Flow of heat conducting fluid in a time dependent domain Aneta - - PowerPoint PPT Presentation

Flow of heat conducting fluid in a time dependent domain

Aneta Wróblewska-Kamińska

Department of Mathematics, Imperial College London Institute of Mathematics, Polish Academy of Sciences

joint work with Ondˇ rej Kreml, Vaclav M´ acha, and ˇ S´ arka Neˇ casov´ a

MathFlows18, Porquerolles, September 2018 Supported by the Royal Society within the International Newton Fellowship

Some motivations

Some motivations

Real life motivation:

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

◮ Navier-Stokes-Fourier system with thermal energy equation – E. Feireisl

(2004)

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

◮ Navier-Stokes-Fourier system with thermal energy equation – E. Feireisl

(2004)

◮ Navier-Stokes-Fourier system with entropy inequality – E. Feireisl,

• A. Novotn´

y (2009)

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

◮ Navier-Stokes-Fourier system with thermal energy equation – E. Feireisl

(2004)

◮ Navier-Stokes-Fourier system with entropy inequality – E. Feireisl,

• A. Novotn´

y (2009)

◮ In a time dependent domain, barotropic case, no-slip boundary

conditions - E. Feireisl, J. Neustupa, J. Stebel - the Brinkman penelization method (2011)

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

◮ Navier-Stokes-Fourier system with thermal energy equation – E. Feireisl

(2004)

◮ Navier-Stokes-Fourier system with entropy inequality – E. Feireisl,

• A. Novotn´

y (2009)

◮ In a time dependent domain, barotropic case, no-slip boundary

conditions - E. Feireisl, J. Neustupa, J. Stebel - the Brinkman penelization method (2011)

◮ Barotropic N-S system with a slip b.c. – E. Feireisl, O. Kreml,

ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel (2013)

Some motivations

Real life motivation:

◮ Motion of a piston in a cylinder filled with a heat conducting gas

Mathematical development - existence of weak solutions for compressible fluids:

◮ A barotropic Navier-Stokes system on a fixed domain – P.-L. Lions

(1998), E. Feireisl, H. Petzetov´ a, A. Novotn´ y (2001)

◮ Navier-Stokes-Fourier system with thermal energy equation – E. Feireisl

(2004)

◮ Navier-Stokes-Fourier system with entropy inequality – E. Feireisl,

• A. Novotn´

y (2009)

◮ In a time dependent domain, barotropic case, no-slip boundary

conditions - E. Feireisl, J. Neustupa, J. Stebel - the Brinkman penelization method (2011)

◮ Barotropic N-S system with a slip b.c. – E. Feireisl, O. Kreml,

ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel (2013)

◮ Navier-Stokes-Fourier system with thermal energy equation, slip b.c. –

• O. Kreml, V. M´

acha, ˇ

• S. Neˇ

casova, A. W-K (2017)

The Navier-Stokes-Fourier system on time dependent domain

◮

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + ∇xp(̺, ϑ) = divxS(∇xu), ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σ,

d dt ̺|u|2 + ̺e dx = 0.

The Navier-Stokes-Fourier system on time dependent domain

◮

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + ∇xp(̺, ϑ) = divxS(∇xu), ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σ,

d dt ̺|u|2 + ̺e dx = 0.

◮ ̺ stands for the density of the fluid, u - the velocity, ϑ - the temperature

The Navier-Stokes-Fourier system on time dependent domain

◮

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + ∇xp(̺, ϑ) = divxS(∇xu), ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σ,

d dt ̺|u|2 + ̺e dx = 0.

◮ ̺ stands for the density of the fluid, u - the velocity, ϑ - the temperature ◮ p - a pressure, s - an internal entropy, e - an internal energy, S - a viscous

stress tensor, q - a heat flux

The Navier-Stokes-Fourier system on time dependent domain

◮

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + ∇xp(̺, ϑ) = divxS(∇xu), ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σ,

d dt ̺|u|2 + ̺e dx = 0.

◮ ̺ stands for the density of the fluid, u - the velocity, ϑ - the temperature ◮ p - a pressure, s - an internal entropy, e - an internal energy, S - a viscous

stress tensor, q - a heat flux

◮ σ - an entropy production rate

σ 1 ϑ

• S : ∇xu − q

ϑ · ∇xϑ

• .

The Navier-Stokes-Fourier system on time dependent domain

◮ We study the above system on a moving domain Ω = Ωt

The boundary of the domain Ωt occupied by the fluid is described by means of a given (regular) velocity field V(t, x), where t ∈ [0, T] and x ∈ R3,

The Navier-Stokes-Fourier system on time dependent domain

◮ We study the above system on a moving domain Ω = Ωt

The boundary of the domain Ωt occupied by the fluid is described by means of a given (regular) velocity field V(t, x), where t ∈ [0, T] and x ∈ R3,

◮

d dt X(t, x) = V

• t, X(t, x)
• , t > 0, X(0, x) = x,

The Navier-Stokes-Fourier system on time dependent domain

◮ We study the above system on a moving domain Ω = Ωt

The boundary of the domain Ωt occupied by the fluid is described by means of a given (regular) velocity field V(t, x), where t ∈ [0, T] and x ∈ R3,

◮

d dt X(t, x) = V

• t, X(t, x)
• , t > 0, X(0, x) = x,

Ωτ = X (τ, Ω0) , where Ω0 ⊂ R3 is a given domain, Γτ = ∂Ωτ, and Qτ = ∪t∈(0,τ){t} × Ωt.

The Navier-Stokes-Fourier system on time dependent domain

◮ We study the above system on a moving domain Ω = Ωt

The boundary of the domain Ωt occupied by the fluid is described by means of a given (regular) velocity field V(t, x), where t ∈ [0, T] and x ∈ R3,

◮

d dt X(t, x) = V

• t, X(t, x)
• , t > 0, X(0, x) = x,

Ωτ = X (τ, Ω0) , where Ω0 ⊂ R3 is a given domain, Γτ = ∂Ωτ, and Qτ = ∪t∈(0,τ){t} × Ωt.

◮ We assume that the volume of the domain can not degenerate in time

there exists V0 > 0 such that |Ωτ| V0 for all τ ∈ [0, T].

The Navier-Stokes-Fourier system on time dependent domain

◮ We study the above system on a moving domain Ω = Ωt

The boundary of the domain Ωt occupied by the fluid is described by means of a given (regular) velocity field V(t, x), where t ∈ [0, T] and x ∈ R3,

◮

d dt X(t, x) = V

• t, X(t, x)
• , t > 0, X(0, x) = x,

Ωτ = X (τ, Ω0) , where Ω0 ⊂ R3 is a given domain, Γτ = ∂Ωτ, and Qτ = ∪t∈(0,τ){t} × Ωt.

◮ We assume that the volume of the domain can not degenerate in time

there exists V0 > 0 such that |Ωτ| V0 for all τ ∈ [0, T].

