ON LOCAL EXISTENCE OF SOLUTIONS OF THE FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE VISCOUS SELF-GRAVITATING FLUID MOTION

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1 APPLICATIONES MATHEMATICAE 27,3(2), pp P.B.MUCHAandW.ZAJA CZKOWSKI(Waszawa) ON LOCAL EXISTENCE OF SOLUTIONS OF THE FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE VISCOUS SELF-GRAVITATING FLUID MOTION Abstact. The local-in-time existence of solutions of the fee bounday poblem fo an incompessible viscous self-gavitating fluid motion is poved. We show the existence of solutions with lowest possible egulaity fo this poblem such that u W ( T ) with > 3. The existence is poved by the method of successive appoximations whee the solvability of the Cauchy Neumann poblem fo the Stokes system is applied. We have to undeline that in the L p -appoach the Lagangian coodinates must be used. We ae looking fo solutions with lowest possible egulaity because this simplifies the poof and deceases the numbe of compatibility conditions. 1. Intoduction. In this pape we conside the motion of a viscous incompessible fluid in a bounded domain t R 3 with a fee bounday S t which is unde the self-gavitational foce. Let v = v(x,t) be the velocity of the fluid, p = p(x,t) the pessue, ν the constant viscosity coefficient and p the extenal pessue. Then the poblem is descibed by the following system: (1.1) v t +v v divt(v,p) = U in T, divv = in T, T(u,p) n = p n on S T, v t= = v in, t t= =, S t t= = S, v n = ϕ t / ϕ on S T, 2 Mathematics Subject Classification: 35Q3, 76D5. Key wods and phases: local existence, Navie Stokes equations, incompessible viscous baotopic self-gavitating fluid, shap egulaity, anisotopic Sobolev space. ReseachsuppotedbyPolishKBNGant2P3A3816. [319]

2 32 P.B.MuchaandW.Zaja czkowski whee T = t T t {t}, S T = t T S t {t}, ϕ(x,t) = descibes S t at least locally, n is the unit outwad vecto nomal to S t, n = ϕ/ ϕ, t is the domain at time t, S t = t, t T. Moeove, the dot denotes the scala poduct in R 3. By T = T(v,p) we denote the stess tenso of the fom (1.2) T(v,p) = {T ij } i,j=1,2,3 = { pδ ij +D ij (v)} i,j=1,2,3 whee (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 defomation tenso. Moeove, U( t,x,t) is the self-gavitational potential (1.4) U( t,x,t) = k t dy x y, whee k is the gavitation constant and some aguments of U ae omitted in evident cases. In view of the equation (1.1) 2 and the kinematic condition (1.1) 6 the total volume is conseved: (1.5) t = t dx = dx =. Let be given. Then we intoduce the Lagangian coodinates ξ as the initial data fo the following Cauchy poblem: x (1.6) t = v(x,t), x t= = ξ, ξ = (ξ 1,ξ 2,ξ 3 ). Integating (1.6), we obtain a tansfomation which connects the Euleian x and the Lagangian ξ coodinates, (1.7) x = x(ξ,t) ξ + t u(ξ,t ) x u (ξ,t), wheeu(ξ,t) = v(x u (ξ,t),t) and the index u in x u (ξ,t) will beomitted when no confusion can aise. Then fom (1.1) 6 we have t = {x R 3 : x = x(ξ,t), ξ } and S t = {x R 3 : x = x(ξ,t), ξ S = S = }. Ou aim is to pove the local-in-time existence of solutions to poblem (1.1) with lowest possible egulaity. Theefoe we apply the L p -appoach. The esult of the pape is the following theoem. Theoem 1.1. Let > 3, v W 2 2/ (), S W 2 1/. Then thee exists T > such that fo all T T thee exists a unique solution (u,p) of (1.1) such that u W ( T ), p W 1, ( T ) W ( S T )

