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

Abstract

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 quasilinear 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 developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the \({{\mathcal {R}}}\)-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; \({{\mathcal {R}}}\) boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (\({{\mathcal {R}}}\)-solvers and their application to periodic \(L_p\) estimates, Preprint in 2019) 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 [3,4,5,6,7,8, 10,11,12,13,14, 20, 23] 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 \({\mathcal T}\), where \({\mathcal T}\) is the period of the data. A fixed point of the Poincaré operator yields an initial value that induces a \({\mathcal T}\)-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 Sects. 1 and 2, 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 \(\Omega _t\), which is also unknown. Usually, \(\Omega _t\) is transformed to a fixed domain \(\Omega \) by introducing an unknown function representing the boundary or the interface of \(\Omega _t\). 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 \({\mathcal R}\)-solver for the resolvent problem of the linearized equations developed in [15,16,17,18,19] and a transference theorem for the \(L_p\) boundedness of the operator-valued Fourier multiplier due to Eiter, Kyed and Shibata in [2]. 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 \(\Omega _t\) be a time-dependent domain in the N-dimensional Euclidean space \({\mathbb R}^N\) (\(N \ge 2\)). Let \(\Gamma _t\) be the boundary of \(\Omega _t\) and \(\mathbf{n}_t\) the unit outer normal to \(\Gamma _t\). We assume that \(\Omega _t\) is occupied by some incompressible viscous fluid of unit mass density whose viscosity coefficient is a positive constant \(\mu \). Let \(\mathbf{u}= {}^\top (u_1(x, t), \ldots , u_N(x,t))\), \(x = (x_1, \ldots , x_N)\in \Omega _t\), and \({\mathfrak {p}}={\mathfrak {p}}(x, t)\) be the velocity field and the pressure field in \(\Omega _t\), respectively, where \({}^\top M\) denotes the transposed of M. We consider the Navier–Stokes equations in \(\Omega _t\) with free boundary condition as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{u}+ \mathbf{u}\cdot \nabla \mathbf{u}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}) - {\mathfrak {p}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }\Omega _t, \\&\qquad \mathrm{div}\,\mathbf{u}= 0&\quad&\text {in }\Omega _t, \\&(\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}_t = \sigma H(\Gamma _t)\mathbf{n}_t&\quad&\text {on }\Gamma _t, \\&\qquad V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t&\quad&\text {on }\Gamma _t \end{aligned} \right. \end{aligned}$$
(1.1)

for \(t \in {\mathbb R}\). Here, \(\mathbf{f}= \mathbf{f}(x, t)\) is a prescribed time-periodic external force with period \(2\pi \); \(H(\Gamma _t)\) denotes the \((N-1)\)-fold mean curvature of \(\Gamma _t\) which is given by \(H(\Gamma _t)\mathbf{n}_t = \Delta _{\Gamma _t}x\) for \(x \in \Gamma _t\), where \(\Delta _{\Gamma _t}\) is the Laplace–Beltrami operator on \(\Gamma _t\); \(V_{\Gamma _t}\) is the evolution speed of \(\Gamma _t\) along \(\mathbf{n}_t\); \(\sigma \) is a positive constant representing the surface tension coefficient; \(\mathbf{D}(\mathbf{u})\) is the doubled deformation tensor given by \(\mathbf{D}(\mathbf{u}) = \nabla \mathbf{u}+ {}^\top \nabla \mathbf{u}\); and \(\mathbf{I}\) is the \((N\times N)\)-identity matrix. Moreover, for any \((N\times N)\)-matrix of functions \(\mathbf{K}\) whose \((i, j)\mathrm{th}\) component is \(K_{ij}\), \(\mathrm{Div}\,K\) is an N-vector whose \(i\mathrm{th}\) component is \(\sum _{j=1}^n\partial _jK_{ij}\) and for any N-vector of functions \(\mathbf{v}= {}^\top (v_1, \ldots , v_N)\), \(\mathbf{v}\cdot \nabla \mathbf{v}\) is an N-vector of functions whose \(i\mathrm{th}\) component is \(\sum _{j=1}^Nv_j\partial _j v_i\), where \(\partial _j=\partial /\partial x_j\).

Our problem is to find \(\Omega _t\), \(\Gamma _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfying the periodic condition:

$$\begin{aligned} \Omega _t = \Omega _{t+2\pi }, \quad \Gamma _t = \Gamma _{t+2\pi }, \quad \mathbf{u}(x, t) = \mathbf{u}(x, t+2\pi ), \quad {\mathfrak {p}}(x, t) = {\mathfrak {p}}(x, t+2\pi ) \end{aligned}$$
(1.2)

for any \(t \in {\mathbb R}\).

To state the main result, we introduce assumptions and some functional spaces. Let \(\mathbf{p}_i = \mathbf{e}_i ={}^T(0, \ldots , 0, \overset{\mathrm{i-th}}{1}, 0, \ldots , 0)\) for \(i=1, \ldots , N\) and \(\mathbf{p}_\ell \) (\(\ell =N+1, \ldots , M)\) be one of \(x_i\mathbf{e}_j - x_j\mathbf{e}_i\) (\(1 \le i< j \le N\)). Notice that \(\mathbf{p}_\ell \) forms a basis of the rigid space \(\{ \mathbf{v}\mid \mathbf{D}(\mathbf{v}) = 0\}\) and the number M is its dimension. We will construct \(\Omega _t\) satisfying the following two conditions:

$$\begin{aligned}&\det \Bigl (\int ^{2\pi }_0(\mathbf{p}_\ell , \mathbf{p}_m)_{\Omega _t}\,\mathrm{d}t\Bigr )_{\ell , m = 1, \ldots , M} \not =0, \end{aligned}$$
(1.3)
$$\begin{aligned}&\int ^{2\pi }_0\Bigl (\int _{\Omega _t} x\,\mathrm{d}x\Bigr )\,\mathrm{d}t = 0, \end{aligned}$$
(1.4)
$$\begin{aligned}&|\Omega _t| = |B_R| \quad \text {for any }t \in (0, 2\pi ). \end{aligned}$$
(1.5)

Here and in the following, \((M_{\ell ,m})_{\ell , m=1,\ldots , N}\) denotes an \((N\times N)\)-matrix whose \((\ell , m)\mathrm{th}\) component is \(M_{\ell , m}\); for any domain G and \((N-1)\)-dimensional hypersurface S, we let

$$\begin{aligned} (f, g)_{G} = \int _G f(x)\cdot \overline{g(x)}\,\mathrm{d}x, \quad (f, g)_{S}= \int _{S} f(x)\cdot \overline{g(x)}\,\mathrm{d}\sigma , \end{aligned}$$

where \(\overline{g(x)}\) denotes the complex conjugate of g(x), and \(\mathrm{d}\sigma \) the surface element of S. |G| denotes the Lebesgue measure of a Lebesgue measurable set G of \({\mathbb R}^N\); and \(B_R\) is the ball with radius R centered at the origin. For \(1< p < \infty \) and any Banach space X with norm \(\Vert \cdot \Vert _X\), let

$$\begin{aligned} L_{p, \mathrm{per}}((0, 2\pi ), X)&= \{f : {\mathbb R}\rightarrow X \mid \Vert f(\cdot )\Vert _X \in L_{1, \mathrm{loc}}({\mathbb R}), \\ f(t+ 2\pi )&= f(t) \quad \text {for any }t \in {\mathbb R}, \\ \Vert f\Vert _{L_p((0,2\pi ),X)}&= \Bigl (\int ^{2\pi }_0\Vert f(t)\Vert _X^p\,{dt}\Bigr )^{1/p}< \infty \}, \\ H^1_{p, \mathrm{per}}(0, 2\pi ), X)&= \{f: {\mathbb R}\rightarrow X \mid \Vert f(t)\Vert _X \in L_{1, \mathrm{loc}}({\mathbb R})\text { and }\Vert \dot{f}(t)\Vert _X \in L_{1, \mathrm{loc}}({\mathbb R}), \\ f(t)&= f(t+2\pi ), \dot{f}(t) = \dot{f}(t+2\pi ) \quad \text {for any }t \in {\mathbb R}, \\ \Vert f\Vert _{H^1_p((0, 2\pi ), X)}&= \Bigl (\int ^{2\pi }_0(\Vert f(t)\Vert _X^p + \Vert \dot{f}(t)\Vert _X^p)\,\mathrm{d}t\Bigr )^{1/p} < \infty \}, \end{aligned}$$

where \(\dot{f}\) denotes the derivative of f with respect to t. Let

$$\begin{aligned} \Vert f\Vert _{L_p((0, 2\pi ), X)}= & {} \Bigl (\int ^{2\pi }_0\Vert f(t)\Vert _X^p\,\mathrm{d}t\Bigr )^{1/p}, \\ \Vert f\Vert _{H^1_p((0, 2\pi ), X)}= & {} \Vert f\Vert _{L_p((0, 2\pi ), X)} + \Vert \dot{f}\Vert _{L_p((0, 2\pi ), X)}. \end{aligned}$$

For any domain G in \({\mathbb R}^N\) and \(1 \le q \le \infty \), \(L_q(G)\), \(H^m_q(G)\), and \(B^s_{q,p}(G)\) denote the standard Lebesgue, Sobolev, and Besov spaces on G, and \(\Vert \cdot \Vert _{L_q(G)}\), \(\Vert \cdot \Vert _{H^m_q(G)}\), and \(\Vert \cdot \Vert _{B^s_{q,p}(G)}\) denote their respective norms. For any integer d, \(X^d\) denotes the d-fold product of the space X, that is \(X^d = \{\mathbf{g}= {}^\top (g_1, \ldots , g_d) \mid g_j \in X (j=1, \ldots , d)\}\), while the norm of \(X^d\) is denoted by \(\Vert \cdot \Vert _X\) instead of \(\Vert \cdot \Vert _{X^d}\) for simplicity.

The following theorem is our main result concerning time-periodic solutions of the one-phase problem for the Navier–Stokes equations.

Theorem 1

Let \(1< p, q < \infty \) and \(2/p + N/q < 1\). Let \(D\subset B_R\) be a domain. Then, there exists a positive constant \(\epsilon \) and an injective map \(x = \Phi (y, t) : B_R \rightarrow {\mathbb R}^N\) for each \(t \in (0, 2\pi )\) with

$$\begin{aligned} \Phi \in L_{p, \mathrm{per}}((0, 2\pi ), H^3_q(B_R)^N) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), H^2_q(B_R)) \end{aligned}$$

for which the following assertion holds: If \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(D)^N)\) satisfies the support condition: \(\mathrm{supp}\,\mathbf{f}(\cdot , t) \subset D\) for any \(t \in (0, 2\pi )\), the orthogonal condition

$$\begin{aligned} \int ^{2\pi }_0 (\mathbf{f}(\cdot , t), \mathbf{p}_\ell )_D \,\mathrm{d}t = 0 \quad \text {for }\ell =1, \ldots , M, \end{aligned}$$
(1.6)

and the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D)^N)} \le \epsilon \), then there exist \(\mathbf{v}(y, t)\), \({\mathfrak {q}}(y, t)\), and \(\rho (y, t)\) with

$$\begin{aligned} \begin{aligned} \mathbf{v}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(B_R)^N) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \\ {\mathfrak {q}}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)), \\ \rho&\in L_{p, \mathrm{per}}((0, 2\pi ), W^{3-1/q}_q(B_R)^N) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)), \end{aligned}\end{aligned}$$
(1.7)

such that

$$\begin{aligned}&\Omega _t = \{x =\Phi (y, t) \mid y \in B_R\}, \quad \mathbf{u}(x, t) = \mathbf{v}(\Phi ^{-1}(x, t), t),\\&\quad {\mathfrak {p}}(x, t) = {\mathfrak {q}}(\Phi ^{-1}(x, t), t), \end{aligned}$$

where \(\Phi ^{-1}(x, t)\) is the inverse map of the correspondence: \(x = \Phi (y, t)\) for any \(t \in (0, 2\pi )\), are solutions of equations (1.2) satisfying the periodicity condition (1.2), and \(\Gamma _t\) is given by

$$\begin{aligned}\Gamma _t = \{x = y + R^{-1}\rho (y, t) y+ \xi (t) \mid y \in S_R\},\end{aligned}$$

where \(\xi (t)\) is the barycenter point of \(\Omega _t\) defined by setting

$$\begin{aligned}\xi (t) = \frac{1}{|\Omega _t|}\int _{\Omega _t} x\,\mathrm{d}x.\end{aligned}$$

Moreover, \(\mathbf{v}\) and \(\rho \) satisfy the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} \\&\quad +\Vert \partial _t \rho \Vert _{L_\infty ((0,2\pi ),W^{1-1/q}_q(S_R))} \le C\epsilon \end{aligned}\end{aligned}$$
(1.8)

for some constant C independent of \(\epsilon \).

Remark 2

In the construction of the map \(\Phi \), we see that \(\Phi (y,t) = y + R^{-1}\rho (y, t) + \xi (t)\) for \(y \in S_R\).

Two-phase problem

Let \(\Omega _{+t}\) be a time-dependent domain in the N-dimensional Euclidean space \({\mathbb R}^N\). Let \(\Gamma _t\) be the boundary of \(\Gamma _t\) and \(\mathbf{n}_t\) its unit outer normal. Let \(\Omega \) be a bounded domain in \({\mathbb R}^N\) and S the boundary of \(\Omega \). We assume that \(\Omega _{+t} \subset \Omega \) and \(\Gamma _t \cap S = \emptyset \). Let \(\Omega _{-t} = \Omega {\setminus }(\Omega _{+t} \cup \Gamma _t)\) and set \(\Omega _t = \Omega _{+t} \cup \Omega _{-t}\). We assume that \(\Omega _{\pm t}\) be occupied by some incompressible viscous fluids of unit mass densities whose viscosity coefficients are positive constants \(\mu _\pm \). Let \(\mathbf{u}= {}^\top (u_1, \ldots , u_N)\) and \({\mathfrak {p}}\) be the velocity field and the pressure field on \(\Omega _t\), respectively. We consider the following Navier–Stokes equations with transmission condition on \(\Gamma _t\) and no-slip condition on S:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{u}_\pm + \mathbf{u}\cdot \nabla \mathbf{u}_\pm - \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}_\pm ) - {\mathfrak {p}}_\pm \mathbf{I}) = \mathbf{f}&\quad&\text {in }\Omega _{\pm t}, \\&\qquad \mathrm{div}\,\mathbf{u}_\pm = 0&\quad&\text {in }\Omega _{\pm t}, \\&[[\mu \mathbf{D}(\mathbf{u}) - {\mathfrak {p}}\mathbf{I}]]\mathbf{n}_t = \sigma H(\Gamma _t)\mathbf{n}_t, \quad [[\mathbf{u}]]=0&\quad&\text {on }\Gamma _t, \\&\qquad V_{\Gamma _t} = \mathbf{u}_+\cdot \mathbf{n}_t&\quad&\text {on }\Gamma _t,\\&\qquad \mathbf{u}_- = 0&\quad&\text {on }S \end{aligned}\right. \end{aligned}$$
(1.9)

for \(t\in {\mathbb R}\), where \(\mathbf{f}=\mathbf{f}(x, t)\) is a prescribed time-periodic external force with period \(2\pi \); \(\mu \) is the viscosity coefficient given by

$$\begin{aligned} \mu = {\left\{ \begin{array}{ll} \mu _+ \quad &{}\text {in }\Omega _{+t}, \\ \mu _-\quad &{}\text {in }\Omega _{-t}; \end{array}\right. } \end{aligned}$$

and [[f]] denotes the jump of \(f_\pm \) defined on \(\Omega _\pm \) along \(\mathbf{n}_t\) defined by setting

$$\begin{aligned} {[}[f]](x_0) = \lim _{x\rightarrow x_0 \atop x \in \Omega _{+t}} f_+(x) - \lim _{x\rightarrow x_0 \atop x \in \Omega _{-t}} f_-(x)\quad \text {for }x_0 \in \Gamma _t. \end{aligned}$$

The purpose of this paper is also to find \(\Omega _{\pm t}\), \(\Gamma _t\), \(\mathbf{u}_\pm \) and \({\mathfrak {p}}_\pm \) which satisfy the periodicity condition:

$$\begin{aligned} \Omega _{\pm t} = \Omega _{\pm t+2\pi }, \Gamma _t = \Gamma _{t+2\pi }, \mathbf{u}_{\pm }(x,t) = \mathbf{u}_{\pm }(x, t+2\pi ), {\mathfrak {p}}_{\pm }(x,t) = {\mathfrak {p}}_{\pm }(x, t+2\pi ). \end{aligned}$$
(1.10)

To state a main result, we introduce the assumptions about \(\Omega _t\) as follows. We assume that \(\Omega \supset B_R\) for some \(R > 0\), and that

$$\begin{aligned}&\int ^{2\pi }_0\Bigl (\int _{\Omega _{+t}} x\,\mathrm{d}x\Bigr )\,\mathrm{d}t = 0, \end{aligned}$$
(1.11)
$$\begin{aligned}&|\Omega _{+t}| = |B_R| \quad \text {for any }t \in (0, 2\pi ). \end{aligned}$$
(1.12)

The following theorem is our main result concerning time-periodic solutions of the two-phase problem for the Navier–Stokes equations.

Theorem 3

Let \(1< p, q < \infty \) and \(2/p + N/q < 1\). \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\). Then, there exist a positive constant \(\epsilon \) and a bijective map \(x = \Phi (y, t)\) from \(\Omega \) onto itself such that for any \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N)\) satisfying the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(\Omega ))} \le \epsilon \), there exist \(\mathbf{v}_\pm (y, t)\), \({\mathfrak {q}}_\pm (y, t)\) and \(\rho (y, t)\) with

$$\begin{aligned} \begin{aligned} \mathbf{v}_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(\Omega _\pm )^N) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega _\pm )^N), \\ {\mathfrak {q}}_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\Omega _\pm )), \\ \rho&\in L_{p, \mathrm{per}}((0, 2\pi ), W^{3-1/q}_q(S_R)) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)) \end{aligned}\end{aligned}$$
(1.13)

such that

$$\begin{aligned}&\Omega _{\pm t} = \{x = \Phi (y, t) \mid y \in \Omega _\pm \}, \quad \mathbf{u}_\pm (x, t) = \mathbf{v}_\pm (\Phi ^{-1}(x, t), t),\\&\quad {\mathfrak {p}}_\pm (x, t) = {\mathfrak {q}}_\pm (\Phi ^{-1}(x, t), t), \end{aligned}$$

where \(y = \Phi ^{-1}(x, y)\) is the inverse map of \(x = \Phi (y, t)\), are solutions of problem (1.9), and \(\Gamma _t\) is given by

$$\begin{aligned}\Gamma _t = \{x = y + R^{-1}\rho (y, t) + \xi (t) \mid y \in S_R\},\end{aligned}$$

where \(\xi (t)\) is the barycenter point of \(\Omega _+\) defined by setting

$$\begin{aligned}\xi (t) = \frac{1}{|\Omega _{+t}|} \int _{\Omega _{+t}} x\,\mathrm{d}x.\end{aligned}$$

Moreover, \(\mathbf{v}_\pm \) and \(\rho \) satisfy the estimate:

$$\begin{aligned} \begin{aligned}&\sum _{\pm }(\Vert \mathbf{v}_\pm \Vert _{L_p((0, 2\pi ), H^2_q(\Omega _\pm ))} + \Vert \partial _t\mathbf{v}_\pm \Vert _{L_p((0, 2\pi ), L_q(\Omega _\pm ))} ) \\&\quad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))}\\&\quad +\Vert \partial _t \rho \Vert _{L_\infty ((0,2\pi ),W^{1-1/q}_q(S_R))} \le C\epsilon \end{aligned} \end{aligned}$$
(1.14)

for some constant C independent of \(\epsilon \).

Method Since the domain \(\Omega _t\) 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 \({\mathcal R}\)-solver to the generalized resolvent problem and applying the transference theorem ([1, 2]) to the solution formula represented by the \({\mathcal R}\)-solver. 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 \({\mathbb C}\), \({\mathbb R}\), \({\mathbb Z}\), and \({\mathbb N}\), respectively. Let \({\mathbb N}_0 = {\mathbb N}\cup \{0\}\). Let X be a Banach space with norm \(\Vert \cdot \Vert _X\). For any X-valued function \(f:{\mathbb R}\rightarrow X\) the functions \({\mathcal F}[f]\) and \({\mathcal F}^{-1}[f]\) denote the Fourier transform and the inverse Fourier transform of f, respectively, defined by setting

$$\begin{aligned} {\mathcal F}[f](\tau ) = \frac{1}{2\pi }\int _{{\mathbb R}} \mathrm{e}^{-i\tau t}f(t)\,\mathrm{d}t, \quad {\mathcal F}^{-1}[f](t) = \int _{{\mathbb R}} \mathrm{e}^{it\tau } f(\tau )\,\mathrm{d}\tau . \end{aligned}$$

Let \(g:{\mathbb T}\rightarrow X\) be an X-valued function defined on the torus \({\mathbb T}= {\mathbb R}/ 2\pi {\mathbb Z}\). We define the Fourier transform \({\mathcal F}_{\mathbb T}\) acting on g by setting

$$\begin{aligned} {\mathcal F}_{\mathbb T}[g](k) = \frac{1}{2\pi }\int ^{2\pi }_0 \mathrm{e}^{-ikt}g(t)\,\mathrm{d}t, \end{aligned}$$

which is regarded as a correspondence \(g \mapsto ({\mathcal F}_{\mathbb T}[g](k)) = \{{\mathcal F}_{\mathbb T}[g](k) \in X \mid k \in {\mathbb Z}\}\). For any sequence \((a_k) = \{a_k \in X \mid k \in {\mathbb Z}\}\), we define the inverse Fourier transform \({\mathcal F}^{-1}_{\mathbb T}\) acting on \((a_k)\) by setting

$$\begin{aligned} {\mathcal F}^{-1}_{\mathbb T}[(a_k)](t) = \sum _{k \in {\mathbb Z}} \mathrm{e}^{ikt}a_k. \end{aligned}$$

For any X-valued periodic function f with period \(2\pi \), we set

$$\begin{aligned}f_S = \frac{1}{2\pi }\int ^{2\pi }_0f(t)\,\mathrm{d}t, \quad f_\perp = f - f_S.\end{aligned}$$

The \(f_S\) and \(f_\perp \) are called stationary part and oscillatory part of f, respectively.

For \(1 \le p \le \infty \), \(L_p({\mathbb R}, X)\) and \(H^1_p({\mathbb R}, X)\) denote the standard Lebesgue and Sobolev spaces of X-valued functions defined on \({\mathbb R}\), and \(\Vert \cdot \Vert _{L_p({\mathbb R}, X)}\), \(\Vert \cdot \Vert _{H^1_p({\mathbb R}, X)}\) denote their respective norms. For \(\theta \in (0, 1)\), \(H^\theta _{p, \mathrm{per}}((0,2\pi ), X)\) denotes the X-valued Bessel potential space of periodic functions defined by

$$\begin{aligned} H^\theta _{p, \mathrm{per}}((0, 2\pi ), X)&= \{f \in L_{p, \mathrm{per}}((0, 2\pi ), X) \mid \Vert f\Vert _{H^\theta _p((0, 2\pi ), X)} < \infty \}, \\ \Vert f\Vert _{H^\theta _p((0, 2\pi ), X)}&= \Bigl (\int ^{2\pi }_0 \Vert {\mathcal F}_{\mathbb T}^{-1}[(1+k^2)^{\theta /2}{\mathcal F}_{\mathbb T}[f](k)](t)\Vert _X^p\,\mathrm{d}t \Bigr )^{1/p}. \end{aligned}$$

As usual, we set \(L_{p, \mathrm{per}}((0, 2\pi ), X)=H^0_{p, \mathrm{per}}((0, 2\pi ), X)\).

For any multi-index \(\alpha = (\alpha _1, \ldots , \alpha _N) \in {\mathbb N}_0^N\) we set \(\partial _x^\alpha h = \partial _1^{\alpha _1}\cdots \partial _N^{\alpha _N} h\) with \(\partial _i = \partial /\partial x_i\). For any scalar function f, we write

$$\begin{aligned}&\nabla f= (\partial _1f, \ldots , \partial _Nf), \quad \bar{\nabla }f=(f, \partial _1f, \ldots , \partial _Nf),\\&\nabla ^nf = (\partial _x^\alpha f \mid |\alpha |=n), \quad \bar{\nabla }^n f = (\partial _x^\alpha f \mid |\alpha | \le n) \quad (n \ge 2), \end{aligned}$$

where \(\partial _x^0f = f\). For any m-vector of functions \(\mathbf{f}={}^\top (f_1, \ldots , f_m)\), we write

$$\begin{aligned}&\nabla \mathbf{f}= (\nabla f_1, \ldots , \nabla f_m), \quad \bar{\nabla }\mathbf{f}=(\bar{\nabla }f_1, \ldots , \bar{\nabla }f_m),\\&\nabla ^n\mathbf{f}= (\nabla ^n f_1, \ldots , \nabla ^nf_m), \quad \bar{\nabla }^n\mathbf{f}= (\bar{\nabla }^nf_1,\ldots , \bar{\nabla }^nf_m). \end{aligned}$$

For any N-vector of functions, \(\mathbf{u}={}^\top (u_1, \ldots , u_N)\), sometimes \(\nabla \mathbf{u}\) is regarded as an \((N\times N)\)-matrix of functions whose \((i, j)\mathrm{th}\) component is \(\partial _ju_i\). For any m-vector \(V=(v_1, \ldots , v_m)\) and n-vector \(W=(w_1, \ldots , w_n)\), \(V\otimes W\) denotes an \((m\times n)\) matrix whose \((i, j)\mathrm{th}\) component is \(V_iW_j\). For any \((mn\times N)\)-matrix \(A=(A_{ij, k} \mid i=1, \ldots , m, j=1, \ldots , n, k=1, \ldots , N)\), \(AV\otimes W\) denotes an N-column vector whose \(k\mathrm{th}\) component is the quantity: \(\sum _{j=1}^m\sum _{j=1}^n A_{ij, k}v_iw_j\).

Let \(\mathbf{a}\cdot \mathbf{b}=<\mathbf{a}, \mathbf{b}>= \sum _{j=1}^Na_jb_j\) for any N-vectors \(\mathbf{a}=(a_1, \ldots , a_N)\) and \(\mathbf{b}=(b_1, \ldots , b_N)\). For any N-vector \(\mathbf{a}\), let \(\Pi _0\mathbf{a}= \mathbf{a}_\tau : = \mathbf{a}- <\mathbf{a}, \mathbf{n}>\mathbf{n}\). For any two \((N\times N)\)-matrices \(\mathbf{A}=(A_{ij})\) and \(\mathbf{B}=(B_{ij})\), the quantity \(\mathbf{A}:\mathbf{B}\) is defined by \(\mathbf{A}:\mathbf{B}= \sum _{i,j=1}^NA_{ij}B_{ji}\). For any domain G with boundary \(\partial G\), we set

$$\begin{aligned} (\mathbf{u}, \mathbf{v})_G = \int _G\mathbf{u}(x)\cdot \overline{\mathbf{v}(x)}\,\mathrm{d}x, \quad (\mathbf{u}, \mathbf{v})_{\partial G} = \int _{\partial G}\mathbf{u}\cdot \overline{\mathbf{v}(x)}\,\mathrm{d}\sigma , \end{aligned}$$

where \(\overline{\mathbf{v}(x)}\) is the complex conjugate of \(\mathbf{v}(x)\) and \(\mathrm{d}\sigma \) denotes the surface element of \(\partial G\). Given \(1< q < \infty \), let \(q' = q/(q-1)\). For \(L > 0\), let \(B_L = \{x \in {\mathbb R}^N \mid |x| < L\}\) and \(S_L = \{x \in {\mathbb R}^N \mid |x| = L \}\).

