Abstract. We consider the motion of a viscous compressible heat conducting fluid in R

POLONICI MATHEMATICI LXIII.3 (1996)

On the global existence theorem for a free boundary problem for equations of a viscous compressible heat conducting fluid

by Ewa Zadrzy´ nska and Wojciech M. Zaja ¸czkowski (Warszawa)

Abstract. We consider the motion of a viscous compressible heat conducting fluid in R

³

bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

1. Introduction. The main result of this paper is the global existence theorem for the following free boundary problem for a viscous compressible heat conducting fluid (see [4], Chs. 2 and 5):

(1.1)

%[v

t

+ (v · ∇)v] + ∇p − µ∆v − ν∇ div v = %f in e Ω

^T

,

%

t

+ div(%v) = 0 in e Ω

^T

,

%c

v

(θ

t

+ v · ∇θ) + θp

θ

div v − κ∆θ

− µ 2

3

X

i,j=1

(v

i,xj

+ v

j,xi

)

²

− (ν − µ)(div v)

²

= %r in e Ω

^T

,

T · n = −p

0

n on e S

^T

,

v · n = − ϕ

t

|∇ϕ| on e S

^T

,

∂θ

∂n = θ

1

on e S

^T

,

%|

t=0

= %

0

, v|

t=0

= v

0

, θ|

t=0

= θ

0

in Ω, where e Ω

^T

= S

t∈(0,T )

Ω

t

× {t}, Ω

_t

⊂ R

³

is a bounded domain depending

1991 Mathematics Subject Classification: 35A05, 35R35, 76N10.

Key words and phrases: viscous compressible heat conducting fluid, global existence, free boundary problem.

[199]

on t, Ω

0

= Ω, e S

^T

= S

t∈(0,T )

S

t

×{t}, S

_t

= ∂Ω

t

, ϕ(x, t) = 0 describes S

t

, n is the unit outward vector normal to the boundary, i.e. n = ∇ϕ/|∇ϕ|. In (1.1), v = v(x, t) is the velocity of fluid, % = %(x, t) the density, θ = θ(x, t) the temperature. Given functions are: f = f (x, t), the external force field per unit mass; r = r(x, t), the heat sources per unit mass; θ

1

= θ

1

(x, t), the heat flow per unit surface; p = p(%, θ), the pressure; c

v

= c

v

(%, θ), the specific heat at constant volume. Moreover, µ and ν denote the viscosity coefficients, κ the coefficient of the heat conductivity, and p

0

the external (constant) pressure.

We assume that µ, ν, κ are constants and thermodynamic considerations imply that c

v

> 0, κ > 0, ν ≥

¹₃

µ > 0. Finally, T = T(v, p) denotes the stress tensor of the form

T = {T

ij

} = {−pδ

_ij

+ µ(v

i,xj

+ v

j,xi

) + (ν − µ)δ

ij

div v}

≡ {−pδ

_ij

+ D

ij

(v)},

where i, j = 1, 2, 3, and D = D(v) = {D

ij

} is the deformation tensor.

Let the domain Ω be given. Then by (1.1)

5

, Ω

t

= {x ∈ R

³

: x = x(ξ, t), ξ ∈ Ω}, where x = x(ξ, t) is the solution of the Cauchy problem

(1.2) ∂x

∂t = v(x, t), x|

t=0

= ξ ∈ Ω, ξ = (ξ

1

, ξ

2

, ξ

3

).

Therefore, we obtain the following relation between the Eulerian x and the Lagrangian ξ coordinates of the same fluid particle:

(1.3) x = ξ +

t

R

0

u(ξ, s) ds ≡ X

u

(ξ, t),

where u(ξ, t) = v(X

u

(ξ, t), t). Moreover, the kinematic boundary condition (1.1)

5

implies that the boundary S

t

is a material surface. Thus, if ξ ∈ S = S

0

then X

u

(ξ, t) ∈ S

t

and S

t

= {x : x = X

u

(ξ, t), ξ ∈ S}.

Equation of continuity (1.1)

2

and (1.1)

5

give the conservation of the total mass, i.e.

(1.4) R

Ωt

%(x, t) dx = M.

In this paper we prove the existence of a global-in-time solution of prob- lem (1.1) near a constant state.

To introduce the definition of the constant state consider the equation

(1.5) p(%

e

, θ

e

) = p

0

,

where θ

e

= (1/|Ω|) R

Ω

θ

0

dξ. We assume that equation (1.5) is solvable with

respect to %

e

> 0.

Definition 1.1. Let f = r = θ

1

= 0. Then by a constant (equilibrium) state we mean a solution (v, θ, %, Ω

t

) of problem (1.1) such that v = 0,

% = %

e

, θ = θ

e

, Ω

t

= Ω

e

for t ≥ 0, where %

e

is a solution of equation (1.5) and |Ω

e

| = M/%

_e

(|Ω

e

| = vol Ω

_e

).

The paper is divided into five sections. In Section 2 we introduce some notation and auxiliary results. In Section 3 we present the local existence theorem (see Theorem 3.1) proved in [16], while in Section 4 we recall the differential inequality (see Theorem 4.1) obtained in [19]. Finally, Section 5 is devoted to the global existence theorem (see Theorem 5.5).

The analogous problem to (1.1) for a viscous compressible barotropic fluid was considered by W. M. Zaj¸ aczkowski in [20]. Hence, in order to prove Theorem 5.5 we apply a method similar to the proof of the global existence theorem in the barotropic case (see [20], Theorem 6.5). We prove Theorem 5.5 under the appropriate choice of %

0

, v

0

, θ

0

, θ

1

, p

0

, κ and the form of the internal energy per unit mass ε = ε(%, θ) (see conditions (5.40)–(5.45)) and under the assumption that ϕ(0) ≤ ε

1

(ϕ(t) is given in (4.5)), where ε

1

is sufficiently small. In Theorem 5.5 we obtain a global solution of (1.1) such that (v, ϑ

0

, ϑ, %

σ

, %

_Ωt

) ∈ M(t) for t ∈ R

¹+

(where ϑ

0

, ϑ, %

σ

, %

_Ωt

are defined in (4.2) and M(t) is defined at the beginning of Section 5) and S

t

∈ W

₂^4−1/2

. The papers [21]–[23] of W. M. Zaj¸ aczkowski and the paper [14] of V. A. Solonnikov and A. Tani are devoted to the motion of a compress- ible barotropic viscous capillary fluid bounded by a free surface.

The motion of a viscous compressible heat conducting fluid in a fixed domain was considered by A. Matsumura and T. Nishida in [5]–[9] and by A. Valli and W. M. Zaj¸ aczkowski in [15], while the papers [11]–[13] of V. A. Solonnikov are concerned with free boundary problems for viscous incompressible fluids.

The papers [1], [2] of J. T. Beale are devoted to the global existence of solutions to free boundary problems, where the free boundary is unbounded and the gravitation is taken into account.

Problem (1.1) is considered also in the papers [16]–[19] of E. Zadrzy´ nska and W. M. Zaj¸ aczkowski. In [16] the local existence of solutions to prob- lem (1.1) is proved. In [18] conservation laws, and in [19] the differential inequality used in the proof of the global existence theorem are derived.

Finally, [17] is a survey of results concerning problem (1.1) and the free boundary problem with surface tension, analogous to problem (1.1).

