ON LOCAL EXISTENCE OF SOLUTIONS OF THE FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE

P. B. M U C H A and W. Z A J A ¸ C Z K O W S K I (Warszawa)

ON LOCAL EXISTENCE OF SOLUTIONS OF THE FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE

VISCOUS SELF-GRAVITATING FLUID MOTION

Abstract. The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved.

We show the existence of solutions with lowest possible regularity for this problem such that u ∈ W

_r^2,1

( e Ω

^T

) with r > 3. The existence is proved by the method of successive approximations where the solvability of the Cauchy–

Neumann problem for the Stokes system is applied. We have to underline that in the L

p

-approach the Lagrangian coordinates must be used. We are looking for solutions with lowest possible regularity because this simplifies the proof and decreases the number of compatibility conditions.

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

t

⊂ R

³

with a free boundary S

t

which is under the self-gravitational force. Let v = v(x, t) be the velocity of the fluid, p = p(x, t) the pressure, ν the constant viscosity coefficient and p

0

the external pressure. Then the problem is described by the following system:

(1.1)

v

t

+ v · ∇v − div T(v, p) = ∇U in e Ω

^T

,

div v = 0 in e Ω

^T

,

T(u, p) · n = −p

0

n on e S

^T

,

v|

t=0

= v

0

in Ω,

Ω

t

|

t=0

= Ω, S

t

|

t=0

= S,

v · n = −ϕ

_t

/|∇ϕ| on e S

^T

,

2000 Mathematics Subject Classification: 35Q30, 76D05.

Key words and phrases: local existence, Navier–Stokes equations, incompressible vis- cous barotropic self-gravitating fluid, sharp regularity, anisotropic Sobolev space.

Research supported by Polish KBN Grant 2 P03A 038 16.

[319]

where e Ω

^T

= S

t≤T

Ω

t

× {t}, e S

^T

= S

t≤T

S

t

× {t}, ϕ(x, t) = 0 describes S

t

at least locally, n is the unit outward vector normal to S

t

, n = ∇ϕ/|∇ϕ|, Ω

t

is the domain at time t, S

t

= ∂Ω

t

, t ≤ T . Moreover, the dot · denotes the scalar product in R

³

.

By T = T(v, p) we denote the stress tensor of the form (1.2) T(v, p) = {T

ij

}

_i,j=1,2,3

= {−pδ

_ij

+ D

_ij

(v)}

_i,j=1,2,3

where

(1.3) D(v) = {D

ij

(v)}

i,j=1,2,3

= {ν(v

i,xj

+ v

j,xi

)}

i,j=1,2,3

is the velocity deformation tensor.

Moreover, U (Ω

t

, x, t) is the self-gravitational potential

(1.4) U (Ω

t

, x, t) = k

\

Ωt

dy

|x − y| ,

where k is the gravitation constant and some arguments of U are omitted in evident cases.

In view of the equation (1.1)

₂

and the kinematic condition (1.1)

₆

the total volume is conserved:

(1.5) |Ω

t

| =

\

Ωt

dx =

\

Ω

dx = |Ω|.

Let Ω be given. Then we introduce the Lagrangian coordinates ξ as the initial data for the following Cauchy problem:

(1.6) ∂x

∂t = v(x, t), x|

_t=0

= ξ, ξ = (ξ

₁

, ξ

₂

, ξ

₃

).

Integrating (1.6), we obtain a transformation which connects the Eulerian x and the Lagrangian ξ coordinates,

(1.7) x = x(ξ, t) ≡ ξ +

t

\

0

u(ξ, t

^′

) dt

^′

≡ x

u

(ξ, t),

where u(ξ, t) = v(x

_u

(ξ, t), t) and the index u in x

_u

(ξ, t) will be omitted when no confusion can arise.

Then from (1.1)

₆

we have Ω

_t

= {x ∈ R

³

: x = x(ξ, t), ξ ∈ Ω} and S

t

= {x ∈ R

³

: x = x(ξ, t), ξ ∈ S = S

0

= ∂Ω}.

Our aim is to prove the local-in-time existence of solutions to problem (1.1) with lowest possible regularity. Therefore we apply the L

p

-approach.

The result of the paper is the following theorem.

Theorem 1.1. Let r > 3, v

0

∈ W

r^2−2/r

(Ω), S ∈ W

r^2−1/r

. Then there exists T

0

> 0 such that for all T ≤ T

0

there exists a unique solution (u, p) of (1.1) such that u ∈ W

_r^2,1

( e Ω

^T

), p ∈ W

_r^1,0

( e Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

( e S

^T

)

and the following estimate holds : (1.8) kuk

_W^2,1

r ( eΩ^T)

+ kpk

_W^1,0

r ( eΩ^T)

+ kpk

_W1−1/r,1/2−1/(2r)

r ( eS^T)

≤ c(T )kv

₀

k

_W2−2/r r (Ω)

. To prove Theorem 1.1 we need solvability of the Cauchy–Neumann prob- lem for the Stokes system from [3]. To recall the result we formulate the problem

(1.9)

u

t

− div T(u, p) = F in Ω

^T

,

div u = G in Ω

^T

,

n · T(u, p) = H on S

^T

, u|

t=0

= u

0

in Ω, where Ω

^T

= Ω × [0, T ] and S

^T

= S × [0, T ].