◮ We assume that divxV = 0 on the neighborhood of Γτ,

The Navier-Stokes-Fourier system on time dependent domain

◮ The stress tensor S is determined by the Newton rheological law

S(ϑ, ∇xu) = µ(ϑ)

• ∇xu + ∇t

xu − 2

3divxuI

• + ζ(ϑ)divxuI, µ > 0, ζ 0

The Navier-Stokes-Fourier system on time dependent domain

◮ The stress tensor S is determined by the Newton rheological law

S(ϑ, ∇xu) = µ(ϑ)

• ∇xu + ∇t

xu − 2

3divxuI

• + ζ(ϑ)divxuI, µ > 0, ζ 0

where µ, ζ ∈ C 1[0, ∞) and satisfy 0 < µ(1 + ϑ) µ(ϑ) µ(1 + ϑ), sup

ϑ∈[0,∞)

|µ′(ϑ)| m,

The Navier-Stokes-Fourier system on time dependent domain

◮ The stress tensor S is determined by the Newton rheological law

S(ϑ, ∇xu) = µ(ϑ)

• ∇xu + ∇t

xu − 2

3divxuI

• + ζ(ϑ)divxuI, µ > 0, ζ 0

where µ, ζ ∈ C 1[0, ∞) and satisfy 0 < µ(1 + ϑ) µ(ϑ) µ(1 + ϑ), sup

ϑ∈[0,∞)

|µ′(ϑ)| m, 0 ζ(ϑ) ζ(1 + ϑ).

The Navier-Stokes-Fourier system on time dependent domain

◮ The stress tensor S is determined by the Newton rheological law

S(ϑ, ∇xu) = µ(ϑ)

• ∇xu + ∇t

xu − 2

3divxuI

• + ζ(ϑ)divxuI, µ > 0, ζ 0

where µ, ζ ∈ C 1[0, ∞) and satisfy 0 < µ(1 + ϑ) µ(ϑ) µ(1 + ϑ), sup

ϑ∈[0,∞)

|µ′(ϑ)| m, 0 ζ(ϑ) ζ(1 + ϑ).

◮ The heat flux q is given by the Fourier law

q = −κ(ϑ)∇xϑ, κ > 0, where 0 < κ(ϑ) ≈ 1 + ϑ + ϑ3

The Navier-Stokes-Fourier system on time dependent domain

The quantities p, e, and s are continuously differentiable functions for positive values of ̺, ϑ and satisfy Gibbs’ equation ϑDs(̺, ϑ) = De(̺, ϑ) + p(̺, ϑ)D

• 1

̺

• for all ̺, ϑ > 0.

The Navier-Stokes-Fourier system on time dependent domain

The quantities p, e, and s are continuously differentiable functions for positive values of ̺, ϑ and satisfy Gibbs’ equation ϑDs(̺, ϑ) = De(̺, ϑ) + p(̺, ϑ)D

• 1

̺

• for all ̺, ϑ > 0.

Further, we assume the following state equation for the pressure and the internal energy and entropy p(̺, ϑ) = pM(̺, ϑ) + pR(ϑ), pR(ϑ) = a 3ϑ4, a > 0,

The Navier-Stokes-Fourier system on time dependent domain

The quantities p, e, and s are continuously differentiable functions for positive values of ̺, ϑ and satisfy Gibbs’ equation ϑDs(̺, ϑ) = De(̺, ϑ) + p(̺, ϑ)D

• 1

̺

• for all ̺, ϑ > 0.

Further, we assume the following state equation for the pressure and the internal energy and entropy p(̺, ϑ) = pM(̺, ϑ) + pR(ϑ), pR(ϑ) = a 3ϑ4, a > 0, e(̺, ϑ) = eM(̺, ϑ) + eR(̺, ϑ), ̺eR(̺, ϑ) = aϑ4, and s(̺, ϑ) = sM(̺, ϑ) + sR(̺, ϑ), ̺sR(̺, ϑ) = 4 3aϑ3.

Hypotheses

Hypotheses

◮

∂pM ∂̺ > 0 for all ̺, ϑ > 0

◮ 0 < ∂eM

∂ϑ c for all ̺, ϑ > 0.

Hypotheses

◮

∂pM ∂̺ > 0 for all ̺, ϑ > 0

◮ 0 < ∂eM

∂ϑ c for all ̺, ϑ > 0.

◮ limϑ→0+ eM(̺, ϑ) = eM(̺) > 0 for any fixed ̺ > 0, ◮

̺ ∂eM(̺,ϑ)

∂̺

• ceM(̺, ϑ) for all ̺, ϑ > 0.

Hypotheses

◮

∂pM ∂̺ > 0 for all ̺, ϑ > 0

◮ 0 < ∂eM

∂ϑ c for all ̺, ϑ > 0.

◮ limϑ→0+ eM(̺, ϑ) = eM(̺) > 0 for any fixed ̺ > 0, ◮

̺ ∂eM(̺,ϑ)

∂̺

• ceM(̺, ϑ) for all ̺, ϑ > 0.

◮ There is a function P satisfying P ∈ C 1[0, ∞), P(0) = 0, P′(0) > 0,

such that pM(̺, ϑ) = ϑ

5 2 P

• ̺

ϑ

3 2

• whenever 0 < ̺ Zϑ

3 2 , or, ̺ > Zϑ 3 2

where 0 < Z < Z

Hypotheses

◮

∂pM ∂̺ > 0 for all ̺, ϑ > 0

◮ 0 < ∂eM

∂ϑ c for all ̺, ϑ > 0.

◮ limϑ→0+ eM(̺, ϑ) = eM(̺) > 0 for any fixed ̺ > 0, ◮

̺ ∂eM(̺,ϑ)

∂̺

• ceM(̺, ϑ) for all ̺, ϑ > 0.

◮ There is a function P satisfying P ∈ C 1[0, ∞), P(0) = 0, P′(0) > 0,

such that pM(̺, ϑ) = ϑ

5 2 P

• ̺

ϑ

3 2

• whenever 0 < ̺ Zϑ

3 2 , or, ̺ > Zϑ 3 2

where 0 < Z < Z

◮ what gives

c̺

5 3 pM c

• ϑ

5 2

if ̺ < Zϑ

3 2

̺

5 3

if ̺ > Zϑ

3 2 ,

Hypotheses

◮

∂pM ∂̺ > 0 for all ̺, ϑ > 0

◮ 0 < ∂eM

∂ϑ c for all ̺, ϑ > 0.

◮ limϑ→0+ eM(̺, ϑ) = eM(̺) > 0 for any fixed ̺ > 0, ◮

̺ ∂eM(̺,ϑ)

∂̺

• ceM(̺, ϑ) for all ̺, ϑ > 0.