3 Fee bounday poblem 321 and the following estimate holds: (1.8) u W ( T ) + p W 1, ( T ) + p W ( S T ) c(t) v W 2 2/ (). To pove Theoem 1.1 we need solvability of the Cauchy Neumann poblem fo the Stokes system fom [3]. To ecall the esult we fomulate the poblem u t divt(u,p) = F in T, (1.9) divu = G in T, n T(u,p) = H on S T, u t= = u in, whee T = [,T] and S T = S [,T]. Theoem 1.2 (see [3]). Let > 3, F L ( T ), G W 1, ( T ), G t divf = divb +A, A,B L ( T ), H W (S T ), u W 2 2/ (), S W 2 2/, and assume the compatibility conditions (1.1) divu = G(x,), n T(u,p ) S = H(x,), whee p = p t=. Then thee exists a unique solution (u,p) to poblem (1.9) such that u W ( T ), p W 1, ( T ) W (S T ) and the following estimate holds: (1.11) u W ( T ) + p W 1, ( T ) + p W (S T ) C(T)[ F L ( T )+ G W 1, ( T ) + B L ( T ) + A L ( T ) + H W (S T ) + u 2 2/ W () ], whee C(T) is a constant inceasing with T which does not depend on the solution (u, p). Poblem (1.1) without the self-gavitation foce is consideed in [4]. Moeove we ecall that the local existence of solutions to poblem (1.1) with suface tension is shown in [5]. 2. Notation. We neeheanisotopic Sobolev spaces W m,n (Q T ) whee m,n R + {}, 1 and Q T = Q (,T), with the nom

4 322 P.B.MuchaandW.Zaja czkowski (2.1) u W m,n (Q T ) = Q + u(x,t) dx m [ m ] Q m =[ m ] n [ n ] Q Q dx D m QQ D [n] D n x u(x,t) dx D m x u(x,t) Dx m u(x,t) dxdx x x s+( m [ m ]) t u(x,t) dx t u(x,t) D [n] t (x,t ), t t 1+(n [n]) whee s = dimq, [α] is the integal pat of α, D l x = l 1 x1... l s xs whee l = (l 1,...,l s ) is a multiindex. In the poof we will use the following esults. and then Poposition 2.1 (see [1]). Let u W m,n ( T ), m,n R +. If q κ = fo all ε (,1). 3 i=1 ( α i ) ( 1 q m + β ) 1 q n < 1 D β t Dα x u L q ( T ) ε 1 κ u W m,n ( T ) +cε κ u L ( T ) Poposition 2.2 (see [1, 2]). Let u W 2m,m ( T ), m R +. If 2m 1/ > then u = u ST is well defined as a function in W 2m 1/,m 1/(2) (S T ) and u 2m 1/,m 1/(2) W (S T ) c u W 2m,m ( T ). Poposition 2.3 (see [1, 2]). Let u W 2m 1/,m 1/(2) (S T ), m R +. If 2m 1/ > then thee exists a function ũ W 2m,m ( T ) such that ũ ST = u and the following estimate holds: ũ W 2m,m ( T ) c u W 2m 1/,m 1/(2) (S T ). In ou consideations we will use well known imbedding theoems fo Sobolev spaces. All constants ae denoted by the same lette c.

5 Fee bounday poblem Poof of Theoem 1.1. To pove local existence of solutions to poblem (1.1) we wite it in the Lagangian coodinates: (3.1) u t div u T u (u,q) = u U u in T, div u u = in T, n u T u (u,q) = p n u on S T, u t= = v on, whee 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) = kj y(ξ,t)dξ x(ξ,t) y(ξ,t), whee J x(ξ,t) is the Jacobian of the tansfomation 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 matix and the summation convention ove epeated indices is used. To pove the existence of solutions to (3.1) we use the following method of successive appoximations: (3.2) u m+1,t div um T um (u m+1,q m+1 ) = um U um in T, div um u m+1 = in T, n um T um (u m+1,q m+1 ) = p n um on S T, u m+1 t= = v on, whee m =,1,2,... and u m is teated as a given function. Assume u = and q =. To apply Theoem 1.2 we wite (3.2) in the fom u m+1,t divt(u m+1,q m+1 ) = div um T um (u m+1,q m+1 ) divt(u m+1,q m+1 )+ um U um in T, (3.3) divu m+1 = divu m+1 div um u m+1 in T, n T(u m+1,q m+1 ) = n T(u m+1,q m+1 ) n um T um (u m+1,q m+1 ) p n um on S T, u m+1 t= = v on, whee the opeatos without index contain deivatives with espect to ξ and n is the unit outwad vecto nomal to S.

