On a differential inequality for equations of a viscous compressible heat conducting fluid bounded

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POLONICI MATHEMATICI LXI.2 (1995)

On a differential inequality for equations of a viscous compressible heat conducting fluid bounded

by a free surface

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

Abstract. We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat conducting fluid. The inequality is essential in proving the global existence of solutions.

1. Introduction. The aim of this paper is to derive a global differential inequality for the following free boundary problem for a viscous compressible heat conducting fluid (see [2], 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

−

¹₂

µ

3

X

i,j=1

(v

i,xj

+ v

j,xi

)

²

− (ν − µ)(div v)

²

= %r in e Ω

^T

,

Tn = −p

0

n on e S

^T

,

v · n = −φ

t

/|∇φ| on e S

^T

,

∂θ/∂n = θ

1

on e S

^T

,

v|

t=0

= v

0

, %|

t=0

= %

0

, θ|

t=0

= θ

0

in Ω, where e Ω

^T

= S

t∈(0,T )

Ω

t

× {t}, Ω

t

is a bounded domain of the drop at time t and Ω

0

= Ω is its initial domain, e S

^T

= S

t∈(0,T )

S

t

× {t}, S

_t

= ∂Ω

t

, φ(x, t) = 0 describes S

t

, and n is the unit outward vector normal to the boundary (i.e. n = ∇φ/|∇φ|).

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

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

[141]

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Moreover, v = v(x, t) is the velocity of the fluid, % = %(x, t) the density, θ = θ(x, t) the temperature, 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, µ and ν the viscosity coefficients, κ the coefficient of heat conductivity, c

v

= c

v

(%, θ) the specific heat at constant volume, and p

0

the external (constant) pressure.

We assume that c

v

> 0, the coefficients µ, ν, κ are constants, and κ > 0, ν ≥ 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

∂x

∂t = v(x, t), x|

t=0

= ξ ∈ Ω, ξ = (ξ

1

, ξ

2

, ξ

3

).

Therefore, the transformation x = x(ξ, t) connects the Eulerian x and the Langrangian ξ coordinates of the same fluid particle. Hence

(1.2) 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}.

By the continuity equation (1.1)

2

and (1.1)

5

the total mass of the drop is conserved and the following relation holds between % and Ω

t

:

R

Ωt

%(x, t) dx = M.

This paper is divided into three sections. In Section 2 we introduce some notation. In Section 3 we derive the main result of the paper, i.e. the differential inequality (3.160) (see Theorem 3.13) which is essential in proving the global existence of a solution of problem (1.1) (see [19]). In order to obtain inequality (3.160) we impose the following assumptions:

a) there exists a sufficiently smooth local solution;

b) the transformation (1.2) together with its inverse exist;

c) the volume and the shape of the domain do not change much in time.

Papers concerning problem (1.1) include [15]–[17] and [20]. In [15] the

local-in-time existence and uniqueness of solution to problem (1.1) in the

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Sobolev–Slobodetski˘ı spaces is proved. In [17] we prove that under an appro- priate 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), var

t

|Ω

t

| is as small as we need. Paper [20] contains the global existence theorem for problem (1.1). In [15], [18], [19], [21] we consider the motion of a viscous compressible heat conducting fluid bounded by a free surface governed by surface tension. Such a motion is described by equations (1.1)

1

–(1.1)

3

with conditions (1.1)

5

–(1.1)

7

and with the condition

(1.3) Tn − σHn = −p

0

n

replacing (1.1)

4

. In (1.3), σ is the constant coefficient of surface tension, and H is the double mean curvature of S

t

.

Similarly to the case σ = 0, in [15] the local motion of a capillary fluid (the case σ 6= 0) is considered, while [18], [19] and [21] give, in that case, analogous to those of [17], the present paper and [20], respectively. In [18]

conservation laws and global estimates for equations (1.1)

1

–(1.1)

3

with conditions (1.3) and (1.1)

5

–(1.1)

7

are presented. Moreover, we prove in [18] that we can choose %

0

, v

0

, θ

0

, θ

1

, p

0

, κ, σ and the form of the internal energy per unit mass ε = ε(%, θ) such that var

t

|Ω

_t

| is as small as we need. This result is used in [21] to prove the global-in-time existence of solutions to problem (1.1)

1

–(1.1)

3

, (1.3), (1.1)

5

–(1.1)

7

. Paper [19] is devoted to a differential inequality for problem (1.1)

1

–(1.1)

3

, (1.3), (1.1)

5

–(1.1)

7

which is analogous to inequality (3.160). In [21] the global existence theorem for problem (1.1)

1

– (1.1)

3

, (1.3), (1.1)

5

–(1.1)

7

is proved. Finally, [16] contains the review of all results from [17]–[21] including the main result proved in this paper.

