BY A FREE BOUNDARY

VOL. 79 1999 NO. 2

ON NONSTATIONARY MOTION OF A FIXED MASS OF A VISCOUS COMPRESSIBLE BAROTROPIC FLUID BOUNDED

BY A FREE BOUNDARY

BY

EWA Z A D R Z Y ´ N S K A

WOJCIECH M. Z A J A ¸ C Z K O W S K I (WARSZAWA)

1. Introduction. In this paper we consider the global motion of a drop of a viscous barotropic fluid in the general case, i.e. without assuming any conditions on the form of the pressure p = p(̺). Here ̺ = ̺(x, t) (where x ∈ Ω t , t ∈ [0, T ], Ω t ⊂ R ³ is a bounded domain of the drop at time t) is the density of the drop.

Next, let v = v(x, t) (v = (v i ) i=1,2,3 ) denote the velocity of the fluid, f = f (x, t) the external force field per unit mass, µ and ν the constant viscosity coefficients, and p 0 the external (constant) pressure. Then the mo- tion of the drop is described by the following system of equations (see [2, Chs. 1, 2]):

̺[v t + (v · ∇)v] − div T (v, p) = ̺f in e Ω ^T ,

̺ t + div(̺v) = 0 in e Ω ^T ,

T _{n = −p 0 n on e S} ^T _,

v · n = − φ t

|∇φ| on e S ^T ,

̺| t=0 = ̺ 0 , v| t=0 = v 0 in Ω, (1.1)

where e Ω ^T = S

t∈(0,T ) Ω t × {t}, e S ^T = S

t∈(0,T ) S t × {t}, S t = ∂Ω t , φ(x, t) = 0 describes S t (at least locally), n is the unit outward vector normal to the boundary, i.e. n = ∇φ/|∇φ|, and Ω = Ω t | t=0 = Ω 0 . In (1.1), T ₌ T _{(v, p) =} {T ij } i,j=1,2,3 = {−pδ ij + µ(v ix

+ v jx

) + (ν − µ)δ ij div v} i,j=1,2,3 is the stress tensor. Moreover, we assume ν > ¹ ₃ µ > 0.

Let the domain Ω be given. Then by (1.1) 4 , Ω t = {x ∈ R ³ _{: x =} x(ξ, t), ξ ∈ Ω}, where x = x(ξ, t) is the solution of the Cauchy problem

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

Key words and phrases: free boundary, compressible viscous barotropic fluid, global existence.

[283]

∂x

∂t = v(x, t), x| t=0 = ξ ∈ Ω, ξ = (ξ 1 , ξ 2 , ξ 3 ).

(1.2)

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

x = ξ +

t

\

0

u(ξ, t ^′ ) dt ^′ ≡ X u (ξ, t), (1.3)

where u(ξ, t) = v(X u (ξ, t), t). Moreover, by (1.1) 4 , S t = {x : x = x(ξ, t), ξ ∈ S = ∂Ω}.

By the continuity equation (1.1) 2 and the kinematic condition (1.1) 4 the total mass is conserved, i.e.

\

Ω

̺(x, t) dx =

\

Ω

̺ 0 (ξ) dξ = M, (1.4)

where M is a given constant.

In [15] the local existence of a unique solution is proved for a problem analogous to (1.1), but describing the motion of a drop of a viscous heat–

conducting fluid.

Let u = u(ξ, t), η = η(ξ, t) denote v and ̺ written in Lagrangian coor- dinates. In the same way as in [15] (see Theorem 4.2 of [15]) one can prove the local existence of a unique solution (v, ̺) of problem (1.1) such that u ∈ A T,Ω , η ∈ B T,Ω , where A T,Ω ≡ A T,Ω

, B T,Ω ≡ B T,Ω

and

B T,Ω

= {f ∈ C(iT, (i + 1)T ; H ² (Ω iT )) : (1.5)

f t ∈ C(iT, (i + 1)T ; H ¹ (Ω iT )) ∩ L 2 (iT, (i + 1)T ; H ² (Ω iT )), f tt ∈ C(iT, (i + 1)T ; L 2 (Ω)) ∩ L 2 (iT, (i + 1)T ; H ¹ (Ω iT ))}, A T,Ω

= B T,Ω

∩ L 2 (iT, (i + 1)T ; H ³ (Ω iT )),

(1.6)

i ∈ N ∪ {0}, T ≤ T ∗ , where T ∗ > 0 is a certain constant.

The aim of this paper is to prove the existence of a global-in-time solu- tion of problem (1.1) near a constant state. Consider the equation

p(̺) = p 0 , (1.7)

where ̺ ∈ R _{+ , p ∈ C} ³ ₍ R _{+ ), and p} ^′ _{> 0.}

We introduce the following definition of a constant state.

Definition 1.1. Let f = 0. Then by a constant (equilibrium) state we mean a solution (v, ̺) of problem (1.1) such that v = 0, ̺ = ̺ e , and Ω t = Ω e

for t ≥ 0, where ̺ e is a solution of (1.7) and |Ω e | = M/̺ e (|Ω e | = vol Ω e ).

First, in Section 2 we derive a differential inequality (2.58) which enables

extending the local solution of (1.1) step by step from the interval [0, T ] to

[0, ∞). To prove the global existence we also use Lemma 2.1, which gives

an energy estimate (2.8), and Lemmas 3.3–3.4. The above lemmas yield in

particular global estimates for kvk ² _L

_(Ω

₎ and kp σ k ² _L

_(Ω

₎ (where p σ = p−p 0 ),

which are used in the proofs of Lemma 3.4 and Theorem 3.9, the main result of the paper.

The global motion of a fluid described by (1.1) has been considered earlier in papers [7] and [17].

In [17] the global existence for problem (1.1) is proved for a special form of p = p(̺):

p = a 0 ̺ ^α , (1.8)

where a 0 > 0 and α > 0 are constants. The global solution obtained in [17]

is more regular than the one obtained in this paper.

A result analogous to that of [17] is proved (under assumption (1.8)) in [18] for the fluid bounded by a free boundary the shape of which is governed by surface tension.

Paper [7] of V. A. Solonnikov and A. Tani is concerned with problem (1.1) with the boundary condition T n − σHn = 0 (where H is the double mean curvature of S t , and σ > 0 is the constant coefficient of surface tension).

In [7] the existence of a solution is proved in some anisotropic Sobolev–

Slobodetski˘ı spaces; it is a little less regular than ours. To prove the local existence the authors of [7] apply potential techniques.

Both in [17] and in [7] the energy conservation law is used in order to derive a global estimate for kvk ² _L

_(Ω

t

) .

Papers [8]–[10] are concerned with the free boundary problem for a vis- cous barotropic self-gravitating fluid with p of the form (1.8).

Next, papers [11]–[14] are devoted to the free boundary problem for a vis- cous heat-conducting fluid under the assumption that the internal energy ε has a special form:

ε = a 0 ̺ ^α + h(̺, θ),

where a 0 > 0, α > 0, h(̺, θ) ≥ h ∗ > 0; a 0 , α and h ∗ are constants.

The free boundary problem for a viscous incompressible fluid was ex- amined by V. A. Solonnikov in [3]–[6].

Finally, we present the notation used in the paper. We denote by k · k l,Q

(where l ≥ 0, Q ⊂ R ³ ) the norms in the Sobolev spaces H ^l (Q), and by Γ _k ^l (Q) (l > 0, k ≥ 0, Q ⊂ R ³ ) the space of functions u = u(x, t) (x ∈ Q, t ∈ (0, T ), T > 0) with the norm

kuk _Γ

l

(Q) = X

i≤l−k

k∂ _t ⁱ uk l−i,Q ≡ |u| l,k,Q .