For two Banach spaces X and Y, \(X+Y = \{x + y \mid x \in X, y\in Y\}\), \({\mathcal L}(X, Y)\) denotes the set of all bounded linear operators from X into Y and \({\mathcal L}(X, X)\) is written simply as \({\mathcal L}(X)\). Moreover, let \({\mathcal R}_{{\mathcal L}(X, Y)}(\{{\mathcal T}(\lambda ) \mid \lambda \in I\})\) be the \({\mathcal R}\)-bound of the operator family \(\{{\mathcal T}(\lambda ) \mid \lambda \in I\}\subset {\mathcal L}(X, Y)\) (see also Definition 7). Let

$$\begin{aligned} i{\mathbb R}&= \{i\lambda \in {\mathbb C}\mid \lambda \in {\mathbb R}\},&\quad i{\mathbb R}_{\lambda _0}&= \{i\lambda \in i{\mathbb R}\mid |\lambda | \ge \lambda _0\}. \end{aligned}$$

The letter C denotes a generic constant and \(C_{a,b,c,\ldots }\) denotes that the constant \(C_{a,b,c,\ldots }\) depends on a, b, \(c, \ldots \); the value of C and \(C_{a,b,c,\ldots }\) 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 \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfies equations (1.1) and the periodicity condition (1.2). We have

$$\begin{aligned}&((\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}_t, \mathbf{e}_i)_{\Gamma _t} =\sigma (\Delta _{\Gamma _t}x, \mathbf{e}_i)_{\Gamma _t} = -\sigma (\nabla _{\Gamma _t}x, \nabla _{\Gamma _t}\mathbf{e}_i)_{\Gamma _t}=0;\\&((\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}_t, x_i\mathbf{e}_j-x_j\mathbf{e}_i)_{\Gamma _t} =\sigma (\Delta _{\Gamma _t}x, x_i\mathbf{e}_j-x_j\mathbf{e}_i)_{\Gamma _t} \\&\quad =-\sigma (\nabla _{\Gamma _t}x_j, \nabla _{\Gamma _t}x_i)_{\Gamma _t} +\sigma (\nabla _{\Gamma _t}x_i, \nabla _{\Gamma _t}x_j)_{\Gamma _t} =0. \end{aligned}$$

Multiplying the first equation in (1.1) with \(\mathbf{p}_\ell \) and integrating the resultant formula on \(\Omega _t\) and using the divergence theorem of Gauss give that

$$\begin{aligned} \frac{d}{\mathrm{d}t}(\mathbf{u}, \mathbf{p}_\ell )_{\Omega _t} = (\mathbf{f}, \mathbf{p}_\ell )_{\Omega _t}. \end{aligned}$$

In fact, we have used the fact that

$$\begin{aligned} \frac{d}{\mathrm{d}t}\int _{\Omega _t}\mathbf{u}(x, t)\cdot \mathbf{p}_\ell (x)\,\mathrm{d}x = \int _{\Omega _t}<\partial _t\mathbf{u}+ \mathbf{u}\cdot \nabla \mathbf{u}, \mathbf{p}_\ell >\,\mathrm{d}x, \end{aligned}$$

which follows from the Reynolds transport theoremFootnote 1 and that \(\mathrm{div}\,\mathbf{u}=0\) in \(\Omega _t\). Thus, the periodicity condition (1.2) yields that

$$\begin{aligned} \int ^{2\pi }_0\Bigl (\int _D \mathbf{f}(x, \cdot )\cdot \mathbf{p}_\ell (x)\,\mathrm{d}x\Bigr )\,\mathrm{d}t = 0 \quad \text {for }\ell =1, \ldots , M, \end{aligned}$$
(2.1)

where we have used the assumption that \(\mathrm{supp}\, \mathbf{f}(\cdot , t) \subset D\) for any \(t \in {\mathbb R}\). 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:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{u}+ \mathbf{u}\cdot \nabla \mathbf{u}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}) - {\mathfrak {p}}\mathbf{I}) +\sum _{k=1}^M\int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_k)_{\Omega _t}\, \mathrm{d}t \,\mathbf{p}_k= \mathbf{f}&\quad&\text {in }\Omega _t, \\&\qquad \mathrm{div}\,\mathbf{u}= 0&\quad&\text {in }\Omega _t, \\&(\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}_t = \sigma H(\Gamma _t)\mathbf{n}_t&\quad&\text {on }\Gamma _t, \\&\qquad V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t&\quad&\text {on }\Gamma _t \end{aligned} \right. \end{aligned}$$
(2.2)

for \(t \in {\mathbb R}\). In fact, if \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (2.2), then we have

$$\begin{aligned} \frac{d}{\mathrm{d}t}(\mathbf{u}(\cdot , t),\mathbf{p}_\ell )_{\Omega _t} + \sum _{k=1}^M\int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_k)_{\Omega _t}\,\mathrm{d}t (\mathbf{p}_k, \mathbf{p}_\ell )_{\Omega _t} =(\mathbf{f}, \mathbf{p}_\ell )_{\Omega _t}, \end{aligned}$$

which, combined with the periodicity condition (1.2), the assumption (1.3) and (2.1), leads to

$$\begin{aligned} \int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_k)_{\Omega _t}\,\mathrm{d}t= 0 \quad \text {for }k=1, \ldots , M. \end{aligned}$$

Thus, \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) 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 \(\xi (t)\) be the barycenter point of \(\Omega _t\) defined by setting

$$\begin{aligned} \xi (t)= \frac{1}{|B_R|}\int _{\Omega _t} x\,\mathrm{d}x, \end{aligned}$$
(2.3)

where we have used the fact that \(|\Omega _t| = |B_R|\), which follows from the assumption (1.5). By the Reynolds transport theorem, we see that

$$\begin{aligned} \frac{d}{\mathrm{d}t} \xi (t) = \frac{1}{|B_R|} \int _{\Omega _t} (\partial _tx + \mathbf{u}\cdot \nabla x)\,\mathrm{d}x = \frac{1}{|B_R|} \int _{\Omega _t} \mathbf{u}(x, t)\,\mathrm{d}x \end{aligned}$$
(2.4)

because \(\mathrm{div}\,\mathbf{u}=0\). Let \(\rho (y, t)\) be an unknown time-periodic function with period \(2\pi \) such that

$$\begin{aligned}\Gamma _t = \{x = y + \rho (y, t)\mathbf{n}+ \xi (t) \mid y \in S_R\}, \end{aligned}$$

where \(S_R = \{x \in {\mathbb R}^N \mid |x| = R\}\) and \(\mathbf{n}\) is the unit outer normal to \(S_R\), that is \(\mathbf{n}= x/|x|\) for \(x \in S_R\). Let \(H_\rho \) be a suitable extension of \(\rho \) to \({\mathbb R}^N\), and then by the K-method in the theory of real interpolation [9, 21], we see that there exist constants \(C_1\) and \(C_2\) such that

$$\begin{aligned}&C_1\Vert H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \le \Vert \rho (\cdot , t)\Vert _{W^{k-1/q}_q(S_R)} \le C_2\Vert H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \quad \text {for }k=1, 2, 3, \nonumber \\&C_1\Vert \partial _t H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \le \Vert \partial _t\rho (\cdot , t)\Vert _{W^{k-1/q}_q(S_R)} \le C_2\Vert \partial _tH_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \quad \text {for }k=1, 2,\qquad \qquad \end{aligned}$$
(2.5)

for any \(t \in (0, 2\pi )\). In the following, we fix the method of this extension. For example, \(\hat{H}_\rho \) is the unique solution of the Dirichlet problem:

$$\begin{aligned} (1-\Delta )\hat{H}_\rho = 0 \quad \text {in }{\mathbb R}^N{\setminus } S_R, \quad \hat{H}_\rho |_{S_R} = \rho . \end{aligned}$$

Let \(\varphi \) be a \(C^\infty ({\mathbb R}^N)\) function which equals one for \(x \in B_{2R}\) and zero for \(x \not \in B_{3R}\), and we set \(H_\rho = \varphi \hat{H}_\rho \). We assume that

$$\begin{aligned} \sup _{t \in {\mathbb R}} \Vert \nabla H_\rho (\cdot , t)\Vert _{H^1_\infty ({\mathbb R}^N)} \le \delta \end{aligned}$$
(2.6)

with some small constant \(\delta > 0\). Notice that \(y/|y| = R^{-1}y\) for \(y \in S_R\) is the unit outer normal to \(S_R\). Let \(\Phi (y, t) = y + R^{-1}H_\rho (y, t)y+ \xi (t)\). We choose \(\delta > 0\) so small that the map \(x = \Phi (y,t)\) is injective. In fact, for any \(y_1\) and \(y_2\)

$$\begin{aligned} |\Phi (y_1, t) - \Phi (y_2, t)|\ge & {} |y_1-y_2| - \sup _{t \in {\mathbb R}}\Vert \nabla H_\rho (\cdot , t)\Vert _{H^1_\infty ({\mathbb R}^N)} |y_1-y_2| \\\ge & {} (1-\delta )|y_1-y_2|, \end{aligned}$$

which leads to the injectivity of the transformation \(x = \Phi (y, t)\) for any \(t \in {\mathbb R}\) provided that \(0< \delta < 1\). Moreover, using the inverse mapping theorem, we see that the map \(x=\Phi (y, t)\) is surjective from \({\mathbb R}^N\) onto \({\mathbb R}^N\).

Let

$$\begin{aligned} \begin{aligned} \Omega _t&= \{x = y+ R^{-1}H_\rho (y, t)y + \xi (t) \mid y \in B_R\}, \\ \Gamma _t&= \{x = y + R^{-1}\rho (y, t)y + \xi (t) \mid y \in S_R\}. \end{aligned} \end{aligned}$$
(2.7)

Let \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy equations (1.1), and let \(\mathbf{v}(y, t) = \mathbf{u}(x, t)\) and \({\mathfrak {q}}(y, t) = {\mathfrak {p}}(x, t)\). We derive an equation for \(\mathbf{v}\) and \(\rho \) from the kinematic condition: \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) on \(\Gamma _t\). From the definition:

$$\begin{aligned} V_{\Gamma _t} = \frac{\partial x}{\partial t}\cdot \mathbf{n}_t = (\frac{\partial \rho }{\partial t} \mathbf{n}+ \xi '(t))\cdot \mathbf{n}_t. \end{aligned}$$

To represent \(\xi '(t)\), we introduce the Jacobian J(t) of the transformation \(x = \Phi (y, t)\), which is written as \(J(t) = 1 + J_0(t)\) with

$$\begin{aligned} J_0(t) = \det \bigl (\delta _{ij} +R^{-1} \frac{\partial }{\partial y_i}(H_\rho (y, t)y_j)\bigr )_{i,j=1, \ldots , N} - 1. \end{aligned}$$

Choosing \(\delta > 0\) small enough in (2.6), we have

$$\begin{aligned} \begin{aligned} |J_0(t)|&\le C\Vert \nabla H_\rho (\cdot , t)\Vert _{L_\infty (B_R)}. \end{aligned} \end{aligned}$$
(2.8)

From (2.4) it follows that

$$\begin{aligned} \xi '(t) = \frac{1}{|B_R|}\int _{B_R} \mathbf{v}(y, t)\,\mathrm{d}y + \frac{1}{|B_R|}\int _{B_R} \mathbf{v}(y, t)J_0(t)\,\mathrm{d}y, \end{aligned}$$
(2.9)

and so noting that \(\mathbf{n}\cdot \mathbf{n}=1\), we have the kinematic equation:

$$\begin{aligned} \partial _t\rho - (\mathbf{v}- \frac{1}{|B_R|}\int _{B_R}\mathbf{v}(y, t)\,\mathrm{d}y)\cdot \mathbf{n}= d(\mathbf{v}, \rho ) \end{aligned}$$
(2.10)

with

$$\begin{aligned} d(\mathbf{v}, \rho ) = \frac{1}{|B_R|}\int _{B_R}\mathbf{v}(y, t)J_0(t)\,\mathrm{d}y \cdot (\mathbf{n}-\mathbf{n}_t)+ \frac{\partial \rho }{\partial t}\mathbf{n}\cdot (\mathbf{n}-\mathbf{n}_t) + \mathbf{v}\cdot (\mathbf{n}_t-\mathbf{n}). \end{aligned}$$
(2.11)

As will be seen in Sect. 3, we have \(<H(\Gamma _t)\mathbf{n}_t, \mathbf{n}_t> = (\Delta _{S_R} + (N-1)/R^2)\rho -(N-1)/R +\) nonlinear terms, and \(-(N-1)/R^2\) is the first eigen-value of the Laplace-Beltrami operator \(\Delta _{S_R}\) on \(S_R\) with eigen-functions \(y_j/R\) for \(y=(y_1, \ldots , y_N) \in S_R\). We need to derive some auxiliary equations to avoid the zero and first eigen-values of \(\Delta _{S_R}\). From the assumption (1.5) and the representation formulas of \(\Omega _t\) and \(\Gamma _t\) in (2.7), by using polar coordinates we have

$$\begin{aligned} |B_R|&= |\Omega _t|=\int _{S_R}\Bigl (\int ^{1+R^{-1}\rho (\omega , t)}_0 r^{N-1}\,dr\Bigr )\,\mathrm{d}\omega = \frac{1}{N}\int _{S_R} (1+R^{-1}\rho (\omega ,t))^N\,\mathrm{d}\omega \\&= |B_R| + R^{-1}\int _{S_R} \rho \,\mathrm{d}\omega + \sum _{k=2}^N\frac{{}_NC_k}{N} R^{-k}\int _{S_R} \rho ^k\,\mathrm{d}\omega , \end{aligned}$$

and so we have

$$\begin{aligned} \int _{S_R}\rho \,\mathrm{d}\omega + \sum _{k=2}^N\frac{{}_NC_k}{N}R^{1-k}\int _{S_R}\rho ^k\,\mathrm{d}\omega =0 \end{aligned}$$
(2.12)

where \(\mathrm{d}\omega \) denotes the surface element of \(S_R\). Moreover, from (2.3) and the assumption (1.5), using polar coordinates centered at \(\xi (t)\), we have

$$\begin{aligned} 0&= \frac{1}{|B_R|}\int _{\Omega _t}(x-\xi (t))\,\mathrm{d}x = \frac{1}{|B_R|} \int _{S_R}\Bigl (\int ^{1+R^{-1}\rho (\omega , t)}_0 r^N\omega \,dr\Bigr )\,\mathrm{d}\omega \\&= \frac{1}{|B_R|}\frac{1}{N+1}\int _{S_R}(1+R^{-1}\rho (\omega , t))^{N+1}\omega \,\mathrm{d}\omega \\&= \frac{1}{|B_R|}\Bigl (R^{-1}\int _{S_R} \rho \omega \,\mathrm{d}\omega + \sum _{k=2}^{N+1}\frac{{}_{N+1}C_k}{N+1}R^{-k}\int _{S_R}\rho ^k \omega \,\mathrm{d}\omega \Bigr ), \end{aligned}$$

from which it follows that

$$\begin{aligned} \int _{S_R} \rho \omega _j\,\mathrm{d}\omega + \sum _{k=2}^{N+1}\frac{{}_{N+1}C_k}{N+1}R^{1-k}\int _{S_R} \rho ^k\omega _j\,\mathrm{d}\omega =0 \end{aligned}$$
(2.13)

for \(j=1, \ldots , N\). Thus, under the assumption (1.5) and the representation of \(\Gamma _t\) and \(\Omega _t\) in (2.7), the kinematic condition (2.10) is equivalent to the equation

$$\begin{aligned}&\partial _t\rho + \int _{S_R} \rho \,\mathrm{d}\omega + \sum _{k=1}^N \Bigl (\int _{S_R}\rho \omega _k\,\mathrm{d}\omega \Bigr )y_k \nonumber \\&-\Bigl (\mathbf{v}- \frac{1}{|B_R|}\int _{B_R}\mathbf{v}\,\mathrm{d}y\Bigr )\cdot \mathbf{n}= \tilde{d}(\mathbf{v}, \rho ) \quad \text {on }S_R\times (0, 2\pi ) \end{aligned}$$
(2.14)

with

$$\begin{aligned} \tilde{d}(\mathbf{v}, \rho ) = d(\mathbf{v}, \rho ) - \sum _{k=2}^N\frac{{}_NC_k}{N}R^{1-k}\int _{S_R}\rho ^k\,\mathrm{d}\omega - \sum _{k=2}^{N+1}\frac{{}_{N+1}C_k}{N+1}R^{1-k}\Bigl (\int _{S_R}\rho ^k \omega \,\mathrm{d}\omega \Bigr )y_k. \end{aligned}$$
(2.15)

Therefore, to prove the existence of \((\Omega _t, \mathbf{u}, {\mathfrak {p}})\), we shall prove the well-posedness of the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{v}+ {\mathcal L}\mathbf{v}_S- \mathrm{Div}\,(\mu (\mathbf{D}(\mathbf{v}) - {\mathfrak {q}}\mathbf{I}) = \mathbf{G}+ \mathbf{F}(\mathbf{v}, \rho )&\quad&\text {in }B_R\times (0, 2\pi ), \\&\mathrm{div}\,\mathbf{v}= g(\mathbf{v}, \rho ) = \mathrm{div}\,\mathbf{g}(\mathbf{v}, \rho )&\quad&\text {in }B_R\times (0, 2\pi ), \\&\partial _t\rho + {\mathcal M}\rho -{\mathcal A}\mathbf{v}\cdot \mathbf{n}= \tilde{d}(\mathbf{v}, \rho )&\quad&\text {on }S_R\times (0, 2\pi ), \\&(\mu \mathbf{D}(\mathbf{v})-{\mathfrak {q}})\mathbf{n}- ({\mathcal B}_R\rho ) \mathbf{n}= \mathbf{h}(\mathbf{v}, \rho )&\quad&\text {on }S_R\times (0, 2\pi ), \end{aligned}\right. \end{aligned}$$
(2.16)

where we have set

$$\begin{aligned} \begin{aligned} {\mathcal L}\mathbf{v}_S&= \sum _{k=1}^M(\mathbf{v}_S, \mathbf{p}_k)_{B_R}\,\mathbf{p}_k; \quad {\mathcal A}\mathbf{v}= \mathbf{v}- \frac{1}{|B_R|}\int _{B_R}\mathbf{v}\,\mathrm{d}y; \\ {\mathcal M}\rho&= \int _{S_R} \rho \,\mathrm{d}\omega + \sum _{k=1}^N \Bigl (\int _{S_R}\rho \omega _k\,\mathrm{d}\omega \Bigr )y_k; \\ {\mathcal B}_R\rho&= (\Delta _{S_R}+\frac{N-1}{R^2})\rho =R^{-2}(\Delta _{S_1} +(N-1))\rho , \end{aligned}\end{aligned}$$
(2.17)

where \(\Delta _{S_1}\) is the Laplace–Beltrami operator on the unit sphere \(S_1\). For the functions on the right side of equations (2.16), \(\mathbf{G}(y, t)\) and \(\mathbf{F}(\mathbf{v}, \rho )\) are given in (3.13) in Sect. 3, \(g(\mathbf{v}, \rho )\) and \(\mathbf{g}(\mathbf{v}, \rho )\) given in (3.6) in Sect. 3, \(\tilde{d}(\mathbf{v}, \rho )\) has been given in (2.15) and \(\mathbf{h}(\mathbf{v}, \rho ) = (\mathbf{h}'(\mathbf{v}, \rho ), h_N(\mathbf{v}, \rho ))\) is given in (3.31) and (3.34) in Sect. 3.

The following theorem is the unique existence theorem of \(2\pi \)-periodic solutions of problem (2.16).

Theorem 4

Let \(1< p, q < \infty \) and \(2/p+N/q < 1\). Then, there exists a small constant \(\epsilon > 0\) such that if \(\mathbf{f}\) satisfies the assumption (1.6) and the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} \le \epsilon \), then problem (2.16) admits \(2\pi \)-periodic solutions \(\mathbf{v}\), \({\mathfrak {q}}\), and \(\rho \) satisfying the regularity condition (1.7) and the estimate (1.8) in Theorem 1.

Proof of Theorem 1

We prove Theorem 1 with the help of Theorem 4. Let \(\xi (t)\) be defined by

$$\begin{aligned}\xi (t) = \int ^t_0\xi '(s)\,\mathrm{d}s + c\end{aligned}$$

where c is chosen in such a way that

$$\begin{aligned} \int ^{2\pi }_0 \xi (s)\,\mathrm{d}s = 0. \end{aligned}$$
(2.18)

Here, \(\xi '(t)\) is given by the formula in (2.9). Then, we define \(\Omega _t\) and \(\Gamma _t\) by the formulas in (2.7). Let \(\Phi (y, t)=y+ R^{-1}H_\rho y+ \xi (t)\). By choosing \(\epsilon \) sufficiently small, estimates (1.8) and (2.5) ensure that the condition (2.6) is satisfied with small \(\delta > 0\). This yields the existence of the inverse map \(y= \Phi ^{-1}(x, t)\) of the map: \(x = \Phi (y, t)\). Thus, the velocity field \(\mathbf{u}(x, t)\) and the pressure \({\mathfrak {p}}(x, t)\) on \(\Omega _t\) are well-defined by setting \(\mathbf{u}(x, t) = \mathbf{v}(y, t)\) and \({\mathfrak {p}}(x, t) = {\mathfrak {q}}(y,t)\). Since \(\mathrm{div}\,\mathbf{u}=0\) in \(\Omega _t\), \(|\Omega _t|\) is a constant, and so \(|\Omega _t| = |B_R|\) by assumption (1.5). Moreover, if we set

$$\begin{aligned}\eta (t) = \frac{1}{|B_R|}\int _{\Omega _t} x\,\mathrm{d}x,\end{aligned}$$

then

$$\begin{aligned}\eta '(t) = \frac{1}{|B_R|}\int _{\Omega _t} \mathbf{u}(x, t)\,\mathrm{d}x = \xi '(t), \end{aligned}$$

and so \(\eta (t) = \xi (t) + d\) with some constant d. We assume that the assumption (1.4) holds, and then by (2.18) we have

$$\begin{aligned}0 = \int ^{2\pi }_0 \eta (t)\,\mathrm{d}t = 2\pi d + \int ^{2\pi }_0\xi (t)\,\mathrm{d}t = 2\pi d, \end{aligned}$$

which leads to \(d=0\), that is

$$\begin{aligned}\xi (t) = \frac{1}{|B_R|} \int _{\Omega _t} x\,\mathrm{d}x. \end{aligned}$$

Combining this with (1.5) gives that

$$\begin{aligned}\int _{S_R}(R+\rho )^N\,\mathrm{d}\omega = 0, \quad \int _{S_R}(R+\rho )^{N+1}\,\mathrm{d}\omega =0, \end{aligned}$$

which yields that \(\rho \) satisfies the equation:

$$\begin{aligned}\partial _t\rho - {\mathcal A}\mathbf{v}\cdot \mathbf{n}= d(\mathbf{v}, \rho ) \quad \text {on }S_R. \end{aligned}$$

Therefore, the kinematic equation: \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) holds on \(\Gamma _t\). So far, we see that \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (2.2). Since \(D \subset B_R\), there exists a constant \(\epsilon _0 > 0\) for which \(D \subset B_{R-3\epsilon _0}\). Since \(\Omega _t\) is a small perturbation of \(B_R\), choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(B_{R-\epsilon _0} \subset \Omega _t\), and so by (1.6) we have

$$\begin{aligned} \int ^{2\pi }_0(\mathbf{f}(\cdot , t), \mathbf{p}_\ell )_{\Omega _t}\,\mathrm{d}t = 0 \quad \text {for }\ell =1, \ldots , N. \end{aligned}$$
(2.19)

Multiplying the first equation in (2.2) with \(\mathbf{p}_\ell \), integrating the resultant formulas with respect to x on \(\Omega _t\) and with respect to t on \((0, 2\pi )\), and using the periodicity (1.2) and (2.19) we have

$$\begin{aligned} \sum _{k=1}^M \int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_k)_{\Omega _t}\,\mathrm{d}t\int ^{2\pi }_0(\mathbf{p}_k, \mathbf{p}_\ell )_{\Omega _t} = \int ^{2\pi }_0(\mathbf{f}(\cdot , t), \mathbf{p}_\ell )_{\Omega _t}\,\mathrm{d}t =0 \end{aligned}$$
(2.20)

for \(\ell =1, \ldots , M\). Since \(\Omega _t\) is a small perturbation of \(B_R\), we may assume that the assumption (1.3) holds, and so by (2.20) we have

$$\begin{aligned}\int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_\ell )_{\Omega _t}\,\mathrm{d}t = 0 \quad \text {for }\ell =1, \ldots , M. \end{aligned}$$

Therefore, \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (1.1), and so we see that Theorem 1 follows immediately from Theorem 4. \(\square \)

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. Let \(\dot{\Omega }= \Omega {\setminus } S_R\), \(\Omega _+=B_R\) and \(\Omega _-=\Omega {\setminus }\overline{B_R}\). We define the barycenter point, \(\xi (t)\), of \(\Omega _{+t}\) by setting

$$\begin{aligned} \xi (t)= \frac{1}{|B_R|}\int _{\Omega _{+t}} x\,\mathrm{d}x, \end{aligned}$$
(2.21)

where we have used the fact that \(|\Omega _{+t}| = |B_R|\), which follows from the assumption (1.12). By the Reynolds transport theorem, we see that

$$\begin{aligned} \frac{d}{\mathrm{d}t} \xi (t) = \frac{1}{|B_R|} \int _{\Omega _t} \mathbf{u}(x, t)\,\mathrm{d}x. \end{aligned}$$
(2.22)

Let \(\rho (y, t)\) be an unknown periodic function with period \(2\pi \) such that

$$\begin{aligned} \Gamma _t = \{x = y + \rho (y, t)\mathbf{n}+ \xi (t) \mid y \in S_R\}, \end{aligned}$$

where \(S_R = \{x \in {\mathbb R}^N \mid |x| = R\}\) and \(\mathbf{n}\) is the unit outer normal to \(S_R\), that is \(\mathbf{n}= y/|y|\) for \(y \in S_R\).

In the following, we fix the method how to extend this to a transformation from \(\dot{\Omega }\) to \(\Omega _t\). Let H be a unique solution of the Dirichlet problem:

$$\begin{aligned} (1-\Delta ) H_\rho = 0 \quad \text {in }{\mathbb R}^N{\setminus } S_R, \quad H_\rho |_{S_R} = \rho . \end{aligned}$$

Let L be a large number for which \(\Omega \subset B_L\). From the K-method in real interpolation theory [9, 21], we see that

$$\begin{aligned}&C_1\Vert H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \le \Vert \rho (\cdot , t)\Vert _{W^{k-1/q}_q(S_R)} \le C_2\Vert H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \quad \text {for }k=1, 2, 3, \nonumber \\&C_1\Vert \partial _t H_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \le \Vert \partial _t\rho (\cdot , t)\Vert _{W^{k-1/q}_q(S_R)} \le C_2\Vert \partial _tH_\rho (\cdot , t)\Vert _{H^k_q({\mathbb R}^N)} \quad \text {for }k=1, 2, \qquad \qquad \end{aligned}$$
(2.23)

for any \(t \in (0, 2\pi )\). We may assume that there exists a small number \(\omega > 0\) for which \(B_{R+3\omega } \subset \Omega \). Let \(\varphi \) be a function in \(C^\infty ({\mathbb R}^N)\) for which equals one for \(x \in B_{R+\omega }\) and zero for \(x\not \in B_{R+2\omega }\). Let \(\Phi (y, t) = y+ \varphi (y)(R^{-1}H_\rho (y, t)y+ \xi (t))\). Notice that \(\Phi (y, t) = y + R^{-1}H_\rho (y, t)y + \xi (t)\) for \(y \in B_R\). Setting \(\Psi (y, t) = \varphi (y)(R^{-1}H_\rho (y, t)y+ \xi (t))\), we assume that

$$\begin{aligned} \sup _{t \in {\mathbb R}} \Vert \Psi (\cdot , t)\Vert _{H^1_\infty ({\mathbb R}^N)} \le \delta \end{aligned}$$
(2.24)

with some small constant \(\delta > 0\). We choose \(\delta > 0\) so small that the map: \(y \mapsto x= \Phi (y, t)\) is bijective from \(\Omega \) onto itself. In fact, for any \(y_1\) and \(y_2\)

$$\begin{aligned}&|\Phi (y_1, t) - \Phi (y_2, t)| \ge |y_1-y_2|\\&- \sup _{t\in {\mathbb R}}\Vert \nabla \Psi (\cdot , t)\Vert _{H^1_\infty ({\mathbb R}^N)} |y_1-y_2| \ge (1-\delta )|y_1-y_2|, \end{aligned}$$

which leads to the injectivity of the map: \(x = \Phi (y, t)\) for any \(t \in {\mathbb R}\) provided that \(0< \delta < 1\). Moreover, using the fact that \(x = \Phi (y, t) = y\) for \(y \in \Omega {\setminus } B_{R+2\omega }\), and the inverse mapping theorem, we see that the map \(x=\Phi (y, t)\) is surjective from \(\Omega \) onto itself. Let

$$\begin{aligned} \begin{aligned} \Omega _{+t}&= \{x = \Phi (y, t) = y+ R^{-1}H_\rho (y, t)y + \xi (t) \mid y \in B_R\}, \\ \Omega _{-t}&= \{x = \Phi (y, t) = y + \varphi (y)(R^{-1}H_\rho (y, t)y + \xi (t)) \mid y \in \Omega {\setminus } (S_R\cup B_R)\}, \\ \Gamma _t&= \{x = y + R^{-1}\rho (y, t)y + \xi (t) \mid y \in S_R\}, \end{aligned}\end{aligned}$$
(2.25)

Notice that \(R^{-1}y\) is the unit outer normal to \(S_R\) for \(y \in S_R\). In the following, the jump quantity of f defined on \(\Omega {\setminus } S_R\) is also denoted by [[f]], which is defined by setting

$$\begin{aligned} {[}[f]](x_0,t) = \lim _{y\rightarrow x_0 \atop y \in \Omega _+} f(y, t) - \lim _{y\rightarrow x_0 \atop y \in \Omega _-}f(y,t)\quad \text {for }x_0 \in S_R, \end{aligned}$$

where we have set \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R \cup S_R)\). Let \(\dot{\Omega }= \Omega _+ \cup \Omega _-\), and for f defined on \(\dot{\Omega }\), we write \(f_\pm = f|_{\Omega _\pm }\). On the other hand, for \(f_\pm \) defined on \(\Omega _\pm \), we define f by \(f|_{\Omega _\pm } = f_\pm \).

Let \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy the equations (1.9), and let \(\Phi ^{-1}(x, t)\) be the inverse map of \(x = \Phi (y, t)\). Let \(\mathbf{v}_\pm (y, t) = \mathbf{u}_\pm (\Phi ^{-1}(y, t), t)\) and \({\mathfrak {q}}_\pm (y, t) = {\mathfrak {p}}_\pm (\Phi ^{-1}(y, t), t)\) for \(y \in \Omega _{\pm t}\). We derive an equation for \(\mathbf{v}_+\) and \(\rho \) from the kinematic condition \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) on \(\Gamma _t\). Noting that \([[\mathbf{u}]] = 0\) on \(\Gamma _t\), we may also assume that \([[\mathbf{v}]]=0\) on \(S_R\), and so \(\mathbf{v}_+ = \mathbf{v}_-\) on \(S_R\).

