ON LOCAL MOTION OF A COMPRESSIBLE BAROTROPIC VISCOUS FLUID WITH THE BOUNDARY SLIP CONDITION. Marek Burnat Wojciech M. ZajĄczkowski. 1.

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1 opological Mehods in Nonlinear Analysis Journal of he Juliusz Schauder Cener Volume 1, 1997, ON LOCAL MOION OF A COMPRESSIBLE BAROROPIC VISCOUS FLUID WIH HE BOUNDARY SLIP CONDIION Marek Burna Wojciech M. ZajĄczkowski (Submied by A. Granas Dedicaed o O. A. Ladyzhenskaya 1. Inroducion We consider he moion of a compressible baroropic viscous fluid in a bounded domain R 3 wih he boundary slip condiion. Le ρ = ρ(x, be he densiy of he fluid, v = v(x, he velociy, p = p(ρ(x, he pressure, f = f(x, he exernal force field per uni mass. hen he moion is described by he following problem (see [3]: (1.1 ρ(v + v v div (v, p = ρf in = (,, ρ + div (ρv = in, ρ = = ρ v = = v in, τ α (v, p n + v τ α =, α = 1, 2, on S = S (,, v n = on S, where (v, p is he sress ensor of he form (1.2 (v, p = { ij (v, p} i,j=1,2,3 = {µ( xi v j + xj v i + (ν µdiv vδ ij pδ ij } i,j=1,2,3, 1991 Mahemaics Subjec Classificaion. 35K5, 35F3, 76N1. Key words and phrases. Compressible baroropic viscous fluid, local exisence, poenial heory, anisoropic Sobolev spaces wih fracional derivaives. he second named auhor was suppored by Polish KBN Gran 2P3A c 1997 Juliusz Schauder Cener for Nonlinear Sudies 195

2 196 M. Burna W. M. Zajączkowski n, τ α, α = 1, 2, are uni orhonormal vecors such ha n is he ouward normal vecor and τ 1, τ 2 are angen o S. Finally, is a posiive consan and µ, ν are consan viscosiy coefficiens. By virue of (1.1 2 and (1.1 5 he oal mass is conserved, so (1.3 ρ (x dx = ρ(x, dx = M. Moreover, from hermodynamic consideraions we have (1.4 ν > µ/3. Le us inroduce he quaniy for vecors u, v H 1 ( (1.5 E (u, v = ( xi u j + xj u i ( xi v j + xj v i dx, where he summaion convenion over he repeaed indices is used. We assume ha E (u = E (u, u. We recall from [7] ha he vecors for which E (u = form a finie dimensional affine space of vecors such ha u = A + B x, where A and B are consan vecors. We define H( = {u : E (u <, u n = on S}. If is a region obained by roaion abou a vecor B, we denoe by H( he space of funcions in H( saisfying he condiion u(xu (x dx =, where u = B x; oherwise we se H( = H( (see [7]. From Lemmas 2.1, 2.2 from [4] (see also Lemma 4 in [7] we infer Lemma 1.1 (Korn inequaliy. Le S H 3+α, α (1/2, 1. hen for any u H(, (1.6 u 2 1, ce (u, where c is a posiive consan. We inroduce Lagrangian coordinaes as he iniial daa for he Cauchy problem (1.7 dx d = v(x,, x = = ξ, hen (1.8 x = ξ + v(ξ, τ dτ x v (ξ,, and v(ξ, = v(x v (ξ,,. We someimes omi he index v in x v.

3 On Local Moion of a Compressible Baroropic Viscous Fluid 197 o prove he exisence of soluions o problem (1.1 we inegrae he equaion of coninuiy using he Lagrangian coordinaes (1.8. herefore we have ( (1.9 ρ(x, = ρ (ξ exp where v = ( ξ k / x i v i,ξk. ( v v(ξ, τ dτ Assuming ha v = u in (1.9 and assuming ha ransformaion (1.8 is generaed by vecor u, we also consider insead of (1.1 he following linearized problem, (1.1 ρ u v div u u (v, p(ρ u = ρ u f u in, τ uα u (v, p(ρ u n u + v τ uα =, α = 1, 2, on S, v n u = on S, v = = v in, where ρ u (ξ, = ρ(x u (ξ,,, f u (ξ, = f(x u (ξ,, and see also (6.3, which assigns he ransformaion (1.11 v = Φ(u. A fixed poin of (1.11 is a soluion o problem (1.1. o prove he exisence of soluions o problem (1.1 we firs consider he following problem (1.12 ω div D(ω = F in, τ α D(ω n = G α, α = 1, 2, on S, ω n = G 3 on S, ω = = ω in. o consider his problem we use he poenial echnique developed by V. A. Solonnikov. he exisence will be proved in Sobolev Slobodeskiĭ spaces. o prove he exisence of soluions o problem (1.1 we need also examine he following problem (1.13 ηω div D(ω = F 1 in, τ α D(ω n = G 1α, α = 1, 2, on S, ω n = G 13 on S, ω = = ω in, where η η >, η is a consan and η is a given funcion.

4 198 M. Burna W. M. Zajączkowski 2. Noaion and auxiliary resuls We use he anisoropic Sobolev Slobodeskiĭ spaces W l,l/2 2 (Q, l R +, Q = Q (,, where Q is eiher (a domain in R 3 or S (he boundary of, wih he norm where for ineger l, and u 2 W l 2 (Q = u 2 = u 2 W l,l/2 2 (Q W2 l (Q d + u 2 dx W l/2 Q 2 (, α [l] u 2 W l, 2 (Q + u 2 W,l/2 2 (Q, u 2 W2 l(q = D α u 2 L, 2(Q α l D α u 2 L 2(Q + α =[l] for nonineger l, where s = dim Q, D α x = α1 x 1 Q Q D α x u(x D α x u(x 2 x x s+2(l [l] dx dx,... αs x s, α = (α 1,..., α s is a muliindex, [l] is he ineger par of l. For Q = S he above norm is inroduced by using local mappings and a pariion of uniy. Finally, for ineger l/2, and u 2 = W l/2 2 (, u i 2 L 2(,, i l/2 u 2 = W l/2 2 (, u i 2 L 2(, + i [l/2] i=[l/2] Moreover, u p, = u Lp(, p [1, ], ( u(x u(x 2 1/2 [u] α, = x x dx dx, 3+2α ( u( u( 2 1/2 [u] α,(, = d d, 1+2α i u( i u( 2 1+2(l/2 [l/2] d d. and ( [u] α,,x = ( [u] α,, = d dx u(x, u(x, 2 1/2 x x 3+2α dx dx, u(x, u(x, 2 1+2α d d 1/2.

