Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension

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1 Report on a local in time solvability of free surface problems for the Navier-Stokes euations with surface tension Yoshihiro SHIBATA Department of Mathematics and Research Institute of Science and Engineering, Waseda University, JST, CREST, Ohkubo 3-4-1, Shinjuku-ku, Tokyo , Japan. address: yshibata@waseda.jp Senjo SHIMIZU Department of Mathematics, Faculty of Science, Shizuoka University, Shizuoka , Japan address: ssshimi@ipc.shizuoka.ac.jp Dedicated to Professor Vsevolod Alekseevich Solonnikov on the occasion of his 75th birthday. Abstract. We consider the free boundary problem of the Navier-Stokes euation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed halfspace or a perturbed layer in R n n 2. We report a local in time uniue existence theorem in the space W,p 2,1 = L p, T, W 2 Ω W 1, T, L Ω with some T >, 2 < p < and n < < for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the L p -L maximal regularity theorem for the corresponding linearized euations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov [15, 16, 18, 19], Beale [3] and Tani [21]. 1 Introduction and Results In this paper we would like to report a result concerning the local in time existence theorem of the free boundary problems of the motion of a viscous, incompressible fluid for the Navier-Stokes euations. In the models the effect of surface tension on free surface is included. Let Ω be an initial domain in R n n 2 and v be an initial velocity. Both are given. Throughout the paper, we assume that Ω is one of the following domains: a bounded domain; an exterior domain, that is the complement of Ω is a bounded domain; a perturbed half space, that is there exist a constant R > and a function ηx, x = x 1,..., x such that Ω B R = {x = x, x n R n x n < ηx } B R, 1.1 where B R = {x R n x > R}; a perturbed layer, that is there exist a constant R > and two functions η 1 x and η 2 x such that Ω B R = {x = x, x n R n η 1 x < x n < η 2 x } B R. 1.2 Partially supported by JSPS Grant-in-aid for Scientific Research C # MOS Subject Classification: 35Q3, 35R35, 76D5. Keywords: Navier-Stokes euation, free boundary problem, surface tension, local in time solvability. 1

2 Concerning the boundary of Ω, we consider the following cases: When Ω is a bounded domain, we consider the two cases: the boundary of Ω consists of two hypersurfaces Γ and Γ b such that Γ Γ b =, the boundary of Ω consists of only one hypersurface Γ. In this case, Γ b =. When Ω is an exterior domain or a perturbed half-space, the boundary of Ω consists of only one hypersurface Γ. In this case, Γ b =. When Ω is a perturbed hypersurface, the boundary of Ω consists of two hypersurfaces Γ and Γ b such that Γ Γ b =. Moreover, we assume that there exists an h such that Γ {x R n x n > 3h} and Γ b {x R n x n < h}. Our problem is to find the domain Ω t for t > occupied by the fluid, the velocity vector field vx, t and the scalar pressure θx, t, x Ω t, satisfying the Navier-Stokes euations: t v + v v Div Sv, θ = fx, t in Ω t, t >, div v = in Ω t, t >, Sv, θν t = σhν t g a x n ν t on Γ t, t >, V n = v ν t on Γ t, t >, v = on Γ b, t >, v t= = v in Ω, 1.3 where the boundary of Ω t is denoted by Ω t = Γ t Γ b with Γ t being the free deformable part. In 1.3, ν t is the unit outward normal to Γ t, Sv, θ = µdv θi is the stress tensor, Dv = Dv ij = v i / x j + v j / x i is a deformation tensor, H is the mean curvature which is given by Hν t = Γt x, Γt is the Laplace-Beltrami operator on Γ t, µ and σ denote the coefficient of viscosity and the coefficient of surface tension, that are positive constants, respectively, and g a > is the acceleration of gravity. V n is the velocity of the evolution of Γ t in the direction of outward normal ν t. For the differentiation, we use the symbols: div v = n D jv j, n n n n v v = v j D j v 1,..., v j D j v n, Div S = D j S 1,j,..., D j S n,j where D j = / x j, M denotes the transposed M, v = v 1,..., v n and S = S ij n n matrix. The problem 1.3 contains the following special cases: If Ω is a bounded domain and Γ b = and g a =, then 1.3 is a drop problem. If Ω is a perturbed layer then 1.3 is an ocean problem. If Ω is a perturbed half-space with x n being the vertical component and Γ b =, then 1.3 is an ocean problem without bottom. Now, we shall discuss some known results of the uniue existence theorem of problem 1.3. Let W,p 2,1 Ω, T = L p, T, W 2 Ω Wp 1, T, L Ω and for simplicity, we write W,p 2,1 = W,p 2,1 Ω, T for some T > and Wp 2,1 = Wp,p 2,1. We shall state mainly the case that the surface tension is taken into account. First we mention about local in time solvability. Solonnikov formulated the drop problem and proved the local in time solvability of 1.3 for arbitrary initial data in the Sobolev-Slobodetskii space W 2+α,1+ α 2 2 with α 1 2, 1 in [15, 18, 19]. Moglilevskiĭ and Solonnikov [5] proved the local in time solvability in Hölder spaces. Schweizer [1] proved the local in time uniue existence for small initial data by using the semigroup approach. Concerning the ocean problems, Allain [2] proved local in time uniue solvability when n = 2. Tani [21] proved the local in time uniue solvability in W 2+α,1+ α 2 2 with α 1 2, 1. Prüss 2