From the definition it follows that

$$\begin{aligned} V_{\Gamma _t} = \frac{\partial x}{\partial t}\cdot \mathbf{n}_t = \left( \frac{\partial \rho }{\partial t} \mathbf{n}+ \xi '(t)\right) \cdot \mathbf{n}_t, \end{aligned}$$

Here and in the following, the unit outer normal to \(S_R\) is denoted by \(\mathbf{n}\), which is given by \(\mathbf{n}(y) = R^{-1}y\) for \(y \in S_R\). To represent the time derivative of \(\xi (t)\) given in (2.21), we introduce the Jacobian \(J_+(t)\) of the transformation: \(x = y + R^{-1}H_\rho y + \xi (t)\) for \(y \in B_R\), which is written as \(J_+(t) = 1 + J_{0,+}(t)\) with

$$\begin{aligned}J_{0,+}(t) = \det \bigl (\delta _{ij} +R^{-1} \frac{\partial }{\partial y_i}(H_\rho (y, t)y_j)\bigr )_{i,j=1, \ldots , N} - 1 \quad \text {for }y \in B_R. \end{aligned}$$

Choosing \(\delta > 0\) small enough in (2.24), we have

$$\begin{aligned} \begin{aligned} \Vert J_{0,+}(t)\Vert _{L_\infty (B_R)}&\le C\Vert \nabla H_\rho (\cdot , t)\Vert _{L_\infty (B_R)}. \end{aligned}\end{aligned}$$
(2.26)

From (2.21) it follows that

$$\begin{aligned} \xi '(t) = \frac{1}{|B_R|}\int _{B_R} \mathbf{v}_+(y, t)\,\mathrm{d}y + \frac{1}{|B_R|}\int _{B_R} \mathbf{v}_+(y, t)J_{0,+}(t)\,\mathrm{d}y, \end{aligned}$$
(2.27)

and noting that \(\mathbf{n}\cdot \mathbf{n}=1\), on \(S_R\) we have the kinematic equation:

$$\begin{aligned} \partial _t\rho - (\mathbf{v}- \frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+(y, t)\,\mathrm{d}y)\cdot \mathbf{n}= d(\mathbf{v}_+, \rho ) \end{aligned}$$
(2.28)

with

$$\begin{aligned}d(\mathbf{v}_+, \rho ) {=} \frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+(y, t)J_{0,+}(t)\,\mathrm{d}y \cdot (\mathbf{n}-\mathbf{n}_t) {+} \frac{\partial \rho }{\partial t}\mathbf{n}\cdot (\mathbf{n}-\mathbf{n}_t) + \mathbf{v}_+\cdot (\mathbf{n}_t-\mathbf{n}). \end{aligned}$$

As was already discussed in Sect. 2.1, from the assumption (1.12) and the representation formulas of \(\Omega _{+t}\) and \(\Gamma _t\) in (2.25), we have (2.12) in Sect. 2.1, too. Moreover, from (2.21) and the assumption (1.12), we have (2.13) in Sect. 2.1, too. Thus, under the assumption (1.12) and the representation of \(\Gamma _t\) and \(\Omega _{+t}\) in (2.25), the kinematic condition is equivalent to the equation:

$$\begin{aligned}&\partial _t\rho + \int _{S_R} \rho \,\mathrm{d}\omega + \sum _{k=1}^N \Bigl (\int _{S_R}\rho \omega _k\,\mathrm{d}\omega \Bigr )y_k \nonumber \\&\quad -\Bigl (\mathbf{v}_+ - \frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+\,\mathrm{d}y\Bigr )\cdot \mathbf{n}= \tilde{d}(\mathbf{v}_+, \rho ) \quad \text {on }S_R\times (0, 2\pi ) \end{aligned}$$
(2.29)

with

$$\begin{aligned} \tilde{d}(\mathbf{v}_+, \rho ) = d(\mathbf{v}_+, \rho ) - \sum _{k=2}^N\frac{{}_NC_k}{N}\int _{S_R}R^{1-k}\rho ^k\,\mathrm{d}\omega - \sum _{k=2}^{N+1}\frac{{}_{N+1}C_k}{N+1}R^{1-k}\Bigl (\int _{S_R}\rho ^k \omega \,\mathrm{d}\omega \Bigr )y_k. \end{aligned}$$
(2.30)

And then, to prove Theorem 3, we shall prove the global well-posedness of the following equations:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{v}_\pm - \mathrm{Div}\,(\mu _\pm (\mathbf{D}(\mathbf{v}_\pm ) - {\mathfrak {q}}_\pm ) = \mathbf{G}_\pm + \mathbf{F}_\pm (\mathbf{v}, \rho )&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\&\mathrm{div}\,\mathbf{v}_\pm = g_\pm (\mathbf{v}, \rho ) = \mathrm{div}\,\mathbf{g}_\pm (\mathbf{v}, \rho )&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\&\partial _t\rho + {\mathcal M}\rho -{\mathcal A}\mathbf{v}_+\cdot \mathbf{n}= \tilde{d}(\mathbf{v}_+, \rho )&\quad&\text {on }S_R\times (0, 2\pi ), \\&[[\mu _\pm \mathbf{D}(\mathbf{v}_\pm )-{\mathfrak {q}}_\pm ]]\mathbf{n}- ({\mathcal B}_R\rho ) \mathbf{n}= \tilde{\mathbf{h}}(\mathbf{v}, \rho )&\quad&\text {on }S_R\times (0, 2\pi ), \\&[[\mathbf{v}]]= 0&\quad&\text {on }S_R\times (0, 2\pi ), \\&\mathbf{v}_- = 0&\quad&\text {on }S\times (0, 2\pi ), \end{aligned}\right. \end{aligned}$$
(2.31)

where we have set

$$\begin{aligned} {\mathcal A}\mathbf{v}_+ = \mathbf{v}_+ - \frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+\,\mathrm{d}y \end{aligned}$$
(2.32)

and \({\mathcal M}\rho \) and \({\mathcal B}_R\rho \) are the same as in (2.17) in Sect. 2.1. For the functions on the right side of equations (2.31), \(\mathbf{G}_\pm \) and \(\mathbf{F}_\pm (\mathbf{v}, \rho )\) are defined in (3.39) of Sect. 3, \(g_\pm (\mathbf{v}, \rho )\) and \(\mathbf{g}_\pm (\mathbf{v}, \rho )\) are defined in (3.38) of Sect. 3, and \(\tilde{\mathbf{h}}(\mathbf{v}, \rho )\) is defined in (3.40) of Sect. 3 .

The following theorem is the unique existence theorem of \(2\pi \)-periodic solutions of problem (2.31).

Theorem 5

Let \(1< p, q < \infty \) and \(2/p+N/q < 1\). Then, there exists a small constant \(\epsilon > 0\) such that for any \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N)\) satisfying the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(\Omega ))} \le \epsilon \), problem (2.31) admits solutions \(\mathbf{v}_\pm \), \({\mathfrak {q}}_\pm \), and \(\rho \) satisfying the regularity condition (1.13) and the estimate (1.14) in Theorem 3.

Employing the same argument as in the proof of Theorem 1 in Sect. 2.1, we see that Theorem 3 immediately follows from Theorem 5.

Derivation of nonlinear terms

One-phase problem case

First, we consider the one-phase problem case and we consider the map

$$\begin{aligned} x = y + \Psi (y, t), \end{aligned}$$
(3.1)

where \(\Psi (y, t)= R^{-1}H_\rho (y,t) y + \xi (t)\) and \(H_\rho \) satisfies the condition (2.5) and (2.6). Recall that \(H_\rho (y, t) = \rho (y, t)\) for \(y \in S_R\). Let \(\Omega _t\), \(\Gamma _t\), \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy the equations (1.1) and

$$\begin{aligned} \Omega _t = \{x = y + \Psi (y, t) \mid y \in B_R\}, \quad \Gamma _t = \{x = y + R^{-1}\rho (y, t)y + \xi (t) \mid y \in S_R\}. \end{aligned}$$

Choose \(\delta > 0\) small in such a way that there exists an inverse map: \(y = \Phi ^{-1}(x, t)\) of the map: \(x = \Phi (y,t)=y+\Psi (y, t)\). Let \(\mathbf{v}(y, t) = \mathbf{u}(\Phi ^{-1}(y, t), t)\) and \({\mathfrak {q}}(y, t) = {\mathfrak {p}}(\Phi ^{-1}(y, t), t)\). By the chain rule, we have

$$\begin{aligned} \nabla _x = (\mathbf{I}+ \mathbf{V}_0(\mathbf{k}))\nabla _y, \quad \frac{\partial }{\partial x_i} = \frac{\partial }{\partial y_i} + \sum _{j=1}^N V_{0ij}(\mathbf{k})\frac{\partial }{\partial y_j} \end{aligned}$$
(3.2)

where \(\nabla _z = {}^\top (\partial /\partial z_1, \ldots , \partial /\partial z_N)\) for \(z \in \{x, y\}\) and \(\mathbf{k}=(k_0, k_1, \ldots , k_N) = (H_\rho , \nabla H_\rho )\). Here, \(\mathbf{V}_0(\mathbf{k})\) is an \((N\times N)\)-matrix of \(C^\infty \) functions defined for \(|\mathbf{k}| \le \delta \) with \(\mathbf{V}_0(0) = 0\) and \(V_{0ij}(\mathbf{k})\) is the \((i, j)\mathrm{th}\) component of \(\mathbf{V}_0(\mathbf{k})\). By (3.2), we can write \(\mathbf{D}(\mathbf{u})\) as \(\mathbf{D}(\mathbf{u}) = \mathbf{D}(\mathbf{v}) + {\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v}\) with

$$\begin{aligned} \begin{aligned} {\mathbf{D}}(\mathbf{v})_{ij}&= \frac{\partial v_i}{\partial y_j} + \frac{\partial v_j}{\partial y_i},\\ ({{\mathcal D}}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij}&= \sum _{k=1}^N \Bigl (V_{0jk}(\mathbf{k})\frac{\partial v_i}{\partial y_k} + V_{0ik}(\mathbf{k})\frac{\partial v_j}{\partial y_k}\Bigr ). \end{aligned} \end{aligned}$$
(3.3)

We next consider \(\mathrm{div}\,\mathbf{v}\). By (3.2), we have

$$\begin{aligned} \mathrm{div}\,_x\mathbf{u}= \sum _{j=1}^N\frac{\partial u_j}{\partial x_j} = \sum _{j,k=1}^N(\delta _{jk} + V_{0jk}(\mathbf{k}))\frac{\partial v_j}{\partial y_k} = \mathrm{div}\,_y\mathbf{v}+ \mathbf{V}_0(\mathbf{k}):\nabla \mathbf{v}. \end{aligned}$$
(3.4)

Let J be the Jacobian of the transformation (3.1). Choosing \(\delta > 0\) small enough, we may assume that \(J = J(\mathbf{k}) = 1 + J_0(\mathbf{k})\), where \(J_0(\mathbf{k})\) is a \(C^\infty \) function defined for \(|\mathbf{k}| < \sigma \) such that \(J_0(0) = 0\).

To obtain another representation formula of \(\mathrm{div}\,_x\mathbf{u}\), we use the inner product \((\cdot , \cdot )_{\Omega _t}\). For any test function \(\varphi \in C^\infty _0(\Omega _t)\), we set \(\psi (y) = \varphi (x)\). We then have

$$\begin{aligned}&(\mathrm{div}\,_x\mathbf{u}, \varphi )_{\Omega _t} = -(\mathbf{u}, \nabla \varphi )_{\Omega _t} = -(J\mathbf{v}, (\mathbf{I}+ \mathbf{V}_0)\nabla _y\psi )_\Omega \\&\quad = (\mathrm{div}\,((\mathbf{I}+ {}^\top \mathbf{V}_0)J\mathbf{v}), \psi )_\Omega = (J^{-1}\mathrm{div}\,((\mathbf{I}+ {}^\top \mathbf{V}_0)J\mathbf{v}), \varphi )_{\Omega _t}, \end{aligned}$$

which, combined with (3.4), leads to

$$\begin{aligned} \mathrm{div}\,_x\mathbf{u}= \mathrm{div}\,_y\mathbf{v}+ \mathbf{V}_0(\mathbf{k}):\nabla \mathbf{v}= J^{-1}(\mathrm{div}\,_y\mathbf{v}+ \mathrm{div}\,_y(J{}^\top \mathbf{V}_0(\mathbf{k})\mathbf{v})). \end{aligned}$$
(3.5)

Recalling that \(J = J(\mathbf{k}) = 1 + J_0(\mathbf{k})\), we define \(g(\mathbf{v}, \rho )\) and \(\mathbf{g}(\mathbf{v}, \rho )\) by letting

$$\begin{aligned} \begin{aligned} g(\mathbf{v}, \rho )&= -(J_0(\mathbf{k})\mathrm{div}\,\mathbf{v}+ (1+J_0(\mathbf{k}))\mathbf{V}_0(\mathbf{k}):\nabla \mathbf{v}), \\ \mathbf{g}(\mathbf{v}, \rho )&= -(1+J_0(\mathbf{k})){}^\top \mathbf{V}_0(\mathbf{k})\mathbf{v}, \end{aligned}\end{aligned}$$
(3.6)

and then by (3.5) we see that the divergence free condition: \(\mathrm{div}\,\mathbf{u}=0\) is transformed to the second equation in the equations (2.16). In particular, it follows from (3.5) that

$$\begin{aligned} J_0(k)\mathrm{div}\,\mathbf{v}+ J(k)\mathbf{V}_0(\mathbf{k}):\nabla \mathbf{v}= \mathrm{div}\,(J(k){}^\top \mathbf{V}_0(\mathbf{k})\mathbf{v}). \end{aligned}$$
(3.7)

To derive \(\mathbf{F}(\mathbf{v}, \rho )\), we first observe that

$$\begin{aligned}&\sum _{j=1}^N\frac{\partial }{\partial x_j}(\mu \mathbf{D}(\mathbf{u})_{ij} - {\mathfrak {p}}\delta _{ij}) \nonumber \\&\quad = \sum _{j,k=1}^N\mu (\delta _{jk} + V_{0jk})\frac{\partial }{\partial y_k} (\mathbf{D}(\mathbf{v})_{ij} + ({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij}) -\sum _{j=1}^N(\delta _{ij} + V_{0ij})\frac{\partial {\mathfrak {q}}}{\partial y_j}, \end{aligned}$$
(3.8)

where we have used (3.3). Since

$$\begin{aligned}\frac{\partial }{\partial t}[u_i(y + \Psi (y, t), t)] = \frac{\partial u_i}{\partial t}(x, t) + \sum _{j=1}^N\frac{\partial \Psi _{j}}{\partial t} \frac{\partial u_i}{\partial x_j}(x, t), \end{aligned}$$

we have

$$\begin{aligned}\frac{\partial u_i}{\partial t} = \frac{\partial v_i}{\partial t}-\sum _{j,k=1}^N \frac{\partial \Psi _{j}}{\partial t} (\delta _{jk} + V_{0jk})\frac{\partial v_i}{\partial y_k}, \end{aligned}$$

and therefore,

$$\begin{aligned} \frac{\partial u_i}{\partial t} + \sum _{j=1}^N u_j\frac{\partial u_i}{\partial x_j} = \frac{\partial v_i}{\partial t} + \sum _{j,k=1}^N(v_j - \frac{\partial \Psi _{j}}{\partial t}) (\delta _{jk} + V_{0jk}(\mathbf{k}))\frac{\partial v_i}{\partial y_k}. \end{aligned}$$
(3.9)

Putting (3.8) and (3.9) together gives

$$\begin{aligned} f_i(x, t)&= \Bigl (\frac{\partial v_i}{\partial t} + \sum _{j,k=1}^N(v_j - \frac{\partial \Psi _{j}}{\partial t}) (\delta _{jk} + V_{0jk}(\mathbf{k}))\frac{\partial v_i}{\partial y_k}\Bigr ) \\&\quad - \mu \sum _{j,k=1}^N(\delta _{jk} + V_{0jk}(\mathbf{k})) \frac{\partial }{\partial y_k}(\mathbf{D}(\mathbf{v})_{ij} + ({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij}) \\&\quad - \sum _{j=1}^N(\delta _{ij}+ V_{0ij}(\mathbf{k}))\frac{\partial {\mathfrak {q}}}{\partial y_j}. \end{aligned}$$

Since \((\mathbf{I}+ \nabla \Psi )(\mathbf{I}+ \mathbf{V}_0) = (\partial x/\partial y)(\partial y/\partial x) = \mathbf{I}\), that is,

$$\begin{aligned} \sum _{i=1}^N(\delta _{mi} + \partial _m\Psi _{i}) (\delta _{ij} + V_{0ij}(\mathbf{k})) = \delta _{mj}, \end{aligned}$$
(3.10)

we have

$$\begin{aligned}&\sum _{i=1}^N(\delta _{mi} + \partial _m\Psi _i)f_i(\Psi (y, t), t)\\&\quad =\sum _{i=1}^N(\delta _{mi} + \partial _m\Psi _{i}) \Bigl (\frac{\partial v_i}{\partial t} + \sum _{j,k=1}^N(v_j - \frac{\partial \Psi _{i}}{\partial t}) (\delta _{jk}+V_{0jk}(\mathbf{k}))\frac{\partial v_i}{\partial y_k}\Bigr ) \\&\qquad -\mu \sum _{i,j,k=1}^N(\delta _{mi} + \partial _m\Psi _{i}) (\delta _{jk}+V_{0jk}(\mathbf{k}))\frac{\partial }{\partial y_k} (\mathbf{D}(\mathbf{v})_{ij} +({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij}) -\frac{\partial {\mathfrak {q}}}{\partial y_m}. \end{aligned}$$

Thus, changing i to \(\ell \) and m to i in the formula above, we define an N-vector of functions \(\mathbf{F}_1(\mathbf{v}, \rho )\) by letting

$$\begin{aligned}&\mathbf{F}_1(\mathbf{v}, \rho )|_i = -\sum _{j,k=1}^N(v_j - \frac{\partial \Psi _{j}}{\partial t}) (\delta _{jk} + V_{0jk}(\mathbf{k}))\frac{\partial v_i}{\partial y_k} \nonumber \\&\quad -\sum _{\ell =1}^N\partial _i\Psi _{\ell } \Bigl (\frac{\partial v_\ell }{\partial t} + \sum _{j,k=1}^N(v_j - \frac{\partial \Psi _{j}}{\partial t}) (\delta _{jk} + V_{0jk}(\mathbf{k}))\frac{\partial v_\ell }{\partial y_k}\Bigr ) \nonumber \\&\quad + \mu \Bigl (\sum _{j=1}^N \frac{\partial }{\partial y_j}({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij} + \sum _{j,k=1}^NV_{0jk}(\mathbf{k})\frac{\partial }{\partial y_k}(\mathbf{D}(\mathbf{v})_{ij} + ({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{ij}) \nonumber \\&\quad + \sum _{j,k,\ell =1}^N\partial _i\Psi _{\ell }(\delta _{jk} + V_{0jk}(\mathbf{k})) \frac{\partial }{\partial y_k}(\mathbf{D}(\mathbf{v})_{\ell j} + ({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})_{\ell j}) \Bigr ), \end{aligned}$$
(3.11)

where \(\mathbf{F}_1(\mathbf{u}, \rho )|_i\) denotes the \(i\mathrm{th}\) component of \(\mathbf{F}_1(\mathbf{u}, \rho )\).

Moreover,

$$\begin{aligned}&(\mathbf{I}+ \nabla \Psi )\sum _{k=1}^M\int ^{2\pi }_0(\mathbf{u}(\cdot , t), \mathbf{p}_k(\cdot ))_{\Omega _t}\,\mathrm{d}t\, \mathbf{p}_k(x)\\&\quad = (\mathbf{I}+ \nabla \Psi )\sum _{k=1}^M\int ^{2\pi }_0\int _{B_R}(\mathbf{v}(y, t)\cdot \mathbf{p}_k(y + \Psi (y, t))(1 + J_0(t))\,\mathrm{d}y\mathrm{d}t\,\mathbf{p}_k(y+\Psi (y, t))\\&\quad = {\mathcal L}\mathbf{v}_S + \mathbf{F}_2(\mathbf{v}, \rho ) \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \mathbf{F}_2(\mathbf{v}, \rho )&= \sum _{k=1}^M\Bigl \{\int ^{2\pi }_0\int _{B_R}(\mathbf{v}(y, t)\cdot (\mathbf{p}_k(y)J_0(t) + \tilde{\mathbf{p}}_k(\Psi (y, t))(1+J_0(t))\,\mathrm{d}y\mathrm{d}t\mathbf{p}_k(y)\\&\quad + \int ^{2\pi }_0\int _{B_R}\mathbf{v}(y, t)\cdot \mathbf{p}_k(y + \Psi (y, t))(1 + J_0(t))\,\mathrm{d}y\mathrm{d}t \,\tilde{\mathbf{p}}_k(\Psi (y, t)) \\&\quad + \nabla \Psi \int ^{2\pi }_0\int _{B_R}\mathbf{v}(y, t)\cdot \mathbf{p}_k(y+\Psi (y, t))(1 + J_0(t))\,\mathrm{d}y\mathrm{d}t \,\mathbf{p}_k(y + \Psi (y, t)), \end{aligned}\end{aligned}$$
(3.12)

where we have set

$$\begin{aligned}\tilde{\mathbf{p}}_k(\Psi (y, t)) = {\left\{ \begin{array}{ll} 0 &{} \quad \text {for }k=1, \ldots , N, \\ c_{ij}(\Psi _i(y, t)\mathbf{e}_j - \Psi _j(y, t)\mathbf{e}_i) &{} \quad \text {for }k = N+1, \ldots , M. \end{array}\right. } \end{aligned}$$

Thus, setting

$$\begin{aligned} \mathbf{G}(y, t) = (\mathbf{I}+ \nabla \Psi (y, t))\mathbf{f}(y+\Psi (y, t), t), \quad \mathbf{F}(\mathbf{v}, \rho ) = \mathbf{F}_1(\mathbf{v}, \rho ) + \mathbf{F}_2(\mathbf{v}, \rho ), \end{aligned}$$
(3.13)

we have the first equation in equations (2.16).

We next consider the transformation of the boundary conditions. Recall that \(\Gamma _t\) is represented by \(x = y + \rho (y, t)\mathbf{n}(y) + \xi (t)\) for \(y \in S_R\) with \(\mathbf{n}(y)=y/|y|\). Let \(x_0\) be any point on \(S_R\) and let \(\Phi (p)\) be a \(C^\infty \) diffeomorphism on \({\mathbb R}^N\) such that—up to a rotation—it holds

$$\begin{aligned}B_R \cap B_\omega (x_0) =\Phi ( \{p \in {\mathbb R}^N \mid 0< p_N< \omega , \quad \mid |p'| < \omega \}) \cap B_\omega (x_0), \end{aligned}$$

where we have set \(B_\omega (x_0) = \{y \in {\mathbb R}^N \mid |y - x_0| < \omega \}\) and \(p' = (p_1, \ldots , p_{N-1})\). Notice that \(y = \Phi (p', 0) \in S_R \cap B_\omega (x_0)\) and \(\rho (y, t) = H_\rho (\Phi (p',0), t)\). Let \(\{x_k\}_{k=1}^K\) and \(\{\zeta _k\}_{k= 1}^K\) be a finite number of points on \(S_R\) and a partition of unity of \(S_R\) such that \(\mathrm{supp}\, \zeta _k \subset B_\omega (x_k)\) and \(\sum _{k=1}^K\zeta _k(y) = 1\) on \(S_R\). In the following, we represent functions on each \(S_R\cap B_\omega (x_k)\), and to represent functions globally, we use the formula:

$$\begin{aligned} f = \sum _{k=1}^K \zeta ^1_kf \quad \text {in }S_R. \end{aligned}$$
(3.14)

Thus, for the detailed calculations, we only consider the domain \(B_R \cap B_\omega (x_\ell )\) (\(\ell =1, \ldots , K\)), and use the local coordinate system: \(y = \Phi _\ell (p)\) for \(p \in U\), where we have written \(\Phi =\Phi _\ell \), and \(U = \{p \in {\mathbb R}^N \mid 0< p_N< \omega , |p'| < \omega \}\).

We write \(\rho = \rho (y(p_1, \ldots , p_{N-1}, 0), t)\) in the following. By the chain rule, we have

$$\begin{aligned} \frac{\partial \rho }{\partial p_i} = \frac{\partial }{\partial p_i}H_\rho (\Phi _\ell (p_1, \ldots , p_{N-1}, 0), t) = \sum _{m=1}^N\frac{\partial H_\rho }{\partial y_m}\frac{\partial \Phi _{\ell , m}}{\partial p_i}|_{p_N=0}, \end{aligned}$$
(3.15)

where we have set \(\Phi _\ell = {}^\top (\Phi _{\ell ,1}, \ldots , \Phi _{\ell ,N})\), and so, \(\partial \rho /\partial p_i\) is defined in \(B_\omega (x_0)\) by letting

$$\begin{aligned} \frac{\partial \rho }{\partial p_i} = \sum _{m=1}^N\frac{\partial H_\rho }{\partial y_m}\circ \Phi _\ell \frac{\partial \Phi _{\ell ,m}}{\partial p_i}. \end{aligned}$$
(3.16)

We first represent \(\mathbf{n}_t\). Since \(\Gamma _t\) is given by \(x = y + \rho (y, t)\mathbf{n}+ \xi (t)\) for \(y \in S_R\),

$$\begin{aligned}\mathbf{n}_t = a(\mathbf{n}+ \sum _{i=1}^{N-1}b_i\tau _i) \quad \text {with }\tau _i = \frac{\partial }{\partial p_i}y =\frac{\partial }{\partial p_i}\Phi _\ell (p', 0). \end{aligned}$$

The vectors \(\tau _i\) (\(i=1, \ldots , N-1\)) form a basis of the tangent space of \(S_R\) at \(y=y(p_1,\ldots , p_{N-1})\). Since \(|\mathbf{n}_t|^2 =1\), we have

$$\begin{aligned} 1 = a^2(1 + \sum _{i,j=1}^{N-1}g_{ij}b_ib_j) \quad \text {with }g_{ij} = \tau _i\cdot \tau _j \end{aligned}$$
(3.17)

because \(\tau _i\cdot \mathbf{n}=0\). The vectors \(\dfrac{\partial x}{\partial p_i}\) \((i=1, \ldots , N-1)\) form a basis of the tangent space of \(\Gamma _t\), and so \(\mathbf{n}_t\cdot \dfrac{\partial x}{\partial p_i}=0\). Thus, we have

$$\begin{aligned} 0 = a\left( \mathbf{n}+ \sum _{j=1}^{N-1}b_j\tau _j\right) \cdot \left( \frac{\partial y}{\partial p_i} + \frac{\partial \rho }{\partial p_i}\mathbf{n}+ \rho \frac{\partial \mathbf{n}}{\partial p_i}\right) . \end{aligned}$$
(3.18)

Since \(\mathbf{n}\cdot \dfrac{\partial y}{\partial p_i} = \mathbf{n}\cdot \tau _i= 0\), \(\dfrac{\partial \mathbf{n}}{\partial p_i}\cdot \mathbf{n}= 0\) (because of \(|\mathbf{n}|^2=1\)), and \(\dfrac{\partial y}{\partial p_i}\cdot \dfrac{\partial y}{\partial p_j} = \tau _i\cdot \tau _j=g_{ij}\), recalling that \(\mathbf{n}= R^{-1}y= R^{-1}\Phi _\ell \), by (3.18) we have

$$\begin{aligned} \frac{\partial \rho }{\partial p_i} + \sum _{j=1}^{N-1}(1+R^{-1}\rho )g_{ij}b_j = 0. \end{aligned}$$

Let \(G=(g_{ij})\) and \(G^{-1} = (g^{ij})\), and then setting \(\nabla _\Gamma '\rho = (\partial \rho /\partial p_1, \ldots , \partial \rho /\partial p_{N-1})\), we have

$$\begin{aligned} b_i = -(1+R^{-1}\rho )^{-1}\sum _{k=1}^{N-1}g^{ik}\frac{\partial \rho }{\partial p_k}, \quad b = -(1+R^{-1}\rho )^{-1}G^{-1}\nabla '_\Gamma \rho , \end{aligned}$$
(3.19)

which leads to

$$\begin{aligned} \mathbf{n}_t = a\Bigl (\mathbf{n}-(1+R^{-1}\rho )^{-1} \sum _{i,j=1}^{N-1}g^{ij}\frac{\partial \rho }{\partial p_j}\tau _i\Bigr ). \end{aligned}$$
(3.20)

Moreover, combining (3.17) and (3.19), we have

$$\begin{aligned} a = (1 + (1+R^{-1}\rho )^{-2}<G^{-1}\nabla _\Gamma '\rho , \nabla _\Gamma '\rho >)^{-1/2}. \end{aligned}$$

Using the formula:

$$\begin{aligned}(1 + f)^{-1/2} = 1 - \frac{1}{2}\int ^1_0(1 + \theta f)^{-3/2}\,\mathrm{d}\theta \,f,\end{aligned}$$

we have

$$\begin{aligned}a = 1 - V_\Gamma (\rho , \nabla '_\Gamma \rho ) \end{aligned}$$

with

$$\begin{aligned} V_\Gamma (\rho , \nabla '_\Gamma \rho )= & {} \frac{1}{2}\int ^1_0(1+\theta (1+R^{-1}\rho )^{-2}<G^{-1}\nabla '_\Gamma \rho , \nabla '_\Gamma \rho> )^{-3/2}\,\mathrm{d}\theta (1+R^{-1}\rho )^{-2}\\< & {} G^{-1}\nabla '_\Gamma \rho , \nabla '_\Gamma \rho >. \end{aligned}$$

Combining these formulas obtained above gives

$$\begin{aligned} \mathbf{n}_t = \mathbf{n}-\sum _{i,j=1}^{N-1}g^{ij}\frac{\partial \rho }{\partial p_j}\tau _i +\mathbf{V}_\mathbf{n}(\rho , \nabla '_\Gamma \rho ) \end{aligned}$$
(3.21)

where we have set

$$\begin{aligned}&\mathbf{V}_\mathbf{n}(\rho , \nabla '_\Gamma \rho ) = \frac{\rho }{R+\rho }\sum _{i,j=1}^{N-1}g^{ij}\frac{\partial \rho }{\partial p_j}\tau _i\\&- \left( \mathbf{n}- \sum _{i,j=1}^{N-1}(1+R^{-1}\rho )^{-1}g^{ij}\frac{\partial \rho }{\partial p_j}\tau _i\right) V_\Gamma (\rho , \nabla '_\Gamma \rho ). \end{aligned}$$

From (3.16), \(\nabla '_\Gamma \rho \) is extended to \({\mathbb R}^N\) by the formula: \(\nabla '_\Gamma \rho = (\nabla \Phi _\ell )\nabla \Psi _\rho \circ \Phi _\ell \), and so we may write

$$\begin{aligned} \mathbf{V}_\mathbf{n}(\rho , \nabla '_\Gamma \rho ) = \mathbf{V}_{\mathbf{n}, \ell }(\mathbf{k})\bar{\nabla }\Psi _\rho \otimes \bar{\nabla }\Psi _\rho \end{aligned}$$

on \(B_\omega (x_\ell )\) with some function \(\mathbf{V}_{\mathbf{n}, \ell }(\mathbf{k}) = \mathbf{V}_{\mathbf{n},\ell }(y, \mathbf{k})\) defined on \(B_\omega (x_\ell )\times \{\mathbf{k}\mid |\mathbf{k}| \le \delta \}\) with \(\mathbf{V}_{\mathbf{n}, \ell }(0) = 0\) possessing the estimate

$$\begin{aligned}\Vert (\mathbf{V}_{\mathbf{n}, \ell }(\cdot ,\mathbf{k}), \partial _{\mathbf{k}} \mathbf{V}_{\mathbf{n}, \ell }(\cdot , \mathbf{k})) \Vert _{H^1_\infty (B_\omega (x_\ell ))} \le C \end{aligned}$$

with some constant C independent of \(\ell \). Here and in the following \(\mathbf{k}\) are the variables corresponding to \(\bar{\nabla }H_\rho = (H_\rho , \nabla H_\rho )\). In view of (3.21), we have

$$\begin{aligned} \mathbf{n}_t = \mathbf{n}- \sum _{i,j=1}^{N-1}g^{ij}\tau _i\frac{\partial \rho }{\partial p_j} +\mathbf{V}_{\mathbf{n}, \ell }(\mathbf{k})\bar{\nabla }\Psi _\rho \otimes \bar{\nabla }\Psi _\rho \quad \text {on }B_\omega (x_\ell ) \cap S_R. \end{aligned}$$
(3.22)