5 On Local Moion of a Compressible Baroropic Viscous Fluid 199 o consider he problems wih vanishing iniial condiions we need a space of funcions which admi a zero exension o <. herefore, for every, we inroduce he space H l,l/2 (Q wih he norm (see [1], [5], [8] For l/2 Z, u 2 = e 2 u 2 H l,l/2 (Q W2 l (Q d + u 2. H,l/2 (Q u 2 = H,l/2 (Q l + e 2 u 2 L 2(Q d e 2 k u (, τ k u (, 2 L d 2(Q dτ, τ 1+2(l/2 k where k = [l/2] < l/2, and u (x, = u(x, for >, u (x, = for <. For l/2 Z, u 2 = H,l/2 (Q e 2 ( l u 2 L + 2(Q l/2 u 2 L d, 2(Q and we assume ha j u = =, j =,..., l/2 1, so u (x, has a generalized derivaive l/2 u in Q (,. For simpliciy we wrie u l,q u l,,q = u l,l/2 W 2 (Q, u l,q = u W l 2 (Q, = u l,l/2 H (Q, u p,q = u Lp(Q. In he above definiions we can use he noaion ( u W l 2 (Q = Dx α u 2 2,Q + l/2 [Dx α u] l [l],q 2. α [l] α =[l] We se R n + = {x R n : x n > }, R n+1 = R n (,, D n+1 = R n + (,, n = 2, 3. For funcions defined in R n+1 and vanishing sufficienly fas a infiniy we define he Fourier ransform wih respec o x and he Laplace ransform wih respec o by he formula (2.1 f(ξ, s = e s d f(x, e ix ξ dx, R n where x ξ = n i=1 x iξ i. Hence we define he norm = dξ ũ(ξ, s 2 ( s + ξ 2 l dξ, s = + iξ, R +. u 2 l,,r n+1 R n Similarly, for funcions defined in D n+1, we have f(ξ, x n, s = e s d f(x, e ix ξ dx, R n 1

6 2 M. Burna W. M. Zajączkowski where x = (x 1,..., x n 1, ξ = (ξ 1,..., ξ n 1, and inroduce he norm u 2 = dξ l,,d n+1 x j n ũ(ξ, x n, s 2,R ( s + ξ 2 l j dξ 1 j [l] R n dξ ũ(ξ,, s 2 l,r dξ 1, s = + iξ, R +. R n 1 + We inroduce a pariion of uniy. Le us define wo collecions of open subses {ω (k } and { (k }, k M N, such ha ω (k (k, k ω(k = k (k =, (k S = for k M and (k S for k N. We assume ha a mos N of he (k have nonempy inersecion, and sup k diam (k 2λ for some λ >. Le ζ (k (x be a smooh funcion such ha ζ (k (x 1, ζ (k (x = 1 for x ω (k, ζ (k (x = for x \ (k and D ν xζ (k (x c/λ ν. hen 1 k (ζ(k (x 2 N. Inroducing he funcion η (k (x = ζ (k (x l (ζ(l (x 2, we have η (k (x = for x \ (k, k η(k (xζ (k (x = 1 and D ν xη (k (x c/λ ν. By ξ (k we denoe he cener of ω (k and (k for k M and he cener of ω (k S and (k S for k N. Considering problems invarian wih respec o ranslaions and roaions we can inroduce a local coordinaes sysem y = (y 1, y 2, y 3 wih he cener a ξ (k such ha he par S (k = S (k of he boundary is described by y 3 = F (y 1, y 2. hen we consider new coordinaes defined by z i = y i, i = 1, 2, z 3 = y 3 F (y 1, y 2. We will denoe his ransformaion by (k ω (k z = Φ k (y, where y ω (k (k ; we assume ha he ses ω (k, (k are described in local coordinaes a ξ (k by he inequaliies respecively. y i < λ, i = 1, 2, < y 3 F (y 1, y 2 < λ, y i < 2λ, i = 1, 2, < y 3 F (y 1, y 2 < 2λ, Le y = Y k (x be a ransformaion from coordinaes x o local coordinaes y, which is a composiion of a ranslaion and a roaion. hen we se û (k (z, = u(φ 1 k Y 1 k (z,, ũ (k (z, = û (k (z, ζ (k (z,. From [5] we recall he necessary for us properies of spaces H l,l/2 H l,l/2 (D n+1, l R +. (R n+1 and

7 R n+1 On Local Moion of a Compressible Baroropic Viscous Fluid 21 Lemma 2.1. Any funcion u H l,l/2 (R n+1, <, can be exended o in such a way ha he exended funcion u H l,l/2 and (2.2 u l,,r n+1 c u l,,r n+1. (R n+1 Any funcion u H l,l/2 (D n+1, <, can be exended o R n+1 in such a way ha he exended funcion u H l,l/2 and (2.3 u l,,r n+1 (R n+1 c u l,,d n+1. Lemma 2.2. here exis consans c 1 and, such ha and c 2, which do no depend on u (2.4 c 1 u l,,r n+1 u l,,r n+1 c 2 u l,,r n+1. Lemma 2.3. here exis consans c 3 and, such ha and c 4, which do no depend on u (2.5 c 3 u l,,d n+1 u l,,d n+1 c 4 u l,,d n+1. From [5] we recall Lemma 2.4. Le u H r,r/2 (. hen for every ε (, 1 and q < r α, (2.6 D α x u q,, ε r α q u r,, + cε q α e u, (ε r α q + c r/2 ε q α u r,,. Lemma 2.5 (see [5]. Le u H l,l/2 (R n+1 and < 2m + α < l. hen m Dx α u H l1,l1/2, where l 1 = l 2m α and (R n+1 (2.7 m D α x u l1,,r n+1 Moreover, for ρ (, l 1 and ε >, (2.8 m D α x u ρ,,r n+1 ε l1 ρ u l,,r n+1 c u l,,r n+1. + cε h e u,r n+1, where h = ρ + 2m + α. Le u H l,l/2 (D n+1 and 2m + α < l 1/2. hen m Dx α u xn= (R n, where l 2 = l 2m α 1/2 and H l2,l2/2 (2.9 m Dx α u xn= l2,,r n c u l,,d n+1. o obain he resuls from Secion 6 we need