◮ There is a function P satisfying P ∈ C 1[0, ∞), P(0) = 0, P′(0) > 0,

such that pM(̺, ϑ) = ϑ

5 2 P

• ̺

ϑ

3 2

• whenever 0 < ̺ Zϑ

3 2 , or, ̺ > Zϑ 3 2

where 0 < Z < Z

◮ what gives

c̺

5 3 pM c

• ϑ

5 2

if ̺ < Zϑ

3 2

̺

5 3

if ̺ > Zϑ

3 2 ,

◮ pM(̺, ϑ) = 2

3̺eM(̺, ϑ) for ̺ > Zϑ

3 2 .

Boundary conditions and initial data

◮ The impermeability of the boundary of the physical domain is described

by the condition (u − V) · n|Γτ = 0 for any τ 0, where n(t, x) denotes the unit outer normal vector to the boundary Γt.

Boundary conditions and initial data

◮ The impermeability of the boundary of the physical domain is described

by the condition (u − V) · n|Γτ = 0 for any τ 0, where n(t, x) denotes the unit outer normal vector to the boundary Γt.

◮ We assume a complete slip boundary conditions in the form

[Sn] × n = 0.

Boundary conditions and initial data

◮ The impermeability of the boundary of the physical domain is described

by the condition (u − V) · n|Γτ = 0 for any τ 0, where n(t, x) denotes the unit outer normal vector to the boundary Γt.

◮ We assume a complete slip boundary conditions in the form

[Sn] × n = 0.

◮ For the heat flux we consider the conservative boundary conditions

q · n = 0 for all t ∈ [0, T], x ∈ Γt.

Boundary conditions and initial data

◮ The impermeability of the boundary of the physical domain is described

by the condition (u − V) · n|Γτ = 0 for any τ 0, where n(t, x) denotes the unit outer normal vector to the boundary Γt.

◮ We assume a complete slip boundary conditions in the form

[Sn] × n = 0.

◮ For the heat flux we consider the conservative boundary conditions

q · n = 0 for all t ∈ [0, T], x ∈ Γt.

◮ Our problem is supplemented by the initial conditions

̺(0, ·) = ̺0 ∈ L

5 3 0,

(̺u)(0, ·) = (̺u)0, (̺u)0 = 0 if ̺0 = 0, ϑ(0, ·) = ϑ0 > 0, (̺s)0 = ̺0s(̺0, ϑ0) ∈ L1(Ω0), E0 =

• Ω0
• 1

2̺0 |(̺u)0|2 + ̺0e(̺0, ϑ0)

• dx < ∞.

Weak formulation

Weak formulation

The continuity equation is satisfied in weak and renormalized sense.

T

• Ωt

̺B(̺)(∂tϕ + u · ∇xϕ) =

T

• Ωt

b(̺)divxuϕ −

• Ω0

̺0B(̺0)ϕ(0) for any ϕ ∈ C 1

c ([0, T) × R3), b ∈ L∞ ∩ C[0, ∞) such that b(0) = 0 and

B(̺) = B(1) + ̺

1 b(z) z2 dz. We suppose that ̺ 0 a.e. in (0, T) × R3.

Weak formulation

The continuity equation is satisfied in weak and renormalized sense.

T

• Ωt

̺B(̺)(∂tϕ + u · ∇xϕ) =

T

• Ωt

b(̺)divxuϕ −

• Ω0

̺0B(̺0)ϕ(0) for any ϕ ∈ C 1

c ([0, T) × R3), b ∈ L∞ ∩ C[0, ∞) such that b(0) = 0 and

B(̺) = B(1) + ̺

1 b(z) z2 dz. We suppose that ̺ 0 a.e. in (0, T) × R3.

The momentum equation is satisfied in a weak sense

T

• Ωt

(̺u · ∂tϕ + ̺[u ⊗ u] : ∇xϕ + p(̺, ϑ)divxϕ − S(ϑ, ∇xu) : ∇xϕ) = −

• Ω0

(̺u)0 · ϕ(0, ·), for any ϕ ∈ C 1

c (QT; R3) such that ϕ(T, ·) = 0, ϕ · n|Γτ = 0 for all τ ∈ [0, T]

Weak formulation

The continuity equation is satisfied in weak and renormalized sense.

T

• Ωt

̺B(̺)(∂tϕ + u · ∇xϕ) =

T

• Ωt

b(̺)divxuϕ −

• Ω0

̺0B(̺0)ϕ(0) for any ϕ ∈ C 1

c ([0, T) × R3), b ∈ L∞ ∩ C[0, ∞) such that b(0) = 0 and

B(̺) = B(1) + ̺

1 b(z) z2 dz. We suppose that ̺ 0 a.e. in (0, T) × R3.

The momentum equation is satisfied in a weak sense

T

• Ωt

(̺u · ∂tϕ + ̺[u ⊗ u] : ∇xϕ + p(̺, ϑ)divxϕ − S(ϑ, ∇xu) : ∇xϕ) = −

• Ω0

(̺u)0 · ϕ(0, ·), for any ϕ ∈ C 1

c (QT; R3) such that ϕ(T, ·) = 0, ϕ · n|Γτ = 0 for all τ ∈ [0, T]

In particular, the impermeability condition is satisfied in the sense of traces, specifically, u, ∇xu ∈ L2(QT; R3) and (u − V) · n(τ, ·)|Γτ = 0 for a.a. τ ∈ [0, T].

Weak formulation

The entropy inequality

T

• Ωt

̺s(∂tϕ + u · ∇xϕ) −

T

• Ωt

κ(ϑ)∇xϑ · ∇xϕ ϑ +

T

• Ωt

ϕ ϑ

• S : ∇xu + κ(ϑ)|∇xϑ|2

ϑ

• −
• Ω0

(̺s)0ϕ(0) holds for all ϕ ∈ C 1

c (QT) such that ϕ(T, ·) = 0 and ϕ 0.

Weak formulation

The entropy inequality

T

• Ωt

̺s(∂tϕ + u · ∇xϕ) −

T

• Ωt

κ(ϑ)∇xϑ · ∇xϕ ϑ +

T

• Ωt

ϕ ϑ

• S : ∇xu + κ(ϑ)|∇xϑ|2

ϑ

• −
• Ω0

(̺s)0ϕ(0) holds for all ϕ ∈ C 1

c (QT) such that ϕ(T, ·) = 0 and ϕ 0.

The energy inequality for the case of a moving domain reads as

• Ωτ

1

2̺|u|2 + ̺e

• (τ, ·)
• Ω0
• 1

2 (̺u)2 ̺0 + ̺0e0−(̺u)0 · V(0)

• −

τ

• Ωt

(̺(u ⊗ u) : ∇xV + p divxV − S : ∇xV + ̺u · ∂tV) dxdt +

• Ωt

̺u · V(τ, ·) for a.a. τ ∈ (0, T).

Remark - weak formulation, the energy inequality

In contrast to [Feireisl and Novotny, 2009] we consider energy inequality rather than energy equation. Although it seems that we are losing a lot of information our definition of weak solution is still sufficient. Namely, if the above defined weak solution is smooth enough it will be a strong

• ne.

