a viscous compressible heat conducting capillary fluid bounded by a free surface

POLONICI MATHEMATICI LXV.1 (1996)

On a differential inequality for

a viscous compressible heat conducting capillary 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 bound- ary problem for a viscous compressible heat concluding capillary fluid. The inequality is essential in proving the global existence of solutions.

1. Introduction. The motion of a viscous compressible heat conducting capillary fluid in a bounded domain Ω

t

⊂ R

³

(which depends on time t ∈ R

¹+

) is described by the following system with the boundary and initial conditions (see [2], Chs. 2 and 5):

(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

X

3 i,j=1

(v

i,xj

+ v

j,xi

)

²

− (ν − µ)(div v)

²

= ̺r in e Ω

^T

,

Tn − σHn = −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 φ(x, t) = 0 describes S

t

, n is the outward vector normal to the bound- ary (i.e. n = ∇φ/|∇φ|), e Ω

^T

= S

t∈(0,T )

Ω

t

×{t}, Ω

0

= Ω is an initial domain, S e

^T

= S

t∈(0,T )

S

t

×{t}. Moreover, v = v(x, t) (v = (v

1

, v

2

, v

3

)) is the velocity

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

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

[23]

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 the coefficients µ, ν, κ are constants such that κ > 0, ν ≥

¹₃

µ > 0 and moreover c

v

> 0, which results from thermodynamic considerations. Further, T = T(v, p) denotes the stress tensor of the form T = {T

ij

} = {−pδ

ij

+ µ(v

i,x_j

+ v

j,x_i

) + (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.

Finally, we denote by H the double mean curvature of S

t

which is neg- ative for convex domains and can be expressed in the form

Hn = ∆

St

(t)x, x = (x

1

, x

2

, x

3

),

where ∆

S_t

(t) is the Laplace–Beltrami operator on S

t

. Let S

t

be determined by x = x(s

¹

, s

²

, t), (s

¹

, s

²

) ∈ R

²

. Then we have

∆

St

(t) = g

^−1/2

∂

∂s

^α

g

^−1/2

b g

αβ

∂

∂s

^β

= g

^−1/2

∂

∂s

^α

g

^1/2

g

^αβ

∂

∂s

^β

(α, β = 1, 2),

where the convention summation over repeated indices is assumed, g = det{g

αβ

}

α,β=1,2

, g

αβ

= x

α

· x

β

, (x

^α

= ∂x/∂s

^α

), {g

^αβ

} is the inverse matrix to {g

αβ

} and {bg

^αβ

} is the matrix of algebraic complements of {g

αβ

}.

Assume that the domain Ω is 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

).

Hence

(1.2) x = ξ +

t

\

0

u(ξ, s) ds ≡ X

u

(ξ, t), where u(ξ, t) = v(X

_u

(ξ, t), t).

Formula (1.2) yields the relation between the Eulerian x and Lagrangian ξ coordinates. 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 equation of continuity (1.1)

2

and (1.1)

5

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

_t

holds:

\

Ωt

̺(x, t) dx = M.

In this paper we prove a global differential inequality for problem (1.1) (see Theorem 3.13) which we shall use in the next paper to prove the global- in-time existence of a solution to problem (1.1) close to a constant state. The paper is divided into three sections. In Section 2 we introduce some notation.

In Section 3 we formulate a series of lemmas (see Lemmas 3.1–3.12) which are used to derive the differential inequalities (3.46) and (3.47).

Problem (1.1) is also considered in [12]–[17]. In [12] we prove the local existence of a solution to problem (1.1) in Sobolev–Slobodetski˘ı spaces in two cases: σ = 0 and σ > 0. [14] and [15] are devoted to conservation laws for problem (1.1) in two cases: without surface tension and with it, respectively.

In [14] and [15] we prove that we can choose ̺

₀

, v

₀

, θ

₀

, θ

₁

, p

₀

, κ, σ (in the case σ > 0) and the form of the internal energy per unit mass ε = ε(̺, θ) in such a way that var

t

|Ω

t

| is as small as we need. In [16] the global differential inequality in the case σ = 0, analogous to inequality (3.46) is obtained. [17]

is concerned with the global-in-time existence of solutions to problem (1.1) when σ = 0. Finally, [13] contains a review of results from [12], [1]–[17] and this paper.

In order to prove the main result of the paper, i.e. Theorem 3.13, we apply the same method as in [18], [19] and [16], which is very close to the methods used in [10] and [11] (see also [4]–[7] and [8]).

Papers [18] and [19] of W. M. Zaj¸aczkowski and [9] of V. A. Solonnikov and A. Tani refer to the problem corresponding to (1.1) for a compressible barotropic fluid.

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

Finally, the motion of a viscous compressible heat conducting fluid in a fixed domain was considered by A. Matsumura and T. Nishida in [3]–[7] and by A. Valli and W. M. Zaj¸aczkowski in [11].

2. Notation. Let Q = Ω

t

or Q = S

t

(t ≥ 0). We denote by k · k

l,Q

(l ≥ 0) and | · |

p,Q

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

₂^l

(Q) and L

p

(Q) spaces, respectively.

Next, we introduce the space Γ

_k^l

(Q) of functions u with the norm kuk

_Γ^l

k(Q)

= X

i≤l−k

k∂

_tⁱ

uk

l−i,Q

≡ |u|

l,k,Q

, where l > 0 and k ≥ 0.

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...xt...t

| {z }

k times

the vector (D

_x^α

∂

ⁱ_t

u)

|α|+i=k

.