Theorem 1.2 (see [3]). Let r > 3, F ∈ L

_r

(Ω

^T

), G ∈ W

_r^1,0

(Ω

^T

), G

t

− div F = div B + A, A, B ∈ L

r

(Ω

^T

),

H ∈ W

1−1/r,1/2−1/(2r)

r

(S

^T

), u

0

∈ W

_r^2−2/r

(Ω), S ∈ W

_r^2−2/r

, and assume the compatibility conditions

(1.10) div u

0

= G(x, 0), n · T(u

0

, p

0

)|

S

= H(x, 0),

where p

0

= p|

t=0

. Then there exists a unique solution (u, p) to problem (1.9) such that

u ∈ W

_r^2,1

(Ω

^T

), p ∈ W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

)

and the following estimate holds : (1.11) kuk

_W^2,1

r (Ω^T)

+ kpk

_W^1,0

r (Ω^T)

+ kpk

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ C(T )[kF k

_Lr(Ω^T₎

+ kGk

_W^1,0

r (Ω^T)

+ kBk

_Lr(Ω^T₎

+ kAk

_Lr(Ω^T₎

+ kHk

_W1−1/r,1/2−1/(2r)

r (S^T)

+ ku

0

k

_W2−2/r r (Ω)

],

where C(T ) is a constant increasing with T which does not depend on the solution (u, p).

Problem (1.1) without the self-gravitation force is considered in [4].

Moreover we recall that the local existence of solutions to problem (1.1) with surface tension is shown in [5].

2. Notation. We need the anisotropic Sobolev spaces W

_r^m,n

(Q

_T

) where

m, n ∈ R

+

∪ {0}, r ≥ 1 and Q

T

= Q × (0, T ), with the norm

(2.1) kuk

^r_W^m,n

r (QT)

=

T\

0

\

Q

|u(x, t)|

^r

dx dt

+ X

0≤|m^′|≤[|m|]

T

\

0

\

Q

|D

_x^m′

u(x, t)|

^r

dx dt

+ X

|m^′|=[|m|]

T

\

0

dt

\

Q

\

Q

|D

_x^m′

u(x, t) − D

^m_x^′

u(x

^′

, t)|

^r

|x − x

^′

|

s+r(|m|−[|m|])

dx dx

^′

+ X

0≤|n^′|≤[|n|]

T\

0

\

Q

|D

_t^n′

u(x, t)|

^r

dx dt

+

\

Q

dx

T

\

0 T

\

0

|D

_t^[n]

u(x, t) − D

_t^[n]

(x, t

^′

)|

^r

|t − t

^′

|

^1+r(n−[n])

dt dt

^′

,

where s = dim Q, [α] is the integral part of α, D

_x^l

= ∂

_x^l1₁

. . . ∂

_x^ls_s

where l = (l

1

, . . . , l

s

) is a multiindex.

In the proof we will use the following results.

Proposition 2.1 (see [1]). Let u ∈ W

_r^m,n

(Ω

T

), m, n ∈ R

+

. If q ≥ r and

κ = X

3 i=1

 α

i

+ 1

r − 1 q

 1 m +

 β + 1

r − 1 q

 1 n < 1 then

kD

^β_t

D

^α_x

uk

L_q(Ω_T)

≤ ε

^1−κ

kuk

_Wr^m,n_(ΩT)

+ cε

^−κ

kuk

L_r(Ω_T)

for all ε ∈ (0, 1).

Proposition 2.2 (see [1, 2]). Let u ∈ W

_r^2m,m

(Ω

T

), m ∈ R

+

. If 2m − 1/r > 0 then u = u|

ST

is well defined as a function in W

2m−1/r,m−1/(2r)

r

(S

T

)

and

kuk

_W2m−1/r,m−1/(2r)

r (ST)

≤ ckuk

_W^2m,m

r (ΩT)

. Proposition 2.3 (see [1, 2]). Let u ∈ W

2m−1/r,m−1/(2r)

r

(S

_T

), m ∈ R

+

. If 2m − 1/r > 0 then there exists a function e u ∈ W

_r^2m,m

(Ω

T

) such that e

u|

ST

= u and the following estimate holds:

ke uk

_W^2m,m

r (ΩT)

≤ ckuk

_W2m−1/r,m−1/(2r)

r (S_T)

.

In our considerations we will use well known imbedding theorems for

Sobolev spaces. All constants are denoted by the same letter c.