8 22 M. Burna W. M. Zajączkowski Lemma 2.6 (see [5]. Le s assume ha a W2 l+1 (, f W2(, l g 2 (, l > 1/2, R 3. hen he following esimaes hold W l+1 (2.1 where ε (, 1. af l, c 1 a, f l, + a l+1, (ε f l, + c 2 (ε f,, af l, c 3 f l, (ε a l+1, + c 4 (ε a,, ag l+1, c 5 a, g l+1, + a l+1, (ε g l+1, + c 6 (ε g,, ag l+1, c 7 a l+1, g l+1,, 3. Exisence of soluions o problem (1.12 wih vanishing iniial daa and in he half space Considering he problem (1.12 in he half space x 3 > and wih vanishing iniial condiions we have (3.1 ω div D(ω = x 3 >, ( ωi µ + ω 3 = b i, i = 1, 2, x 3 =, x 3 x i ω 3 = b 3 x 3 =, ω = = x 3 >. By applying he Fourier Laplace ransformaion (3.2 f(ξ, x 3, s = e s d f(x, e ix dx, R 2 Res >, s = + iξ, where ξ = (ξ 1, ξ 2, x = (x 1, x 2, x ξ = x 1 ξ 1 + x 2 ξ 2, we obain problem (3.1 in he form (3.3 µ d2 ω k dx νiξ k d ω 3 dx 3 (s + µξ 2 ω k νξ k ξ j ω j =, k = 1, 2, x 3 >, (µ + ν d2 ω 3 dx νiξ j d ω j dx 3 (s + µξ 2 ω 3 =, x 3 >, (3.4 µ d ω k dx 3 + µiξ k ω 3 = b k, k = 1, 2, x 3 =, ω 3 = b 3, x 3 =, ω as x 3, where ξ = (ξ 1, ξ 2, ξ 2 = ξ 2 1 +ξ 2 2 and he summaion convenion over he repeaed indices is assumed. Every soluion o (3.3 vanishing a infiniy has he form (3.5 ω = Φ(ξ, se τ1x3 + ψ(ξ, s(ξ 1, ξ 2, iτ 2 e τ2x3, where Φ(ξ, s = (φ 1, φ 2, (i/τ 1 ξ φ, φ j = φ j (ξ, s, j = 1, 2, τ 1 = s/µ + ξ 2, τ 2 = s/(µ + ν + ξ2, arg τ j ( π/4, π/4, j = 1, 2, ξ φ = ξ 1 φ 1 +ξ 2 φ 2, φ = (φ 1, φ 2.

9 On Local Moion of a Compressible Baroropic Viscous Fluid 23 Insering (3.5 ino (3.4 yields (3.6 τ 1 φ k + τ 2 ξ k ψ = i b 3 ξ k b k /µ, k = 1, 2, ξ φ + τ 1 τ 2 ψ = iτ 1 b3. From (3.6 we have (3.7 τ 1 φ ξ + τ 2 ξ 2 ψ = i b 3 ξ 2 b ξ/µ, ξ φ + τ 1 τ 2 ψ = iτ 1 b3. Solving (3.7, we obain (3.8 φ ξ = τ ( 1 τ1 2 2ξ 2 i b 3 1 µ b ξ ξ2 where, ψ = (3.9 τ 2 1 ξ 2 = s/µ. ( 1 τ 2 (τ1 2 (τ ξ2 1 2 ξ 2 i b 3 + 1, µ b ξ Using (3.8 and (3.9 in (3.6 implies (3.1 φ k = 1 ξξk 1, k = 1, 2, ψ = 1 ( i b ξ. sτ 1 b µτ 1 bk τ 2 s b Le (3.11 e i = e τix3, i = 1, 2, e = e 1 e 2 τ 1 τ 2. hen we wrie (3.5 in he form (3.12 ω = ψ ξ 2 (e 2 e 1 + iτ 2 where (3.13 where we used and (3.14 ξ 1 V j = 1 (i b ξ (τ 2 τ 1 ξ j τ 2 s b c φ 1 + ψξ 1 φ 2 + ψξ 2 iφ ξ/τ 1 + iτ 2 ψ e 1 V e + W e 1, = τ 2 (τ 1 + τ 2 (is b 3 + b ξξ j, j = 1, 2, ( V 3 = i i b ξ (τ 2 τ 1 = s b ic (is b 3 + τ 1 + τ b ξ, 2 W j τ 2 1 τ 2 2 = ν µ(µ + ν s c s, = φ j + ψξ j = 1 ( 1 1 b ξξj 1 + i ξ j s τ 2 τ 1 µτ 1 bj τ 2 b3 c = ξξ j τ 1 τ 2 (τ 1 + τ 2 b 1 + i ξ j, j = 1, 2, µτ 1 bj τ 2 b3 W 3 = b 3.

10 24 M. Burna W. M. Zajączkowski We need he following resul (see [1], [5], [8]. Lemma 3.1. For ξ R 2, s = + iξ, R +, ξ R, >, and for any nonnegaive ineger j and κ (, 1, (3.15 j x 3 e i 2,R + c 1 τ i 2j 1, i = 1, 2, j x 3 e i (x 3 + z j x 3 e i (x 3 2 dx 3 dz z 1+2κ c 2 τ i 2(j+κ 1, i = 1, 2, where c 1, c 2 do no depend on τ i, i = 1, 2 and ξ. Moreover, (3.16 j x 3 e 2,R + c 3 τ 1 2j 1 + τ 2 2j 1 τ 1 2, j x 3 e (x 3 + z j x 3 e (x 3 2 dx 3 dz z 1+2κ where c 3, c 4 do no depend on τ i, i = 1, 2 and ξ. Using Lemma 3.1, we obain Lemma 3.2. We assume ha c 4 τ 1 2(j+κ 1 + τ 2 2(j+κ 1 τ 1 2, b i H 1/2+α,1/4+α/2 (R 3, i = 1, 2, b 3 H 3/2+α,3/4+α/2 (R 3, >, α >. hen here exiss a soluion o problem (3.1 in H 2+α,1+α/2 (D 4 and he following esimae holds 3 ( 2 (3.17 ω i 2+α,,D 4 c( b i 1/2+α,,R 3 + b 3 3/2+α,,R 3, i=1 i=1 where c( remains bounded for >. Proof. he exisence follows from consrucions (3.12 (3.14. o obain (3.17 we consider ω 2 2+α,,D = dξ 4 x j 3 ω(ξ, x 3, s 2,R ( s + ξ 2 2+α j dξ 1 j 2 R dξ ω(ξ, x 3, s 2 2+α,R dξ 1 R 2 + dξ j 2 R ( V 2 x j 3 e 2,R + W 2 1 x j e 1 2,R 1 + ( s + ξ 2 2+α j dξ + dξ R ( V 2 e 2 2+α,R + W 2 e α,R dξ 1 I. +

11 On Local Moion of a Compressible Baroropic Viscous Fluid 25 Using he inequaliies (see [1], [5] c 1 τ 1 τ 2 c 2, τ 1 + τ ( τ 1 + τ 2, c 3 ( s + ξ 2 τ i 2 c 4 ( s + ξ 2, i = 1, 2, we obain (3.18 V 2 c( τ 2 b 3 + b 2, ( 1 W 2 c τ 2 b 2 + b 3, where τ replaces eiher τ 1 or τ 2, and b = ( b 1, b 2. In view of (3.18 and Lemma 3.1 we obain I c dξ ( b 2 (1 + τ 1+2α + b 3 2 (1 + τ 3+2α dξ R 2 c( b 2 1/2+α,,R 3 + b 3 2 3/2+α,,R 3. his concludes he proof. Now we consider he nonhomogeneous problem (3.19 ω div D(ω = f x 3 >, ( ωi µ + ω 3 = b i, i = 1, 2, x 3 =, x 3 x i ω 3 = b 3 x 3 =, ω = = x 3 >. Lemma 3.3. Le α >, >. If we assume ha f H α,α/2 (D 4, b = (b 1, b 2 H α+1/2,α/2+1/4 (R 3, b 3 H α+3/2,α/2+3/4 (R 3, >, hen he problem (3.19 has a unique soluion for ν > µ/3 such ha ω H 2+α,1+α/2 (D 4 and (3.2 ω 2+α,,D 4 c( f α,,d 4 + b α+1/2,,r 3 + b 3 α+3/2,,r 3. Proof. We exend f o a funcion f on R 4 in such a way ha f (R 4 (see (2.2, (2.3 and H α,α/2 (3.21 f α,,r 4 c f α,,d 4. Le ω be a soluion of he problem (3.22 L( x, ω ω div D(ω = f in R 4. Applying he Fourier Laplace ransformaion (2.1 o (3.22 yields (3.23 L(iξ, s ω = f,