The motion of a viscous compressible heat conducting fluid in a fixed domain was considered by A. Matsumura and T. Nishida [3]–[7], A.Valli [13], and A. Valli and W. M. Zaj¸ aczkowski [14]. Papers [3] and [4] are concerned with the initial value problem for equations (1.1)

1

–(1.1)

3

considered in R

³

× (0, ∞). In [4] the existence and uniqueness of a global-in-time clas- sical solution of system (1.1)

1

–(1.1)

3

is proved for the initial conditions (1.4) v|

t=0

= v

0

, %|

t=0

= %

0

, θ|

t=0

= θ

0

in R

³

.

The solution is obtained in a neighbourhood of a constant state (v, %, θ) = (0, %, θ), where % and θ are positive constants. In [3] the same type of result is obtained for a polytropic gas, i.e. under the assumption that ε = c

v

θ, where ε is the internal energy. In [7] the global existence theorem is proved for system (1.1)

1

–(1.1)

3

considered in Ω × (0, ∞) (where Ω is a halfspace or an exterior domain of any bounded region with smooth boundary) with initial conditions (1.4) and with the boundary conditions of Dirichlet or Neumann type. Papers [5], [6], [13] and [14] are concerned with the global motion of a viscous compressible heat conducting fluid in a bounded domain Ω ⊂ R

³

.

For a compressible barotropic fluid (i.e. when the temperature of the

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fluid is constant) the problem corresponding to (1.1) has been examined by W. M. Zaj¸ aczkowski [22]–[25] and V. A. Solonnikov and A. Tani [12]. In [23]–[24] the local motion of a compressible barotropic fluid bounded by a free surface is considered, while [22], [25] and [12] are devoted to the global motion of such a fluid.

In [8] K. Pileckas and W. M. Zaj¸ aczkowski proved the existence of a stationary motion of a viscous compressible barotropic fluid bounded by a free surface governed by surface tension.

Finally, papers of V. A. Solonnikov [9]–[11] concern free boundary prob- lems for viscous incompressible fluids. In the case of an incompressible fluid

% = const, so the continuity equation (1.1)

2

reduces to

(1.5) div v = 0.

Therefore, the problem examined by V. A. Solonnikov [9]–[11] is described by the Navier–Stokes equations (1.1)

1

(where p = p(x, t)) and by (1.5) with the initial condition v|

t=0

= v

0

and with the boundary condition being either (1.1)

4

or (1.3).

2. Notation. Let Q = Ω

t

or Q = S

t

(t ≥ 0). By k · k

l,Q

(l ≥ 0) and

| · |

_p,Q

(1 ≤ p ≤ ∞) we denote the norms in the usual Sobolev spaces W

₂^l

(Q) and in the L

p

(Q) spaces, respectively.

Next, we introduce the space Γ

_k^l

(Ω) of functions u with the norm kuk

_Γl

k(Ω)

= X

i≤l−k

k∂

ⁱ_t

uk

l−i,Ω

≡ |u|

_l,k,Ω

, where l > 0, k ≥ 0.

In the sequel we shall use the following notation for derivatives of u.

If u is a scalar-valued function we denote by D

_x,t^k

u or u

x...x t...t

| {z }

k times

the vector (D

_x^α

∂

_tⁱ

u)

_|α|+i=k

.

Similarly, if u = (u

1

, u

2

, u

3

) we denote by D

_x,t^k

u or u

x...x t...t

| {z }

k times

the vector (D

_x^α

∂

_tⁱ

u

j

)

|α|+i=k,j=1,2,3

. Hence |D

^k_x,t

u| = P

|α|+i=k

|D

^α_x

∂

_tⁱ

u|.

We use the following lemma.

Lemma 2.1. The following imbedding holds: W

r^l

(Q) ⊂ L

^α_p

(Q) (Q ⊂ R

³

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

^αp

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

^α_x

u|

p,Ω

< ∞, and W

_r^l

(Q) is the Sobolev space.

Moreover , the following interpolation inequalities hold :

|D

_x^α

u|

p,Q

≤ cε

^1−κ

|D

^l_x

u|

r,Q

+ cε

^−κ

|u|

_r,Q

,

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

|D

^α_x

u|

q,∂Q

≤ cε

^1−κ

|D

_x^l

u|

r,Q

+ cε

^−κ

|u|

_r,Q

,

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where κ = |α|/l + 3/(lr) − 2/(lq) < 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 [1].

3. Global differential inequality. Assume that the existence of a sufficiently smooth local solution of problem (1.1) has been proved. To show the differential inequality we consider the motion near the constant state v

e

= 0, p

e

= p

0

, θ

e

= θ

0

=

_|Ω|¹

R

Ω

θ

0

dξ and %

e

, where %

e

is a solution of the equation

(3.1) p(%

e

, θ

e

) = p

0

.

Let

(3.2) p

σ

= p − p

0

, %

σ

= % − %

0

, ϑ

0

= θ − θ

e

, ϑ = θ − θ

Ωt

, where

θ

Ωt

= 1

|Ω

_t

|

R

Ωt

θ dx.