3 and Simonett [9] proved local in time uniue solvability in Wp 2,1 p > n + 2 for two phase free boundary problem under the assumption of the smallness of the first derivative of a height function. Concerning the global in time solvability, Solonnikov [16] proved the global in time solvability of the drop problem in W 2+α,1+ α 2 2 with α 1 2, 1 for f = provided that initial data are sufficiently small and the initial domain Ω is sufficiently close to a ball. Padula and Solonnikov [8] proved the global in time uniue solvability of the drop problem in Hölder spaces by using the mapping of Ω t on a ball instead of Lagrangean coordinates. In [3], Beale proved the global in time uniue solvability of the ocean problem in H l, l 2 2 with 3 < l < 7 2 for σ >, n = 3 and f = provided that the initial data are sufficiently small. Beale and Nishida [4] obtained the asymptotic power-like in time decay of the global solutions of the ocean problem. Tani and Tanaka [22] proved the global in time solvability of the ocean problem in W2 2+α with α 1 2, 1 for σ and n = 3 provided that initial data are sufficiently small by using Solonnikov s method. Concerning the problem without surface tension, the local in time uniue solvability for any initial data and the global in time uniue solvability for small initial data of the drop problem were proved by Solonnikov [17] in Wp 2,1 n < p <, and by Shibata and Shimizu [11], [12] in W,p 2,1 2 < p < and n < <. Also the local in time solvability in Wp 2,1 n < p < was proved by Mucha and Zaj aczkowski [6, 7] for the drop problem, and Abels [1] for the ocean problem. In this paper, we shall report a local in time uniue existence theorem of 1.3 in the space W,p 2,1 2 < p < and n < < for any initial data which satisfy compatibility condition. In fact, all the results mentioned above, except for Moglilevskiĭ and Solonnikov [5] and Prüss and Simonett [9], are obtained in some Sobolev-Slobodetskii space W 2+α,1+ α 2 2 with α 1 2, 1, that is in the L 2 framework. So far, there were no results in W,p 2,1 in the case that the surface tension is taken into account. But, to solve the problem 1.3 in the space W,p 2,1 is important from the viewpoint of lower regularity condition on the initial data. Before stating our main results, first of all we shall discuss the formulation of the problem 1.3 by the Lagrange coordinate, instead of the Euler coordinate. Aside from the dynamical boundary condition, a further kinematic condition for Γ t is satisfied which gives Γ t as a set of points x = xξ, t, ξ Γ, where xξ, t is the solution of the Cauchy problem: dx dt = vx, t, x t= = ξ. 1.4 This expresses the fact that the free surface Γ t consists for all t > of the same fluid particles, which do not leave it and are not incident on it from Ω t. From now on, we write Ω = Ω and Γ = Γ. The problem 1.3 can therefore be written as an initial boundary value problem in the given region Ω if we go over the Euler coordinates x Ω t to the Lagrange coordinates ξ Ω connected with x by 1.4. If a velocity vector field uξ, t = u 1,..., u n is known as a function of the Lagrange coordinates ξ, then this connection can be written in the form: x = ξ + uξ, τ dτ X u ξ, t. 1.5 Passing to the Lagrange coordinates in 1.3 and setting θx u ξ, t, t = πξ, t, we obtain t u Div Su, π = Div Qu + Ru π + fx u ξ, t, t in Ω, t >, div u = Eu = divẽu in Ω, t >, Su, π + Quν tu σhν tu + g a X u,n ν tu = on Γ, t >, u = on Γ b, t >, u t= = u ξ in Ω, 1.6 where u ξ = v x and X u,n stands for the n-th component of X u. Moreover, ν tu stands for the unit outer normal to Γ t given by ν tu = t A 1 ν / t A 1 ν, where A is the matrix whose element {a jk } is the Jacobian of 1.5: a jk = x j u j = δ jk + dτ, ξ k ξ k 3