6 324 P.B.MuchaandW.Zaja czkowski Fist we obtain a unifom bound fo the sequence {u m } m= detemined by (3.3). Lemma 3.1. Assume that S W 2 1/, v W 2 2/ (). Then (3.4) u m W ( T ) + q m W 1, ( T ) + q m W (S T ) if T is small enough. c( v W 2 2/ Poof. Applying Theoem 1.2 to poblem (3.3) yields (), S W 2 2/ (3.5) u m+1 W ( T ) + q m+1 W 1, ( T ) + q m+1 W (S T ) c divt(u m+1,q m+1 ) div um T um (u m+1,q m+1 ) L ( T ) +c um U um L ( T ) +c divu m+1 div um u m+1 W 1, ( T ) +c n T(u m+1,q m+1 ) n um T um (u m+1,q m+1 ) W (S T ) +c n um W +c ((I A (u m ))u m+1 ) t L ( T ), (S T ) +c v 2 2/ W () whee ((I A (u Ì m ))u m+1 ) t is teated as B fom Theoem 1.2 (Ã = ) and t A ij (u m ) = δ ij + u mi,ξ j dτ, A kl (u m) = A 1 lk (u m). Hee 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 fom 3 k=1 ξ k A lk (u m )(ξ,t) =. All the above elations hold unde the assumption that div um 1 u m =. To continue the induction we need to have div um u m+1 =, but this is given by (3.3) 2. Now we estimate the paticula tems fom the Ì.h.s. of (3.5). Define a m = T ( 1)/ u m W ( T ), α t m(t) = {α ij (u m )} = { u mi,ξ j dτ}. Fo > 3 we have α m L () ca m. To estimate the fist tem on the.h.s. of (3.5) we calculate div um T um (u m+1,q m+1 ) divt(u m+1,q m+1 ) = {ν(ξ lxj ξ kxj x s x sξl δ σi +ξ lxj ξ kxi x s x sξl δ σj )u m+1σ,ξk +ν(ξ lxi ξ kxj δ jk δ jl )u m+1i,ξl ξ k +ν(ξ lxj ξ kξi δ lj δ ki )u m+1j,ξl ξ k (ξ lxj δ lj q m+1,ξl )}, whee the matix ξ,x depends on u m. )

7 Fee bounday poblem 325 Since x iξj = δ ij + Ì t u iξ j (τ)dτ = δ ij +α ij and ξ jxi is the invese matix to x iξj, we have ξ jxi = δ ij +φ ij (α), whee φ ij is a polynomial matix-valued function which contains tems of α and α 2 (α = {α ij }). Then ξ j,xi x k = φ ij,αs α s,ξσ ξ σ,xk, whee α s,ξσ = Ì t u,ξ s ξ σ (τ)dτ. Then we wite the fist tem of the.h.s. of (3.5) in the fom ψ 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 L ( T ) φ(a m )a m ( u m+1 W ( T ) + q m+1 W 1, ( T ) ), whee ψ i ae some functions with ψ i () and φ always denotes an inceasing positive function. We estimate the thid tem by the same quantity. The fouth tem can be expessed in the fom whee ψ 3 (α m )α m u m+1,ξ +ψ 4 (α m )α m q m+1 W (S T ) I Next we have L ( ( ψ 3 (α m )α m u m+1,ξ W (S T ) + ψ 4 (α m )α m q m+1 W (S T ) I +J, ψ 3 (α m )α m u m+1,ξ W 1()dτ ( + ψ 3 (α m )α m u m+1,ξ W 1/2 (,T) L+K. ψ 3 (α m )α m u m+1,ξ L () dτ ( ( ( ψ 3,αm (α m ) ψ 3 (α m ) t t u m,ξξ dτ α m u m+1,ξ u m,ξξ dτ u m+1,ξ L () L () ψ 3 (α m )α m u m+1,ξξ L () dτ L1 +L 2 +L 3 +L 4.