Then problem (1.1) takes the form

(3.3)

%[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

− 1 2 µ X

i,j

(∂

xi

v

j

+ ∂

xj

v

i

)

²

− (ν − µ)(div v)

²

= %r in Ω

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 local existence.

In the sequel we shall use the following Taylor formula for p

σ

: p

σ

= p(%, θ) − p(%

e

, θ

e

) = p(%, θ) − p(%

e

, θ) + p(%

e

, θ) − p(%

e

, θ

e

) (3.4)

= (% − %

e

)

1

R

0

p

%

(%

e

+ s(% − %

e

), θ) ds

+ (θ − θ

e

)

1

R

0

p

θ

(%

e

, θ

e

+ s(θ − θ

e

)) ds ≡ p

1

%

σ

+ p

2

ϑ

0

.

We shall also use the formula

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p

σ

= p(%, θ) − p(%

Ωt

, θ

Ωt

) (3.5)

= (% − %

Ωt

)

1

R

0

p

%

(%

Ωt

+ s(% − %

Ωt

), θ) ds

+ (θ − θ

Ωt

)

1

R

0

p

θ

(%

Ωt

, θ

Ωt

+ s(θ − θ

Ωt

)) ds ≡ p

3

%

_Ω_t

+ p

4

ϑ, where the function %

Ωt

= %

Ωt

(t) is a solution of the problem

(3.6) p(%

Ωt

, θ

Ωt

) = p

0

, %

Ωt

|

_t=0

= %

e

and

(3.7) %

_Ω_t

= % − %

Ωt

.

The functions p

i

(i = 1, 2, 3, 4) in (3.4) and (3.5) are positive and p

1

= p

1

(%, θ), p

2

= p

2

(%

e

, θ), p

3

= p

3

(%

Ωt

, %, θ), p

4

= p

4

(%

Ωt

, θ

Ωt

, θ).

Now we point out the following facts concerning the estimates in Lemmas 3.1–3.12 and Theorem 3.13:

• By ε we denote small constants and for simplicity we do not distinguish them.

• By C

₁

and C

2

we denote constants which depend on %

∗

, %

^∗

, θ

∗

, θ

^∗

, T , R

T

0

kvk

²_3,Ω

t0

dt

⁰

, kSk

4−1/2

, on the parameters which guarantee the existence of the inverse transformation to x = x(ξ, t) and also the constants of the imbedding theorems and the Korn inequalities. C

1

is always the coefficient of a linear term, while C

2

is the coefficient of a nonlinear term. For simplicity we do not distinguish different C

1

’s and C

2

’s.

• By c we denote absolute constants which may depend on µ, ν, κ, and by c

0

< 1 we denote positive constants which may depend on µ, ν, κ, %

∗

,

%

^∗

, θ

∗

, θ

^∗

. For simplicity we do not distinguish different c’s and c

0

’s.

• We underline that all the estimates are obtained under the assumption that there exists a local-in-time solution of (1.1), so all the quantities

%

∗

, %

^∗

, θ

∗

, θ

^∗

, T , R

T 0

kvk

²_3,Ω

t0

dt

⁰

, kSk

4−1/2

are estimated by the data functions. Moreover, the existence of the inverse transformation to x = x(ξ, t) is guaranteed by the estimates for the local solution (see [14]).

Lemma 3.1. Let v, %, ϑ

0

be a sufficiently smooth solution of (3.3). Then (3.8) 1

2 d dt

R

Ωt

%v

²

+ p

1

% %

²_σ

+ %

²_Ω_t

+ p

2

%c

v

p

θ

θ ϑ

²₀

dx

+ c

0

kvk

²_1,Ω_t

+ (ν − µ)kdiv vk

²_0,Ω_t

+ c

0

kϑ

_0x

k

²_0,Ω_t

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≤ ε(kp

_σ

k

²_0,Ω_t

+ kϑ

0tx

k

²_0,Ω_t

)

+ C

1

(kvk

²_0,Ω_t

+ krk

²_0,Ω_t

+ krk

0,Ωt

+ kθ

1

k

²_1,Ω_t

+ kθ

1

k

_1,Ω_t

+ kf k

²_0,Ω_t

) + C

2

(k%

σ

k

⁴_2,Ω_t

+ k%

_Ω_t

k

⁴_2,Ω_t

+ kvk

⁴_2,Ω_t

+ kϑ

0

k

⁴_2,Ω_t

),

where ε > 0 is sufficiently small.

P r o o f. Multiplying (3.3)

1

by v, integrating over Ω

t

and using the continuity equation (3.3)

2

and (3.4) we obtain

(3.9) 1 2

d dt

R

Ωt

%v

²

dx + µ

2 E

Ωt

(v) + (ν − µ)kdiv vk

²_0,Ω_t

− R

Ωt

p

1

%

σ

div v dx − R

Ωt

p

2

ϑ

0

div v dx = R

Ωt

%f v dx,

where E

Ωt

(v) = R

Ωt

P

3

i,j=1

(∂

xi

v

j

+ ∂

xj

v

i

)

²

dx.