Thus, in view of (3.14) and (3.16), we may write

$$\begin{aligned} \mathbf{n}_t = \mathbf{n}- \sum _{i,j=1}^{N-1}g^{ij}\partial '_j\rho \tau _i + \mathbf{V}_{\mathbf{n}}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \otimes \bar{\nabla }H_\rho \quad \text {on }S_R, \end{aligned}$$
(3.23)

where \(\partial '_j\rho = \partial \rho /\partial p_j\) locally on \(B_\omega (x_\ell ) \cap S_R\), \(\bar{\nabla }H_\rho = (H_\rho , \nabla H_\rho )\), and \(\mathbf{V}_{\mathbf{n}}(\mathbf{k})\) is a matrix of functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:

$$\begin{aligned} \Vert (\mathbf{V}_{\mathbf{n}}, \partial _{\mathbf{k}}\mathbf{V}_{\mathbf{n}})(\cdot , \mathbf{k})\Vert _{H^1_\infty (B_R)} \le C \quad \text {for }|\bar{\mathbf{k}}| \le \delta . \end{aligned}$$
(3.24)

And also we may write

$$\begin{aligned} \mathbf{n}_t = \mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \end{aligned}$$
(3.25)

where \(\tilde{\mathbf{V}}_{\mathbf{n}}(\mathbf{k})\) is a matrix of functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:

$$\begin{aligned} \Vert (\tilde{\mathbf{V}}_\mathbf{n}(\cdot , \mathbf{k}), \partial _{\mathbf{k}}\tilde{\mathbf{V}}_\mathbf{n}(\cdot , \mathbf{k}))\Vert _{H^1_\infty (B_R)} \le C \quad \text {for }|\mathbf{k}| \le \delta . \end{aligned}$$
(3.26)

We now consider the boundary condition:

$$\begin{aligned} (\mu \mathbf{D}(\mathbf{u}) - {\mathfrak {p}}\mathbf{I})\mathbf{n}_t = \sigma H(\Gamma _t)\mathbf{n}_t-p_0\mathbf{n}_t \end{aligned}$$
(3.27)

It is convenient to divide the formula in (3.27) into the tangential part and normal part on \(\Gamma _t\) as follows:

$$\begin{aligned}&\displaystyle \Pi _t\mu \mathbf{D}(\mathbf{u})\mathbf{n}_t = 0, \end{aligned}$$
(3.28)
$$\begin{aligned}&\displaystyle<\mu \mathbf{D}(\mathbf{v})\mathbf{n}_t, \mathbf{n}_t> - {\mathfrak {p}}= \sigma <H(\Gamma _t)\mathbf{n}_t, \mathbf{n}_t>-p_0 = h_N(\mathbf{v}, \rho ) \end{aligned}$$
(3.29)

Here, \(\Pi _t\) is defined by \(\Pi _t\mathbf{d}= \mathbf{d}- < \mathbf{d}, \mathbf{n}_t>\mathbf{n}_t\) for any N-vector of functions \(\mathbf{d}\). In the last equation in equations (2.16), we set \(\mathbf{h}'(\mathbf{v}, \rho ) = \mathbf{h}(\mathbf{v}, \rho ) - <\mathbf{h}(\mathbf{v}, \rho ), \mathbf{n}>\mathbf{n}\) and \(h_N(\mathbf{v}, \rho ) = <\mathbf{h}(\mathbf{v}, \rho ), \mathbf{n}>\). By (3.25) and (3.3), we see that the boundary condition (3.28) is transformed to the following formula:

$$\begin{aligned} (\mu \mathbf{D}(\mathbf{v})\mathbf{n})_\tau = \mathbf{h}'(\mathbf{v}, \rho ) \quad \text {on }\Gamma \times (0, T), \end{aligned}$$
(3.30)

where we have set \(\mathbf{d}_\tau = \mathbf{d}- <\mathbf{d}, \mathbf{n}>\mathbf{n}\) and

$$\begin{aligned} \begin{aligned}&\mathbf{h}'(\mathbf{v}, \rho ) = - \mu \mathbf{D}(\mathbf{v})\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \\&\quad +\mu \{<\mathbf{D}(\mathbf{v})\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho , \mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho> (\mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho )\\&\quad +<\mathbf{D}(\mathbf{v})\mathbf{n}, \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho> (\mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ) \\&\quad +<\mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}>\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \} -\mu ({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})(\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ) \\&\quad -\mu <({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v})(\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ), \mathbf{n}\\&\quad +\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho > (\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ). \end{aligned} \end{aligned}$$
(3.31)

Finally, we derive the nonlinear term \(h_N(\mathbf{u}, \rho )\) in (3.29). Recall that \(\Gamma _t\) is represented by \(x = (R + \rho )\mathbf{n}(y) + \xi (t)\) for \(y \in S_R\), where \(\mathbf{n}= y/|y| \in S_1\). Then, we have

$$\begin{aligned} \frac{\partial x}{\partial p_j}= (R+\rho )\tau _j + \frac{\partial \rho }{\partial p_j}\mathbf{n}\end{aligned}$$

where \(\tau _j = \frac{\partial \mathbf{n}}{\partial p_j}\), which forms a basis of the tangent space of \(S_1\). Since \(\tau _j\cdot \mathbf{n}= 0\), the \((i, j)\mathrm{th}\) component of the first fundamental form \(G_t=(g_{tij})\) of \(\Gamma _t\) is given by

$$\begin{aligned} g_{tij}=\frac{\partial x}{\partial p_i}\cdot \frac{\partial x}{\partial p_j} = (R+\rho )^2g_{ij} + \frac{\partial \rho }{\partial p_i}\frac{\partial \rho }{\partial p_j}, \end{aligned}$$

where \(g_{ij} = \tau _i\cdot \tau _j\) is the \((i, j)\mathrm{th}\) element of the first fundamental form, G, of \(S_1\), and so

$$\begin{aligned} G_t&= (R+\rho )^2(G+(R+\rho )^{-2}\nabla '_\Gamma \rho \otimes \nabla '_\Gamma \rho ) \\&= (R+\rho )^2G(\mathbf{I}+ (R+\rho )^{-2}(G^{-1}\nabla '_\Gamma \rho )\otimes \nabla '_\Gamma \rho ). \end{aligned}$$

Since

$$\begin{aligned} \det (\mathbf{I}+ \mathbf{a}'\otimes \mathbf{b}') = 1 + \mathbf{a}'\cdot \mathbf{b}', \quad (\mathbf{I}+ \mathbf{a}'\otimes \mathbf{b}')^{-1} = \mathbf{I}- \frac{\mathbf{a}'\otimes \mathbf{b}'}{1+\mathbf{a}'\cdot \mathbf{b}'} \end{aligned}$$
(3.32)

for any \((N-1)\)-vectors \(\mathbf{a}'\) and \(\mathbf{b}' \in {\mathbb R}^{N-1}\), we have

$$\begin{aligned}G_t^{-1}&=(R+\rho )^{-2} \Bigl (\mathbf{I}- \frac{(R+\rho )^{-2}(G^{-1}\nabla '_\Gamma \rho )\otimes \nabla '_\Gamma \rho }{1 + (R+\rho )^{-2}<G^{-1}\nabla '_\Gamma \rho , \nabla '_\Gamma \rho >} \Bigr )G^{-1} \\&= (R+\rho )^{-2}G^{-1} + O_2. \end{aligned}$$

Here and in the following, \(O_2\) denotes a symbol defined by setting

$$\begin{aligned}O_2 = a_0H_\rho ^2 + \sum _{j=1}^Nb_jH_\rho \frac{\partial H_\rho }{\partial y_j} + \sum _{i,j=1}^N c_{ij}\frac{\partial H_\rho }{\partial y_i}\frac{\partial H_\rho }{\partial y_j} \end{aligned}$$

with some coefficients \(a_0\), \(b_j\) and \(c_{ij}\) defined on \(\overline{B_R}\) satisfying the estimate: \(|(a_0, b_j, c_{ij})(y, t)| \le C\) and \(|\nabla (a_0, b_j, c_{ij})(y, t)| \le C|\nabla ^2 H_\rho (y, t)|\) provided that \(\Vert H_\rho \Vert _{L_\infty ((0, 2\pi ), H^1_\infty (B_R))} \le \delta \). In particular,

$$\begin{aligned}g_t^{ij} = (R+\rho )^{-2}g^{ij} + O_2, \end{aligned}$$

componentwise.

We next calculate the Christoffel symbols of \(\Gamma _t\). Since

$$\begin{aligned} \tau _{ti}&= (R+\rho )\tau _i + \frac{\partial \rho }{\partial p_i}\mathbf{n}, \\ \tau _{tij}&= (R+\rho )\tau _{ij} + \frac{\partial \rho }{\partial p_j}\tau _i + \frac{\partial \rho }{\partial p_i}\tau _j + \frac{\partial ^2\rho }{\partial p_i\partial p_j}\mathbf{n}, \end{aligned}$$

we have

$$\begin{aligned}<\tau _{tij}, \tau _{t\ell }>&= (R+\rho )^2<\tau _{ij}, \tau _\ell > + (R+\rho )\left( \frac{\partial \rho }{\partial p_\ell }\ell _{ij} + g_{i\ell }\frac{\partial \rho }{\partial p_j} + g_{j\ell }\frac{\partial \rho }{\partial p_i}\right) \\&\quad + \frac{\partial ^2\rho }{\partial p_i\partial p_j}\frac{\partial \rho }{\partial p_\ell }, \end{aligned}$$

where \(\ell _{ij} = <\tau _{ij}, \mathbf{n}>\), and so

$$\begin{aligned} \Lambda ^k_{tij}&= g_t^{k\ell }<\tau _{tij}, \tau _{t\ell }> \\&=\bigg ((R+\rho )^{-2}g^{k\ell } + O_2\bigg ) \bigg ( (R+\rho )^2<\tau _{ij}, \tau _\ell >\\&\quad + (R+\rho )(\frac{\partial \rho }{\partial p_\ell }\ell _{ij} + g_{i\ell }\frac{\partial \rho }{\partial p_j} + g_{j\ell }\frac{\partial \rho }{\partial p_i}) + \frac{\partial ^2\rho }{\partial p_i\partial p_j}\frac{\partial \rho }{\partial p_\ell }\bigg )\\&= \Lambda ^k_{ij} + (R+\rho )^{-1}g^{k\ell }(\frac{\partial \rho }{\partial p_\ell }\ell _{ij} + \delta ^k_i\frac{\partial \rho }{\partial p_j} + \delta ^k_j\frac{\partial \rho }{\partial p_i})\\&\quad +((R+\rho )^{-2}g^{k\ell }\frac{\partial \rho }{\partial p_\ell } + O_2)\frac{\partial ^2\rho }{\partial p_i \partial p_j} + O_2. \end{aligned}$$

Thus,

$$\begin{aligned}&\Delta _{\Gamma _t}f = g_t^{ij}(\partial _i\partial _jf- \Lambda ^k_{tij}\partial _kf) \\&\quad =(R+\rho )^{-2}g^{ij}(\partial _i\partial _jf - \Lambda ^k_{ij}\partial _kf) +(A^k(\nabla '_p\rho ,\nabla '^2_p\rho )\partial _kf + O_2\otimes (\bar{\nabla }'^2f) \end{aligned}$$

where \(\bar{\nabla }'^2f\) is an \(((N-1)^2+N)\)-vector of the form: \(\bar{\nabla }'^2f = (\partial _i\partial _jf, \partial _if, f \mid i, j=1, \ldots , N-1)\), \(\partial _i = \partial /\partial p_i\), \(\nabla '^2_p = (\partial _i\partial _j\rho \mid i, j=1, \ldots , N-1)\), and

$$\begin{aligned} A^k(\nabla '_p\rho , \nabla '^2_p\rho )&= -(R+\rho )^{-3}g^{ij}g^{k\ell }\left( \frac{\partial \rho }{\partial p_\ell }\ell _{ij} + \delta ^k_i\frac{\partial \rho }{\partial p_j} +\delta ^k_j\frac{\partial \rho }{\partial p_i}\right) \\&\quad -(R+\rho )^{-2}\left( (R+\rho )^{-2}g^{ij}g^{k\ell }\frac{\partial \rho }{\partial p_\ell } + g^{ij}O_2\right) \frac{\partial ^2\rho }{\partial p_i\partial p_j}, \end{aligned}$$

and so

$$\begin{aligned} H(\Gamma _t)\mathbf{n}_t&= \Delta _{\Gamma _t}[(R+\rho )\mathbf{n}+ \xi (t)] \\&=(R+\rho )^{-2}g^{ij}(\partial _i\partial _j - \Lambda ^k_{ij}\partial _k)((R+\rho )\mathbf{n}) + (A^k\nabla ^2_p\rho )\partial _k((R+\rho )\mathbf{n}) \\&\qquad + O_2\otimes \bar{\nabla }'^2((R+\rho )\mathbf{n})\\&= (R+\rho )^{-1}g^{ij}(\partial _i\partial _j\mathbf{n}-\Lambda ^k_{ij}\partial _k\mathbf{n}) + (R+\rho )^{-2}g^{ij}(\partial _i\rho \partial _j\mathbf{n}+ \partial _j\rho \partial _i\mathbf{n})\\&\quad + (R+\rho )^{-2}g^{ij}(\partial _i\partial _j\rho -\Lambda ^k_{ij}\partial _k\rho )\mathbf{n}+ A^k(\nabla '_p\rho , \nabla '^2_p\rho )(\partial _k\rho )\mathbf{n}\\&\quad + A^k(\nabla '_p\rho , \nabla '^2_p\rho )(R+\rho )\partial _k\mathbf{n}+ O_2\otimes \bar{\nabla }'^2(R+\rho ) \end{aligned}$$

Combining this formula with (3.21), using \(<\partial _i\mathbf{n}, \mathbf{n}> = 0\), \(<\mathbf{n}, \tau _\ell > = 0\), \(\Delta _{S_1}\mathbf{n}= -(N-1)\mathbf{n}\), and (3.15) gives

$$\begin{aligned}&<H(\Gamma _t)\mathbf{n}_t, \mathbf{n}_t> \\&\quad = -(R+\rho )^{-1}(N-1) + (R+\rho )^{-2} \Delta _{S_1}\rho + (O_1+O_2)\otimes \nabla ^2_p\rho + O_2, \end{aligned}$$

where \(O_1\) denotes a symbol defined by setting

$$\begin{aligned}O_1 = a_0'H_\rho + \sum _{j=1}^Nb_j' \frac{\partial H_\rho }{\partial y_j}\end{aligned}$$

with some coefficients \(a_0'\) and \(b_j'\) defined on \(\overline{B_R}\) satisfying the estimate: \(|(a_0', b_j')(y, t)|\le C\) and \(|\nabla (a_0', b_j')(y, t)| \le C|\nabla ^2H_\rho (y, t)|\) provided that \(\Vert H_\rho \Vert _{L_\infty ((0, 2\pi ), H^1_\infty (B_R))} \le \delta \). Since

$$\begin{aligned} (R+\rho )^{-1}&=R^{-1} - \rho R^{-2} + O(\rho ^2), \\ (R+\rho )^{-2}\Delta _{S_1}\rho&= R^{-2}\Delta _{S_1}\rho +2R^{-3}\rho \Delta _{S_1}\rho + O_2\otimes \nabla ^2_p\rho , \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned}&<H(\Gamma _t)\mathbf{n}_t, \mathbf{n}_t> = -\frac{N-1}{R} + {\mathcal B}\rho + (O_1+ O_2)\otimes \nabla ^2_p\rho + O_2. \end{aligned} \end{aligned}$$
(3.33)

Setting \(p_0 = -(N-1)/R\), from (3.27) we have

$$\begin{aligned} <\mu \mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}> - {\mathfrak {q}}- \sigma {\mathcal B}\rho = h_N(\mathbf{v}, \rho ) \end{aligned}$$

on \(S_R\times (0, 2\pi ) \). Here, in view of (3.3) and (3.33), we have defined \(h_N(\mathbf{v}, \rho )\) by letting

$$\begin{aligned} h_N(\mathbf{v}, \rho ) = \mathbf{V}_{h,N}(\bar{\nabla }H_\rho ) \bar{\nabla }H_\rho \otimes \nabla \mathbf{v}+ \sigma \tilde{\mathbf{V}}'_\Gamma (\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \otimes \bar{\nabla }^2 H_\rho , \end{aligned}$$
(3.34)

where \(\mathbf{V}_{h, N}(\mathbf{k})\) and \(\tilde{\mathbf{V}}'_\Gamma (\mathbf{k})\) are functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:

$$\begin{aligned}&\sup _{|\mathbf{k}|< \delta }\Vert (\mathbf{V}_{h, N}(\cdot , \mathbf{k}), \partial _{\mathbf{k}}\mathbf{V}_{h, N}(\cdot , \mathbf{k}))\Vert _{H^1_\infty (B_R)} \le C, \\&\sup _{|\mathbf{k}| < \delta }\Vert (\tilde{\mathbf{V}}'_{\Gamma }(\cdot , \mathbf{k}), \partial _{\mathbf{k}}\tilde{\mathbf{V}}'_{\Gamma }(\cdot , \mathbf{k}))\Vert _{H^1_\infty (B_R)} \le C \end{aligned}$$

for some constant C.

Two-phase problem case

Let \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\). In the two-phase case, we let

$$\begin{aligned}\Psi _+(y, t) =R^{-1}H_\rho (y, t)y + \xi (t), \quad \Psi _-(y, t) = \varphi (y)(R^{-1}H_\rho (y,t) y + \xi (t)). \end{aligned}$$

Let \(J_\pm (t)\) be the Jacobian of the map: \(x = y+ \Psi _\pm (y, t)\) for \(y \in \Omega _\pm \), which are defined by setting

$$\begin{aligned} \left\{ \begin{aligned} J_+(t)&= \det (I + R^{-1}\nabla _y (H_\rho (y, t)y)) \quad&\text {for }y \in \Omega _+, \\ J_-(t)&=\det (I + \nabla _y(\varphi (y)(R^{-1}(H_\rho (y, t)y + \xi (t))) \quad&\text {for }y \in \Omega _-. \end{aligned}\right. \end{aligned}$$
(3.35)

Notice that

$$\begin{aligned}\xi (t) = \int ^t_0\int _{B_R}\mathbf{v}_+(y, s)J_{+}(s)\,\mathrm{d}y\mathrm{d}s + c\end{aligned}$$

where c is the unique constant for which the following equality holds:

$$\begin{aligned}\int ^{2\pi }_0\xi (t) = 0.\end{aligned}$$

We assume that

$$\begin{aligned} \sup _{t \in (0, 2\pi )}\Vert H_\rho (\cdot , t)\Vert _{H^1_\infty (\Omega _\pm )} \le \delta , \quad \sup _{t \in (0, 2\pi )} |\xi (t)| \le \delta \end{aligned}$$
(3.36)

with suitably small constant \(\delta > 0\). Since

$$\begin{aligned}|\xi (t)| \le C\sup _{t \in (0, 2\pi )}\Vert \mathbf{v}(\cdot , t)\Vert _{L_q(B_R)}\sup _{t \in (0, 2\pi )} |J_+(t)||B_R|, \end{aligned}$$

there exists a constant \(\delta _1 > 0\) such that if

$$\begin{aligned} \sup _{t \in (0, 2\pi )}\Vert \mathbf{v}_+(\cdot , t)\Vert _{L_q(B_R)} \le \delta _1 \end{aligned}$$
(3.37)

then the condition for \(\xi (t)\) in (3.36) holds. Thus, in the proof of Theorem 5, we assume that the conditions (3.36) and (3.37) hold.

Set \(J_{0\pm }(t) = J_\pm (t) - 1\). By the chain rule, we have

$$\begin{aligned}\nabla _x = (\mathbf{I}+ \mathbf{V}_{\pm 0}(\mathbf{k}_\pm ))\nabla _y, \quad \frac{\partial }{\partial x_i} + \sum _{j=1}^N V_{\pm 0ij}(\mathbf{k}_\pm )\frac{\partial }{\partial y_j} \end{aligned}$$

where \(\mathbf{V}_{\pm 0}(\mathbf{k}_\pm )\) is given by

$$\begin{aligned}\mathbf{V}_{\pm 0}(\mathbf{k}_\pm ) = \left\{ \begin{aligned}&(\mathbf{I}+ \nabla _y (R^{-1}H_\rho (y, t) y)^{-1} - \mathbf{I}&\quad&\text {for }y \in \Omega _+, \\&(\mathbf{I}+ \nabla _y\Psi _{-, \rho }(y, t))^{-1} - \mathbf{I}&\quad&\text {for }y \in \Omega _-. \end{aligned}\right. \end{aligned}$$

Here and in the following, \(\mathbf{k}_+\) and \(\mathbf{k}_-\) denote the variables corresponding to \((H_\rho , \nabla H_\rho )\) and \((\Psi _{-, \rho }, \nabla \Psi _{-, \rho })\).

Employing the same argument as for obtaining the formulas in (3.6), we have

$$\begin{aligned} \begin{aligned} g_\pm (\mathbf{v}, \rho )&= -(J_{0\pm }(\mathbf{k}_\pm )\mathrm{div}\,\mathbf{v}_\pm + (1 + J_{0\pm }(\mathbf{k}_\pm ))\mathbf{V}_{0\pm }(\mathbf{k}_\pm ) :\nabla \mathbf{v}_\pm ), \\ \mathbf{g}_\pm (\mathbf{v}, \rho )&= -(1+J_{0\pm }(\mathbf{k}_\pm )){}^\top \mathbf{V}_{0\pm }(\mathbf{k}_\pm )\mathbf{v}_\pm . \end{aligned}\end{aligned}$$
(3.38)

And also, from (3.13) we have

$$\begin{aligned}&\mathbf{G}_\pm (y, t) = (\mathbf{I}+ \nabla \Psi _\pm (y, t))\mathbf{f}(y + \Psi _\pm (y, t), t),\nonumber \\&\quad \mathbf{F}_\pm (\mathbf{v}, \rho ) = {}^\top (F_{1\pm }(\mathbf{v}, \rho ), \ldots , F_{N\pm }(\mathbf{v}, \rho )) \end{aligned}$$
(3.39)

with

$$\begin{aligned} F_{i\pm }(\mathbf{v}, \rho )&= -\sum _{j,k=1}^N(v_{\pm j} - \frac{\partial \Psi _{\pm j}}{\partial t}) (\delta _{jk} + V_{0jk}(\mathbf{k}_\pm ))\frac{\partial v_{\pm i}}{\partial y_k} \\&\quad -\sum _{\ell =1}^N\partial _i\Psi _{\pm \ell } \Bigl (\frac{\partial v_{\pm \ell }}{\partial t} + \sum _{j,k=1}^N(v_{\pm j} - \frac{\partial \Psi _{\pm j}}{\partial t}) (\delta _{jk} + V_{\pm 0jk}(\mathbf{k}_\pm ))\frac{\partial v_{\pm \ell }}{\partial y_k}\Bigr ) \\&\quad + \mu \Bigl (\sum _{j=1}^N \frac{\partial }{\partial y_j}({\mathcal D}_\mathbf{D}(\mathbf{k}_\pm )\nabla \mathbf{v}_\pm )_{ij} + \sum _{j,k=1}^NV_{0jk}(\mathbf{k}_\pm )\frac{\partial }{\partial y_k}(\mathbf{D}(\mathbf{v}_\pm )_{ij} + ({\mathcal D}_\mathbf{D}(\mathbf{k}_\pm )\nabla \mathbf{v}_\pm )_{ij}) \\&\quad + \sum _{j,k,\ell =1}^N\partial _i\Psi _{\pm \ell }(\delta _{jk} + V_{\pm 0jk}(\mathbf{k})) \frac{\partial }{\partial y_k}(\mathbf{D}(\mathbf{v}_\pm )_{\ell j} + ({\mathcal D}_\mathbf{D}(\mathbf{k}_\pm )\nabla \mathbf{v}_\pm )_{\ell j}) \Bigr ). \end{aligned}$$

Here and in the following, we have set \(\Psi _{\pm }(y, t) = {}^\top (\Psi _{\pm 1}(y, t), \ldots , \Psi _{\pm N}(y, t))\), \(\mathbf{v}_\pm = {}^\top (v_{\pm 1}, \ldots , v_{\pm N})\), and

$$\begin{aligned} ({\mathcal D}_\mathbf{D}(\mathbf{k}_\pm )\nabla \mathbf{v}_\pm )_{ij} = \sum _{k=1}^N \Bigl (V_{\pm 0jk}(\mathbf{k}_\pm )\frac{\partial v_{\pm i}}{\partial y_k} + V_{\pm 0ik}(\mathbf{k}_\pm )\frac{\partial v_{\pm j}}{\partial y_k}\Bigr ). \end{aligned}$$

To define the right hand side of the transmission condition, we use (3.31) and (3.34). We first introduce a symbol \(((\cdot ))\). For \(f_\pm \), let \([f_\pm ]\) be a suitable extension of \(f_\pm \) to \(\Omega _{\mp }\) such that

$$\begin{aligned}\Vert [f_\pm ]\Vert _{H^k_q(\Omega _{\mp })} \le C_k\Vert f_\pm \Vert _{H^k_q(\Omega _\pm )}, \quad \Vert \partial _t[f_\pm ]\Vert _{H^k_q(\Omega _{\mp })} \le C_k\Vert \partial _tf_\pm \Vert _{H^k_q(\Omega _\pm )} \end{aligned}$$

with some constant \(C_k\). Here, if the right-hand side is finite, then \([f_\pm ]\) and \(\partial _t[f_\pm ]\) exist and the estimates above hold. In particular, we set \(H^0_q(\Omega _\pm ) = L_q(\Omega _\pm )\). We set

$$\begin{aligned} ex[f_\pm ](y, t) = {\left\{ \begin{array}{ll} f_\pm (y, t) &{} \quad \text {for }y \in \Omega _\pm , \\ {[}f_\pm ](y, t) &{} \quad \text {for }y \in \Omega _{\mp }. \end{array}\right. } \end{aligned}$$

And then, ((f)) is defined by setting

$$\begin{aligned}((f)) = ex[f_+] - ex[f_-]. \end{aligned}$$

Using this symbol, we can proceed as for the derivation of (3.31) and (3.34) and define \(\tilde{\mathbf{h}}'(\mathbf{v}, \rho )\) and \(\tilde{h}_N(\mathbf{v}, \rho )\) by setting

$$\begin{aligned} \begin{aligned}&\tilde{\mathbf{h}}'(\mathbf{v}, \rho ) = - \mu ((\mathbf{D}(\mathbf{v})))\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \\&\quad +\mu \{<((\mathbf{D}(\mathbf{v})))\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho , \mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho> (\mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho )\\&\quad +<((\mathbf{D}(\mathbf{v})))\mathbf{n}, \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho> (\mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ) \\&\quad +<((\mathbf{D}(\mathbf{v})))\mathbf{n}, \mathbf{n}>\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \} -\mu (({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v}))(\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ) \\&\quad -\mu <(({\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v}))(\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ), \mathbf{n}+\tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho > (\mathbf{n}+ \tilde{\mathbf{V}}_\mathbf{n}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho ) \\&\tilde{h}_N(\mathbf{v}, \rho ) = \mathbf{V}_{h,N}(\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \otimes ((\nabla \mathbf{v})) + \sigma \tilde{\mathbf{V}}'_\Gamma (\bar{\nabla }H_\rho )\bar{\nabla }H_\rho \otimes \bar{\nabla }^2 H_\rho . \end{aligned}\end{aligned}$$
(3.40)

And then, we set \(\tilde{\mathbf{h}}(\mathbf{v}, \rho ) =(\tilde{\mathbf{h}}'(\mathbf{v}, \rho ), \tilde{h}_N(\mathbf{v}, \rho ))\).

On periodic solutions of the linearized equations

In this section, we shall prove the \(L_p\)\(L_q\) maximal regularity of \(2\pi \)-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:

$$\begin{aligned} \begin{aligned} \partial _t\mathbf{u}+ {\mathcal L}\mathbf{u}_S-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I}) = \mathbf{F}&\quad&\text {in }B_R\times (0, 2\pi ), \\ \mathrm{div}\,\mathbf{u}=G = \mathrm{div}\,\mathbf{G}&\quad&\text {in }B_R\times (0, 2\pi ), \\ \partial _t\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{u})\cdot \mathbf{n}=D&\quad&\text {on }S_R\times (0, 2\pi ),\\ (\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}- ({\mathcal B}_R\rho )\mathbf{n}=\mathbf{H}&\quad&\text {on }S_R\times (0, 2\pi ), \end{aligned} \end{aligned}$$
(4.1)

where \({\mathcal L}\), \({\mathcal M}\), and \({\mathcal A}\) are the linear operators defined in (2.17). We shall prove the unique existence theorem of \(2\pi \)-periodic solutions of equations (4.1). Our main result is this section is stated as follows.

Theorem 6

Let \(1< p, q < \infty \). Then, for any \(\mathbf{F}\), D, G, \(\mathbf{G}\) and \(\mathbf{H}\) with

$$\begin{aligned} \mathbf{F}&\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \quad D \in L_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)) \\ G&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)) \cap H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)), \quad \mathbf{G}\in H^1_{p, \mathrm{per}}((0, 2\pi ). L_q(B_R)^N), \\ \mathbf{H}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)^N) \cap H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \end{aligned}$$

problem (4.1) admits unique solutions \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) with

$$\begin{aligned} \mathbf{u}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(B_R)^N) \cap H^1_{p, \mathrm{per}} ((0, 2\pi ), L_q(B_R)^N), \\ {\mathfrak {p}}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)), \\ \rho&\in L_{p, \mathrm{per}}((0, 2\pi ), W^{3-1/q}_q(S_R)) \cap H^1_{p, \mathrm{per}} ((0, 2\pi ), W^{2-1/q}_q(S_R)) \end{aligned}$$

possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{u}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{u}\Vert _{L_p((0, 2\pi ), L_q(B_R))} + \Vert \nabla {\mathfrak {p}}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad \qquad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} \\&\qquad \le C\{\Vert \mathbf{F}\Vert _{L_p((0, 2\pi ), L_q(B_R))} + \Vert D\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \Vert \partial _t\mathbf{G}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad \qquad + \Vert (G, \mathbf{H})\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} + \Vert (G, \mathbf{H})\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))}\} \end{aligned}\end{aligned}$$
(4.2)

for some constant \(C > 0\).

To prove Theorem 6, our approach is to use the \({\mathcal R}\)-solver, Weis’ operator-valued Fourier multiplier theorem [22] and a transference theorem, which is created in Eiter, Kyed and Shibata [2]. To introduce the notion of \({\mathcal R}\)-solver, we introduce the \({\mathcal R}\)-boundedness of operator families.

