1. Introduction. In this paper we consider the motion of a viscous heat conducting fluid in a bounded domain Ω

POLONICI MATHEMATICI LIX.2 (1994)

On local motion of a general compressible

viscous heat conducting fluid bounded by a free surface by Ewa Zadrzy´ nska ( L´ od´ z)

and Wojciech M. Zaja ¸czkowski (Warszawa)

Abstract. The motion of a viscous compressible heat conducting fluid in a domain in R³bounded by a free surface is considered. We prove local existence and uniqueness of solutions in Sobolev–Slobodetski˘ı spaces in two cases: with surface tension and without it.

1. Introduction. In this paper we consider the motion of a viscous heat conducting fluid in a bounded domain Ω

⊂ R

with a free boundary S

. Let v = v(x, t) be the velocity of the fluid (i.e. v = (v

, v

, v

)), % = %(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 efficiency of heat sources per unit mass, p = p(%, ϑ) the pressure, µ and ν the constant viscosity coefficients, σ the constant coefficient of surface tension, κ the constant coefficient of heat conductivity, c

= c

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

the external (constant) pressure. Then the problem is described by the following system (see [4], Chs.

2 and 5):

(1.1)

%(v

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

,

%

+ div(%v) = 0 in e Ω

,

%c

(ϑ

+ v · ∇ϑ) + ϑp

div v − κ∆ϑ

− µ 2

3

X

i,j=1

(v

+ v

)

− (ν − µ)(div v)

= %r in e Ω

,

T · n − σHn = −p

n on e S

,

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

Key words and phrases: free boundary, compressible viscuous heat conducting fluid, local existence, anisotropic Sobolev spaces.

Research supported by KBN grant no. 211419101.

(1.1)

[cont.]

v · n = − φ

|∇φ| on e S

,

∂ϑ

∂n = e ϑ on e S

, v|

= v

in Ω ,

%|

= %

in Ω , ϑ|

= ϑ

in Ω ,

where φ(x, t) = 0 describes S

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

= S

t∈(0,T )

Ω

× {t}, Ω

is the domain of the drop at time t and Ω

= Ω is its initial domain, e S

= S

t∈(0,T )

S

× {t}.

Finally, T = T(v, p) denotes the stress tensor of the form

T = {T

} = {−pδ

+ µ(v

+ v

) + (ν − µ)δ

div v}

(1.2)

≡ {−pδ

+ D

(v)} ,

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

} is the deformation tensor. Moreover, thermodynamic considerations imply that c

> 0, κ > 0, ν ≥

µ > 0. By H we denote the double mean curvature of S

which is negative for convex domains and can be expressed in the form

(1.3) Hn = ∆

(t)x, x = (x

, x

, x

) ,

where ∆

(t) is the Laplace–Beltrami operator on S

. Let S

be given by x = x(s

, s

, t), (s

, s

) ∈ U ⊂ R

, where U is an open set. Then

∆

(t) = g

∂

∂s

g

b g

∂

∂s

(1.4)

= g

∂

∂s

g

g

∂

∂s

(α, β = 1, 2) ,

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

}

, g

= x

· x

(x

= ∂x/∂s

), {g

} is the inverse matrix to {g

} and { b g

} is the matrix of algebraic complements of {g

}.

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

, Ω

= {x ∈ R

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

(1.5) ∂x

∂t = v(x, t), x|

= ξ ∈ Ω, ξ = (ξ

, ξ

, ξ

) .

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

(1.6) x = ξ +

t

R

0

u(ξ, s) ds ≡ X

(ξ, t) ,

where u(ξ, t) = v(X

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

implies that the boundary S

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

then X

(ξ, t) ∈ S

and S

= {x : x = X

(ξ, t), ξ ∈ S}.

By the equation of continuity (1.1)

and (1.1)

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

holds:

(1.7) R

Ωt

%(x, t) dx = M .