By the continuity equation (3.3)

2

, energy equation (3.3)

3

and condition (3.3)

5

we have

− R

Ωt

p

1

%

σ

div v dx = R

Ωt

p

1

% %

σ

(%

σt

+ v · ∇%

σ

) dx (3.10)

= 1 2

d dt

R

Ωt

p

1

%

²_σ

% dx + I

1

, where

|I

1

| ≤ ε(kv

x

k

²_0,Ω_t

+ kϑ

0x

k

²_0,Ω_t

) + C

1

(krk

²_0,Ω_t

+ kθ

1

k

²_1,Ω_t

) (3.11)

+ C

2

(k%

σ

k

⁴_1,Ω_t

+ kvk

²_1,Ω_t

k%

σ

k

²_2,Ω_t

+ kvk

²_1,Ω_t

kϑ

0

k

²_2,Ω_t

+ kvk

²_2,Ω_t

k%

σ

k

²_1,Ω_t

+ k%

σ

k

²_2,Ω_t

k%

σ

k

²_1,Ω_t

).

Next, dividing equation (3.3)

3

by θ%

θ

, multiplying the result by p

2

ϑ

0

and integrating over Ω

t

we get

(3.12) R

Ωt

p

2

%c

v

θp

θ

∂

t

ϑ

²₀

2 + v · ∇ ϑ

²₀

2 dx + R

Ωt

p

2

ϑ

0

div v dx − R

Ωt

p

2

κ∆ϑ

0

θp

θ

ϑ

0

dx

− R

Ωt

p

2

µ 2θp

θ

X

i,j

(∂

xi

v

j

+ ∂

xj

v

i

)

²

ϑ

0

dx − R

Ωt

p

2

(ν − µ) θp

θ

(div v)

²

ϑ

0

dx

= R

Ωt

p

2

%r θp

θ

ϑ

0

dx.

Hence applying the boundary condition (3.3)

5

we have

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(3.13) R

Ωt

p

2

%c

v

θp

θ

∂

t

ϑ

²₀

2 +v ·∇ ϑ

²₀

2 dx+ R

Ωt

p

2

ϑ

0

div v dx+ R

Ωt

p

2

κ θp

θ

|ϑ

_0x

|

²

dx

= I

2

+ R

Ωt

p

2

%r θp

θ

ϑ

0

dx + R

St

p

2

κ θp

θ

1

ϑ

0

dx, where

|I

₂

| ≤ ε(kv

_x

k

²_0,Ω

t

+ kϑ

0x

k

²_0,Ω

t

) (3.14)

+ C

2

kϑ

₀

k

²_1,Ω

t

(kvk

²_2,Ω_t

+ k%

σ

k

²_2,Ω

t

+ kϑ

0

k

²_2,Ω

t

).

Moreover,

R

Ωt

p

2

%r θp

θ

ϑ

0

dx

≤

R

Ωt

p

2

%r θp

θ

ϑ dx

+

R

Ωt

p

2

%r θp

θ

(θ

Ωt

− θ

e

) dx (3.15)

≤ εkϑk

²_0,Ω_t

+ C

1

(krk

²_0,Ω_t

+ krk

0,Ωt

) and

(3.16)

R

St

p

2

κ θp

θ

1

ϑ

0

ds

≤ ε(kϑk

²_0,Ω

t

+ kϑ

0x

k

²_0,Ω

t

) + C

1

(kθ

1

k

²_1,Ω

t

+ kθ

1

k

_1,Ω_t

).

Next, using equations (3.3)

2

, (3.3)

3

and condition (3.3)

5

yields (3.17) R

Ωt

p

2

%c

v

θp

θ

∂

t

ϑ

²₀

2 + v · ∇ ϑ

²₀

2 dx = 1 2

d dt

R

Ωt

p

2

%c

v

θp

θ

ϑ

²₀

dx + I

3

, where

|I

₃

| ≤ ε(kv

_x

k

²_0,Ω_t

+ kϑ

0x

k

²_0,Ω_t

) + C

1

(krk

²_0,Ω_t

+ kθ

1

k

²_1,Ω_t

) (3.18)

+ C

2

kϑ

₀

k

²_1,Ω_t

(kϑ

0

k

²_2,Ω_t

+ kvk

²_2,Ω_t

+ k%

σ

k

²_2,Ω_t

).