8 326 P.B.MuchaandW.Zaja czkowski Continuing, we have L 1 φ(a m (T))a m (T) u m+1 W L 2 +L 3 φ(a m (T))a m (T) Next we examine ( K dξ ( t ( T ), u m,ξξ dτ u m+1,ξ φ(a m (T))a m (T)a 2 m(t) u m+1 W ( T ), L 4 φ(a m (T))a m (T) u m+1 W ( T ). ( + dξ + ( dξ L () ψ 3(α m (t)) ψ 3 (α m (t )) α m (t) u m+1,ξ (t) t t 1+/2 K 1 +K 2 +K 3. Using the fomula (3.6) ψ 3 (α m (t)) ψ 3 (α m (α m (t )) we obtain K 1 +K 2 ( φ(a m (T)) dξ ψ 3(α m (t )) α m (t) α m (t ) u m+1,ξ (t) t t 1+/2 ψ 3(α m (t)) α m (t) u m+1,ξ (t) u m+1,ξ (t ) t t 1+/2 = (α m (t) α m (t )) ( φ(a m (T)) dξ 1 ( φ(a m (T)) u m W ( T ) dξ ψ 3,αm (t )(α m (t )+s(α m (t) α m (t ))ds Ì t u t m,ξ dτ u m+1,ξ (t) t t 1+/2 t t t /2 2 Integating with espect to t we get (/2 > 1) t u m,ξ dτ u m+1,ξ (t) t t /2 2 u m+1,ξ K4. K 4 φ(a m (T))T /2 1 u m W ( T ) u m+1 W ( T ).

9 Finally Fee bounday poblem 327 K 3 φ(a m (T))a m (T) u m+1 W ( T ). Summaizing the above consideations we obtain Similaly, we obtain I φ(a m (T))a m (T) u m+1 W ( T ) +φ(a m (T))T /2 1 u m W ( T ) u m+1 W ( T ). J φ(a m (T))a m (T) q m+1 W 1,1/2 ( T ) +φ(a m (T))T /2 1 u m W ( T ) q m+1 W 1,1/2 ( T ), whee q m+1 is an extension of q m+1 W (S T ). The fifth tem on the.h.s. of (3.5) is estimated by ψ 5 (α m ) W whee M 1 φ(a m (T)) ( M 2 dξ S ( using (3.6) we have ( φ(a m (T)) dξ S ( dξ S (S T ) ( t ψ 5 (α m (t)) 1 1/ W (S) ( + ψ 5 (α m (t)) 1/2 1/(2) W (,T) dξ S M 1 +M 2, u m,ξ dτ W 1 () φ(am (T))T 1/ a m (T), ψ 5(α m (t)) ψ 5 (α m (t )) t t 1+(1/2 1/(2)) Ì t u t m,ξ (ξ,τ)dτ t t 1+(1/2 1/(2)) u m,ξ (ξ,τ) dτ ( φ(a m (T))T 1/2+/2 u m W ( T ). ; t t /2 3/2 The seventh tem of the.h.s. of (3.5) will be consideed in the fom ((I A (u m ))u m+1 ) t L ( T ) ψ 6 (α m )α m u m+1,t L ( T ) + ψ 7 (α m )u m,ξ u m+1 L ( T ) N 1 +N 2

10 328 P.B.MuchaandW.Zaja czkowski and we have (3.7) N 1 φ(a m (T)) α m u m+1 W ( T ), ( N 2 φ(a m (T)) (u m,ξ v,ξ +v,ξ ) v + φ(a m (T))T 1/ v 2 W 2 2/ () +φ(a m (T)) u m W ( T ) a m+1 +φ(a m (T)) v 2 2/ W () Tβ ( u m W ( T ) + v W 2 2/ () ). t u m+1,t ) L(T) In the last tem of the.h.s. of (3.7) 2 we have applied the imbedding W 1,1/2 ( T ) C β (,T;L ()) with < β < 1/2 1/. This enables us to get u m,ξ v,ξ L T β ( u m W ( T ) + v W 2 2/ () ). Finally we conside the second tem of the.h.s. of (3.5). We have J xu m um U um = (ξ,t) um x um (ξ,t) x um (ξ,t) dξ = = Assuming that um x um (ξ,t) (x um (ξ,t) x um (ξ,t)) x um (ξ,t) x um (ξ,t) 3 J xu m (ξ,t)dξ um x um (ξ,t) (ξ ξ Ì Ì 1 )(1+ dsì t su m (ξ +s(ξ ξ ),τ)dτ) ξ ξ dsì t su m (ξ +s(ξ ξ ),τ)dτ 3 J xu m (ξ,t)dξ. t u mξ (,τ) L ()dτ < 1, we obtain um U um L ( T ) φ(a m (T))T 1/ dξ ξ ξ 2 φ(a m (T))T 1/. L () Fo simplicity we intoduce X k = u k W ( T ) + q k W 1, ( T ) + q m W (S T ).