The aim of this paper is to prove the local-in-time existence and unique- ness of solutions to problem (1.1) in Sobolev–Slobodetski˘ı spaces (see def- inition in Sect. 2). In the case of compressible barotropic fluid the corre- sponding drop problem has been considered by W. M. Zaj¸ aczkowski in [13]

and [16], while papers [14] and [15] refer to the global existence of solution to the same drop problem. Local existence of solutions in the compressible barotropic case was also considered in [5], [6], [12], while in the incompress- ible barotropic case local existence is proved in [2] and [10].

This paper consists of four sections. In Section 2 notation and auxiliary results are presented. In Section 3 we prove the local existence and unique- ness of solution to problem (1.1) in the case σ = 0. In this case there is no surface tension. Finally, Section 4 concerns the local existence and unique- ness of solution to problem (1.1) in the case σ 6= 0, i.e. when the shape of the free boundary S

of Ω

is governed by surface tension.

2. Notations and auxiliary results. In Sections 3 and 4 of this paper we use the anisotropic Sobolev–Slobodetski˘ı spaces W

(Q

), l ∈ R

(see [3]), of functions defined in Q

where Q

= Ω

≡ Ω × (0, T ) (Ω ⊂ R

is a domain, T ≤ ∞) or Q

= S

≡ S × (0, T ), S = ∂Ω.

We define W

(Ω

) as the space of functions u such that kuk

=

 X

|α|+2i≤[l]

kD

∂

uk

(2.1)

+ X

|α|+2i=[l]



R

0

R

Ω

R

Ω

|D

∂

u(ξ, t) − D

∂

u(ξ

, t)|

|ξ − ξ

|

dξ dξ

dt

+ R

Ω T

R

0 T

R

0

|D

∂

u(ξ, t) − D

∂

u(ξ, t

)|

|t − t

|

dt dt

dξ



≡ h X

|α|+2i≤[l]

|D

∂

u|

+ X

|α|+2i=[l]

([D

∂

u]

+ [D

∂

u]

) i

< ∞ ,

where we use generalized (Sobolev) derivatives, D

= ∂

1

∂

2

∂

3

, ∂

j

=

∂

/∂ξ

(j = 1, 2, 3), α = (α

, α

, α

) is a multiindex, |α| = α

+ α

+ α

,

∂

= ∂

/∂t

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

Similarly to W

(Ω

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

(S

) of functions defined on S

= S × (0, T ), where S = ∂Ω.

We also use the usual Sobolev spaces W

(Q), where l ∈ R

, Q = Ω (Ω ⊂ R

is a bounded domain) or Q = S. In the case Q = Ω the norm in W

(Ω) is defined as follows:

kuk

2(Ω)

=

 X

|α|≤[l]

kD

uk

+ X

|α|=[l]

R

Ω

R

Ω

|D

u(ξ) − D

u(ξ

)|

|ξ − ξ

|

dξ dξ



≡  X

|α|≤[l]

|D

u|

+ [D

u]



,

where the last term is omitted when l is an integer. Similarly, by using local mappings and a partition of unity we define W

(S).

To simplify notation we write kuk

= kuk

2 (Q)

if Q = Ω

or Q = S

, l ≥ 0 ; kuk

= kuk

2(Q)

if Q = Ω or Q = S, l ≥ 0 , and W

(Q) = W

(Q) = L

(Q). Moreover,

kuk

= |u|

, 1 ≤ p ≤ ∞ . Next introduce the space Γ

(Ω) with the norm

kuk

= X

|α|+2i≤l

kD

∂

uk

≡ |u|

and the space L

(0, T ; Γ

(Ω)) with the norm

kuk

≡ u

, where 1 ≤ p ≤ ∞ .

Moreover, let C

(Ω

) (α ∈ (0, 1)) denote the H¨ older space with the norm

kuk

= sup

Ω^T

|u(ξ, t) − u(ξ

, t

)|

(p|ξ − ξ

|

+ |t − t

|

)

;

let C

(Ω

) be the space of continuous bounded functions on Ω

with the norm

kuk

B(Ω^T)

= sup

Ω^T

|u(ξ, t)|

and let C

(Q) (Q ⊂ R

× (0, T )) denote the space of functions u such that D

∂

u ∈ C

(Q) for |α| + 2i ≤ 2.