4 and Qu, Rπ, Eu and Ẽu are nonlinear terms of the following forms: Qu = µv 1 Eu = V 3 u dτ u, Ru = V 2 u dτ u, Ẽu = V 4 u dτ u dτu 1.7 with some polynomials V j of u dτ, j = 1, 2, 3, 4, such as V j = cf. Appendix in [11]. We shall discuss a local in time uniue existence theorem for the problem 1.6 instead of the problem 1.3. In order to state our main results precisely, at this point we introduce the function spaces. For any domain D in R n, integer m and 1, L D and W m D denote the usual Lebesgue space and Sobolev space of functions defined on D with norms: L D and W m, respectively. For any D Banach space X with norm X, X n stands for the n product space defined by X n = {f = f 1,..., f n f i X i = 1,..., n}. The norm of X n denotes also X which is defined by the formula: f X = n f j X for any f = f 1,..., f n X n. For the space of the pressure term, we introduce Ŵ 1 D which is defined by Ŵ 1 D = {u L,loc D u L D n }. For any interval I R, integer l and 1 p, L p I, X and W l pi, X denote the usual Lebesgue space and Sobolev space of the X-valued functions defined on I with norms: L pi,x and W l p I,X, respectively. Set W m,l,p D I = L p I, W m D W l pi, L D, = u + u u W,p m,l D I LpI,W m u W 2,1 D Wp li,l D,p D I. 1.8 Note that W D = L D and W p I, X = L p I, X. The following theorem will be proved in the forthcoming paper [14] based on Theorem 4.1 below. Theorem 1.1. Let Ω R n n 2 be one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer. Let Γ W 3 and Γ b W 2. Let 2 < p < and n < <. Then, for any initial data u [L Ω, W 2 Ω] 1 1/p,p which satisfies the compatibility conditions: div u = in Ω, Du ν Du ν, ν ν = on Γ, u = on Γ b, 1.9 and f L p R +, L R n n such that D j f L R n R + n for j = 1,..., n, there exists a T > such that the problem 1.6 admits a uniue solution u, π W 2,1,p Ω, T L p, T, Ŵ 1 Ω. Here, [, ] 1 1/p,p denotes the real interpolation functor. In the rest of the paper, we shall give a sketch of our idea of the proof of Theorem Reduction of the boundary condition to linearized problems In this section, we shall discuss the reduction of the boundary condition: Su, π + Quν tu σhν tu + g a X u,n ν tu =, 2.1 which is the first key step in our proof of Theorem 1.1. hyperplanes of Γ t and Γ, which are defined by Let Π t and Π be projections to tangent Π t d = d d, ν tu ν tu, Π d = d d, ν ν. 2.2 for an arbitrary vector field d defined on Γ t and Γ, respectively. We know the following fact cf. Solonnikov [2] and also the appendix below. 4

5 Lemma 2.1. If ν t ν, then for arbitrary vector d, d = is euivalent to We apply Lemma 2.1 for 2.1. Since we obtain Π Π t d =, ν d =. 2.3 Π t µdu + Quν tu = 2.4 by applying Π t to the left hand side of 2.1, the first euation of 2.3 for 2.1 is given by Π µduν = Π Π t Π µduν tu Π µduν tu ν Π Π t Quν tu, 2.5 where we have used Π Π = Π. On the other hand, we shall consider the innerproduct of the boundary condition with ν. Using the fact that Hν tu = Γt X u and substituting 1.5 for 2.1, we obtain ν Su, π + Quν tu σν Γt Γ ξ + σν Γ ξ + uξ, τ dτ + g a ν ξ n + Taking a commutator between Γ and ν, we have ν Γ u dτ = Γ By 2.6 and 2.7, we obtain ν Su, πν σ Γ ν u dτ [ + σ Γ ν ν u dτ Γ ν u dτ + ν Γ Γt uξ, τ dτ u n ξ, τ dτ u dτ 2 Γ ] u dτ + ν Γ Γt ξ = ν Su, πν ν tu ν Quν tu + σh Γ g a ξ n g a u n dτ ν ν tu g a ξ n ν ν tu ν 2σ Γ ν tu =. 2.6 u dτ Γ ν. 2.7 u dτ Γ ν, 2.8 where we have used ν Γ ξ = H Γ. We denote the terms in the bracket of the left hand side of 2.8 by F u, that is F u = Γ ν u dτ + ν Γ Γt u dτ + ν Γ Γt ξ. In 2.8, since Γt and Γ contain the second order tangential derivatives of X u, in order to avoid the loss of regularity we apply the inverse operator m Γ 1 with sufficiently large number m to F u. Namely, we proceed as follows: ν Su, πν + σm Γ ν u dτ + m Γ 1 F u σmν = ν Su, πν ν tu ν Quν tu + σh Γ g a ξ n g a u n dτ ν ν tu g a ξ n ν ν tu ν 2σ Γ We introduce a function η by the formula: η = ν u dτ u dτ Γ ν. 2.9 u dτ + m Γ 1 F u on Γ