Definition 7

Let X and Y be two Banach spaces. A family of operators \({\mathcal T}\subset {\mathcal L}(X, Y)\) is called \({\mathcal R}\)-bounded on \({\mathcal L}(X, Y)\), if there exist a constant \(C > 0\) and \(p \in [1, \infty )\) such that for each \(n \in {\mathbb N}\), \(\{T_j\}_{j=1}^n \in {\mathcal T}^n\), and \(\{f_j\}_{j=1}^n \in X^n\), we have

$$\begin{aligned}\Vert \sum _{k=1}^n r_kT_kf_k\Vert _{L_p((0,1), Y)} \le C\Vert \sum _{k=1}^nr_kf_k\Vert _{L_p((0, 1), X)}. \end{aligned}$$

Here, the Rademacher functions \(r_k\), \(k \in {\mathbb N}\), are given by \(r_k : [0, 1] \rightarrow \{-1, 1\}\), \(t \mapsto \mathrm{sign}\,(\sin 2^k\pi t)\). The smallest such C is called \({\mathcal R}\)-bound of \({\mathcal T}\) on \({\mathcal L}(X, Y)\), which is denoted by \({\mathcal R}_{{\mathcal L}(X, Y)}{\mathcal T}\).

We quote Weis’ operator-valued Fourier multiplier theorem and the transference theorem for operator-valued Fourier multipliers.

Theorem 8

[Weis] Let X and Y be two UMD Banach spaces. Let \(m \in C^1({\mathbb R}{\setminus }\{0\}, {\mathcal L}(X, Y))\) satisfies the multiplier condition:

$$\begin{aligned} {\mathcal R}_{{\mathcal L}(X, Y)}\{ (\tau \partial _\tau )^\ell m(\tau ) \mid \tau \in {\mathbb R}{\setminus }\{0\}\} \le r_b \end{aligned}$$

for \(\ell = 0, 1\) with some constant \(r_b\). Let \(T_m\) be a multiplier defined by \(T_m[f] = {\mathcal F}^{-1}[m{\mathcal F}[f]]\). Then, \(T_m \in {\mathcal L}(L_p({\mathbb R}, X), L_p({\mathbb R}, Y))\) with

$$\begin{aligned}\Vert T_m[f]\Vert _{L_p({\mathbb R}, Y)} \le C_pr_b\Vert f\Vert _{L_p({\mathbb R}, X)}\end{aligned}$$

for any \(p \in (1, \infty )\) with some constant \(C_p\) depending on p but independent of \(r_b\).

The transference theorem for operator-valued Fourier multipliers obtained in [2] is stated as follows.

Theorem 9

Let X and Y be two Banach spaces and \(p \in (1, \infty )\). Assume that Y is reflexive. Let

$$\begin{aligned}m \in L_\infty ({\mathbb R}, {\mathcal L}(X, Y)) \cap C({\mathbb R}, {\mathcal L}(X, Y)),\end{aligned}$$

and let \(m|_{\mathbb T}\) denote the restriction of m on \({\mathbb T}\). We define multipliers on \({\mathbb R}\) and \({\mathbb T}\) associated with m by setting

$$\begin{aligned} T_{m, {\mathbb R}}[f](t) = {\mathcal F}^{-1}[m{\mathcal F}[f]], \quad T_{m, {\mathbb T}}[f] = {\mathcal F}^{-1}_{\mathbb T}[m|_{\mathbb T}{\mathcal F}_{\mathbb T}[f]]. \end{aligned}$$

If \(T_{m, {\mathbb R}} \in {\mathcal L}(L_p({\mathbb R}, X), L_p({\mathbb R}, Y))\) possessing the estimate:

$$\begin{aligned}\Vert T_{m, {\mathbb R}}[f]\Vert _{L_p({\mathbb R}, Y)} \le M\Vert f\Vert _{L_p({\mathbb R}, X)}\end{aligned}$$

for any \(f \in L_p({\mathbb R}, X)\) with some constant M, then \(T_{m, {\mathbb T}} \in {\mathcal L}(L_p({\mathbb T}, X), L_p({\mathbb T}, Y))\) and

$$\begin{aligned}\Vert T_{m, {\mathbb T}}[f]\Vert _{L_p({\mathbb T}, Y)} \le C_pM\Vert f\Vert _{L_p({\mathbb T}, X)} \end{aligned}$$

for any \(f \in L_{p}({\mathbb T}, X)\) 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 [1], and so this theorem is an extension to the operator-valued case.

We now consider the \({\mathcal R}\)-solver of the generalized resolvent problem:

$$\begin{aligned} \begin{aligned} ik\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v})-{\mathfrak {q}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ \mathrm{div}\,\mathbf{v}=g = \mathrm{div}\,\mathbf{g}&\quad&\text {in }B_R, \\ ik\eta + {\mathcal M}\eta -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=d&\quad&\text {on }S_R,\\ (\mu \mathbf{D}(\mathbf{v})-{\mathfrak {q}}\mathbf{I})\mathbf{n}- ({\mathcal B}_R\eta )\mathbf{n}=\mathbf{h}&\quad&\text {on }S_R \end{aligned}\end{aligned}$$
(4.3)

for \(k \in {\mathbb R}\). From Theorem 4.8 in Shibata [18] (cf. also Shibata [15, 16]) we know the following theorem concerned with the existence of an \({\mathcal R}\)-solver of problem (4.1).

Theorem 11

Let \(1< q < \infty \) and let \({\mathbb R}_{k_0} = {\mathbb R}{\setminus }(-k_0, k_0)\). Let

$$\begin{aligned} X_q(B_R)&= \{(\mathbf{f}, d, \mathbf{h}, g, \mathbf{g}) \mid \mathbf{f}\in L_q(B_R)^N, d \in W^{2-1/q}_q(S_R), \\&\mathbf{h}\in H^1_q(B_R)^N, g \in H^1_q(B_R), \mathbf{g}\in L_q(B_R)^N\}, \\ {\mathcal X}_q(B_R)&= \{F=(F_1, F_2, \ldots , F_7) \mid F_1, F_3, F_7 \in L_q(B_R)^N,\\&F_2 \in W^{2-1/q}_q(S_R), F_4 \in H^1_q(B_R)^N, \\&\,\, F_5 \in L_q(B_R), F_6 \in H^1_q(B_R)\}. \end{aligned}$$

Then, there exist a constant \(k_0 > 0\) and operator families \({\mathcal A}(ik)\), \({\mathcal P}(ik)\), and \({\mathcal H}(ik)\) with

$$\begin{aligned} {\mathcal A}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(B_R), H^2_q(B_R)^N)), \\ \quad {\mathcal P}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(B_R), H^1_q(B_R))), \\ \quad {\mathcal H}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(B_R), W^{3-1/q}_q(S_R))) \end{aligned}$$

such that for any \((\mathbf{f}, d, \mathbf{h}, g, \mathbf{g})\) and \(k \in {\mathbb R}_{k_0}\), \(\mathbf{v}= {\mathcal A}(ik){\mathcal F}_k\), \({\mathfrak {q}}= {\mathcal P}(ik){\mathcal F}_k\) and \(\eta = {\mathcal H}(ik){\mathcal F}_k\), where

$$\begin{aligned} {\mathcal F}_k = (\mathbf{f}, d, (ik)^{1/2}\mathbf{h}, \mathbf{h}, (ik)^{1/2}g, g, ik \mathbf{g}), \end{aligned}$$

are unique solutions of equations (4.3), and

$$\begin{aligned} \begin{aligned} {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), H^{2-m}_q(B_R)^N)} (\{(k\partial _k)^\ell ((ik)^{m/2}{\mathcal A}(ik))\mid k \in {\mathbb R}_{k_0}\})&\le r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), L_q(B_R)^N)}(\{(k\partial _k)^\ell \nabla {\mathcal P}(ik) \mid k \in {\mathbb R}_{k_0}\})&\le r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), W^{3-n-1/q}_q(S_R))} (\{(k\partial _k)^\ell ((ik)^n{\mathcal H}(ik))\mid k \in {\mathbb R}_{k_0}\})&\le r_b \end{aligned}\end{aligned}$$
(4.4)

for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\) with some constant \(r_b\).

Remark 12

  1. (1)

    Here and in the following, for \(\theta \in (0,1)\) we set

    $$\begin{aligned}(ik)^\theta = {\left\{ \begin{array}{ll} \mathrm{e}^{i\pi \theta /2}| k|^\theta &{}\quad \text {for }k > 0, \\ \mathrm{e}^{-i\pi \theta /2}| k|^\theta &{}\quad \text {for }k < 0. \end{array}\right. } \end{aligned}$$
  2. (2)

    The functions \(F_1\), \(F_2\), \(F_3\), \(F_4\), \(F_5\), \(F_6\), and \(F_7\) are variables corresponding to \(\mathbf{f}\), d, \((ik)^{1/2}\mathbf{h}\), \(\mathbf{h}\), \((ik)^{1/2}g\), g, and \(ik\, \mathbf{g}\), respectively.

  3. (3)

    We define the norm \(\Vert \cdot \Vert _{{\mathcal X}_q(B_R)}\) by setting

    $$\begin{aligned}&\Vert (F_1, \ldots , F_7)\Vert _{{\mathcal X}_q(B_R)} = \Vert (F_1, F_3, F_5, F_7)\Vert _{L_q(B_R)}\\&\qquad + \Vert F_2\Vert _{W^{2-1/q}_q(S_R)} + \Vert (F_4, F_6)\Vert _{H^1_q(B_R)}. \end{aligned}$$

Let \(\varphi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+2}\) and zero for \(k \not \in {\mathbb R}_{k_0+1}\), and let \(\psi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+4}\) and zero for \(k \not \in {\mathbb R}_{k_0+3}\). Notice that \(\varphi (ik)\psi (ik) = \varphi (ik)\). Let \({\mathcal A}(ik)\), \({\mathcal P}(ik)\) and \({\mathcal H}(ik)\) be the \({\mathcal R}\)-solvers given in Theorem 11. Then, we have

$$\begin{aligned} \begin{aligned} {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), H^{2-m}_q(B_R)^N)} (\{(k\partial _k)^\ell ((ik)^{m/2}(\varphi (ik){\mathcal A}(ik)))\mid k \in {\mathbb R}_{k_0}\})&\le C\Vert \varphi \Vert _{H^1_\infty ({\mathbb R})}r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), L_q(B_R)^N)}(\{(k\partial _k)^\ell \nabla (\varphi (ik){\mathcal P}(ik))\mid k \in {\mathbb R}_{k_0}\})&\le C\Vert \varphi \Vert _{H^1_\infty ({\mathbb R})}r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), W^{3-n-1/q}_q(S_R))} (\{(k\partial _k)^\ell ((ik)^n(\varphi (ik){\mathcal H}(ik)))\mid k \in {\mathbb R}_{k_0}\})&\le C\Vert \varphi \Vert _{H^1_\infty ({\mathbb R})}r_b \end{aligned}\end{aligned}$$
(4.5)

for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\). To prove (4.5), we use the following lemma concerning the fundamental properties of the \({\mathcal R}\)-bound and scalar-valued Fourier multipliers.

Lemma 13

(a) Let X and Y be Banach spaces, and let \({\mathcal T}\) and \({\mathcal S}\) be \({\mathcal R}\)-bounded families in \({\mathcal L}(X,Y)\). Then, \({\mathcal T}+ {\mathcal S}= \{T + S \mid T \in {\mathcal T}, S \in {\mathcal S}\}\) is also an \({\mathcal R}\)-bounded family in \({\mathcal L}(X,Y)\) and

$$\begin{aligned}{\mathcal R}_{{\mathcal L}(X,Y)}({\mathcal T}+ {\mathcal S}) \le {\mathcal R}_{{\mathcal L}(X,Y)}({\mathcal T}) + {\mathcal R}_{{\mathcal L}(X,Y)}({\mathcal S}).\end{aligned}$$

(b) Let X, Y and Z be Banach spaces, and let \({\mathcal T}\) and \({\mathcal S}\) be \({\mathcal R}\)-bounded families in \({\mathcal L}(X, Y)\) and \({\mathcal L}(Y, Z)\), respectively. Then, \({\mathcal S}{\mathcal T}= \{ST \mid T \in {\mathcal T}, S \in {\mathcal S}\}\) is also an \({\mathcal R}\)-bounded family in \({\mathcal L}(X, Z)\) and

$$\begin{aligned}{\mathcal R}_{{\mathcal L}(X, Z)}({\mathcal S}{\mathcal T}) \le {\mathcal R}_{{\mathcal L}(X,Y)}({\mathcal T}){\mathcal R}_{{\mathcal L}(Y, Z)}({\mathcal S}).\end{aligned}$$

(c) Let \(1< p, \, q < \infty \) and let D be a domain in \({\mathbb R}^N\). Let \(m=m(\lambda )\) be a bounded function defined on a subset U of \({\mathbb C}\) and let \(M_m(\lambda )\) be a map defined by \(M_m(\lambda )f = m(\lambda )f\) for any \(f \in L_q(D)\). Then, \({\mathcal R}_{{\mathcal L}(L_q(D))}(\{M_m(\lambda ) \mid \lambda \in U\}) \le C_{N,q,D}\Vert m\Vert _{L_\infty (U)}\).

(d) Let \(n=n(\tau )\) be a \(C^1\)-function defined on \({\mathbb R}{\setminus }\{0\}\) that satisfies the conditions \(|n(\tau )| \le \gamma \) and \(|\tau n'(\tau )| \le \gamma \) with some constant \(c > 0\) for any \(\tau \in {\mathbb R}{\setminus }\{0\}\). Let \(T_n\) be an operator-valued Fourier multiplier defined by \(T_n f = {\mathcal F}^{-1}[n {\mathcal F}[f]]\) for any f with \({\mathcal F}[f] \in {\mathcal D}({\mathbb R}, L_q(D))\). Then, \(T_n\) is extended to a bounded linear operator from \(L_p({\mathbb R}, L_q(D))\) into itself. Moreover, denoting this extension also by \(T_n\), we have

$$\begin{aligned} \Vert T_n\Vert _{{\mathcal L}(L_p({\mathbb R}, L_q(D)))} \le C_{p,q,D}\gamma . \end{aligned}$$

Here, we only prove the \({\mathcal R}\)-boundedness of \(\varphi (ik)ik{\mathcal A}(ik)\). The \({\mathcal R}\)-boundedness of the other terms can be proved by the same argument. Let \(n \in {\mathbb N}\), \(\{k_\ell \}_{\ell =1}^n \in {\mathbb R}^n\), \(\{F_\ell \}_{\ell =1}^n \in {\mathcal X}_q(B_R)^n\). Changing the labeling of indices if necessary, we may assume that \(\varphi (k_\ell )\not =0\) for \(k = 1, \ldots , m\) and \(\varphi (k_\ell ) = 0\) for \(\ell =m+1,\ldots , n\). And then, using Lemma 13, we have

$$\begin{aligned}&\Vert \sum _{\ell =1}^n r_\ell \varphi (ik_\ell )(ik_\ell ){\mathcal A}(ik_\ell )F_\ell \Vert _{L_q((0, 1), L_q(B_R))} \\&\quad = \Vert \sum _{\ell =1}^m r_\ell \varphi (ik_\ell )(ik_\ell ){\mathcal A}(ik_\ell )F_\ell \Vert _{L_q((0, 1), L_q(B_R))} \\&\quad \le r_b\Vert \sum _{\ell =1}^m r_\ell \varphi (ik_\ell )F_\ell \Vert _{L_q((0, 1), L_q(B_R))} \\&\quad = r_b\Vert \sum _{\ell =1}^n r_\ell \varphi (ik_\ell )F_\ell \Vert _{L_q((0, 1), L_q(B_R))} \\&\quad \le C_{q,R} \Vert \varphi \Vert _{H^1_\infty (B_R)}r_b\Vert \sum _{\ell =1}^n r_\ell F_\ell \Vert _{L_q((0, 1), L_q(B_R))}, \end{aligned}$$

which shows that

$$\begin{aligned} {\mathcal R}_{{\mathcal L}({\mathcal X}_q(B_R), L_q(B_R)^N)} (\{ik\varphi (ik){\mathcal A}(ik) \mid k \in {\mathbb R}_{k_0}\}) \le C_{q,R}\Vert \varphi \Vert _{H^1_\infty ({\mathbb R})}r_b. \end{aligned}$$

For \(f \in \{\mathbf{F}, G, \mathbf{G}, D, \mathbf{H}\}\), let

$$\begin{aligned} f_\psi = {\mathcal F}^{-1}_{\mathbb T}[\psi {\mathcal F}_{\mathbb T}[f]]. \end{aligned}$$

We consider the high frequency part of the equations (4.1):

$$\begin{aligned} \begin{aligned} \partial _t\mathbf{u}_\psi -\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}_\psi )-{\mathfrak {p}}_\psi \mathbf{I}) = \mathbf{F}_\psi&\quad&\text {in }B_R\times (0, 2\pi ), \\ \mathrm{div}\,\mathbf{u}_\psi =G_\psi = \mathrm{div}\,\mathbf{G}_\psi&\quad&\text {in }B_R\times (0, 2\pi ), \\ \partial _t\rho _\psi + {\mathcal M}\rho _\psi -({\mathcal A}\mathbf{u}_\psi )\cdot \mathbf{n}=D_\psi&\quad&\text {on }S_R\times (0, 2\pi ),\\ (\mu \mathbf{D}(\mathbf{u}_\psi )-{\mathfrak {p}}_\psi \mathbf{I})\mathbf{n}- ({\mathcal B}_R\rho _\psi )\mathbf{n}=\mathbf{H}_\psi&\quad&\text {on }S_R\times (0, 2\pi ). \end{aligned}\end{aligned}$$
(4.6)

By Theorem 8, Theorem 9, and (4.5), we have immediately the following theorem.

Theorem 14

Let \(1< p, q < \infty \). Then, for any functions \(\mathbf{F}\), G, \(\mathbf{G}\), D, and \(\mathbf{H}\) with

$$\begin{aligned} \mathbf{F}&\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \quad D \in L_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(B_R)), \quad \\ \mathbf{H}&\in H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)^N), \\ G&\in H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)),\\&\quad \mathbf{G}\in H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \end{aligned}$$

We let

$$\begin{aligned} \mathbf{u}_\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal A}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \\ {\mathfrak {p}}_\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal P}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \\ \rho _\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal A}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \end{aligned}$$

where we have set

$$\begin{aligned} {\mathcal F}_k(\mathbf{F}_\psi , D_\pi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi ) =\,&\psi (ik)({\mathcal F}_{\mathbb T}[\mathbf{F}](ik), {\mathcal F}_{\mathbb T}[D](ik), (ik)^{1/2}{\mathcal F}_{\mathbb T}[\mathbf{H}](ik), {\mathcal F}_{\mathbb T}[\mathbf{H}](ik), \\&\quad (ik)^{1/2}{\mathcal F}_{\mathbb T}[G](ik), {\mathcal F}_{\mathbb T}[G](ik), ik{\mathcal F}_{\mathbb T}[\mathbf{G}](ik)). \end{aligned}$$

Then, \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) are the unique solutions of equations (4.6), which possess the following estimate:

$$\begin{aligned}&\Vert \mathbf{u}_\psi \Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{u}_\psi \Vert _{L_p((0, 2\pi ), L_q(B_R))} + \Vert \nabla {\mathfrak {p}}_\psi \Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad \quad + \Vert \rho _\psi \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho _\psi \Vert _{H^1_p((0, 2\pi ), W^{2-1/q}_q(S_R))}\\&\quad \le C\{\Vert \mathbf{F}_\psi \Vert _{L_p((0, 2\pi ), L_q(B_R))}\\&\qquad + \Vert D_\psi \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \Vert \Lambda ^{1/2}(G_\psi , \mathbf{H}_\psi )\Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad \quad + \Vert (G_\psi , \mathbf{H}_\psi )\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} + \Vert \partial _t\mathbf{G}_\psi \Vert _{L_p((0, 2\pi ), L_q(B_R))}\} \end{aligned}$$

for some constant \(C > 0\). Here, we have set

$$\begin{aligned} \Lambda ^{1/2}(G_\psi , \mathbf{H}_\psi ) = {\mathcal F}^{-1}_{\mathbb T}[(ik)^{1/2}\psi (ik)({\mathcal F}_{\mathbb T}[G](ik), {\mathcal F}_{\mathbb T}[\mathbf{H}] (ik))]. \end{aligned}$$

We now consider the lower frequency part of solutions of equations (4.1). Namely, we consider equations (4.3) for \(k \in {\mathbb R}\) with \(1 \le |k| < k_0+4\). We shall show the following theorem.

Theorem 15

Let \(1< q < \infty \) and \(k \in {\mathbb Z}\) with \(1 \le |k| \le k_0+3\). Then, for any \(\mathbf{f}\in L_q(B_R)^N\), \(g \in H^1_q(B_R)\), \(d \in W^{2-1/q}_q(S_R)\), \(\mathbf{h}\in H^1_q(B_R)^N\), and \(\mathbf{g}\in L_q(B_R)^N\), problem (4.3) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {q}}\in H^1_q(B_R)\), and \(\eta \in W^{3-1/q}_q(S_R)\) possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{v}\Vert _{H^2_q(B_R)} + \Vert \nabla {\mathfrak {q}}\Vert _{L_q(B_R)} + \Vert \eta \Vert _{W^{3-1/q}_q(S_R)} \\&\quad \le C(\Vert \mathbf{f}\Vert _{L_q(B_R)} + \Vert d\Vert _{W^{2-1/q}_q(S_R)} + \Vert (g, \mathbf{h})\Vert _{H^1_q(B_R)} + \Vert \mathbf{g}\Vert _{L_q(B_R)}) \end{aligned}\end{aligned}$$
(4.7)

for some constant \(C > 0\).

Proof

From Theorem 11, problem (4.3) with \(k = k_0+4\) admits unique solutions \(\mathbf{v}_{k_0} \in H^2_q(B_R)^N\), \({\mathfrak {q}}_{k_0} \in H^1_q(B_R)\), and \(\eta _{k_0} \in W^{3-1/q}_q(S_R)\) possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{v}_{k_0}\Vert _{H^2_q(B_R)} + \Vert \nabla {\mathfrak {q}}_{k_0}\Vert _{L_q(B_R)} + \Vert \eta _{k_0}\Vert _{W^{3-1/q}_q(S_R)} \\&\quad \le C(\Vert \mathbf{f}\Vert _{L_q(B_R)} + \Vert d\Vert _{W^{2-1/q}_q(S_R)} + \Vert (g, \mathbf{h})\Vert _{H^1_q(B_R)} + \Vert \mathbf{g}\Vert _{L_q(B_R)}) \end{aligned}\end{aligned}$$
(4.8)

for some constant C. Thus, for any \(k \in {\mathbb R}\) with \(|k| < k_0+4\), we consider the unique solvability of the equations:

$$\begin{aligned} \begin{aligned} ik\mathbf{w}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w}) - {\mathfrak {r}}\mathbf{I}) = \mathbf{f}, \quad \mathrm{div}\,\mathbf{w}= 0&\quad&\text {in }B_R, \\ ik\zeta + {\mathcal M}\zeta -({\mathcal A}\mathbf{w})\cdot \mathbf{n}= d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{w}) - {\mathfrak {r}}\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\zeta )\mathbf{n}= 0&\quad&\text {on }S_R, \end{aligned}\end{aligned}$$
(4.9)

where we have set \(\mathbf{f}= i(k-k_0)\mathbf{v}_{k_0}\) and \(d=i(k_0-k)\eta _{k_0}\). In fact, if we set \(\mathbf{v}= \mathbf{v}_{k_0}+\mathbf{w}\), \({\mathfrak {q}}={\mathfrak {q}}_{k_0}+{\mathfrak {r}}\), and \(\eta = \eta _{k_0}+ \zeta \), then \(\mathbf{v}\), \({\mathfrak {q}}\) and \(\eta \) are unique solutions of equations (4.3).

In what follows, we study the unique solvability of equations (4.9) in the case where \(\mathbf{f}\in L_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\) 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 \({\mathfrak {r}}\) and the divergence condition \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Given \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\), let \(K=K(\mathbf{v}, \zeta ) \in H^1_q(B_R)\) be the unique solution of the weak Dirichlet problem:

$$\begin{aligned} (\nabla K, \nabla \varphi )_{B_R} = (\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}))-\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{B_R} \quad \text {for any }\varphi \in \hat{H}^1_{q',0}(B_R) \end{aligned}$$
(4.10)

subject to

$$\begin{aligned} K = <\mu \mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}> - \sigma {\mathcal B}\zeta - \mathrm{div}\,\mathbf{v}\quad \text {on }S_R, \end{aligned}$$
(4.11)

where we have set

$$\begin{aligned}\hat{H}^1_{q', 0}(B_R) = \{\varphi \in L_{q, \mathrm{loc}}(B_R) \mid \nabla \varphi \in L_q(B_R)^N, \quad \varphi |_{S_R}=0\}\end{aligned}$$

and \(q' = q/(q-1)\). In view of Poincaré’s inequality, \(\hat{H}^1_{q',0}(B_R) = H^1_{q', 0}(B_R) = \{\varphi \in H^1_{q'}(B_R) \mid \varphi |_{S_R}=0\}\). Instead of (4.9), we consider the equations:

$$\begin{aligned} \begin{aligned} ik\mathbf{w}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w}) - K(\mathbf{w}, \zeta )\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ ik\zeta + {\mathcal M}\zeta -({\mathcal A}\mathbf{w})\cdot \mathbf{n}= d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{w}) - K(\mathbf{w}, \zeta )\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\zeta )\mathbf{n}= 0&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.12)

In view of the boundary condition (4.11) for \(K(\mathbf{w}, \zeta )\), that \(\mathbf{w}\) and \(\zeta \) satisfy the third equation of equations (4.12) is equivalent to

$$\begin{aligned} (\mu \mathbf{D}(\mathbf{w})\mathbf{n})_\tau = 0 \quad \text {and}\quad \mathrm{div}\,\mathbf{w}= 0 \quad \text {on }S_R, \end{aligned}$$
(4.13)

where \(\mathbf{d}_\tau = \mathbf{d}- <\mathbf{d}, \mathbf{n}>\mathbf{n}\) for any N-vector \(\mathbf{d}\). Let \(J_q(B_R)\) be a solenoidal space defined by setting

$$\begin{aligned} J_q(B_R) = \{\mathbf{v}\in L_q(B_R) \mid (\mathbf{v}, \nabla \varphi )_{B_R} = 0 \quad \text {for any }\varphi \in \hat{H}^1_{q',0}(B_R)\}. \end{aligned}$$

Obviously, for \(\mathbf{v}\in H^1_q(B_R)\), in order that \(\mathrm{div}\,\mathbf{v}= 0\) in \(B_R\), it is necessary and sufficient that \(\mathbf{v}\in J_q(B_R)\). For any \(\mathbf{f}\in L_q(B_R)^N\), let \(\psi \in H^1_{q,0}(B_R)\) be a unique solution of the weak Dirichlet problem:

$$\begin{aligned} (\nabla \psi , \nabla \varphi )_{B_R} = (\mathbf{f}, \nabla \varphi )_{B_R} \quad \text {for any }\varphi \in \hat{H}^1_{q', 0}(B_R). \end{aligned}$$

Let \(\mathbf{g}=\mathbf{f}-\nabla \psi \) and inserting this formula into equations (4.9), we have

$$\begin{aligned} \begin{aligned} ik\mathbf{w}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w}) - ({\mathfrak {r}}-\psi )\mathbf{I}) = \mathbf{g}, \quad \mathrm{div}\,\mathbf{w}= 0&\quad&\text {in }B_R, \\ ik\zeta + {\mathcal M}\zeta -({\mathcal A}\mathbf{w})\cdot \mathbf{n}= d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{w}) - ({\mathfrak {r}}-\psi )\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\zeta )\mathbf{n}= 0&\quad&\text {on }S_R. \end{aligned} \end{aligned}$$

where we have used the fact that \(\psi |_{S_R}=0\). Therefore, we shall solve equations (4.9) for \(\mathbf{f}\in J_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\). When \(\mathbf{f}\in J_q(B_R)\), the equations (4.9) and (4.12) are equivalent. In fact, if \(\mathbf{w}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\) satisfy equations (4.9) with some \({\mathfrak {r}}\in H^1_q(B_R)\). Then, for any \(\varphi \in \hat{H}^1_{q',0}(B_R)\), we have

$$\begin{aligned} 0&= (\mathbf{f}, \nabla \varphi )_{B_R} = (ik\mathbf{w}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w})), \nabla \varphi )_{B_R} + (\nabla {\mathfrak {r}}, \nabla \varphi )_{B_R}\\&= (\nabla ({\mathfrak {r}}- K(\mathbf{w}, \zeta )), \nabla \varphi )_{B_R}, \end{aligned}$$

where we have used the fact that \(\mathrm{div}\,\mathbf{w}= 0\). Moreover, from the boundary conditions in equations (4.9) and (4.11), it follows that

$$\begin{aligned}{\mathfrak {r}}- K(\mathbf{w}, \zeta ) = <\mu \mathbf{D}(\mathbf{w}) \mathbf{n}, \mathbf{n}> - \sigma {\mathcal B}_R\zeta - K(\mathbf{w}, \zeta ) = \mathrm{div}\,\mathbf{w}= 0 \end{aligned}$$

on \(S_R\) because \(\mathrm{div}\,\mathbf{w}=0\). Thus, the uniqueness of the solutions to his weak Dirichlet problem yields that \({\mathfrak {r}}= K(\mathbf{w}, \zeta )\), and so \(\mathbf{w}\) and \(\zeta \) satisfy equations (4.12). Conversely, let \(\mathbf{w}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\) be solutions of equations (4.12). For any \(\varphi \in \hat{H}^1_{q', 0}(B_R)\), we have

$$\begin{aligned} 0&= (\mathbf{f}, \nabla \varphi )_{B_R} = ik(\mathbf{w}, \nabla \varphi )_{B_R} -(\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w})), \nabla \varphi )_{B_R} + (\nabla K(\mathbf{w}, \zeta ), \nabla \varphi )_{B_R} \\&= -ik(\mathrm{div}\,\mathbf{w}, \varphi )_{B_R} - (\nabla \mathrm{div}\,\mathbf{w}, \nabla \varphi )_{B_R} \end{aligned}$$

Moreover, from the boundary condition (4.13) it follows that \(\mathrm{div}\,\mathbf{w}=0\) on \(S_R\). The uniqueness implies that \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Thus, \(\mathbf{w}\), \({\mathfrak {r}}= K(\mathbf{w}, \zeta )\) and \(\zeta \) are solutions of equations (4.9). In particular, for solutions \(\mathbf{w}\) and \(\zeta \) of equations (4.12), we see that \(\mathbf{w}\) satisfies the divergence condition: \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\) automatically.