Finally, the following seminorms are used:

u

=



R

0

|u|

t

dt



,

where Q = Ω (Ω ⊂ R

is a bounded domain) or Q = S, and κ ∈ (0, 1);

[u]

= [u]

+ [u]

, where

[u]

= X

|α|+2i=[l]

[D

∂

u]

, [u]

= X

|α|+2i=[l]

[D

∂

u]

,

Q = Ω × J (Ω ⊂ R

is a domain, J = (0, T ) or J = (−∞, T )) or Q = S × J . In the case when J = (0, T ) the seminorms [D

∂

u]

and [D

∂

u]

are defined in (2.1). In the case when J = (−∞, T ) we define the above seminorms in the same way.

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

, . . . , u

) belongs to X if u

∈ X for any 1 ≤ i ≤ ν.

In the sequel we shall use various notations for derivatives of u (where u is a scalar- or vector-valued function u = (u

, u

, u

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

u (where ξ ∈ Ω ⊂ R

) the vector of all derivatives of u of order k, i.e. D

u = (D

u)

. Similarly, if u = (u

, u

, u

) we denote by D

u the vector (D

u

)

. By D

u we denote the vector (D

∂

u

)

in the case when u = (u

, u

, u

) and the vector (D

∂

u)

in the scalar case. Hence

|D

u| = X

|α|=k

|D

u| and |D

u| = X

|α|+2i=k

|D

∂

u| .

We also use the notation ∇

u ≡ D

u or u

≡ D

u.

Next, we denote by u · v either the scalar product of vectors u and v, or

the product of matrices u and v.

Finally, we denote by D

uD

v the following number:

(2.2) D

uD

v = X

|α|+2i=k

|β|+2j=l p

X

m=1 s

X

n=1

D

∂

u

D

∂

v

,

where u = (u

, . . . , u

), v = (v

, . . . , v

) (k ≥ 0, l ≥ 0, p > 1, s > 1). The product of more than two such factors is defined similarly.

We use the following lemmas.

Lemma 2.1. The following imbedding holds:

(2.3) W

(Ω) ⊂ L

(Ω) (Ω ⊂ R

) ,

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

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

u|

< ∞; W

(Ω) is the Sobolev space.

Moreover , the following interpolation inequalities hold : (2.4) |D

u|

≤ cε

|D

u|

+ cε

|u|

,

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

(2.5) |D

u|

≤ cε

|D

u|

+ cε

|u|

,

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 [3].

Lemma 2.2 (see [7]). For sufficiently regular u we have k∂

u(t)k

≤ c(kuk

+ k∂

u(0)k

) , where 0 ≤ 2i ≤ 2l − 1, l ∈ N and c > 0 is a constant independent of T .

Lemma 2.3. Let u(ξ, t) = 0 for t ≤ 0. Then

T

R

−∞

dt

T

R

−∞

dt

|u(ξ, t) − u(ξ, t

)|

|t − t

|

≤

T

R

0

dt

T

R

0

dt

|u(ξ, t) − u(ξ, t

)|

|t − t

|

+ 1

α

T

R

0

|u(t)|

t

dt , where Q = Ω (Ω ⊂ R

is a domain) or Q = S = ∂Ω, and α ∈ R.

Lemma 2.4. Let τ ∈ (0, 1). Then for all u ∈ W

(Ω

), (2.6)

T

R

0

|u|

dt t

≤ c

T

R

0

dt

T

R

0

dt

|u(·, t) − u(·, t

)|

|t − t

|

+ c

T

T

R

0

|u|

dt ,

where c

, c

do not depend on T and u. For T = ∞ the last term in (2.6) vanishes.

This was shown in [11], Lemma 6.3.

