Phase field models in fluid dynamics

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Phase field models in fluid dynamics

Eduard Feireisl

Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague

joint work with

H.Abels (Regensburg), H.Petzeltov´a (Praha) E.Rocca (Milano), G.Schimperna (Pavia) Villa Clythia, Fr´ejus, April 7th - 12th, 2013

Eduard Feireisl Phase field models

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Motto

Johann von Neumann [1903-1957]

In mathematics you don’t understand things.

You just get used to them.

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Fluid equations

Mass conservation - equation of continuity

∂_t% + div_x(%u) = 0 Momentum balance

∂t(%u) + divx(%u ⊗ u) = divxT+ %g

% = %(t, x ) . . . mass density u = u(t, x ) . . . velocity field T . . . Cauchy stress g . . . external (volume) force

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Phase field models

Phase field function - concentration difference

−1 ≤ c(t, x) ≤ 1

∂t(%c) + divx(%cu) = divx∇_xµ

µ = µ(t, x ) . . . chemical potential Cahn-Hillard type equation

%µ = %∂f (%, c)

∂c − ∆c

∂_t(%c) + div_x(%cu) = ∆ ∂f (%, c)

∂c −1

%∆c

f = f (%, c) . . . free energy

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Stress relation

Young-Laplace law

[_T· n] = σHn on interface Viscous stress

S= ν(c)

∇_xu + ∇^t_xu −2 3div_xu_I

+ η(c)div_xu_I

Pressure

p(%, c) = %²∂f (%, c)

∂%

Extra stress

T=_S− p_I−

∇_xc ⊗ ∇_xc − 1

2|∇_xc|²_I

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Anderson, McFadden, and Wheeler [1998]

Equation of continuity

∂t% + divx(%u) = 0 Momentum balance

∂_t(%u) + div_x(%u ⊗ u) + ∇_xp(%, c)

= div_x

ν(c)

+ η(c)div_xu_I

−div_x

∇_xc ⊗ ∇xc −1

2|∇_xc|²I

+ %g

Phase field equation

∂t(%c) + divx(%cu) = ∆ ∂f (%, c)

∂c −1

%∆c

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Boundary conditions

x ∈ Ω ⊂ R³

Ω . . . regular (bounded) domain No-slip

u|_∂Ω= 0 No-flux

∇_xc · n|∂Ω= 0, ∇xµ · n|∂Ω= 0

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A bit of history of global existence for large data

Jean Leray [1906-1998]

Global existence of weak solutions for the

incompressible Navier-Stokes system (3D)

Olga Aleksandrovna Ladyzhenskaya [1922-2004] Global existence of classical solutions for the incompressible 2D Navier-Stokes system

Pierre-Louis Lions[*1956] Global existence of weak solutions for the compressible barotropic Navier-Stokes system (2,3D)

and many, many others...

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Constitutive relations

Free energy

f (%, c) = f_e(%) + f_mix(%, c), f_mix(%, c) = H(c) log(%) + G (c) Pressure

p(%, c) = %²∂f (%, c)

∂% = p_e(%) + %H(c), f_e(%) = Z %

1

p_e(z) z² dz p_e(0) = 0, p₁%^γ−1− p₂ ≤ p_e⁰(%) ≤ p₃(1 + %^γ−1), γ > 3

2 Mixing

−H₁ ≤ H⁰(c), H(c) ≤ H₂, G₁c − G₂≤ G⁰(c) ≤ G₃(1 + c)

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Energy estimates

Total energy E (t) =

Z

Ω

1

2%|u|² + 1

2|∇_xc|²+ %f (%, c) dx Total mass

M = Z

Ω

% dx , M(t) = Z

Ω

%(t, ·) dx = Z

Ω

%0 dx , %(0, ·) = %0

Energy (in)equality d

dtE (t) + Z

Ω

S: ∇_xu + |∇_xµ|² dx ≤

Z

Ω

%g · u dx

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Kinetic energy

∂t

Z

Ω

1

2%|u|² dx + Z

Ω

S: ∇xu dx

= Z

Ω

p(%)divxu dx + Z

Ω

∆c∇xc · u dx

µ = ∂f (%, c)

∂c − 1

%∆c, p(%, c) = %²∂f (%, c)

