On periodic solutions for one-phase and two-phase problems of the Navier-Stokes equations

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier-Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasiliner systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value $0$, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $L_p$-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of $\mathcal{R}$-solvers to the resolvent problem for the linearized equations and the transference theorem for the $L_p$ boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.


Introduction
This paper is concerned with time-periodic solutions of one-phase and two-phase problems for the Navier-Stokes equations. The periodic solutions for the Navier-Stokes equations have been studied in many articles [2,3,4,5,6,7,12,13,15,16,17,23,26] and references therein. One well-known approach to prove the existence of periodic solutions is the utilization of the Poincaré operator, which maps an initial value into the solution of the PDE at time T , where T is the period of the data. A fixed point of the Poincaré operator yields an initial value that induces a T -time-periodic solution. Such a utilization of the Poincaré operator is naturally carried out under the global well-posedness of the corresponding initial-boundary value problem for the bounded data on the right hand side of the equations. In the bounded domain case, this is deeply related with the situation where 0 does not belong to the spectrum of the system of the linearized equations. However, in many interesting problems in mathematical physics, we meet the situation that 0 is in the spectrum. One-phase or two-phase problems for the Navier-Stokes equations are typical examples. As explained in Sections 1 and 2 below, the one-phase and two-phase problems we treat in this paper are formulated by the Navier-Stokes equations with free boundary condition or transmission condition on the interface in a time-dependent domain Ω t , which is also unknown. Usually, Ω t is transformed to a fixed domain Ω by introducing an unknown function representing the boundary or the interface of Ω t . Thus, the problem treated here becomes a quasilinear system of equations with nonlinear boundary or transmission conditions. The first of our key approaches is to separate solutions into stationary part and oscillatory part. Then, the zero eigen-value of the linearized equations appears only in the equations for the stationary problem. We change the linearized equations by using some necessary conditions for the unique existence of solutions to avoid eigen-value 0 for the linearized problem. This technique is possible under the separation of the stationary part and the oscillatory part, which does not appear when working with the Poincaré operator. The second is to introduce a systematic approach to the maximal L p -L q regularity for the oscillatory part based solely on the R-solver for the resolvent problem of the linearized equations developed in [18,19,21,20,22] and a transference theorem for the L p boundedness of the operator-valued Fourier multiplier due to Eiter, Kyed and Shibata in [10]. The L p -L q maximal regularity for the oscillatory part of solutions is necessary because our problem is a quasilinear system with non-homogeneous boundary conditions. Since the maximal regularity for the oscillatory part of the periodic solutions does not seem to be well-studied, our systematic approach gives a quite important contribution to the study of systems of parabolic equations with non-homogeneous boundary conditions, which is the novelty of this paper.

One-phase problem
Let Ω t be a time-dependent domain in the N -dimensional Euclidean space R N (N ≥ 2). Let Γ t be the boundary of Ω t and n t the unit outer normal to Γ t . We assume that Ω t is occupied by some incompressible viscous fluid of unit mass density whose viscosity coefficient is a positive constant µ. Let u = ⊤ (u 1 (x, t), . . . , u N (x, t)), x = (x 1 , . . . , x N ) ∈ Ω t , and p = p(x, t) be the velocity field and the pressure field in Ω t , respectively, where ⊤ M denotes the transposed of M . We consider the Navier-Stokes equations in Ω t with free boundary condition as follows: in Ω t , (µD(u) − pI)n t = σH(Γ t )n t on Γ t , V Γt = u · n t on Γ t (1.1) for t ∈ R. Here, f = f (x, t) is a prescribed time-periodic external force with period 2π; H(Γ t ) denotes the (N −1)-fold mean curvature of Γ t which is given by H(Γ t )n t = ∆ Γt x for x ∈ Γ t , where ∆ Γt is the Laplace-Beltrami operator on Γ t ; V Γt is the evolution speed of Γ t along n t ; σ is a positive constant representing the surface tension coefficient; D(u) is the doubled deformation tensor given by D(u) = ∇u + ⊤ ∇u; and I is the (N × N )-identity matrix. Moreover, for any (N × N )-matrix of functions K whose (i, j) th component is K ij , Div K is an N -vector whose i th component is n j=1 ∂ j K ij and for any N -vector of functions v = ⊤ (v 1 , . . . , v N ), v · ∇v is an N -vector of functions whose i th component is Our problem is to find Ω t , Γ t , u and p satisfying the periodic condition: Ω t = Ω t+2π , Γ t = Γ t+2π , u(x, t) = u(x, t + 2π), p(x, t) = p(x, t + 2π) (1.2) for any t ∈ R.
To state the main result, we introduce assumptions and some functional spaces. Let p i = e i = T (0, . . . , 0, i−th 1 , 0, . . . , 0) for i = 1, . . . , N and p ℓ (ℓ = N +1, . . . , M ) be one of x i e j −x j e i (1 ≤ i < j ≤ N ). Notice that p ℓ forms a basis of the rigid space {v | D(v) = 0} and the number M is its dimension. We will construct Ω t satisfying the following two conditions: |Ω t | = |B R | for any t ∈ (0, 2π). (1.5) Here and in the following, (M ℓ,m ) ℓ,m=1,...,N denotes an (N × N )-matrix whose (ℓ, m) th component is M ℓ,m ; for any domain G and (N − 1)-dimensional hypersurface S, we let where g(x) denotes the complex conjugate of g(x), and dσ the surface element of S. |G| denotes the Lebesgue measure of a Lebesgue measurable set G of R N ; and B R is the ball with radius R centered at the origin. For 1 < p < ∞ and any Banach space X with norm · X , let L p,per ((0, 2π), X) = {f : R → X | f (·) X ∈ L 1,loc (R), H 1 p,per (0, 2π), X) = {f : R → X | f (t) X ∈ L 1,loc (R) and ḟ (t) X ∈ L 1,loc (R), f (t) = f (t + 2π),ḟ (t) =ḟ (t + 2π) for any t ∈ R, whereḟ denotes the derivative of f with respect to t. Let For any domain G in R N and 1 ≤ q ≤ ∞, L q (G), H m q (G), and B s q,p (G) denote the standard Lebesgue, Sobolev, and Besov spaces on G, and · Lq(G) , · H m q (G) , and · B s q,p (G) denote their respective norms. For any integer d, X d denotes the d-fold product of the space X, that is X d = {g = ⊤ (g 1 , . . . , g d ) | g j ∈ X (j = 1, . . . , d)}, while the norm of X d is denoted by · X instead of · X d for simplicity.