Similarly, if u = (u

1

, u

2

, u

3

) we denote by D

_x,t^k

u or u

x...xt...t

| {z }

k times

the vector (D

^α_x

∂

_tⁱ

u

j

)

|α|+i=k, j=1,2,3

. Hence

|D

^k_x,t

u| = X

|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

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

_x^α

u|

_p,Q

< ∞, and W

_r^l

(Q) is the Sobolev space.

Moreover , the following interpolation inequalities are true:

(2.1) |D

^α_x

u|

p,Q

≤ cε

^1−κ

|D

_x^l

u|

r,Q

+ cε

^−κ

|u|

r,Q

,

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

(2.2) |D

_x^α

u|

_q,∂Q

≤ cε

^1−κ

|D

^l_x

u|

_r,Q

+ cε

^1−κ

|u|

r,Q

,

where κ = |α|/l + 3/(lr) − 2/(lq) < 1, ε is a parameter , c > 0 is a constant independent of u and ε.

Lemma 2.1 follows from Theorem 10.2 of [1].

3. Global differential inequality. In this section we assume that ν >

1

3

µ. Further, assume that the existence of a sufficiently smooth local solution of problem (1.1) has been proved. To prove the desired differential inequality we assume that Ω

t

(t ≤ T , T is the time of local existence) is diffeomorphic to a ball, so S

t

can be described by

|x| ≡ r = R(ω, t), ω ∈ S

¹

, where S

¹

is the unit sphere.

Moreover, we consider the motion near the constant state v

e

= 0, p

e

= p

0

+2σ/R

e

, θ

e

= (1/|Ω|)

T

Ω

θ

0

dξ, ̺

e

= M/((4π/3)R

³_e

), where R

e

is a solution of the equation

p

 M

(4π/3)R

³_e

, θ

e



= p

e

.

(Obviously, we assume that the above equation is solvable with respect to R

e

> 0.)

Let

p

σ

= p − p

0

− q

0

, ̺

σ

= ̺ − ̺

e

, ϑ

0

= θ − θ

e

, ϑ = θ − θ

Ωt

,

where q

0

= 2σ/R

e

and θ

Ω_t

= (1/|Ω

t

|)

T

Ωt

θ dx. Then problem (1.1) takes the form

(3.1)

̺[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

(∂

x_i

v

j

+ ∂

x_j

v

i

)

²

− (ν − µ)(div v)

²

= ̺r in Ω

t

, t ∈ [0, T ], T(v, p

σ

)n = σ∆

St

x · n n + q

0

n on S

t

, t ∈ [0, T ],

∂ϑ

0

/∂n = θ

1

on S

t

, t ∈ [0, T ],

where T(v, p

σ

) = {µ(∂

x_i

v

j

+ ∂

x_j

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

)

(3.2)

= p(̺, θ) − p(̺

_e

, θ) + p(̺

_e

, θ) − p(̺

_e

, θ

_e

)

= (̺ − ̺

_e

)

1\

0

p

_̺

(̺

_e

+ s(̺ − ̺

_e

), θ) ds

+ (θ − θ

e

)

1\

0

p

θ

(̺

e

, θ

e

+ s(θ − θ

e

)) ds

≡ p

1

̺

_σ

+ p

₂

ϑ

₀

. We shall also use the formula:

p

_σ

= p(̺, θ) − p(̺

_Ωt

, θ

_Ωt

) (3.3)

= (̺ − ̺

Ωt

)

1\

0

p

̺

(̺

Ωt

+ s(̺ − ̺

Ωt

), θ) ds

+ (θ − θ

Ωt

)

1\

0

p

θ

(̺

Ωt

, θ

Ωt

+ s(θ − θ

Ωt

)) ds

≡ p

3

̺

_Ωt

+ p

₄

ϑ,

where the function ̺

_Ωt

= ̺

_Ωt

(t) is a solution of the problem (3.4) p(̺

_Ωt

, θ

_Ωt

) = p

_e

, ̺

_Ωt

|

t=0

= ̺

_e

and

̺

_Ωt

= ̺ − ̺

Ω_t

.

The functions p

i

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

Set

̺

∗

= min e

Ω^T

̺(x, t), ̺

^∗

= max e

Ω^T

̺(x, t), θ

∗

= min

e

Ω^T

θ(x, t), θ

^∗

= max e

Ω^T

θ(x, t).

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

1. We denote by ε small constants and for simplicity we do not distinguish them.

2. We denote by C

1

and C

2

constants which depend on ̺

∗

, ̺

^∗

, θ

∗

, θ

^∗

, T ,

TT

0

kvk

²_3,Ω

t′

dt

^′

, kSk

4+1/2

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

3. We denote by c absolute constants which may depend on such param- eters as µ, ν, κ, and by c

0

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

∗

, ̺

^∗

, θ

∗

, θ

^∗

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

0

’s.

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

̺

∗

, ̺

^∗

, θ

∗

, θ

^∗

, T ,

TT

0

kvk

²_3,Ω

t′

dt

^′

, kSk

_4+1/2

are estimated by the data func- tions. Moreover, the existence of the inverse transformation to x = x(ξ, t) is guaranteed by the estimates for the local solutions (see [12]).

Lemma 3.1. Let v, ̺, ϑ

0

be a sufficiently smooth solution of (3.1).