2. Differential inequality. Assume that the existence of a sufficiently smooth local solution of problem (1.1) has been proved and let

f = 0.

(2.1)

In this section we obtain a special differential inequality which enables us to prove the global existence. To get the inequality we consider the motion near the constant state. Let

p σ = p − p 0 , ̺ σ = ̺ − ̺ e , (2.2)

where ̺ e is introduced in Definition 1.1. Then problem (1.1) takes the form

̺[v t + (v · ∇)v] − div T _{(v, p σ ) = 0 in Ω t} , t ∈ (0, T ),

̺ σt + div(̺v) = 0 in Ω t , t ∈ (0, T ), T _{(v, p σ )n = 0 on S t} , t ∈ (0, T ),

̺ σ | t=0 = ̺ σ0 = ̺ 0 − ̺ e , v| t=0 = v 0 , in Ω.

(2.3)

In the sequel we use the following Taylor formula for p σ :

p σ = (̺ − ̺ e )

1

0

p ^′ (̺ e + s(̺ − ̺ e )) ds = p 1 ̺ σ , (2.4)

where the function p 1 is positive.

Now, let ̺ ∗ and ̺ ^∗ be positive constants such that

̺ ∗ < ̺ < ̺ ^∗ for x ∈ Ω t , t ∈ [0, T ].

(2.5)

In the lemmas below we denote by ε small constants, by c 0 < 1 a positive constant depending on µ, ν, and by c a positive constants depending on T (the time of local existence), ̺ ∗ , ̺ ^∗ ,

T

t 0 kvk ² _3,Ω

t′

dt ^′ , kSk 5/2 , on the parameters which guarantee the existence of the inverse transformation to x = x(ξ, t) and on the constants of imbedding theorems and Korn inqualities. We do not distinguish different ε’s or c’s.

We underline that all the estimates below are obtained under the as- sumption that there exists a local-in-time solution of problem (1.1), so all the quantities ̺ ∗ , ̺ ^∗ , T ,

T

t 0 kvk ² _3,Ω

t′

dt ^′ , kSk 5/2 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 solution (see [15]).

Now, assume the relations

\

Ω

̺v dx = 0, (2.6)

\

Ω

̺v · η dx = 0, (2.7)

where η = a + b × x and a and b are arbitrary vectors.

Lemma 2.1. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v ² + p 1

̺ ̺ ² _σ



dx + c 0 kvk ² _1,Ω

≤ cX ₁ ² (1 + X 1 ), (2.8)

where X 1 = kvk ² _2,Ω

+ k̺ σ k ² _2,Ω

.

P r o o f. Multiplying (2.3) 1 by v, integrating over Ω t and using the con- tinuity equation (2.3) 2 , boundary condition (2.3) 4 and (2.4) we obtain

1 2

d dt

\

Ω

̺v ² dx + µ

2 E Ω

(v) + (ν − µ)kdiv vk ² _0,Ω

−

\

Ω

p 1 ̺ σ div v dx = 0, (2.9)

where E Ω

(v) =

T

Ω

P 3

i,j=1 (v ix

+ v jx

) ² dx.

In [13] it is proved that µ

2 E Ω

(v) + (ν − µ)kdiv vk ² _0,Ω

≥ cE Ω

(v), where c > 0 is a constant.

Next, by the continuity equation (2.3) 2 we have

−

\

Ω

p 1 ̺ σ div v dx = 1 2

d dt

\

Ω

p 1 ̺ ² _σ

̺ dx + J, (2.10)

where

|J| ≤ ε(k̺ σt k ² _0,Ω

+ kvk ² _1,Ω

) + cX ₁ ² (1 + X 1 ).

(2.11)

Moreover, in view of assumptions (2.6) and (2.7), Lemma 5.2 of [17]

yields

kvk ² _1,Ω

≤ c(E Ω

(v) + k̺ σ k ² _0,Ω

kvk ² _0,Ω

) (2.12)

and by the continuity equation (2.3) 2 ,

k̺ σt k ² _0,Ω

≤ ckvk ² _1,Ω

+ ckvk ² _1,Ω

k̺ σ k ² _2,Ω

. (2.13)

Taking into account (2.9)–(2.13) we get estimate (2.8).

Lemma 2.2. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v _t ² + p ̺σ

̺ ̺ ² _σt



dx + c 0 (kv t k ² _1,Ω

+ k̺ σt k ² _0,Ω

) (2.14)

≤ ckvk ² _1,Ω

+ cY ₁ ² (1 + X 2 ), where

X 2 = |v| ² _2,0,Ω

+ |̺ σ | ² _2,0,Ω

+

t

\

0

kvk ² _3,Ω

t′

dt ^′ , (2.15)

Y 1 = X 2 −

t

\

0

kvk ² _3,Ω

t′

dt ^′ . (2.16)

P r o o f. Differentiating (2.3) 1 with respect to t, multiplying by v t and

integrating over Ω t yields

1 2

d dt

\

Ω

̺v ² _t dx + µ

2 E Ω

(v t ) + (ν − µ)kdiv v t k ² _0,Ω

(2.17)

−

\

Ω

p σ̺ ̺ σt div v t dx ≤ cY ₁ ² (1 + X 2 ),

where we have used the boundary condition (2.3) 4 .

By Lemma 5.3 of [17] we have the following Korn type inequality:

kv t k ² _1,Ω

≤ c[E Ω

(v t ) + Y ₁ ² (1 + Y 1 )].

(2.18)

Finally, using the continuity equation (2.3) 3 we get

−

\

Ω

p σ̺ ̺ σt div v t dx = 1 2

d dt

\

Ω

p σ̺

̺ ̺ ² _σt dx + J, (2.19)

where

|J| ≤ ε(kv t k ² _1,Ω

+ k̺ σt k ² _0,Ω

) + cY ₁ ² (1 + Y 1 ).

(2.20)

In view of inequalities (2.17)–(2.20) and (2.13) we obtain (2.14).

Lemmas 2.1 and 2.2 yield

Lemma 2.3. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺(v ² + v ² _t ) + p 1

̺ ̺ ² _σ + p σ̺

̺ ̺ ² _σt

 dx (2.21)

+ c 0 (kvk ² _1,Ω

+ kv t k ² _1,Ω

+ k̺ σt k ² _0,Ω

) ≤ cY ₁ ² (1 + X 2 ), where X 2 and Y 1 are given by (2.15) and (2.16), respectively.

Next, we obtain

Lemma 2.4. Let v, ̺ σ be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v _tt ² + p σ̺

̺ ̺ ² _σtt



dx + c 0 (kv tt k ² _1,Ω

+ k̺ σtt k ² _0,Ω

)

≤ c(kvk ² _1,Ω

+ kv t k ² _1,Ω

) + cX 2 Y 2 (1 + X ₂ ² ), where X 2 is given by (2.15) and

Y 2 = |v| ² _3,1,Ω

+ k̺ σ k ² _2,Ω

+ k̺ σt k ² _2,Ω

+ k̺ σtt k ² _1,Ω

. (2.22)

The above lemma can be proved in the same way as Lemmas 2.1 and 2.2. To estimate E Ω

(v tt ) we use here Lemma 5.4 of [17].