The following theorem is our main result concerning time-periodic solutions of the one-phase problem for the Navier-Stokes equations. for which the following assertion holds: If f ∈ L p,per ((0, 2π), L q (D) N ) satisfies the support condition: supp f (·, t) ⊂ D for any t ∈ (0, 2π), the orthogonal condition such that where Φ −1 (x, t) is the inverse map of the correspondence: x = Φ(y, t) for any t ∈ (0, 2π), are solutions of equations (1.2) satisfying the periodicity condition (1.2), and Γ t is given by where ξ(t) is the barycenter point of Ω t defined by setting Moreover, v and ρ satisfy the estimate: v Lp((0,2π),H 2 q (BR)) + ∂ t v Lp((0,2π),Lq(BR)) + ρ Lp((0,2π), for some constant C independent of ǫ.

Two-phase problem
Let Ω +t be a time-dependent domain in the N -dimensional Euclidean space R N . Let Γ t be the boundary of Γ t and n t its unit outer normal. Let Ω be a bounded domain in R N and S the boundary of Ω. We assume that Ω +t ⊂ Ω and Γ t ∩ S = ∅. Let Ω −t = Ω \ (Ω +t ∪ Γ t ) and set Ω t = Ω +t ∪ Ω −t . We assume that Ω ±t be occupied by some incompressible viscous fluids of unit mass densities whose viscosity coefficients are positive constants µ ± . Let u = ⊤ (u 1 , . . . , u N ) and p be the velocity field and the pressure field on Ω t , respectively. We consider the following Navier-Stokes equations with transmission condition on Γ t and no-slip condition on S: for t ∈ R, where f = f (x, t) is a prescribed time-periodic external force with period 2π; µ is the viscosity coefficient given by and [[f ]] denotes the jump of f ± defined on Ω ± along n t defined by setting The purpose of this paper is also to find Ω ±t , Γ t , u ± and p ± which satisfy the periodicity condition: To state a main result, we introduce the assumptions about Ω t as follows. We assume that Ω ⊃ B R for some R > 0, and that |Ω +t | = |B R | for any t ∈ (0, 2π). (1.12) The following theorem is our main result concerning time-periodic solutions of the two-phase problem for the Navier-Stokes equations.
Method Since the domain Ω t is unknown, using the Hanzawa transform, we reduce the equations onto a fixed domain, which results in a system of quasilinear equations. Thus, we cannot use the analytic C 0 -semi-group approach. Our main tool is to use the L p -L q maximal regularity for periodic solutions to the linearized equations, which can be obtained by using the R-solver to the generalized resolvent problem and applying the transference theorem ( [9,10]) to the solution formula represented by the Rsolver. This is a quite new and more direct approach and a completely different idea than exploiting the Poincaré operator.
Further notation This section is ended by explaining further notation used in this paper. We denote the sets of all complex numbers, real numbers, integers, and natural numbers by C, R, Z, and N, respectively. Let N 0 = N ∪ {0}. Let X be a Banach space with norm · X . For any X-valued function f : R → X the functions F [f ] and F −1 [f ] denote the Fourier transform and the inverse Fourier transform of f , respectively, defined by setting Let g : T → X be an X-valued function defined on the torus T = R/2πZ. We define the Fourier transform F T acting on g by setting For any sequence (a k ) = {a k ∈ X | k ∈ Z}, we define the inverse Fourier transform F −1 T acting on (a k ) by setting For any X-valued periodic function f with period 2π, we set The f S and f ⊥ are called stationary part and oscillatory part of f , respectively. For 1 ≤ p ≤ ∞, L p (R, X) and H 1 p (R, X) denote the standard Lebesgue and Sobolev spaces of Xvalued functions defined on R, and · Lp(R,X) , · H 1 p (R,X) denote their respective norms. For θ ∈ (0, 1), H θ p,per ((0, 2π), X) denotes the X-valued Bessel potential space of periodic functions defined by As usual, we set L p,per ((0, 2π), X) = H 0 p,per ((0, 2π), X).
For any scalar function f , we write For any N -vector of functions, u = ⊤ (u 1 , . . . , u N ), sometimes ∇u is regarded as an (N × N )-matrix of functions whose (i, j) th component is ∂ j u i . For any m-vector V = (v 1 , . . . , v m ) and n-vector W = (w 1 , . . . , w n ), V ⊗ W denotes an (m × n) matrix whose (i, j) th component is V i W j . For any (mn × N )matrix A = (A ij,k | i = 1, . . . , m, j = 1, . . . , n, k = 1, . . . , N ), AV ⊗ W denotes an N -column vector whose k th component is the quantity: where v(x) is the complex conjugate of v(x) and dσ denotes the surface element of ∂G. Given 1 < q < ∞, For two Banach spaces X and Y , X + Y = {x + y | x ∈ X, y ∈ Y }, L(X, Y ) denotes the set of all bounded linear operators from X into Y and L(X, X) is written simply as L(X). Moreover, let The letter C denotes a generic constant and C a,b,c,... denotes that the constant C a,b,c,... depends on a, b, c, . . .; the value of C and C a,b,c,... may change from line to line.