Then (3.6) 1

2 d dt

\

Ωt



̺v

²

+ p

₁

̺ ̺

²_σ

+ ̺

²_Ωt

+ p

₂

̺c

_v

p

θ

θ ϑ

²₀

 dx

+ σ 2

d dt

\

St

 g

^αβ

n ·

t

\

0

v

s^α

dt

^′

n ·

t

\

0

v

_sβ

dt

^′



ds + c

0

kvk

²_1,Ωt

+ (ν − µ)kdiv vk

²_0,Ωt

+ c

0

kϑ

0x

k

²_0,Ωt

≤ ε 

kp

σ

k

²_0,Ωt

+ kϑ

0tx

k

²_0,Ωt

+

t

\

0

v

s

dt

^′

2 0,St

+ kH(·, 0) + 2/R

e

k

²_0,S¹

 + C

1

(kvk

²_0,Ωt

+ krk

²_0,Ωt

+ krk

0,Ω_t

+ kθ

1

k

²_1,Ωt

+ kθ

1

k

1,Ω_t

+ kf k

²_0,Ωt

) + C

2

X

1

Y

1

,

where ε > 0 is sufficiently small, s = (s

¹

, s

²

) and

X

1

= kvk

²_2,Ωt

+ k̺

σ

k

²_2,Ωt

+ kϑ

0

k

²_2,Ωt

+ k̺

_Ωt

k

²_0,Ωt

, Y

1

= X

1

+

1

\

0

v dt

^′

2 2,St

.

P r o o f. Multiplying (3.1)

1

by v, integrating over Ω

t

and using the con- tinuity equation (3.1)

₂

, boundary condition (3.1)

₄

and (3.2) we obtain

1 2

d dt

\

Ω_t

̺v

²

dx + µ

2 E

Ω_t

(v) + (ν − µ)kdiv vk

²_0,Ωt

(3.7)

−

\

Ωt

p

1

̺

σ

div vdx −

\

Ωt

p

2

ϑ

0

div v dx

− σ

\

St

(∆

St

x · n + 2/R

e

)n · v ds =

\

Ωt

̺f v dx,

where E

Ω_t

(v) =

T

Ωt

P

3

i,j=1

(∂

x_i

v

j

+ ∂

x_j

v

i

)

²

dx.

First, we consider the sum of the second and third terms on the left-hand side of (3.7). We have

(3.8) µ

2 E

Ωt

(v) + (ν − µ)kdiv vk

²_0,Ωt

= µ 2

\

Ωt

(v

_i,xj

+ v

_j,xi

)

²

dx + (ν − µ)

\

Ωt

(div v)

²

dx

= µ 2

X

i6=j

\

Ω_t

(v

i,xj

+ v

j,xi

)

²

dx + µ 2

X

i=j

\

Ω_t

(v

i,xj

+ v

j,xi

)

²

dx

+ (ν − µ)

\

Ωt

(div v)

²

dx

= µ 2

X

i6=j

\

Ω_t

(v

i,xj

+ v

j,xi

)

²

dx + µ 2 ε

1

X

i=j

\

Ω_t

(v

i,xj

+ v

j,xi

)

²

dx

+ µ

2 (1 − ε

1

) · 4 X

i

\

Ω_t

(v

i,x_j

)

²

dx + (ν − µ)

\

Ω_t

(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]

\

Ωt

(div v)

²

dx.

Assuming that ν = (1 + 2ε

1

)µ/3 we obtain ε

1

=

_2µ³

(ν − µ/3), so (3.9) I ≥ µ

2 ε

1

\

Ωt

(v

i,x_j

+ v

j,x_i

)

²

dx = 3 4

 ν − µ

3



\

Ωt

(v

i,x_j

+ v

j,x_i

)

²

dx.

By the continuity equation (3.1)

₂

, energy equation (3.1)

₃

and boundary condition (3.1)

5

we have

−

\

Ωt

p

1

̺

σ

div v dx =

\

Ωt

p

1

̺ ̺

σ

(̺

σt

+ v · ∇̺

σ

) dx (3.10)

= 1 2

d dt

\

Ωt

p

1

̺

²_σ

̺ dx + I

1

, where

|I

1

| ≤ ε(kv

x

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

) + C

1

(krk

²_0,Ωt

+ kθ

1

k

²_1,Ωt

) (3.11)

+ C

2

(k̺

σ

k

⁴_1,Ωt

+ kvk

²_1,Ωt

k̺

σ

k

²_2,Ωt

+ kvk

²_1,Ωt

kϑ

0

k

²_2,Ωt

+ kvk

²_2,Ωt

k̺

σ

k

²_1,Ωt

+ k̺

σ

k

²_2,Ωt

k̺

σ

k

²_1,Ωt

).

Now, we consider the boundary term in (3.7). In the same way as in Lemma 4.1 of [19] we obtain

(3.12) −

\

St

(∆

_St

x · n + 2/R

_e

)v · n ds

= 1 2

d dt

\

St

 g

^αβ

n ·

t

\

0

v

s^α

dt

^′

n ·

t

\

0

v

_sβ

dt

^′



ds + I

1

, where

|I

1

| ≤ ε 

t

\

0

v

s

dt

^′

2 0,St

+ kH(·, 0) + 2/R

e

k

²_0,S¹

+ kvk

²_1,Ωt

 (3.13)

+ C

1

kvk

²_0,Ωt

+ C

2

t

\

0

v dt

^′

2 2,St

kvk

²_2,Ωt

.

Next, dividing (3.1)

3

by θp

θ

, multiplying the result by p

2

ϑ

0

and inte- grating over Ω

_t

we get

(3.14)

\

Ωt

p

₂

̺c

_v

θp

θ



∂

_t

ϑ

²₀

2 + v · ∇ ϑ

²₀

2

 dx +

\

Ωt

p

₂

ϑ

₀

div v dx

−

\

Ωt

p

₂

κ∆ϑ

₀

θp

θ

ϑ

0

dx −

\

Ωt

p

₂

µ 2θp

θ

X

i,j

(∂

xj

v

i

+ ∂

xi

v

j

)

²

ϑ

0

dx

−

\

Ωt

p

₂

(ν − µ) θp

θ

(div v)

²

ϑ

₀

dx =

\

Ωt

p

₂

̺r θp

θ

ϑ

₀

dx.