3. Proof of Theorem 1.1. To prove local existence of solutions to problem (1.1) we write it in the Lagrangian coordinates:

(3.1)

u

t

− div

u

T

u

(u, q) = ∇

u

U

u

in Ω

^T

,

div

u

u = 0 in Ω

^T

,

n

_u

· T

u

(u, q) = −p

₀

n

_u

on S

^T

,

u|

t=0

= v

0

on Ω,

where u(ξ, t) = v(x(ξ, t), t), q(ξ, t) = p(x(ξ, t), t), ∇

u

= ξ

i,x

∇

ξi

, T

u

(u, q) = D

u

(u) − qI, D

u

(u) = ν{ξ

k,xi

u

j,ξk

+ ξ

k,xj

u

i,ξk

}

i,j=1,2,3

,

U

u

(ξ, t) =

\

Ω

kJ

y(ξ^′,t)

dξ

^′

|x(ξ, t) − y(ξ

^′

, t)| ,

where J

x(ξ,t)

is the Jacobian of the transformation x = x(ξ, t), div

u

u = ξ

_k,xi

u

_i,xk

, n

_u

(ξ, t) = n(x(ξ, t), t), I is the unit matrix and the summation convention over repeated indices is used.

To prove the existence of solutions to (3.1) we use the following method of successive approximations:

(3.2)

u

m+1,t

− div

um

T

um

(u

m+1

, q

m+1

) = ∇

um

U

um

in Ω

^T

,

div

_um

u

_m+1

= 0 in Ω

^T

,

n

_um

· T

um

(u

_m+1

, q

_m+1

) = −p

₀

n

_um

on S

^T

,

u

m+1

|

t=0

= v

0

on Ω,

where m = 0, 1, 2, . . . and u

m

is treated as a given function. Assume u

0

= 0 and q

0

= 0.

To apply Theorem 1.2 we write (3.2) in the form

(3.3)

u

m+1,t

− div T(u

m+1

, q

m+1

)

= div

um

T

um

(u

m+1

, q

m+1

)

− div T(u

m+1

, q

m+1

) + ∇

u_m

U

u_m

in Ω

^T

, div u

m+1

= div u

m+1

− div

um

u

m+1

in Ω

^T

, n

₀

· T(u

m+1

, q

_m+1

)

= n

0

· T(u

m+1

, q

m+1

)

− n

um

· T

um

(u

m+1

, q

m+1

) − p

0

n

um

on S

^T

,

u

m+1

|

t=0

= v

0

on Ω,

where the operators without index contain derivatives with respect to ξ and

n

0

is the unit outward vector normal to S.

First we obtain a uniform bound for the sequence {u

m

}

^∞_m=0

determined by (3.3).

Lemma 3.1. Assume that S ∈ W

r^2−1/r

, v

0

∈ W

r^2−2/r

(Ω). Then (3.4) ku

m

k

_W^2,1

r (ΩT)

+ kq

m

k

_W^1,0

r (Ω^T)

+ kq

m

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ c(kv

0

k

_W2−2/r

r (Ω)

, kSk

_W2−2/r

r

)

if T is small enough.

P r o o f. Applying Theorem 1.2 to problem (3.3) yields (3.5) ku

m+1

k

_W^2,1

r (ΩT)

+ kq

m+1

k

_W^1,0

r (Ω^T)

+ kq

m+1

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ ckdiv T(u

m+1

, q

m+1

) − div

um

T

um

(u

m+1

, q

m+1

)k

_Lr(Ω^T₎

+ ck∇

um

U

um

k

_Lr(Ω^T₎

+ ckdiv u

m+1

− div

um

u

m+1

k

_W^1,0

r (Ω^T)

+ ckn

0

· T(u

m+1

, q

m+1

)

− n

um

· T

um

(u

m+1

, q

m+1

)k

_W1−1/r,1/2−1/(2r)

r (S^T)

+ ckn

um

k

_W1−1/r,1/2−1/(2r)

r (S^T)

+ ckv

0

k

_W^2−2/r

r (Ω)

+ ck((I − A

^∗

(u

_m

))u

_m+1

)

_t

k

_Lr(ΩT)

,

where ((I − A

^∗

(u

m

))u

m+1

)

t

is treated as e B from Theorem 1.2 ( e A = 0) and A

ij

(u

m

) = δ

ij

+

Tt

0

u

mi,ξj

dτ , A

^∗_kl

(u

m

) = A

⁻¹_lk

(u

m

). Here we note that div

um

u

m+1

= A

⁻¹_kl

∂

ξ_l

u

^l_m+1

= div

ξ

(A

^∗

u

m+1

),

which follows from P

3 k=1 ∂

∂ξk

A

lk

(u

m

)(ξ, t) = 0. All the above relations hold under the assumption that div

_um−1

u

_m

= 0.

To continue the induction we need to have div

um

u

m+1

= 0, but this is given by (3.3)

2

.

Now we estimate the particular terms from the r.h.s. of (3.5). Define a

m

= T

^(r−1)/r

ku

m

k

_Wr^2,1_(ΩT)

, α

m

(t) = {α

ij

(u

m

)} = {

Tt

0

u

mi,ξj

dτ }.

For r > 3 we have kα

m

k

_L∞(Ω)

≤ ca

m

.