In order to obtain estimates for derivatives with respect to x we rewrite

problem (2.3) in Lagrangian coordinates. We have

ηu it − ∇ u

T uij (u, p σ ) = 0 (i = 1, 2, 3) in Ω ^T ≡ Ω × (0, T ),

η σt + η∇ u · u = 0 in Ω ^T ,

T _{u (u, p σ )n u = 0 on S} ^T ≡ S × (0, T ), u| t=0 = v 0 , η σ | t=0 = ̺ σ0 , in Ω,

(2.23)

where η(ξ, t) = ̺(X u (ξ, t), t), u(ξ, t) = v(X u (ξ, t), t) (X u is given by (1.3)), η σ = η − ̺ e , ̺ σ0 = ̺ 0 − ̺ e , T _{u (u, p σ ) = {T uij (u, p σ )} i,j=1,2,3 = {−p σ δ ij +} µ(∂ x

ξ k ∂ ξ

u j + ∂ x

ξ k ∂ ξ

u i ) + (ν − µ)δ ij div u u} i,j=1,2,3 , div u u = ∇ u · u =

∂ x

ξ k ∂ ξ

u i , ∇ u = (ξ kx

∂ ξ

) i=1,2,3 , ∇ u

= ξ kx

∂ ξ

, ∂ x

ξ k are the elements of the matrix ξ x which is inverse to x ξ = I +

T

t

0 u ξ (ξ, t ^′ ) dt ^′ , I = {δ ij } i,j=1,2,3

is the unit matrix, n u = n(X u (ξ, t), t) = ∇ x φ(x, t)/|∇ x φ(x, t)| x=X

(ξ,t) (S t

is determined at least locally by the equation φ(x, t) = 0) and summation over repeated indices is assumed.

By (2.4) we have p σ = p 1 η σ , where p 1 = p 1 (η).

Now, 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 6= ∅, e ω ∩ S 6= ∅, e ω ⊂ e Ω. Take any β ∈ e ω ∩ S = e S and introduce local coordinates {y} associated with {ξ} by

y k = α kl (ξ l − β l ), α 3k = n k (β), k = 1, 2, 3, (2.24)

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 ∈ H ^5/2 and

Ω = {y : |y e i | < d, i = 1, 2, F (y ^′ ) < y 3 < F (y ^′ ) + d, y ^′ = (y 1 , y 2 )}.

Next, we introduce u ^′ , η ^′ , η _σ ^′ by

u ^′ _i (y) = α ij u j (ξ)| ξ=ξ(y) (i = 1, 2, 3), η ^′ (y) = η(ξ)| ξ=ξ(y) , η ^′ _σ (y) = η ^′ (y) − ̺ e ,

where ξ = ξ(y) is the inverse transformation to (2.24).

Next, 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 ∈ H ³ is an extension of F ).

Let

Ω = Φ( e b Ω) = {z : |z i | < d, i = 1, 2, 0 < z 3 < d} and S = Φ( e b S).

(2.25) Define

b

u(z) = u ^′ (y)| y=Φ

(z) , η(z) = η b ^′ (y)| y=Φ

(z) , η b σ (z) = b η(z) − ̺ e .

Set b ∇ k = ξ lx

z iξ

∇ z

| _ξ=χ

(z) , where χ(ξ) = Φ(ψ(ξ)) and y = ψ(ξ) is descri- bed by (2.24). We also introduce the following notation:

e

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

(z) .

Using the above notation we rewrite problem (2.23) in the following form in an interior subdomain:

ηe u it −∇ u

T uij (e u, e p σ ) = −∇ u

B uij (u, ζ)−T uij (u, p σ )∇ u

ζ ≡ k 1 , i = 1, 2, 3, e

η σt + η∇ u · e u = ηu · ∇ u ζ ≡ k 2 ,

where e p σ = p σ ζ and B _u (u, ζ) = {B uij (u, ζ)} i,j=1,2,3 = {µ(u i ∇ u

ζ + u j ∇ u

ζ) + (ν − µ)δ ij u · ∇ u ζ} i,j=1,2,3 .

In boundary subdomains we have b

η e u it − b ∇ j T b ij = − b ∇ j B b ij (b u, b ζ) − b T ij (b u, p σ ) b ∇ j ζ ≡ k b 3i , i = 1, 2, 3, e

η σt + b η b ∇ · e u = b η b u · b ∇b ζ ≡ k 4 , b

T _{(e u, e p σ )b n = k 5 ,} (2.26)

where k 5i = b B ij (b u, b ζ)b n j , b ∇ = ( b ∇ j ) j=1,2,3 and b T _{and b} B indicate that the operator ∇ u is replaced by b ∇.

In Lemmas 2.5–2.7 below we denote z 1 , z 2 , by τ , i.e. τ = (z 1 , z 2 ), and z 3 by n.

Lemma 2.5. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v _x ² + p σ̺

̺ ̺ ² _σx



dx + c 0 (kvk ² _2,Ω

+ k̺ σx k ² _0,Ω

) (2.27)

≤ c(kvk ² _1,Ω

+ kv t k ² _1,Ω

+ k̺ σt k ² _0,Ω

+ kp σ k 0,Ω

) + cX ₂ ² (1 + X 2 ), where X 2 is given by (2.15), v _x ² = P 3

i,j=1 v ² _ix

, and ̺ ² _σx = P 3 i=1 ̺ ² _σx

. P r o o f. First, we consider the following elliptic problem:

µ∇ ² _u u + ν∇ u ∇ u · u − p ση ∇ u η = ηu t in Ω,

div u u = div u u in Ω,

T _{u (u, p σ )n u = 0 on S.}

(2.28)

Since the complementarity condition for (2.28) is satisfied we can apply to problem (2.28) the Agmon–Douglis–Nirenberg theory (see [1]). Thus, we get

kuk ² _2,Ω + kη σ k ² _1,Ω ≤ c(kηu t k ² _0,Ω + kdiv u uk ² _1,Ω ) (2.29)

≤ c(ku t k ² _0,Ω + kdiv uk ² _1,Ω + cX ₂ ² (Ω)(1 + X 2 (Ω))),

where we have used the fact that kdiv u u − div uk ² _1,Ω ≤ εkuk ² _2,Ω (ε > 0 is sufficiently small), and

X 2 (Ω) = |u| ² _2,0,Ω + |η σ | ² _2,0,Ω +

t

\

0

kuk ² _3,Ω dt ^′ . (2.30)

In view of (2.29) we see that in order to obtain inequality (2.27) it remains to get appropriate estimates for kdiv uk ² _1,Ω and for ¹ _{2 dt} ^d

T

Ω

(̺v _x ² + (p σ̺ /̺)̺ ² _σx ) dx. To do this, consider first boundary subdomains. Differen- tiate (2.26) 1 with respect to τ , multiply the result by e u τ J (J is the Jacobian of the transformation x = x(z)) and integrate over b Ω. Hence using the Korn inequality and equation (2.26) 2 we obtain

1 2

d dt

\

b

Ω

b

η e u ² _τ J dz + c 0 ke u τ k ²

1, ^Ω b (2.31)

−

\

b

S

(b T _{(e u, e p σ )b n) ,τ u e τ J dz −}

\

b

Ω

e

p στ ∇ · e u τ J dz

≤ ε(kb η σ k ²

0, ^Ω b + ke u τ k ²

1, ^Ω b ) + c(kb uk ²

1, ^Ω b + kp σ k ²

0, ^Ω b ) + cX ₂ ² ( b Ω)(1 + X 2 ( b Ω)), where

X 2 ( b Ω) = |b u| ²

2,0, ^Ω b + |b η σ | ²

2,0, ^Ω b +

t

\

0

kb uk ²

3, ^Ω b dt ^′ , u e ² _τ = X 3 i=1

X 2 j=1

e u iz

. (2.32)

Using the boundary condition (2.26) 3 we have

−

\

b

S

(b T _{(e u, e p σ )b n) ,τ e u τ J dτ = −}

\

b

S

( b B ij (b u, b ζ)b n j ) ,τ u e iτ J dτ (2.33)

=

\

b

S

∂ _τ ^1/2 ( b B ij (b u, b ζ)b n j )∂ _τ ^1/2 (e u iτ J) dτ ≤ εke u τ k ²

1, ^Ω b + kb uk ²

1, ^Ω b + cX ₂ ² ( b Ω), where to use the derivative ∂ τ ^1/2 we have to apply the Fourier transformation.