Taking into account (3.9)–(3.11), (3.13)–(3.18), using Lemma 5.2 of [21]

and the Poincar´ e inequality

(3.19) kϑk

_0,Ω_t

≤ C

₁

kϑ

_0x

k

_0,Ω_t

we obtain for sufficiently small ε,

(3.20) 1 2

d dt

R

Ωt

%v

²

+ p

1

%

²_σ

% + p

2

%c

v

θp

θ

ϑ

²₀

dx + c

0

kvk

²_1,Ω

t

+ (ν − µ)kdiv vk

²_0,Ω_t

+ c

0

kϑ

_0x

k

²_0,Ω_t

≤ C

1

(kvk

²_0,Ω_t

+ krk

²_0,Ω_t

+ krk

0,Ωt

+ kθ

1

k

²_1,Ω_t

+ kθk

1,Ωt

+ kf k

²_0,Ω_t

) + C

2

[k%

σ

k

²_1,Ω

t

(k%

σ

k

²_1,Ω

t

+ kvk

²_2,Ω_t

) + k%

σ

k

²_2,Ω

t

(kvk

²_1,Ω_t

+ kϑ

0

k

²_2,Ω

t

) + kϑ

0

k

²_1,Ω

t

(kϑ

0

k

²_2,Ω

t

+ kvk

²_2,Ω_t

)].

(9)

Finally, by (3.3)

2

and (3.7) we have

(3.21) ∂

t

%

_Ω_t

+ v · ∇%

_Ω_t

+ % div v + ∂

t

%

Ωt

= 0, where in view of (3.6),

(3.22) ∂

t

%

Ωt

= − p

θ_Ωt

p

%_Ωt

∂

t

θ

Ωt

. Using the definition of θ

Ωt

we calculate

∂

t

θ

Ωt

= 1

|Ω

_t

|

R

Ωt

ϑ

0t

dx + 1

|Ω

_t

|

R

Ωt

θ div v dx (3.23)

− 1

|Ω

t

|

²

R

Ωt

θ dx R

Ωt

div v dx . Consider now

1 2

d dt

R

Ωt

%

²_Ω_t

dx = − R

Ωt

%

²_Ω_t

% div v dx − R

Ωt

%

_Ω_t

∂

t

%

Ωt

dx (3.24)

+ 1 2

R

Ωt

%

²_Ω_t

div v dx, where we have used equation (3.21). Since by (3.3)

3

, (3.25) kϑ

0t

k

²_0,Ω_t

≤ εkϑ

_0xt

k

²_0,Ω

t

+ C

1

(krk

²_0,Ω_t

+ kθ

1

k

²_1,Ω

t

+ kv

x

k

²_0,Ω

t

+ kϑ

0x

k

²_0,Ω

t

) + C

2

(kvk

²_1,Ω_t

kϑ

₀

k

²_2,Ω

t

+ kvk

⁴_1,Ω_t

+ k%

σ

k

²_2,Ω

t

kϑ

₀

k

²_2,Ω

t

+ kϑ

0

k

⁴_2,Ω

t

), relations (3.22)–(3.24) give the estimate

1 2

d dt

R

Ωt

%

²_Ω_t

dx ≤ ε(k%

_Ω_t

k

²_0,Ω

t

+ kϑ

0tx

k

²_0,Ω

t

) (3.26)

+ C

1

(krk

²_0,Ω_t

+ kθ

1

k

²_1,Ω_t

+ kv

x

k

²_0,Ω_t

+ kϑ

0x

k

²_0,Ω_t

) + C

2

(kvk

²_1,Ω_t

kϑ

0

k

²_2,Ω_t

+ kvk

⁴_2,Ω_t

+ k%

_Ω_t

k

⁴_1,Ω_t

+ k%

σ

k

²_2,Ω_t

kϑ

0

k

²_2,Ω_t

+ kϑ

0

k

⁴_2,Ω_t

).

By (3.5) and the Poincar´ e inequality (3.19) we have

(3.27) k%

_Ω

t

k

_0,Ω_t

≤ C

₁

(kϑ

0x

k

_0,Ω_t

+ k%

σ

k

_0,Ω_t

).

The estimates (3.20), (3.26) and (3.27) imply (3.8).

Lemma 3.2. Let v, %, ϑ

0

be a sufficiently smooth solution of (3.3). Then

(10)

(3.28) 1 2

d dt

R

Ωt

%v

t

+ p

σ%

% %

²_σt

+ %c

v

θ ϑ

²_0t

dx

+ c

0

kv

t

k

²_1,Ω_t

+ (ν − µ)k div v

t

k

²_0,Ω_t

+ c

0

kϑ

0t

k

²_1,Ω_t

≤ ε(kvk

²_1,Ω_t

+ kϑ

0x

k

²_0,Ω_t

)

+ C

1

(|f |

²_1,0,Ω_t

+ |r|

²_1,0,Ω_t

+ |θ

1

|

²_2,1,Ω_t

) + C

2

X

₁²

(1 + X

1

), where X

1

= |v|

²_2,1,Ω_t

+ |%

σ

|

²_2,1,Ω

t

+ |ϑ

0

|

²_2,1,Ω

t

.