(L.Poul, 2009)

Main result

Main result

Definition

We say that the trio (̺, u, ϑ) is a variational solution of problem NSF with slip boundary conditions and initial conditions give above if

◮ ̺ ∈ L∞(0, T; L

5 3 (R3)), ̺ 0, ̺ ∈ Lq(QT) for certain q > 5

3,

◮ u, ∇xu ∈ L2(QT), ̺u ∈ L∞(0, T; L1(R3)), ◮ ϑ > 0 a.a. on QT, ϑ ∈ L∞(0, T; L4(R3)), ϑ, ∇xϑ ∈ L2(QT), and

log ϑ, ∇x log ϑ ∈ L2(QT),

◮ ̺s, ̺su,

q ϑ ∈ L1(QT),

◮ weak formulation is satisfied.

Main result

Definition

We say that the trio (̺, u, ϑ) is a variational solution of problem NSF with slip boundary conditions and initial conditions give above if

◮ ̺ ∈ L∞(0, T; L

5 3 (R3)), ̺ 0, ̺ ∈ Lq(QT) for certain q > 5

3,

◮ u, ∇xu ∈ L2(QT), ̺u ∈ L∞(0, T; L1(R3)), ◮ ϑ > 0 a.a. on QT, ϑ ∈ L∞(0, T; L4(R3)), ϑ, ∇xϑ ∈ L2(QT), and

log ϑ, ∇x log ϑ ∈ L2(QT),

◮ ̺s, ̺su,

q ϑ ∈ L1(QT),

◮ weak formulation is satisfied.

Theorem

Let Ω0 ⊂ R3 be a bounded domain of class C 2+ν with some ν > 0, and let V ∈ C 1([0, T]; C 3

c (R3; R3)) be given. Let assumptions on V, p, e, s, µ, ζ, κ be

• satisfied. Let proper initial data be given.

Then the problem NSF with slip boundary condition admits a variational solution in the sense of Definition 1 on any finite time interval (0, T).

The strategy - our penalization scheme

The strategy - our penalization scheme

◮ Letus consider our problem on a large fixed time independent domain

B ⊃ Ωτ for all τ ∈ [0, T]

The strategy - our penalization scheme

◮ Letus consider our problem on a large fixed time independent domain

B ⊃ Ωτ for all τ ∈ [0, T]

◮ To deal with the slip boundary condition, we introduce to the weak

formulation of the momentum equation a term 1 ε

T

• Γt

(u − V) · n ϕ · n dSxdt (originally proposed by Stokes and Carey)

The strategy - our penalization scheme

◮ Letus consider our problem on a large fixed time independent domain

B ⊃ Ωτ for all τ ∈ [0, T]

◮ To deal with the slip boundary condition, we introduce to the weak

formulation of the momentum equation a term 1 ε

T

• Γt

(u − V) · n ϕ · n dSxdt (originally proposed by Stokes and Carey)

◮ In the limit ε → 0 the above term yields the boundary condition

(u − V) · n = 0

• n Γt

and thus the large domain (0, T) × B becomes divided by an impermeable interface ∪t∈(0,T){t} × Γt to a fluid domain QT and a solid domain Qc

T.

The strategy - our penalization scheme

◮ Letus consider our problem on a large fixed time independent domain

B ⊃ Ωτ for all τ ∈ [0, T]

◮ To deal with the slip boundary condition, we introduce to the weak

formulation of the momentum equation a term 1 ε

T

• Γt

(u − V) · n ϕ · n dSxdt (originally proposed by Stokes and Carey)

◮ In the limit ε → 0 the above term yields the boundary condition

(u − V) · n = 0

• n Γt

and thus the large domain (0, T) × B becomes divided by an impermeable interface ∪t∈(0,T){t} × Γt to a fluid domain QT and a solid domain Qc

T.

◮ Then we need to find a way how to get rid of terms outside of a fluid

domain.

Penalisation in terms outside of a fluid domain

We introduce:

Penalisation in terms outside of a fluid domain

We introduce:

◮ A variable shear viscosity coefficient µ = µω and bulk viscosity coefficient

ζ = ζω µω(ϑ, ·) ∈ C ∞

c ,

µω(ϑ, τ, ·)|Ωτ = µ(ϑ) ζω(ϑ, ·) ∈ C ∞

c ,

ζω(ϑ, τ, ·)|Ωτ = η(ϑ) and µω, ζω → 0 a.e. in ((0, T) × B) \ QT as ω → 0.

Penalisation in terms outside of a fluid domain

We introduce:

◮ A variable shear viscosity coefficient µ = µω and bulk viscosity coefficient

ζ = ζω µω(ϑ, ·) ∈ C ∞

c ,

µω(ϑ, τ, ·)|Ωτ = µ(ϑ) ζω(ϑ, ·) ∈ C ∞

c ,

ζω(ϑ, τ, ·)|Ωτ = η(ϑ) and µω, ζω → 0 a.e. in ((0, T) × B) \ QT as ω → 0.

◮ A variable heat conductivity coefficient κν(t, x, ϑ)

κν(ϑ, t, x) = χν(t, x)κ(ϑ), where χν = 1 in QT, χν = ν in ((0, T) × B) \ QT

Penalisation in terms outside of a fluid domain

We introduce:

◮ A variable shear viscosity coefficient µ = µω and bulk viscosity coefficient

ζ = ζω µω(ϑ, ·) ∈ C ∞

c ,

µω(ϑ, τ, ·)|Ωτ = µ(ϑ) ζω(ϑ, ·) ∈ C ∞

c ,

ζω(ϑ, τ, ·)|Ωτ = η(ϑ) and µω, ζω → 0 a.e. in ((0, T) × B) \ QT as ω → 0.

◮ A variable heat conductivity coefficient κν(t, x, ϑ)

κν(ϑ, t, x) = χν(t, x)κ(ϑ), where χν = 1 in QT, χν = ν in ((0, T) × B) \ QT

◮ A variable coefficient a := aη(t, x) which represents the radiative part of

pressure, internal energy and entropy. Namely, we assume that aη(t, x) = χη(t, x)a, a > 0, where χη = 1 in QT, χη = η in ((0, T) × B \ QT.

Artificial pressure and extra temperature term

We introduce:

Artificial pressure and extra temperature term

We introduce:

◮ The artificial pressure

pη,δ(̺, ϑ) = pM(̺, ϑ) + aη 3 ϑ4+δ̺β, β 4, δ > 0

Artificial pressure and extra temperature term

We introduce:

◮ The artificial pressure

pη,δ(̺, ϑ) = pM(̺, ϑ) + aη 3 ϑ4+δ̺β, β 4, δ > 0

◮ An extra term in the energy inequality

λϑ5 and in the entropy inequality formulation λϑ4

Existence of solutions to our new penalization on a fixed domain, uniform estimates

◮ There exists solution {̺ε, uε, ϑε}ε to our penalized problem for each fixed

ε > 0 and fixed η, ω, ν, λ, δ > 0.