Next,

−

\

b

Ω

e

p στ ∇ u · e u τ J dz = −

\

b

Ω

p _σ b ^η η e στ ∇ · e b u τ J dz + J 1 , (2.34)

where |J 1 | ≤ εke u τ k ²

1, ^Ω b + ckp σ k ²

0, ^Ω b and

−

\

b

Ω

p _σ b ^η e η στ ∇ · e b u τ J dz = 1 2

d dt

\

b

Ω

p _σ b ^η b

η e η _στ ² J dz + J 2 , (2.35)

where

|J 2 | ≤ εke η στ k ²

0, ^Ω b + ckb uk ²

1, ^Ω b + cX ₂ ² ( b Ω).

(2.36)

Taking into account (2.31), (2.33)–(2.36) and assuming that ε is sufficiently small we obtain

1 2

d dt

\

b

Ω

 b

η e u _τ ² + p _σ b ^η b η η e _στ ²



J dz + c 0 ke u τ k ²

1, ^Ω b (2.37)

≤ εkb η στ k ²

0, ^Ω b + c(kb uk ²

1, ^Ω b + kp σ k ²

0, ^Ω b ) + cX ₂ ² ( b Ω)(1 + X 2 ( b Ω)).

Now, applying the operator (µ + ν)∇ z

to (2.26) 2 , dividing the result by b

η, adding to (2.26) 1 and multiplying both sides of the result by p _σ b ^η gives µ + ν

b

η p _σ b ^η ∇ z

e η σt + p ² _σ b ^η ∇ z

e η σ

(2.38)

= p ² _σ b ^η η b σ ∇ z

ζ − p b 1 p _σ b ^η η b σ ∇ z

ζ + p b _σ b ^η k 3i + µp _σ b ^η ( b ∇ ² u e i − b ∇ i ∇ · e b u) + (µ + ν)p _σ b ^η ( b ∇ i − ∇ z

) b ∇ · e u + µ + ν

b

η p _σ b ^η ∇ z

(b ηb u · b ∇b ζ)

− p _σ b ^η η e b u it − µ + ν b

η p _σ b ^η ∇ z

b η b ∇ · e u, i = 1, 2, 3.

Multiplying the normal component of (2.38) by η σn J and integrating over Ω we obtain b

1 2

d dt

\

b

Ω

p _σ b ^η b

η η e _σn ² J dz + c 0 ke η σn k ²

0, ^Ω b (2.39)

≤ (ε + cd)ke u nn k ²

0, ^Ω b + εke η σn k ²

0, ^Ω b + c(ke u zτ k ²

0, ^Ω b + kb uk ²

1, ^Ω b + ke u t k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cX ₂ ² ( b Ω)(1 + X 2 ( b Ω)), where d is from formula (2.25).

Now, we write (2.26) 1 in the form b

η e u it − µ∆e u i − ν∇ z

∇ · e u = b ∇ i p e σ + k 3i − k 6i , (2.40)

where k 6i = (µ∆e u i + ν∇ z

∇ · e u) − (µ b ∇ ² e u i + ν b ∇ i ∇ · e b u).

Multiplying the third component of (2.40) by e u 3nn J and integrating over Ω yields b

1 2

d dt

\

b

Ω

b

η e u _3n ² J dz + c 0 ke u 3nn k ²

0, ^Ω b (2.41)

≤ (ε + cd)ke u nn k ²

0, ^Ω b + c(ke u zτ k ²

0, ^Ω b + kb uk ²

1, ^Ω b + ke u t k ²

1, ^Ω b + ke η σn k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cX ₂ ² ( b Ω)(1 + X 2 ( b Ω)).

For an interior subdomain the following estimate is obtained in the same

way as (2.37):

1 2

d dt

\

e

Ω



ηe u ² _ξ + p ση

η e η _σξ ²



A dξ + c 0 ke uk ²

2, ^Ω e (2.42)

≤ ε(ke η σξ k ²

0, ^Ω e + ke u ξξ k ²

0, ^Ω e ) + c(kuk ²

1, ^Ω e + kp σ k ² _0,Ω

) + cX ₂ ² ( e Ω)(1 + X 2 ( e Ω)), where

X 2 ( e Ω) = |u| ²

2,0, ^Ω e + |η σ | ²

2,0, ^Ω e +

t

\

0

kuk ²

3, ^Ω e dt ^′ (2.43)

and A is the Jacobian of the transformation x = x(ξ).

Finally, we have 1 2

d dt

\

Ω

ηu ² _ξ A dξ ≤ c(kuk ²

1, ^Ω e + ku t k ²

1, ^Ω e ), (2.44)

where we have used (2.23) 1 .

Going back to the old variables ξ in estimates (2.37), (2.39), (2.41) and summing them and (2.42) over all neighbourhoods of the partition of unity, using (2.29) and (2.44), assuming that ε and d are sufficiently small and passing to the variables x we obtain (2.27).

Lemma 2.6. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v ² _xt + p σ̺

̺ ̺ ² _xt



dx + c 0 (kv t k ² _2,Ω

+ k̺ σt k ² _1,Ω

)

≤ c(kvk ² _1,Ω

+ kv t k ² _1,Ω

+ kv tt k ² _1,Ω

+ k̺ σt k ² _0,Ω

+ kp σ k ² _0,Ω

) + cX 2 Y 2 (1 + X ₂ ² ),

where X 2 is given by (2.15) and Y 2 is given by (2.22).

P r o o f. Differentiating problem (2.28) with respect to t we get the fol- lowing elliptic problem:

µ∇ ² _u u t + ν∇ u ∇ u · u t − p ση ∇ u η σt = η σt u t + ηu tt − ν(∇ u ∇ u ) ,t · u

−µ(∇ ² _u ) ,t u + p σηη η σt ∇ u η σ + p ση (∇ u ) ,t η σ ≡ K 1 in Ω,

div u u t = div u u t in Ω,

T _{u (u t , p σt )n u = −(} T _{u ) ,t (u, p σ )n u −} T _{u (u, p σ )(n u ) ,t ≡ K 2 on S.}

By the Agmon–Douglis–Nirenberg theory (see [1]) we have the estimate ku t k ² _2,Ω + kη σt k ² _1,Ω ≤ c(kK 1 k ² _0,Ω + kK 2 k ² _1/2,S + kdiv u u t k ² _1,Ω ), where

kK 1 k ² _0,Ω + kK 2 k ² _1/2,S ≤ c(kη σζ k ² _0,Ω + ku tt k ² _0,Ω + kp σ k ² _0,Ω )

+ X 2 (Ω)Y 2 (Ω)(1 + X ₂ ² (Ω)),

with X 2 (Ω) given by (2.30) and

Y 2 (Ω) = |u| ² _3,1,Ω + kη σ k ² _2,Ω + kη σt k ² _2,Ω + kη σtt k ² _1,Ω . (2.45)

The remaining part of the proof is analogous to that in Lemma 2.5.