12 26 M. Burna W. M. Zajączkowski so ω = L 1 (iξ, s f. Since de L(iξ, s = (s+µξ 2 2 (s+(µ+νξ 2 and s = +iξ wih >, one obains easily ω 2+α,,R 4 c f α,,r 4, and so (3.24 ω 2+α,,D 4 c f α,,d 4. (3.25 where (3.26 (3.27 Now v = ω ω is a soluion o he problem Using (3.26 we have v div D(v =, ( vi µ + v 3 = h i, i = 1, 2, x 3 x i v 3 = h 3, v = =, ( ω h i = b i µ i + ω 3, i = 1, 2, x 3 x i h 3 = b 3 ω 3. h i 1/2+α/2,,R 3 c( b i 1/2+α/2,,R 3 + ω 2+α,,D 4, i = 1, 2, h 3 3/2+α/2,,R 3 c( b 3 3/2+α/2,,R 3 + ω 2+α,,D 4, so applying Lemma 3.2 o problem (3.25 we see ha ω = v + ω is a soluion o (3.19 and he esimae (3.2 holds. ( Exisence of soluions o problem (1.12 Firs we consider he problem (1.12 wih vanishing iniial daa L( x, u u div D(u = f in (,, B 1i (x, x u τ i D(u n = b i, i = 1, 2, on S (,, B 2 (xu u n = b 3 on S (,. We wrie shorly B(x, x u = (B 1 (x, x u, B 2 (xu. Le f (k (x, = ζ (k (x, f(x,. We denoe by R (k, k M, he operaor (4.2 u (k (x, = R (k f (k (x,, where u (k (x, is a soluion of he Cauchy problem (4.3 L( x, u (k (x, = f (k (x,. For k N we define R (k o be he operaor (4.4 û (k (z, = R (k ( f (k (z,, b (k (z,, where û (k (z, is a soluion o he boundary value problem (4.5 L( z, û (k (z, = f (k (z,, B(z, z û (k (z, = b (k (z,,

13 On Local Moion of a Compressible Baroropic Viscous Fluid 27 where û (k (z, = Z 1 k u(k (x, and Z k describes he ransformaion beween û (k (z, and u (k (x,. hen we define an operaor R (called a regularizer by he formula (see [2], [6] (4.6 Rh = k η (k (xu (k (x,, where h (k (x, = { f (k (x, k M, Z k { f (k (z,, b (k (z, } k N, and { R (k f (k (x, k M, u (k (x, = Z k R (k (Z 1 k f (k (x,, Z 1 k b(k (x, k N. Lemma 3.3 implies he exisence of soluions of problems (4.3, (4.5 and he esimaes (4.7 u (k 2+α,,R 4 c f (k α,,r 4, k M, and (4.8 û (k 2+α,,R 4 c( f (k α,,d 4 + b (k 1/2+α/2,,R 3 + b (k 3 3/2+α/2,,R 3, k N. Le h = (f, b, b H α,α/2 and le V α ( H 1/2+α,1/4+α/2 (S H 3/2+α,3/4+α/2 (S H α = H 2+α,1+α/2 (. Inequaliies (4.7 and (4.8 imply Lemma 4.1 (see [1], [11]. Le S C 2+α, h H α, α > 1/2 and le be sufficienly large. hen here exiss a bounded linear operaor R : H α V α (defined by (4.6 such ha (4.9 Rh V α c h H α, where c does no depend on or h. We wrie problem (4.1 in he following shor form (4.1 Au = h, A = (L, B. Lemma 4.2. Le S H 3+α, h H α wih sufficienly large and α > 1/2. hen (4.11 ARh = h + h,

14 28 M. Burna W. M. Zajączkowski where is a bounded operaor in H α wih a small norm for small λ and large. and Proof. We have LRh = k M N + k N + (L( x, η (k u (k η (k L( x, u (k η (k Z k (L( z F z3, L( z, Z 1 k u(k (x, η (k L( x, u (k (x, + k M k N = f + 1 h BRh = k N(B(x, x η (k u (k η (k B(x, x u (k + k N + k N + k N η (k (B(x, x B(ξ (k, x u (k η (k Z k L( z, Z 1 k u(k (x, η (k Z k (B(ξ (k, z F z3 B(ξ (k, z Z 1 k u(k (x, η (k Z k B(ξ (k, z Z 1 k u(k (x, = b + 2 h. Now, we esimae operaors 1 and 2. By Lemmas 2.4, 2.5 and Lemma 3.3 he firs erm in 1 h is esimaed in he following way k M N (Lη (k u (k η (k Lu (k c(ε δ1 + c (ε δ2 α,, k M N c(ε δ1 + c (ε δ2 h H α, c k M N u (k 2+α,,Q (k u (k 1+α,,Q (k where δ i >, i = 1, 2, Q (k = (k (, and c (ε is a decreasing funcion. he second erm in 1 h is bounded by c k N( F 2 F û (k z=φk (y(x α,,q (k + ( F (1 + F 2 û (k z=φk (y(x α,,q (k + ( 2 F û (k z=φk (y(x α,,q (k c k N(p( F 1+α,Q (k u (k 1+α,,Q (k + F,Q (k(1 + F,Q (k u (k 2+α,,Q (k I,

15 On Local Moion of a Compressible Baroropic Viscous Fluid 29 where p is an increasing funcion. Using F, (k cλ a F 1+α, (k, a >, he inerpolaion inequaliies from Lemma 2.4 and Lemma 3.3, we have I c(ε δ1 + c (ε(λ δ2 + δ3 h H α, δ i >, i = 1, 2, 3, and c (ε is a decreasing funcion. Now, we consider he second erm in 2 h. herefore, we have o esimae he expression by ( η (k (B 1 (x, x B 1 (ξ (k, x u (k 1/2+α,,Q (k S k N + η (k (B 2 (x, x B 2 (ξ (k, x u (k 3/2+α,,Q (k S k N p( F 1+α, (k F 2+α, (k u (k 2+α,,Q (k c(ε δ1 + c (ελ δ2 h H α, δ i >, i = 1, 2, where p = p( is a polynomial and c (ε is a decreasing funcion. Similar consideraions can be applied o he oher erms of 1 and 2. Summarizing we have (4.12 h H α c[ε δ1 + c (ε(λ δ2 + δ3 ] h H α. his concludes he proof. Lemma 4.3. Le S H 3+α, α > 1/2. hen for every v V α, (4.13 RAv = v + W v, where W is a bounded operaor in V α and large, because whose norm can be made small for small λ (4.14 W v V α c(ε δ1 + c (ε(λ δ2 + δ3 v V α, ε (, 1, δ i, i = 1, 2, 3, and c (ε has he same properies as before. Proof. We have (4.15 RAv = η (k Z k R (k [Z 1 k ζ(k L( x, v, Z 1 k ζ(k B(x, x v S ] k N + η (k R (k ζ (k L( x, v = k N k M η (k Z k R (k [Z 1 k (ζ(k L( x, v L( x, ζ (k v, Z 1 k (ζ(k B(x, x v B(x, x ζ (k v S ] + k N + k N η (k Z k R (k [, Z 1 k (B(x, x B(ξ (k, x ζ (k v S ] η (k Z k R (k [Z 1 k L( x, ζ (k v, Z 1 k B(ξ(k, x ζ (k v S ]