Existence of solutions to our new penalization on a fixed domain, uniform estimates

◮ There exists solution {̺ε, uε, ϑε}ε to our penalized problem for each fixed

ε > 0 and fixed η, ω, ν, λ, δ > 0.

◮ Is based on the proof given by Feireisl, Novotn´

y (2009)

Existence of solutions to our new penalization on a fixed domain, uniform estimates

◮ There exists solution {̺ε, uε, ϑε}ε to our penalized problem for each fixed

ε > 0 and fixed η, ω, ν, λ, δ > 0.

◮ Is based on the proof given by Feireisl, Novotn´

y (2009)

◮ The term 1 ε

τ

• Γt ((u − V) · n ϕ · n) dSxdt and

1 ε

T

• Γt (u − V) · n u · nψdSx dt can be treated as a ”compact”

perturbation.

Existence of solutions to our new penalization on a fixed domain, uniform estimates

◮ There exists solution {̺ε, uε, ϑε}ε to our penalized problem for each fixed

ε > 0 and fixed η, ω, ν, λ, δ > 0.

◮ Is based on the proof given by Feireisl, Novotn´

y (2009)

◮ The term 1 ε

τ

• Γt ((u − V) · n ϕ · n) dSxdt and

1 ε

T

• Γt (u − V) · n u · nψdSx dt can be treated as a ”compact”

perturbation.

◮ On the level of the Galerkin approximation and strong solutions we

need to adjust a proof to the case of variable coefficients µ, ζ, κ and a.

Uniform estimates on the set (0, T) × B given by the total dissipation inequality

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

• λϑ5
• L1 c(λ)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

• λϑ5
• L1 c(λ)

uL2 + ∇xuL2 c(λ, ω)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

• λϑ5
• L1 c(λ)

uL2 + ∇xuL2 c(λ, ω) ess sup

τ∈(0,T)

aηϑ4(τ, ·)L1 + ess sup

τ∈(0,T)

̺(τ, ·)

L

5 3 c(λ)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

• λϑ5
• L1 c(λ)

uL2 + ∇xuL2 c(λ, ω) ess sup

τ∈(0,T)

aηϑ4(τ, ·)L1 + ess sup

τ∈(0,T)

̺(τ, ·)

L

5 3 c(λ)

T

• B
• |∇x log(ϑ)|2 + |∇xϑ

3 2 |2

dxdt c(λ)

Uniform estimates on the set (0, T) × B given by the total dissipation inequality T

• Γt

|(u − V) · n|2 dSxdt εc(λ) ess sup

τ∈(0,T)

δ̺β(τ, ·)L1 + ess sup

τ∈(0,T)

√̺u(τ, ·)L2 c(λ)

• λϑ5
• L1 c(λ)

uL2 + ∇xuL2 c(λ, ω) ess sup

τ∈(0,T)

aηϑ4(τ, ·)L1 + ess sup

τ∈(0,T)

̺(τ, ·)

L

5 3 c(λ)

T

• B
• |∇x log(ϑ)|2 + |∇xϑ

3 2 |2

dxdt c(λ) ̺s(̺, ϑ)Lq+̺s(̺, ϑ)uLq+

• κν(ϑ)

ϑ ∇xϑ

• Lq

+̺e(̺, ϑ)L1 c(λ) with some q > 1

Passing with ε → 0

Passing with ε → 0

◮ First of all, directly from uniform estimates we derive that

(u − V) · n(τ, ·)|Γτ = 0 for a.a. τ ∈ [0, T]. in the limit as ε → 0.

Passing with ε → 0

◮ First of all, directly from uniform estimates we derive that

(u − V) · n(τ, ·)|Γτ = 0 for a.a. τ ∈ [0, T]. in the limit as ε → 0.

◮ More demanding (but already known strategy):

◮

ϑε → ϑ a.a. in (0, T) × B.

◮

̺ε → ̺ a.a. in (0, T) × B.

◮ Div-Curl Lemma, Young measure theory; effective viscous pressure,

• scillations defect measure

Passing with ε → 0 - delicate issues

Passing with ε → 0 - delicate issues

◮ We have at hand only the local estimates on the pressure,

K

• p(̺, ϑ)̺π + δ̺β+π

dxdt c(K) for certain π > 0 and for any compact K ⊂ ((0, T) × B) such that K ∩

• ∪τ∈[0,T]
• {τ} × Γτ
• = ∅,

Passing with ε → 0 - delicate issues

◮ We have at hand only the local estimates on the pressure,

K

• p(̺, ϑ)̺π + δ̺β+π

dxdt c(K) for certain π > 0 and for any compact K ⊂ ((0, T) × B) such that K ∩

• ∪τ∈[0,T]
• {τ} × Γτ
• = ∅,

◮ We have to restrict ourselves to the class of test functions for a

momentum equation ϕ ∈ C 1([0, T); W 1,∞ (B; R3)), supp[divxϕ(τ, ·)] ∩ Γτ = ∅, ϕ · n|Γτ = 0 for all τ ∈ [0, T].

Passing with ε → 0 - delicate issues

◮ We have at hand only the local estimates on the pressure,

K

• p(̺, ϑ)̺π + δ̺β+π

dxdt c(K) for certain π > 0 and for any compact K ⊂ ((0, T) × B) such that K ∩

• ∪τ∈[0,T]
• {τ} × Γτ
• = ∅,

◮ We have to restrict ourselves to the class of test functions for a

momentum equation ϕ ∈ C 1([0, T); W 1,∞ (B; R3)), supp[divxϕ(τ, ·)] ∩ Γτ = ∅, ϕ · n|Γτ = 0 for all τ ∈ [0, T].

◮ But this can be extended to the class

ϕ ∈ C ∞

c ([0, T] × B; R3),

ϕ(τ, ·) · n|Γτ = 0 for any τ ∈ [0, T].

Passing with ε → 0 - delicate issues

◮ Internal energy and ”pressure” terms in the energy inequality

Passing with ε → 0 - delicate issues

◮ Internal energy and ”pressure” terms in the energy inequality ◮ To pass to the limit in

pη,δ(̺ε, ϑε)divxVψ we need to assume that divxV = 0 in the neighbourhood of Γt

Passing with ε → 0 - delicate issues

◮ Internal energy and ”pressure” terms in the energy inequality ◮ To pass to the limit in

pη,δ(̺ε, ϑε)divxVψ we need to assume that divxV = 0 in the neighbourhood of Γt

◮ We do not have uniform integrability of the sequence

{̺εeη(̺ε, ϑε)}ε>0

Passing with ε → 0 - delicate issues

◮ Internal energy and ”pressure” terms in the energy inequality ◮ To pass to the limit in

pη,δ(̺ε, ϑε)divxVψ we need to assume that divxV = 0 in the neighbourhood of Γt

◮ We do not have uniform integrability of the sequence

{̺εeη(̺ε, ϑε)}ε>0

◮ Since the sequence {̺εeη(̺ε, ϑε)}ε is nonnegative, the sequence is

integrable, density and temperature converges a.e., by the Fatou lemma we deduce lim sup

T

• B

̺εeη(̺ε, ϑε)∂tψdxdt

T

• B

̺eη(̺, ϑ)∂tψdxdt as far as ψ ∈ C 1

c ([0, T)) and ∂tψ 0.