3. Local existence in the case σ = 0. In order to prove local existence of solutions of (1.1) we rewrite it in the Lagrangian coordinates introduced by (1.5) and (1.6):

(3.1)

ηu

− µ∇

u − ν∇

∇

· u + ∇

p(η, γ) = ηg in Ω

,

η

+ η∇

· u = 0 in Ω

,

ηc

(η, γ)γ

− κ∇

γ = −γp

(η, γ)∇

· u + µ

2

3

X

i,j=1

(ξ

· ∇

u

+ ξ

· ∇

u

)

+ (ν − µ)(∇

· u)

+ ηk in Ω

,

T

(u, p) · n = −p

n on S

,

n · ∇

γ = e γ on S

,

u|

= v

in Ω ,

η|

= ρ

in Ω ,

γ|

= ϑ

in Ω ,

where u(ξ, t) = v(X

(ξ, t), t), γ(ξ, t) = ϑ(X

(ξ, t), t), η(ξ, t) = ρ(X

(ξ, t), t), g(ξ, t) = f (X

(ξ, t), t), k(ξ, t) = r(X

(ξ, t), t), ∇

= ξ

∇

≡ {ξ

∂

},

T

(u, p) = −p1 + D

(u) ,

D

(u) = {µ(ξ

∂

u

+ ξ

∂

u

) + (ν − µ)δ

∇

· u}

(here the summation convention over repeated indices is assumed and 1 is the unit matrix) and γ(ξ, t) = e e ϑ(X

(ξ, t), t).

Let A = {a

} be the Jacobi matrix of the transformation x = X

(ξ, t), where a

= δ

+ R

0

∂

u

(ξ, τ ) dτ . Assuming that |∇

u|

≤ M we obtain (3.2) 0 < c

(1 − M t)

≤ det{x

} ≤ c

(1 + M t)

, t ≤ T ,

where c

, c

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



R

0

∇

· u dτ



= ρ

/η .

Let S

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

≡ e φ(ξ) = 0. Thus, we have

n(x(ξ, t), t) = − ∇

φ(x, t)

|∇

φ(x, t)|

and n

(ξ) = − ∇

φ(ξ) e

|∇

φ(ξ)| e . First we consider the linear problems

(3.3)

Lu ≡ u

− µ∇

u − ν∇

∇

· u = F in Ω

,

D

(u) · n

= G on S

,

u|

= u

in Ω

(where D

(u) = {µ(∂

u

+ ∂

u

) + (ν − µ)δ

∂

u

}) and

(3.4)

γ

− κ∇

γ = K in Ω

, n

· ∇

γ = e γ on S

, γ|

= γ

in Ω . We assume

(3.5) F ∈ W

(Ω

), G ∈ W

2

(S

),

D

G

< ∞, u

∈ W

(Ω) and

(3.6) K ∈ W

(Ω

), e γ ∈ W

2

(S

),

D

e γ

< ∞, γ

∈ W

(Ω) . Moreover, we assume the following compatibility conditions:

(3.7) D

(D

(u(0)) · n

− G(0)) = 0, |α| ≤ 1, on S , and

(3.8) D

(n

· ∇

γ(0) − e γ(0)) = 0, |α| ≤ 1, on S .

First we consider problem (3.3). Define functions ψ

for i = 0, 1 by (3.9) ψ

= ∂

u|

in Ω .

Hence

(3.10) ψ

= u

, ψ

= µ∆u

+ ν∇ div u

+ F (0) .

Lemma 3.1. Let Ω ⊂ R

be either a halfspace or a bounded domain with smooth boundary ∂Ω and let T ≤ ∞. Assume that

(3.11) ψ

∈ W

(Ω), ψ

∈ W

(Ω) . Then there exists v ∈ W

(Ω

) such that

(3.12) ∂

v|

= ψ

in Ω (i = 0, 1)

and

(3.13) kvk

≤ c(kψ

k

+ kψ

k

) , where c > 0 is a constant independent of T .

