ON SOME TWO PHASE PROBLEM FOR COMPRESSIBLE AND COMPRESSIBLE VISCOUS FLUID FLOW SEPARATED BY SHARP INTERFACE

. In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic vis- cous ﬂuid ﬂow without surface tension in the L p in time and the L q in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W 2 − 1 /q q domain in R N ( N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L p - L q regularity of the generalized Stokes operator for the compressible viscous ﬂuid ﬂow with free boundary condition. The key step of our method is to prove the existence of R -bounded solution operator to resolvent problem corresponding to linearized problem. The R -boundedness combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L p - L q regularity theorem.


(Communicated by Eduard Feireisl)
Abstract. In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the Lp in time and the Lq in space framework with 2 < p < ∞ and N < q < ∞ under the assumption that the initial domain is a uniform W 2−1/q q domain in R N (N ≥ 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal Lp-Lq regularity theorem.
The kinematic condition for Γ t and Γ −.t is satisfied, namely they give where x = x(ξ, t) is the solution to the Cauchy problem This fact means that the interface Γ t and free surface Γ −,t consist of the same fluid particles, which do not leave them and are not incident of them from inside Ω +,t ∪ Ω −,t for t > 0. It is clear that Ω ±,t = {x = x(ξ, t) | ξ ∈ Ω ± }. A free boundary problem for a viscous compressible barotrophic fluid has been studied by some mathematicians. For the results for one phase problem, local in time unique existence of solutions to the free boundary problem without surface tension in the multi-dimensional case was proved by Secchi and Valli [9] in L 2 framework and by Tani [14] in the Hölder space, respectively. Later on, the same problem with surface tension was studied by Solonnikov and Tani [12] in the L 2 framework and Dennisova and Solonnikov [2], [3] in Hölder spaces.
For two phase problem of compressible and incompressible viscous fluids, in [1] Denisova first showed a local in time existence theorem with surface tension on Γ t under the assumption µ + 1 < µ − 1 and µ + 1 + µ + 2 < µ + 1 /R ∞ with some positive constant R ∞ and Γ −,t and Γ + are empty sets. Recently Kubo, Shibata and Soga [8] considered corresponding resolvent problem and showed the existence of its Rbounded solution operator, which implies maximal L p -L q regularity theorem for linearlized problem and the local in time existence theorem for two phase problem. 1T M denotes the transposed M .
On the other hand, for two phase problem of compressible and compressible viscous fluid, Tani [14], [15] studied a local in time existence theorem under the natural condition in Hölder space framework. As far as we know, there are no literatures concerning the L p approach to the two phase problem of compressible and compressible viscous fluid. In this paper, we shall consider the two phase problem of compressible and compressible fluid in L p -L q framework and prove the local in time existence theorem of our problem in a similar way as Enomoto, Below and Shibata [5] and Kubo, Shibata and Soga [8]. As we shall explain later, after transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve our problem by contraction mapping principle with maximal L p -L q regularity theorem for the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. Maximal L p -L q regularity theorem follows from the R-boundedness of solution operator to the generalized resolvent equation corresponding to our liearized problem with the help of Weis's operator valued Fourier multiplier theorem. Therefore our goal of this paper is to prove the existence of R-bounded solution operator.
We shall now go back to our approach. As we mentioned, we transfer Ω ±,t to some fixed domain. Our problem can be written as an initial boundary value problem in the given domain Ω ± if we transfer the Euler coordinates x ∈ Ω ±,t to Lagrange coordinates ξ ∈ Ω ± . If velocity field u ± (ξ, t) defined on Ω ± is known as functions of the Lagrange coordinates ξ ∈ Ω ± , then this connection can be written in the form where u ± (ξ, t) = v ± (X u± (ξ, t), t) are the velocity vector fields defined on Ω ± known as functions of the Lagrange coordinates ξ ∈ Ω ± . Let A ± be the Jacobi matrix of the transformation x = X u± (ξ, t) with element a ± ij = δ ij + t 0 (∂ ξj u ±,i )(ξ, s)ds. There exists a small number σ such that A ± are invertible, that is det A ± = 0 whenever max i,j=1,...,N t 0 (∂ ξj u ±,i )(·, s)ds In this case, we have ∇ s) ds, defined on |w ± | < 2σ and V 0 (0) = 0. For the unit outer normal vector n and n − to Γ and Γ − , by (1.2), we see that the relation between ( n t , n −,t ) and ( n, n − ) is given by and withŵ(ξ, t) = w(X u± (ξ, t), t), where cof A denotes the cofactor matrix of A. Setting ρ(X u± (ξ, t), t) = ρ 0,± + θ 0,± (ξ) + θ ± (ξ, t) and using (1.5)-(1.7), we write dynamical for 0 < t < T , subject to the initial condition (θ ± , u ± )| t=0 = (0, v 0,± ). Here where V div (w ± ),V D (w ± ) and V Div (w ± ) are some matrices of C ∞ functions with respect to w ± defined on |w ± | < 2σ, which satisfy conditions V div (0) = V D (0) = V Div (0) = 0 and relations: div v ± = div u ± + tr( To state our theorem on the local well-posedness of problem (P), we introduce some functional spaces and the definition of uniform W 2−1/r r domain. For any domain D and 1 ≤ q ≤ ∞, L q (D) and W m q (D) denote the usual Lebesgue space and Sobolev space. We set W 0 q (D) = L q (D). For any Banach space X and 1 ≤ p ≤ ∞, L p ((a, b), X) and W m p ((a, b), X) denote the usual Lebesgue space and Sobolev space of X-valued functions defined on an interval (a, b). For 0 < θ < 1 and = 1, 2, B θ q,p (D) denotes the real interpolation space defined by B θ q,p (D) = (L q (D), W q (D)) θ,p with real interpolation functor (·, ·) θ,p . We set W θ For the simplicity of notations we use · Y as its norm instead of · Y d .