16 21 M. Burna W. M. Zajączkowski + η (k R (k (ζ (k L( x, v L( x, ζ (k v k M + η (k R (k L( x, ζ (k v. k M By uniqueness, for he Cauchy problem (4.3 we have (4.16 R (k L( x, ζ (k v = ζ (k v, k M, so he las erm on he r.h.s. of (4.15 is equal o v. he hird erm on he r.h.s. of (4.15 has he form (4.17 η (k Z k R (k [L( z F (z z3, ṽ (k, k N B (k (ξ (k, z F (z z3 ṽ (k z3=] = k N η (k (Z k R (k [(L( z F (z z3, L( z, ṽ (k, (B (k (ξ (k, z F (z z3 B (k (ξ (k, z ṽ (k z3=] + Z k ṽ (k, where ṽ (k is a soluion o problem (4.5, so Z k R (k [L( z, ṽ (k, B (k (ξ (k, z ṽ (k z3=] = Z k ṽ (k = ζ (k (xv(x,, for k N and B (k is obained from B by applying Z 1 k. herefore, he operaor W is deermined by he firs, second and fourh expressions on he r.h.s. of (4.15 and he firs erm on he r.h.s. of (4.17. he fourh erm on he r.h.s. of (4.15 is esimaed in he following way (4.18 η (k R (k (ζ (k L( x, v L( x, ζ (k v 2+α,, c( ζ (k v α,, + 2 ζ (k v α,, c v 1+α,,Q (k where (2.6 has been used and δ i >, i = 1, 2. c(ε δ1 + c (ε δ2 v 2+α,,Q (k, Nex, we consider he firs erm on he r.h.s. of (4.17. he firs par of his erm is bounded by (4.19 c k N( z F 2 z F z ṽ (k α,,q (k + z F (1 + z F 2 z ṽ (k α,,q (k cp( F 2+α, (ε δ1 + c (ε(λ δ2 + δ3 v V α. Coninuing we obain (4.14. his concludes he proof. Summarizing we have

17 On Local Moion of a Compressible Baroropic Viscous Fluid 211 heorem 4.4. Le s assume ha S H 3+α, f H α,α/2 ( (,, b H 1/2+α,1/4+α/2 (S (,, b H 3/2+α,3/4+α/2 (S (,, α > 1/2 and is sufficienly large. hen here exiss a unique soluion o problem (4.1 such ha u H 2+α,1+α/2 ( (, and (4.2 u 2+α,, c( f α,, + b 1/2+α,,S + b 3/2+α,,S, where c does no depend on u and. Now, we consider problem (4.1 wih nonvanishing iniial daa. herefore, we formulae i in he form (4.21 u div D(u = f 1 in, τ D(u n = b 1 on S, u n = b 1 on S, u = = u in. We have Lemma 4.5. We assume ha f 1 W α,α/2 2 (, b 1 W 1/2+α/2,1/4+α/4 2 (S, b 1 W 3/2+α/2,3/4+α/4 2 (S, u W2 1+α (, S W2 3+α, α (1/2, 1, and use he following compaibiliy condiions (4.22 b 1 = τ D(u n =, b 1 = u n =. hen here exiss a soluion o problem (4.21 such ha u W 2+α,1+α/2 2 ( and he esimae holds (4.23 u 2+α, c( f 1 α, + b 1 1/2+α/2,S + b 1 3/2+α/2,S + u 1+α,. Proof. Since u W2 1+α ( here exiss a funcion ũ W 2+α,1+α/2 2 ( such ha ũ = = u and ũ 2+α, c u 1+α,. Inroducing he funcion v = u ũ we see ha i is a soluion o he problem (4.24 v div D(v = f 2 in, τ D (v n = b 2 on S, v n = b 2 on S, v = = in, where (4.25 f 2 = f 1 (ũ div D(ũ W α,α/2 2 (, b 2 = b 1 τ D(ũ n W 1/2+α/2,1/4+α/4 2 (S, b 2 = b 1 ũ n W 3/2+α/2,3/4+α/4 2 (S,

18 212 M. Burna W. M. Zajączkowski and we have he following esimaes (4.26 f 2 α, c( f 1 α, + u 1+α,, b 2 1/2+α/2,S c( b 1 1/2+α/2,S + u 1+α,, b 2 3/2+α/2,S c( b 1 3/2+α/2,S + u 1+α,. Since he compaibiliy condiions (4.22 are saisfied he funcions f 2, b 2, b 2 can be exended by zero for <, and he exended funcions f 2, b 2, b 2 are such ha (4.27 and (4.28 f 2 H α,α/2 (, b 2 H 1/2+α/2,1/4+α/4 (S, b 2 H 3/2+α/2,3/4+α/4 (S, f 2 α,, c f 2 α,, b 2 1/2+α/2,,S c b 2 1/2+α/2,S, b 2 3/2+α/2,,S c b 2 3/2+α/2,S. Since < he norms H α,α/2 ( and H α,α/2 ( are equivalen (and similarly for boundary norms we have ha (4.29 and (4.3 f 2 H α,α/2 (, b 2 H 1/2+α/2,1/4+α/4 (S, b 2 H 3/2+α/2,3/4+α/4 (S, f 2 α,,s c( f 2 α,,, b 2 1/2+α/2,,S c( b 2 1/2+α/2,,S, b 2 3/2+α/2,,S c( b 2 3/2+α/2,,S. By virue of (4.23 (4.3 and heorem 4.4 we obain he exisence of soluions o problem (4.24 such ha v H 2+α,1+α/2 ( and he following esimae holds (4.31 v 2+α,, c( f 1 α, + b 1 1/2+α/2,S + b 1 3/2+α/2,S + u 1+α,. Now, in view of he definiion of v, we obain he exisence of soluions o (4.21 such ha u W 2+α,1+α/2 2 ( and by (4.31 we have (4.32 u 2+α, c( v 2+α, + u 1+α, c( v 2+α,, + u 1+α,, hence, (4.23 holds. his concludes he proof.