Therefore, using the same argument as in Lemma 3.1 of [16], by (3.7)–(3.14) we have

(3.15) 1 2

d dt



̺v

²

+ p

₁

̺

²_σ

̺ + p

₂

̺c

_v

θp

θ

ϑ

²₀

 dx

+ σ 2

d dt

\

S_t

g

^αβ

n ·

t

\

0

v

s^α

dt

^′

n ·

t

\

0

v

_s^β

dt

^′

ds + c

0

kvk

²_1,Ωt

+ (ν − µ)kdiv vk

²_0,Ωt

+ c

₀

kϑ

0x

k

²_0,Ωt

≤ ε 

t

\

0

v

_s

dt

^′

2 0,St

+ kH(·, 0) + 2/R

₀

k

²_0,S¹

 + C

1

(kvk

²_0,Ωt

+ krk

²_0,Ωt

+ krk

0,Ωt

+ kθ

1

k

²_1,Ωt

+ kθ

1

k

1,Ωt

+ kf k

²_0,Ωt

) + C

2

X

1

Y

1

. Finally, by (3.1)

₂

and (3.5) we have

(3.16) ∂

t

̺

_Ωt

+ v · ∇̺

_Ωt

+ ̺ div v + ∂

t

̺

Ωt

= 0, where in view of (3.4) we get

(3.17) ∂

t

̺

Ω_t

= − p

_θΩt

p

_̺Ωt

∂

t

θ

Ω_t

. Using the definition of θ

Ω_t

we calculate

∂

t

θ

Ω_t

= 1

|Ω

t

|

\

Ωt

ϑ

0t

dx + 1

|Ω

t

|

\

Ωt

θ div v dx (3.18)

− 1

|Ω

t

|

²



^\

Ωt

θ dx 

^\

Ωt

div v dx  . Equation (3.16) and formulas (3.17), (3.18) yield

1 2

d dt

\

Ω_t

̺

²_Ωt

dx ≤ ε(k̺

_Ωt

k

²_0,Ωt

+ kϑ

0tx

k

²_0,Ωt

) (3.19)

+ C

1

(kv

x

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

+ krk

²_0,Ωt

+ kθ

1

k

²_1,Ωt

) + C

2

(kvk

²_1,Ωt

kϑ

0

k

²_2,Ωt

+ kvk

⁴_2,Ωt

+ k̺

_Ωt

k

⁴_1,Ωt

+ k̺

_σ

k

²_2,Ωt

kϑ

0

k

²_2,Ωt

+ kϑ

₀

k

⁴_2,Ωt

).

By (3.3) and the Poincar´e inequality

(3.20) kϑk

0,Ω_t

≤ kϑ

0x

k

0,Ω_t

we get

(3.21) k̺

_Ωt

k

0,Ωt

≤ C

1

(kϑ

_0x

k

0,Ωt

+ kp

_σ

k

0,Ωt

).

The estimates (3.15), (3.19) and (3.21) yield (3.6).

Lemma 3.2. Let v, ̺, ϑ

0

be a sufficiently smooth solution of (3.1). Then (3.22) 1

2 d dt

\

Ωt



̺v

_t²

+ p

_σ̺

̺ ̺

²_σt

+ ̺c

_v

θ ϑ

²_0t

 dx

+ σ 2

d dt

\

St

g

^αβ

v

_s^α

· nv

s^β

· n ds + c

0

kv

t

k

²_1,Ωt

+ (ν − µ)kdivv

t

k

²_0,Ωt

+ c

0

kϑ

0t

k

²_1,Ωt

≤ ε(kvk

²_1,Ωt

+ kv

xx

k

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

)

+ C

1

(|f |

²_1,0,Ωt

+ |r|

²_1,0,Ωt

+ krk

0,Ωt

+ |θ

1

|

²_2,1,Ωt

+ kθ

1

k

1,Ωt

) + C

2

X

₂²

(1 + X

2

),

where X

2

= |v|

²_2,1,Ω

t

+ |̺

σ

|

²_2,1,Ωt

+ |ϑ

0

|

²_2,1,Ωt

.

P r o o f. By the same argument as in Lemma 3.2 of [16] we have (3.23) 1

2 d dt

\

Ωt



̺v

_t²

+ p

σ̺

̺ ̺

²_σt

+ ̺c

v

θ ϑ

²_0t



dx + kv

t

k

²_1,Ωt

+ (ν − µ)kdivv

t

k

²_0,Ωt

+ kϑ

0t

k

²_1,Ωt

−

\

S_t

[T(v, p

σ

)]

,t

n · v

t

ds

≤ ε(k̺

σt

k

²_0,Ωt

+ kv

t

k

²_0,Ωt

+ kϑ

0t

k

²_0,Ωt

+ kϑ

0tx

k

²_0,Ωt

) + C

₁

(krk

²_0,Ω

t

+ kr

_t

k

²_0,Ωt

+ |θ

₁

|

²_2,1,Ωt

) + C

₂

X

₂²

(1 + X

₂

).