11 Fee bounday poblem 329 Summing up the estimates fo all tems of the.h.s. of (3.5) we get X m+1 a m φ(a m )X m+1 +φ(a m )T /2 1 X m X m+1 Putting we have (3.8) +φ(a m )a m T 1/ +φ(a m )T 1/2+/2 X m +φ(a m )X m a m+1 (T) +(T 1/ +T β )φ(a m )+φ(a m (T))T β X m. { 1 a = min, 2 2, 1, } 2,β X m+1 T a φ(a m )X m X m+1 +T a φ(a m )X m +φ(a m )T a X m X m+1 +T a φ(a m ). By induction we pove that X k 1 (X = ). Taking T such that T 1 and T a φ(1) 1/4, inseting X m 1 in (3.8), we obtain X m X m X m , which gives X m+1 1. The poof of the lemma is complete. Lemma 3.2. Assume that S W 2 1/ exist u W ( T ) and p W 1,, v W 2 2/ ( T ) W (). Then thee (S T ) such that (S T ) u m u in W ( T ) and q m p in W 1, ( T ) W as m fo T small enough. Poof. We show that {(u m,q m )} n=1 is convegent. Fo this pupose we conside v m = u m+1 u m, m = q m+1 q m which satisfy the system (3.9) v m,t divt(v m, m ) = div um T um (u m+1,q m+1 ) div um 1 T um 1 (u m,q m ) divt(v m,q m ) + um U um um 1 U um 1 I in T, divv m = divv m div um u m+1 +div um 1 u m J in T, n T(v m, m ) = n T(v m, m ) n um T(u m+1,q m+1 ) +n um 1 (u m,q m ) p n um +p n um 1 K on S T, v m t= = on. By Theoem 1.2 we obtain an estimate on solutions of (3.9): (3.1) v m W ( T ) + m W 1, ( T ) + m W (S T ) c( I L ( T ) + J W 1, ( T ) + K + B W (S T ) L ( )), T whee B is defined by the elation J t = divb.

12 33 P.B.MuchaandW.Zaja czkowski Fist we estimate the tems of the.h.s. of (3.9) 1 in L ( 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 ) divt(v m,q m ) = (div um T um divt)(v m, 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 fist tem of the.h.s. of (3.5): I 11 L ( T ) φ(a m )a m ( v W ( T ) + m W 1, ( T ) ). Fo the second tem we have I 12 L ( T ) = (div um (T um T um 1 ) Next we conside +(div um div um 1 )T um 1 )(u m,q m ) L ( T ) φ(a m )T ( 1)/ v m 1 W ( T ). 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 fist tem is I 21 = A(u m ) ( xm (ξ,t) y m (ξ,t) x m (ξ,t) y m (ξ,t) 3J y m (ξ,t) x m 1(ξ,t) y m 1 (ξ,t) x m 1 (ξ,t) y m 1 (ξ,t) 3 ) dξ x m 1(ξ,t) y m 1 (ξ,t) x m 1 (ξ,t) y m 1 (ξ,t) 3J y m 1 (ξ,t) x m (ξ,t) y m (ξ,t) = A(u m ) x m (ξ,t) y m (ξ,t) 3(J y m (ξ,t) J ym 1 (ξ,t))dξ ( xm (ξ,t) y m (ξ,t) A(u m ) x m (ξ,t) y m (ξ,t) 3 ) J ym 1 (ξ,t)dξ I 211 +I 212, whee x k (ξ,t) = ξ + Ì t u k(ξ,t ) and y k (ξ,t) = ξ + Ì t u k(ξ,t ). Since we have J ym (ξ,t) J ym 1 (ξ,t) c u m u m 1 W ( T ), The same holds fo I 212. Since I 211 L ( T ) φ(a m )T 1/ v m 1 W ( T ).