2. Notation and auxiliary results. In Section 3 we use the anisotropic

Sobolev–Slobodetski˘ı spaces W

₂^l,l/2

(Q

T

), l ∈ R

¹+

(see [3]), of functions de-

fined in Q

T

, where Q

T

= Ω

^T

≡ Ω × (0, T ) (Ω ⊂ R

³

is a domain, T < ∞ or

T = ∞) or Q

T

= S

^T

≡ S × (0, T ), S = ∂Ω.

We define W

₂^l,l/2

(Ω

^T

) as the space of functions u such that kuk

W₂^l,l/2(Ω^T)

=

 X

|α|+2i≤[l]

kD

^α_ξ

∂

_tⁱ

uk

²_L2(ΩT)

+ X

|α|+2i=[l]



T

R

0

R

Ω

R

Ω

|D

_ξ^α

∂

ⁱ_t

u(ξ, t) − D

^α_ξ0

∂

_tⁱ

u(ξ

⁰

, t)|

²

|ξ − ξ

⁰

|

^3+2(l−[l])

dξ dξ

⁰

dt

+ R

Ω T

R

0 T

R

0

|D

^α_ξ

∂

_tⁱ

u(ξ, t) − D

_ξ^α

∂

_tⁱ0

u(ξ, t

⁰

)|

²

|t − t

⁰

|

1+2(l/2−[l/2])

dt dt

⁰

dξ



1/2

< ∞, where we use generalized (Sobolev) derivatives, D

^α_ξ

= ∂

_ξ^α₁¹

∂

_ξ^α₂²

∂

_ξ^α₃³

, ∂

_ξ^α_j^j

=

∂

^αj

/∂ξ

_j^αj

(j = 1, 2, 3), α = (α

1

, α

2

, α

3

) is a multiindex, |α| = α

1

+ α

2

+ α

3

,

∂

_tⁱ

= ∂

ⁱ

/∂t

ⁱ

and [l] is the integer part of l. In the case when l is an integer the second terms in the above formulae must be omitted, while in the case of l/2 being an integer the last terms in the above formulae must be omitted as well.

Similarly to W

₂^l,l/2

(Ω

^T

), using local mappings and a partition of unity we introduce the normed space W

₂^l,l/2

(S

^T

) of functions defined on S

^T

= S × (0, T ), where S = ∂Ω. We also use the ordinary Sobolev spaces W

₂^l

(Q), where l ∈ R

¹+

, Q = Ω (Ω ⊂ R

³

is a bounded domain) or Q = S. To simplify notation we write

kuk

_l,Q

= kuk

_W^l,l/2

2 (Q)

if Q = Ω

^T

or Q = S

^T

, kuk

_l,Q

= kuk

_W^l

2(Q)

if Q = Ω or Q = S.

Moreover, kuk

Lp(Q)

= |u|

p,Q

, 1 ≤ p ≤ ∞.

Now, we introduce the spaces Γ

_k^l

(Ω) and Γ

_k^l,l/2

(Ω) of functions u defined on Ω × (0, T ) (T < ∞ or T = ∞) such that

|u|

_l,k,Ω

≡ kuk

_Γl

k(Ω)

= X

i≤l−k

k∂

_tⁱ

uk

l−i,Ω

< ∞ and

u

l,k,Ω

≡ kuk

Γ_k^l,l/2(Ω)

= X

2i≤l−k

k∂

_tⁱ

uk

l−2i,Ω

< ∞, where l ∈ R

¹+

, k ≥ 0.

Next, define the space L

p

(0, T ; Γ

₀^l,l/2

(Ω)) (where 1 ≤ p ≤ ∞) with the norm kuk

_L

p(0,T ;Γ₀^l,l/2(Ω))

= u

_l,0,p,Ω^T

.

Moreover, let C

^2,1

(Q) (resp. C

_B^2,1

(Q)) (Q ⊂ R

³

× [0, ∞)) denote the space of functions u such that D

^α_x

∂

_tⁱ

u ∈ C

⁰

(Q) (resp. D

^α_x

∂

_tⁱ

u ∈ C

_B⁰

(Q)) for

|α| + 2i ≤ 2 (C

_B⁰

(Q) is the space of continuous bounded functions on Q).

Finally, the following seminorm is used:

u

_κ,Q^T

=



T

R

0

|u|

²_2,Q

t

^2κ

dt



1/2

, where Q = ∂Ω.

Let X be whichever of the function spaces mentioned above. We say that a vector-valued function u = (u

1

, . . . , u

ν

) belongs to X if u

i

∈ X for any 1 ≤ i ≤ ν.

Moreover, we use the following lemmas.

Lemma 2.1. The following imbedding holds:

(2.1) W

_r^l

(Ω) ⊂ L

^α_p

(Ω) (Ω ⊂ R

³

),

where |α| + 3/r − 3 ≤ l, l ∈ Z, 1 ≤ p, r ≤ ∞; L

^αp

(Ω) is the space of functions u such that |D

_x^α

u|

p,Ω

< ∞, and W

_r^l

(Ω) is the Sobolev space.

Moreover , the following interpolation inequalities are true:

(2.2) |D

^α_x

u|

p,Ω

≤ cε

^1−κ

|D

_x^l

u|

r,Ω

+ cε

^−κ

|u|

_r,Ω

,

where κ = |α|/l + 3/(lr) − 3/(lp) < 1, ε is a parameter , and c > 0 is a constant independent of u and ε.

Lemma 2.1 follows from Theorem 10.2 of [3].

Lemma 2.2 (see [10]). For a sufficiently regular u we have k∂

ⁱ_t

u(t)k

2l−1−2i,Ω

≤ c(kuk

_2l,Ω^T

+ k∂

_tⁱ

u(0)k

2l−1−2i,Ω

),

where 0 ≤ 2i ≤ 2l − 1, l ∈ N, and c > 0 is a constant independent of T . Now, consider problem (1.1). For (1.1) the energy conservation law is satisfied (see [4], Ch. 5).

Assume that the internal energy per unit mass ε = ε(%, θ) has the form (2.3) ε(%, θ) = a

0

%

^α

+ h(%, θ),

where a

0

> 0, α > 0, h(%, θ) ≥ h

∗

> 0, a

0

, α, h

∗

are constants and h(%, θ) is a sufficiently regular function of its arguments. Moreover, we assume that h(%, θ) has at (%

e

, θ

e

) (%

e

and θ

e

are introduced in Definition 1.1) the only minimum point equal to h

∗

, i.e. min

%,θ

h(%, θ) = h(%

e

, θ

e

) = h

∗

.

In [19] it is shown that assumption (2.3) and the thermodynamical rela- tion

dε = θds + p

%

²

d%

(where s is the density of entropy per unit mass) imply the following relations between h, p and c

v

:

(2.4) αa

0

%

^α+1

+ %

²

h

%

= p − θp

θ

and

(2.5) c

v

= ∂ε

∂θ = h

θ

. In [18] (Corollary 1) the following result is proved.

Lemma 2.3. Let conditions (2.3)–(2.5) be satisfied. Let

(2.6) f = 0, θ

1

≥ 0.

