞, 0), u0 ≥ a in (0,∞).

The proof is essentially based on a L1 stability argument

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Existence of two phase solutions

Local existence.Consider separately the cases of moving/steady transition layer.

moving boundary: two-phase Stefan type problem(free boundary, iteration/fixed point arguments);

steady boundary: transmission problem(two diffusion equations connected so that φ(u) is C 1 at x = 0)

...piecewise linear case X...general nonlinear case

Global existence.Glue together the two cases.

delicate point: to show that when one type of evolution leadsto a non-entropic evolution, the other type is admissible

...see Terracina (2011)

(to appear in SIAM J Math Anal).

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The pseudo-parabolic regularization Numerical experiments

Numerical experiments

Lafitte, Mascia (2010): comparison of two approaches

1. first, fixed ε > 0, space discretization; then, pass to the limit ε→ 0.

dU

dt= −

(h2 I + εA

)−1 Aφ(U) dU

dt= −h−2Aφ(U)

where A is the matrix of the discrete Laplace operator.

2. first, pass to the limit ε→ 0; then, space discretization.two main ingredients:

i. diffusion equation to be solved in the region of the stable phases;ii. description of the interface motion ξ, based on entropy condition.

Framework:– space interval I = (0, 1);– homogenous Neumann boundary conditions;

– test for the Riemann problem (self-similar solutions available).

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Comparison of the two approaches, 1/3

exact sol.

explicit

two-phase

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56x

u

– both of order 1 in h (space discretization);– the interface ξ is better approximated by the two phase scheme.

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Evolution of the relative error of interface ξ for data (u−, u+) = (−2, 4):– explicit scheme for the pseudo-parabolic equation (blue),

– two phase scheme (red).

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Two phase solutions do not take values in the spinodal region (b, a),while the movement of the boundary for solutions to

dU

dt= −h−2Aφ(U)

is due to a transition of a point from on stable phase to the other,

by passing all along through the unstable region (b, a).

S-S+

-2

-1

0

1

2

3

4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

explicit schem e - u at t im e 0.0005498 dx= 1/255

x

u

L

S-S+

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56

explicit schem e - u at t im e 0.0005498 dx= 1/255

x

u

(...movie...)

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Links to hyperbolic equations

Plan





29 / 34



Coming back to the fifth-order equation and neglecting the term vs relative

to the cost of the transitions, for ψ(u) = ε u and D = 1, we get the equation

τ∂2u

∂t2+∂u

∂t= ∆

(φ(u) + ε

∂u

∂t

)equivalent to the hyperbolic-parabolic system:

∂u

∂t+ div J = 0

τ∂J

∂t+ gradφ(u) = ε∆J− J

As ε→ 0+, the system becomes hyperbolic-elliptic,with ellipticity region given by the valures u such that φ′(u) < 0.

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Links to hyperbolic equations The van der Waals fluids

The van der Waals fluids

J.D. van der Waals (1873): taking into accountthe volume of fluid particles, we geta non-monotone state equation for the pressure

( Nobel prize 1910)

The equations for isentropic fluids become∂u

∂t+ div J = 0

∂J

∂t+ gradφ(u) = 0

where the pressure φ as changing monotonicity.

The system is hyperbolic-elliptic and possesses a natural entropy

η(u, J) = φ(u) +1

2|J|2,

that is not convex in the region where φ is decreasing.31 / 34


Selection criteria for transitions

One-dimensional casewith (∗) to be madeprecise

∂u

∂t+∂J

∂x= 0

∂J

∂t+∂φ(u)

∂x=∂2(∗)∂x2

Slemrod (1983): A phase transition is admissible for the system with(∗) = 0 if it can be obtained as limit of traveling wave (U, J)(x − c t)of the system with (∗) 6= 0.

In the context of van der Waals fluids, two (physical) selection criteria:

– viscosity: (∗) = ε J;

– visco-capillarity: (∗) = ε J + γ∂u

∂x.

In the theory of nonlinear elasticity, James (1980) considers

an analogous problem, examing only the viscosity criterium.

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Viscosity and visco-capillarity

Slemrod shows that the two criteria, viscosity and visco-capillarity,give raise to different admissibility conditions.

The choice of the “right” criterium is dictated by the specific problem.

...wide progeny with keywords:

non-classical shocks, kinetic relations...

Note that

viscosity∂2u

∂t2=

∂2

∂x2

(φ(u) + ε

∂u

∂t

)visco-capillarity

∂2u

∂t2=

∂2

∂x2

(φ(u) + ε

∂u

∂t− γ ∂

2u

∂x2

)With respect to the case previously discussed, at the the right-hand side

there is a second order time derivative in place of the first order one.

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Links to hyperbolic equations Which synthesis?

Which synthesis?

Parabolic and hyperbolic versions have different rescaling properties

self-similar solution (scaling x/√

t) vs. traveling waves

consequence of the presence of different time-scales.

...are they different “stories”?

Going back to the derivation of the equation, as ε→ 0, we obtain ahyperbolic-elliptic system with damping

∂u

∂t+ div J = 0

τ∂J

∂t+ gradφ(u) = −J

Is the parabolic dynamics the time-asymptotics of thehyperbolic one? (relation between heat and telegraph equation)

Role of the other higher order terms?34 / 34

two phase solutions for a forward-backward equation

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