Lemma 2.7. Let (v, ̺ σ ) be a sufficiently smooth solution of (2.3). Then 1

2 d dt

\

Ω



̺v _xx ² + p σ̺

̺ ̺ ² _σxx



dx + c 0 (kvk ² _3,Ω

+ k̺ σx k ² _1,Ω

) (2.46)

≤ c(kvk ² _2,Ω

+ kv t k ² _1,Ω

+ k̺ σx k ² _0,Ω

+ kp σ k ² _0,Ω

) + εkv t k ² _2,Ω

+ cX 2 Y 2 (1 + X ₂ ² ),

where X 2 and Y 2 are given by (2.15) and (2.22), respectively, and

v xx ² = X 3 i,j,k=1

v ² ix

x

, ̺ ² σxx = X 3 j,k=1

̺ ² σx

x

.

P r o o f. First, we consider problem (2.28). By the Agmon–Douglis–

Nirenberg theory (see [1]) we have

kuk ² _3,Ω + kη σ k ² _2,Ω ≤ c(ku t k ² _1,Ω + kdiv uk ² _2,Ω ) (2.47)

+cX 2 (Ω)Y 2 (Ω)(1 + X ₂ ² (Ω)),

where X 2 (Ω) and Y 2 (Ω) are given by (2.30) and (2.45), respectively. Thus, to obtain (2.46) we have to estimate kdiv uk ² _2,Ω and ¹ _{2 dt} ^d

T

Ω

(̺v ² _xx + ^p

_̺ ̺ ² _σxx ) dx.

To do this, consider first boundary subdomains. Differentiate (2.26) 1 twice with respect to τ , multiply the result by e u τ τ J and integrate over b Ω. Using the Korn inequality, the continuity equation (2.26) 2 , and the boundary condi- tion (2.26) 3 we get

1 2

d dt

\

b

Ω

 b

η e u _{τ τ} ² + p _σ b ^η b η η e _{στ τ} ²



J dz + c 0 ke u τ τ k ²

1, ^Ω b (2.48)

≤ ε(kb η στ τ k ²

0, ^Ω b + ke u τ τ k ²

1, ^Ω b ) + c(kb uk ²

2, ^Ω b + kb η σz k ²

0, ^Ω b ) + cX 2 ( b Ω)Y 2 ( b Ω)(1 + X ₂ ² ( b Ω)),

where X 2 ( b Ω) is given by (2.32) and Y 2 ( b Ω) = |b u| ²

3,1, ^Ω b + kb η σ k ²

2, ^Ω b + kb η σt k ²

2, ^Ω b + kb η σtt k ²

1, ^Ω b .

In the same way we obtain the following estimate in an interior subdomain:

1 2

d dt

\

e

Ω



ηe u _ξξ ² + p ση

η η e _σξξ ²



A dξ + c 0 ke uk ²

3, ^Ω e (2.49)

≤ ε(ke η σξξ k ²

0, ^Ω e + ke u ξξξ k ²

0, ^Ω e ) + c(kuk ²

2, ^Ω e + kη σξ k ²

0, ^Ω b + kp σ k ²

0, ^Ω e ) + cX 2 ( e Ω)Y 2 ( e Ω)(1 + X ₂ ² ( e Ω)),

where X 2 ( e Ω) is given by (2.43) and Y 2 ( e Ω) = |u| ²

3,1, ^Ω e + kη σ k ²

2, ^Ω e + kη σt k ²

2, ^Ω e + kη σtt k ²

1, ^Ω e .

Now, differentiate the third component of (2.38) in τ , multiply the result by e η σnτ J and integrate over b Ω to get

1 2

d dt

\

b

Ω

p _σ b ^η b

η η e _σnτ ² J dz +

\

b

Ω

p ²

σ b ^η e η _σnτ ² J dz (2.50)

≤ εke η σnτ k ²

0, ^Ω b + c(kb uk ²

2, ^Ω b + kb u t k ²

1, ^Ω b + kb η σz k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cdke uk ²

3, ^Ω b + cke u zτ τ k ²

0, ^Ω b + cX 2 ( b Ω)Y 2 ( b Ω)(1 + X ₂ ² ( b Ω)), where d is from formula (2.25).

In the same way we obtain 1

2 d dt

\

b

Ω

p _σ b ^η b

η η e _σnn ² J dz +

\

b

Ω

p ² _σ b ^η η e _σnn ² J dz (2.51)

≤ εke η σnn k ²

0, ^Ω b + c(kb uk ²

2, ^Ω b + kb u t k ²

1, ^Ω b + kb η σz k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cdke uk ²

3, ^Ω b + cke u znτ k ²

0, ^Ω b + cX 2 ( b Ω)Y 2 ( b Ω)(1 + X ₂ ² ( b Ω)).

Next, differentiating the third component of (2.40) in τ , multiplying by e

u 3nnτ J and integrating over b Ω we have 1

2 d dt

\

b

Ω

b

η e u _3nτ ² J dz + c 0 ke u 3nnτ k ²

0, ^Ω b (2.52)

≤ εke u 3nnτ k ²

0, ^Ω b + εke u t k ²

2, ^Ω b + c(ke uk ²

2, ^Ω b + ke u t k ²

1, ^Ω e + ke u zτ τ k ²

0, ^Ω b + kb η σnτ k ²

0, ^Ω b + kb η σz k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cdkb uk ²

3, ^Ω b + cX 2 ( b Ω)Y 2 ( b Ω)(1 + X ₂ ² ( b Ω)).

In order to estimate k(div e u) ,nn k ²

0, ^Ω b rewrite equation (2.26) 1 in the form (ν + µ)∇ z

div e u = −µ(∆e u i − ∇ z

div e u) + b η e u it − k 3i

(2.53)

+(µ∆e u i + ν∇ z

div e u − µ b ∇ ² u e i − ν b ∇ i ∇ · e b u) +p 1 b η σ ∇ b i b ζ + b ζp _σ b ^η ∇ b i b η σ , i = 1, 2, 3.

Differentiating the third component of (2.53) with respect to n gives k(div e u) ,nn k ²

0, ^Ω b ≤ cdke u nnn k ²

0, ^Ω b + c(ke u τ k ²

2, ^Ω b + kb uk ²

2, ^Ω b + ke u t k ²

1, ^Ω b (2.54)

+kb η σn k ²

1, ^Ω b + kp σ k ²

0, ^Ω b ) + cX 2 ( b Ω)Y 2 ( b Ω).