P r o o f. Using the Hestenes–Whitney method (see [1]) we can extend ψ

and ψ

to functions e ψ

∈ W

(R

) and e ψ

∈ W

(R

) such that k e ψ

k

≤ ckψ

k

, k e ψ

k

≤ ckψ

k

,

where c = c(Ω). Then from [16] (Lemma 6.5) we deduce that there exists e v ∈ W

(R

× R

) such that

(3.14) ∂

e v|

= e ψ

, i = 0, 1 , and

(3.15) k e vk

+

≤ c(k e ψ

k

+ k e ψ

k

) ≤ c(kψ

k

+ kψ

k

) . Therefore v = e v|

satisfies conditions (3.12) and (3.13).

Now we prove the following theorem.

Theorem 3.1. Let S ∈ W

and let assumptions (3.5) and (3.7) be satisfied (T < ∞). Then there exists a solution of (3.3) such that u ∈ W

(Ω

) and

(3.16) kuk

≤ c(T )(kF k

+ kGk

+ D

G

+ |u(0)|

) , where c is an increasing continuous function of T and

(3.17) |u(0)|

= ku

k

+ k∂

u|

k

. P r o o f. Introduce the function

(3.18) w = u − v ,

where v is the function from Lemma 3.1. Then instead of (3.3) we obtain the problem

(3.19)

w

− µ∆w − ν∇ div w = F − Lv ≡ f in Ω

, D

(w) · n

= G − D

(v) · n

≡ g on S

,

w|

= 0 in Ω ,

where in view of the compatibility condition (3.7) we have

(3.20) f (0) = g(0) = 0 .

It is sufficient to consider problem (3.19) in Ω

≡ R

≡ R

× [0, T ] because using a partition of unity and appropriate norms (see [8], Sects.

20, 21) we obtain the existence and the appropriate estimate of solutions of

(3.19) in a bounded domain Ω.

Thus, consider problem (3.19) in R

and extend functions f , g, w by zero for t < 0 to functions f

, g

, w

. Then instead of (3.19) we get the following boundary value problem:

(3.21) Lw

= f

in e D

(T ) = R

× (−∞, T ] , D

(w

) · n

= g

in e E

(T ) = R

× (−∞, T ] .

Next using the Hestenes–Whitney method extend f

and g

to functions f

and g

defined on R

× (−∞, ∞) and R

× (−∞, ∞), respectively. Then instead of (3.21) we have the problem

(3.22)

Lw

= f

in e D

= R

× (−∞, ∞) , D

(w

) · n

= g

in e E

= R

× (−∞, ∞) , and the estimate

(3.23) kf

k

2,^D

e

≤ ckf

k

2,^D

e

.

Next we extend f

by the Hestenes–Whitney method to a function f

∈ W

(R

) such that

(3.24) kf

k

≤ ckf

k

2,^D

e

. Consider now the system

(3.25) Lw

= f

in R

.

By potential techniques (see [8], Sections 12 and 21) and (3.23), (3.24) there exists a solution w

∈ W

(R

) of (3.25) and

(3.26) kw

k

≤ c(T )kf k

, where c(T ) is an increasing function of T .

Introduce the function

(3.27) w

= w

− w

.

By (3.22) and (3.25) we have (3.28)

Lw

= 0 in e D

,

D

(w

) · n

= g

− D

(w

) · n

≡ g

in e E

.

Again by potential techniques there exists a solution w

∈ W

( e D

) of (3.28) and

kw

k

4,^D

e

≤ c[g

]

E

e

≤ c([g

]

E

e

+ kw

k

4,^D

e

)

≤ c(T )([g

]

E e

(T ) + kf k

) ,

where we used (3.26) and the Hestenes–Whitney method for g

. Hence

(3.29) kw

k

≤ c(T )([g

]

3−1/2,2,^E

e

+ kf k

) . Further, using Lemma 2.3 we have

[g

]

E

e

= X

|α|=2

[D

g

]

E

e

+ [∂

g

]

E

e

(3.30)

+ X

|α|=2

[D

g

]

E

e

+ [∂

g

]

E

e

≤ c

 X

|α|=2

[D

g]

+ [∂

g]

+ X

|α|=2

[D

g]

+ [∂

g]

+



R

0

|D

g|

t

dt





≤ c(kgk

+ D

g

) , where S

= R

× [0, T ].