To estimate the first term on the r.h.s. of (3.5) we calculate div

um

T

um

(u

m+1

, q

m+1

) − div T(u

m+1

, q

m+1

)

= {ν(ξ

lxj

ξ

kxjxs

x

sξl

δ

σi

+ ξ

lxj

ξ

kxixs

x

sξl

δ

σj

)u

m+1σ,ξk

+ ν(ξ

lxi

ξ

kxj

− δ

jk

δ

jl

)u

m+1i,ξlξk

+ ν(ξ

lxj

ξ

kξi

− δ

lj

δ

ki

)u

m+1j,ξlξk

− (ξ

_lxj

− δ

_lj

q

_m+1,ξl

)},

where the matrix ξ

,x

depends on u

m

.

Since x

iξ_j

= δ

ij

+

Tt

0

u

iξ_j

(τ ) dτ = δ

ij

+ α

ij

and ξ

jx_i

is the inverse matrix to x

iξj

, we have

ξ

jxi

= δ

ij

+ φ

ij

(α),

where φ

ij

is a polynomial matrix-valued function which contains terms of α and α

²

(α = {α

ij

}). Then ξ

j,xixk

= φ

ij,αrs

α

rs,ξσ

ξ

σ,xk

, where α

rs,ξσ

=

Tt

0

u

r,ξ_sξ_σ

(τ ) dτ .

Then we write the first term of the r.h.s. of (3.5) in the form

kψ

1

(α

m

)(I − A(u

m

))(u

m+1,ξξ

+ q

m+1,ξ

) + ψ

2

(α

m

)A(u

m

)

,ξ

u

m+1,ξ

k

_Lr(Ω^T₎

≤ φ(a

_m

)a

_m

(ku

_m+1

k

_W^2,1

r (Ω^T)

+ kq

_m+1

k

_W^1,0

r (Ω^T)

), where ψ

_i

are some functions with ψ

_i

(0) 6= 0 and φ always denotes an in- creasing positive function.

We estimate the third term by the same quantity.

The fourth term can be expressed in the form

kψ

3

(α

m

)α

m

u

m+1,ξ

+ ψ

4

(α

m

)α

m

q

m+1

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ kψ

₃

(α

_m

)α

_m

u

_m+1,ξ

k

_W1−1/r,1/2−1/(2r)

r (S^T)

+ kψ

4

(α

m

)α

m

q

m+1

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≡ I + J, where

I ≤ 

^T\

0

kψ

3

(α

m

)α

m

u

m+1,ξ

k

^r_W1

r(Ω)

dτ 

1/r

+ 

^\

Ω

kψ

3

(α

m

)α

m

u

m+1,ξ

k

^r_W1/2 r (0,T )



1/r

≡ L + K.

Next we have L ≤ 

^T\

0

kψ

3

(α

m

)α

m

u

m+1,ξ

k

^r_Lr(Ω)

dτ 

1/r

+ 

^T\

0

ψ

^3,αm

(α

m

)

t

\

0

u

m,ξξ

dτ α

m

u

m+1,ξ

r

Lr(Ω)

dt 

1/r

+ 

^T\

0

ψ

³

(α

m

)

t

\

0

u

m,ξξ

dτ u

m+1,ξ

r

Lr(Ω)

dt 

1/r

+ 

^T\

0

kψ

3

(α

m

)α

m

u

m+1,ξξ

k

^r_Lr(Ω)

dτ 

1/r

≡ L

1

+ L

2

+ L

3

+ L

4

.

Continuing, we have

L

₁

≤ φ(a

_m

(T ))a

_m

(T )ku

_m+1

k

_W^2,1

r (Ω^T)

, L

2

+ L

3

≤ φ(a

m

(T ))a

m

(T ) 

^T\

0

t

\

0

u

m,ξξ

dτ u

m+1,ξ

r

Lr(Ω)

dt 

1/r

≤ φ(a

m

(T ))a

m

(T )a

²_m

(T )ku

m+1

k

_Wr^2,1_(ΩT)

, L

4

≤ φ(a

m

(T ))a

m

(T )ku

m+1

k

_W^2,1

r (Ω^T)

. Next we examine

K ≤



\

Ω

dξ

T\

0

dt

T\

0

dt

^′

|ψ

₃

(α

_m

(t)) − ψ

₃

(α

_m

(t

^′

))|

^r

|α

_m

(t)|

^r

|u

_m+1,ξ

(t)|

^r

|t − t

^′

|

^1+r/2



1/r

+



\

Ω

dξ

T\

0

dt

T\

0

dt

^′

|ψ

3

(α

m

(t

^′

))|

^r

|α

m

(t) − α

m

(t

^′

)|

^r

|u

m+1,ξ

(t)|

^r

|t − t

^′

|

^1+r/2



1/r

+



\

Ω

dξ

T

\

0

dt

T

\

0

dt

^′

|ψ

3

(α

m

(t))|

^r

|α

m

(t)|

^r

|u

m+1,ξ

(t)−u

m+1,ξ

(t

^′

)|

^r

|t − t

^′

|

^1+r/2



1/r

≡ K

1

+ K

2

+ K

3

. Using the formula

(3.6) ψ

₃

(α

_m

(t)) − ψ

₃

(α

_m

(α

_m

(t

^′

))

= (α

m

(t) − α

m

(t

^′

))

1

\

0

ψ

3,αm(t^′)

(α

m

(t

^′

) + s(α

m

(t) − α

m

(t

^′

)) ds we obtain

K

1

+ K

2

≤ φ(a

m

(T ))