To obtain an estimate for ke u τ k ²

2, ^Ω b consider the following elliptic problem:

µ b ∇ ² u + ν b e ∇ b ∇ · e u − p _σ b ^η η b σ = b η e u t + (p 1 − p _σ b ^η )b η σ ∇b b ζ (2.55)

+ b ∇ · b B _{(b u, b ζ) + b} T _{(b u, p σ ) · b ∇b ζ,}

∇ · e b u = b ∇ · e u, b

T _{(e u, p σ )b n = k 5 ,}

where b ∇·b B _{(b u, b ζ)={ b ∇ j} B b _{ij (b u, b ζ)} i=1,2,3 , b} T _{(b u, p σ )· b ∇b ζ = { b T ij (b u, p σ ) b ∇ j} ζ} b i=1,2,3 . Differentiating (2.55) with respect to τ and next using the Agmon–

Douglis–Nirenberg theory we get ke u τ k ²

2, ^Ω b + ke η στ k ²

1, ^Ω b (2.56)

≤ c(ke u τ τ k ²

1, ^Ω b + ke u 3nnτ k ²

0, ^Ω b + kb uk ²

2, ^Ω b + ke u t k ²

1, ^Ω b + kb η σz k ²

0, ^Ω b + kp σ k ²

0, ^Ω b ) + cX 2 ( b Ω)Y 2 ( b Ω)(1 + X 2 ( b Ω)).

Finally, we have 1 2

d dt

\

Ω

ηu ² _ξξ A dξ ≤ ckuk ² _2,Ω + εku t k ² _2,Ω . (2.57)

Going back to the old variables ξ in estimates (2.48), (2.50)–(2.52), (2.54), (2.56) and summing them and (2.49) over all neighbourhoods of the partition of unity, using (2.47) and (2.57), assuming that ε and d are sufficiently small and passing to the variables x we obtain (2.46).

Lemmas 2.1–2.7 and the estimates

k̺ σtt k ² _1,Ω

≤ ckv t k ² _2,Ω

+ c(k̺ σt k ² _2,Ω

kvk ² _2,Ω

+ k̺ σ k ² _2,Ω

kv t k ² _2,Ω

) and

k̺ σt k ² _2,Ω

≤ ckvk ² _3,Ω

+ cX 2 Y 2 (1 + X 2 )

(which follow from equations (2.3) 2 and (2.23) 2 , respectively) imply the following theorem.

Theorem 2.8. Let ν > ¹ ₃ µ > 0 and let relations (2.6) and (2.7) be satisfied. Then for a sufficiently smooth solution (v, ̺ σ ) of problem (2.3) we have

dφ

dt + c 0 Φ ≤ c 1

 φ +

t

\

0

kvk ² _3,Ω

t′

dt ^′  (2.58)

· h 1 + 

φ +

t

\

0

kvk ² _3,Ω

t′

dt ^′  2 i

Φ + c 2 Ψ for t ≤ T,

where

φ(t) =

\

Ω

̺ X

0≤|α|+i≤2

|D _x ^α ∂ _t ⁱ v| ² dx +

\

Ω

p 1

̺ ̺ ² _σ dx +

\

Ω

p σ̺

̺

X

1≤|α|+i≤2

|D ^α _x ∂ _t ⁱ ̺ σ | ² dx, φ(t) = |v| ² _2,0,Ω

+ |̺ σ | ² _2,0,Ω

,

Φ(t) = |v| ² _3,1,Ω

+ k̺ σ k ² _2,Ω

+ k̺ σt k ² _2,Ω

+ k̺ σtt k ² _1,Ω

, Ψ (t) = kp σ k ² _0,Ω

,

(2.59)

c i (i = 1, 2) are positive constants depending on ̺ ∗ , ̺ ^∗ , µ, ν,

T

t 0 kvk ² _3,Ω

t′

dt ^′ , kSk 5/2 , T and on the constants of imbedding theorems and Korn inequalities;

c 0 < 1 is a positive constant depending on µ and ν; and ̺ σ and p σ are given by (2.2).

3. Global existence. Assume (2.1) and rewrite problem (1.1) in La- grangian coordinates as follows (see problem (2.23)):

ηu t − µ∇ ² _u u − ν∇ u ∇ u · u + ∇p = 0 in Ω ^T ,

η t + η∇ u · u = 0 in Ω ^T ,

T _{u (u, p)n u = −p 0 n u on S} ^T _, u| t=0 = v 0 , η| t=0 = ̺ 0 , in Ω.

(3.1)

The local existence of a solution of problem (3.1) can be proved by the method of successive approximations (see [15]), taking as a zero step function the solution u ⁰ ∈ A T,Ω (A T,Ω is given by (1.6)) of the following parabolic problem:

u ⁰ _t − div D _(u ⁰ _{) = 0 in Ω} ^T _, D _(u ⁰ _{)n 0 = (p(̺ 0 ) − p 0 )n 0 on S} ^T _,

u ⁰ | t=0 = v 0 in Ω,

(3.2)

where D _(u ⁰ _{) = {µ(u} ⁰ _iξ

j

+ u ⁰ _jξ

) + (ν − µ)δ ij div u ⁰ } i,j=1,2,3 and n 0 is the unit outward vector normal to S.

Assume that

l > 0 is a constant such that ̺ e − l > 0 and ̺ 1 < ̺ 0 < ̺ 2 , (3.3)

where ̺ 1 = ̺ e − l, ̺ 2 = ̺ e + l, and ̺ e is given in Definition 1.1.

The function u ⁰ satisfies the estimate (see [15], estimate (4.3)) ku ⁰ k ² _A

(3.4)

≤ C 1 (T )(k(p(̺ 0 ) − p 0 )n 0 k ² _3/2,S + kv 0 k ² _2,Ω + ku ⁰ _t (0)k ² _1,Ω + ku ⁰ _tt (0)k ² _0,Ω )

< C 1 (T )(ecφ(0) + kv ⁰ k ² _2,Ω + ku ⁰ _t (0)k ² _1,Ω + ku ⁰ _tt (0)k ² _0,Ω ) ≡ A 0 ,

where C 1 (T ) is a positive constant; ec > 0 is a constant depending on ̺ 1 , ̺ 2

and on the volume and shape of Ω; φ is defined in (2.59); u ⁰ _t (0), u ⁰ _tt (0) are calculated from (3.2); and to obtain A 0 in (3.4) we have used (2.4).

Next, define H 0 = 1

̺ 1

+ k̺ 0 k ² _2,Ω + kv 0 k ² _2,Ω + ku t (0)k ² _1,Ω + ku tt (0)k ² _0,Ω (3.5)

≤ 1

̺ 1

+ cφ(0) + |Ω|̺ ² _e < e H 0 ,

where u t (0), u tt (0) are calculated from (3.1) 1 ; c > 0 is a constant depending on ̺ 1 , ̺ 2 ; and e H 0 > 0 is a constant. Then the following theorem holds.

Theorem 3.1. (see [15, Theorem 4.2]). Assume that ̺ 0 , v 0 ∈ H ² (Ω),

̺ 0 > 0, u t (0), u ⁰ t (0) ∈ H ¹ (Ω), u tt (0), u ⁰ tt (0) ∈ L 2 (Ω) (where u t (0), u tt (0) are calculated from (3.1)), S ∈ H ^5/2 , and p ∈ C ³ ( R ² ₊ ). Let assumption (3.3) and the following compatibility conditions be satisfied :

D _{(v 0 )n 0 = (p(̺ 0 ) − p 0 )n 0 on S.}

(3.6)

Assume that A 0 < A, where A > 0 is a constant depending also on e H 0 (i.e.

there exists a positive continuous increasing function F = F ( e H 0 ) satisfying F ( e H 0 ) < A). Then there exists T ∗ > 0 (depending on A) such that for T ≤ T ∗ there exists a unique solution of (1.1) such that u ∈ A T,Ω , η ∈ B T,Ω

and

kuk ² _A

≤ A, (3.7)

kηk ² _B

≤ ψ 1 (A), (3.8)

where ψ 1 is a positive continuous increasing function of A (A T,Ω and B T,Ω

are given by (1.6) and (1.5), respectively).