Taking into account the right-hand side of (3.19) and (3.18), (3.21), (3.22), (3.25)–(3.30) we conclude that there exists a solution u ∈ W

(Ω

) of (3.3) satisfying

(3.31) kuk

≤ c(T )(kF k

+ kvk

+ kGk

+ D

G

+ kv

n

k

+ D

(v

n

)

) , where

v

n

=

3

X

i,j,k=1

v

n

. By Lemma 3.1 we have

(3.32) kvk

≤ c(kφ

k

+

+ kφ

k

+

) and

kv

n

k

≤ k e v

n

k

≤ ck v e

k

(3.33)

≤ ckvk

≤ c(kφ

k

+

+ kφ

k

) , where we have used the fact that S ∈ W

.

It remains to estimate D

(v

n

)

. We have

(3.34) D

(v

n

)

=



R

0

|D

n

D

v + D

n

D

v + n

D

v + n

∂

D

v|

t

dt



≤ c



R

0

|D

n

|

|D

v|

+ |D

v|

+ |D

v|

+ |∂

D

v|

t

dt



, where we have used the fact that S ∈ W

and Lemma 2.1, and where the products are understood in the sense of (2.2).

Next, by Lemma 2.1 we get

T

R

0

|D

n

|

|D

v|

t

dt ≤ c

T

R

0

kD

vk

t

dt . Hence in view of Lemma 2.4, (3.34) yields

D

(v

n

)

≤ c



R

0

|D

e v|

+ |D

e v|

+ |D

e v|

+ |D

∂

e v|

t

dt



≤ c



R

0

dt

∞

R

0

dt

 |D

e v(ξ, t) − D

e v(ξ, t

)|

|t − t

|

+ |D

e v(ξ, t) − D

e v(ξ, t

)|

|t − t

|

+ |∂

D

e v(ξ, t) − ∂

D

v(ξ, t e

)|

|t − t

|

+ |D

e v(ξ, t) − D

e v(ξ, t

)|

|t − t

|



. Therefore

(3.35) D

(v

n

)

≤ ckvk

, where we have used Theorem 5.1 from [8].

Taking into account (3.31)–(3.33) and (3.35) we get (3.16). This com- pletes the proof of the theorem.

In the same way we can prove

Theorem 3.2. Let S ∈ W

and let assumptions (3.6) and (3.8) be satisfied (T < ∞). Then there exists a solution of (3.4) such that γ ∈ W

(Ω

) and

(3.36) kγk

≤ c(T )(kKk

+ k e γk

+ D

e γ

+ |γ(0)|

) ,

where c is an increasing function of T and

(3.37) |γ(0)|

= kγ

k

+ k∂

γ|

k

. Now we have to consider the following problems:

(3.38)

ηu

− µ∇

u − ν∇

∇

· u = F in Ω

, D

(u) · n

= G on S

,

u|

= u

in Ω

and

(3.39)

ηc

(η, β)γ

− κ∇

γ = K in Ω

, n

· ∇

γ = e γ on S

,

γ|

= γ

in Ω .

First we consider (3.38). The following theorem is proved in [15].

Theorem 3.3. Assume that

S ∈ W

, F ∈ W

(Ω

), G ∈ W

2

(S

),

D

G

< ∞, u

∈ W

(Ω),

η ∈ L

(0, T ; Γ

(Ω)) ∩ C

(Ω

) (α ∈ (0, 1)), 1/η ∈ L

(Ω

).