From now on, we study the unique existence theorem for equations (4.12) for any \(\mathbf{f}\in J_q(B_R)\) and \(d\in W^{2-1/q}_q(S_R)\). To formulate problem (4.12) in a functional analytic setting, we define the spaces \({\mathcal H}_q\), \({\mathcal D}_q\) and the operator \(\mathbf{A}\) by setting

$$\begin{aligned} {\mathcal H}_q&= \{(\mathbf{f}, d) \mid \mathbf{f}\in J_q(B_R), \quad d \in W^{2-1/q}_q(S_R)\}, \\ {\mathcal D}_q&= \{(\mathbf{w}, \zeta ) \in {\mathcal H}_q \mid \mathbf{w}\in H^2_q(B_R)^N, \quad \zeta \in W^{3-1/q}_q(S_R), \quad (\mu \mathbf{D}(\mathbf{w}))_\tau |_{S_R} = 0\}, \\ \mathbf{A}U&= (\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w})-K(\mathbf{w}, \zeta )\mathbf{I}), (-{\mathcal M}\zeta +({\mathcal A}\mathbf{w})\cdot \mathbf{n})|_{S_R}) \quad \text {for }U = (\mathbf{w}, \zeta ) \in {\mathcal D}_q, \end{aligned}$$

where we have used (4.13) and \(\mathrm{div}\,\mathbf{w}=0\) in the definition of \({\mathcal D}_q\). We write equations (4.12) as

$$\begin{aligned} ik U - \mathbf{A}U = F \quad \text {in }{\mathcal H}_q. \end{aligned}$$
(4.14)

In view of Theorem 11, we see that \(k=k_0+4\) is an element of the resolvent set of the operator \(\mathbf{A}\), and so \((i(k_0+4)\mathbf{I}- \mathbf{A})^{-1}\) exists in \({\mathcal L}({\mathcal H}_q, {\mathcal D}_q)\). Since \(B_R\) is a compact set, it follows from the Rellich compactness theorem that \((i(k_0+4)\mathbf{I}- \mathbf{A})^{-1}\) is a compact operator from \({\mathcal H}_q\) 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 \(U = (\mathbf{w}, \zeta ) \in {\mathcal D}_q\) satisfy the homogeneous equations:

$$\begin{aligned} ik U - \mathbf{A}U = 0 \quad \text {in }{\mathcal H}_q. \end{aligned}$$
(4.15)

Namely, \((\mathbf{w}, \zeta ) \in {\mathcal D}_q\) satisfies equations:

$$\begin{aligned} \begin{aligned} ik\mathbf{w}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{w}) - K(\mathbf{w}, \zeta )\mathbf{I}) = 0&\quad&\text {in }B_R, \\ ik\zeta + {\mathcal M}\zeta -({\mathcal A}\mathbf{w})\cdot \mathbf{n}= 0&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{w}) - K(\mathbf{w}, \zeta )\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\zeta )\mathbf{n}= 0&\quad&\text {on }S_R. \end{aligned} \end{aligned}$$
(4.16)

We first prove that

$$\begin{aligned} (\zeta , 1)_{S_R} = 0, \quad (\zeta , x_j)_{S_R} = 0 \quad \text {for }j=1, \ldots , N. \end{aligned}$$
(4.17)

Integrating the second equation of equations (4.16) and applying the divergence theorem of Gauss gives that

$$\begin{aligned}0 = ik(\zeta , 1)_{S_R} + (\zeta , 1)_{S_R}|S_R|-\int _{B_R} \mathrm{div}\,{\mathcal A}\mathbf{w}\,\mathrm{d}x =(ik + |S_R|)(\zeta , 1)_{S_R},\end{aligned}$$

where we have set \(|S_R| = \int _{S_R}\,\mathrm{d}\omega \) and we have used the fact that \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Thus, we have \((\zeta , 1)_{S_R} = 0\). 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

$$\begin{aligned} 0 = ik(\zeta , x_\ell )_{S_R} + (\zeta , x_\ell )_{S_R}(x_\ell , x_\ell )_{S_R} -\int _{B_R} \mathrm{div}\,(x_\ell {\mathcal A}\mathbf{w})\,\mathrm{d}x, \end{aligned}$$
(4.18)

because \((x_j, x_\ell )_{S_R}=0\) for \(j\not =\ell \). Since

$$\begin{aligned}\int _{B_R} \mathrm{div}\,(x_\ell {\mathcal A}\mathbf{w})\,\mathrm{d}x = \int _{B_R} (\mathbf{w}_\ell - \frac{1}{|B_R|}\int _{B_R} \mathbf{w}_\ell \,\mathrm{d}x)\,\mathrm{d}x = 0, \end{aligned}$$

we have \((\zeta , x_\ell )_{S_R} = 0\), because \((x_\ell , x_\ell )_{S_R} = (R^2/N)|S_R| > 0\). Thus, we have proved (4.17). In particular, \({\mathcal M}\zeta = 0\) in (4.16).

We now prove that \(\mathbf{w}= 0\). For this purpose, we first consider the case where \(2 \le q < \infty \). Since \(B_R\) is bounded, \({\mathcal D}_q \subset {\mathcal D}_2\). Multiplying the first equation of (4.16) with \(\mathbf{w}\) and integrating the resultant formula over \(B_R\) and using the divergence theorem of Gauss gives that

$$\begin{aligned} 0 = ik\Vert \mathbf{w}\Vert _{L_2(B_R)}^2 - \sigma ({\mathcal B}_R\zeta , \mathbf{n}\cdot \mathbf{w})_{S_R} + \frac{\mu }{2} \Vert \mathbf{D}(\mathbf{w})\Vert _{L_2(B_R)}^2, \end{aligned}$$

because \(\mathrm{div}\,\mathbf{w}= 0\) in \(B_R\). By the second equation of (4.16) with \({\mathcal M}\zeta =0\), we have

$$\begin{aligned}\sigma ({\mathcal B}_R\zeta , \mathbf{n}\cdot \mathbf{w})_{S_R} = \sigma ({\mathcal B}_R\zeta , ik\zeta )_{S_R} + \sum _{k=1}^N\frac{1}{|B_R|}\int _{B_R}w_j\,\mathrm{d}t({\mathcal B}_R\zeta , R^{-1}x_j)_{S_R} \end{aligned}$$

where we have used \(\mathbf{n}=R^{-1}x= R^{-1}(x_1,\ldots , x_N)\) for \(x \in S_R\). Thus,

$$\begin{aligned}({\mathcal B}_R\zeta , x_j)_{S_R} = (\zeta , (\Delta _{S_R}+\frac{N-1}{R^2})x_j)_{S_R} =0. \end{aligned}$$

Moreover, since \(\zeta \) satisfies (4.17), we know that

$$\begin{aligned}-({\mathcal B}_R\zeta , \zeta )_{S_R} \ge c\Vert \zeta \Vert _{L_2(S_R)}^2 \end{aligned}$$

for some positive constant c, and therefore (4.18) implies \(\mathbf{w}=0\).

Now the first equation of (4.16) yields \(\nabla K(\mathbf{w},\zeta )=0\), so that \(K(\mathbf{w},\zeta )\) 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, \(K(\mathbf{w},\zeta )=0\).

Finally, the third equation of (4.16) yields that \({\mathcal B}_R\zeta =0\) on \(S_R\), and so by (4.17) we have \(\zeta =0\). This completes the proof of the uniqueness in the case where \(2 \le q < \infty \). In particular, we have the unique existence theorem of solutions to equation (4.14).

We now consider the case where \(1< q < 2\). Let \(\mathbf{f}\) be any element in \(J_{q'}(B_R)\) and let \(V = (\mathbf{v}, \eta ) \in {\mathcal D}_{q'}\) be a solution of the equation:

$$\begin{aligned}-ik V - \mathbf{A}V= (\mathbf{f}, 0) \quad \text {in }{\mathcal H}_{q'}. \end{aligned}$$

The existence of such V has already been proved above. Since \(d=0\), we see that \(\eta \) satisfies the relations:

$$\begin{aligned}(\eta , 1)_{S_R} = 0, \quad (\eta , x_j)_{S_R} = 0 \quad \text {for }j=1, \ldots , N, \end{aligned}$$

and so \({\mathcal M}\eta = 0\). Using the divergence theorem of Gauss, we have

$$\begin{aligned} (\mathbf{w}, \mathbf{f})_{B_R}&= (\mathbf{w}, -ik \mathbf{v}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \eta )\mathbf{I}))_{B_R} \\&= (ik\mathbf{w}, \mathbf{v})_{B_R} -(\mathbf{w}, (\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \eta )\mathbf{I})\mathbf{n})_{S_R} +\frac{\mu }{2}(\mathbf{D}(\mathbf{w}), \mathbf{D}(\mathbf{v}))_{B_R} \\&= (\mathrm{Div}\,(\mu (\mathbf{D}(\mathbf{w}) - K(\mathbf{w}, \zeta )\mathbf{I}), \mathbf{v})_{B_R} -\sigma (\mathbf{w}\cdot \mathbf{n}, {\mathcal B}_R\eta )_{S_R} +\frac{\mu }{2} (\mathbf{D}(\mathbf{w}), \mathbf{D}(\mathbf{v}))_{B_R}\\&= \sigma ({\mathcal B}_R \zeta , \mathbf{n}\cdot \mathbf{v})_{S_R} - \sigma (\mathbf{w}\cdot \mathbf{n}, {\mathcal B}_R\eta )_{S_R} \\&= \sigma ({\mathcal B}_R\zeta , -ik\eta + \frac{1}{|B_R|}\int _{B_R}\mathbf{v}\,\mathrm{d}y\cdot \mathbf{n})_{S_R} -\sigma (ik\zeta \\&\quad + \frac{1}{|B_R|}\int _{B_R}\mathbf{w}\,\mathrm{d}y\cdot \mathbf{n}, {\mathcal B}_R\eta )_{S_R}. \end{aligned}$$

Using the fact that \(({\mathcal B}_R\zeta , x_j)_{S_R} = (x_j, {\mathcal B}_R\eta )_{S_R} =0\), we have

$$\begin{aligned} (\mathbf{w}, \mathbf{f})_{B_R}&= \sigma ik({\mathcal B}_R\zeta , \eta )_{S_R} - \sigma ik(\zeta , {\mathcal B}_R\eta )_{S_R} \\&=\sigma ik\Bigl \{\frac{N-1}{R^2}(\zeta , \eta )_{S_R} - (\nabla _{S_R}\zeta , \nabla _{S_R}\eta )_{S_R}\\&\quad -\frac{N-1}{R^2}(\zeta , \eta )_{S_R} + (\nabla _{S_R}\zeta , \nabla _{S_R}\eta )_{S_R}\Bigr \} = 0. \end{aligned}$$

For any \(\mathbf{g}\in L_{q'}(B_R)^N\), let \(\psi \in \hat{H}^1_{q', 0}(B_R)\) be a unique solution of the weak Dirichlet problem:

$$\begin{aligned}(\nabla \psi , \nabla \varphi )_{B_R} = (\mathbf{g}, \nabla \varphi )_{B_R} \quad \text {for any }\varphi \in \hat{H}^1_{q,0}(B_R).\end{aligned}$$

Let \(\mathbf{f}= \mathbf{g}- \nabla \psi \), and then \(\mathbf{f}\in J_{q'}(B_R)\), and so using the fact that \(\mathbf{w}\in J_q(B_R)\), we have \((\mathbf{w}, \mathbf{g})_{B_R} = (\mathbf{w}, \mathbf{f})_{B_R} + (\mathbf{w}, \nabla \psi )_{B_R} = 0\). The arbitrariness of \(\mathbf{g}\in L_{q'}(B_R)^N\) implies that \(\mathbf{w}=0\). Thus, the second equation of (4.16) and (4.17) leads to \(\zeta =0\). This completes the proof of the uniqueness in the case where \(1< q < 2\), and therefore the proof of Theorem 15. \(\square \)

We now consider the linearized stationary problem:

$$\begin{aligned} \begin{aligned} {\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ \mathrm{div}\,\mathbf{v}= g = \mathrm{div}\,\mathbf{g}&\quad&\text {in }B_R, \\ {\mathcal M}\rho -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}=\mathbf{h}&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.19)

We shall prove the following theorem.

Theorem 16

Let \(1< q < \infty \). Then, for any \(\mathbf{f}\in L_q(B_R)^N\), \(d \in W^{2-1/q}_q(S_R)\), \(g \in H^1_q(B_R)\), \(\mathbf{g}\in L_q(B_R)^N\), and \(\mathbf{h}\in H^1_q(B_R)^N\), problem (4.19) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\), and \(\rho \in W^{3-1/q}_q(S_R)\) possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{v}\Vert _{H^2_q(B_R)} + \Vert {\mathfrak {p}}\Vert _{H^1_q(B_R)} + \Vert \rho \Vert _{W^{3-1/q}_q(S_R)}\\&\quad \le C(\Vert \mathbf{f}\Vert _{L_q(B_R)} + \Vert d\Vert _{W^{2-1/q}_q(S_R)} + \Vert (g, \mathbf{h})\Vert _{H^1_q(B_R)} + \Vert \mathbf{g}\Vert _{L_q(B_R)}) \end{aligned}\end{aligned}$$
(4.20)

for some constant \(C > 0\).

Proof

The strategy of the proof is the same as that of Theorem 15. Since \({\mathcal L}\mathbf{v}\), \({\mathcal M}\rho \), and \(|B_R|^{-1}\int _{B_R}\mathbf{v}\,\mathrm{d}y\) are lower order perturbations, choosing \(k_0 > 0\) large enough, the generalized resolvent problem:

$$\begin{aligned} \begin{aligned} ik_0\mathbf{v}+ {\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ \mathrm{div}\,\mathbf{v}= g = \mathrm{div}\,\mathbf{g}&\quad&\text {in }B_R, \\ ik_0\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}=\mathbf{h}&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.21)

admits unique solutions: \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\), and \(\rho \in W^{3-1/q}_q(S_R)\) 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 \(g=\mathbf{g}=\mathbf{h}=0\), that is

$$\begin{aligned} \begin{aligned} {\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ \mathrm{div}\,\mathbf{v}= 0&\quad&\text {in }B_R, \\ {\mathcal M}\rho -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{v}) - {\mathfrak {p}}\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}=0&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.22)

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 \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\rho \in W^{3-1/q}_q(S_R)\), let \(K = K(\mathbf{v}, \rho ) \in H^1_q(B_R)\) be the unique solution of the weak Dirichlet problem:

$$\begin{aligned} (\nabla K, \nabla \varphi )_{B_R} = (\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v})-{\mathcal L}\mathbf{v}- \nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{B_R} \quad \text {for any }\varphi \in \hat{H}^1_{q',0}(B_R), \end{aligned}$$
(4.23)

subject to the boundary condition:

$$\begin{aligned} K = <\mu \mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}> - \sigma {\mathcal B}_R\rho - \mathrm{div}\,\mathbf{v}\quad \text {on }B_R. \end{aligned}$$
(4.24)

Then, the reduced problem corresponding to problem (4.19) with \(g=\mathbf{g}=\mathbf{h}=0\) is given by the following equations:

$$\begin{aligned} \begin{aligned} {\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}) = \mathbf{f}&\quad&\text {in }B_R, \\ {\mathcal M}\rho -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=d&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}=0&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.25)

Then, for \(\mathbf{f}\in J_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\), problems (4.22) and (4.25) are equivalent. In fact, if problem (4.22) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\) and \(\rho \in W^{3-1/q}_q(S_R)\), then for any \(\varphi \in \hat{H}^1_{q',0}(B_R)\), we have

$$\begin{aligned} 0&= (\mathbf{f}, \nabla \varphi )_{B_R} = ({\mathcal L}\mathbf{v}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v})), \nabla \varphi )_{B_R} + (\nabla {\mathfrak {p}}, \nabla \varphi )_{B_R}\\&= (\nabla ({\mathfrak {p}}- K(\mathbf{v}, \rho )), \nabla \varphi )_{B_R} \end{aligned}$$

because \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). Moreover, from the boundary conditions in (4.22) and (4.24) it follows that

$$\begin{aligned} {\mathfrak {p}}-K(\mathbf{v}, \rho ) = <\mu \mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}> - \sigma {\mathcal B}_R\rho - K(\mathbf{v}, \rho ) = \mathrm{div}\,\mathbf{v}= 0 \end{aligned}$$

on \(S_R\). The uniqueness of the weak Dirichlet problem leads to \({\mathfrak {p}}= K(\mathbf{v}, \rho )\), and therefore \(\mathbf{v}\) and \(\rho \) satisfy equations (4.25). Conversely, if \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\rho \in W^{3-1/q}_q(S_R)\) satisfy the equations (4.25), then for any \(\varphi \in \hat{H}^1_{q', 0}(B_R)\) we have

$$\begin{aligned} 0 = (\mathbf{f}, \nabla \varphi )_{B_R} = ({\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}), \nabla \varphi )_{B_R} + (\nabla K(\mathbf{v}, \rho ), \nabla \varphi )_{B_R} = (\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{B_R}. \end{aligned}$$

Moreover, the boundary conditions of (4.25) and (4.24) gives that

$$\begin{aligned} \mathrm{div}\,\mathbf{v}= <\mu \mathbf{D}(\mathbf{v})\mathbf{n}, \mathbf{n}>- \sigma {\mathcal B}_R\rho - K(\mathbf{v}, \rho ) = 0. \end{aligned}$$

The uniqueness of the weak Dirichlet problem yields that \(\mathrm{div}\,\mathbf{v}=0\), and therefore, \(\mathbf{v}\), \({\mathfrak {p}}= K(\mathbf{v}, \rho )\) and \(\rho \) 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 \(\mathbf{v}\in H^2_2(B_R)^N\) and \(\rho \in W^{5/2}_2(S_R)\) satisfy the homogeneous equations:

$$\begin{aligned} \begin{aligned} {\mathcal L}\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}) = 0&\quad&\text {in }B_R, \\ {\mathcal M}\rho -({\mathcal A}\mathbf{v})\cdot \mathbf{n}=0&\quad&\text {on }S_R, \\ (\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I})\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}=0&\quad&\text {on }S_R. \end{aligned}\end{aligned}$$
(4.26)

Note that \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). Employing the same argument as in the proof of Theorem 15, we have

$$\begin{aligned} (\rho , 1)_{S_R} = 0, \quad (\rho , x_j)_{S_R}=0 \quad \text {for }j=1, \ldots , N. \end{aligned}$$
(4.27)

In particular, \({\mathcal M}\rho =0\). Multiplying the first equation with \(\mathbf{v}\), integrating the resultant formula on \(B_R\) and using the divergence theorem of Gauss gives that

$$\begin{aligned}0 = ({\mathcal L}\mathbf{v}, \mathbf{v})_{B_R} + \sigma ({\mathcal B}_R\rho , \mathbf{n}\cdot \mathbf{v})_{S_R} + \frac{\mu }{2}\Vert \mathbf{D}(\mathbf{v})\Vert _{L_2(B_R)}^2, \end{aligned}$$

because \((K(\mathbf{v}, \rho ), \mathrm{div}\,\mathbf{v}) = 0\) as follows from \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). From (2.17) it follows that

$$\begin{aligned}({\mathcal L}\mathbf{v}, \mathbf{v})_{B_R} = \sum _{k=1}^M|(\mathbf{v}, \mathbf{p}_k)_{B_R}|^2. \end{aligned}$$

From the second equation of (4.26) with \({\mathcal M}\rho =0\) it follows that

$$\begin{aligned}({\mathcal B}_R\rho , \mathbf{n}\cdot \mathbf{v})_{S_R} = \sum _{j=1}^N R^{-1} ({\mathcal B}_R\rho , x_j)_{S_R}\frac{1}{|B_R|}\int _{B_R}v_j\,\mathrm{d}y = 0. \end{aligned}$$

Combining these formulas yields that

$$\begin{aligned}0 = \sum _{k=1}^M|(\mathbf{v}, \mathbf{p}_k)_{B_R}|^2 + \frac{\mu }{2}\Vert \mathbf{D}(\mathbf{v})\Vert _{L_2(B_R)}^2, \end{aligned}$$

which leads to \(\mathbf{D}(\mathbf{v}) = 0\) and \((\mathbf{v}, \mathbf{p}_k)_{B_R} = 0\) for \(k = 1, \ldots , M\). Thus, we have \(\mathbf{v}=0\). From the first equation of (4.26), we have \(\nabla K(\mathbf{v}, \rho ) = 0\), and so \(K(\mathbf{v}, \rho ) = c\) with some constant c. From the boundary condition of (4.26), we have \(\sigma {\mathcal B}\rho =-c\) on \(B_R\). Integrating this formula on \(S_R\) and using the fact \((\rho , 1)_{S_R}=0\) in (4.27) gives that \(c=0\). Thus, \({\mathcal B}_R\rho =0\) on \(S_R\), but we know (4.27), and so

$$\begin{aligned}0 = -({\mathcal B}_R\rho , \rho )_{S_R} \ge c\Vert \rho \Vert _{L_2(S_R)}^2 \end{aligned}$$

for some constant \(c > 0\), which shows that \(\rho =0\). This completes the proof of the uniqueness in the \(L_2\) framework, the proof of Theorem 16. \(\square \)

Proof of Theorem 6

We now prove Theorem 6. Let \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) be functions given in Theorem 14 which are solutions of equations (4.6). Notice that \(\psi (ik) = 1\) for \(|k| \ge k_0+4\) and \(\psi (ik) = 0\) for \(|k| \le k_0+3\). For \(k \in {\mathbb Z}\) with \(1\le |k| \le k_0+3\), let

$$\begin{aligned}&\mathbf{f}= {\mathcal F}_{\mathbb T}[\mathbf{F}](ik), \quad g={\mathcal F}_{\mathbb T}[G](ik), \quad \mathbf{g}= {\mathcal F}_{\mathbb T}[\mathbf{G}](ik),\\&\quad d = {\mathcal F}_{\mathbb T}[D](ik), \quad \mathbf{h}= {\mathcal F}_{\mathbb T}[\mathbf{H}](ik) \end{aligned}$$

in equations (4.3), and we write solutions \(\mathbf{v}\), \({\mathfrak {q}}\) and \(\eta \) as \(\mathbf{v}_k=\mathbf{v}\), \({\mathfrak {q}}_k={\mathfrak {q}}\) and \(\eta _k = \eta \). Let

$$\begin{aligned} \mathbf{u}_k&= \mathrm{e}^{ikt}\mathbf{v}_k, \quad {\mathfrak {p}}_k = \mathrm{e}^{ikt}{\mathfrak {q}}_k, \quad \rho _k = \mathrm{e}^{ikt}\eta _k, \end{aligned}$$

and then, \(\mathbf{u}_k\), \({\mathfrak {p}}_k\) and \(\rho _k\) satisfy the equations:

$$\begin{aligned} \begin{aligned} \partial _t\mathbf{u}_k -\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}_k)-{\mathfrak {p}}_k\mathbf{I}) = \mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[\mathbf{F}](ik)&\quad&\text {in }B_R, \\ \mathrm{div}\,\mathbf{u}_k =\mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[G](ik) = \mathrm{div}\,(\mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[\mathbf{G}](ik))&\quad&\text {in }B_R, \\ \partial _t\rho _k + {\mathcal M}\rho _k-({\mathcal A}\mathbf{u}_k)\cdot \mathbf{n}=\mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[D](ik)&\quad&\text {on }S_R,\\ (\mu \mathbf{D}(\mathbf{u}_k)-{\mathfrak {p}}_k\mathbf{I})\mathbf{n}- ({\mathcal B}_R\rho _k)\mathbf{n}=\mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[\mathbf{H}](ik)&\quad&\text {on }S_R. \end{aligned} \end{aligned}$$
(4.28)

Let \(\mathbf{f}= \mathbf{F}_S\), \(d=D_S\), \(g=G_S\), \(\mathbf{g}= \mathbf{G}_S\) and \(\mathbf{h}= \mathbf{H}_S\) in equations (4.19), and let \(\mathbf{v}\), \({\mathfrak {p}}\) and \(\rho \) be unique solutions of equations (4.19). We write \(\mathbf{u}_S = \mathbf{v}\), \({\mathfrak {p}}_S = {\mathfrak {p}}\) and \(\rho _S = \rho \). Under these preparations, we set

$$\begin{aligned} \mathbf{u}&= \mathbf{u}_S + \sum _{1 \le |k| \le k_0+3} \mathbf{u}_k + \mathbf{u}_\psi , \\ {\mathfrak {p}}&= {\mathfrak {p}}_S + \sum _{1 \le |k| \le k_0+3} {\mathfrak {p}}_k + {\mathfrak {p}}_\psi , \\ \rho&= \rho _S + \sum _{1 \le |k| \le k_0+3} \rho _k + \rho _\psi \end{aligned}$$

and then \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) are unique solutions of equations (4.1). Moreover, by Theorem 14, Theorem 15, and Theorem 16, we see that \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) satisfy the estimate (4.2). In fact, for \(f = f_S + \sum _{1\le |k| \le k_0+3} \mathrm{e}^{ikt}f_k + f_\psi \), we have the following estimates:

$$\begin{aligned} \Vert f\Vert _{L_p((0, 2\pi ), X)}&\le \Vert f_S\Vert _{L_p((0, 2\pi ), X)} + \sum _{1 \le |k| \le k_0+3}\Vert \mathrm{e}^{ikt}f_k\Vert _{L_p((0, 2\pi ), X)} + \Vert f_\psi \Vert _{L_p((0, 2\pi ), X)}\\&\le (2\pi )^{1/p}\Vert f_S\Vert _X + (2\pi )^{1/p}\sum _{1 \le |k| \le k_0+3}\Vert f_k\Vert _X + \Vert f_\psi \Vert _{L_p((0, 2\pi ), X)}, \\ \Vert \partial _t f\Vert _{L_p((0, 2\pi ), X)}&\le \sum _{1 \le |k| \le k_0+3}\Vert (ik)\mathrm{e}^{ikt}f_k\Vert _{L_p((0, 2\pi ), X)} + \Vert \partial _tf_\psi \Vert _{L_p((0, 2\pi ), X)}\\&\le (2\pi )^{1/p}(k_0+3)\sum _{1 \le |k| \le k_0+3}\Vert f_k\Vert _X + \Vert \partial _tf_\psi \Vert _{L_p((0, 2\pi ), X)}. \end{aligned}$$

By Hölder’s inequality, we have

$$\begin{aligned}&\Vert f_S\Vert _{L_p((0, 2\pi ), X)} \le 2\pi \Vert f\Vert _{L_p((0, 2\pi ), X)},\\&\quad \Vert \mathrm{e}^{ikt}{\mathcal F}_{\mathbb T}[f](ik)\Vert _{L_p((0, 2\pi ), X)} \le 2\pi \Vert f\Vert _{L_p((0, 2\pi ), X)}, \end{aligned}$$

and for any UMD Banach space X, using Lemma 13 and transference theorem, Theorem 9, we have

$$\begin{aligned} \Vert f_\psi \Vert _{L_p((0, 2\pi ), X)},&\le C\Vert \psi \Vert _{H^1_\infty }\Vert f\Vert _{L_p((0, 2\pi ), X)}, \\ \Vert \partial _t f_\psi \Vert _{L_p((0, 2\pi ), X)}&\le C\Vert \psi \Vert _{H^1_\infty }\Vert \partial _tf\Vert _{L_p((0, 2\pi ), X)}, \\ \Vert \Lambda ^{1/2}f_\psi \Vert _{L_p((0, 2\pi ), X)}&\le \Vert {\mathcal F}^{-1}_{\mathbb T}[((ik)^{1/2}/(1+k^2)^{1/4})\psi (ik)(1+k^2)^{1/4}\\&\quad {\mathcal F}_{\mathbb T}[f](ik)]\Vert _{L_p ((0, 2\pi ), X)} \\&\le C\bigl (\sum _{\ell =0,1}\sup _{\lambda \in {\mathbb R}}|\bigl (\lambda \frac{d}{\mathrm{d}\lambda }\bigr )^\ell ( ((i\lambda )^{1/2}/(1+\lambda ^2)^{1/4})\psi (i\lambda ))|\bigr )\\&\quad \Vert f\Vert _{H^{1/2}_p((0, 2\pi ), X)}. \end{aligned}$$

\(\square \)

On linearized problem of two-phase problem

In this subsection, we consider the linear equations:

$$\begin{aligned} \begin{aligned} \partial _t\mathbf{u}_\pm -\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}_\pm )-{\mathfrak {p}}_\pm \mathbf{I}) = \mathbf{F}_\pm&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\ \mathrm{div}\,\mathbf{u}_\pm =G_\pm = \mathrm{div}\,\mathbf{G}_\pm&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\ \partial _t\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{u})\cdot \mathbf{n}=D&\quad&\text {on }S_R\times (0, 2\pi ),\\ [[\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})]]\mathbf{n}- ({\mathcal B}_R\rho )\mathbf{n}=\mathbf{H}&\quad&\text {on }S_R\times (0, 2\pi ), \\ [[\mathbf{u}]]=0&\quad&\text {on }S_R\times (0, 2\pi ), \\ \mathbf{u}_-=0&\quad&\text {on }S\times (0, 2\pi ). \end{aligned}\end{aligned}$$
(4.29)

where \(\Omega _+ = B_R\), \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\), and \({\mathcal M}\), \({\mathcal A}\) and \({\mathcal B}_R\) are the linear operators defined in (2.17). We shall prove the unique existence theorem of \(2\pi \)-periodic solutions of equations (4.29). Our main result in this section is stated as follows.

Theorem 17

Let \(1< p, q < \infty \). Then, for any \(\mathbf{F}_\pm \), D, \(G_\pm \), \(\mathbf{G}_\pm \) and \(\mathbf{H}\) with

$$\begin{aligned} \mathbf{F}_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega _\pm )^N), \quad D \in L_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)) \\ G_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\Omega _\pm )) \cap H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega _\pm )),\\&\quad \mathbf{G}_\pm \in H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega _\pm )^N), \\ \mathbf{H}&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\Omega )^N) \cap H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N), \end{aligned}$$

problem (4.1) admits unique solutions \(\mathbf{u}_\pm \), \({\mathfrak {p}}_\pm \) and \(\rho \) with

$$\begin{aligned} \mathbf{u}_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(\Omega _\pm )^N) \cap H^1_{p, \mathrm{per}} ((0, 2\pi ), L_q(\Omega _\pm )^N), \\ {\mathfrak {p}}_\pm&\in L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\Omega _\pm )), \quad \sum _{\pm }\int _{\Omega _\pm }{\mathfrak {p}}_\pm (x,t)\,\mathrm{d}x=0 \ \text { for }t\in (0,2\pi ), \\ \rho&\in L_{p, \mathrm{per}}((0, 2\pi ), W^{3-1/q}_q(S_R)) \cap H^1_{p, \mathrm{per}} ((0, 2\pi ), W^{2-1/q}_q(S_R)) \end{aligned}$$

possessing the estimate:

$$\begin{aligned} \begin{aligned}&\sum _\pm \{ \Vert \mathbf{u}_\pm \Vert _{L_p((0, 2\pi ), H^2_q(\Omega _\pm ))} + \Vert \partial _t\mathbf{u}_\pm \Vert _{L_p((0, 2\pi ), L_q(\Omega _\pm ))} + \Vert \nabla {\mathfrak {p}}_\pm \Vert _{L_p((0, 2\pi ), L_q(\Omega _\pm ))}\} \\&\qquad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} \\&\quad \le C\{\sum _{\pm }\Vert \mathbf{F}_\pm \Vert _{L_p((0, 2\pi ), L_q(\Omega _\pm ))} + \Vert D\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))}\\&\qquad + \sum _\pm \Vert \partial _t\mathbf{G}_\pm \Vert _{L_p((0, 2\pi ), L_q(\Omega _\pm ))} \\&\qquad + \sum _\pm \Vert G_\pm \Vert _{L_p((0, 2\pi ), H^1_q(\Omega _\pm ))} + \Vert G_\pm \Vert _{H^{1/2}_p((0, 2\pi ), L_q(\Omega _\pm ))} \\&\qquad + \Vert \mathbf{H}\Vert _{L_p((0, 2\pi ), H^1_q(\Omega ))} + \Vert \mathbf{H}\Vert _{H^{1/2}_p((0, 2\pi ), L_q(\Omega ))} \}\end{aligned} \end{aligned}$$
(4.30)

for some constant \(C > 0\).