Assume that (2.7) R

Ω

%

0

v

₀²

2 dξ + κ sup

t t

R

0

dt

⁰

R

S_t0

θ

1

(s, t

⁰

) ds

+ R

Ω

%

0

h(%

0

, θ

0

) dξ − inf

t

R

Ωt

%h(%, θ) dx ≤ δ

0

,

(2.8) R

Ω

|%

₀

− %

_e

| dξ ≤ δ

₀

,

(2.9) (β − 1)

^β−1

β

^β

p

^β−1₀

(a

0

%

^β_e

+ p

0

)

^β

− a

₀

%

^β_e

≤ δ

₀

,

(2.10) a

0

 R

Ω

%

^β₀

dξ − inf

t

R

Ωt

%

^β

dx 

≤ δ

₀

,

where δ

0

> 0 is a sufficiently small constant and β = α + 1. Then 1

2

R

Ωt

%v

²

dx + a

0

 R

Ωt

%

^β

dx − inf

t

R

Ωt

%

^β

dx



+ R

Ωt

%h(%, θ) dx − inf

t

R

Ωt

%h(%, θ) dx + p

0

(|Ω

t

| − |Ω

_∗

|) ≤ ce δ, where |Ω

∗

| = inf

_t

|Ω

_t

|, c = const > 0 is a constant and e δ = e δ(δ

0

), e δ → 0 as δ

0

→ 0.

R e m a r k 2.4 (see [18], Theorem 2.7). Assumptions (2.3)–(2.9) imply that var

t

|Ω

_t

| ≤ cδ, where c > 0 is a constant, var

_t

|Ω

_t

| = sup

_t

|Ω

_t

|−inf

_t

|Ω

_t

|, δ

²

= e cδ

0

and e c is a constant.

R e m a r k 2.5. Since

(2.11) R

Ωt

%h(%, θ) dx ≥ h

∗

R

Ωt

% dx = h

∗

M assumption (2.7) is satisfied if

(2.12) R

Ω

%

0

v

²₀

2 dξ + κ sup

t t

R

0

dt

⁰

R

S_t0

θ

1

(s, t

⁰

) ds

+ R

Ω

%

0

(h(%

0

, θ

0

) − h

∗

) dξ ≤ δ

0

.

R e m a r k 2.6. By Remark 3 of [18] we have (d − inf

t

R

Ωt

%h(%, θ) dx)(β − 1) βp

0

− cδ

≤ |Ω

t

| ≤ cδ + (d − inf

t

R

Ωt

%h(%, θ) dx)(β − 1) βp

0

where c > 0 is some constant, δ = δ(δ

0

) with δ → 0 as δ

0

→ 0, and d = R

Ω

%

0

 v

₀²

2 + a

0

%

²₀

+ h(%

0

, θ

0

)



dξ + p

0

|Ω| + κ sup

t t

R

0

dt

⁰

R

S_t0

θ

1

(s, t

⁰

) ds.

Therefore, using (2.11) and the estimate

R

Ωt

%

^β

dx ≥ |Ω

t

|

^1−β

 R

Ωt

% dx 

β

= M

^β

|Ω

t

|

^β−1

we obtain

inf

t

R

Ωt

%

^β

dx ≥ M

^β

(βp

0

)

^β−1

[cδβ%

0

+ (d − h

∗

M )(β − 1)]

^β−1

. Hence, assumption (2.10) is satisfied if

(2.13) a

0



R

Ω

%

^β₀

dx − M

^β

(βp

0

)

^β−1

[cδβ%

0

+ (d − h

∗

M )(β − 1)]

^β−1



≤ δ

₀

. We see that the left-hand side of (2.13) tends to 0 as β → 1, so for β sufficiently close to 1, it is as small as we wish.

3. Local existence. To prove the local existence for (1.1) we rewrite it in the Lagrangian coordinates introduced by (1.2) and (1.3):

(3.1)

ηu

t

− µ∇

²_u

u − ν∇

u

∇

_u

· u + ∇

_u

p(η, Γ ) = ηg in Ω

^T

≡ Ω × (0, T ),

η

t

+ η∇

u

· u = 0 in Ω

^T

,

ηc

v

(η, Γ )Γ

t

− κ∇

²_u

Γ = −Γ p

Γ

(η, Γ )∇

u

· u + µ

2

3

X

i,j=1

(ξ

xi

∇

_ξ

u

j

+ ξ

xj

∇

_ξ

u

i

)

²

+ (ν − µ)(∇

u

· u)

²

+ ηk in Ω

^T

,

T

u

(u, p) · n = −p

0

n on S

^T

,

n · ∇

u

Γ = Γ

1

on S

^T

,

u|

t=0

= v

0

in Ω,

η|

t=0

= %

0

in Ω,

Γ |

t=0

= θ

0

in Ω,

where u(ξ, t) = v(X

u

(ξ, t), t), Γ (ξ, t) = θ(X

u

(ξ, t), t), η(ξ, t) = %(X

u

(ξ, t), t), g(ξ, t) = f (X

u

(ξ, t), t), k(ξ, t) = r(X

u

(ξ, t), t), ∇

u

= ξ

x

∇

_ξ

≡ {ξ

_ix

∂

ξi

}, T

u

(u, p) = −pI +D

^u

(u), D

^u

(u) = {µ(ξ

kxi

∂

ξk

u

j

+ξ

kxj

∂

ξk

u

i

)+(ν −µ)δ

ij

∇

u

u}

(here the summation convention over the repeated indices is assumed), and Γ

1

(ξ, t) = θ

1

(X

u

(ξ, t), t).

Let A = {a

ij

} be the Jacobi matrix of the transformation x = X

u

(ξ, t), where a

ij

= δ

ij

+ R

t

0

∂

ξj

u

i

(ξ, t

⁰

) dt

⁰

. Assuming that |∇

ξ

u|

_∞,Ω^T

≤ M we obtain

0 < c

1

(1 − M t)

³

≤ det{x

_ξ

} ≤ c

₂

(1 + M t)

³

, t ≤ T,

where c

1

, c

2

> 0 are constants and T > 0 is sufficiently small. Moreover, det A = exp( R

t

0

∇

_u

u dt

⁰

) = %

0

/η.

Let S

t

be determined (at least locally) by the equation ϕ(x, t) = 0. Then S is described by ϕ(x(ξ, t), t)|

t=0

= ϕ(ξ) = 0. Thus, we have e

n(x(ξ, t), t) = − ∇

x

ϕ(x, t)

|∇

_x

ϕ(x, t)|

  



x=x(ξ,t)

and n

0

(ξ) = − ∇

ξ

ϕ(ξ) e

|∇

_ξ

ϕ(ξ)| e . Now, we are able to formulate the local existence theorem.

Theorem 3.1 (see [16], Theorem 3.7). Let S ∈ W

2^4−1/2

, f ∈ C

^2,1

(R

³

× [0, T ]), r ∈ C

^2,1

(R

³

× [0, T ]), θ

₁

∈ C

^2,1

(R

³

× [0, T ]), v

₀

∈ W

₂³

(Ω), θ

0

∈ W

₂³

(Ω), 1/θ

0

∈ L

_∞

(Ω), θ

0

> 0, %

0

∈ W

₂³

(Ω), 1/%

0

∈ L

_∞

(Ω), %

0

> 0, c

v

∈ C

²

(R

²+

), c

v

> 0, p ∈ C

³

(R

²+

). Moreover , assume that the following compatibility conditions are satisfied :

(3.2) D

_ξ^α

(D

ξ

(v

0

) · n

0

− p(%

₀

, θ

0

)n

0

) = −D

_ξ^α

(p

0

n

0

), |α| ≤ 1, on S and

(3.3) D

^α_ξ

(n

0

· ∇

ξ

θ

0

) = D

_ξ^α

(θ

1

(ξ, 0)), |α| ≤ 1, on S.