By the boundary condition (3.1)

4

and the same argument as in Lemma 4.2 of [19] we get

(3.24) −

\

St

[T(v, p

σ

)]

_,t

n · v

_t

ds = σ 2

d dt

\

St

g

^αβ

v

_s^α

· nv

s^β

· n ds + I

2

, where

(3.25) |I

2

| ≤ ε(kv

t

k

²_1,Ωt

+ kv

xx

k

²_0,Ωt

) + C

1

kvk

²_0,Ωt

+ C

2

X

₂²

. From the continuity equation (3.1)

2

it follows that

(3.26) k̺

σt

k

²_0,Ωt

≤ C

1

kvk

²_1,Ωt

+ C

2

kvk

²_1,Ωt

k̺

σ

k

²_2,Ωt

and equation (3.1)

3

yields

kϑ

0t

k

²_0,Ωt

≤ εkϑ

0xt

k

²_0,Ωt

(3.27)

+ C

₁

(kv

_x

k

²_0,Ωt

+ kϑ

_0x

k

²_0,Ωt

+ krk

²_0,Ωt

+ kθ

₁

k

²_1,Ωt

) + C

₂

(kvk

²_1,Ωt

kϑ

0

k

²_2,Ωt

+ kvk

⁴_1,Ωt

+ k̺

σ

k

²_2,Ωt

kϑ

0

k

²_2,Ωt

+ kϑ

0

k

⁴_2,Ωt

).

Therefore, taking into account (3.23)–(3.27) we obtain (3.22).

Lemmas 3.1 and 3.2 imply

Lemma 3.3. Let v, ̺, ϑ

0

be a sufficiently smooth solution of (3.1). Then 1

2 d dt

\

Ωt



̺(v

²

+ v

²_t

) + 1

̺ (p

₁

̺

²_σ

+ p

_σ̺

̺

²_σt

) + ̺

²_Ωt

+ ̺c

_v

θ

 p

₂

p

θ

ϑ

²₀

+ ϑ

²_0t



dx

+ σ 2

d dt

\

St

g

^αβ

 n ·

t

\

0

v

s^α

dt

^′

n ·

t

\

0

v

_sβ

dt

^′

+ n · v

s^α

n · v

_sβ

 ds + c

0

(kvk

²_1,Ωt

+ kv

t

k

²_1,Ωt

) + (ν − µ)(kdiv vk

²_0,Ωt

+ kdiv v

t

k

²_0,Ωt

) + c

0

(kϑ

0x

k

²_0,Ωt

+ kϑ

0t

k

²_1,Ωt

)

≤ ε 

k̺

σ

k

²_0,Ωt

+ kv

xx

k

²_0,Ωt

+

t

\

0

v

s

dt

^′

2 0,St

+ kH(·, 0) + 2/R

e

k

²_0,S¹

 + C

1

(kvk

²_0,Ωt

+ |r|

²_1,0,Ωt

+ krk

0,Ω_t

+ |θ

1

|

²_2,1,Ωt

+ kθ

1

k

1,Ω_t

+ |f |

²_1,0,Ωt

) + C

2

[X

1

Y

1

+ X

₂²

(1 + X

2

)].

In order to obtain an inequality for derivatives with respect to x we rewrite problem (3.1) in the Lagrangian coordinates and next we introduce a partition of unity in the fixed domain Ω. Thus we have

(3.28)

ηu

_it

− ∇

uj

T

^iju

(u, p

_σ

) = ηg

_i

, i = 1, 2, 3, η

σ_t

+ 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 X

3 i,j=1

(ξ

kx_i

∂

ξ_k

u

j

+ ξ

kx_j

∂

ξ_k

u

i

)

²

+ (ν − µ)(∇

u

· u)

²

,

T

u

(u, p

σ

)n = σ∆

S_t

x(ξ, t) · n n + q

0

n, n · ∇

_u

γ

₀

= 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

(ξ, t) = θ

1

(x(ξ, t), t), n = n(ξ, t) and

T

u

(u, p

σ

) = {T

_u^ij

(u, p

σ

)} = {−p

σ

δ

ij

+ µ(∇

ui

u

j

+ ∇

uj

u

i

) + (ν − µ)δ

ij

∇

u

· u},

∇

u_i

= ξ

kx_i

∂

ξ_k

and div T

u

(u, p

σ

) = ∇

u

· T

u

(u, p

σ

).

By (3.4) and (3.5) we have respectively

p

σ

= p

1

η

σ

+ p

2

γ

0

and

p

σ

= p

3

η

_Ωt

+ p

4

γ,

where η

σ

= η − ̺

e

, γ

0

= Γ − θ

e

, η

_Ωt

= η − ̺

Ω_t

, p

1

=

T1

0

p

η

(̺

e

+ sη

σ

, Γ ) ds, p

₂

=

T1

0

p

_Γ

(̺

_e

, θ

_e

+ sγ

₀

) ds, p

₃

=

T1

0

p

_η

(̺

_Ωt

+ sη

_Ωt

, Γ ) ds, p

₄

=

T1

0

p

_Γ

(̺

_Ωt

, θ

Ω_t

+ sγ) ds, p

i

> 0 (i = 1, 2, 3, 4).

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 e ω be a set such that e ω ⊂ e Ω and ζ(ξ) = 1 for ξ ∈ e ω. Otherwise we assume that e Ω ∩ S = ∅, e ω ∩ S 6= ∅, e ω ⊂ e Ω. Take any β ∈ e ω ∩ S ⊂ e Ω ∩ S = e S∂ and introduce local coordinates {y} associated with {ξ} by the relation

(3.29) y

k

= α

kl

(ξ

l

− β

l

), α

3k

= n

k

(β), k = 1, 2, 3,

where {α

_kl

} is a constant orthogonal matrix such that e S is described 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

^′

) < y

₃

< F (y

^′

) + d, y

^′

= (y

₁

, y

₂

)}.