Linearization principle
We now formulate the problems (1.1) and (1.9) in a fixed domain and state main results in this setting. Theorems 1 and 3 follow from the main theorems of this section.

One-phase problem
Let Ω t , u and p satisfies equations (1.1) and the periodicity condition (1.2). We have Multiplying the first equation in (1.1) with p ℓ and integrating the resultant formula on Ω t and using the divergence theorem of Gauss give that In fact, we have used the fact that which follows from the Reynolds transport theorem * and that div u = 0 in Ω t . Thus, the periodicity condition (1.2) yields that where we have used the assumption that supp f (·, t) ⊂ D for any t ∈ R. Thus, the condition (1.6) is a necessary one to prove Theorem 1. From this observation, instead of problem (1.2), we consider the following equations: for t ∈ R. In fact, if Ω t , u and p satisfy equations (2.2), then we have which, combined with the periodicity condition (1.2), the assumption (1.3) and (2.1), leads to Thus, Ω t , u and p satisfy the first equation in (1.1). Therefore, under the stated assumptions, a solution to problem (2.2) is a solution to the original problem (1.1). However, as we shall see below, the condition (2.1) is not necessary to find a solution to (2.2). From now on, we consider problem (2.2). We reduce problem (2.2) to some nonlinear equations on B R by using the Hanzawa transform, which we explain below. Let ξ(t) be the barycenter point of Ω t defined by setting 3) * For any f (x, t) defined on Ωt, we have which is called the Reynolds transport theorem.
where we have used the fact that |Ω t | = |B R |, which follows from the assumption (1.5). By the Reynolds transport theorem, we see that because div u = 0. Let ρ(y, t) be an unknown time-periodic function with period 2π such that where S R = {x ∈ R N | |x| = R} and n is the unit outer normal to S R , that is n = x/|x| for x ∈ S R . Let H ρ be a suitable extension of ρ to R N , and then by the K-method in the theory of real interpolation [11,24], we see that there exist constants C 1 and C 2 such that for any t ∈ (0, 2π). In the following, we fix the method of this extension. For example,Ĥ ρ is the unique solution of the Dirichlet problem: Let ϕ be a C ∞ (R N ) function which equals one for x ∈ B 2R and zero for x ∈ B 3R , and we set H ρ = ϕĤ ρ . We assume that sup with some small constant δ > 0. Notice that y/|y| = R −1 y for y ∈ S R is the unit outer normal to S R . Let Φ(y, t) = y + R −1 H ρ (y, t)y + ξ(t). We choose δ > 0 so small that the map x = Φ(y, t) is injective. In fact, for any y 1 and y 2 which leads to the injectivity of the transformation x = Φ(y, t) for any t ∈ R provided that 0 < δ < 1. Moreover, using the inverse mapping theorem, we see that the map x = Φ(y, t) is surjective from R N onto R N . Let Let u(x, t) and p(x, t) satisfy equations (1.1), and let v(y, t) = u(x, t) and q(y, t) = p(x, t). We derive an equation for v and ρ from the kinematic condition: V Γt = u · n t on Γ t . From the definition: To represent ξ ′ (t), we introduce the Jacobian J(t) of the transformation x = Φ(y, t), which is written as Choosing δ > 0 small enough in (2.6), we have and so noting that n · n = 1, we have the kinematic equation: As will be seen in Sect. 3, we have < H(Γ t )n t , n t >= (∆ SR + (N − 1)/R 2 )ρ − (N − 1)/R+ nonlinear terms, and −(N − 1)/R 2 is the first eigen-value of the Laplace-Beltrami operator ∆ SR on S R with eigenfunctions y j /R for y = (y 1 , . . . , y N ) ∈ S R . We need to derive some auxiliary equations to avoid the zero and first eigen-values of ∆ SR . From the assumption (1.5) and the representation formulas of Ω t and Γ t in (2.7), by using polar coordinates we have and so we have where dω denotes the surface element of S R . Moreover, from (2.3) and the assumption (1.5), using polar coordinates centered at ξ(t), we have for j = 1, . . . , N . Thus, under the assumption (1.5) and the representation of Γ t and Ω t in (2.7), the kinematic condition (2.10) is equivalent to the equation Therefore, to prove the existence of (Ω t , u, p), we shall prove the well-posedness of the following equations: where we have set where ∆ S1 is the Laplace-Beltrami operator on the unit sphere S 1 . For the functions on the right side of equations (2.16), G(y, t) and F(v, ρ) are given in (3.13) in Sect. 3 below, g(v, ρ) and g(v, ρ) given in (3.6) in Sect. 3 below,d(v, ρ) has been given in (2.15) and  Here, ξ ′ (t) is given by the formula in (2.9). Then, we define Ω t and Γ t by the formulas in (2.7). Let Φ(y, t) = y + R −1 H ρ y + ξ(t). By choosing ǫ sufficiently small, estimates (1.8) and (2.5) ensure that the condition (2.6) is satisfied with small δ > 0. This yields the existence of the inverse map y = Φ −1 (x, t) of the map: x = Φ(y, t). Thus, the velocity field u(x, t) and the pressure p(x, t) on Ω t are well-defined by setting u(x, t) = v(y, t) and p(x, t) = q(y, t). Since div u = 0 in Ω t , |Ω t | is a constant, and so |Ω t | = |B R | by assumption (1.5). Moreover, if we set and so η(t) = ξ(t) + d with some constant d. We assume that the assumption (1.4) holds, and then by which leads to d = 0, that is Combining this with (1.5) gives that which yields that ρ satisfies the equation: Therefore, the kinematic equation: V Γt = u · n t holds on Γ t . So far, we see that Ω t , u and p satisfy equations Since Ω t is a small perturbation of B R , choosing ǫ > 0 smaller if necessary, we may assume that B R−ǫ0 ⊂ Ω t , and so by (1.6) we have Multiplying the first equation in (2.2) with p ℓ , integrating the resultant formulas with respect to x on Ω t and with respect to t on (0, 2π), and using the periodicity (1.2) and (2. Since Ω t is a small perturbation of B R , we may assume that the assumption (1.3) holds, and so by (2.20) we have Therefore, Ω t , u and p satisfy equations (1.1), and so we see that Theorem 1 follows immediately from Theorem 4.