Let T

^∗

> 0 be so small that 0 < c

1

(1 − CK

0

T

^∗

)

³

≤ det{x

_ξ

} ≤ c

₂

(1 + CK

0

T

^∗

)

³

(where x(ξ, t) = ξ + R

t

0

u

0

(ξ, t

⁰

) dt

⁰

for t ≤ T

^∗

, u

0

is given by (3.74) of [16], K

0

≤ c(k%

₀

k

_3,Ω

+ |%

0

|

_∞,Ω

+ |1/%

0

|

_∞,Ω

+ kv

0

k

_3,Ω

+ kθ

0

k

_3,Ω

+ ku

t

(0)k

1,Ω

+ kΓ

t

(0)k

1,Ω

), c > 0 is a constant , C = C(K

0

) is a nondecreasing continuous function of K

0

satisfying (3.94) of [16]). Then there exists T

^∗∗

with 0 <

T

^∗∗

≤ T

^∗

such that for T ≤ T

^∗∗

there exists a unique solution (u, Γ, η) ∈ W

₂^4,2

(Ω

^T

) × W

₂^4,2

(Ω

^T

) × C

⁰

(0, T ; Γ

₀^3,3/2

(Ω)) of problem (3.1). Moreover , η

t

∈ C

⁰

(0, T ; W

₂²

(Ω)) ∩ L

2

(0, T ; W

₂³

(Ω)), η

tt

∈ L

₂

(0, T ; W

₂¹

(Ω)) and

kuk

_4,Ω^T

+ kΓ k

4,Ω^T

≤ CK

₀

, sup

t

kηk

_3,Ω

+ sup

t

kη

_t

k

_2,Ω

+ kη

t

k

_{L2(0,T ;W}³

2(Ω))

+ kη

tt

k

_{L2(0,T ;W}¹

2(Ω))

≤ Φ

1

(T, T

^a

K

0

)k%

0

k

3,Ω

, (3.4)

|1/η|

_∞,ΩT

+ |η|

_∞,Ω^T

≤ Φ

₂

(T

^1/2

K

0

)|1/%

0

|

_∞,Ω

+ Φ

3

(T

^1/2

K

0

)|%

0

|

_∞,Ω

,

where Φ

1

, Φ

2

and Φ

3

are increasing continuous functions, and a > 0.

In order to consider the global existence we need R e m a r k 3.2. Assume that g = 0 and define

p

σ

= p − p

0

, γ

0

= Γ − θ

e

, η

σ

= η − %

e

(where θ

e

and %

e

are introduced in Definition 1.1). Then problem (3.1) can be written in the form

(3.5)

ηu

t

− µ∇

²_u

u − ν∇

u

∇

_u

· u + ∇

_u

p

σ

= 0, η

σt

+ η∇

u

· u = 0,

ηc

v

(η, Γ )γ

0t

− κ∇

²_u

γ

0

+ Γ p

Γ

(η, Γ )∇

u

· u

− µ 2

3

X

i,j=1

(ξ

xi

∇

ξ

u

j

+ ξ

xj

∇

ξ

u

i

)

²

− (ν − µ)(∇

u

· u)

²

= ηk, T

u

(u, p

σ

) · n = 0,

n · ∇

u

γ

0

= Γ

1

,

u|

t=0

= v

0

, η

σ

|

_t=0

= %

σ0

, γ

0

|

_t=0

= ϑ

00

, where %

σ0

= %

0

− %

_e

and ϑ

00

= θ

0

− θ

_e

.

Let the assumptions of Theorem 3.1 be satisfied and let (u, Γ, η) be the corresponding local solution of problem (3.1). Then by Theorems 3.5, 3.6 and Lemma 3.3 of [16] for a solution (u, γ

0

, η

σ

) of (3.5) such that

T

^a

(kuk

_4,Ω^T

+ kv

0

k

_3,Ω

+ kϑ

00

k

_3,Ω

+ k%

σ0

k

_3,Ω

)ϕ

1

(T, K

0

) ≤ δ

(where a > 0 is a constant, ϕ

1

is an increasing continuous function of its arguments, δ > 0 is sufficiently small) the following estimate holds:

(3.6) kuk

_4,Ω^T

+ kη

σ

k

_3,Ω^T

+ η

_{σ 3,0,∞,Ω}^T

+ kγ

0

k

_4,Ω^T

≤ ϕ

₂

(T, K

0

)(kv

0

k

_3,Ω

+ k%

σ0

k

_3,Ω

+ kϑ

00

k

_3,Ω

+ kkk

_2,Ω^T

+ kk(0)k

1,Ω

+ kΓ

1

k

_3−1/2,ST

+ D

²_ξ,t

Γ

_{1 1/4,S}^T

), where ϕ

2

is an increasing continuous function of its arguments.

4. Differential inequality. In order to prove the global existence of solutions we need the differential inequality derived in [19] (Theorem 3.13).

Assume that the existence of a sufficiently smooth local solution of problem (1.1) has been proved and consider the motion near the constant state (see Definition 1.1) v

e

= 0, p

e

= p

0

, θ

e

= (1/|Ω|) R

Ω

θ

0

dξ and %

e

, where %

e

is a solution of the equation

(4.1) p(%

e

, θ

e

) = p

0

.

Let

(4.2) p

σ

= p − p

0

, %

σ

= % − %

e

, ϑ

0

= θ − θ

e

, ϑ = θ − θ

Ωt

, %

_Ωt

= % − %

Ωt

,

where θ

Ωt

= (1/|Ω

t

|) R

Ωt

θ dx, and %

Ωt

= %

Ωt

(t) is a solution of the problem (4.3) p(%

Ωt

, θ

Ωt

) = p

0

, %

Ωt

|

_t=0

= %

e

.

Then problem (1.1) takes the form

(4.4)

%[v

t

+ (v · ∇)v] − div T(v, p

σ

) = %f in Ω

t

, t ∈ [0, T ],

%

t

+ div(%v) = 0 in Ω

t

, t ∈ [0, T ],

%c

v

(%, θ)(ϑ

0t

+ v · ∇ϑ

0

) + θp

θ

(%, θ) div v

− κ∆ϑ

₀

− µ 2

X

i,j

(∂

xi

v

j

+ ∂

xj

v

i

)

²

− (ν − µ)(div v)

²

= %r inΩ

t

, t ∈ [0, T ], T(v, p

σ

) · n = 0 on S

t

, t ∈ [0, T ],

∂ϑ

0

/∂n = θ

1

on S

t

, t ∈ [0, T ],

where T(v, p

σ

) = {µ(∂

xi

v

j

+ ∂

xj

v

i

) + (ν − µ)δ

ij

div v − p

σ

δ

ij

} and T is the time of the local existence.