Next introduce functions u

^′

, η

^′

, Γ

^′

, γ

^′₀

, γ

^′

, Γ

₁^′

by means of the formulas u

^′_i

(y) = α

ij

u

j

(ξ)|

ξ=ξ(y)

, η

^′

(y) = η(ξ)|

ξ=ξ(y)

,

Γ

^′

(y) = Γ (ξ)|

_ξ=ξ(y)

, γ

₀^′

(y) = γ

0

(ξ)|

_ξ=ξ(y)

, γ

^′

(y) = γ(ξ)|

_ξ=ξ(y)

, Γ

₁^′

(y) = Γ

1

(ξ)|

_ξ=ξ(y)

,

where ξ = ξ(y) is the inverse transformation to (3.29). Further, we introduce new variables by

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 , so e F ∈ W

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

b

u(z) = u

^′

(y)|

y=Φ⁻¹(z)

, η(z) = η b

^′

(y)|

y=Φ⁻¹(z)

, Γ (z) = Γ b

^′

(y)|

_y=Φ−1(z)

, b γ

0

(z) = γ

₀^′

(y)|

_y=Φ−1(z)

,

bγ(z) = γ

^′

(y)|

y=Φ⁻¹(z)

, Γ b

1

(z) = Γ

1

(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 (3.29). We also introduce the following notation:

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

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

0

(z) = bγ

⁰

(z)b ζ(z),

eγ(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= ∅.

Using the above notation we can rewrite problem (3.28) in the following form in an interior subdomain :

ηe u

it

− ∇

uj

T

_u^ij

(e u, e p

σ

) = ηeg

ⁱ

− ∇

uj

B

_u^ij

(u, ζ) − T

_u^ij

(u, p

σ

)∇

uj

ζ

≡ ηeg

ⁱ

+ k

1

, i = 1, 2, 3, η e

σt

+ η∇

u

· e u = ηu · ∇

u

ζ ≡ k

2

,

ηc

v

(η, Γ )eγ

^t

− κ∇

²_u

eγ + Γ p

Γ

(η, Γ )∇

u

· e u

= ηe k +

 µ 2

X

3 i,j=1

(ξ

kxi

∂

ξ_k

u

j

+ ξ

kxj

∂

ξ_k

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 e p

σ

= p

σ

ζ and

B

u

(u, ζ) = {B

_u^ij

(u, ζ)} = {µ(u

_i

∇

uj

ζ + u

_j

∇

ui

ζ) + (ν − µ)δ

_ij

u · ∇

_u

ζ}.

In boundary subdomains we have

ηe b u

it

− b ∇

j

T b

^ij

(e u, e p

σ

) = b ηeg

ⁱ

− b ∇

j

B b

^ij

(b u, b ζ) − b T

^ij

(b u, p

σ

) b ∇

j

ζ b (3.30)

≡ b ηeg

i

+ k

₄ⁱ

, e

η

σt

+ b η b ∇ · e u = b ηb u · b ∇b ζ ≡ k

5

, b

ηc

v

(b η, b Γ )eγ

^t

− κ b ∇

²

e γ + b Γ p

Γˆ

(b η, b Γ ) b ∇ · e u

= b ηe k +

 µ 2

X

3 i,j=1

( b ∇

i

b u

j

+ b ∇

j

b u

i

)

²

+ (ν − µ)( b ∇ · b u)

²

 ζ b

+ b Γ p

Γˆ

(b η, b Γ )b u · b ∇b ζ − κ( b ∇

²

ζbγ + b b ∇b ζ · b ∇bγ)

− b ηc

v

(b η, b Γ )∂

t

θ

Ωt

b ζ ≡ b ηe k + k

6

, T(e b u, e p

_σ

)b n − σ b ∆

Sˆ

ξ · b b nb nb ζ − σ b ∆

Sˆ

t

\

0

e u dt

^′

· bnbn = 2σ R

0

ζ b b n + k

₇

+ k

₈

, b

n · b ∇eγ = e Γ

1

+ k

9

,

where k

₇ⁱ

= b B

^ij

(b u, b ζ)b n

j

, k

8

= −σ(2 b ∇

Tt

0

b u dt

^′

∇b b ζ +

Tt

0

b u dt

^′

∇ b

²

ζ) · b b nb n, k

9

= b

n · b ∇b ζbγ and b T, 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 (3.1). Then 1

2 d dt

\

Ωt



̺v

_x²

+ p

σ̺

̺ ̺

²_x

+ ̺c

v

θ ϑ

²_0x

 dx

+ σ 2

d dt

\

S_t

1 2 δ e

^αβ

n ·

t

\

0

v

pp^α

dt

^′

n ·

t

\

0

v

_pp^β

dt

^′

ds

+ σ 2

d dt

\

St

  n ·

t

\

0

v

p¹p²

dt

^′

  

2

ds

+ σ 2

d dt

\

St

X

2 i=1

 1 2 n ·

t

\

0

v

pⁱpⁱ

dt

^′

+ 2(H(·, 0) + 2/R

e

)



2

ds

+ c

0

(kv

x

k

²_1,Ωt

+ k̺

_Ωt

k

²_0,Ωt

+ k̺

σx

k

²_0,Ωt

+ k̺

σt

k

²_0,Ωt

+ kϑ

0xx

k

²_0,Ωt

)