\

Ω

dξ

T

\

0

dt

T

\

0

dt

^′

|

Tt

t^′

u

m,ξ

dτ |

^r

|u

m+1,ξ

(t)|

^r

|t − t

^′

|

^1+r/2



1/r

≤ φ(a

m

(T )) 

^\

Ω

dξ

T

\

0

dt

T

\

0

dt

^′

|t − t

^′

|

^r/2−2

t

\

t^′

|u

m,ξ

|

^r

dτ |u

m+1,ξ

(t)|

^r



1/r

≤ φ(a

m

(T ))ku

m

k

_Wr^2,1_(ΩT)



^\

Ω

dξ

T\

0

dt

T\

0

dt

^′

|t − t

^′

|

^r/2−2

|u

m+1,ξ

|

^r



1/r

≡ K

4

. Integrating with respect to t

^′

we get (r/2 > 1)

K

4

≤ φ(a

m

(T ))T

^r/2−1

ku

m

k

_Wr^2,1_(ΩT)

ku

m+1

k

_Wr^2,1_(ΩT)

.

Finally

K

3

≤ φ(a

m

(T ))a

m

(T )ku

m+1

k

_W^2,1

r (Ω^T)

. Summarizing the above considerations we obtain

I ≤ φ(a

_m

(T ))a

_m

(T )ku

_m+1

k

_W^2,1

r (Ω^T)

+ φ(a

_m

(T ))T

^r/2−1

ku

_m

k

_W^2,1

r (Ω^T)

ku

_m+1

k

_W^2,1

r (Ω^T)

. Similarly, we obtain

J ≤ φ(a

_m

(T ))a

_m

(T )ke q

_m+1

k

_W1,1/2 r (Ω^T)

+ φ(a

_m

(T ))T

^r/2−1

ku

_m

k

_W^2,1

r (Ω^T)

ke q

_m+1

k

_W1,1/2 r (Ω^T)

, where e q

m+1

is an extension of q

m+1

∈ W

1−1/r,1/2−1/(2r)

r

(S

^T

).

The fifth term on the r.h.s. of (3.5) is estimated by kψ

5

(α

m

)k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ 

^T\

0

kψ

5

(α

m

(t))k

_W^1−1/r

r (S)

dt 

1/r

+ 

^\

S

kψ

₅

(α

_m

(t))k

_W1/2−1/(2r)

r (0,T )

dξ 

1/r

≡ M

₁

+ M

₂

, where

M

1

≤ φ(a

m

(T )) 

^T\

0

dt

t

\

0

u

m,ξ

dτ

r W_r¹(Ω)



1/r

≤ φ(a

m

(T ))T

^1/r

a

m

(T ),

M

₂

≤



\

S

dξ

T

\

0

dt

T

\

0

dt

^′

|ψ

5

(α

m

(t)) − ψ

5

(α

m

(t

^′

))|

^r

|t − t

^′

|

1+r(1/2−1/(2r))



1/r

; using (3.6) we have

φ(a

m

(T ))



\

S

dξ

T\

0

dt

T\

0

dt

^′

|

Tt

t^′

u

_m,ξ

(ξ, τ ) dτ |

^r

|t − t

^′

|

1+r(1/2−1/(2r))



1/r

≤ 

^\

S

dξ

T\

0

|u

m,ξ

(ξ, τ )|

^r

dτ 

1/r



^T\

0

dt

T\

0

|t − t

^′

|

^r/2−3/2

dt

^′



1/r

≤ φ(a

m

(T ))T

^1/2+r/2

ku

m

k

_Wr^2,1_(ΩT)

.

The seventh term of the r.h.s. of (3.5) will be considered in the from k((I − A

^∗

(u

m

))u

m+1

)

t

k

_Lr(Ω^T₎

≤ kψ

6

(α

m

)α

m

u

m+1,t

k

_Lr(Ω^T₎

+ kψ

₇

(α

_m

)u

_m,ξ

u

_m+1

k

_Lr(ΩT)

≡ N

₁

+ N

₂

and we have

N

1

≤ φ(a

m

(T ))|α

m

| ku

m+1

k

_W^2,1

r (Ω^T)

, N

2

≤ φ(a

m

(T ))

(u

^m,ξ

− v

0,ξ

+ v

0,ξ

)  v

0

+

t

\

0

u

m+1,t

dt 

Lr(Ω^T)

(3.7)

≤ φ(a

m

(T ))T

^1/r

kv

0

k

²_W2−2/r

r (Ω)

+ φ(a

m

(T ))ku

m

k

_W^2,1

r (Ω^T)

a

m+1

+ φ(a

m

(T ))kv

0

k

_W2−2/r r (Ω)

T

^β

× (ku

m

k

_W^2,1

r (Ω^T)

+ kv

0

k

_W^2−2/r

r (Ω)

).

In the last term of the r.h.s. of (3.7)

2

we have applied the imbedding W

r^1,1/2

(Ω

^T

) ⊂ C

^β

(0, T ; L

_r

(Ω)) with 0 < β < 1/2 − 1/r. This enables us to get

ku

m,ξ

− v

0,ξ

k

L_r

≤ T

^β

(ku

m

k

_W^2,1

r (Ω^T)

+ kv

0

k

_W2−2/r r (Ω)

).