19 On Local Moion of a Compressible Baroropic Viscous Fluid Exisence of soluions o (1.13 Using a pariion of uniy in we have (see [12]: Lemma 5.1. If we assume ha η C β (, β >, 1/η L (, η >, F 1 W α,α/2 2 (, G 1j W 1/2+α,1/4+α/2 2 (S, j = 1, 2, G 13 W 3/2+α,3/4+α/2 (S, w W 1+α 2 (, S H 3+α. hen here exiss a soluion o problem (1.13 such ha w W 2+α,1+α/2 2 ( and he esimae holds (5.1 w 2+α, c( η C β (, 1/η, ( 2 F 1 α, + G 1j 1/2+α,S + G 13 3/2+α,S + w 1+α,. j=1 6. Exisence of soluions o problem (1.1 o prove he exisence of soluions o problem (1.1 we formulae i in Lagrangian coordinaes. o simplify he consideraions we inroduce hem once again using anoher noaion. By Lagrangian coordinaes we mean he iniial daa for he following Cauchy problem (6.1 dx d = v(x,, x = = ξ, ξ = (ξ 1, ξ 2, ξ 3. Inegraing (6.1 we obain he following ransformaion beween he Eulerian x and he Lagrangian ξ coordinaes of he same fluid paricle. Hence (6.2 x = ξ + where u(ξ, = v(x(ξ,,. he condiion (1.1 5 implies ha u(ξ, s ds x(ξ,, x = x(ξ, for ξ, S x = x(ξ, for ξ S. Using he Lagrangian coordinaes, we can formulae he problem (1.1 in he form (6.3 ηu div u D u (u + u q = ηg in, η + η u u = in, τ uα D u (u n u + u τ uα =, α = 1, 2, on S, n u u = on S, u = = v, η = = ρ in,

20 214 M. Burna W. M. Zajączkowski where η(ξ, = ρ(x(ξ,,, q(ξ, = p(x(ξ,,, g(ξ, = f(x(ξ,,, u = x ξ i ξi, ξi = ξi, D u (u = {µ( ui u j + uj u i + (ν µδ ij u u}, u u = xi ξ j ξj u i, ui = xi ξ k ξk and he summaion convenion over he repeaed indices is assumed. Moreover, n u (ξ, = n(x(ξ,,, τ u (ξ, = τ(x(ξ,,. Le A x(ξ, be he Jacobi marix of he ransformaion x = x(ξ, wih elemens a ij (ξ, = δ ij + u iξ j dτ and le J x(ξ, = de{a ij } i,j=1,2,3 be he Jacobian. hen J x(ξ, = a ij A ij = A ij u i ξ j, J x(ξ, = 1, where A ij are he algebraic complemens of a ij and A = {A ij } i,j=1,2,3. Hence (6.4 J x(ξ, = 1 + A ij u i ξ j d τ = 1 + A u dτ. Moreover, since A ij u i / ξ j = A ij a kj v i / x k = v(x, x=x(ξ, J x(ξ, i follows ha ( ( J x(ξ, = exp v x(ξ, dτ = exp u u dτ, where u = J 1 x(ξ, A. herefore from (6.3 2,5 we have ( (6.5 η(ξ, = ρ (ξ exp From (6.4 and (6.5 we obain also ( (6.6 η(ξ, = ρ (ξ 1 + u u dτ = ρ (ξj 1 x(ξ, (ξ,. A u dτ 1. Lemma 6.1. We assume ha ρ W 1+α 2 (, α (1/2, 1, ρ (ξ ρ >, and (6.7 1/2 u 2+α, δ, where δ is sufficienly small. hen soluions of problem (6.3 2,5 saisfy η(, 1+α, φ(a ρ 1+α,, (6.8 [η ξ ] α/2,, ρ 1+α, φ(a, a, η ξ ( η ξ ( 2 sup 1+α dξ d ( ρ 2 1+α,φ(a 1 α u 2 2+α, d 1 + u 2 2+α, d,

21 On Local Moion of a Compressible Baroropic Viscous Fluid 215 where a = a u 2+α,, a > and φ is an increasing posiive funcion. Proof. (6.8 1 follows easily from (6.5. o show (6.8 2 we ge from (6.5 η ξ ( η ξ ( 2 1+α dξ d d ( ρ 2 u ξ dτ 2 1+α,φ(a 1+α + u ξ dτ 2 u ξξ dτ 2 1+α + ρ 2 1+α,φ(a 2 α dξ d d u ξξ dτ 2 1+α ( dξ (u 2 ξ + u 2 ξξ dτ + ρ 2 1+α,φ(a 2 α u 2 2+α, dτ u 2 ξ dτ ( 1 + u 2 2+α, dτ. u 2 ξξ dτ herefore, (6.8 2 has been proved. A proof of (6.8 3 is similar o he proof of ( o prove he exisence of soluions o problem (1.1 we use he following mehod of successive approximaions (6.9 η m u m+1 div um D um (u m+1 + um q(η m = η m g in, τ umα D um (u m+1 n um = u m τ umα, α = 1, 2, on S, n um u m+1 = on S, u m+1 = = v in, where η m, u m are given, and (6.1 η m + η m div um u m = in, η m = = ρ in, where u m is given. o apply Lemma 5.1 we wrie (6.9 in he form (6.11 η m u m+1 divd(u m+1 = (div D(u m+1 div um D um (u m+1 um q(η m + η m g l 1 + l 2 + l 3 in, τ α D(u m+1 n = (τ α D(u m+1 n τ umαd um (u m+1 n um u m τ umα l 4 + l 5, α = 1, 2, on S, n u m+1 = (n n um u m+1 l 6 on S, u m+1 = = v in, where n = n(ξ,, τ α = τ α (ξ,, α = 1, 2. Firs we show

22 216 M. Burna W. M. Zajączkowski Lemma 6.2. We assume ha S W2 3+α, ρ W2 1+α (, v W2 1+α (, α (1/2, 1, ρ (ξ ρ >, where ρ is a consan, and u m W 2+α,1+α/2 2 ( and (6.7 holds wih δ sufficienly small. hen for soluions of he problem (6.11 such ha u m+1 W 2+α,1+α/2 2 ( he following inequaliy holds (6.12 u m+1 2+α, G(, a u m 2+α,,,, a >, where G is an increasing posiive funcion of is argumens and and G(,, = G ( >. = ρ 1+α, + g α, + v 1+α,, Proof. Applying Lemma (5.1 o (6.11 we have (6.13 u m+1 2+α, φ 1 ( η m Cβ (, 1/η m, ( 3 5 l i α, + l i 1/2+α,S + l 6 3/2+α,S + v 1+α,, i=1 i=4 where φ 1 is an increasing posiive funcion and β < α 1/2. Now we have o esimae he norms from he r.h.s. of (6.13. o esimae l 1 we wrie i in he qualiaive form l 1 ψ 1 (bu m+1ξξ + ψ 2 (bb ξ u m+1ξ, where b = {b ij } = { u iξ j (ξ, τ dτ}, and ψ 1, ψ 2 are marix funcions such ha ψ 1 = ξ x ξ x, ψ 2 = ξ x ξ x,b, where he producs are he marix producs. We shall resric our consideraions o esimae he highes derivaives in he considered norms only. where ( [l 1 ] 2 α,,x φ 2(a m (u mξ u mξ dτ 2 ξ ξ 3+2α u m+1ξξ 2 + u m+1ξξ u m+1ξ ξ 2 u mξ dτ 2 ξ ξ 3+2α + (u mξ u mξ dτ 2 u mξξ dτ 2 u m+1ξ 2 ξ ξ 3+2α + (u mξξ u mξ ξ dτ 2 u m+1ξ 2 ξ ξ 3+2α + u mξξ dτ 2 u m+1ξ u m+1ξ 2 ξ ξ 3+2α dξ dξ d φ 2 ( 1/2 a m = a u m 2 2+α dτ, a >. 5 I i, i=1