Moreover , let the compatibility condition (3.7) be satisfied. Then there exists a unique solution u ∈ W

(Ω

) to problem (3.38) satisfying the estimate

kuk

≤ φ

(|1/η|

, |η|

, T )[kF k

+ kGk

(3.40)

+ D

G

+ φ

( η

, kηk

)kuk

] + φ

(|1/η|

, T )|u(0)|

,

where φ

(i = 1, 2, 3) are nonnegative increasing continuous functions of their arguments.

Now we consider problem (3.39). Assume that η ∈ L

(0, T ; Γ

(Ω)) and β ∈ W

(Ω

). Applying Theorems 10.2 and 10.4 of [3] we find that η ∈ C

(Ω

) and β ∈ C

(Ω

). Therefore, there exists a bounded domain V ⊂ R

such that (η(ξ, t), β(ξ, t)) ∈ V for any (ξ, t) ∈ Ω

.

Lemma 3.2. Assume that η ∈ L

(0, T ; Γ

(Ω)) ∩ C

(Ω

) (where α ∈ (0, 1/2)), 1/η ∈ L

(Ω

), η > 0, β ∈ W

(Ω

), 1/β ∈ L

(Ω

), β > 0, c

∈ C

(R

), c

> 0. Then ηc

(η, β) ∈ L

(0, T ; Γ

(Ω)), ηc

∈ C

(Ω

), 1/(ηc

(η, β)) ∈ L

(Ω

) and

(3.41) ηc

(η, β)

≤ ψ

(kc

k

, k1/c

k

, η

, kβk

, |β(0)|

) ,

(3.42) |ηc

(η, β)|

≤ kc

k

|η|

,

(3.43) |1/(ηc

(η, β))|

≤ k1/c

k

|1/η|

, (3.44) kηc

(η, β)k

≤ ψ

(kc

k

, kηk

, kβk

, |β(0)|

, T ) , where ψ

(i = 1, 2) are nonnegative increasing continuous functions of their arguments.

P r o o f. By the assumptions we can choose V such that V ⊂ R

. Thus (3.42) and (3.43) are obviously satisfied. In order to obtain (3.41) we cal- culate the derivatives D

∂

(ηc

) (where |α| + 2i ≤ 2) and next we apply Lemmas 2.1 and 2.2. To prove (3.44) we also use Lemmas 2.1 and 2.2 and the Sobolev imbedding theorem.

Theorem 3.4. Assume that

S ∈ W

, K ∈ W

(Ω

), e γ ∈ W

2

(S

), γ

∈ W

(Ω), η ∈ L

(0, T ; Γ

(Ω)), 1/η ∈ L

(Ω

), η ∈ C

(Ω

) (0 < α < 1/2), η > 0, β ∈ W

(Ω

), 1/β ∈ L

(Ω

), β > 0, c

∈ C

(R

), c

> 0.

Moreover , let the compatibility condition (3.8) be satisfied. Then there exists a unique solution γ ∈ W

(Ω

) to problem (3.39) satisfying the estimate

kγk

≤ φ

(kc

k

, k1/c

k

, |η|

, |1/η|

, T ) (3.45)

× [kKk

+ k e γk

+ D

e γ

+ φ

(kc

k

, k1/c

k

, η

, kηk

, kβk

, |β(0)|

, T )kγk

] + φ

(k1/c

k

, |1/η|

, T )|γ(0)|

,

where φ

(i = 4, 5, 6) are nonnegative increasing continuous functions of their arguments.

P r o o f. Since by Lemma 3.2, ηc

(η, β) ∈ L

(0, T ; Γ

(Ω)), ηc

(η, β) ∈ C

(Ω

) (α ∈ (0, 1/2)) and 1/(ηc

(η, β)) ∈ L

(Ω

), using the same argu- ment as in Theorem 3.3 and inequalities (3.41)–(3.44) we prove the existence of a unique solution γ ∈ W

(Ω

) of problem (3.39) satisfying the estimate (3.45).

Now consider the problems

(3.46)

ηu

− µ∇

u − ν∇

∇

· u = F in Ω

,

D

(u) · n = G on S

,

u|

= u

in Ω ,