Two-phase problem
We now formulate problem (1.9) in the fixed domain. The idea is essentially the same as in the one-phase case. LetΩ = Ω \ S R , Ω + = B R and Ω − = Ω \ B R . We define the barycenter point, ξ(t), of Ω +t by setting where we have used the fact that |Ω +t | = |B R |, which follows from the assumption (1.12). By the Reynolds transport theorem, we see that Let ρ(y, t) be an unknown periodic function with period 2π such that where S R = {x ∈ R N | |x| = R} and n is the unit outer normal to S R , that is n = y/|y| for y ∈ S R . In the following, we fix the method how to extend this to a transformation fromΩ to Ω t . Let H be a unique solution of the Dirichlet problem: Let L be a large number for which Ω ⊂ B L . From the K-method in real interpolation theory [11,24], we see that for any t ∈ (0, 2π). We may assume that there exists a small number ω > 0 for which B R+3ω ⊂ Ω. Let ϕ be a function in C ∞ (R N ) for which equals one for x ∈ B R+ω and zero for with some small constant δ > 0. We choose δ > 0 so small that the map: y → x = Φ(y, t) is bijective from Ω onto itself. In fact, for any y 1 and y 2 which leads to the injectivity of the map: x = Φ(y, t) for any t ∈ R provided that 0 < δ < 1. Moreover, using the fact that x = Φ(y, t) = y for y ∈ Ω \ B R+2ω , and the inverse mapping theorem, we see that the map x = Φ(y, t) is surjective from Ω onto itself. Let , Notice that R −1 y is the unit outer normal to S R for y ∈ S R . In the following, the jump quantity of f defined on Ω \ S R is also denoted by [[f ]], which is defined by setting From the definition it follows that Here and in the following, the unit outer normal to S R is denoted by n, which is given by n(y) = R −1 y for y ∈ S R . To represent the time derivative of ξ(t) given in (2.21), we introduce the Jacobian J + (t) of the transformation: x = y + R −1 H ρ y + ξ(t) for y ∈ B R , which is written as J + (t) = 1 + J 0,+ (t) with Choosing δ > 0 small enough in (2.24), we have and noting that n · n = 1, on S R we have the kinematic equation: As was already discussed in Subsec.
And then, to prove Theorem 3, we shall prove the global well-posedness of the following equations: and Mρ and B R ρ are the same as in (2.17) in Subsec. 2.1. For the functions on the right side of equations The following theorem is the unique existence theorem of 2π-periodic solutions of problem (2.31).
Employing the same argument as in the proof of Theorem 1 in Subsec. 2.1, we see that Theorem 3 immediately follows from Theorem 5.
3 Derivation of nonlinear terms 3.1 One-phase problem case First, we consider the one-phase problem case and we consider the map and H ρ satisfies the condition (2.5) and (2.6). Recall that H ρ (y, t) = ρ(y, t) for y ∈ S R . Let Ω t , Γ t , u(x, t) and p(x, t) satisfy the equations (1.1) and Choose δ > 0 small in such a way that there exists an inverse map: By the chain rule, we have Let J be the Jacobian of the transformation (3.1). Choosing δ > 0 small enough, we may assume that To obtain another representation formula of div x u, we use the inner product (·, ·) Ωt . For any test function ϕ ∈ C ∞ 0 (Ω t ), we set ψ(y) = ϕ(x). We then have and then by (3.5) we see that the divergence free condition: div u = 0 is transformed to the second equation in the equations (2.16). In particular, it follows from (3.5) that and therefore, Putting (3.8) and (3.9) together gives Thus, changing i to ℓ and m to i in the formula above, we define an N -vector of functions F 1 (v, ρ) by letting where we have setp Thus, setting we have the first equation in equations (2.16). We next consider the transformation of the boundary conditions. Recall that Γ t is represented by x = y + ρ(y, t)n(y) + ξ(t) for y ∈ S R with n(y) = y/|y|. Let x 0 be any point on S R and let Φ(p) be a C ∞ diffeomorphism on R N such that-up to a rotation-it holds be a finite number of points on S R and a partition of unity of S R such that supp ζ k ⊂ B ω (x k ) and K k=1 ζ k (y) = 1 on S R . In the following, we represent functions on each S R ∩ B ω (x k ), and to represent functions globally, we use the formula: (3.14) Thus, for the detailed calculations, we only consider the domain B R ∩ B ω (x ℓ ) (ℓ = 1, . . . , K), and use the local coordinate system: We write ρ = ρ(y(p 1 , . . . , p N −1 , 0), t) in the following. By the chain rule, we have We first represent n t . Since Γ t is given by The vectors τ i (i = 1, . . . , N − 1) form a basis of the tangent space of S R at y = y(p 1 , . . . , p N −1 ). Since |n t | 2 = 1, we have because τ i · n = 0. The vectors ∂x ∂p i (i = 1, . . . , N − 1) form a basis of the tangent space of Γ t , and so Let G = (g ij ) and G −1 = (g ij ), and then setting ∇ ′ which leads to Moreover, combining (3.17) and (3.19), we have Using the formula: Combining these formulas obtained above gives where we have set with some constant C independent of ℓ. Here and in the following k are the variables corresponding tō ∇H ρ = (H ρ , ∇H ρ ). In view of (3.21), we have Thus, in view of (3.14) and (3.16), we may write where ∂ ′ j ρ = ∂ρ/∂p j locally on B ω (x ℓ ) ∩ S R ,∇H ρ = (H ρ , ∇H ρ ), and V n (k) is a matrix of functions defined on B R × {k | |k| < δ} possessing the estimate: And also we may write n t = n +Ṽ n (∇H ρ )∇H ρ (3.25) whereṼ n (k) is a matrix of functions defined on B R × {k | |k| < δ} possessing the estimate: We now consider the boundary condition: It is convenient to divide the formula in (3.27) into the tangential part and normal part on Γ t as follows: where we have set d τ = d− < d, n > n and (3.31) Finally, we derive the nonlinear term h N (u, ρ) in (3.29). Recall that Γ t is represented by x = (R + ρ)n(y) + ξ(t) for y ∈ S R , where n = y/|y| ∈ S 1 . Then, we have where τ j = ∂n ∂pj , which forms a basis of the tangent space of S 1 . Since τ j · n = 0, the (i, j) th component of the first fundamental form G t = (g tij ) of Γ t is given by where g ij = τ i · τ j is the (i, j) th element of the first fundamental form, G, of S 1 , and so for any (N − 1)-vectors a ′ and b ′ ∈ R N −1 , we have Here and in the following, O 2 denotes a symbol defined by setting with some coefficients a 0 , b j and c ij defined on B R satisfying the estimate: In particular, componentwise. We next calculate the Christoffel symbols of Γ t . Since where ℓ ij =< τ ij , n >, and so Thus, and so Combining this formula with (3.21), using < ∂ i n, n >= 0, < n, τ ℓ >= 0, ∆ S1 n = −(N − 1)n, and (3.15) gives where O 1 denotes a symbol defined by setting with some coefficients a ′ 0 and b ′ j defined on B R satisfying the estimate: Here, in view of (3.3) and (3.33), we have defined h N (v, ρ) by letting where V h,N (k) andṼ ′ Γ (k) are functions defined on B R × {k | |k| < δ} possessing the estimate: for some constant C.

On periodic solutions of the linearized equations
In this section, we shall prove the L p -L q maximal regularity of 2π-periodic solutions of the linearized equations.

On linearized problem of one-phase problem
In this subsection, we consider the L p -L q maximal regularity of periodic solutions to linearized equations: where L, M, and A are the linear operators defined in (2.17). We shall prove the unique existence theorem of 2π-periodic solutions of equations (4.1). Our main result is this section is stated as follows.
To prove Theorem 6, our approach is to use the R-solver, Weis' operator-valued Fourier multiplier theorem [25] and a transference theorem, which is created in Eiter, Kyed and Shibata [10]. To introduce the notion of R-solver, we introduce the R-boundedness of operator families. Here, the Rademacher functions r k , k ∈ N, are given by r k : [0, 1] → {−1, 1}, t → sign (sin 2 k πt). The smallest such C is called R-bound of T on L(X, Y ), which is denoted by R L(X,Y ) T .
We quote Weis' operator-valued Fourier multiplier theorem and the transference theorem for operatorvalued Fourier multipliers.
Theorem 8 (Weis). Let X and Y be two UMD Banach spaces. Let m ∈ C 1 (R \ {0}, L(X, Y )) satisfies the multiplier condition: for any p ∈ (1, ∞) with some constant C p depending on p but independent of r b .
The transference theorem for operator-valued Fourier multipliers obtained in [10] is stated as follows.
Theorem 9. Let X and Y be two Banach spaces and p ∈ (1, ∞). Assume that Y is reflexive. Let m ∈ L ∞ (R, L(X, Y )) ∩ C(R, L(X, Y )), and let m| T denote the restriction of m on T. We define multipliers on R and T associated with m by setting If T m,R ∈ L(L p (R, X), L p (R, Y )) possessing the estimate: for any f ∈ L p (R, X) with some constant M , then T m,T ∈ L(L p (T, X), L p (T, Y )) and for any f ∈ L p (T, X) with some constant C p depending solely on p and independent of M .

Remark 10.
In the usual scalar-valued multiplier case, the transference theorem was proved by de Leeuw [9], and so this theorem is an extension to the operator-valued case.
We now consider the R-solver of the generalized resolvent problem: for k ∈ R. From Theorem 4.8 in Shibata [21] (cf. also Shibata [18,19]) we know the following theorem concerned with the existence of an R-solver of problem (4.1).
(3) We define the norm · Xq(BR) by setting Let ϕ(ik) be a function in C ∞ (R) which equals one for k ∈ R k0+2 and zero for k ∈ R k0+1 , and let ψ(ik) be a function in C ∞ (R) which equals one for k ∈ R k0+4 and zero for k ∈ R k0+3 . Notice that ϕ(ik)ψ(ik) = ϕ(ik). Let A(ik), P(ik) and H(ik) be the R-solvers given in Theorem 11. Then we have (d) Let n = n(τ ) be a C 1 -function defined on R \ {0} that satisfies the conditions |n(τ )| ≤ γ and |τ n ′ (τ )| ≤ γ with some constant c > 0 for any τ ∈ R \ {0}. Let T n be an operator-valued Fourier multiplier defined by T n f = F −1 [nF [f ]] for any f with F [f ] ∈ D(R, L q (D)). Then, T n is extended to a bounded linear operator from L p (R, L q (D)) into itself. Moreover, denoting this extension also by T n , we have T n L(Lp(R,Lq(D))) ≤ C p,q,D γ.