To prove Theorem 17, the strategy is the same as in the proof of Theorem 6. Therefore, we first consider the \({\mathcal R}\)-solver of the generalized resolvent problem:

$$\begin{aligned} \begin{aligned} ik\mathbf{v}_\pm -\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}_\pm )-{\mathfrak {q}}_\pm \mathbf{I}) = \mathbf{f}_\pm&\quad&\text {in }\Omega _\pm , \\ \mathrm{div}\,\mathbf{v}_\pm =g_\pm = \mathrm{div}\,\mathbf{g}_\pm&\quad&\text {in }\Omega _\pm , \\ ik\eta + {\mathcal M}\eta -({\mathcal A}\mathbf{v}_+)\cdot \mathbf{n}=d&\quad&\text {on }S_R,\\ [[\mu \mathbf{D}(\mathbf{v})-{\mathfrak {q}}\mathbf{I}]]\mathbf{n}- ({\mathcal B}_R\eta )\mathbf{n}=\mathbf{h}&\quad&\text {on }S_R, \\ [[\mathbf{v}]]=0&\quad&\text {on }S_R, \\ \mathbf{v}_- = 0&\quad&\text {on }S \end{aligned}\end{aligned}$$
(4.31)

for \(k \in {\mathbb R}\). From Theorem 2.1.4 in Shibata and Saito [19] we know the following theorem concerned with the existence of an \({\mathcal R}\)-solver of problem (4.29).

Theorem 18

Let \(1< q < \infty \) and let \({\mathbb R}_{k_0} = {\mathbb R}{\setminus }(-k_0, k_0)\). Let

$$\begin{aligned} X_q(\dot{\Omega })&= \{(\mathbf{f}, d, \mathbf{h}, g, \mathbf{g}) \mid \mathbf{f}\in L_q(\dot{\Omega }), d \in W^{2-1/q}_q(S_R),\\&\mathbf{h}\in H^1_q(\Omega )^N, g \in H^1_q(\dot{\Omega }), \mathbf{g}\in L_q(\dot{\Omega })^N\}, \\ {\mathcal X}_q(\dot{\Omega })&= \{F=(F_1, F_2, \ldots , F_7) \mid F_1, F_7 \in L_q(\dot{\Omega })^N, F_2 \in W^{2-1/q}_q(S_R),\\&F_3 \in L_q(\Omega )^N, F_4 \in H^1_q(\Omega )^N, \\&\,\, F_5 \in L_q(\dot{\Omega }), F_6 \in H^1_q(\dot{\Omega })\}. \end{aligned}$$

Then, there exist a constant \(k_0 > 0\) and operator families \({\mathcal A}(ik)\), \({\mathcal P}(ik)\), and \({\mathcal H}(ik)\) with

$$\begin{aligned} {\mathcal A}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(\dot{\Omega }), H^2_q(\dot{\Omega })^N)), \\ \quad {\mathcal P}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(\dot{\Omega }), \dot{H}^1_q(\dot{\Omega }))), \\ \quad {\mathcal H}(ik)&\in C^1({\mathbb R}_{k_0}, {\mathcal L}({\mathcal X}_q(\dot{\Omega }), W^{3-1/q}_q(S_R))) \end{aligned}$$

such that for any \((\mathbf{f}, d, \mathbf{h}, g, \mathbf{g})\) and \(k \in {\mathbb R}_{k_0}\), \(\mathbf{v}= {\mathcal A}(ik){\mathcal F}_k\), \({\mathfrak {q}}= {\mathcal P}(ik){\mathcal F}_k\) and \(\eta = {\mathcal H}(ik){\mathcal F}_k\), where

$$\begin{aligned} {\mathcal F}_k = (\mathbf{f}, d, (ik)^{1/2}\mathbf{h}, \mathbf{h}, (ik)^{1/2}g, g, ik \mathbf{g}), \end{aligned}$$

are unique solutions of equations (4.31), and

$$\begin{aligned} \begin{aligned} {\mathcal R}_{{\mathcal L}({\mathcal X}_q(\dot{\Omega }), H^{2-m}_q(\dot{\Omega })^N)} (\{(k\partial _k)^\ell ((ik)^{m/2}{\mathcal A}(ik))\mid k \in {\mathbb R}_{k_0}\})&\le r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(\dot{\Omega }), L_q(\dot{\Omega })^N)}(\{(k\partial _k)^\ell \nabla {\mathcal P}(ik) \mid k \in {\mathbb R}_{k_0}\})&\le r_b, \\ {\mathcal R}_{{\mathcal L}({\mathcal X}_q(\dot{\Omega }), W^{3-n-1/q}_q(S_R))} (\{(k\partial _k)^\ell ((ik)^n{\mathcal H}(ik))\mid k \in {\mathbb R}_{k_0}\})&\le r_b \end{aligned}\end{aligned}$$
(4.32)

for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\) with some constant \(r_b\).

Remark 19

  1. (1)

    Here \(f \in L_q(\dot{\Omega })\) means that \(f_\pm \in L_q(\Omega _\pm )\), and \(f \in H^1_q(\dot{\Omega })\) means that \(f_\pm \in H^1_q(\Omega _\pm )\), and we set

    $$\begin{aligned}\Vert f\Vert _{L_q(\dot{\Omega })} = \sum _\pm \Vert f_\pm \Vert _{L_q(\Omega _\pm )}, \quad \Vert f\Vert _{H^1_q(\dot{\Omega })} = \sum _\pm \Vert f_\pm \Vert _{H^1_q(\Omega _\pm )}. \end{aligned}$$

    Moreover, we define

    $$\begin{aligned} \dot{H}^1_q(\dot{\Omega }) = \Big \{\theta \in H^1_q(\dot{\Omega }) \mid \int _{\dot{\Omega }} \theta \,\mathrm{d}x = 0\Big \}. \end{aligned}$$
  2. (2)

    For f defined on \(\dot{\Omega }\), we set \(f_\pm = f|_{\Omega _\pm }\) and for \(f_\pm \) defined on \(\Omega _\pm \), we set \(f = f_\pm \) on \(\Omega _\pm \). The functions \(F_1\), \(F_2\), \(F_3\), \(F_4\), \(F_5\), \(F_6\), and \(F_7\) are variables corresponding to \(\mathbf{f}\), d, \((ik)^{1/2}\mathbf{h}\), \(\mathbf{h}\), \((ik)^{1/2}g\), g, and \(ik\, \mathbf{g}\), respectively.

  3. (3)

    We define the norm \(\Vert \cdot \Vert _{{\mathcal X}_q(\Omega )}\) by setting

    $$\begin{aligned} \Vert (F_1, \ldots , F_7)\Vert _{{\mathcal X}_q(\Omega )}= & {} \Vert (F_1, F_5, F_7)\Vert _{L_q(\dot{\Omega })} + \Vert F_2\Vert _{W^{2-1/q}_q(S_R)} + \Vert F_6\Vert _{H^1_q(\dot{\Omega })} \\&+ \Vert F_3\Vert _{L_q(\Omega )} + \Vert F_4\Vert _{H^1_q(\Omega ))}. \end{aligned}$$

Let \(\varphi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+2}\) and zero for \(k \not \in {\mathbb R}_{k_0+1}\), and let \(\psi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+4}\) and zero for \(k \not \in {\mathbb R}_{k_0+3}\). For \(f \in \{\mathbf{F}_\pm , G_\pm , \mathbf{G}_\pm , D, \mathbf{H}\}\), we set

$$\begin{aligned}f_\psi = {\mathcal F}^{-1}_{\mathbb T}[\psi {\mathcal F}_{\mathbb T}[f]].\end{aligned}$$

We consider the high frequency part of the equations (4.29):

$$\begin{aligned} \begin{aligned} \partial _t\mathbf{u}_{\pm \psi } -\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{u}_{\pm \psi })-{\mathfrak {p}}_{\pm \psi }\mathbf{I}) = \mathbf{F}_{\pm \psi }&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\ \mathrm{div}\,\mathbf{u}_{\pm \psi } =G_{\pm \psi } = \mathrm{div}\,\mathbf{G}_{\pm \psi }&\quad&\text {in }\Omega _\pm \times (0, 2\pi ), \\ \partial _t\rho _\psi + {\mathcal M}\rho _\psi -({\mathcal A}\mathbf{u}_{+\psi })\cdot \mathbf{n}=D_\psi&\quad&\text {on }S_R\times (0, 2\pi ),\\ {[}[\mu \mathbf{D}(\mathbf{u}_\psi )-{\mathfrak {p}}_\psi \mathbf{I})\mathbf{n}- ({\mathcal B}_R\rho _\psi )\mathbf{n}=\mathbf{H}_\psi&\quad&\text {on }S_R\times (0, 2\pi ), \\ {[}[\mathbf{u}_\psi ]]=0&\quad&\text {on }S_R\times (0, 2\pi ), \\ \mathbf{u}_{-\psi } = 0&\quad&\text {on }S\times (0, 2\pi ). \\ \end{aligned} \end{aligned}$$
(4.33)

By Theorem 8, Theorem 9, and the analogue of (4.5) resulting from (4.35), we have immediately the following theorem.

Theorem 20

Let \(1< p, q < \infty \). Then, for any functions \(\mathbf{F}\), G, \(\mathbf{G}\), D, and \(\mathbf{H}\) with

$$\begin{aligned} \mathbf{F}&\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\dot{\Omega })^N), \quad D \in L_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)), \quad \\ \mathbf{H}&\in H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\Omega )^N), \\ G&\in H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(\dot{\Omega })) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(\dot{\Omega })),\\&\quad \mathbf{G}\in H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(\dot{\Omega })^N), \end{aligned}$$

We let

$$\begin{aligned} \mathbf{u}_\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal A}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \\ {\mathfrak {p}}_\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal P}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \\ \rho _\psi&= {\mathcal F}^{-1}_{\mathbb T}[\varphi (ik){\mathcal A}(ik){\mathcal F}_k(\mathbf{F}_\psi , D_\psi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi )](\cdot , t), \end{aligned}$$

where we have set

$$\begin{aligned} {\mathcal F}_k(\mathbf{F}_\psi , D_\pi , \mathbf{H}_\psi , G_\psi , \mathbf{G}_\psi ) =\,&\psi (ik)({\mathcal F}_{\mathbb T}[\mathbf{F}](ik), {\mathcal F}_{\mathbb T}[D](ik),\\&(ik)^{1/2}{\mathcal F}_{\mathbb T}[\mathbf{H}](ik), {\mathcal F}_{\mathbb T}[\mathbf{H}](ik), \\&\quad (ik)^{1/2}{\mathcal F}_{\mathbb T}[G](ik), {\mathcal F}_{\mathbb T}[G](ik), ik{\mathcal F}_{\mathbb T}[\mathbf{G}](ik)). \end{aligned}$$

Then, \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) are the unique solutions of equations (4.33), which possess the following estimate:

$$\begin{aligned}&\Vert \mathbf{u}_\psi \Vert _{L_p((0, 2\pi ), H^2_q(\dot{\Omega }))} + \Vert \partial _t\mathbf{u}_\psi \Vert _{L_p((0, 2\pi ), L_q(\dot{\Omega }))} + \Vert \nabla {\mathfrak {p}}_\psi \Vert _{L_p((0, 2\pi ), L_q(\dot{\Omega }))} \\&\qquad + \Vert \rho _\psi \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho _\psi \Vert _{H^1_p((0, 2\pi ), W^{2-1/q}_q(S_R))}\\&\quad \le C\{\Vert \mathbf{F}_\psi \Vert _{L_p((0, 2\pi ), L_q(\dot{\Omega }))} + \Vert D_\psi \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \Vert \partial _t\mathbf{G}_\psi \Vert _{L_p((0, 2\pi ), L_q(\dot{\Omega }))} \\&\qquad + \Vert \Lambda ^{1/2}G_\psi \Vert _{L_p((0, 2\pi ), L_q(\dot{\Omega }))} + \Vert G_\psi \Vert _{L_p((0, 2\pi ), H^1_q(\dot{\Omega }))}\\&\qquad +\Vert \Lambda ^{1/2}\mathbf{H}_\psi \Vert _{L_p((0, 2\pi ), L_q(\Omega ))} + \Vert \mathbf{H}_\psi \Vert _{L_p((0, 2\pi ), H^1_q(\Omega ))}\} \end{aligned}$$

for some constant \(C > 0\). Here, we have set

$$\begin{aligned} \Lambda ^{1/2}(G_\psi , \mathbf{H}_\psi ) = {\mathcal F}^{-1}_{\mathbb T}[(ik)^{1/2}\psi (ik)({\mathcal F}_{\mathbb T}[G](ik), {\mathcal F}_{\mathbb T}[\mathbf{H}] (ik))]. \end{aligned}$$

We now consider the lower frequency part of solutions of equations (4.29). Namely, we consider equations (4.31) for \(k \in {\mathbb R}\) with \(1 \le |k| < k_0+4\). We shall show the following theorem.

Theorem 21

Let \(1< q < \infty \) and \(k \in {\mathbb Z}\) with \(|k| \le k_0+3\). Then, for any \(\mathbf{f}_\pm \in L_q(\Omega _\pm )^N\), \(g_\pm \in H^1_q(\Omega _\pm )\), \(d \in W^{2-1/q}_q(S_R)\), \(\mathbf{h}\in H^1_q(\Omega )^N\), and \(\mathbf{g}_\pm \in L_q(\Omega _\pm )^N\), problem (4.31) admits unique solutions \(\mathbf{v}_\pm \in H^2_q(\Omega _\pm )^N\), \({\mathfrak {q}}_\pm \in H^1_q(\Omega _\pm )\) with \(\int _\Omega {\mathfrak {q}}\,\mathrm{d}x=0\), and \(\eta \in W^{3-1/q}_q(S_R)\) possessing the estimate:

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{v}\Vert _{H^2_q(\dot{\Omega })} + \Vert \nabla {\mathfrak {q}}\Vert _{L_q(\dot{\Omega })} + \Vert \eta \Vert _{W^{3-1/q}_q(S_R)} \\&\quad \le C(\Vert \mathbf{f}\Vert _{L_q(\dot{\Omega })} + \Vert d\Vert _{W^{2-1/q}_q(S_R)} + \Vert g\Vert _{H^1_q(\dot{\Omega })} + \Vert \mathbf{g}\Vert _{L_q(\dot{\Omega })} + \Vert \mathbf{h}\Vert _{H^1_q(\Omega )}) \end{aligned} \end{aligned}$$
(4.34)

for some constant \(C > 0\).

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:

$$\begin{aligned} \begin{aligned} ik\mathbf{v}-\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v})-{\mathfrak {p}}\mathbf{I}) = \mathbf{f}&\quad&\text {in }\dot{\Omega }, \\ \mathrm{div}\,\mathbf{v}=0&\quad&\text {in }\dot{\Omega }, \\ ik\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{v}_+)\cdot \mathbf{n}=d&\quad&\text {on }S_R,\\ {[}[\mu \mathbf{D}(\mathbf{v})-{\mathfrak {p}}\mathbf{I}]]\mathbf{n}- ({\mathcal B}_R\rho )\mathbf{n}=0&\quad&\text {on }S_R, \\ [[\mathbf{v}]]=0&\quad&\text {on }S_R, \\ \mathbf{v}_- = 0&\quad&\text {on }S. \end{aligned} \end{aligned}$$
(4.35)

For any \(\mathbf{v}_\pm \in H^2_q(\Omega _\pm )^N\) and \(\rho \in W^{3-1/q}_q(S_R)\), let \(K=K(\mathbf{v}, \rho ) \in \dot{H}^1_q(\dot{\Omega })\) be the unique solution of the weak Neumann problem:

$$\begin{aligned} (\nabla K, \nabla \varphi )_{\dot{\Omega }} = (\mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}))-\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{\dot{\Omega }} \quad \text {for any }\varphi \in \dot{H}^1_{q'}(\Omega ) \end{aligned}$$
(4.36)

subject to the transmission condition:

$$\begin{aligned} {[}[K]]=<[[\mu \mathbf{D}(\mathbf{v})]]\mathbf{n}, \mathbf{n}> - \sigma ({\mathcal B}_R\zeta )\mathbf{n}- [[\mathrm{div}\,\mathbf{v}]] \quad \text {on }S_R, \end{aligned}$$
(4.37)

where \(\mu \) is piecewise constant defined by \(\mu |_{\Omega _\pm } = \mu _\pm \). Here and in the following, \(\dot{H}^1_q(\Omega )\) is defined by setting

$$\begin{aligned} \dot{H}^1_q(\Omega ) = \Big \{\varphi \in H^1_q(\Omega ) \mid \int _\Omega \varphi \, \mathrm{d}x = 0\Big \}. \end{aligned}$$

The reduced problem corresponding to equations (4.35) is

$$\begin{aligned} \begin{aligned} ik\mathbf{v}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}) = \mathbf{f}&\quad&\text {in }\dot{\Omega }, \\ ik\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{v}_+)\cdot \mathbf{n}= d&\quad&\text {on }S_R, \\ [[\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}]]\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}= 0&\quad&\text {on }S_R, \\ {[}[\mathbf{v}]]=0&\quad&\text {on }S_R, \\ \mathbf{v}_-=0&\quad&\text {on }S. \end{aligned} \end{aligned}$$
(4.38)

Let \(J_q(\dot{\Omega })\) be the solenoidal space defined by setting

$$\begin{aligned} J_q(\dot{\Omega }) = \{\mathbf{u}\in L_q(\dot{\Omega }) \mid (\mathbf{u}, \nabla \varphi )_{\dot{\Omega }} = 0 \quad \text {for any }\varphi \in \dot{H}^1_{q'}(\Omega )\}. \end{aligned}$$

For any \(\mathbf{f}\in J_q(\dot{\Omega })\) and \(d \in W^{2-1/q}_q(S_R)\), problems (4.35) and (4.38) are equivalent. In fact, if problem (4.35) admits unique solutions \(\mathbf{v}\in H^2_q(\dot{\Omega })^N\), \({\mathfrak {p}}\in \dot{H}^1_q(\dot{\Omega })\) and \(\rho \in W^{3-1/q}_q(S_R)\), then using the divergence theorem of Gauss and noting that \([[\varphi ]]=0\) on \(S_R\) gives that for any \(\varphi \in \dot{H}^1_{q'}(\Omega )\),

$$\begin{aligned} 0&= (\mathbf{f}, \nabla \varphi )_{\dot{\Omega }} = ik(\mathbf{v}, \nabla \varphi )_{\dot{\Omega }} -(\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{\dot{\Omega }}\\&\quad +(\nabla ({\mathfrak {p}}- K(\mathbf{v}, \rho )), \nabla \varphi )_{\dot{\Omega }} = (\nabla ({\mathfrak {p}}- K(\mathbf{v}, \rho )), \nabla \varphi )_{\dot{\Omega }} \end{aligned}$$

because \(\mathrm{div}\,\mathbf{v}=0\) on \(\dot{\Omega }\). Moreover, the transmission conditions in (4.35) and (4.37) gives that

$$\begin{aligned} {[}[{\mathfrak {p}}-K(\mathbf{v}, \rho )]] = [[\mathrm{div}\,\mathbf{v}]]=0 \quad \text {on }S_R. \end{aligned}$$

Thus, the uniqueness of the weak Neumann problem in \(\dot{H}^1_q(\dot{\Omega })\) yields that \({\mathfrak {p}}- K(\mathbf{v}, \rho ) = 0\) in \(\Omega \). Thus, \(\mathbf{v}\) and \(\rho \) satisfy the equations (4.38).

Conversely, if \(\mathbf{v}\in H^2_q(\dot{\Omega })^N\) and \(\rho \in W^{3-1/q}_q(S_R)\) satisfy equations (4.38), then the divergence theorem of Gauss gives that for any \(\varphi \in \dot{H}^1_{q'}(\Omega )\) we have

$$\begin{aligned} 0 = (\mathbf{f}, \nabla \varphi )_{\dot{\Omega }} = ik(\mathbf{v}, \nabla \varphi )_{\dot{\Omega }} -(\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{\dot{\Omega }} = -\{ik(\mathrm{div}\,\mathbf{v}, \varphi )_{\dot{\Omega }} +(\nabla \mathrm{div}\,\mathbf{v}, \nabla \varphi )_{\dot{\Omega }}\}. \end{aligned}$$

Moreover, the transmission conditions in (4.38) and (4.37) give that

$$\begin{aligned} {[}[\mathrm{div}\,\mathbf{v}]]= <[[\mu \mathbf{D}(\mathbf{v})]]\mathbf{n}, \mathbf{n}> - \sigma ({\mathcal B}_R\zeta ) - [[K]]=0 \quad \text {on }S_R. \end{aligned}$$

Thus, the uniqueness of this weak Neumann problem yields that \(\mathrm{div}\,\mathbf{v}=c\) in \(\dot{\Omega }\) for some global constant c. Now the divergence theorem of Gauss and the boundary conditions in (4.38) yield \(c=0\), that is, \(\mathrm{div}\,\mathbf{v}=0\), which shows that \(\mathbf{v}\), \({\mathfrak {p}}= K(\mathbf{v}, \rho )\) and \(\rho \) 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 \(\mathbf{v}\in H^2_2(\dot{\Omega })^N\) and \(\rho \in W^{5/2}_2(S_R)\) be solutions of the homogeneous equations:

$$\begin{aligned} \begin{aligned} ik\mathbf{v}- \mathrm{Div}\,(\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}) = 0&\quad&\text {in }\dot{\Omega }, \\ ik\rho + {\mathcal M}\rho -({\mathcal A}\mathbf{v}_+)\cdot \mathbf{n}= 0&\quad&\text {on }S_R, \\ [[\mu \mathbf{D}(\mathbf{v}) - K(\mathbf{v}, \rho )\mathbf{I}]]\mathbf{n}- \sigma ({\mathcal B}_R\rho )\mathbf{n}= 0&\quad&\text {on }S_R, \\ [[\mathbf{v}]]=0&\quad&\text {on }S_R, \\ \mathbf{v}_-=0&\quad&\text {on }S, \end{aligned} \end{aligned}$$
(4.39)

and we shall show that \(\mathbf{v}=0\) and \(\rho =0\). Notice that \(\mathrm{div}\,\mathbf{v}= 0\) on \(\dot{\Omega }\). Moreover, by \([[\mathbf{v}]]=0\), we have \(\mathbf{v}\in H^1_q(\Omega ) \cap H^2_q(\dot{\Omega })\). Integrating the second equation in (4.39) over \(S_R\) and using the divergence theorem of Gauss on \(\Omega _+ = B_R\) gives that

$$\begin{aligned} 0&=ik(\rho , 1)_{S_R} + \int _{S_R}\rho \,\mathrm{d}\omega |S_R|\\&\quad - \int _{B_R}\mathrm{div}\,(\mathbf{v}_+-\frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+\,\mathrm{d}y)\,\mathrm{d}x = (ik+|S_R|)\int _{S_R}\rho \,\mathrm{d}\omega |S_R| \end{aligned}$$

because \(\mathrm{div}\,\mathbf{v}_+ = 0\) on \(B_R\), and so \((\rho , 1)_{S_R}=0\). Moreover, multiplying the second equation in (4.39) by \(x_j\) and integrating over \(S_R\), similar arguments lead to

$$\begin{aligned} 0&=ik(\rho , x_j)_{S_R} + \int _{S_R}\rho x_j\,\mathrm{d}\omega (x_j, x_j)_{S_R} - \int _{B_R}\mathrm{div}\,\{x_j(\mathbf{v}_+(x)-\frac{1}{|B_R|}\int _{B_R}\mathbf{v}_+\,\mathrm{d}y)\}\,\mathrm{d}x \\&=ik(\rho , x_j)_{S_R} + \int _{S_R}\rho x_j\,\mathrm{d}\omega (x_j, x_j)_{S_R} -\int _{B_R}(v_{+j}(x) - \frac{1}{|B_R|}\int _{B_R}v_{+j}\,\mathrm{d}y)\,\mathrm{d}x\\&= ik(\rho , x_j)_{S_R} + \int _{S_R}\rho x_j\,\mathrm{d}\omega (x_j, x_j)_{S_R}, \end{aligned}$$

because \((1, x_j)_{S_R}=0\), and \((x_k, x_j)_{S_R} = 0\) for \(j\not =k\). Since \((x_j, x_j)_{S_R} = (R^2/N)|S_R| > 0\), we have \((\rho , x_j) = 0\). Summing up, we have proved

$$\begin{aligned} (\rho , 1)_{S_R} = 0, \quad (\rho , x_j)_{S_R} = 0 \quad (j=1\ldots , N). \end{aligned}$$
(4.40)

In particular, \({\mathcal M}\rho =0\).

We now prove that \(\mathbf{v}= 0\). Multiplying the first equation of (4.39) with \(\mathbf{v}\) and integrating the resultant formula over \(\dot{\Omega }\) and using the divergence theorem of Gauss gives that

$$\begin{aligned} 0 = ik\Vert \mathbf{v}\Vert _{L_2(\dot{\Omega })}^2 - \sigma ({\mathcal B}_R\rho , \mathbf{n}\cdot \mathbf{v})_{S_R} + \frac{\mu }{2} \Vert \mathbf{D}(\mathbf{v})\Vert _{L_2(\dot{\Omega })}^2, \end{aligned}$$

because \(\mathrm{div}\,\mathbf{v}= 0\) in \(\dot{\Omega }\). By the second equation of (4.39) with \({\mathcal M}\rho =0\), we have

$$\begin{aligned} \sigma ({\mathcal B}_R\rho , \mathbf{n}\cdot \mathbf{v})_{S_R} = \sigma ({\mathcal B}_R\rho , ik\rho )_{S_R} + \sum _{k=1}^N\frac{1}{|B_R|}\int _{B_R}w_j\,\mathrm{d}t({\mathcal B}_R\rho , R^{-1}x_j)_{S_R} \end{aligned}$$

where we have used \(\mathbf{n}=R^{-1}x= R^{-1}(x_1,\ldots , x_N)\) for \(x \in S_R\). This also yields

$$\begin{aligned} ({\mathcal B}_R\rho , x_j)_{S_R} = (\rho , (\Delta _{S_R}+\frac{N-1}{R^2})x_j)_{S_R} =0. \end{aligned}$$

Moreover, since \(\rho \) satisfies (4.40), we know that

$$\begin{aligned} -({\mathcal B}_R\rho , \rho )_{S_R} \ge c\Vert \rho \Vert _{L_2(S_R)}^2 \end{aligned}$$

for some positive constant c, and therefore we have \(\mathbf{D}(\mathbf{v})=0\). Since \(\mathbf{v}\in H^1_q(\Omega )\) and \(\mathbf{v}=0\) on \(S_-\), we have \(\mathbf{v}=0\).