∂%

“Phase field” stored energy

∂t

Z

Ω

1

2|∇_xc|² dx + Z

Ω

|∇_xµ|² dx

= − Z

Ω

%∂f (%, c)

∂c ∂_tc + %∂f (%, c)

∂c u · ∇_xc

dx

− Z

Ω

∆c∇_xc · u dx

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Dissipative bounds and Korn’s inequality

Z

Ω

S: ∇_xu dx ≥ Z

Ω

ν(c) 2

2

dx

Korn’s inequality Z

Ω

ν(c) 2

2

dx ≥ νk∇_xuk²_L2(Ω)

Transport coefficients

ν ≤ ν(c) ≤ ν

0 ≤ η(c) ≤ η

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Available a priori bounds

ess sup

t∈(0,T )

k%(t, ·)k_Lγ(Ω) ≤ C (data)

ess sup

t∈(0,T )

k∇_xc(t, ·)k_L2(Ω;R³) ≤ C (data)

ess sup

t∈(0,T )

k√

%u(t, ·)k_L2(Ω;R³)≤ C (data)

Z T 0

h

k∇_xu(t, ·)k²_L2(Ω;R^3×3)+ k∇xµ(t, ·)k²_L2(Ω;R³)

i

dt ≤ C (data)

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Div-Curl lemma

Hypotheses

U_n→ U weakly in L^p(R^N; R^N), V_n→ V weakly in L^q(R^N; R^N) 1

p + 1 q = 1

r < 1

{DIV[U_n]}^∞_n=1, {CURL[Vn]}^∞_n=1 precompact in W^−1,s for a certain s ≥ 1

Conclusion

U_n· V_n→ U · V weakly in L^r(R^N)

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Weak compactness of the convective terms

General principle

∂_tS_n+ div_xF_n= r_n∈ precompat set in W^−1,s

∇_xH_n∈ bounded in L^p Application of DIV-CURL lemma

Un= [Sn, Fn], Vn= [Hn, 0, 0, 0]

Conclusion

SH = S H

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Convective terms

Direct application of DIV-CURL lemma

%u = %u, %u ⊗ u = %u ⊗ u

%c = %c, %c² = %c² Strong convergence outside vacuum

c_n→ c in L² on the set {% > 0}

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Strong convergence of ∇

_x

c

−∆c_n= %_n∂f (%_n, c_n)

∂c − %_nµ_n

−∆c = %∂f (%, c)

∂c − %µ

Application of previous observations

%∂f (%, c)

∂c c = %∂f (%, c)

∂c c

%µc = %µc Conclusion

∇_xc_n→ ∇_xc (strongly) in L²((0, T ) × Ω; R³)

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Renormalized equation of continuity

b ∈ C¹[0, ∞), b⁰(z) = 0 for all z ≥ z0

Renormalized equation

∂tb(%) + divx(b(%)u) +

b⁰(%)% − b(%)

divxu = 0

∂t(% log(%)) + divx

% log(%)u

+ %divxu = 0

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Density oscillations - first approximation

Limit of renormalized equations

∂_t% log(%) + div_x

% log(%)u

+ %div_xu = 0

Renormalized limit equation

∂_t

% log(%)

+ div_x

% log(%)u

+ %div_xu = 0

Density oscillations d dt

Z

Ω

% log(%) − % log(%)

dx

+ Z

Ω

%divxu − %divxu

dx = 0

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DiPerna-Lions’ regularization method

v_δ= κ_δ∗ v Regularized equation

∂t%δ+ divx(%δu) = divx

%δu

− [div_x(%u)]_δ ≡ r_δ

∇_xu ∈ L^p, % ∈ L^p⁰ 1 p + 1

p⁰ = 1

⇒ r_δ → 0 as δ → 0

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Effective viscous flux identity

“Weak continuity” of the effective viscous flux p(%)% − 4

3ν(c) + η(c)

%divxu

= p(%)% − 4

3ν(c) + η(c)

%div_xu

%divxu − %divxu ≥ 0

⇒

% log(%) = % log(%)

⇒

%ε→ % a.a. in (0, T ) × Ω

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Effective viscous flux - Step I

“Approximate equation”

p(%_n)