Define

ϕ(t) = R

Ωt

% X

1≤|α|+i≤3

|D

_x^α

∂

_tⁱ

v|

²

dx

+ R

Ωt

 p

1

% %

²_σ

+ %

²_Ωt

+ p

2

%c

v

p

θ

θ ϑ

²₀

 dx

+ R

Ωt

p

σ%

%

X

1≤|α|+i≤3

|D

^α_x

∂

_tⁱ

%

σ

|

²

dx

+ R

Ωt

%c

v

θ

X

1≤|α|+i≤3

|D

_x^α

∂

_tⁱ

ϑ

0

|

²

dx,

ϕ(t) = |v|

²_3,0,Ωt

+ |ϑ

0

|

²_3,0,Ωt

+ |%

σ

|

²_3,0,Ωt

+ k%

_Ωt

k

²_0,Ωt

, (4.5)

Φ(t) = |v|

²_4,1,Ωt

+ |ϑ

0

|

²_4,1,Ωt

− kϑ

₀

k

²_0,Ωt

+ kϑk

²_0,Ωt

+ |%

σ

|

²_3,0,Ω

t

− k%

_σ

k

²_0,Ω

t

+ k%

_Ωt

k

²_0,Ω

t

, F (t) = kf

ttt

k

²_0,Ωt

+ |f |

²_2,0,Ωt

+ kr

ttt

k

²_0,Ωt

+ |r|

²_2,0,Ωt

+ krk

0,Ωt

+ |θ

1

|

²_4,1,Ωt

+ kθ

1

k

_1,Ωt

, ψ(t) = kvk

²_0,Ωt

+ kp

σ

k

²_0,Ω

t

.

The following theorem is proved in [19] (see Theorem 3.13).

Theorem 4.1. Let ν >

¹₃

µ. Then for a sufficiently smooth solution (v, ϑ

0

, %) of (4.4) we have

dϕ

dt + c

0

Φ ≤ c

1

P (ϕ)

 ϕ +

t

R

0

kvk

²_4,Ω

t0

dt

⁰



(1 + ϕ

³

)(ϕ + Φ) (4.6)

+ c

2

F + c

3

ψ,

where P is an increasing continuous function; 0 < c

0

< 1 is a constant de- pending on %

^∗

, %

∗

, θ

^∗

, θ

∗

, µ, ν, κ; and c

i

(i = 1, 2, 3) are positive constants depending on %

∗

, %

^∗

, θ

∗

, θ

^∗

, R

t

0

kvk

_3,Ωt0

dt

⁰

, kSk

_4−1/2

, T and the constants from the imbedding Lemma 2.1 and the Korn inequalities from [20] (Sec- tion 5).

R e m a r k 4.2. Theorem 3.13 of [20] was proved under the assumption that ν ≥ µ. This assumption implies that

(4.7) µ

2 E

Ωt

(v) + (ν − µ)kdiv vk

²_0,Ωt

≥ 0, where E

Ωt

(v) = R

Ωt

P

3

i,j=1

(v

i,xj

+ v

j,xi

)

²

.

It turns out that the condition ν ≥ µ is too restrictive and we can now show that (4.7) is satisfied for ν >

¹₃

µ, which is assumed in Theorem 4.1. In fact, we have

µ

2 E

Ωt

(v) + (ν − µ)kdiv vk

²_0,Ωt

= µ 2

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx + (ν − µ) R

Ωt

(div v)

²

dx

= µ 2

X

i6=j

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx + µ 2

X

i=j

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx

+ (ν − µ) R

Ωt

(div v)

²

dx

= µ 2

X

i6=j

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx

+ µ 2 ε

1

X

i=j

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx

+ µ

2 (1 − ε

1

) · 4 X

i

R

Ωt

(v

i,xj

)

²

dx

+ (ν − µ) R

Ωt

(div v)

²

dx ≡ I,

where ε

1

∈ (0, 1).

Since (ξ

1

+ξ

2

+ξ

3

)

²

≤ 3(ξ

₁²

+ξ

₂²

+ξ

₃²

) the last two terms in I are estimated from below by



ν − (1 + 2ε

1

) µ 2



R

Ωt

(div v)

²

dx.

Assuming that ν = (1 + 2ε

1

)µ/3 we obtain ε

1

=

_2µ³

(ν − µ/3), so I ≥ µ

2 ε

1

R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx = 3 4

 ν − µ

3



R

Ωt

(v

i,xj

+ v

j,xi

)

²

dx

> 0 for ν >

¹₃

µ.

5. Global existence. We assume that

(5.1) f = 0, θ

1

≥ 0,

and

(5.2) krk

²_0,Ω

t

+ |r|

²_2,0,Ωt

+ krk

0,Ωt

+ |θ

1

|

²_4,1,Ω

t

+ kθ

1

k

_1,Ωt

≤ η

₁

e

^−η2t

, where η

1

> 0 is sufficiently small and η

2

> 1.

Let ϕ(t) and Φ(t) be defined by (4.5). We introduce the spaces N(t) = {(v, ϑ

0

, %

σ

, %

_Ωt

) : ϕ(t) < ∞},

M(t) = n

(v, ϑ

0

, ϑ, %

σ

, %

_Ωt

) : ϕ(t) +

t

R

0

Φ(t

⁰

) dt

⁰

< ∞ o .

Notice that (v, ϑ

0

, %

σ

, %

_Ωt

) ∈ N(t) iff ϕ(t) < ∞ and (v, ϑ

0

, ϑ, %

σ

, %

_Ωt

) ∈ M(t) iff ϕ(t) + R

t

0

Φ(t

⁰

) dt

⁰

< ∞. Moreover, c

⁰

ϕ(t) ≤ ϕ(t) ≤ c

⁰⁰

ϕ(t), where c

⁰

, c

⁰⁰

> 0 are constants.

Lemma 5.1. Let the assumptions of Theorem 3.1 be satisfied. Let the initial data v

0

, %

0

, θ

0

, S of problem (1.1) be such that (v, ϑ

0

, %

σ

, %

_Ωt

) ∈ N(0) and S ∈ W

₂^4−1/2

. Let

R

Ω

%

0

v

0

dξ = 0, R

Ω

%

0

ξ dξ = 0.

Moreover , assume

(5.3) ϕ(0) ≤ ε

1

,

where ε

1

is sufficiently small. Then the local solution (v, θ, %) of problem (1.1 ) is such that (v, ϑ

0

, ϑ, %

σ

, %

_Ωt

) ∈ M(t) for t ≤ T , where T is the time of local existence and

(5.4) ϕ(t) +

t

R

0

Φ(t

⁰

) dt

⁰

≤ cε

₁

.

P r o o f. Take (v, ϑ

0

, %

σ

, %

_Ωt

) ∈ N(0), S ∈ W

₂^4−1/2

. Then (v

0

, ϑ

00

, %

σ0

) ∈ W

₂³

(Ω) (%

σ0

= %

0

− %

e

, ϑ

00

= θ

0

− θ

e

) and by Theorem 3.1, Remark 3.2 and (5.2) there exists a solution of problem (1.1) such that

(5.5) u ∈ W

₂^4,2

(Ω

^T

), ϑ

0

∈ W

₂^4,2

(Ω

^T

), η

σ

∈ W

₂^3,3/2

(Ω

^T

) ∩ C

⁰

(0, T ; Γ

₀^3,3/2

(Ω)) and

(5.6) kuk

²_4,ΩT

+ kη

σ

k

²_3,ΩT

+ η

σ 2

3,0,∞,Ω^T

+ kγ

0

k

²_4,ΩT

≤ c(kv

₀

k

²_3,Ω

+ k%

σ0

k

²_3,Ω

+ kϑ

00

k

²_3,Ω

) ≤ cϕ(0) ≤ cε

1

, where u = v(x(ξ, t), t), η

σ

= %

σ

(x(ξ, t), t), γ

0

= ϑ

0

(x(ξ, t), t).