6 Then, from 2.9 and 2.1, we obtain the two euations on the boundary Γ as follows: ν Su, πν + σm Γ η = ν Su, πν ν tu ν Quν tu + σh Γ g a ξ n g a u n dτ g a + σmν u n dτ ν ν tu g a ξ n ν ν tu ν 2σ Γ u dτ Γ ν u dτ 2.11 t η ν u = m Γ 1 F u, 2.12 where F u denotes the derivative of F u with respect to t. Finally we arrive at the euivalent euation to 1.6 as follows: where t u Div Su, π = Div Qu + Ru π + fx u ξ, t, t in Ω, t >, div u = Eu = divẽu in Ω, t >, t η ν u = Gu on Γ, t >, Π µduν = H t u on Γ, t >, ν Su, πν + σm Γ η = H n u, π + σh Γ g a ξ n on Γ, t >, u = on Γ b, t >, u t= = u ξ in Ω, η t= = on Γ, 2.13 Gu = m σ Γ 1 F u, F u = Γ ν u ν Γt u dτ + ν Γ Γt u ν Γt ξ, H t u = Π Π t Π µduν tu Π µduν tu ν Π Π t Quν tu, H n u, π = ν Su, πν ν tu ν Quν tu g a u n dτ g a u n dτ ν ν tu g a ξ n ν ν tu ν 2σ Γ and Qu, Ru, Eu and Ẽu are nonlinear terms defined by Initial flow u dτ Γ ν + σmν In this section, we shall discuss the initial flow to reduce the problem 2.13 to the case where u ξ = and σh Γ g a ξ n =. We study the problem in the two steps. Step 1 Let u 1, π 1 be a solution to the problem: λu 1 Div Su 1, π 1 = in Ω, div u 1 = in Ω, Su 1, π 1 ν = σh Γ g a ξ n ν on Γ, u dτ, u 1 = on Γ b. 3.1 If positive number λ is large enough, then we know that 3.1 admits a uniue solution u 1, π 1 W 2 Ω Ŵ 1 Ω 6

7 which satisfies the estimate λ u 1 LΩ + 2 u 1 LΩ + π 1 LΩ C 1 σ H Γ 1 1/ W Γ + g a ξ n 1 1/ W. 3.2 Γ Step 2 We consider the linear time-dependent problem in the time interval, 2: t u 2 Div Su 2, π 2 = λu 1 in Ω, 2 div u 2 = in Ω, 2 Su 2, π 2 ν = on Γ, 2 u 2 = on Γ b, 2 u 2 t= = u ξ u 1 ξ in Ω. 3.3 If the initial data u [L Ω, W 2 Ω] 1 1/p,p satisfies the compatibility condition 1.9, then problem 3.3 admits a uniue solution u 2 W 2,1,p Ω, 2, π 2 L p, 2, Ŵ 1 Ω. Moreover, π 2 has an additional information about the regularity at boundary with respect to time variable t. To state this fact, we give a functional space. Given α, we set < D t > α ut = F 1 [1 + s 2 α/2 Fus]t, H α p R, X = {u L p R, X : < D t > α u L p R, X}, u H α p R,X = < D t > α u L pr,x + u LpR,X. Here and hereafter, F and F 1 denote the Fourier transform and its inverse formula, respectively. Set,p D R = Hp 1/2 R, L D L p R, W 1 D, H 1,1/2 H 1,1/2,p D, T = {u v H 1,1/2,p D R, u = v on D, T }, = inf { v u H 1,1/2,p, D,T H,p 1,1/2 D R v H 1,1/2,p D R, v = u on D, T }. By using these symbols, we can state the additional property of π 2 on the boundary Γ as follows: There exists π 2 H,p 1,1/2 Ω R such that π 2 Γ = π 2 Γ. Moreover, u 2, π 2 and π 2 satisfy the estimate: u 2 W 2,1,p Ω,2 + π 2 Lp,2,Ŵ 1Ω + π 2 1,1/2 H,p C 2 u B 21 1/p,p Ω, Ω + σ H Γ W 1 1/ Γ + g a ξ n 1 1/ W. 3.4 Γ If we set z = u 1 + u 2 and τ = π 1 + π 2, then z, τ satisfies the time-dependent linear euation in the time interval, 2: t z Div Sz, τ = in Ω, 2 div z = in Ω, 2 Sz, τν = σh Γ g a ξ n ν on Γ, 2 z = on Γ b, 2 z t= = u in Ω. 3.5 z is our initial flow. Now, we look for a solution u, π of the euation 2.13 of the form: u = z + w and π = τ + κ in the time interval, T with < T 1. Setting τ = π 1 + π 2, we see that v, θ and η should satisfy the euations: t w Div Sw, κ = Div Qz + w + Rz + w τ + κ + fx z+w ξ, t, t in Ω, T div w = Ez + w = divẽz + w in Ω, T 7