= divx∆⁻¹divxSn− ∂_t∆⁻¹divx[%nun] − divx∆⁻¹divx[%nun⊗ u_n] +“compact terms”

Limit equation

p(%) = div_x∆⁻¹div_x_S− ∂_t∆⁻¹div_x[%u] − div_x∆⁻¹div_x[%u ⊗ u]

+“compact terms”

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Effective viscous flux - Step II

ν > 0, η = 0 constant

“Approximate equation”

p(%n)%n−4ν

3 %ndivxun

= −∆⁻¹divx[%nun]divx(%nun) − divx∆⁻¹divx[%nun⊗ u_n]%n

Limit equation

p(%)% − 4ν

3 %div_xu

= −∆⁻¹div_x[%u]div_x(%u) − div_x∆⁻¹div_x[%u ⊗ u]%

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Effective viscous flux - Step III

Effective viscous flux identity revisited p(%)% −4ν

3 %divxu ≈ p(%)% − 4ν 3 %divxu +weak − lim_n→∞u_n·h

%_ndiv_x∆⁻¹div_x[%_nu_n] − %_nu_ndiv_x∆⁻¹div_x[%_n]i

−u · h

%div_x∆⁻¹div_x[%u] − %udiv_x∆⁻¹div_x[%]i

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Div-Curl lemma revisited

Hypotheses

U_n→ U weakly in L^p(R^N; R^N), V_n→ V weakly in L^q(R^N; R^N) 1

p + 1 q = 1

r < 1 R_{i ,j} = ∂xi∆⁻¹∂xj

Conclusion

X

i ,j

U_nⁱR_{i ,j}[V_n^j] − V_nⁱR_{i ,j}[U_n^j] →X

i ,j

UⁱR_{i ,j}[V^j] − VⁱR_{i ,j}[U^j]

weakly in L^r(R^N)

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Conclusion

u_n→ u weakly in L²(0, T ; W₀^1,2(Ω; R³))

%_n→ % in C_weak([0, T ]; L^γ(Ω))

%nun→ %u inC_weak([0, T ]; L^2γ/γ+1(Ω; R³)) Conclusion

un·h

%ndivx∆⁻¹divx[%nun] − %nundivx∆⁻¹divx[%n] i

→ u ·h

%div_x∆⁻¹div_x[%u] − %udiv_x∆⁻¹div_x[%]i

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Non-constant viscosity coefficients

Commutator R_{i ,j}

ν(cn)

∇_xun+ ∇xu^t_n−2

3divxun∈ I

= R_{i ,j}

ν(c_n)

∇_xu_n+ ∇_xu^t_n−2

3div_xu_n∈ I

−4

3ν(c_n)div_xu_n +4

3ν(cn)divxun

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Commutator lemma [Coifman and Meyer]

Lemma

Let w ∈ W^1,r(R^N), V ∈ L^p(R^N; R^N) be given, where 1 < r < N, 1 < p < ∞, 1

r + 1 p − 1

N < 1.

The for any s satisfying 1 r + 1

p − 1 N < 1

s < 1 there exists β > 0 such that

N

X

j =1

R_{i ,j}[wVj] −

N

X

j =1

w Ri ,j[Vj]

W^β,s(R^N,R^N)

≤ ckw k_W1,rkVk_L^p,

i = 1, . . . , n

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General constitutive relations

Oscillations defect measure

ωq[%n→ %](Q) = sup

k>0

lim sup

n→∞

Z

Q

|T_k(%n) − Tk(%)|^q dx dt

T_k(r ) = min{r , k}

Claim

ω_q[%_n→ %]((0, T ) × Ω) < ∞, q = γ + 1 > 2

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Renormalized equation revisited

Lemma

ω_q[%_n→ %]((0, T ) × Ω) < ∞, q > 2

⇒

%, u satisfy the renormalized equation

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Conclusion - possible extensions

Sir Winston Churchill, [1874–1965]

However beautiful the strategy, you should occasionally look at the results

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Allen-Cahn model

Allen-Cahn equation

∂_t(%c) + div_x(%cu) = −µ

µ = µ(t, x ) . . . chemical potential

%µ = %∂f (%, c)

∂c − ∆c

Singular elastic pressure

pe(%) ≈ (% − %)⁻³ as % → %−