Now, we shall derive an estimate for the local solution (u, η σ ) of problem (2.23). Using (3.7) and (3.8) and the interpolation inequality we have

k∇p σ k ² _1,2,2,Ω

+ k∇p σt k ² _0,Ω

+ ε ∗ k∇p σtt k ² _0,Ω

(3.9)

+ sup

t k∇p σ k ² _0,Ω + kp σ n u k ² _3/2,2,2,S

+ k(p σ n u ) ,t k ² _1/2,2,2,S

+ ε ∗ k(p σ n u ) ,tt k ² _0,S

+ sup

t

kp σ n u k ² _0,S

≤ ψ ^′ (A, T )(k̺ σ0 k ² _2,Ω + kv 0 k ² _2,Ω + ku t (0)k ² _1,Ω ) + (ε + T )ψ ^′′ (A, T )kuk ² _A

,

where ψ ^′ and ψ ^′′ are positive continuous increasing functions of their argu-

ments, and ε ∗ , ε ∈ (0, 1) are sufficiently small constants.

By estimate (3.9), Lemmas 3.5 and 2.3 of [15] and by Theorem 3.1 the local solution (u, η σ ) of problem (2.23) satisfies, for sufficiently small ε and T ,

kuk ² _A

+ kη σ k ² _B

(3.10)

≤ ψ 2 (A, T )(k̺ σ0 k ² _2,Ω + kv 0 k ² _2,Ω + ku t (0)k ² _1,Ω + ku tt (0)k ² _0,Ω ), where ψ 2 is a positive continuous function.

Now, let φ(t), φ(t) and Φ(t) be defined by (2.59). Introduce the spaces N (t) = {(v, ̺ σ ) : φ(t) < ∞},

M _{(t) =} ⁿ _{(v, ̺ σ ) : φ(t) +}

t

\

0

Φ(t ^′ ) dt ^′ < ∞ o .

Notice that (v, ̺ σ ) ∈ N (t) iff φ(t) < ∞, and (v, ̺ σ ) ∈ M (t) iff φ(t) +

T

t

0 Φ(t ^′ ) dt ^′ ≤ ∞. Moreover,

c ^′ φ(t) ≤ φ(t) ≤ c ^′′ φ(t), (3.11)

where c ^′ , c ^′′ > 0 are constants depending on ̺ ∗ , ̺ ^∗ given by (2.5).

From inequality (3.10) and from the definitions of N _{(t) and} M _{(t) it} follows that the local solution satisfies the estimate

φ(t) +

t

\

0

Φ(t ^′ ) dt ^′ ≤ c 3 φ(0), (3.12)

where c 3 > 0 is a constant depending on the same quantities as c 1 and c 2

from Theorem 2.8.

Hence we obtain the following lemma.

Lemma 3.2. Let (v, ̺ σ ) ∈ N _{(0), S ∈ H} ^5/2 _{, u} ⁰ _{t (0) ∈ H} ¹ _{(Ω), u} ⁰ _{tt (0) ∈} L 2 (Ω) (u ⁰ is the solution of problem (3.2)), and p ∈ C ³ ( R ² ₊ ). Let assumption (3.3) and the compatibility condition (3.6) be satisfied. Moreover , assume

φ(0) ≤ α, (3.13)

where α > 0 is sufficiently small. Then the local solution (v, ̺) of problem (1.1) is such that (v, ̺ σ ) ∈ M (t) for t ≤ T , where T > 0 is the time of local existence, and the following estimate holds:

φ(t) +

t

\

0

Φ(t ^′ ) dt ^′ ≤ c 3 α,

where c 3 > 0 is a constant depending on the same quantities as c 1 and c 2

from Theorem 2.8.

Next, we prove

Lemma 3.3. Let the assumptions of Lemma 3.2 be satisfied. Then there

exist constants µ 1 > 1 and µ 2 > 0 (depending on the same quantities as c 1

and c 2 from (2.58)) such that

φ(t) ≤ µ 1 φ(0)e ^−µ

^t for t ≤ T, (3.14)

where T > 0 is the time of local existence.

P r o o f. Consider inequality (2.58) and assume that α from (3.13) is so small that

c 1

 φ +

t

\

0

kvk ² _3,Ω

t′

dt ^′ h 1 + 

φ +

t

\

0

kvk ² _3,Ω

t′

dt ^′  2 i

< c 0

4 . (3.15)

Then inequality (2.58) implies dφ

dt + 3

4 c 0 Φ < c 2 kp σ k ² _0,Ω

. (3.16)

Applying the same argument as in the proof of Lemma 6.2 of [17] yields kp σ k ² _0,Ω

≤ ε(kp σx k ² _0,Ω

+ kv xx k ² _0,Ω

) + c(ε)(kvk ² _0,Ω

+ kv t k ² _0,Ω

).

(3.17)

Since kp σx k ² _0,Ω

≤ c 4 k̺ σx k ² _0,Ω

, inequalities (3.16) and (3.17) imply, for suf- ficiently small ε,

dφ dt + 3

4 c 0 Φ < c 5 (kvk ² _0,Ω

+ kv t k ² _0,Ω

).

(3.18)

Now, multiplying (2.21) by a constant c 6 so large that c 0 c 6 − c 5 > 0 and c 6 > 1, adding to (3.18) and using Lemma 3.2 we obtain

d

dt (φ + c 6 J) + 3 4 c 0 Φ (3.19)

+ (c 0 c 6 − c 5 )(kvk ² _1,Ω

+ kv t k ² _1,Ω

+ k̺ σt k ² _0,Ω

) < c 7 αφ, where

J = 1 2

\

Ω



̺(v ² + v _t ² ) + p 1

̺ ̺ ² _σ + p σ̺

̺ ̺ ² _σt

 dx.

Since φ/c ^′′ ≤ φ ≤ Φ and φ ≥ J for sufficiently small α (so small that c 7 α < ¹ ₄ c 0 ), inequality (3.19) implies

d

dt (φ + c 6 J) + c 8 (φ + c 6 J) < 0, (3.20)

where c 8 = c 0 /(4c ^′′ c 6 ) (c ^′′ > 0 is the constant from (3.11)).

Inequality (3.20) yields (3.14) with µ 1 = c 6 + 1 and µ 2 = c 8 . By using Lemma 3.3 we prove

Lemma 3.4. Let the assumptions of Lemma 3.2 be satisfied. Moreover , assume

C 0 ≡ kv 0 k ² _0,Ω + k̺ σ0 k ² _0,Ω ≤ δ,

(3.21)

where ̺ σ0 = ̺ 0 − ̺ e . Then

kvk ² _0,Ω

+ k̺ σ k ² _0,Ω

≤ c 9 α ² + c 10 c 11 δ for t ≤ T, (3.22)

where c 9 = ^c

^µ

2 1

c

µ

c 3 c(1 + c 3 α); c ^′ is the constant from inequality (3.11); α and c 3 are the constants from Lemma 3.2; µ 1 , µ 2 are the constants from Lemma 3.3; c is the constant from Lemma 2.1 and c 10 , c 11 > 0 are constants depending on ̺ ∗ , ̺ ^∗ such that

1 c 11

(kvk ² _0,Ω

+ k̺ σ k ² _0,Ω

) ≤ 1 2

\

Ω



̺v ² + p 1

̺ ̺ ² _σ

 dx

≤ c 10 (kvk ² _0,Ω

+ k̺ σ k ² _0,Ω

) for t ≤ T ; and T > 0 is the time of local existence. Moreover ,

kp σ k ² _0,Ω

≤ c 12 (c 9 α ² + c 10 c 11 δ), (3.23)

where c 12 > 0 is a constant depending on p, ̺ ∗ , ̺ ^∗ .