23 On Local Moion of a Compressible Baroropic Viscous Fluid 217 Now, we esimae he expression I i, i = 1,..., 5. ( I 1 a u mξ u mξ 2p1 1/p1 dξ ξ ξ dξ d 3+2p1(3µ1/2 3/2p1+α ( u m+1ξξ (ξ 2p2 1/p2 ξ ξ dξ dξ d I 1, 3µ2p2 where a >, 1/p 1 + 1/p 2 = 1, µ 1 + µ 2 = 1 and µ 2 p 2 < 1. Using he imbeddings W2 2+α ( W 1+α+3µ1/2 3/2p1 2p 1 ( and W2 2+α ( L 2p2 (, which holds simulaneously because 3/2 3/2p 1 + 3µ 1 /2 3/2p 1 + α 1 + α, 3/2 3/2p 2 α are valid for α > 1/2, we obain I 1 a u m 2+α, u m+1 2+α,. Moreover, we have I 2 [u m+1ξξ ] 2 α,,x u mξ 2, dτ u m 2 2+α, dτ[u m+1ξξ ] 2 α,,x. Coninuing we see ha I 3, I 5 can be esimaed in he similar way as I 1 and I 4 as I 2. Summarizing we have [l 1 ] 2 α,,x φ 3(a m a u m 2+α, u m+1 2+α,, where a >. Nex, we esimae ( [l 1 ] 2 α/2,, φ u mξ dτ 2 4(a m 1+α u m+1ξξ 2 + u m+1ξξ( u m+1ξξ ( α u mξ dτ + u mξ dτ 2 u mξξ dτ 2 u m+1ξ 2 1+α + + u mξξ dτ 2 u m+1ξ ( u m+1ξ ( 2 1+α dξ dξ d φ 4 (a m ( 1 α dξ u mξ 2 dτ u m+1ξξ 2 dτ + [u m+1ξξ ] 2 α/2,, u mξ 2, dτ + 2 α dξ u mξ 2 dτ u 2 mξξ dτ + 1 α + 1 Ii 2, i=6 dξ dξ u 2 mξξ dτ u 2 mξξ dτ u 2 m+1ξ dτ u mξξ dτ 2 u m+1ξ 2 1+α u 2 m+1ξ dτ u m+1ξ ( u m+1ξ ( 2 1+α d d

24 218 M. Burna W. M. Zajączkowski where I 6 + I 8 + I 9 φ 5 (a m, a m u m+1 2+α,, and ( I 1 φ 6 (a m 1/2 u mξξ 2 2p 1, d ( 1/2 u m+1ξ ( u m+1ξ ( 2 2p 2, 1+α φ 6 (a m 1/2 u m 2+α, u m+1 2+α,, 1/2 where we used imbeddings D 2 W 2+α 2 ( L 2p1 (, D 2 W 2+α 2 ( L 2p2 (, which hold because 3/2 3/2p 1 α, 3/2 3/2p 2 1, 1/p 1 + 1/p 2 = 1 are saisfied ogeher for α > 1/2. Summarizing, we have shown [l 1 ] α/2,, φ 7 (a m a u m 2+α, u m+1 2+α,. Now, we consider l 2 = ψ 3 (b q(η m η mξ, where he do denoes he derivaive wih respec o he argumen, ( [ [l 2 ] α,,x φ 8 (a m, sup η m 1+α, (u mξ u mξ dτ 2 η mξ 2 ξ ξ 3+2α + η m(ξ η m (ξ 2 η mξ 2 ξ ξ 3+2α φ 9 (a m, sup d dξ dξ + [η mξ ] 2 α,,x η m 1+α, a sup η m 1+α,, where o esimae he firs wo inegrals on he r.h.s of he firs inequaliy we used he same mehod as in he case of I 1. Nex, ( u mξ dτ 2 η mξ 2 [l 2 ] α/2,, φ 1 (a m, sup η m 1+α, 1+α Coninuing, + η m( η m ( 2 1+α η mξ 2 + η mξ( η mξ ( 2 1+α φ 1 I 11. I 11 2 α u m 2 2+α, dτ sup η m 2 1+α, + dξ d d η m ( η m ( 2 2p 1, 1+α η mξ 2 2p 2, d d + [η mξ ] α 2,, I 12, where 1/p 1 +1/p 2 = 1. o esimae he middle erm in I 12 we use he imbeddings W2 1 ( L 2p1 (, DW2 1+α ( L 2p2 (, which hold ogeher because he relaions 3/2 3/2p 1 1, 3/2 3/2p α are saisfied for α > 1/2. Using (6.8 we obain [l 2 ] α/2,, φ 11 (a m, sup η m 1+α, a m.

25 On Local Moion of a Compressible Baroropic Viscous Fluid 219 In view of Lemma 2.6 we have Nex we have l 3 α, sup η m 1+α, g α,. l 4 1/2+α,S c l 4 1+α,, which can be esimaed in he same way as l 1. Nex l 5 1/2+α,S c l 5 1+α, c u m 1+α, ε u m 2+α, + c(ε u m, Finally, ε u m 2+α, + c(ε 1/2 ( u m 2+α, + v 1+α,. l 6 3/2+α,S c l 6 2+α, a u m 2+α, u m+1 2+α,. Summarizing he above consideraions we obain (6.12. his concludes he proof. Le (6.14 α m ( = u m 2+α,. Le A > be sufficienly large such ha G ( < A, α m ( A. hen here exiss a ime such ha for we have (6.15 α m+1 ( G(, a A, A. Hence we obain (6.16 α m ( A, for m =, 1,..., and. Finally, we define he zero approximaion funcion u as a soluion o he following problem u div D(u = in, τ α D(u n + u τ α =, α = 1, 2, on S, u n = on S, u = = v in. herefore, we have proved Lemma 6.3. If we assume ha daa are such ha <, and ha α > 1/2, hen he sequence consruced by (6.9 and (6.1 is bounded for. Now we show ha he sequence {η m, u m } converges. o his end we consider he differences (6.17 H m = η m η m 1, U m = u m u m 1,