Vanishing density outside of fluid part

In order to get rid of the density dependent terms supported by the ”solid” part ((0, T) × B) \ QT we use result of [Feireisl, Kreml, Necasova, Neustupa, Stebel, JDE, 2013] which reads as

Vanishing density outside of fluid part

In order to get rid of the density dependent terms supported by the ”solid” part ((0, T) × B) \ QT we use result of [Feireisl, Kreml, Necasova, Neustupa, Stebel, JDE, 2013] which reads as

Lemma

Let ̺ ∈ L∞(0, T; L2(B)), ̺ 0, u ∈ L2(0, T; W 1,2 (B; R3)) be a weak solution

• f the equation of continuity, specifically,
• B
• ̺(τ, ·)ϕ(τ, ·) − ̺0ϕ(0, ·)
• dx =

τ

• B
• ̺∂tϕ + ̺u · ∇xϕ
• dxdt

for any τ ∈ [0, T] and any test function ϕ ∈ C 1

c ([0, T] × R3).

In addition, assume that (u − V)(τ, ·) · n|Γτ = 0 for a.a. τ ∈ (0, T), and that ̺0 ∈ L2(R3), ̺0 0, ̺0|B\Ω0 = 0. Then ̺(τ, ·)|B\Ωτ = 0 for any τ ∈ [0, T].

The limit system ε → 0 and with ̺0 = 0 on B \ Ω0

The limit system ε → 0 and with ̺0 = 0 on B \ Ω0

The continuity equation reads

T

• Ωt

̺B(̺)(∂tϕ + u · ∇xϕ) =

T

• Ωt

b(̺)divxuϕ −

• Ω0

̺0,δB(̺0,δ)ϕ(0) for any ϕ ∈ C 1

c ([0, T) × R3), and any b ∈ L∞ ∩ C[0, ∞) such that b(0) = 0

and B(̺) = B(1) + ̺

1 b(z) z2 dz.

The limit system ε → 0 and with ̺0 = 0 on B \ Ω0

The continuity equation reads

T

• Ωt

̺B(̺)(∂tϕ + u · ∇xϕ) =

T

• Ωt

b(̺)divxuϕ −

• Ω0

̺0,δB(̺0,δ)ϕ(0) for any ϕ ∈ C 1

c ([0, T) × R3), and any b ∈ L∞ ∩ C[0, ∞) such that b(0) = 0

and B(̺) = B(1) + ̺

1 b(z) z2 dz.

Next the momentum equation reduces to

T

• Ωt

(̺u · ∂tϕ + ̺[u ⊗ u] : ∇xϕ + pη,δ(̺, ϑ)divxϕ − Sω(ϑ, ∇xu) : ∇xϕ) = −

• Ω0

(̺u)0,δ · ϕ(0, ·)+

T

• B\Ωt

Sω(ϑ, ∇xu) : ∇ϕ −

T

• B\Ωt

aη 3 ϑ4divxϕ for any ϕ ∈ C ∞

c ([0, T] × B; R3), ϕ(τ, ·) · n|Γτ = 0 for any τ ∈ [0, T].

The limit system ε → 0 and with ̺0 = 0 on B \ Ω0

The balance of entropy takes the following form

T

• Ωt

̺s(̺, ϑ)(∂tϕ + u · ∇xϕ)+

T

• B\Ωt

4 3aηϑ3(∂tϕ + u · ∇xϕ) −

T

• Ωt

κν(ϑ)∇xϑ · ∇xϕ ϑ −

T

• B\Ωt

κν(ϑ)∇xϑ · ∇xϕ ϑ +

T

• Ωt

ϕ ϑ

• Sω : ∇xu + κν(ϑ)|∇xϑ|2

ϑ

• +

T

• B\Ωt

ϕ ϑ

• Sω : ∇xu + κν(ϑ)|∇xϑ|2

ϑ

• −

T

• B

λϑ4ϕ −

• Ω0

(̺s)0,δϕ(0)−

• B\Ω0

4 3aηϑ3

0,δϕ(0)

for all ϕ ∈ C 1

c ([0, T) × B), ϕ 0.

The limit system ε → 0 and with ̺0 = 0 on B \ Ω0

The total energy balance reads

T

• Ωt
• 1

2̺|u|2 + ̺e(̺, ϑ) + δ β − 1̺β

• ∂tψ+

T

• B\Ωt

aηϑ4∂tψ −

T

• B

λϑ5ψ −

• Ω0
• 1

2 (̺u)2

0,δ

̺0,δ + ̺0,δe0,δ + δ β − 1̺β

0,δ − (̺u)0,δ · V(0, ·)

• ψ(0)

−

• B\Ω0

aηϑ4

0,δψ(0)

−

T

• Ωt

(Sω : ∇xVψ − ̺u · ∂t(Vψ) − ̺(u ⊗ u) : ∇xVψ − pδ(̺, ϑ)divxVψ) −

τ

• B\Ωt
• Sω : ∇xV − 1

3aηϑ4divxV

• ψ

for all ψ ∈ C 1

c ([0, T)), ∂tψ 0.

Passing with η → 0

Passing with η → 0

◮ Let us denote by {̺η, uη, ϑη}η>0 solutions to the system obtained as

ε → 0

Passing with η → 0

◮ Let us denote by {̺η, uη, ϑη}η>0 solutions to the system obtained as

ε → 0

◮ Let aη = ηa on B \ Ωt

Passing with η → 0

◮ Let us denote by {̺η, uη, ϑη}η>0 solutions to the system obtained as

ε → 0

◮ Let aη = ηa on B \ Ωt ◮ Since we have

• λϑ5

η

• L1 c(λ),

uηL2(0,T;W 1,2(B)) c

Passing with η → 0

◮ Let us denote by {̺η, uη, ϑη}η>0 solutions to the system obtained as

ε → 0

◮ Let aη = ηa on B \ Ωt ◮ Since we have

• λϑ5

η

• L1 c(λ),

uηL2(0,T;W 1,2(B)) c

◮ for η → 0 we get

T

• B\Ωt

1 3aηϑ4

ηdivxϕ → 0

T

• B\Ωt

1 3aηϑ4

ηdivxV → 0

T

• B\Ωt

4 3aηϑ3

η∂tϕ → 0

T

• B\Ωt

4 3aηϑ3

ηuη · ∇xϕ → 0

Passing with η → 0

Passing with η → 0

◮ Since ϑη → ϑ weakly in L1((0, T) × B), we obtain

T

• B

λϑ5dxdt lim inf

η→0

T

• B

λϑ5

ηdxdt.