13 Fee bounday poblem 331 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 we obtain and c u m u m 1 W ( T ), I 212 L ( T ) φ(a m )T 1/ v m 1 W ( T ). We estimate J in W 1, ( T ) by the same quantity. Let K = K 1 +K 2, whee K 1 = n T(v m, m ) n um T(u m+1,q m+1 )+n um 1 (u m,q m ) = (n T(v m, m ) n um T(v m, m )) +(n um T um n um 1 T um 1 )(u m,q m ) K 11 +K 12 K 2 = p (n um n um 1 ). The tem K 11 is estimated just as the fouth tem of the.h.s. of (3.5): K 11 W (S T ) φ(a m(t))a m (T) v m W ( T ) The second tem is +φ(a m (T))T /2 1 u m W ( T ) v m W ( T ). 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. Fo K 121 we have K 121 W (S T ) φ(a m(t))t ( 1)/ v m 1 W ( T ). Since n um n um 1 W (S T ) φ(a m(t))t ( 1)/ v m 1 W ( T ), we conclude that K 122 W (S T ) + K 2 W (S T ) φ(a m (T))T ( 1)/ v m 1 W ( T ). Finally we have to examine B which is defined by J t = divb. We have J = divv m div um u m 1 +div um 1 u m = (divv m div um v m )+(div um 1 u m div um u m ) J 1 +J 2. To examine J 1 we poceed as in the case of the seventh tem of the.h.s. of (3.5). By the same agument 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 fo N 2 in the poof of Lemma 3.1, the following estimate:

14 332 P.B.MuchaandW.Zaja czkowski B 2 L ( T ) φ(a m (T))(T ( 1)/ +T β ) v m 1 W ( T ), whee < β < 1/2 1/. Define Y m = v m W ( T ) + m W 1, ( T ) + m W (S T ). Summing up the estimates fo all tems of the.h.s. of (3.1) we obtain Y m φ(a m (T))(T ( 1)/ +T /2 1 +T β )Y m +φ(a m (T))(T ( 1)/ +T β )Y m 1. Taking T so small that φ(a m (T))(T ( 1)/ +T /2 1 +T β ) 1/2 we get Y m φ(a m (T))(T ( 1)/ +T β )Y m 1. Thus if φ(a m (T))(T ( 1)/ +T β ) < 1 we have a contaction, hence Y m as m. This yields the existence of u W ( T ) and p W 1, ( T ) (S T ) such that W u m u in W ( T ), q m p in W 1, ( T ) W (S T ). The poof of the lemma is complete. By Lemma 3.2 we see that system (3.1) has a unique solution (u,q) in W ( T ) W 1, ( T ) W (S T ). By Lemma 3.1 we get the estimate (3.11) u W ( T ) + q W 1, ( T ) + q W (S T ) c( v W 2 2/ (), S W 2 2/ Since fo u W ( T ) the tansfomation (1.7) is invetible, fom (3.11) we obtain estimate (1.8). Theoem 1.1 is poved. ). Refeences [1] O.V.Besov,V.P.Il inands.m.nikol skiĭ,integalrepesentationsoffunctions and Imbedding Theoems, Nauka, Moscow, 1975(in Russian). [2] O.A.Ladyzhenskaya,V.A.SolonnikovandN.N.Ual tseva,lineaand Quasilinea Equations of Paabolic Type, Ame. Math. Soc., Povidence, RI, [3] P.B.MuchaandW.M.Zaj aczkowski,ontheexistencefothecauchy Neumann poblemfothestokessysteminthel p-famewok,studiamath.,toappea. [4] V.A.Solonnikov,Onnonstationaymotionofanisolatedvolumeofaviscous incompessible fluid, Izv. Akad. Nauk SSSR 51(1987), (in Russian).

15 Fee bounday poblem 333 [5] V.A.Solonnikov,Solvabilityonafinitetimeintevalofthepoblemofevolution of a viscous incompessible fluid bounded by a fee suface, Algeba Anal. 3(1991), (in Russian). Piot Bogus law Mucha Institute of Applied Mathematics and Mechanics Wasaw Univesity Banacha Waszawa, Poland mucha@hyda.mimuw.edu.pl Wojciech Zaja czkowski Institute of Mathematics Polish Academy of Sciences Śniadeckich 8-95 Waszawa, Poland and Institute of Mathematics and Opeations Reseach Militay Univesity of Technology Kaliskiego Waszawa, Poland wz@impan.gov.pl Received on ; evised vesion on