. Let 1 < r < ∞ and let Ω be a domain in R N with boundary ∂Ω. We say that Ω is a uniform W 2−1/r r domain, if there exist positive constants α, β and K such that for any (1.9) Here (x 1 , . . . ,x j , . . . , x N ) = (x 1 , . . . , The following theorem is concerned with the local well-posedness of problem (P).
(Ω) N be initial data for (1.1), which satisfy compatibility condition (1.10), range condition (1.11) and θ 0, (Ω) ≤ R. Then there exists a T > 0 depending on R such that (1.1) with kinematic condition (1.2) admits a unique solution (ρ ± , v ± ) with Finally we introduce more symbols and functional spaces used throughout this paper. N and C denote the sets of all natural numbers and complex numbers, respectively. We set N 0 = N∪{0}. For 1 < q < ∞, let q = q/(q −1). For any multi- . , x N ) and ∂ j = ∂/∂x j . For the differentations of a scalar function f and N -vector g = (g 1 , . . . , g N ), we use the following symbols: We set W m, and set L p,γ (R, X) = W 0 p,γ (R, X), L p,γ,0 (R, X) = W 0 p,γ,0 (R, X). Let D(R, X) and S(R, X) be the set of all X-valued C ∞ -functions having compact support and the Schwartz space of rapidly decreasing X-valued functions, respectively, while S (R, X) = L(S(R, C), X). Given M ∈ L 1,loc (R \ {0}, X), we define the operator T M : Here F x and F −1 ξ denote the Fourier transform and its inversion defined by respectively. Let F x and F −1 ξ denote the partial Fourier transform with respect to x = (x 1 , . . . , x N −1 ) and its inversion defined by 13) respectively. Let L and L −1 denote the Laplace transform and its inversion, which are defined by with λ = γ + iτ ∈ C, respectively. Given s ∈ R and X-valued function f (t), we set The Bessel potential space of X-valued functions of order s > 0 are defined by Following the argument due to Enomoto, Below and Shibata [5], we can prove Theorem 1.2 by contraction mapping principle with the help of the maximal L p -L q results, Theorems 2.1 and 2.2 (see [5] for detail). Thus, this paper consists of the following four sections. In Sect. 2, we present the maximal L p -L q regularity theorem (Theorems 2.1 and 2.2) and the theorem concerning the existence of R-bounded solution operator for linearized problem (Theorem 2.6). As was seen in Enomoto, Below and Shibata [5], the maximal L p -L q regularity theorem is direct consequence of Theorem 2.6 concerning the generalized resolvent problem for the linearized equations with the help of Weis' operator valued Fourier multiplier theorem, so that the main part of this paper is to show Theorem 2.6. In Sect 3, we consider the generalized resolvent problem for the linearized problem in the half-space and we show the existence of its R-bounded solution operator. In Sect 4, following the argumentation due to Enomoto, Below and Shibata [5], we show Theorem 2.6.
The following two theorems are maximal L p -L q regularity theorem for linear problem (LP). First theorem is the maximal L p -L q regularity theorem for (LP) and that Ω ± are uniformly W 2−1/r r domains. Then there exists a positive number λ 1 such that the following assertion is valid: for any initial data θ 0,± ∈ W 1 q (Ω ± ) and u 0,± ∈ B (Ω ± ) satisfying the compatibility conditions: for γ ≥ λ 1 with some constant C. Here λ 1 and C depend on µ ± 1 , µ ± 2 , q, r, Ω ± , N , ρ 0,± and ρ 2,± .
In order to prove our main results for (LP) (Theorem 2.1 and Theorem 2.2), we introduce the definition of R-bounded operator family and operator valued Fourier multiplier theorem due to Weis [16]. The definition of R-boundedness which is the key word in our method is the following.
The following theorem is given by Weis [16] Theorem 2.4. Let X and Y be two UMD spaces and 1 < p < ∞. Let M be a function in with some constant κ. Then the operator T M defined by (1.12) may uniquely be extended to a bounded linear operator from L p (R, X) into L p (R, Y ). Moreover denoting this extension by T M , we have for some positive constant C depending on p, X and Y .