P r o o f. Differentiating (3.3)

1

with respect to t, multiplying by v

t

and integrating over Ω

t

we obtain

(3.29) R

Ωt

%∂

t

v

_t²

2 + %v · ∇ v

_t²

2 dx + µ

2 E

Ωt

(v

t

) + (ν − µ)kdiv v

t

k

²_0,Ω_t

− R

Ωt

p

%

σt

div v

t

dx − R

Ωt

p

θ

ϑ

0t

div v

t

dx

+ R

Ωt

(%

t

v

_t²

+ %

t

v

t

· (v∇v) + %v

_t

· (v

_t

∇v)) dx + R

St

(T(v, p

σ

)n

t

) · v

t

ds

= R

Ωt

∂

t

(%f ) · v

t

dx,

where we have used the boundary condition (3.3)

4

. The continuity equation (3.3)

2

yields

(3.30) − R

Ωt

p

%

σt

div v

t

dx

= R

Ωt

p

%

% %

σt

%

σtt

+ p

%

% %

²_σt

div v + p

%

% %

σt

v

t

∇%

σ

+ p

%

% %

σt

v∇%

σt

dx

= 1 2

d dt

R

Ωt

p

%

% %

²_σt

dx + I

4

, where

(3.31) |I

₄

| ≤ ε(kv

_t

k

²_1,Ω_t

+ k%

σt

k

²_0,Ω_t

) + C

2

[k%

σt

k

²_1,Ω

t

(k%

σ

k

²_2,Ω

t

+ kϑ

0t

k

²_1,Ω

t

+ k%

σt

k

²_1,Ω

t

+ kϑ

0

k

²_2,Ω

t

)].

Using (3.30), (3.31) and Lemma 5.3 of [21] we obtain from (3.29) the

inequality

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(3.32) 1 2

d dt

R

Ωt

%v

_t²

+ p

%

% %

²_σt

dx

+ c

0

kv

_t

k

²_1,Ω_t

+ (ν − µ)kdiv v

t

k

²_0,Ω_t

− R

Ωt

p

θ

ϑ

0t

div v

t

dx

≤ εk%

_σt

k

²_0,Ω

t

+ C

1

|f |

²_1,0,Ω

t

+ C

2

X

₁²

(1 + X

1

),

Dividing now (3.3)

3

by θ, differentiating with respect to t, multiplying by ϑ

0t

, integrating over Ω

t

and next applying the H¨ older and Young inequalities and the Sobolev lemma gives

(3.33) 1 2

d dt

R

Ωt

%c

v

θ ϑ

²_0t

dx + R

Ωt

p

θ

ϑ

0t

div v

t

dx + κ θ

^∗

R

Ωt

|ϑ

_0tx

|

²

dx

≤ ε(k%

σt

k

²_0,Ω_t

+ kv

t

k

²_0,Ω_t

+ kϑ

0t

k

²_0,Ω_t

+ kϑ

0tx

k

²_0,Ω_t

) + C

1

(krk

²_0,Ω_t

+ kr

t

k

²_0,Ω_t

+ |θ

1

|

²_2,1,Ω_t

) + C

2

X

₁²

. From the continuity equation (3.3)

2

it follows that

(3.34) k%

σt

k

²_0,Ω_t

≤ C

1

kvk

²_1,Ω_t

+ C

2

kvk

²_1,Ω_t

k%

σ

k

²_2,Ω_t

.

Finally, adding inequalities (3.32)–(3.33) and using (3.25) and (3.34) we obtain (3.28).

Lemmas 3.1 and 3.2 imply

Lemma 3.3. Let v, %, ϑ

⁰

be a sufficiently smooth solution of (3.3). Then (3.35) 1

2 d dt

R

Ωt

%(v

²

+ v

_t²

) dx + 1 2

d dt

R

Ωt

1 % (p

1

%

²_σ

+ p

%

²_σt

) dx + 1

2 d dt

R

Ωt

%

²_Ω_t

dx+ 1 2

d dt

R

Ωt

%c

v

θ

p

2

p

θ

ϑ

²₀

+ϑ

²_0t

dx+c

0

(kvk

²_1,Ω_t

+kv

t

k

²_1,Ω

t

) + (ν − µ)(kdiv vk

²_0,Ω_t

+ kdiv v

t

k

²_0,Ω_t

)

+ c

0

(kϑ

0x

k

²_0,Ω_t

+ kϑ

0t

k

²_1,Ω_t

)

≤ εkp

_σ

k

²_0,Ω

t

+C

1

(kvk

²_0,Ω_t

+|r|

²_1,0,Ω_t

+krk

0,Ωt

+|θ

1

|

²_2,1,Ω

t

+kθ

1

k

_0,Ω_t

+|f |

²_1,0,Ω_t

) + C

2

[k%

_Ω_t

k

⁴_2,Ω

t

+ X

₁²

(1 + X

1

)], where X

1

is defined in Lemma 3.2.