Finally we consider the second term of the r.h.s. of (3.5). We have

∇

um

U

um

= ∇

um

\

Ω

J

_xum(ξ^′_,t)

|x

um

(ξ, t) − x

um

(ξ

^′

, t)| dξ

^′

= −

\

Ω

∇

_um

x

_um

(ξ, t) · (x

_um

(ξ, t) − x

_um

(ξ

^′

, t))

|x

um

(ξ, t) − x

um

(ξ

^′

, t)|

³

J

_xum(ξ^′_,t)

dξ

^′

= −

\

Ω

∇

um

x

um

(ξ, t) · (ξ −ξ

^′

)(1+

T1 0

ds

Tt

0

∂

s

u

m

(ξ

^′

+s(ξ − ξ

^′

), τ ) dτ )

|ξ −ξ

^′

|

³

|1 +

T1 0

ds

Tt

0

∂

s

u

m

(ξ

^′

+ s(ξ − ξ

^′

), τ ) dτ |

³

× J

_xum(ξ^′_,t)

dξ

^′

. Assuming that

t

\

0

ku

mξ

(·, τ )k

L∞(Ω)

dτ < 1, we obtain

k∇

um

U

um

k

_Lr(Ω^T₎

≤ φ(a

m

(T ))T

^1/r

\

Ω

dξ

^′

|ξ − ξ

^′

|

²

Lr(Ω)

≤ φ(a

m

(T ))T

^1/r

.

For simplicity we introduce X

k

= ku

k

k

_W^2,1

r (Ω^T)

+ kq

k

k

_W^1,0

r (Ω^T)

+ kq

m

k

_W1−1/r,1/2−1/(2r)

r (S^T)

.

Summing up the estimates for all terms of the r.h.s. of (3.5) we get X

m+1

≤ a

m

φ(a

m

)X

m+1

+ φ(a

m

)T

^r/2−1

X

m

X

m+1

+ φ(a

_m

)a

_m

T

^1/r

+ φ(a

_m

)T

^1/2+r/2

X

_m

+ φ(a

_m

)X

_m

a

_m+1

(T ) + (T

^1/r

+ T

^β

)φ(a

m

) + φ(a

m

(T ))T

^β

X

m

.

Putting

a = min

 r − 1 r , r − 2

2 , 1 r , 1

2 + r 2 , β



we have

X

m+1

≤ T

^a

φ(a

m

)X

m

X

m+1

+ T

^a

φ(a

m

)X

m

(3.8)

+ φ(a

_m

)T

^a

X

_m

X

_m+1

+ T

^a

φ(a

_m

).

By induction we prove that X

_k

≤ 1 (X

₀

= 0). Taking T such that T ≤ 1 and T

^a

φ(1) ≤ 1/4, inserting X

m

≤ 1 in (3.8), we obtain

X

m+1

≤ 1

4 X

m+1

+ 1 4 + 1

4 X

m+1

+ 1 4 , which gives X

m+1

≤ 1.

The proof of the lemma is complete.

Lemma 3.2. Assume that S ∈ W

r^2−1/r

, v

0

∈ W

r^2−2/r

(Ω). Then there exist u ∈ W

_r^2,1

(Ω

^T

) and p ∈ W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

) such that

u

_m

→ u in W

_r^2,1

(Ω

^T

) and q

_m

→ p in W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

)

as m → ∞ for T small enough.

P r o o f. We show that {(u

m

, q

m

)}

^∞_n=1

is convergent. For this purpose we consider v

_m

= u

_m+1

− u

_m

, r

_m

= q

_m+1

− q

_m

which satisfy the system

(3.9)

v

m,t

− div T(v

m

, r

m

) = div

um

T

um

(u

m+1

, q

m+1

)

− div

um−1

T

um−1

(u

m

, q

m

) − div T(v

m

, q

m

)

+∇

_um

U

_um

− ∇

_um−1

U

_um−1

≡ I in Ω

^T

, div v

_m

= div v

_m

− div

_um

u

_m+1

+ div

_um−1

u

_m

≡ J in Ω

^T

, n

0

· T(v

m

, r

m

) = n

0

T(v

m

, r

m

) − n

um

T(u

m+1

, q

m+1

)

+n

um−1

(u

m

, q

m

) − p

0

n

um

+ p

0

n

um−1

≡ K on S

^T

,

v

_m

|

_t=0

= 0 on Ω.

By Theorem 1.2 we obtain an estimate on solutions of (3.9):

(3.10) kv

m

k

_W^2,1

r (Ω^T)

+ kr

m

k

_W^1,0

r (Ω^T)

+ kr

m

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ c(kIk

_Lr(Ω^T₎

+ kJk

_W^1,0

r (Ω^T)

+ kKk

_W1−1/r,1/2−1/(2r)

r (S^T)

+ kBk

_Lr(Ω^T₎

),

where B is defined by the relation J

t

= div B.