P r o o f. Integrating (2.8) with respect to t over (0, t) (t ≤ T ) we get kvk ² _0,Ω

+ k̺ σ k ² _0,Ω

(3.24)

≤ c 11 c sup

0≤t

≤t

φ(t ^′ )

t

\

0

φ(t ^′ ) dt ^′ (1 + sup

0≤t

≤t

φ(t ^′ )) + c 10 c 11 C 0 .

Using Lemmas 3.2–3.3 and assumption (3.21) we obtain

kvk ² _0,Ω

+ k̺ σ k ² _0,Ω

≤ c 11 cµ 1

c ^′ c 3 α ² (1 + c 3 α)

t

\

0

e ^−µ

^t

dt ^′ + c 10 c 11 C 0

(3.25)

≤ c 9 α ² + c 10 c 11 δ.

Estimate (3.23) follows from (3.22) and (2.4).

Remark 3.5. Estimate (3.12) and assumption (3.13) yield  

t

\

0

u(ξ, t ^′ ) dt ^′    < c ¹³ T ^1/2  ^T

0

kuk ² _2,Ω dt ^′  1/2

(3.26)

≤ c 13 ψ 3 (A, T )T ^1/2 α ^1/2 ≡ c 14 T ^1/2 α ^1/2 ,

where ψ 3 is a positive continuous function; c 13 > 0 is a constant from the

imbedding theorem depending on Ω. Hence, relation (1.3) implies that both

the shape and the volume of Ω t do not change much for t ≤ T and the

constants c i (i = 1, . . . , 12), µ i (i = 1, 2) (from Lemma 3.3) and c (from

Lemma 3.4) can be chosen independent of time for t ≤ T .

Remark 3.6. Under assumption (2.1) one can prove the following mo- mentum conservation law (see [18]):

d dt

\

Ω

̺v · η dx = 0, (3.27)

where η = a + b × x and a, b are arbitrary constant vectors. Moreover, d

dt

\

Ω

̺x dx =

\

Ω

̺v dx.

(3.28)

Assuming

\

Ω

̺ 0 v 0 · η dξ = 0,

\

Ω

̺ 0 ξ dξ = 0, (3.29)

in view of (3.27) and (3.28) we get (2.6) and (2.7), respectively. Condition (2.6) guarantees that the barycentre of Ω t coincides with the origin of coor- dinates.

Now, we can prove

Lemma 3.7. Let the assumptions of Lemma 3.2 and estimate (3.22) be satisfied. Then

φ(t) ≤ α for t ≤ T, (3.30)

where α is sufficiently small (so that (3.15) and (3.32) are satisfied ), and T > 0 is the time of local existence.

P r o o f. For α so small that (3.15) is satisfied, the differential inequality (2.58) implies (3.16). Hence by estimate (3.23) of Lemma 3.4 we have

dφ dt + 3

4 c 0 Φ < c 2 c 12 (c 9 α ² + c 10 c 11 δ).

Therefore, since φ/c ^′′ ≤ Φ (where c ^′′ is the constant from inequality (3.11)) we obtain

dφ dt + 3

4 c 0

c ^′′ φ < c 2 c 12 (c 9 α ² + c 10 c 11 δ).

(3.31)

Now, assume that t ∗ = inf{t ∈ [0, T ] : φ(t) > α} and consider (3.31) in the interval (0, t ∗ ]. From the definition of t ∗ we have φ(t ∗ ) = α. Therefore (3.31) yields

dφ

dt (t ∗ ) < − 3 4

c 0

c ^′′ α + c 2 c 12 (c 9 α ² + c 10 c 11 δ).

Let α and δ be so small that

c 2 c 12 (c 9 α ² + c 10 c 11 δ) < 3 4

c 0

c ^′′ α.

(3.32)

Then (dφ/dt)(t ∗ ) < 0, a contradiction. Therefore, (3.30) holds.

Lemma 3.7 suggests that the solution can be continued to the interval [T, 2T ]. However, to do this we also need the analogous lemma for the so- lution of (3.2), to have the sum on the right-hand side of (3.4) with initial condition at T estimated by A.

Set

φ 1 (t) = |u ⁰ (t)| ² _2,0,Ω , Φ 1 (t) = |u ⁰ (t)| ² _3,1,Ω − ku ⁰ (t)k ² _3,Ω , where u ⁰ is the solution of (3.2).

Lemma 3.8. Let the assumptions of Lemma 3.7 and (3.21) be satisfied.

Moreover , assume that φ 1 (0) ≤ α 1 , where α 1 > 0 is a constant. Then if the constants δ from Lemma 3.4 and α are sufficiently small we have

φ 1 (t) ≤ α 1 for t ≤ T.

(3.33)

P r o o f. First, we shall obtain a differential inequality similar to (2.58).

Multiplying (3.2) 1 by u ⁰ , integrating over Ω and using the boundary condi- tion (3.2) 2 and (2.4) (where p 1 = p 1 (̺ 0 )) we get

1 2

d dt

\

Ω

(u ⁰ ) ² dξ + µ

2 E Ω (u ⁰ ) +

\

S

p 1 ̺ σ0 n 0 u ⁰ dξ s = 0, (3.34)

where E Ω (u ⁰ ) =

T

Ω

P 3

i,j=1 (u ⁰ _ix

+ u ⁰ _jx

) ² dξ.

In view of assumptions (3.29), Lemma 5.2 of [14] and the interpolation inequality, equality (3.34) yields

1 2

d dt

\

Ω

(u ⁰ ) ² dξ + c 0 ku ⁰ k ² _1,Ω (3.35)

≤ ck̺ σ0 k ² _0,Ω ku ⁰ k ² _0,Ω + εk̺ σ0 k ² _1,Ω + c(ε)k̺ σ0 k ² _0,Ω , where ε ∈ (0, 1).

Next, differentiating (3.2) 1 with respect to t, multiplying by u ⁰ _t , integra- ting over Ω and using the Korn inequality we get

1 2

d dt

\

Ω

(u ⁰ _t ) ² dξ + c 0 ku ⁰ _t k ² _1,Ω ≤ cku ⁰ _t k ² _0,Ω (3.36)

and from (3.2) 1 we obtain

ku ⁰ _t k ² _0,Ω ≤ εku ⁰ _t k ² _1,Ω + εk̺ σ0 k ² _1,Ω + c(ε)k̺ σ0 k ² _0,Ω + cku ⁰ k ² _1,Ω . (3.37)

By (3.36) and (3.37) we have 1

2 d dt

\

Ω

(u ⁰ _t ) ² dξ + c 0 ku ⁰ _t k ² _1,Ω ≤ εk̺ σ0 k ² _1,Ω + c(ε)k̺ σ0 k ² _0,Ω + cku ⁰ k ² _1,Ω . (3.38)

In the same way we obtain 1

2 d dt

\

Ω

(u ⁰ _tt ) ² dξ + c 0 ku tt k ² _1,Ω ≤ cku ⁰ _t k ² _1,Ω .

(3.39)