In order to obtain an inequality for derivatives with respect to x we

rewrite problem (3.3) in the Lagrangian coordinates and next we introduce

(12)

a partition of unity in the fixed domain Ω. Thus we have

(3.36)

ηu

it

− ∇

_u_j

T

_u^ij

(u, p

σ

) = ηg

i

, i = 1, 2, 3, η

σt

+ η∇

u

· u = 0,

ηc

v

(η, Γ )γ

0t

− κ∇

²_u

γ

0

= ηk − Γ p

Γ

(η, Γ )∇

u

· u + µ

2

3

X

i,j=1

(ξ

kxi

∂

ξk

u

j

+ ξ

kxj

∂

ξk

u

i

)

²

+ (ν − µ)(∇

u

· u)

²

, T

u

(u, p

σ

)n(ξ, t) = 0,

n · ∇

u

Γ = Γ

1

, where

η(ξ, t) = %(x(ξ, t), t), u(ξ, t) = v(x(ξ, t), t), g(ξ, t) = f (x(ξ, t), t), Γ (ξ, t) = θ(x(ξ, t), t), γ

0

(ξ, t) = ϑ

0

(x(ξ, t), t), Γ

1

= θ

1

(x(ξ, t), t) and

(3.37)

T

u

(u, p

σ

) = {T

_u^ij

(u, p

σ

)}

= {−p

σ

δ

ij

+ µ(∇

uj

u

i

+ ∇

ui

u

j

) + (ν − µ)δ

ij

∇

_u

· u},

∇

_u

= ξ

x

∂

ξ

≡ (ξ

_ix_k

∂

ξi

)

k=1,2,3

, ∇

_u_i

= ξ

kxi

∂

ξk

, div

u

T

u

(u, p

σ

) = ∇

u

T

u

(u, p

σ

).

By (3.4), (3.5) we have respectively

(3.38) p

σ

= p

1

η

σ

+ p

2

γ

0

and

(3.39) p

σ

= p

3

η

_Ω_t

+ p

4

γ, where

η

σ

= η − %

e

, γ

0

= Γ − θ

e

, η

_Ω_t

= η − %

Ωt

, γ = Γ − θ

Ωt

, p

1

= p

1

(η, Γ ), p

2

= p

2

(η, Γ ), p

3

= p

3

(%

Ωt

, η, Γ ),

p

4

= p

4

(%

Ωt

, θ

Ωt

, Γ ), p

i

> 0 (i = 1, 2, 3).

Let us introduce a partition of unity ({ e Ω

i

}, {ζ

_i

}), Ω = S

i

Ω e

i

. Let e Ω be one of the e Ω

i

’s and ζ(ξ) = ζ

i

(ξ) be the corresponding function. If e Ω is an interior subdomain then let ω be a set such that e ω ⊂ e e Ω and ζ(ξ) = 1 for ξ ∈ ω. Otherwise we assume that e e Ω ∩ S 6= ∅, ω ∩ S 6= ∅, e ω ⊂ e e Ω. Take any β ∈ ω ∩ S ⊂ e e Ω ∩ S = e S and introduce local coordinates {y} associated with {ξ} by

(3.40) y

k

= α

kl

(ξ

l

− β

_l

), α

3k

= n

k

(β), k = 1, 2, 3,

(13)

where α

kl

is a constant orthogonal matrix such that e S is determined by the equation y

3

= F (y

1

, y

2

), F ∈ W

₂^4−1/2

and

Ω = {y : |y e

i

| < d, i = 1, 2, F (y

⁰

) < y

3

< F (y

⁰

) + d, y

⁰

= (y

1

, y

2

)}.

Next, we introduce functions u

⁰

, η

⁰

, Γ

⁰

, γ

₀⁰

, γ

⁰

, Γ

₁⁰

by

(3.41)

u

⁰_i

(y) = α

ij

u

j

(ξ)|

ξ=ξ(y)

, η

⁰

(y) = η(ξ)|

ξ=ξ(y)

, Γ

⁰

(y) = Γ (ξ)|

ξ=ξ(y)

, γ

₀⁰

(y) = γ

0

(ξ)|

ξ=ξ(y)

,

γ

⁰

(y) = γ(ξ)|

ξ=ξ(y)

, Γ

₁⁰

(y) = Γ

1

(ξ)|

ξ=ξ(y)

,

where ξ = ξ(y) is the inverse transformation to (3.40). Further, we introduce new variables by

(3.42) z

i

= y

i

(i = 1, 2), z

3

= y

3

− e F (y), y ∈ e Ω,

which will be denoted by z = Φ(y) (where e F is an extension of F with F ∈ W e

₂⁴

).

Let b Ω = Φ( e Ω) = {z : |z

i

| < d, i = 1, 2, 0 < z

₃

< d} and b S = Φ( e S).

Define

(3.43)

u(z) = u b

⁰

(y)|

y=Φ⁻¹(z)

, η(z) = η b

⁰

(y)|

y=Φ⁻¹(z)

, Γ (z) = Γ b

⁰

(y)|

y=Φ⁻¹(z)

, b γ

0

(z) = γ

₀⁰

(y)|

y=Φ⁻¹(z)

, b γ(z) = γ

⁰

(y)|

y=Φ⁻¹(z)

, Γ b

1

(z) = Γ

₁⁰

(y)|

y=Φ⁻¹(z)

.