26 22 M. Burna W. M. Zajączkowski which are soluions of he following problems η m U m+1 div D(U m+1 = (div D(U m+1 div um D um (U m+1 + (div um D um (u m div um 1 D um 1 (u m q(η m um H m + q(η m ( um um 1 (η m 1 (6.18 ( q(η m q(η m 1 um 1 η m 1 + H m g H m u m τ α D(U m+1 n = (τ α D(U m+1 n τ umα D um (U m+1 n um 7 L i, i=1 and (6.19 (τ umα D um (u m n um τ um 1α D um 1 (u m n um 1 (u m τ umα u m 1 τ um 1α = 1 i=8 L i, α = 1, 2, n U m+1 = (n n um U m+1 (n um n um 1 u m U m+1 = =, H m + H m div um u m = η m 1 (div um u m div um 1 u m, H m = =. o show he convergence of he sequence {u m, η m } we need 12 i=11 Lemma 6.4. Le he assumpions of Lemma 6.3 be saisfied. hen (6.2 U m+1 2+α, c(a a U m 2+α,, where a > and A is he bound from (6.16. Proof. From Lemma 5.1 we have (6.21 U m+1 2+α, φ 1 ( η m C β (, 1/η m L( ( 7 1 L i α, + L i 1/2+α,S + i=1 i=8 Since L 1 = ψ 1 (b m b m U m+1ξξ + ψ 2 (b m b mξ U m+1ξ we have L 1 α, φ 2 (A, a U m+1 2+α,. 12 i=11 L i, L i 3/2+α,S Since L 2 = ψ 3 (b m, b m 1 U mξ dτu mξξ +ψ 4 (b m, b m 1 U mξξ dτu mξ we obain L 2 α, φ 3 (A, a U m 2+α,..

27 On Local Moion of a Compressible Baroropic Viscous Fluid 221 Nex, we have ha L 3 = q(η m ψ 5 (b m H mξ. herefore ( [L 3 ] 2 α,,x φ ηm (ξ η m (ξ 2 H mξ 2 4(A, ξ ξ 3+2α + (u mξ u mξ dτ 2 H mξ 2 ξ ξ 3+2α + H mξ H mξ 2 ξ ξ 3+2α d dξ dξ and φ 5 (A, sup H m 1+α,, [L 3 ] 2 α,, φ 6(A, ( ηm ( η m ( 2 H mξ 2 1+α + u mξ dτ 2 H mξ 2 1+α + H mξ( H mξ ( 2 1+α dξ d d φ 7 (A, a sup H m 1+α, + φ 6 (A, H mξ ( H mξ ( 2 1+α dξ d d. o esimae he las expression we inegrae (6.19 o ge ( (6.22 H m (ξ, = exp div um u m dτ ( exp div um u m dτ η m 1 (div um u m div um 1 u m 1 d. Using he fac ha H m has he following qualiaive form H m = ψ 6 (b m ( we obain ha (6.23 Similarly, we have [ψ 7 (b m (τu mξ (τ + ψ 8 (b m (τ H mξ ( H mξ ( 2 1+α dξ d φ 7 (A, a U mτ (τ dτu m 1 (τ] dτ (6.24 sup H m 2 1+α, φ 8 (A, a U m 2 2+α, d. Using (6.23 and (6.24 in he esimaion of L 3 we obain Similarly L 3 we esimae L 5. Moreover, L 3 α, φ 9 (A, a U m 2+α,. L 4 α, φ 1 (A, a U m 2+α,. U m 2 2+α, d.

28 222 M. Burna W. M. Zajączkowski Le us consider L 6. We have and [L 6 ] 2 α/2., [L 6 ] 2 α,,x L 6 2 α, d sup H m 2 1+α, g 2 α,, ( Hm ( H m ( 2 g( 2 1+α + H m( 2 g( g( 2 1+α dξ d d ( Hm ( H m ( 2 1, 1+α g 2 1+α, + H m( 2 1+α, g( g( 2 2, 1+α d d ( sup d H m( H m ( 2 1, 1+α + sup Using (6.23 and (6.24 we have he esimae Similarly, we have H m 2 1+α, g 2 α, d H m 2 1+α, L 6 α, φ 11 (A, a U m 2+α, g α,. L 7 α, φ 12 (A, a U m 2+α, u m α,. g 2 α,. Coninuing he consideraions we prove he lemma. Summarizing we have heorem 6.5. Le he assumpions of Lemma 6.3 be saisfied. hen here exiss a ime such ha for here exiss a soluion o problem (1.1 such ha v W 2+α,1+α/2 2 (, ρ W 1+α,1/2+α/2 2 (. References [1] M. S. Agranovich, M. I. Vishik, Ellipic problems wih parameers and parabolic problems of general ype, Uspiechi Ma. Nauk 19 (117 (1964, [2] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralseva, Linear and Quasilinear Equaions of Parabolic ype, Nauka, Moscow, (Russian [3] L. Landau and E. Lifschiz, Mechanics of Coninuum Media, Nauka, Moscow, 1984 (Russian; English ransl.: Pergamon Press, Oxford 1959; new ediion: Hydrodynamics, Nauka, Moscow 1986 (Russian. [4] K. Pileckas and W. M. Zajączkowski, On free boundary problem for saionary compressible Navier Sokes equaions, Comm. Mah. Phys. 129 (199, [5] V. A. Solonnikov, An iniial boundary value problem for Sokes sysem ha arises in he sudy of a problem wih free boundary, rudy Ma. Ins. Seklov. 188 (199, (Russian; English ranslaion: Proc. Seklov Ins. Mah. 3 (1991,

29 On Local Moion of a Compressible Baroropic Viscous Fluid 223 [6], On boundary problems for linear parabolic sysems of differenial equaions of general ype, rudy Ma. Ins. Seklov. 83 (1965. (Russian [7] V. A. Solonnikov and V. E. Shchadilov, On a boundary value problem for a saionary sysem of Navier Sokes equaions, rudy Ma. Ins. Seklov. 125 (1973, (Russian; English ransl.: Proc. Seklov Ins. Mah. 125 (1973, [8] V. A. Solonnikov and A. ani, Free boundary problem for a viscous compressible flow wih a surface ension, Zap. Nauchn. Sem. LOMI 182 (199, ; Consanin Caraheodory: an Inernaional ribue, (M. Rassias, ed., vol. 2, World Sci., 1991, pp [9] G. Sröhmer and W. M. Zajączkowski, Local exisence of soluions of free boundary problem for he equaions of compressible baroropic viscous self graviaing fluids (o appear. [1] W. M. Zajączkowski, On an iniial boundary value problem for parabolic sysem which appears in free boundary problems for compressible Navier Sokes equaions, Disseraiones Mah. 34 (199, [11], Exisence of local soluions for free boundary problems for viscous compressible baroropic fluids, Ann. Polon. Mah. 6 (3 (1995, [12], On local moion of a compressible baroropic viscous fluid bounded by a free surface, Banach Cener Publ. 27 (1992, Manuscrip received April 27, 1997 Marek Burna Insiue of Applied Mahemaics and Mechanics Warsaw Universiy Warsaw, POLAND Wojciech ZajĄczkowski Insiue of Mahemaics Polish Academy of Sciences Śniadeckich 8-95 Warsaw, POLAND address: wz@impan.gov.pl MNA : Volume N o 2