8 t η ν w = Gz + w + ν z on Γ, T Π µ Dwν = H t z + w on Γ, t > ν Sw, κν + σm Γ η = H n z + w, τ + κ on Γ, t > w = on Γ b, t > w t= = in Ω, η t= = on Γ L p -L maximal regularity In order to solve 3.6 locally in time, we consider the following time-dependent problem: t u Div Su, π = f, in Ω, t >, div u = f d = div f d in Ω, t >, t η ν u = d on Γ, t >, Su, πν + σm Γ η ν = h on Γ, t >, u = on Γ b, t >, u t= =, η t= =. 4.1 Our L p -L maximal regularity result about the problem 4.1 is the following, which will be proved in a forthcoming paper based on [13]. Theorem 4.1. Let Ω R n n 2 be one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer. Let 1 < p, < and r be a number such that n < r < and r. Assume that Γ Wr 3 1/r and Γ b Wr 2 1/r, respectively. Let T be any positive number and < T T. Then, for any f, f d, f d, d and h in 4.1 satisfying the regularity conditions: and compatibility conditions: f L p, T, L Ω n, f d L p, T, W 1 Ω, fd W 1 p, T, L Ω n, d L p, T, W 2 1/ Γ, h H,p 1,1/2 Ω, T n f d t= =, h t= =, the problem 4.1 admits a uniue solution u, π, η which satisfies the regularity condition: u W 2,1,p Ω, T, π L p, T, Ŵ 1 Ω, t η L p, T, W 2 1/ Γ, η L p, T, W 3 1/ Γ. Moreover there exists a π Γ = π Γ such that π H,p 1,1/2 Ω, T, and u, π, η and π satisfy the estimate: C u W 2,1,p Ω,T + π L p,t,l Ω + π H 1,1/2,p Ω,T + t η Lp,T,W 2 1/ Γ + η L p,t,w 3 1/ Γ f Lp,T,L Ω + d Lp,T,W 2 1/ Γ + f d Lp,T,W 1 Ω + f d W 1 p,t,l Ω + h H 1,1/2,p Ω,T with some constant C independent of T whenever < T T. Using the standard fixed point argument based on Theorem 4.1 cf. [11], we can solve 3.6 locally in time. Concerning the estimates of the nonlinear terms appearing in 3.6, we use the following facts. Lemma 4.2. Let n < <. Then, for any f, g W 1 Ω and u W 1 1 1/ Γ and v W 1 1 1/ Γ, we have. fg W 1 Ω C Ω, f W 1 Ω g W 1 Ω, for some constants C Ω, and C Γ,. uv W 1 1/ Γ C Γ, u W 1 1/ Γ v W 1 1/ Γ 8