The following lemma concerning the R-boundedness of the summation and composition of operator is known (see Denk, Hieber and Prüß [4]).
Lemma 2.5. (1) Let X and Y be Banach spaces, and let T and S be R-bounded families in L(X, Y ). Then T + S = {T + S | T ∈ T , S ∈ S} is also an R-bounded family in L(X, Y ) and (2) Let X, Y and Z be Banach spaces, and let T and S be R-bounded families in L(X, Y ) and L(Y, Z), respectively. Then ST = {ST | T ∈ T , S ∈ S} is also an R-bounded family in L(X, Z) and In order to prove the maximal L p -L q regularity theorem with the help of Theorem 2.4, we need the R-boundedness for solution operator to the following generalized resolvent problem: We can show the existence of the R-bounded solution operator to (RP) as follows: Then, there exist operator families such that for any (f + , f − , g + , g − , h, h − , k) ∈ X q and λ ∈ Λ ε,λ0 , solve problem (RP) uniquely. Moreover, there exists a constant C depending on ε, λ 0 , q and N such that Let B be the linear operator defined by . Since Definition 2.3 with n = 1 implies that the boundedness of the operator family T , it follows from Theorem 2.6 that Λ ε,λ0 is contained in the resolvent set of B and for any λ ∈ Λ ε,λ0 and (f + , f − , g + , g − , h, h − , k) ∈ X q , (ρ ± , u ± ) given by (2.3) and (2.4) satisfies the estimate: we have the following theorem: Moreover there exists constant λ 3 > 0 and M > 0 such that for any , u − (t)) = T (t)F satisfies the following estimate: for t > 0. Here λ 3 and M depend solely on µ ± 1 , µ ± 2 , q, r, Ω ± , N , λ 0 , ρ 0,± and ρ 2,± .
Following the argument due to Enomoto, Below and Shibata [5], we can obtain Theorem 2.1 and Theorem 2.2 as direct consequence of Theorem 2.6 with the help of Theorem 2.4. Moreover we can prove Theorem 1.2 by contraction mapping principle with Theorem 2.1 and 2.2 (see [5] in detail). Thus we omit the proof of Theorem 1.2, Theorem 2.1 and Theorem 2.2 and show the proof of only Theorem 2.6 in this paper.
3. R-bounded solution operators for model problem.
In the second step, applying the partial Fourier transform defined by (1.13), we derive a solution formula of the problem (3.1). We consider the following generalized resolvent problem: Here and hereafter, j and J run from 1 through N − 1 and N .

ON TWO PHASE PROBLEM FOR COMPRESSIBLE FLUID FLOW 3757
Here setting (3.20) If det L = 0, the inverse of L exists and we see In this section, we assume det L = 0 and continue to obtain the solution formula. We shall prove det L = 0 when λ ∈ Γ ε,λ0 in next section. By (3.21), we obtain we see Moreover we have Summing up, we obtain the following representation formula of solutions: where we have set By the Volevich trick: where we set DM ± (z) = (M ± (z)) = e ∓B±(z) + A ± M ± (z) for simplicity.
3.3. Proof of Theorem 2.8. In this section, we shall show the proof of Theorem 2.8. In order to prove Theorem 2.8, we use the following lemmas which is proven by Götz and Shibata [7].  where G λ is an operator defined by G λ u = (λu, γu, λ 1/2 ∇u, ∇ 2 u).

4.
Proof of Theorem 2.6. We shall prove Theorem 2.6. First step is to show the existence of R-bounded solution operator to a bent space problem by using the change of variable and Theorem 2.8. Let Φ : R N → R N be two bijections of C 1 class and let Φ −1 be their inverse maps. We set ∇Φ ± = A + B(x) and ∇Φ −1 ± = A −1 + B −1 (x) and assume that A and A −1 are orthogonal matrices with constant coefficients and B(x) and B −1 (x) are matrices of function in W 1 r (R N ) with N < r < ∞ satisfying Set Ω ± = Φ(R N ± ) and Γ = Φ(R N 0 ). Let γ ± 0 (x) and γ ± 3 (x) be real-valued functions defined on R N satisfying the following conditions:
Finally we give the proof of Theorem 2.6. We start with introducing the following proposition concerning some important properties of a uniform W 2−1/r r domain that was proved in Enomoto and Shibata [6]. This proposition will be used to construct a solution operator in Ω. Proposition 1. Let N < r < ∞ and let Ω ± be uniform W 2−1/r r domains in R N . Let M 1 be the number given in (4.1). Then, there exist constants M 2 > 0, 0 < d 0 , d 1 ± , d 2 ± < 1, at most countably many N -vector of functions Φ 0 j , Φ 1 j,± ∈ W 2 r (R N ) N and points x 0 j ∈ Γ, x 1 j,± ∈ Γ ± , x 2 j,± ∈ Ω ± such that the following assertions hold. (i) The maps: R N x → Φ 0 j (x) ∈ R N and R N x → Φ 1 j,± (x) ∈ R N (j ∈ N) are bijective.