Passing with η → 0

◮ Since ϑη → ϑ weakly in L1((0, T) × B), we obtain

T

• B

λϑ5dxdt lim inf

η→0

T

• B

λϑ5

ηdxdt.

◮ In the energy inequality we need to restrict ourself to test functions

ψ ∈ C 1

c ([0, T)),

ψ 0, ∂tψ 0

Passing with η → 0

◮ Since ϑη → ϑ weakly in L1((0, T) × B), we obtain

T

• B

λϑ5dxdt lim inf

η→0

T

• B

λϑ5

ηdxdt.

◮ In the energy inequality we need to restrict ourself to test functions

ψ ∈ C 1

c ([0, T)),

ψ 0, ∂tψ 0

◮ To pass to the limit in remaining terms outside of the fluid part and in all

terms in the fluid part we use the same arguments as for passing with ε → 0.

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

◮ Let us denote by {̺ω, uω, ϑω}ω>0 solutions to the system obtained as

η → 0.

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

◮ Let us denote by {̺ω, uω, ϑω}ω>0 solutions to the system obtained as

η → 0.

◮ Since µω and ζω ≈ ω on B \ Ωt, and we control

T

• B\Ωt

1 ϑω Sω(ϑω, ∇xuω) : ∇xuω c,

T

• B\Ωt

λϑ5

ω c

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

◮ Let us denote by {̺ω, uω, ϑω}ω>0 solutions to the system obtained as

η → 0.

◮ Since µω and ζω ≈ ω on B \ Ωt, and we control

T

• B\Ωt

1 ϑω Sω(ϑω, ∇xuω) : ∇xuω c,

T

• B\Ωt

λϑ5

ω c

◮ In the momentum equation

T

• B\Ωt

Sω(ϑω, ∇xuω) : ∇xϕ → 0 as ω → 0.

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

◮ Let us denote by {̺ω, uω, ϑω}ω>0 solutions to the system obtained as

η → 0.

◮ Since µω and ζω ≈ ω on B \ Ωt, and we control

T

• B\Ωt

1 ϑω Sω(ϑω, ∇xuω) : ∇xuω c,

T

• B\Ωt

λϑ5

ω c

◮ In the momentum equation

T

• B\Ωt

Sω(ϑω, ∇xuω) : ∇xϕ → 0 as ω → 0.

◮ In the total energy inequality

T

• B\Ωt

Sω(ϑω, ∇xuω) : ∇xVdxdt → 0 as ω → 0.

Passing with ω → 0

We would like to get rid of terms related to the viscous stress tensor outside of the fluid domain.

◮ Let us denote by {̺ω, uω, ϑω}ω>0 solutions to the system obtained as

η → 0.

◮ Since µω and ζω ≈ ω on B \ Ωt, and we control

T

• B\Ωt

1 ϑω Sω(ϑω, ∇xuω) : ∇xuω c,

T

• B\Ωt

λϑ5

ω c

◮ In the momentum equation

T

• B\Ωt

Sω(ϑω, ∇xuω) : ∇xϕ → 0 as ω → 0.

◮ In the total energy inequality

T

• B\Ωt

Sω(ϑω, ∇xuω) : ∇xVdxdt → 0 as ω → 0.

◮ In the entropy inequality we skip the term T

• B\Ωt

ϕ ϑ Sω : ∇xu

since it is positive for ϕ 0.

Passing with ν → 0

◮ Let {̺ν, uν, ϑν}ν>0 denote solutions to the system obtained in the

previous step

Passing with ν → 0

◮ Let {̺ν, uν, ϑν}ν>0 denote solutions to the system obtained in the

previous step

◮ For each fixed ν > 0 we still know that

• χνκ(ϑν)|∇xϑν|2

ϑ2

ν

• L1((0,T)×B)

c(λ) and λϑ5

νL1((0,T)×B) c(λ)

Passing with ν → 0

◮ Let {̺ν, uν, ϑν}ν>0 denote solutions to the system obtained in the

previous step

◮ For each fixed ν > 0 we still know that

• χνκ(ϑν)|∇xϑν|2

ϑ2

ν

• L1((0,T)×B)

c(λ) and λϑ5

νL1((0,T)×B) c(λ)

◮ In the internal entropy inequality

T

• B\Ωt

νκ(ϑν)∇xϑν ϑν · ∇xϕ → 0 as ν → 0

Passing with ν → 0

◮ Let {̺ν, uν, ϑν}ν>0 denote solutions to the system obtained in the

previous step

◮ For each fixed ν > 0 we still know that

• χνκ(ϑν)|∇xϑν|2

ϑ2

ν

• L1((0,T)×B)

c(λ) and λϑ5

νL1((0,T)×B) c(λ)

◮ In the internal entropy inequality

T

• B\Ωt

νκ(ϑν)∇xϑν ϑν · ∇xϕ → 0 as ν → 0

◮ Since T

• B\Ωt

ϕ ϑν

• κν(ϑν)|∇x ϑν|2

ϑν

• 0 for ϕ 0, we can skip this term

as ν → 0

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

◮ Let {̺λ, uλ, ϑλ}λ∈(0,1) be solution to the limit system obtained in

previous step.

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

◮ Let {̺λ, uλ, ϑλ}λ∈(0,1) be solution to the limit system obtained in

previous step.

◮ We need to build uniform estimates independent of λ

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

◮ Let {̺λ, uλ, ϑλ}λ∈(0,1) be solution to the limit system obtained in

previous step.

◮ We need to build uniform estimates independent of λ ◮ We keep that

• λϑ5
• L1((0,T)×B) c so in the entropy inequality

T

• B

λϑ4

λ → 0

as λ → 0.

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

◮ Let {̺λ, uλ, ϑλ}λ∈(0,1) be solution to the limit system obtained in

previous step.

◮ We need to build uniform estimates independent of λ ◮ We keep that

• λϑ5
• L1((0,T)×B) c so in the entropy inequality

T

• B

λϑ4

λ → 0

as λ → 0.

◮ Next notice that the term τ

• B λϑ5

λψ 0 for all λ > 0 and all

ψ ∈ C 1

c ([0, T)), ψ 0 and therefore can be skipped in the total energy

inequality as λ → 0

Passing with λ → 0

Now we get rid the term related to coefficient λ - the only terms which are left also outside of a fluid domain.

◮ Let {̺λ, uλ, ϑλ}λ∈(0,1) be solution to the limit system obtained in

previous step.

◮ We need to build uniform estimates independent of λ ◮ We keep that

• λϑ5
• L1((0,T)×B) c so in the entropy inequality

T

• B

λϑ4

λ → 0

as λ → 0.