Here, we only prove the R-boundedness of ϕ(ik)ikA(ik). The R-boundedness of the other terms can be proved by the same argument. Let n ∈ N, {k ℓ } n ℓ=1 ∈ R n , {F ℓ } n ℓ=1 ∈ X q (B R ) n . Changing the labeling of indices if necessary, we may assume that ϕ(k ℓ ) = 0 for k = 1, . . . , m and ϕ(k ℓ ) = 0 for ℓ = m+1, . . . , n. And then, using Lemma 13, we have which shows that . We consider the high frequency part of the equations (4.1): (4.6) By Theorem 8, Theorem 9, and (4.5), we have immediately the following theorem.
Theorem 14. Let 1 < p, q < ∞. Then, for any functions F, G, G, D, and H with We let where we have set Then, u ψ , p ψ and ρ ψ are the unique solutions of equations (4.6), which possess the following estimate: Lp((0,2π),Lq(BR)) } for some constant C > 0. Here, we have set We now consider the lower frequency part of solutions of equations (4.1). Namely, we consider equations (4.3) for k ∈ R with 1 ≤ |k| < k 0 + 4. We shall show the following theorem.
for some constant C. Thus, for any k ∈ R with |k| < k 0 + 4, we consider the unique solvability of the equations: where we have set f = i(k − k 0 )v k0 and d = i(k 0 − k)η k0 . In fact, if we set v = v k0 + w, q = q k0 + r, and η = η k0 + ζ, then v, q and η are unique solutions of equations (4.3).
In what follows, we study the unique solvability of equations (4.9) in the case where f ∈ L q (B R ) and d ∈ W 2−1/q q (S R ) are arbitrary. To solve (4.9), it is convenient to study the functional analytic form of (4.9), and so we eliminate the pressure term r and the divergence condition div be the unique solution of the weak Dirichlet problem: (∇K, ∇ϕ) BR = (Div (µD(v)) − ∇div v, ∇ϕ) BR for any ϕ ∈Ĥ 1 q ′ ,0 (B R ) (4.10) subject to K =< µD(v)n, n > −σBζ − div v on S R , (4.11) where we have setĤ and q ′ = q/(q −1). In view of Poincaré's inequality, . Instead of (4.9), we consider the equations: In view of the boundary condition (4.11) for K(w, ζ), that w and ζ satisfy the third equation of equations (4.12) is equivalent to (µD(w)n) τ = 0 and div w = 0 on S R , (4.13) where d τ = d− < d, n > n for any N -vector d. Let J q (B R ) be a solenoidal space defined by setting Obviously, for v ∈ H 1 q (B R ), in order that div v = 0 in B R , it is necessary and sufficient that v ∈ J q (B R ). For any f ∈ L q (B R ) N , let ψ ∈ H 1 q,0 (B R ) be a unique solution of the weak Dirichlet problem: (∇ψ, ∇ϕ) BR = (f , ∇ϕ) BR for any ϕ ∈Ĥ 1 q ′ ,0 (B R ).
Let g = f − ∇ψ and inserting this formula into equations (4.9), we have where we have used the fact that ψ| SR = 0. Therefore, we shall solve equations (4.9) for f ∈ J q (B R ) and d ∈ W 2−1/q q (S R ). When f ∈ J q (B R ), the equations (4.9) and (4.12) are equivalent. In fact, if w ∈ H 2 q (B R ) N and ζ ∈ W 3−1/q q (S R ) satisfy equations (4.9) with some r ∈ H 1 q (B R ). Then, for any ϕ ∈Ĥ 1 q ′ ,0 (B R ), we have where we have used the fact that div w = 0. Moreover, from the boundary conditions in equations (4.9) and (4.11), it follows that on S R because div w = 0. Thus, the uniqueness of the solutions to his weak Dirichlet problem yields that r = K(w, ζ), and so w and ζ satisfy equations (4.12). Conversely, let w ∈ H 2 q (B R ) N and ζ ∈ W 3−1/q q (S R ) be solutions of equations (4.12). For any ϕ ∈Ĥ 1 q ′ ,0 (B R ), we have Moreover, from the boundary condition (4.13) it follows that div w = 0 on S R . The uniqueness implies that div w = 0 in B R . Thus, w, r = K(w, ζ) and ζ are solutions of equations (4.9). In particular, for solutions w and ζ of equations (4.12), we see that w satisfies the divergence condition: div w = 0 in B R automatically.