≤ ε 

kv

xt

k

²_0,Ωt

+ kϑ

0xt

k

²_0,Ωt

+

t

\

0

v dt

^′

2 0,Ω_t

+ kH(·, 0) + 2/R

_e

k

²_0,S¹

+ kR(·, t) − R(·, 0)k

²_2,S¹

 + C

1

(|v|

²_1,0,Ωt

+ k̺

_Ωt

k

²_0,Ωt

+ kϑk

²_0,Ωt

+ kϑ

0x

k

²_0,Ωt

+ kϑ

0t

k

²_0,Ωt

+ kf k

²_1,Ωt

+ krk

²_1,Ωt

+ kθ

1

k

²_2,Ωt

) + C

₂

(X

₃

Y

₃

+ kH(·, 0) + 2/R

_e

k

⁴_0,S¹

),

where the summation over the repeated indices (α, β = 1, 2) and coordinates (x, p = (p

¹

, p

²

)) is assumed, e δ

^αβ

on each boundary part Σ

t

= S

t

∩{ζ(x) 6= 0}

(ζ belongs to a partition of unity of Ω

_t

) is of the form e δ

^αβ

= δ

^αβ

+ 2ε

^αβ

, ε

^αβ

= −F

p^α

F

_p^β

(1 + F

_p²1

+ F

_p²2

)

⁻¹

, F is the function such that in the local coordinates {y}, P

t

is described by the formula

(3.31) y

i

= p

ⁱ

(i = 1, 2), y

3

= F (p

¹

, p

²

, t) and supp ζ is so small that |F

p

| ≤ 1/2. Moreover ,

X

₃

= |v|

²_2,1,Ω

t

+ |̺

_σ

|

²_2,1,Ωt

+ |ϑ

₀

|

²_2,1,Ωt

+ k̺

_Ωt

k

²_0,Ωt

, Y

₃

= X

₃

+ kvk

²_3,Ω

t

+ kϑ

_0x

k

²_2,Ωt

+ kϑk

²_0,Ω

t

+ k̺

_Ωt

k

²_0,Ωt

+

t

\

0

kvk

²_3,Ω

t′

dt

^′

.

P r o o f. Similarly to [16] (see the proof of Lemma 3.4) we obtain the following estimate for interior subdomains:

(3.32) 1 2

d dt

\

e

Ω



ηe u

²_ξ

+ p

ση

η eη

²_Ωtξ

+ ηc

v

Γ γ e

_ξ²

 A dξ

+ µ

2 ke u

ξ

k

²_1,^Ω

e + κ

θ

^∗

keγ

^ξξ

k

²_0,^Ω

e + ke η

_Ωt

k

²_1,^Ω

e

≤ ε(ke u

ξξ

k

²

0,^Ω

e + kη

σξ

k

²

0,^Ω

e + keγ

^ξξ

k

²

0,^Ω

e ) + C

1

(|u|

²

1,0,^Ω

e + kvk

²_1,Ωt

+ kγ

0ξ

k

²_0,^Ω

e + kγk

²

0,^Ω

e + kϑ

_0t

k

²_0,Ωt

+ kη

_Ω

t

k

²_0,

e

Ω

+ kegk

²_0,^Ω

e + ke kk

²

0,^Ω

e ) + C

2

h

X

3

( e Ω) +

t

\

0

kuk

²

3,^Ω

e dt

^′



Y

3

( e Ω) + kγk

²

2,^Ω

e (kϑ

0t

k

²_0,Ωt

+ kvk

²_1,Ωt

) i , where

X

3

( e Ω) = |u|

²

2,1,^Ω

e + |̺

σ

|

²_2,1,^Ω

e + |γ

0

|

²_2,1,^Ω

e + kη

_Ωt

k

²_0,^Ω

e , Y

3

( e Ω) = X

3

( e Ω) + kuk

²

3,^Ω

e + kγk

²

3,^Ω

e + kη

_Ωt

k

²_0,^Ω

e +

t

\

0

kuk

²_3,^Ω

e dt

^′

. Now, we consider subdomains near the boundary. Differentiate (3.30)

1

with respect to τ , multiply the result by e u

τ

J and integrate over b Ω (J is the Jacobian of the transformation x = x(z)). Next, divide (3.30)

3

by b Γ , differentiate the result with respect to τ , multiply by eγ

^τ

J and integrate over Ω. Hence using Lemma 5.1 of [18] we get b

1 2

d dt

\

Ωˆ



b ηe u

²_τ

+ p

σ ˆη

b

η eη

²_Ωtτ

+ ηc b

v

Γ b eγ

τ²



J dz + µ

2 ke u

τ

k

²_1,^Ω

b + κ

θ

^∗

keγ

τ z

k

²

0,^Ω

b −

\

Sˆ

(b nb T(e u, e p

σ

))

,τ

u e

τ

J dz

^′

− κ

\

Sˆ

 b n 1

Γ b ∇eγ b



,τ

eγ

τ

J dz

^′

≤ ε(ke u

_zz

k

²

0,^Ω

b + kb η

_σz

k

²

0,^Ω

b + kbγ

0zz

k

²

0,^Ω

b ) + C

1

(|b u|

²

1,0,^Ω

b + kvk

²_1,Ωt

+ kbγ

^0τ

k

²

0,^Ω

b

+ kbγk

²_0,^Ω

b + kϑ

0t

k

²_0,Ωt

+ kb η

_Ωt

k

²_0,^Ω

b + kegk

²_1,^Ω

b + ke kk

²

1,^Ω

b ) + C

₂

h

X

₂

( b Ω) +

t

\

0

kb uk

²

3,^Ω

b dt

^′



Y

₂

( b Ω) + kbγk

²_2,^Ω

b (kϑ

_0t

k

²_0,Ωt

+ kvk

²_1,Ω

t

) i

,

where X

2

( b Ω) and Y

2

( b Ω) are defined analogously to X

2

( e Ω) and Y

2

( e Ω).