Using estimate (5.6) for the local solution and the imbeddings (see Lem- mas 2.2 and 2.1)

sup

t

(kuk

²_3,Ω

+ ku

t

k

²_1,Ω

) ≤ c(kuk

²_4,ΩT

+ ku(0)k

²_3,Ω

+ |u(0)|

²_1,0,Ω

) ≤ cϕ(0) ≤ cε

1

and

t

R

0

|u

_ξ

|

_∞,Ω

dt

⁰

≤ cT

^1/2

kuk

_4,ΩT

≤ cT

^1/2

ϕ(0)

we have the following estimate for the solution η

σ

of (3.5)

2

(see [22], Lemma 6.1):

N

1

≡ sup

t

(kη

σtt

k

²_0,Ω

+ kη

σt

k

²_2,Ω

+ kη

σ

k

²_3,Ω

) (5.7)

+ kη

σtt

k

²_L

2(0,T ;W₂¹(Ω))

+ kη

σt

k

²_L

2(0,T ;W₂³(Ω))

≤ ϕ

1

(T, ϕ(0)) ≤ cε

1

,

where ϕ

1

is an increasing continuous function of its arguments.

Repeating the proof of Lemma 3.10 of [19] we get (5.8) 1

2 d dt

R

Ωt



%v

²_xxt

+ p

σ%

% %

²_σxxt

+ %c

v

θ ϑ

²_0xxt

 dx + C(kv

xxt

k

²_1,Ωt

+ k%

σxxt

k

²_0,Ωt

+ kϑ

0xxt

k

²_1,Ωt

)

≤ (ε

⁰₁

+ cN )(kv

xttt

k

²_0,Ωt

+ kv

xxxt

k

²_0,Ωt

+ kv

xxtt

k

²_0,Ωt

+ kϑ

0xttt

k

²_0,Ωt

+ kϑ

0xxtt

k

²_0,Ω

t

+ kϑ

0xxxt

k

²_0,Ω

t

) + cM (1 + N )

²

+ cF (t),

where C, c > 0 are constants, N = N

1

+ N

2

, N

2

= sup

_t

(kuk

²_3,Ω

+ ku

t

k

²_1,Ω

+ kγ

₀

k

²_3,Ω

+ kγ

0t

k

²_1,Ω

) and M is such that R

T

0

M dt

⁰

≤ cϕ(0) holds in view of

the estimates for the local solution.

Similarly, using Lemma 3.11 of [19] yields (5.9) 1

2 d dt

R

Ωt



%v

²_xxt

+ p

σ%

% %

²_σxxt

+ %c

v

θ ϑ

²_0xxt

 dx + C(kv

xtt

k

²_1,Ωt

+ k%

σxtt

k

²_0,Ωt

+ kϑ

0xtt

k

²_1,Ωt

)

≤ (ε

⁰₂

+ cN )(kv

xttt

k

²_0,Ωt

+ kv

xxtt

k

²_0,Ωt

+ kv

xxxt

k

²_0,Ωt

+ kϑ

0xttt

k

²_0,Ωt

+ kϑ

0xxtt

k

²_0,Ωt

+ kϑ

0xxxt

k

²_0,Ωt

) + c(1 + N )

²

× (kv

xxt

k

²_1,Ωt

+ kϑ

0xxt

k

²_1,Ωt

) + cM (1 + N )

²

+ cF (t).

Next, Lemma 3.12 of [19] implies (5.10) 1

2 d dt

R

Ωt



%v

_ttt²

+ p

σ%

% %

²_σttt

+ %c

v

θ ϑ

²_0ttt

 dx

+ C(kv

ttt

k

²_1,Ωt

+ k%

σttt

k

²_0,Ωt

+ kϑ

0ttt

k

²_1,Ωt

)

≤ (ε

⁰₃

+ N + M )(kv

ttt

k

²_0,Ωt

+ kϑ

0ttt

k

²_0,Ωt

)

+ cN (kv

xtt

k

²_1,Ωt

+ kv

xttt

k

²_0,Ωt

+ kv

xxt

k

²_1,Ωt

+ kϑ

0xtt

k

²_1,Ωt

+ kϑ

0xttt

k

²_0,Ωt

+ kϑ

0xxt

k

²_1,Ωt

+ k%

σttt

k

²_0,Ωt

) + ckv

xtt

k

²_1,Ωt

+ ckϑ

0xtt

k

²_1,Ω

t

+ cM (1 + N )

²

+ cF (t), where in virtue of the continuity equation (4.4)

2

we have (5.11) k%

_σttt

k

²_0,Ωt

≤ c(1 + N )kv

_xtt

k

²_0,Ωt

+ cM (1 + N )

²

.

Finally, to estimate k%

_Ωt

k

²_0,Ω

t

+ kϑk

²_0,Ωt

rewrite the equation p(%

Ωt

, θ

Ωt

) − p(%

e

, θ

e

) = 0

using the Taylor formula as

p

%

(%

Ωt

− %

_e

) + p

θ

(θ

Ωt

− θ

_e

) = 0.

Hence (5.12) k%

_Ω

t

k

²_0,Ω

t

+ kϑk

²_0,Ωt

≤ k%

_σ

k

²_0,Ωt

+ k%

e

− %

_Ωt

k

²_0,Ωt

+ kϑ

0

k

²_0,Ωt

+ kθ

e

− θ

_Ωt

k

²_0,Ωt

≤ c(k%

_σ

k

²_0,Ω

t

+ kϑ

0

k

²_0,Ω

t

+ kθ

e

− θ

_Ωt

k

²_0,Ω

t

)

≤ c



k%

_σ

k

²_0,Ωt

+ kϑ

0

k

²_0,Ωt

+

1

|Ω

_t

|

R

Ωt

ϑ

0

dx

2

0,Ωt



≤ cϕ(0) ≤ cε

₁

,

where to estimate k%

σ

k

²_0,Ω

t

and kϑ

0

k

²_0,Ω

t

we have used (5.6).

From (5.8)–(5.12) and (5.1)–(5.2), for sufficiently small ε

⁰₁

, ε

⁰₂

, ε

⁰₃

, N , R

T

0

M dt

⁰

and η

1

from (5.2), we deduce that (v(t), %

σ

(t), %

_Ωt

(t), ϑ

0

(t), ϑ(t)) ∈ M(t) for t ≤ T and (5.4) is satisfied. Of course to prove the last statement the standard technique of mollifiers or differences should be used. This concludes the proof.

Lemma 5.2. Assume that there exists a local solution to problem (1.1) which belongs to M(t) for t ≤ T , i.e. let the assumptions of Lemma 5.1 be satisfied. Let the assumptions of Lemma 2.3 be satisfied. Then there exist δ

1

, δ

2

∈ (0, 1) sufficiently small such that

kp

σ

k

²_0,Ωt

≤ δ

1

, (5.13)

kϑ

₀

k

²_0,Ωt

+ k%

σ

k

²_0,Ωt

≤ δ

₂

, (5.14)

where δ

1

= cε

1

δ

⁰

+ c(δ

⁰

)e δ, δ

2

= cε

1

δ

⁰

+ c(δ

⁰

)(δ

0

+ e δ), δ

⁰

∈ (0, 1) is as small as needed , c(δ

⁰

) is a decreasing function of δ

⁰

, and δ

0

and e δ are taken from Lemma 2.3.

P r o o f. Estimate (5.13) can be proved in exactly the same way as esti- mate (6.13) in [20]. In order to prove (5.14) we use the relation

|Ω| − |Ω

e

| = 1

%

e

R

Ω

(%

e

− %

0

) dξ.