Set b ∇

k

= ξ

lxk

(ξ)z

iξl

∇

zi

|

_ξ=χ⁻¹_(z)

, where χ(ξ) = Φ(ψ(ξ)) and y = ψ(ξ) is described by (3.40). We also introduce the following notation:

(3.44)

e u(ξ) = u(ξ)ζ(ξ), e η(ξ) = η(ξ)ζ(ξ), Γ (ξ) = Γ (ξ)ζ(ξ), e e γ

0

(ξ) = γ

0

(ξ)ζ(ξ),

e γ(ξ) = γ(ξ)ζ(ξ), Γ e

1

(ξ) = Γ

1

(ξ)ζ(ξ) for ξ ∈ e Ω, e Ω ∩ S = ∅, and

(3.45)

e u(z) = b u(z)b ζ(z), e η(z) = η(z)b b ζ(z), Γ (z) = b e Γ (z)b ζ(z), e γ

0

(z) = b γ

0

(z)b ζ(z),

e γ(z) = b γ(z)b ζ(z), Γ e

1

(z) = b Γ

1

(z)b ζ(z) for z ∈ b Ω = Φ( e Ω), e Ω ∩ S 6= ∅, where b ζ(z) = ζ(ξ)|

ξ=χ⁻¹(z)

.

Using the above notation and (3.2) we can rewrite problem (3.36) in the

following form in an interior subdomain:

(14)

(3.46)

η u e

it

− ∇

_u_j

T

_u^ij

( e u, p e

σ

) = η g e

i

− ∇

_u_j

B

_u^ij

(u, ζ) − T

_u^ij

(u, p

σ

)∇

uj

ζ

≡ η g e

i

+ k

1

, i = 1, 2, 3, e η

σt

+ η∇

u

· u = ηu · ∇ e

u

ζ ≡ k

2

,

ηc

v

(η, Γ ) e γ

t

− κ∇

²_u

e γ + Γ p

Γ

(η, Γ )∇

u

· u e

= ηe k + 1 2 µ

3

X

i,j=1

(ξ

kxi

∂

ξk

u

j

+ ξ

kxj

∂

ξk

u

i

)

²

+ (ν − µ)(∇

u

· u)

²

ζ + Γ p

Γ

(η, Γ )u · ∇

u

ζ − κ(∇

²_u

ζγ + 2∇

u

ζ · ∇

u

γ)

− ηc

_v

(η, Γ )ζ∂

t

θ

Ωt

≡ ηe k + k

3

, where p

σ

= p

σ

ζ and

B

u

(u, ζ) = {B

_u^ij

(u, ζ)} = {µ(u

i

∇

_u_j

ζ + u

j

∇

_u_i

ζ) + (ν − µ)∂

ij

u · ∇

u

ζ}.

In boundary subdomains we have

(3.47)

b η u e

it

− b ∇

_j

T b

^ij

( u, e p e

σ

) = η b e g

i

− b ∇

_j

B b

^ij

( b u, b ζ) − b T

^ij

( u, p b

σ

) b ∇

_j

ζ b

≡ η b e g

i

+ k

ⁱ₄

, η e

σt

+ η b b ∇ · u = e η b b u · b ∇b ζ ≡ k

5

,

ηc b

v

( η, b b Γ ) e γ

t

− κ b ∇

²

e γ + b Γ p

Γ

( η, b b Γ ) b ∇ · u e

= ηe b k + 1 2 µ

3

X

i,j=1

( b ∇

_i

b u

j

+ b ∇

_j

u b

i

)

²

+ (ν − µ)( b ∇ · u) b

²

ζ b + b Γ p

Γ

( η, b b Γ ) b u · b ∇b ζ − κ( b ∇

²

ζ · b b γ + 2 b ∇b ζ · b ∇ b γ)

− b ηc

v

( b η, b Γ )∂

t

θ

Ωt

ζ ≡ b ηe b k + k

6

, T(e b u, p e

σ

) n = k b

7

,

b n · b ∇ e γ = e Γ

1

+ k

8

,

where k

ⁱ₇

= b B

^ij

( u, b b ζ) n b

j

, k

8

= b n· b ∇b ζ b γ, b ∇ = ( b ∇

_j

)

j=1,2,3

, and b T and b B indicate that the operator ∇

u

is replaced by b ∇.

In the considerations below we denote z

1

, z

2

by τ and z

3

by n.

Lemma 3.4. Let v, %, ϑ

0

be a sufficiently smooth solution of problem (3.3). Then

(3.48) 1 2

d dt

R

Ωt

%v

²_x

+ p

σ%

% %

²_σx

+ %c

v

θ ϑ

²_0x

dx

+ c

0

(kv

x

k

²_1,Ω_t

+ k%

_Ω_t

k

²_0,Ω_t

+ k%

σx

k

²_0,Ω_t

+ k%

σt

k

²_0,Ω_t

+ kϑ

0x

k

²_1,Ω_t

)