From now on, we study the unique existence theorem for equations (4.12) for any f ∈ J q (B R ) and d ∈ W 2−1/q q (S R ). To formulate problem (4.12) in a functional analytic setting, we define the spaces H q , D q and the operator A by setting where we have used (4.13) and div w = 0 in the definition of D q . We write equations (4.12) as (4.14) In view of Theorem 11, we see that k = k 0 + 4 is an element of the resolvent set of the operator A, and so (i(k 0 + 4)I − A) −1 exists in L(H q , D q ). Since B R is a compact set, it follows from the Rellich compactness theorem that (i(k 0 + 4)I − A) −1 is a compact operator from H q into itself. Thus, in view of Riesz-Schauder theory, in particular, Fredholm alternative principle, that k belongs to the resolvent set if and only if uniqueness holds for k. Thus, our task is to prove the uniqueness of solutions to equations (4.14). Let U = (w, ζ) ∈ D q satisfy the homogeneous equations: ikU − AU = 0 in H q .  Integrating the second equation of equations (4.16) and applying the divergence theorem of Gauss gives that where we have set |S R | = SR dω and we have used the fact that div w = 0 in B R . Thus, we have (ζ, 1) SR = 0. Multiplying the second equation of equations (4.16) with x j , integrating the resultant formula over S R and using the divergence theorem of Gauss gives that Thus, we have proved (4.17). In particular, Mζ = 0 in (4.16). We now prove that w = 0. For this purpose, we first consider the case where 2 ≤ q < ∞. Since B R is bounded, D q ⊂ D 2 . Multiplying the first equation of (4.16) with w and integrating the resultant formula over B R and using the divergence theorem of Gauss gives that because div w = 0 in B R . By the second equation of (4.16) with Mζ = 0, we have where we have used n = R −1 x = R −1 (x 1 , . . . , x N ) for x ∈ S R . Thus, Moreover, since ζ satisfies (4.17), we know that for some positive constant c, and therefore (4.18) implies w = 0. Now the first equation of (4.16) yields ∇K(w, ζ) = 0, so that K(w, ζ) is constant. Integration of the third equation of (4.16) over S R combined with (4.17) shows that this constant is 0, that is, K(w, ζ) = 0.
Finally, the third equation of (4.16) yields that B R ζ = 0 on S R , and so by (4.17) we have ζ = 0. This completes the proof of the uniqueness in the case where 2 ≤ q < ∞. In particular, we have the unique existence theorem of solutions to equation (4.14).
We now consider the case where 1 < q < 2. Let f be any element in J q ′ (B R ) and let V = (v, η) ∈ D q ′ be a solution of the equation: The existence of such V has already been proved above. Since d = 0, we see that η satisfies the relations: (η, 1) SR = 0, (η, x j ) SR = 0 for j = 1, . . . , N , and so Mη = 0. Using the divergence theorem of Gauss, we have Using the fact that (B R ζ, For any g ∈ L q ′ (B R ) N , let ψ ∈Ĥ 1 q ′ ,0 (B R ) be a unique solution of the weak Dirichlet problem: (∇ψ, ∇ϕ) BR = (g, ∇ϕ) BR for any ϕ ∈Ĥ 1 q,0 (B R ). Let f = g − ∇ψ, and then f ∈ J q ′ (B R ), and so using the fact that w ∈ J q (B R ), we have (w, g) BR = (w, f ) BR + (w, ∇ψ) BR = 0. The arbitrariness of g ∈ L q ′ (B R ) N implies that w = 0. Thus, the second equation of (4.16) and (4.17) leads to ζ = 0. This completes the proof of the uniqueness in the case where 1 < q < 2, and therefore the proof of Theorem 15.
We now consider the linearized stationary problem: We shall prove the following theorem.
for some constant C > 0.
Proof. The strategy of the proof is the same as that of Theorem 15. Since Lv, Mρ, and |B R | −1 BR v dy are lower order perturbations, choosing k 0 > 0 large enough, the generalized resolvent problem: , and ρ ∈ W 3−1/q q (S R ) possessing the estimate (4.20). Of course, the constant C in (4.20) depends on k 0 in this case, but k 0 is fixed, and so we can say that C in (4.20) is some fixed constant. The essential part of the proof is to show the unique existence of solutions to equations (4.19) with g = g = h = 0, that is And then, the uniqueness of the reduced problem in the L 2 framework implies the unique existence of solutions as was studied in the proof Theorem 15. Thus, we define the reduced problem corresponding to equations (4.19). For v ∈ H 2 q (B R ) N and ρ ∈ W 3−1/q q (S R ), let K = K(v, ρ) ∈ H 1 q (B R ) be the unique solution of the weak Dirichlet problem: (∇K, ∇ϕ) BR = (Div (µD(v) − Lv − ∇div v, ∇ϕ) BR for any ϕ ∈Ĥ 1 q ′ ,0 (B R ), (4.23) subject to the boundary condition: Then the reduced problem corresponding to problem (4.19) with g = g = h = 0 is given by the following equations: (4.25) Then, for f ∈ J q (B R ) and d ∈ W Moreover, from the boundary conditions in (4.22) and (4.24) it follows that Moreover, the boundary conditions of (4.25) and (4.24) gives that The uniqueness of the weak Dirichlet problem yields that div v = 0, and therefore, v, p = K(v, ρ) and ρ are solutions of equations (4.22). Finally, we show the uniqueness of equations (4.21) in the L 2 -framework, which yields Theorem 16. Let v ∈ H 2 2 (B R ) N and ρ ∈ W 5/2 2 (S R ) satisfy the homogeneous equations: In particular, Mρ = 0. Multiplying the first equation with v, integrating the resultant formula on B R and using the divergence theorem of Gauss gives that From the second equation of (4.26) with Mρ = 0 it follows that Combining these formulas yields that for some constant c > 0, which shows that ρ = 0. This completes the proof of the uniqueness in the L 2 framework, the proof of Theorem 16.
Proof of Theorem 6. We now prove Theorem 6. Let u ψ , p ψ and ρ ψ be functions given in Theorem 14 which are solutions of equations (4.6). Notice that ψ(ik) = 1 for for |k| ≥ k 0 + 4 and ψ(ik) = 0 for |k| ≤ k 0 + 3. For k ∈ Z with 1 ≤ |k| ≤ k 0 + 3, let in equations (4.3), and we write solutions v, q and η as v k = v, q k = q and η k = η. Let u k = e ikt v k , p k = e ikt q k , ρ k = e ikt η k , and then, u k , p k and ρ k satisfy the equations: (4.28) Let f = F S , d = D S , g = G S , g = G S and h = H S in equations (4.19), and let v, p and ρ be unique solutions of equations (4.19). We write u S = v, p S = p and ρ S = ρ. Under these preparations, we set By Hölder's inequality, we have and for any UMD Banach space X, using Lemma 13 and transference theorem, Theorem 9, we have

On linearized problem of two-phase problem
In this subsection, we consider the linear equations:  where Ω + = B R , Ω − = Ω \ (B R ∪ S R ), and M, A and B R are the linear operators defined in (2.17). We shall prove the unique existence theorem of 2π-periodic solutions of equations (4.29). Our main result in this section is stated as follows.