Hence, by assumption (2.8) of Lemma 2.3 we have (5.15) ||Ω| − |Ω

_e

|| ≤ cδ

₀

. Using (5.15) and Remark 2.4 we obtain (5.16) ||Ω

t

| − |Ω

e

|| ≤ cδ, where δ = δ(δ

0

) → 0 as δ

0

→ 0.

If % > %

e

then (1/%

e

) R

Ωt

|% − %

_e

| dx = |Ω

_e

| − |Ω

_t

|.

If % < %

e

then (1/%

e

) R

Ωt

|% − %

_e

| dx = |Ω

_t

| − |Ω

_e

|.

Therefore, from (5.16) it follows that

(5.17) R

Ωt

(% − %

e

)

²

dx ≤ c R

Ωt

|% − %

_e

| dx ≤ cδ.

Hence

(5.18) k%

_σ

k

_0,Ωt

≤ δ

⁰₂

. Using the Taylor formula we have

(5.19) p

σ

= p

1

%

σ

+ p

2

ϑ

0

,

where p

1

= p

1

(%, θ) and p

2

= p

2

(%

e

, θ) (see (3.4) of [19]). Now estimates (5.13), (5.18) and formula (5.19) yield

(5.20) kϑ

₀

k

_0,Ωt

≤ δ

⁰⁰₂

.

By (5.19) and (5.20) we get (5.14).

Lemma 5.3. Assume that there exists a local solution of (1.1) in M(t) for 0 ≤ t ≤ T . Let the assumptions of Lemma 2.3 be satisfied. Assume that the initial data are in N(0) and

(5.21) ϕ(0) ≤ γ, γ ∈ (0, 1/2],

where γ is sufficiently small. Then the solution at t ∈ [0, T ] belongs to N(t) and

(5.22) ϕ(t) ≤ γ.

P r o o f. Assumption (5.21) and Lemma 5.1 imply that the estimate (4.6) can be written as

(5.23) dϕ

dt + c

0

Φ ≤ c

⁰₁

 ϕ +

t

R

0

Φ(t

⁰

) dt

⁰



(1 + ϕ

³

)(ϕ + Φ) + c

2

F + c

3

ψ.

Since Φ + kϑ

0

k

²_0,Ω

t

+ k%

σ

k

²_0,Ω

t

≥ ϕ, using Lemma 5.2 we obtain

(5.24) Φ + δ

2

≥ ϕ,

where δ

2

is independent of γ.

Next, by Lemmas 5.2 and 2.3 and assumptions (5.1)–(5.2) we have

(5.25) F + ψ ≤ η + δ

1

+ ce δ,

where η = η

1

e

^−η2t

and δ

1

, e δ are sufficiently small. Using (5.24) in (5.23) gives

(5.26) dϕ

dt + c

0

Φ ≤ c

⁰₁

 ϕ +

t

R

0

Φ(t

⁰

) dt

⁰



(1 + ϕ

³

)(2Φ + δ

2

) + c

2

F + c

3

ψ.

Assuming that the initial data are so small that 2c

⁰₁

 ϕ +

t

R

0

Φ(t

⁰

) dt

⁰



(1 + ϕ

³

)Φ ≤ c

0

Φ 2 , instead of (5.26) we get

(5.27) dϕ

dt + c

0

Φ 2 ≤ c

⁰₁



ϕ +

t

R

0

Φ(t

⁰

) dt

⁰



(1 + ϕ

³

)δ

2

+ c

2

F + c

3

ψ.

Now using (5.24), (5.25), (5.21) and Lemma 5.1 in (5.27) yields

(5.28) dϕ

dt + c

0

Φ 2 ≤ c

₄



γδ

2

+ δ

2

2 + η + δ

1

+ ce δ

 .

By (5.21), ϕ(0) ≤ γ, γ ∈ (0, 1/2]. Assume that t

_∗

= inf{t ∈ [0, T ] : ϕ(t) >

γ}. Consider (5.28) in the interval [0, t

∗

]. From the definition of t

∗

we have

ϕ(t

∗

) = γ. Therefore (5.28) implies (5.29) ϕ

_t

(t

∗

) ≤ − γ

2 + c

4



γδ

2

+ δ

2

2 + η + δ

1

+ ce δ

 . Assume that δ

2

, η, δ

1

and e δ are so small that

c

4



γδ

2

+ δ

2

2 + η + δ

1

+ ce δ



< γ 4 .

Hence (5.29) yields ϕ

_t

(t

∗

) < 0, a contradiction. Therefore (5.22) holds.

Lemma 5.3 suggests that the solution can be continued to the interval [T, 2T ], but to do this we need the following facts:

(5.30) (a) The existence of the transformation x = x(ξ, t) and its inverse for t ∈ [T, 2T ].

(b) The validity of the Korn inequality with the same constant for the whole interval [0, 2T ].

(c) The variations of the shape of Ω

t

for t ∈ [0, 2T ] are so small that the constants in Lemma 2.1 (imbedding (2.1)) can be chosen independently of t.

Generally, to prove the global existence we need these facts for all t.

Theorem 2.7 of [18] implies that the volume of Ω

t

does not change much but we have not shown yet any restriction on the variations of its shape.

It is sufficient to show (c), because (a) and (b) follow.

Lemma 5.4. Assume that there exists a local solution of (1.1) in M(t) for 0 ≤ t ≤ T with initial data in N(0) sufficiently small (see (5.3)). Then there exist constants µ

1

> 0 and µ

2

> 0 (µ

2

is sufficiently small ) such that (5.31) ϕ(t) ≤ ce

^−µ1t

(ϕ(0) + µ

2

), t ≤ T,

where c > 0 is a constant and T is the time of local existence. Moreover , if we assume (5.2) with η

1

= 0, then (5.31) holds with µ

2

= 0.

P r o o f. Inequalities (3.20) and (3.28) of [19] (see the proof of Lemma 3.1 and the assertion of Lemma 3.2 of [19]) imply

(5.32) d dt

R

Ωt



%(v

²

+ v

²_t

) + 1

% (p

1

%

²_σ

+ p

%

%

²_σt

) + %c

v

θ

 p

₂

p

θ

ϑ

²₀

+ ϑ

²_0t



dx + c

0

(kvk

²_1,Ωt

+ kv

t

k

²_1,Ωt

+ kdiv vk

²_0,Ωt

+ kdiv v

t

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

+ kϑ

0t

k

²_1,Ωt

+ k%

σt

k

²_0,Ωt

)

≤ C

₂

ϕ

²

(t)(1 + ϕ(t)) + C

1

F (t).

Multiplying both sides of (5.32) by e

^−αt

(where 0 < α < 1) we obtain (5.33) d

dt

 e

^−αt

R

Ωt



%(v

²

+ v

_t²

) + 1

% (p

1

%

²_σ

+ p

%

%

²_σt

) + %c

v

θ

 p

2

p

θ

ϑ

²₀

+ ϑ

²_0t



dx



+ αe

^−αt

R

Ωt



%(v

²

+ v

²_t

)

+ 1

% (p

1

%

²_σ

+ p

%

%

²_σt

) + %c

v

θ

 p

2

p

θ

ϑ

²₀

+ ϑ

²_0t



dx

+ c

0

e

^−αt

(kvk

²_1,Ωt

+ kv

t

k

²_1,Ωt

+ kdiv vk

²_0,Ωt

+ kdiv v

t

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

+ kϑ

0t

k

²_1,Ωt

+ k%

σt

k

²_0,Ωt

)

≤ C

₂

e

^−αt

ϕ

²

(t)(1 + ϕ(t)) + C

1

e

^−αt

F (t).