9 Lemma 4.3. BUC, 2, [L Ω, W 2 Ω] 1 1/p,p is continuously imbedded into W 2,1,p Ω, 2. Here, BUC, 2, X denotes the set of all bounded uniformly continuous X-valued function on, 2. Lemma 4.4. Let 1 < p <, n < < and < T 1. Set Ŵ 1,1,p Ω I = {f W 1,1, Ω I : t f L p I, W 1 Ω}. If f Ŵ,p 1,1 Ω R, g H,p 1,1/2 Ω R and f vanishes when t [, 2T ], then we have C fg H,p 1,1/2 p, [ f + T n/p Ω R L R,W 1 f Ω t 1 n/2 f L R,L Ω t n/2 LpR,W 1Ω] g H,p 1,1/2 Ω R Lemma 4.5. Let Ω R n n 2 be one of a bounded domain, an exterior domain, a perturbed halfspace or a perturbed layer. Assume that Γ C 2. Let 1 < < and let W 1/ Γ be the dual space of Γ with 1/ + 1/ = 1. Then, we have the following two assertions: W 1 1/ 1 There exists an m 1 such that m Γ is a bijection from W 2 1/ any f W 1/ Γ and g W 1 1/ Γ we have fg W 1/ A A proof of Lemma 2.1 Γ C Γ, f 1/ W Γ g W 1 1/ Γ. Γ onto W 1/ Γ. 2 For Solonnikov used the fact formulated in Lemma 2.1 without proof to formulate his linearization of the nonlinear problem, that is different from ours given in Section 2. We did not find any proof of Lemma 2.1 and it is important to our formulation, so that we will give its proof below. To prove Lemma 2.1, we use the following fact about the determinant which can be proved by mathematical induction. Lemma A.1. Let a 1,..., a n be n real numbers such that a a 2 n < 1 and let δ ij be the Kronecker s delta symbol, that is δ ii = 1 and δ ij = for i j. Let A be an n n matrix whose i, j component is δ ij a i a j, that is 1 a 2 1 a 1 a 2 a 1 a n a 2 a 1 1 a 2 2 a 2 a n A = a n a 1 a n a 2 1 a 2 n Then, det A = 1 n a2 j. Now we prove Lemma 2.1. If d =, then obviously Π Π t d = and ν d =. Therefore, we shall prove the opposite direction. Let e 1,..., e, ν be an orthonormal basis of R n and set a j = e j ν t j = 1,..., n 1. Here, stands for the standard inner-product of R n. Then, we have ν t = a j e j + ν t ν ν.. Since ν t ν and ν t ν t = 1, we have a 2 j < 1. Let d be an n-vector such that Π Π t d = and ν d =. Then, we have A.1 d = e j de j A.2 9

10 because ν d =. Since d ν t = e j de j ν t = a je j d, we have Π t d = d d ν t ν t = d n a je j dν t. Therefore, we have n n Π Π t d = d a j e j dν t ν d a j e j dν t ν = d a j e j dν t + a j e j dν ν t ν k=1 = d a j e j ν t e k ν t e k, where we have used ν t = k=1 e k ν t e k + ν ν t ν in the final step. Now, substituting A.2 into the last formula, we have Π Π t d = e k d a j a k e j de k. Since Π Π t d = and ν d =, we have k=1 e k d a j a k e j d = k = 1,..., n 1. A.3 If we define the n 1 n 1 matrix A by the formula: 1 a 2 1 a 1 a 2 a 1 a a 2 a 1 1 a 2 2 a 2 a A =......, a a 1 a a 2 1 a 2 then noting A.1 and using Lemma A.1 we have det A. On the other hand, the euation A.3 is written in the matrix form: Ae 1 d,..., e d =, which implies that e 1 d = = e d =. Since ν d =, we have d =, which completes the proof of Lemma 2.1. References [1] H. Abels, The initial-value problem for the Navier-Stokes euations with a free surface in L Sobolev spaces, Adv. Differential Euations, [2] G. Allain, Small-time existence for the Navier-Stokes euations with a free surface, Appl. Math. Optim., [3] J. T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal., [4] J. T. Beale and T. Nishida, Large time behavior of viscous surface waves, Lecture Notes in Numer. Appl. Anal., [5] I. Sh. Mogilevskiĭ and V. A. Solonnikov, On the solvability of a free boundary problem for the Navier-Stokes euations in the Hölder spaces of functions, Nonlinear Analysis. A Tribute in Honour of Giovanni Prodi, Quaderni, Pisa [6] P. B. Mucha and W. Zaj aczkowski, On the existence for the Cauchy-Neumann problem for the Stokes system in the L p -framework, Studia Mathematica, [7] P. B. Mucha and W. Zaj aczkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Applicationes Mathematicae, [8] M. Padula, V. A. Solonnikov On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation, Quad. Mat., [9] J. Prüss and G. Simonett, On the two-phase Navier-Stokes euations with surface tension, to appear in Proc. Banach Center Publication. 1

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