Finally, the first equation of (4.39) yields that \(\nabla K(\mathbf{v}, \rho ) = 0\), which shows that \(K(\mathbf{v}, \rho )\) is constant in \(\dot{\Omega }\). Thus, \([[K(\mathbf{v}, \rho )]]\) is constant. Integrating the third equation of (4.39) yields that

$$\begin{aligned} {[}[K(\mathbf{v}, \rho )]]\int _{S_R}\,\mathrm{d}\omega = \sigma (\Delta _{S_R}\rho ,1)_{S_R} + \frac{N-1}{R^2}(\rho , 1)_{S_R} = 0 \end{aligned}$$

where we have used (4.40). In particular, \(K(\mathbf{v}, \rho )\) is a constant globally in \(\Omega \). Finally, we have \({\mathcal B}_R\rho = 0\) on \(S_R\), which, combined with (4.40) leads to \(\rho =0\). This completes the proof of uniqueness for equations (4.38) in the \(L_2\) framework. Therefore, we have proved Theorem 21. \(\square \)

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. \(\square \)

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 \({\mathcal I}_\epsilon \) with some small constant \(\epsilon > 0\) determined later by setting

$$\begin{aligned}&{\mathcal I}_\epsilon = \{(\mathbf{v}, h) \mid \quad \mathbf{v}\in L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(B_R)^N) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)^N), \nonumber \\&\quad h \in L_{p, \mathrm{per}}((0, 2\pi ), W^{3-1/q}_q(S_R)) \cap H^1_{p, \mathrm{per}}((0, 2\pi ), W^{2-1/q}_q(S_R)) \cap H^1_{\infty , \mathrm{per}}((0, 2\pi ),\nonumber \\&W^{1-1/q}_q(S_R)), \nonumber \\&\sup _{t \in (0, 2\pi )}\Vert H_h(\cdot , t)\Vert _{H^1_\infty (B_R)} \le \delta , \quad E(\mathbf{v}, h ) \le \epsilon \}, \end{aligned}$$
(5.1)

where we have set

$$\begin{aligned} E(\mathbf{v}, h)&= \Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \mathbf{v}\Vert _{H^1_p((0, 2\pi ), L^2_q(B_R))}\\&\quad + \Vert h\Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(B_R))} + \Vert h\Vert _{H^1_p((0, 2\pi ), W^{2-1/q}_q(B_R))} \\&\quad +\Vert \partial _th\Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))}. \end{aligned}$$

In view of (2.9), we define \(\xi (t)\) by setting

$$\begin{aligned} \xi (t) = \int ^t_0\xi '(s)\,\mathrm{d}s + c = \frac{1}{|B_R|} \int ^t_0\int _{B_R}\mathbf{v}(x, s)(1+ J_0(x,s))\,\mathrm{d}x\mathrm{d}s + c \end{aligned}$$
(5.2)

where c is a constant for which

$$\begin{aligned}&\int ^{2\pi }_0 \xi (s)\,\mathrm{d}s = 0, \quad \text {that is},\nonumber \\&\quad c = -\frac{1}{2\pi |B_R|}\int ^{2\pi }_0\Bigl (\int ^t_0\int _{B_R}(\mathbf{v}(x, s)(1+ J_0(x,s))\,\mathrm{d}x\mathrm{d}s\Bigr )\,\mathrm{d}t. \end{aligned}$$
(5.3)

We choose \(\delta > 0\) so small that the map \(x =\Phi (y, t)= y + \Psi (y, t)\) with \(\Psi (y, t)=\Psi _h(y, t) = R^{-1}H_h(y, t)y + \xi (t)\) is one to one. In particular, we may assume that \(\delta > 0\) and the inverse map: \(y = \Xi (y, t)\) is well-defined and has the same regularity property as \(\Phi (y, t)\). In particular, we may assume that

$$\begin{aligned} \Xi (D) \subset B_R. \end{aligned}$$
(5.4)

Since \(\epsilon > 0\) will be chosen small eventually, we may assume that \(0< \epsilon < 1\), and so for example, we estimate \(\epsilon ^2 < \epsilon \) if necessary. Let \((\mathbf{v}, h) \in {\mathcal I}_\epsilon \) and let \(\mathbf{u}\) and \(\rho \) be solutions of linearized equations:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t\mathbf{u}+ {\mathcal L}\mathbf{u}_S- \mathrm{Div}\,(\mu (\mathbf{D}(\mathbf{u}) - {\mathfrak {p}}\mathbf{I}) = \mathbf{G}+ \mathbf{F}(\mathbf{v}, h)&\quad&\text {in }B_R\times (0, 2\pi ), \\&\mathrm{div}\,\mathbf{u}= g(\mathbf{v}, h) = \mathrm{div}\,\mathbf{g}(\mathbf{v}, h)&\quad&\text {in }B_R\times (0, 2\pi ), \\&\partial _t\rho + {\mathcal M}\rho -{\mathcal A}\mathbf{u}\cdot \mathbf{n}= \tilde{d}(\mathbf{v}, h)&\quad&\text {on }S_R\times (0, 2\pi ), \\&(\mu \mathbf{D}(\mathbf{u})-{\mathfrak {p}}\mathbf{I})\mathbf{n}- ({\mathcal B}_R\rho ) \mathbf{n}= \mathbf{h}(\mathbf{v}, h)&\quad&\text {on }S_R\times (0, 2\pi ). \end{aligned}\right. \end{aligned}$$
(5.5)

In view of Theorem 6, we shall show that

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{F}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))} \\&\quad + \Vert \tilde{d}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \Vert (g(\mathbf{v}, h), \mathbf{h}(\mathbf{v}, h)\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} \\&\quad + \Vert (g(\mathbf{v}, h), \mathbf{h}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} + \Vert \partial _t\mathbf{g}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon ^2, \end{aligned} \end{aligned}$$
(5.6)

for some constant \(C > 0\) independent of \(\epsilon > 0\). In the following, C denotes generic constants independent of \(\epsilon > 0\), the value of which may change from line to line. Before starting with the estimates of the nonlinear terms, we summarize some inequalities which are useful for our estimations. The following inequalities follow from Sobolev’s inequality and the estimate of the boundary trace:

$$\begin{aligned} \begin{aligned} \Vert f\Vert _{L_\infty (B_R)}&\le C\Vert f\Vert _{H^1_q(B_R)}, \\ \Vert fg\Vert _{H^1_q(B_R)}&\le C\Vert f\Vert _{H^1_q(B_R)}\Vert g\Vert _{H^1_q(B_R)}, \\ \Vert fg\Vert _{H^2_q(B_R)}&\le C(\Vert f\Vert _{H^2_q(B_R)}\Vert g\Vert _{H^1_q(B_R)} + \Vert f\Vert _{H^1_q(B_R)}\Vert g\Vert _{H^2_q(B_R)}), \\ \Vert fg\Vert _{W^{1-1/q}_q(S_R)}&\le C\Vert f\Vert _{W^{1-1/q}_q(S_R)}\Vert g\Vert _{W^{1-1/q}_q(S_R)}, \\ \Vert fg\Vert _{W^{2-1/q}_q(S_R)}&\le C(\Vert f\Vert _{W^{2-1/q}_q(S_R)}\Vert g\Vert _{W^{1-1/q}_q(S_R)} + \Vert f\Vert _{W^{1-1/q}_q(S_R)}\Vert g\Vert _{W^{2-1/q}_q(S_R)}) \end{aligned}\end{aligned}$$
(5.7)

for \(N< q < \infty \) with some constant C. The following inequalities follow from real interpolation theorem and the periodicity of functions, which will be used to estimate the \(L_\infty \) norm with respect to the time variable of lower order regularity terms with respect to the space variable x:

$$\begin{aligned} \begin{aligned} \Vert \mathbf{v}\Vert _{L_\infty ((0, 2\pi ), B^{2(1-1/q)}_{q,p}(B_R))}&\le C(\Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))}), \\ \Vert h\Vert _{L_\infty ((0, 2\pi ), B^{3-1/p-1/q}_{q,p}(S_R))}&\le C(\Vert h\Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _th\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))}). \end{aligned} \end{aligned}$$
(5.8)

In fact, to obtain (5.8) we use the following well-known result: Let X and Y be two Banach spaces such that Y is continuously embedded into X, and then \(C([0, \infty ), (X, Y)_{1-1/p,p})\) is continuously embedded into \(H^1_p((0, \infty ), X) \cap L_p((0, \infty ), Y)\) and

$$\begin{aligned} \Vert f\Vert _{L_\infty ((0, \infty ), (X, Y)_{1-1/p,p})} \le \Vert f\Vert _{L_p((0, \infty ), Y)} + \Vert f\Vert _{H^1_p((0, \infty ), X)}. \end{aligned}$$

For its proof, we refer to [9, 21].

We start with the estimate of \(\mathbf{F}(\mathbf{v}, h)\). From (3.11), we have

$$\begin{aligned}&\Vert \mathbf{F}_1(\mathbf{v}, h)\Vert _{L_q(B_R)} \le C\{\Vert \mathbf{v}\Vert _{L_\infty (B_R)}\Vert \nabla \mathbf{v}\Vert _{L_q(B_R)} + \Vert \partial _t\Psi _h\Vert _{L_\infty (B_R)}\Vert \nabla \mathbf{v}\Vert _{L_q(B_R)} \\&\quad + \Vert \nabla \Psi _h\Vert _{L_\infty (B_R)}\Vert \partial _t\mathbf{v}\Vert _{L_q(B_R)} + \Vert \nabla \Psi _h\Vert _{L_\infty (B_R)}\Vert \nabla ^2\mathbf{v}\Vert _{L_q(B_R)}\\&\quad + \Vert \nabla ^2\Psi _h\Vert _{L_q(B_R)}\Vert \nabla \mathbf{v}\Vert _{L_\infty (B_R)}). \end{aligned}$$

By (5.7) and (2.5), we have

$$\begin{aligned} \Vert \mathbf{F}_1(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))}&\le C\{\Vert \mathbf{v}\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))} \Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \\&\quad +\Vert \partial _th\Vert _{L_p((0, 2\pi ), W^{1-1/q}_q(S_R))}\Vert \mathbf{v}\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}\\&\quad + \Vert h\Vert _{L_\infty ((0, 2\pi ), W^{2-1/q}_q(S_R))}(\Vert \partial _t\mathbf{v}\Vert _{L_q((0, 2\pi ), L_q(B_R))}\\&\quad + \Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))}), \end{aligned}$$

which, combined with (5.8) and (5.1), leads to

$$\begin{aligned} \Vert \mathbf{F}_1(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon ^2, \end{aligned}$$
(5.9)

because \(1 < 2(1-1/p)\) and \(2-1/q < 3-1/p-1/q\). From (3.12), it follows that

$$\begin{aligned}&\Vert \mathbf{F}_2(\mathbf{v}, h)(\cdot , t)\Vert _{L_q(B_R)} \\&\quad \le C\int ^{2\pi }_0\Vert \mathbf{v}(\cdot ,t)\Vert _{L_q(B_R)}(\Vert J_0(\cdot ,t)\Vert _{L_\infty (B_R)}\\&\qquad +\Vert \Psi (\cdot , t)\Vert _{L_\infty (B_R)}(1+\Vert J_0(\cdot ,t)\Vert _{L_\infty (B_R)}))\,\mathrm{d}t \\&\qquad + \int ^{2\pi }_0\Vert \mathbf{v}(\cdot , t)\Vert _{L_q(B_R)}(1+\Vert \Psi (\cdot , t)\Vert _{L_\infty (B_R)}) (1+\Vert J_0(\cdot ,t)\Vert _{L_\infty (B_R)})\,\mathrm{d}t\Vert \Psi (\cdot , t)\Vert _{L_q(B_R)} \\&\qquad + \Vert \nabla \Psi (\cdot , t)\Vert _{L_q(B_R)}\int ^{2\pi }_0 \Vert \mathbf{v}(\cdot , t)\Vert _{L_q(B_R)}(1 + \Vert \Psi (\cdot , t)\Vert _{L_\infty (B_R)}) (1 + \Vert J_0(\cdot ,t)\Vert _{L_\infty (B_R)})\,\mathrm{d}y\mathrm{d}t\\&\qquad \times (1+\Vert \Psi (\cdot , t)\Vert _{L_\infty (B_R)}). \end{aligned}$$

To estimate \(\mathbf{F}_2(\mathbf{v}, h)\), we recall

$$\begin{aligned} J_0(y,t) = \det \Bigl (\delta _{ij} + R^{-1}\frac{\partial }{\partial y_j}H_h(y, t)y_i\Bigr ) - 1 \end{aligned}$$

and that \(\Psi (y, t) = R^{-1}H_h(y, t)y + \xi (t)\), where \(\xi (t)\) is given by

$$\begin{aligned} \begin{aligned} \xi (t)&= \int ^t_0\frac{1}{|B_R|}\int _{B_R}(\mathbf{v}(y, s)(1 + J_0(y,s))\,\mathrm{d}y\mathrm{d}s + c, \\ c&=-\int ^{2\pi }_0\int ^t_0\frac{1}{|B_R|}\int _{B_R}(\mathbf{v}(y, s)(1 + J_0(y,s))\,\mathrm{d}y\mathrm{d}s\mathrm{d}t. \end{aligned} \end{aligned}$$
(5.10)

By (5.7) and (2.5) we obtain

$$\begin{aligned} \begin{aligned} \Vert H_h(\cdot , t)\Vert _{L_\infty (B_R)} \le C\Vert h(\cdot , t)\Vert _{W^{1-1/q}_q(S_R)} \le C\epsilon , \\ \Vert \nabla H_h(\cdot , t)\Vert _{L_\infty (B_R)} \le C\Vert h(\cdot , t)\Vert _{W^{2-1/q}_q(S_R)} \le C\epsilon , \end{aligned} \end{aligned}$$
(5.11)

By (5.7), (2.5), (5.8), the fact that \(2-1/q < 3-1/p-1/q\), and (5.1), we have

$$\begin{aligned} \begin{aligned} \Vert J_0(\cdot ,t)\Vert _{L_\infty (B_R)}&\le C\Vert \nabla H_h(\cdot , t)\Vert _{L_\infty (B_R)}(1 + \Vert \nabla H_h(\cdot , t)\Vert _{L_\infty (B_R)})^{N-1} \\&\le C\Vert h(\cdot , t)\Vert _{W^{2-1/q}_q(S_R)}(1+ \Vert h(\cdot , t)\Vert _{W^{2-1/q}_q(S_R)})^{N-1} \\&\le C\epsilon . \end{aligned} \end{aligned}$$
(5.12)

From (5.10) and (5.1), it follows that

$$\begin{aligned} |\xi (t)| \le C\Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon . \end{aligned}$$
(5.13)

In particular, by (5.11) and (5.13), we have

$$\begin{aligned} \Vert \Psi (\cdot , t)\Vert _{L_\infty (B_R)} \le C\epsilon , \quad \Vert \nabla \Psi (\cdot , t)\Vert _{L_\infty (B_R)} \le C\epsilon . \end{aligned}$$
(5.14)

Combining (5.1) and (5.14) gives that

$$\begin{aligned} \Vert \mathbf{F}_2(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon \Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon ^2, \end{aligned}$$

which, combined with (5.9), leads to

$$\begin{aligned} \Vert \mathbf{F}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\epsilon ^2. \end{aligned}$$
(5.15)

By (5.4) and (5.14), we have

$$\begin{aligned} \Vert \mathbf{G}\Vert _{L_p((0, 2\pi ), L_q(B_R))} \le C\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))}. \end{aligned}$$
(5.16)

We next estimate \(\tilde{d}(\mathbf{v}, h)\). By (3.25) and (5.1),

$$\begin{aligned} \Vert \mathbf{n}_t-\mathbf{n}\Vert _{W^{1-1/q}_q(S_R)}&\le C\Vert H_h(\cdot , t)\Vert _{H^2_q(B_R)} \le C\epsilon , \\ \Vert \mathbf{n}_t-\mathbf{n}\Vert _{W^{2-1/q}_q(S_R)}&\le C(\Vert H_h(\cdot , t)\Vert _{H^3_q(B_R)} + \Vert H_h(\cdot , t)\Vert _{H^2_q(B_R)}\Vert H_h(\cdot , t)\Vert _{H^2_\infty (B_R)}). \end{aligned}$$

Since we assume that \(2/p + N/q < 1\), we can choose \(\kappa > 0\) so small that \(2+N/q + \kappa -1/q < 3-1/p-1/q\) and \(1 + N/q + \kappa < 2(1-1/p)\), and then by Sobolev’s inequality and (5.8) we have

$$\begin{aligned} \begin{aligned}&\sup _{t \in (0, 2\pi )}\Vert \mathbf{v}(\cdot , t)\Vert _{H^{1}_\infty (B_R)} \le C\sup _{t \in (0, 2\pi )}\Vert \mathbf{v}(\cdot , t)\Vert _{B^{2(1-1/p)}_{q,p}(B_R)} \le C\epsilon ; \\&\sup _{t \in (0, 2\pi )}\Vert H_h(\cdot , t)\Vert _{H^2_\infty (B_R)} \le C\sup _{t \in (0, 2\pi )}\Vert h(\cdot , t)\Vert _{B^{3-1/p-1/q}_{q,p}(S_R)} \le C\epsilon , \end{aligned}\end{aligned}$$
(5.17)

where we have used (2.5) in the last inequality. Then, in particular, using again (2.5), we have

$$\begin{aligned} \Vert \mathbf{n}_t-\mathbf{n}\Vert _{W^{2-1/q}_q(S_R)} \le C\Vert H_h(\cdot , t)\Vert _{H^3_q(B_R)} \le C\Vert h(\cdot , t)\Vert _{W^{3-1/q}_q(S_R)}. \end{aligned}$$

Thus, applying (5.12) to the formula in (2.11) and using (5.1) and (5.7) gives that

$$\begin{aligned} \begin{aligned} \Vert d(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))}&\le C\epsilon (\Vert \mathbf{v}\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))} +\Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))}\\&\qquad + \Vert \partial _th\Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))}\\&\qquad +\Vert \partial _t h\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))})\\&\le C\epsilon ^2. \end{aligned} \end{aligned}$$
(5.18)

On the other hand, by (5.11),

$$\begin{aligned} \Vert h(\cdot , t)\Vert _{L_\infty (S_R)} \le C \Vert H_h(\cdot , t)\Vert _{L_\infty (B_R)} \le C\epsilon , \end{aligned}$$

and so

$$\begin{aligned} \Bigl |\int _{S_R}h^k\,\mathrm{d}\omega \Bigr | \le C\epsilon ^2, \quad \Bigl |\int _{S_R}h^k\omega \,\mathrm{d}\omega \Bigr | \le C\epsilon ^2 \quad \text {for }k \ge 2, \end{aligned}$$

which, combined with (5.18), leads to

$$\begin{aligned} \Vert \tilde{d}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} \le C\epsilon ^2. \end{aligned}$$
(5.19)

We next consider \(\mathbf{g}(\mathbf{v}, h)\) given in (3.6), where \(\rho \) is replaced by h. We may write

$$\begin{aligned} \mathbf{g}(\mathbf{v}, h) = \mathbf{V}_\mathbf{g}(\mathbf{k})(H_h, \nabla H_h)\otimes \mathbf{v}. \end{aligned}$$

where \(\mathbf{k}\) denotes variables corresponding to \((H_h, \nabla H_h)\) and \(\mathbf{V}_\mathbf{g}\) is a \(C^\infty \) function defined on \(|\mathbf{k}| < \delta \). We write

$$\begin{aligned} \partial _t\mathbf{g}(\mathbf{v}, h)= & {} \mathbf{V}_\mathbf{g}'(\mathbf{k})\partial _t(H_h, \nabla H_h)\otimes (H_h, \nabla H_h)\otimes \mathbf{v}+ \mathbf{V}_\mathbf{g}(\mathbf{k})\partial _t(H_h, \nabla H_h)\otimes \mathbf{v}\\&+ \mathbf{V}_\mathbf{g}(\mathbf{k})(H_h, \nabla H_h)\otimes \partial _t\mathbf{v}, \end{aligned}$$

and so, by (5.11), (2.5), we have

$$\begin{aligned} \begin{aligned} \Vert \partial _t\mathbf{g}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), L_q(B_R))}&\le C(\Vert \mathbf{v}\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}+\Vert h\Vert _{L_\infty ((0, 2\pi ), W^{2-1/q}_q(S_R))}) \\&\qquad \times (\Vert h\Vert _{H^1_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \Vert \partial _t\mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))}) \\&\quad \le C\epsilon ^2. \end{aligned} \end{aligned}$$
(5.20)

We next estimate \(g(\mathbf{v}, h)\) and \(\mathbf{h}(\mathbf{v}, h)=(\mathbf{h}'(\mathbf{v},h),h_N(\mathbf{v},h))\) given in (3.6), (3.31) and (3.34), where \(\rho \) is replaced by h. We may write

$$\begin{aligned} g(\mathbf{v}, h) = V_g(\mathbf{k})(H_h, \nabla H_h)\otimes \nabla \mathbf{v}, \end{aligned}$$

where \(\mathbf{k}\) are variables corresponding to \((H_h, \nabla H_h)\) and \(V_g(\mathbf{k})\) is some matrix of \(C^\infty \) functions defined on \(|\mathbf{k}| < \delta \). To estimate g, we use the following two lemmas.

Lemma 22

Let \(1<p < \infty \) and \(N< q < \infty \). Let

$$\begin{aligned} f&\in H^1_{\infty , \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{\infty , \mathrm{per}}((0, 2\pi ), H^1_q(B_R)),\\ g&\in H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R)). \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned}&\Vert fg\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert fg\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \\&\quad \le C(\Vert f\Vert _{H^1_\infty ((0, 2\pi ), L_q(B_R))} + \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))})^{1/2} \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}^{1/2}\\&\qquad \times (\Vert g\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert g\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}) \end{aligned} \end{aligned}$$
(5.21)

for some constant \(C > 0\).

Proof

By (5.7), we have

$$\begin{aligned} \Vert fg\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \le \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}\Vert g\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}. \end{aligned}$$
(5.22)

To estimate the \(H^{1/2}\) norm, we use the complex interpolation relation:

$$\begin{aligned} \begin{aligned}&H^{1/2}_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^{1/2}_q(B_R))\\&\quad = \big (L_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)), H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^1_q(B_R))\big )_{1/2} \end{aligned} \end{aligned}$$
(5.23)

where \((\cdot , \cdot )_{1/2}\) denotes a complex interpolation of order 1/2. By (5.7), we have

$$\begin{aligned} \Vert fg\Vert _{H^1_p((0, 2\pi ), L_q(B_R))}&\le C(\Vert \partial _tf\Vert _{L_\infty ((0, 2\pi ), L_q(B_R))} + \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))})\\&\quad \times (\Vert g\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} + \Vert \partial _t g\Vert _{L_p((0, 2\pi ), L_q(B_R))}), \\ \Vert fg\Vert _{L_p((0, 2\pi ), L_q(B_R))}&\le C\Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}\Vert g\Vert _{L_p((0, 2\pi ), L_q(B_R))}.&\end{aligned}$$

Thus, by (5.23), we have

$$\begin{aligned} \begin{aligned} \Vert fg\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))}&\le C (\Vert f\Vert _{H^1_\infty ((0, 2\pi ), L_q(B_R))} \\&\quad + \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))})^{1/2} \Vert f\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))}^{1/2}\\&\quad \times (\Vert g\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert g\Vert _{L_p((0, 2\pi ), H^{1/2}_q(B_R))}) \end{aligned} \end{aligned}$$
(5.24)

Since \(\Vert g\Vert _{L_p((0, 2\pi ), H^{1/2}_q(B_R))} \le C\Vert g\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}\), combining (5.22) and (5.24) leads to (5.21), which completes the proof of Lemma 22. \(\square \)

Lemma 23

Let \(1< p, q < \infty \). Then, there exists a constant C such that for any u with

$$\begin{aligned} u \in H^1_{p, \mathrm{per}}((0, 2\pi ), L_q(B_R)) \cap L_{p, \mathrm{per}}((0, 2\pi ), H^2_q(B_R)), \end{aligned}$$

we have

$$\begin{aligned} \Vert u\Vert _{H^{1/2}_p((0, 2\pi ), H^1_q(B_R))} \le C(\Vert u\Vert _{H^{1}_p((0, 2\pi ), L_q(B_R))} + \Vert u\Vert _{L_p((0, 2\pi ), H^2_q(B_R))}) \end{aligned}$$
(5.25)

for some constant \(C > 0\).

Proof

As was proved in the proof of Proposition 1 in Shibata [17], there exist two operators \(\Phi _1\) and \(\Phi _2\) with

$$\begin{aligned} \Phi _1 \in C^1({\mathbb R}{\setminus }\{0\}, {\mathcal L}(L_q(B_R), L_q(B_R)^N)), \quad \Phi _2 \in C^1({\mathbb R}{\setminus }\{0\}, {\mathcal L}(H^2_q(B_R), L_q(B_R)^N) \end{aligned}$$

such that for any \(g \in H^2_q(B_R)\), we have

$$\begin{aligned} (1+\lambda ^2)^{1/4}\nabla g =\Phi _1(\lambda )(1+\lambda ^2)^{1/2}g + \Phi _2(\lambda )g, \end{aligned}$$

and

$$\begin{aligned}&{\mathcal R}_{{\mathcal L}(L_q(B_R), L_q(B_R)^N)} (\{(\lambda \partial _\lambda )^{\ell }\Phi _1(\lambda ) \mid \lambda \in {\mathbb R}{\setminus }\{0\}\}) \le r_b, \\&{\mathcal R}_{{\mathcal L}(H^2_q(B_R), L_q(B_R)^N)} (\{(\lambda \partial _\lambda )^{\ell }\Phi _1(\lambda ) \mid \lambda \in {\mathbb R}{\setminus }\{0\}\}) \le r_b, \end{aligned}$$

for \(\ell =0,1\) with some constant \(r_b\). Thus, by Weis’ operator-valued Fourier multiplier theorem, Theorem 8, and transference theorem, Theorem 9, we have (5.25), which completes the proof of Lemma 23. \(\square \)

By (5.1), (2.5), (5.7) and (5.17), we have

$$\begin{aligned}&\Vert \partial _tV_g(\mathbf{k})(H_h, \nabla H_h)\Vert _{L_\infty ((0, 2\pi ), L_q(B_R))} \le C\Vert h\Vert _{H^1_p((0, 2\pi ), W^{1-1/q}_q(B_R))} \le C\epsilon , \\&\Vert V_g(\mathbf{k})(H_h, \nabla H_h)\Vert _{L_\infty ((0, 2\pi ), H^1_q(B_R))} \le C\Vert H_h\Vert _{L_\infty ((0, 2\pi ), H^2_q(B_R))} \le C\epsilon . \end{aligned}$$

Thus, by Lemma 22, Lemma 23, and (5.1), we have

$$\begin{aligned} \begin{aligned}&\Vert g(\mathbf{v}, h)\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert g(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}\\&\quad \le C\epsilon (\Vert \nabla \mathbf{v}\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert \nabla \mathbf{v}\Vert _{L_p((0, 2\pi ), H^1_q(B_R))})\\&\quad \le C\epsilon (\Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{v}\Vert _{L_p((0, 2\pi ), L_q(B_R))}) \\&\quad \le C\epsilon ^2. \end{aligned} \end{aligned}$$
(5.26)

Analogously, recalling the definition of \(\mathbf{h}(\mathbf{v}, h)=(\mathbf{h}'(\mathbf{v},h),h_N(\mathbf{v},h))\) given in (3.31) and (3.34), where \(\rho \) is replaced by h, and using Lemma 22, we have

$$\begin{aligned}&\Vert \mathbf{h}(\mathbf{v}, h)\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert \mathbf{h}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \\&\quad \le C\epsilon (\Vert \nabla \mathbf{v}\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert \nabla \mathbf{v}\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \\&\qquad + \Vert \overline{\nabla }^2H_h\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert \overline{\nabla }^2H_h\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}). \end{aligned}$$

Since \(H^{1/2}_p((0, 2\pi ), L_q(B_R)) \supset H^1_p((0, 2\pi ), L_q(B_R))\), we have

$$\begin{aligned} \Vert \overline{\nabla }^2H_h\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} \le C\Vert \overline{\nabla }^2H_h\Vert _{H^1_p((0, 2\pi ), L_q(B_R))}, \end{aligned}$$

and so using Lemma 23, (2.5), and (5.1), we have

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{h}(\mathbf{v}, h)\Vert _{H^{1/2}_p((0, 2\pi ), L_q(B_R))} + \Vert \mathbf{h}(\mathbf{v}, h)\Vert _{L_p((0, 2\pi ), H^1_q(B_R))} \\&\quad \le C\epsilon (\Vert \mathbf{v}\Vert _{H^1_p((0, 2\pi ), L_q(B_R))} + \Vert \mathbf{v}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} \\&\qquad + \Vert \partial _t H_h\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert H_h\Vert _{L_p((0, 2\pi ), H^3_q(B_R))})\\&\quad \le C\epsilon ^2. \end{aligned} \end{aligned}$$
(5.27)

Combining (5.15), (5.19), (5.20), (5.26), and (5.27) gives (5.6). Applying Theorem 6 to equations (5.5) and using (5.6) and (5.16) gives that

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{u}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf{u}\Vert _{L_p((0, 2\pi ), L_q(B_R))}\\&\qquad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))} + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} \\&\quad \le M_1\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} + M_2\epsilon ^2 \end{aligned} \end{aligned}$$
(5.28)

for some constants \(M_1\) and \(M_2\) independent of \(\epsilon \in (0, 1)\). Finally, we estimate \(\Vert \partial _t\rho \Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))}\). From the third equation in equations (5.5), we have

$$\begin{aligned} \Vert \partial _t\rho \Vert _{W^{1-1/q}_q(S_R)} \le \Vert {\mathcal M}\rho \Vert _{W^{1-1/q}_q(S_R)} + \Vert {\mathcal A}\mathbf{u}\Vert _{W^{1-1/q}_q(S_R)} +\Vert \tilde{d}(\mathbf{v}, h)\Vert _{W^{1-1/q}_q(S_R)}. \end{aligned}$$

Therefore, by (5.1), (5.7), (5.8), (5.11), (5.12), and (5.13), we have

$$\begin{aligned} \Vert \partial _t\rho \Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))}&\le C(\Vert \mathbf {u}\Vert _{L_p((0, 2\pi ), H^2_q(B_R))} + \Vert \partial _t\mathbf {u} \Vert _{L_p((0, 2\pi ), L_q(B_R))}\\&\qquad + \Vert \rho \Vert _{L_p((0, 2\pi ), W^{3-1/q}_q(S_R))}\\&\qquad + \Vert \partial _t\rho \Vert _{L_p((0, 2\pi ), W^{2-1/q}_q(S_R))} + \epsilon ^2), \end{aligned}$$

which, combined with (5.28), leads to

$$\begin{aligned} \begin{aligned} E(\mathbf{u}, \rho ) \le M_1'\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} + M_2'\epsilon ^2 \end{aligned} \end{aligned}$$
(5.29)

for some constants \(M_1'\) and \(M_2'\) independent of \(\epsilon \in (0, 1)\). We choose \(\epsilon > 0\) so small that \(M'_2\epsilon < 1/2\) and we assume that \(M_1'\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} \le \epsilon /2\). Then, we have

$$\begin{aligned} E(\mathbf{u}, \rho ) \le \epsilon . \end{aligned}$$
(5.30)

Moreover, by (2.5) and (5.8), we have

$$\begin{aligned} \sup _{t \in (0, 2\pi )} \Vert H_\rho \Vert _{H^1_\infty (B_R))} \le C\Vert \rho \Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))} \le M_3E(\mathbf{u}, \rho ) \le M_3\epsilon . \end{aligned}$$

Choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(0< M_3\epsilon < \delta \), and so \((\mathbf{u}, \rho ) \in {\mathcal I}_\epsilon \). If we define a map \(\Phi \) acting on \((\mathbf{v}, h) \in {\mathcal I}_\epsilon \) by setting \(\Phi (\mathbf{v}, h) = (\mathbf{u}, \rho )\), and then \(\Phi \) is a map from \({\mathcal I}_\epsilon \) into itself. Employing a similar argument as for proving (5.30), we see that for any \((\mathbf{v}_i, h_i) \in {\mathcal I}_\epsilon \) (\(i = 1,2\)),

$$\begin{aligned} E(\Phi (v_1, h_1) - \Phi (v_2, h_2)) \le M_4\epsilon E((\mathbf{v}_1, h_1) - (\mathbf{v}_2, h_2)). \end{aligned}$$

Choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(M_4\epsilon \le 1/2\), and so \(\Phi \) is a contraction map on \({\mathcal I}_\epsilon \). The Banach fixed-point theorem yields the unique existence of a fixed point \((\mathbf{v}, \rho ) \in {\mathcal I}_\epsilon \) of the map \(\Phi \), that is \((\mathbf{v}, \rho ) = \Phi (\mathbf{v}, \rho )\), which is the required solution of equations (2.16). This completes the proof of Theorem 4.

Notes

  1. 1.

    For any f(xt) defined on \(\Omega _t\), we have

    $$\begin{aligned} \frac{d}{\mathrm{d}t}\int _{\Omega _t}f(x, t)\,\mathrm{d}x = \int _{\Omega _t} (\partial _tf + \mathrm{div}\,(f\mathbf{u}))\,\mathrm{d}x, \end{aligned}$$

    which is called the Reynolds transport theorem.

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Mads Kyed: Joint Appointment Professor of Waseda University. Y. Shibata: Partially supported by JSPS Grant-in-aid for Scientific Research (A) 17H0109, and Top Global University Project. Adjunct Faculty member in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh.

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Eiter, T., Kyed, M. & Shibata, Y. On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00619-5

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