To prove Theorem 17, the strategy is the same as in the proof of Theorem 6. Therefore, we first consider the R-solver of the generalized resolvent problem: for k ∈ R. From Theorem 2.1.4 in Shibata and Saito [22] we know the following theorem concerned with the existence of an R-solver of problem (4.29).
(3) We define the norm · Xq(Ω) by setting Let ϕ(ik) be a function in C ∞ (R) which equals one for k ∈ R k0+2 and zero for k ∈ R k0+1 , and let ψ(ik) be a function in C ∞ (R) which equals one for k ∈ R k0+4 and zero for k ∈ R k0+3 . For f ∈ {F ± , G ± , G ± , D, H}, we set . We consider the high frequency part of the equations (4.29): in Ω ± × (0, 2π), (4.33) By Theorem 8, Theorem 9, and the analogue of (4.5) resulting from (4.35), we have immediately the following theorem.
Proof. The strategy of the proof is the same as that in Theorem 15. The only difference is the reduced problem. First, we can reduce equations (4.31) to equations: For any v ± ∈ H 2 q (Ω ± ) N and ρ ∈ W be the unique solution of the weak Neumann problem: subject to the transmission condition: where µ is piecewise constant defined by µ| Ω± = µ ± . Here and in the following,Ḣ 1 q (Ω) is defined by settingḢ The reduced problem corresponding to equations (4.35) is Thus, the uniqueness of the weak Neumann problem inḢ 1 q (Ω) yields that p − K(v, ρ) = 0 in Ω. Thus, v and ρ satisfy the equations (4.38).
Conversely, if v ∈ H 2 q (Ω) N and ρ ∈ W 3−1/q q (S R ) satisfy equations (4.38), then the divergence theorem of Gauss gives that for any ϕ ∈Ḣ 1 q ′ (Ω) we have Moreover, the transmission conditions in (4.38) and (4.37) give that Thus, the uniqueness of this weak Neumann problem yields that div v = c inΩ for some global constant c. Now the divergence theorem of Gauss and the boundary conditions in (4.38) yield c = 0, that is, div v = 0, which shows that v, p = K(v, ρ) and ρ satisfy equations (4.35).
Employing the same argument as that in the proof of Theorem 15, we see that to prove Theorem 21, it is sufficient to prove the uniqueness of solutions to equations (4.38) in the L 2 framework. Thus, we choose v ∈ H 2 2 (Ω) N and ρ ∈ W 5/2 2 (S R ) be solutions of the homogeneous equations: because div v + = 0 on B R , and so (ρ, 1) SR = 0. Moreover, multiplying the second equation in (4.39) by x j and integrating over S R , similar arguments lead to In particular, Mρ = 0. We now prove that v = 0. Multiplying the first equation of (4.39) with v and integrating the resultant formula overΩ and using the divergence theorem of Gauss gives that because div v = 0 inΩ. By the second equation of (4.39) with Mρ = 0, we have where we have used n = R −1 x = R −1 (x 1 , . . . , x N ) for x ∈ S R . This also yields Moreover, since ρ satisfies (4.40), we know that for some positive constant c, and therefore we have D(v) = 0. Since v ∈ H 1 q (Ω) and v = 0 on S − , we have v = 0.
Finally, the first equation of (4.39) yields that ∇K(v, ρ) = 0, which shows that K(v, ρ) is constant inΩ. Thus, [[K(v, ρ)]] is constant. Integrating the third equation of (4.39) yields that where we have used (4.40). In particular, K(v, ρ) is a constant globally in Ω. Finally, we have B R ρ = 0 on S R , which, combined with (4.40) leads to ρ = 0. This completes the proof of uniqueness for equations (4.38) in the L 2 framework. Therefore, we have proved Theorem 21.
Proof of Theorem 17. Employing the same argument as in the proof of Theorem 6 and using Theorem 20 and Theorem 21, we can prove Theorem 17. We may omit the detailed proof.

Proofs of main results
In this section, we shall prove Theorem 4. The proof of Theorem 5 is parallel to that of Theorem 4, and so we may omit it. We prove Theorem 4 with the help of the usual Banach fixed-point argument, and we define an underlying space I ǫ with some small constant ǫ > 0 determined later by setting where we have set In view of (2.9), we define ξ(t) by setting where c is a constant for which (v(x, s)(1 + J 0 (x, s)) dxds dt.
Choosing ǫ > 0 smaller if necessary, we may assume that 0 < M 3 ǫ < δ, and so (u, ρ) ∈ I ǫ . If we define a map Φ acting on (v, h) ∈ I ǫ by setting Φ(v, h) = (u, ρ), and then Φ is a map from I ǫ into itself. Employing a similar argument as for proving (5.30), we see that for any (v i , h i ) ∈ I ǫ (i = 1, 2), Choosing ǫ > 0 smaller if necessary, we may assume that M 4 ǫ ≤ 1/2, and so Φ is a contraction map on I ǫ . The Banach fixed-point theorem yields the unique existence of a fixed point (v, ρ) ∈ I ǫ of the map Φ, that is (v, ρ) = Φ(v, ρ), which is the required solution of equations (2.16). This completes the proof of Theorem 4.