First we estimate the terms of the r.h.s. of (3.9)

1

in L

r

(Ω

^T

). Let I = I

1

+ I

2

. We examine

I

1

= div

um

T

um

(u

m+1

, q

m+1

) − div

um−1

T

um−1

(u

m

, q

m

) − div T(v

m

, q

m

)

= (div

um

T

um

− div T)(v

m

, r

m

)

+ (div

um

T

um

− div

um−1

T

um−1

)(u

m

, q

m

) ≡ I

11

+ I

12

.

I

11

is estimated in the same way as the first term of the r.h.s. of (3.5):

kI

11

k

_Lr(Ω^T₎

≤ φ(a

m

)a

m

(kvk

_Wr^2,1_(ΩT)

+ kr

m

k

_Wr^1,0_(ΩT)

).

For the second term we have

kI

12

k

_Lr(Ω^T₎

= k(div

um

(T

um

− T

um−1

)

+ (div

um

− div

um−1

)T

um−1

)(u

m

, q

m

)k

_Lr(Ω^T₎

≤ φ(a

m

)T

^(r−1)/r

kv

m−1

k

_Wr^2,1_(ΩT)

. Next we consider

I

2

= ∇

um

U

um

− ∇

um−1

U

um−1

= ∇

um

(U

um

− U

um−1

) + (∇

um

− ∇

um−1

)U

um−1

≡ I

21

+ I

22

. The first term is

I

₂₁

= − A(u

_m

)

\

Ω

 x

_m

(ξ, t) − y

_m

(ξ

^′

, t)

|x

m

(ξ, t) − y

m

(ξ

^′

, t)|

³

J

_ym(ξ^′_,t)

− x

m−1

(ξ, t) − y

m−1

(ξ

^′

, t)

|x

_m−1

(ξ, t) − y

_m−1

(ξ

^′

, t)|

³

J

_ym−1(ξ^′_,t)

 dξ

^′

= − A(u

m

)

\

Ω

x

_m

(ξ, t) − y

_m

(ξ

^′

, t)

|x

m

(ξ, t) − y

m

(ξ

^′

, t)|

³

(J

ym(ξ^′,t)

− J

ym−1(ξ^′,t)

) dξ

^′

−A(u

m

)

\

Ω

 x

m

(ξ, t) − y

m

(ξ

^′

, t)

|x

_m

(ξ, t) − y

_m

(ξ

^′

, t)|

³

− x

m−1

(ξ, t) − y

m−1

(ξ

^′

, t)

|x

m−1

(ξ, t) − y

m−1

(ξ

^′

, t)|

³



J

ym−1(ξ^′,t)

dξ

^′

≡ I

211

+ I

212

, where x

_k

(ξ, t) = ξ +

Tt

0

u

_k

(ξ, t

^′

) dt

^′

and y

_k

(ξ

^′

, t) = ξ

^′

+

Tt

0

u

_k

(ξ

^′

, t

^′

) dt

^′

. Since

|J

_ym(ξ^′_,t)

− J

_ym−1(ξ^′_,t)

| ≤ cku

m

− u

m−1

k

_W^2,1

r (Ω^T)

, we have

kI

211

k

_Lr(Ω^T₎

≤ φ(a

m

)T

^1/r

kv

m−1

k

_W^2,1

r (Ω^T)

.

The same holds for I

212

. Since

\

   

x

m

(ξ, t) − y

m

(ξ

^′

, t)

|x

m

(ξ, t) − y

m

(ξ

^′

, t)|

³

− x

m−1

(ξ, t) − y

m−1

(ξ

^′

, t)

|x

m−1

(ξ, t) − y

m−1

(ξ

^′

, t)|

³

    dξ

^′

≤ cku

m

− u

m−1

k

_W^2,1

r (Ω^T)

, we obtain

kI

212

k

_Lr(Ω^T₎

≤ φ(a

m

)T

^1/r

kv

m−1

k

_W^2,1

r (Ω^T)

. We estimate J in W

_r^1,0

(Ω

^T

) by the same quantity.

Let K = K

₁

+ K

₂

, where

K

1

= n

0

T(v

m

, r

m

) − n

um

T(u

m+1

, q

m+1

) + n

um−1

(u

m

, q

m

)

= (n

0

T(v

m

, r

m

) − n

um

T(v

m

, r

m

))

+ (n

um

T

um

− n

um−1

T

um−1

)(u

m

, q

m

) ≡ K

11

+ K

12

and

K

2

= −p

0

(n

um

− n

um−1

).