◮ Next notice that the term τ

• B λϑ5

λψ 0 for all λ > 0 and all

ψ ∈ C 1

c ([0, T)), ψ 0 and therefore can be skipped in the total energy

inequality as λ → 0

◮ We pass to the limit with in other term in the same way as for ε → 0. The

bound ess supτ∈(0,T) aϑ4(τ, ·)L1(Ωτ ) < c provides enough information.

Almost the final step

Almost the final step

◮ The only term which left is artificial pressure δ̺β.

Almost the final step

◮ The only term which left is artificial pressure δ̺β. ◮ The passage with δ → 0 is a kind of ”classic” now and we can follow

[Feireisl, Novotn´ y, 2009]

Almost the final step step

Almost the final step step

◮ In particular, for the energy inequality we worked with ”weak” form and

we need to choose proper test function to obtain formulation from our definition of variational solution In particular, the energy inequality has the form

T

• Ωτ

1

2̺|u|2 + ̺e

• ∂tψ −
• Ω0
• 1

2 (̺u)2 ̺0 + ̺0e0 − (̺u)0 · V(0, ·)

• ψ(0)

−

T

• Ωt

(S : ∇xVψ − ̺u · ∂t(Vψ) − ̺(u ⊗ u) : ∇xVψ − p(̺, ϑ)divxVψ) for all ψ ∈ C 1

c ([0, T)), ψ 0, ∂tψ 0.

Almost the final step step

◮ In particular, for the energy inequality we worked with ”weak” form and

we need to choose proper test function to obtain formulation from our definition of variational solution In particular, the energy inequality has the form

T

• Ωτ

1

2̺|u|2 + ̺e

• ∂tψ −
• Ω0
• 1

2 (̺u)2 ̺0 + ̺0e0 − (̺u)0 · V(0, ·)

• ψ(0)

−

T

• Ωt

(S : ∇xVψ − ̺u · ∂t(Vψ) − ̺(u ⊗ u) : ∇xVψ − p(̺, ϑ)divxVψ) for all ψ ∈ C 1

c ([0, T)), ψ 0, ∂tψ 0.

◮ We choose ψξ such that ψξ ∈ C 1

c ([0, T)) is non-increasing function which

fulfils ψξ(t) =

• 1 for t < τ − ξ

0 for t τ for some τ ∈ (0, T) and arbitrary ξ > 0 as a test function in the total energy inequality and we pass with ξ → 0. Then obtain the formulation required in the definition of variational solution

What can one do next? Ongoing work

What can one do next? Ongoing work

◮ Relative entropy inequality (like for weak-strong uniqueness)

What can one do next? Ongoing work

◮ Relative entropy inequality (like for weak-strong uniqueness) ◮ Low Mach number limit - speed of sound dominates characteristic speed

• f the fluid - for ill-prepared initial data the system is driven to the

incompressible Oberback-Boussinesq approximation

What can one do next? Ongoing work

◮ Relative entropy inequality (like for weak-strong uniqueness) ◮ Low Mach number limit - speed of sound dominates characteristic speed

• f the fluid - for ill-prepared initial data the system is driven to the

incompressible Oberback-Boussinesq approximation

◮ Barotropic case - E. Feireisl, O. Kreml, ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel (SIAM, 2014)

What can one do next? Ongoing work

◮ Relative entropy inequality (like for weak-strong uniqueness) ◮ Low Mach number limit - speed of sound dominates characteristic speed

• f the fluid - for ill-prepared initial data the system is driven to the

incompressible Oberback-Boussinesq approximation

◮ Barotropic case - E. Feireisl, O. Kreml, ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel (SIAM, 2014)

◮ time evolution of a Helmholtz projection operator associated with the

domain Ωτ and its gradient counterpart

What can one do next? Ongoing work

◮ Relative entropy inequality (like for weak-strong uniqueness) ◮ Low Mach number limit - speed of sound dominates characteristic speed

• f the fluid - for ill-prepared initial data the system is driven to the

incompressible Oberback-Boussinesq approximation

◮ Barotropic case - E. Feireisl, O. Kreml, ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel (SIAM, 2014)

◮ time evolution of a Helmholtz projection operator associated with the

domain Ωτ and its gradient counterpart

◮ propagation of the acoustic wave is governed by the Neumann Laplacian

whose spectral properties and their dependence of τ need to be examined

Ongoing work. Low Mach number limit

◮ NSF with Ma= ε and Fr= √ε

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + 1 ε2 ∇xp = divxS + 1 ε ̺∇xF, ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σε,

d dt ε2 2 ̺|u|2 + ̺e − ε̺F

• dx = 0.

(NSFε)

Ongoing work. Low Mach number limit

◮ NSF with Ma= ε and Fr= √ε

∂t̺ + divx(̺u) = 0, ∂t(̺u) + divx(̺u ⊗ u) + 1 ε2 ∇xp = divxS + 1 ε ̺∇xF, ∂t(̺s) + divx(̺su) + divx

q

ϑ

• = σε,

d dt ε2 2 ̺|u|2 + ̺e − ε̺F

• dx = 0.

(NSFε)

◮ OB approximation where G(t) :=

Ωt V · ∇xFdx.

divxU = 0 ̺(∂tU + divx(U ⊗ U)) + ∇xΠ − divxµ(ϑ)[∇xU + ∇T

x U] = r∇xF

̺ cp (∂tΘ + divx(ΘU)) − κ(ϑ)∆xΘ − α ρ ϑU · ∇xF = −α̺ϑG r + ̺ αΘ = 0 (OB)

◮ ̺ε → ̺, uε → U, ϑε−ϑ

ε

→ Θ, ̺ε−̺

ε

→ r in a certain sense

Bibliography

◮ O. Kreml, V. M´

acha, ˇ

• S. Neˇ

casova, A. Wróblewska-Kamińska. Flow of heat conducting fluid in a time dependent domain. ZAMP 2018.

◮ O. Kreml, V. M´

acha, ˇ

• S. Neˇ

casova, A. Wróblewska-Kamińska. Weak solutions to the full Navier-Stokes-Fourier system with slip boundary conditions in time dependent domains. Journal de Math´ ematiques Pures et Appliqu´ ees, 2017.

◮ E. Feireisl and A. Novotn´

• y. Singular limits in thermodynamics of viscous
• fluids. Birkh¨

auser-Verlag, Basel, 2009.

◮ E. Feireisl, J. Neustupa, and J. Stebel. Convergence of a Brinkman-type

penalization for compressible fluid flows. J. Differential Equations 250(1) (2011) 596–606.

◮ E. Feireisl, O. Kreml, ˇ

• S. Neˇ

casov´ a, J. Neustupa, J. Stebel. Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains. J. Differential Equations 254 (1) (2013) 125–140.

◮ O. A. Ladyzhenskaja. An initial-boundary value problem for the

Navier-Stokes equations in domains with boundary changing in time. Zap. Nauˇ

• cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 11 (1968)

97–128.