Next multiplying inequality (4.6) by e

^−αt

we have (5.34) dϕ

₁

dt + αϕ

1

+ c

0

Φ

1

≤ c

₁

P (ϕ)  ϕ +

t

R

0

kvk

²_4,Ω

t0

dt

⁰



(1 + ϕ

³

)(ϕ

1

+ Φ

1

) + c

2

F

1

+ c

3

ψ

1

,

where ϕ

1

= ϕe

^−αt

, ϕ

₁

= ϕe

^−αt

, Φ

1

= Φe

^−αt

, F

1

= F e

^−αt

, and ψ

1

= ψe

^−αt

. From the assumption that the initial data are sufficiently small we de- duce that ϕ + R

t

0

kvk

²_4,Ω

t0

dt

⁰

is also small (see Lemma 5.1). Therefore, for sufficiently small data from (5.34) we get

(5.35) dϕ

₁

dt + c

4

(ϕ

1

+ Φ

1

) ≤ c

5

F

1

+ c

6

(kp

σ

k

²_0,Ω

t

+ kvk

²_0,Ωt

)e

^−αt

. Now, applying the same argument as in the proof of Lemma 6.2 of [20] we obtain

(5.36) kp

_σ

k

²_0,Ωt

≤ ε(kp

_σx

k

²_0,Ωt

+ kv

xx

k

²_0,Ωt

) + c(ε)(kvk

²_0,Ωt

+ kv

t

k

²_0,Ωt

).

Moreover,

(5.37) kp

_σx

k

²_0,Ωt

≤ c(k%

_σx

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

).

Using (5.36) and (5.37) in (5.35) we have (5.38) dϕ

₁

dt + c

4

(ϕ

1

+ Φ

1

) ≤ c

5

F

1

+ c

7

(kvk

²_0,Ωt

+ kv

t

k

²_0,Ωt

)e

^−αt

.

Multiplying (5.33) by a sufficiently large constant c

8

, adding the result to

(5.38) and using the fact that ϕ(0) is sufficiently small we obtain

(5.39) de ϕ

dt + c

9

( ϕ + e e Φ) ≤ c

10

F

1

, where

ϕ = ϕ e

1

+ c

8

e

^−αt

R

Ωt



%(v

²

+ v

²_t

) + 1

% (p

1

%

²_σ

+ p

σ

%

²_σt

) + %c

v

θ

 p

2

p

θ

ϑ

²₀

+ ϑ

²_0t



dx,

ϕ = ϕ e

₁

+ c

8

e

^−αt

R

Ωt



%(v

²

+ v

²_t

) + 1

% (p

1

%

²_σ

+ p

σ

%

²_σt

) + %c

v

θ

 p

2

p

θ

ϑ

²₀

+ ϑ

²_0t



dx,

Φ = Φ e

1

+ c

8

e

^−αt

(kvk

²_1,Ωt

+ kv

t

k

²_1,Ωt

+ k div vk

²_0,Ωt

+ kdiv v

t

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

+ kϑ

0t

k

²_1,Ωt

+ k%

σt

k

²_0,Ωt

).

There exist constants c

⁰₀

, c

⁰⁰₀

> 0 such that

c

⁰₀

ϕ

₁

≤ ϕ e ≤ c

⁰⁰₀

ϕ

₁

and c

⁰₀

Φ

1

≤ e Φ ≤ c

⁰⁰₀

Φ

1

. Hence by assumptions (5.1) and (5.2) inequality (5.39) implies (5.40) ϕ

₁

≤ c

₁₂

e

^−c10t

(ϕ(0) + c

11

),

where c

11

> 0 is sufficiently small.

For α sufficiently small, from (5.40) we obtain (5.31).

Finally, we prove the main result of this paper.

Theorem 5.5. Let ν >

¹₃

µ. Let (5.1), (5.2) with η

1

= 0 and the as- sumptions of Theorem 3.1 with r ∈ C

_B^2,1

(R

³

× [0, +∞)) and θ

₁

∈ C

_B^2,1

(R

³

× [0, +∞)) be satisfied. Furthermore, let (v, ϑ

0

, %

σ

, %

_Ωt

) ∈ N(0) and

(5.41) ϕ(0) ≤ ε

1

,

where ε

1

∈ (0, 1) is sufficiently small. Let the following compatibility condi- tions be satisfied :

(5.42) D

^α

∂

_tⁱ

(T · n + p

0

n)|

t=0,S

= 0, |α| + i ≤ 2, D

^α

∂

_tⁱ

(n · ∇θ − θ

1

)|

t=0,S

= 0, |α| + i ≤ 2.

Assume also that the internal energy per unit mass ε = ε(%, θ) has the form

(2.3) and conditions (2.4)–(2.5) hold. Moreover , assume that

R

Ω

%

0

v

₀²

2 dξ + κ sup

t t

R

0

dt

⁰

R

S_t0

θ

1

(s, t

⁰

) ds (5.43)

+ R

Ω

%

0

(h(%

0

, θ

0

) − h

∗

) dξ ≤ ε

2

,

R

Ω

|%

₀

− %

_e

| dξ ≤ ε

₂

, (5.44)

(β − 1)

^β−1

β

^β

p

^β−1₀

(a

0

%

^β_e

+ p

0

)

^β

− a

₀

%

^β_e

≤ ε

₂

, (5.45)

a

0



R

Ω

%

^β₀

dx − M

^β

(βp

0

)

^β−1

[cδβ%

0

+ (d − h

∗

M )(β − 1)]

^β−1



≤ ε

₂

, (5.46)

where ε

2

> 0 is a sufficiently small constant , β = α + 1, and c > 0 and δ > 0 are the constants from Remark 2.6. Assume, finally, that

(5.47) R

Ω

%

0

v

0

(a + b × ξ) dξ = 0, R

Ω

%

0

ξ dξ = 0, R

Ω

%

0

dξ = M, where a, b are arbitrary constant vectors. Then there exists a global solution of (1.1) such that (v, ϑ

0

, ϑ, %

σ

, %

_Ωt

) ∈ M(t) for t ∈ R

¹+

and S

t

∈ W

₂^4−1/2

.

P r o o f. The theorem is proved step by step using the local existence in a fixed time interval. Under the assumption that

(5.48) (v, ϑ

0

, %

σ

, %

_Ωt

) ∈ N(0),

Theorem 3.1 and Remark 3.2 yield the local existence of solutions of (1.1) such that

(5.49) u ∈ W

₂^4,2

(Ω

^T

), ϑ

0

∈ W

₂^4,2

(Ω

^T

), η

σ

∈ W

₂^3,3/2

(Ω

^T

) ∩ C

⁰

(0, T ; Γ

₀^3,3/2

(Ω)),

where T is the time of the existence. By (5.48) and (5.49), Lemma 5.1 implies that the local solution belongs to M(t) for t ≤ T . For small ε

1

the existence time T is correspondingly large, so we can assume it is a fixed positive number.

To prove the last result we needed the Korn inequalities (see [20]) and Lemma 2.1 (imbedding (2.2)).The constants in those theorems depend on Ω

t

and the shape of S

t

, so generally they are functions of t. In view of (5.41), Lemma 5.1 gives

ϕ(t) +

t

R

0

Φ(t

⁰

) dt

⁰

≤ cε

₁

.