The term K

11

is estimated just as the fourth term of the r.h.s. of (3.5):

kK

11

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ φ(a

m

(T ))a

m

(T )kv

m

k

_Wr^2,1_(ΩT)

+ φ(a

m

(T ))T

^r/2−1

ku

m

k

_Wr^2,1_(ΩT)

kv

m

k

_Wr^2,1_(ΩT)

. The second term is

K

12

= (n

um

(T

um

− T

um−1

) + (n

um

− n

um−1

)T

um−1

)(u

m

, q

m

) ≡ K

121

+ K

122

. For K

₁₂₁

we have

kK

121

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ φ(a

m

(T ))T

^(r−1)/r

kv

m−1

k

_W^2,1

r (Ω^T)

. Since

kn

_um

− n

_um−1

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ φ(a

_m

(T ))T

^(r−1)/r

kv

_m−1

k

_W^2,1

r (Ω^T)

, we conclude that

kK

122

k

_W1−1/r,1/2−1/(2r)

r (S^T)

+ kK

2

k

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ φ(a

m

(T ))T

^(r−1)/r

kv

m−1

k

_W^2,1

r (Ω^T)

. Finally we have to examine B which is defined by J

t

= div B. We have

J = div v

m

− div

um

u

m−1

+ div

um−1

u

m

= (div v

m

− div

u_m

v

m

) + (div

u_m−1

u

m

− div

u_m

u

m

) ≡ J

1

+ J

2

. To examine J

1

we proceed as in the case of the seventh term of the r.h.s.

of (3.5). By the same argument as in Lemma 3.1 we have J

2

= (div

um−1

− div

um

)u

m

= div ·((A

^∗_m−1

− A

^∗_m

)u

m

).

Hence we put B

₂

= ((A

^∗_m−1

− A

^∗_m

)u

_m

)

_t

and we get, just as for N

₂

in the

proof of Lemma 3.1, the following estimate:

kB

2

k

_Lr(Ω^T₎

≤ φ(a

m

(T ))(T

^(r−1)/r

+ T

^β

)kv

m−1

k

_W^2,1

r (Ω^T)

, where 0 < β < 1/2 − 1/r.

Define

Y

m

= kv

m

k

_Wr^2,1_(ΩT)

+ kr

m

k

_Wr^1,0_(ΩT)

+ kr

m

k

_W1−1/r,1/2−1/(2r)

r (S^T)

.

Summing up the estimates for all terms of the r.h.s. of (3.10) we obtain Y

m

≤ φ(a

m

(T ))(T

^(r−1)/r

+T

^r/2−1

+T

^β

)Y

m

+φ(a

m

(T ))(T

^(r−1)/r

+T

^β

)Y

m−1

. Taking T so small that φ(a

m

(T ))(T

^(r−1)/r

+ T

^r/2−1

+ T

^β

) ≤ 1/2 we get

Y

m

≤ φ(a

m

(T ))(T

^(r−1)/r

+ T

^β

)Y

m−1

.

Thus if φ(a

m

(T ))(T

^(r−1)/r

+ T

^β

) < 1 we have a contraction, hence Y

m

→ 0 as m → ∞. This yields the existence of u ∈ W

_r^2,1

(Ω

^T

) and p ∈ W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

) such that

u

m

→ u in W

_r^2,1

(Ω

^T

),

q

m

→ p in W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

).

The proof of the lemma is complete.

By Lemma 3.2 we see that system (3.1) has a unique solution (u, q) in W

_r^2,1

(Ω

^T

) × W

_r^1,0

(Ω

^T

) ∩ W

1−1/r,1/2−1/(2r)

r

(S

^T

). By Lemma 3.1 we get the estimate

(3.11) kuk

_Wr^2,1_(ΩT)

+ kqk

_Wr^1,0_(ΩT)

+ kqk

_W1−1/r,1/2−1/(2r)

r (S^T)

≤ c(kv

0

k

_W^2−2/r

r (Ω)

, kSk

_W^2−2/r

r

).

Since for u ∈ W

_r^2,1

(Ω

^T

) the transformation (1.7) is invertible, from (3.11) we obtain estimate (1.8). Theorem 1.1 is proved.

References

[1] O. V. B e s o v, V. P. I l ’ i n and S. M. N i k o l ’ s k i˘ı, Integral Representations of Func- tions and Imbedding Theorems , Nauka, Moscow, 1975 (in Russian).

[2] O. A. L a d y z h e n s k a y a, V. A. S o l o n n i k o v and N. N. U r a l ’ t s e v a, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1975.

[3] P. B. M u c h a and W. M. Z a j¸a c z k o w s k i, On the existence for the Cauchy–Neumann problem for the Stokes system in the L

p

-framework , Studia Math., to appear.

[4] V. A. S o l o n n i k o v, On nonstationary motion of an isolated volume of a viscous

incompressible fluid , Izv. Akad. Nauk SSSR 51 (1987), 1065–1087 (in Russian).

[5] V. A. S o l o n n i k o v, Solvability on a finite time interval of the problem of evolution of a viscous incompressible fluid bounded by a free surface, Algebra Anal. 3 (1991), 222–257 (in Russian).

Piotr Bogus law Mucha Institute of Applied

Mathematics and Mechanics Warsaw University

Banacha 2

02-097 Warszawa, Poland

E-mail: mucha@hydra.mimuw.edu.pl

Wojciech Zaja¸czkowski Institute of Mathematics Polish Academy of Sciences Sniadeckich 8 ´ 00-950 Warszawa, Poland and Institute of Mathematics and Operations Research Military University of Technology Kaliskiego 2 01-489 Warszawa, Poland E-mail: wz@impan.gov.pl

Received on 28.5.1999;

revised version on 17.1.2000