Using the boundary condition (3.30)

4

we have (3.33) −

\

b

S

(b nb T(e u, e p

σ

))

,τ

u e

τ

J dz

^′

≤ − σ 2

d dt

\

b

S

g

^αβ

n · b

t

\

0

u e

pp^α

dt

^′

b n ·

t

\

0

u e

_pp^β

dt

^′

J dz

^′

− σ

\

b

S

( b H(·, 0) + 2/R

e

)b ζ · e u

pp

· bnJ dz

^′

+ ε 

t

\

0

e u dt

^′

2

2,^S

b + ke u

zz

k

²

0,^Ω

b + k( b H(·, 0) + 2/R

e

)b ζk

²

0,^S

b + kR(·, t) − R(·, 0)k

²_2,S¹



+ C

₂

 kb uk

²

0,^Ω

b + kb uk

²

2,^Ω

b

t

\

0

u dt e

^′

2 3,^Ω

b

 .

By the boundary condition (3.30)

5

we get (3.34) − κ

\

b

S

 b n · 1

Γ b ∇bγ b



,τ

e γ

τ

J dz

^′

≤ εkbγ

0zz

k

²

0,^Ω

b + C

₁

(kbγk

²_0,^Ω

b + kbγ

0z

k

²

0,^Ω

b + k e Γ

₁

k

²

2,^Ω

b ) + C

2

kbγk

²_2,^Ω

b

 kbγ

0

k

²

2,^Ω

b + kbγk

²_2,^Ω

b + kb η

σ

k

²

2,^Ω

b +

t

\

0

b u dt

^′

2 3,^Ω

b

 . To obtain (3.33) and (3.34) we have applied the interpolation inequality (2.2) (see Lemma 2.1).

For the quantities 1

2 d dt

\

Ωˆ

p

_{σ ˆη}

b

η eη

²_Ωtn

J dz + c

₀

ke η

_Ωtn

k

²

0,^Ω

b , 1

2 d dt

\

Ωˆ

ηe b u

²_3n

Jdz + c

0

ke u

3nn

k

²_0,^Ω

b ,

ke η

_Ωt

k

²_0,Ωt

, ke u

^′_zτ

k

²_0,^Ω

b , ke η

_Ωtτ

k

²_0,^Ω

b , ke u

^′_nn

k

²_0,^Ω

b , 1 2

d dt

\

Ωˆ

ηe b u

²_n

J dz, 1

2 d dt

\

Ωˆ

b ηc

v

Γ b γ e

_n²

Jdz + κ

θ

^∗

keγ

ⁿⁿ

k

²_0,^Ω

b

we obtain the same estimates as in the proof of Lemma 3.4 of [16]. Therefore, we have

(3.35) 1 2

d dt

\

Ωˆ



ηe b u

²_z

+ p

_σ

b

^η

b

η eη

²_Ωtz

+ ηc b

_v

Γ b e γ

_z²

 J dz

+ σ 2

d dt

\

Sˆ

h g

^αβ

b n ·

t

\

0

e

u

pp^α

dt

^′

n · b

t

\

0

e u

_pp^β

dt

^′

+ 2( b H(·, 0) + 2/R

e

)b ζ b n ·

t

\

0

e u

pp

dt

^′

i

J dz

^′

+ µ 2 ke u

z

k

²

1,^Ω

b + κ

θ

^∗

keγ

^zz

k

²_0,^Ω

b + c

0

ke η

_Ωt

k

²_1,^Ω

b

≤ ε  ke u

zz

k

²

0,^Ω

b + kb η

σz

k

²

0,^Ω

b + kbγ

^0zz

k

²

0,^Ω

b + ke u

zt

k

²

0,^Ω

b + keγ

^0zt

k

²

0,^Ω

b +

t

\

0

u dt e

^′

2

2,^S

b + k( b H(·, 0) + 2/R

e

)b ζk

²

0,^S

b + kR(·, t) − R(·, 0)k

²_2,S¹

 + C

1

(|b u|

²

1,0,^Ω

b + kvk

²_1,Ωt

+ kbγ

^0τ

k

²

0,^Ω

b + kbγk

²_0,^Ω

b + kϑ

0t

k

²_0,Ωt

+ kb η

_Ωt

k

²_0,^Ω

b + kegk

²_1,^Ω

b + ke kk

²

1,^Ω

b ) + C

₂

h

X

₂

( b Ω) +

t

\

0

kb uk

²

3,^Ω

b dt

^′



Y

₂

( b Ω) + kbγk

²_2,^Ω

b (kϑ

_0t

k

²_0,Ωt

+ kvk

²_1,Ωt

) i . We estimate the second term on the left-hand side of (3.35) in the same way as in the proof of Lemma 4.4 of [19]. Going back to the variables ξ in (3.35), next from the resulting estimate and (3.32), after summing over all neighbourhoods of the partition of unity and finally going back to the variables x and using (3.26) we get

(3.36) 1 2

d dt

\

Ω_t



̺v

_x²

+ p

σ̺

̺ ̺

²_σx

+ ̺c

v

θ ϑ

²_0x

 dx

+ σ 2

d dt

\

S_t

1 2 e δ

^αβ

n ·

t

\

0

v

pp^α

dt

^′

n ·

t

\

0

v

_pp^β

dt

^′

ds

+ σ 2

d dt

\

St

  n ·

t

\

0

v

p¹p²

dt

^′

  

2

ds

+ σ 2

d dt

\

St

X

2 i=1

 1 2 n ·

t

\

0

v

_pipⁱ

dt

^′

+ 2( b H(·, 0) + 2/R

_e

)



2

ds