---
abstract: 'Consider a free boundary problem of compressible-incompressible two-phase flows with surface tension and phase transition in bounded domains $\Omega_{t+}, \Omega_{t-} \subset {\mathbb{R}}^N$ where the domains are separated by a sharp compact interface $\Gamma_t \subset {\mathbb{R}}^{N - 1}$. We prove a global in time unique existence theorem for such free boundary problem under the assumption that the initial data are near the equilibrium and initial domains are bounded. In particular, we obtain the solution in the maximal $L_p$-$L_q$ regularity class with $2 < p <\infty$ and $N < q < \infty$ and exponential stability of the corresponding analytic semigroup on the infinite time interval.'
address: 'Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Ookubo,Shinjuku-ku, Tokyo,169-8555, Japan'
author:
- KEIICHI WATANABE
title: |
    Global solvability of compressible-incompressible\
    two-phase flows with phase transitions\
    in bounded domains
---

[^1]

Introduction
============

The purpose of this paper is to prove the global solvability of the free boundary problem of compressible-incompressible two-phase flows in bounded domains. Two fluids are separated by a free boundary $\Gamma_t$ and a surface tension and phase transitions are taken into account. Our problem is formulated as follows: Let $\Omega$ be a ball with radius $R+R'$ in $N$-dimensional Euclidean space ${\mathbb{R}}^N$ with boundary either $\Gamma_{\mathrm{I}}$ or $\Gamma_{\mathrm{II}}$. The sharp interface $\Gamma_t$ separates $\Omega$ into $\Omega_{t+}$ and $\Omega_{t-}$ $(t \geq 0)$, where $\Omega_{t+}$ denotes the domain of compressible fluid and $\Omega_{t-}$ denotes the domain of incompressible fluid. We assume that $\mathrm{dist}\,(\Gamma_t, \Gamma_\pm) \in (0,\infty)$, $\Omega_{t+} \cap \Omega_{t-} = \emptyset$, $\Omega \backslash \Gamma_t = \Omega_{t+} \cup \Omega_{t-}$. For any function $f$ defined on $\Omega_{t\pm}$, we write $f_\pm = f|_{\Omega_{t\pm}}$. Then our problem is described by the following system: $$\begin{aligned}
\label{eq-1.1}
& \left\{\begin{aligned}
{\partial}_t \varrho_+ + {\mathrm{div}\,}(\varrho_+ {\mathbf{v}}_+) & = 0 & \qquad & \text{in $\Omega_{t+} \times (0, T)$}, \\
\varrho_+ ({\partial}_t {\mathbf{v}}_+ + {\mathbf{v}}_+ \cdot \nabla {\mathbf{v}}_+)
- {\mathrm{Div}\,}{\mathbf{T}}_+ & = 0 & \qquad & \text{in $\Omega_{t+} \times (0, T)$}, \\
{\mathrm{div}\,}{\mathbf{v}}_- & = 0 & \qquad & \text{in $\Omega_{t-} \times (0, T)$}, \\
\varrho_- ({\partial}_t {\mathbf{v}}_- + {\mathbf{v}}_- \cdot \nabla {\mathbf{v}}_-) - {\mathrm{Div}\,}{\mathbf{T}}_- & = 0 & \qquad & \text{in $\Omega_{t-} \times(0, T)$}    
\end{aligned}\right.
\intertext{with the interface conditions on $\Gamma_t \times (0, T)$:}
\label{eq-1.2}
& \left\{\begin{aligned}
& V_{\Gamma_t} = {\mathbf{v}}_{\Gamma_t} \cdot {\mathbf{n}}_t = \cfrac{{\llbracket}\varrho {\mathbf{v}}{\rrbracket}\cdot {\mathbf{n}}_t}{{\llbracket}\varrho {\rrbracket}}, \\
& {\llbracket}{\mathbf{v}}{\rrbracket}= \j \bigg {\llbracket}\frac{1}{\varrho} \bigg {\rrbracket}{\mathbf{n}}_t, \quad
\j {\llbracket}{\mathbf{v}}{\rrbracket}- {\llbracket}{\mathbf{T}}{\rrbracket}{\mathbf{n}}_t = - \sigma H_{\Gamma_t} {\mathbf{n}}_t, \\
& {\llbracket}\psi {\rrbracket}+ \cfrac{\j^2}{2} \bigg {\llbracket}\cfrac{1}{\varrho^2} \bigg {\rrbracket}- \bigg {\llbracket}\cfrac{1}{\varrho} ({\mathbf{T}}{\mathbf{n}}_t \cdot {\mathbf{n}}_t) \bigg {\rrbracket}= 0, \\
& (\nabla \varrho_+) \cdot {\mathbf{n}}_t |_+ = 0,   
\end{aligned}\right.\end{aligned}$$ and the boundary conditions on $\Gamma_{\mathrm{I}}$ and $\Gamma_{\mathrm{II}}$: $$\label{cond-1.3}
\begin{aligned}
{\mathbf{v}}_+ & = 0, & \quad \nabla \varrho_+ \cdot \omega_+ & = 0 & \qquad & \text{on $\Gamma_{\mathrm{I}}\times (0, T)$}, \\
& & {\mathbf{v}}_- & = 0 & \qquad & \text{on $\Gamma_{\mathrm{II}}\times (0, T)$,}    
\end{aligned}$$ and the initial conditions: $$\label{cond-1.4}
\begin{aligned}
(\varrho_+, {\mathbf{v}}_+) |_{t = 0} & = (\rho_{*+} + \varrho_{0+}, {\mathbf{v}}_{0+}) & \qquad & \text{in $\Omega_{0+}$}, \\
{\mathbf{v}}_- |_{t = 0} & = {\mathbf{v}}_{0-} & \qquad & \text{in $\Omega_{0-}$}
\end{aligned}$$ with $0 < T \leq \infty$. In our model, the *Navier-Stokes-Korteweg equations* and *Navier-Stokes equations* are used to describe the motion of the compressible fluid and incompressible fluid, respectively. Namely, the stress tensor fields ${\mathbf{T}}_\pm$ are defined by $$\begin{aligned}
{\mathbf{T}}_+ = & \mu_+ {\mathbf{D}}({\mathbf{v}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{v}}_+ {\mathbf{I}}- {\mathfrak{p}}_+ {\mathbf{I}}\\
& + \bigg( \frac{\kappa_+ + \varrho_+ \kappa'_+}{2} | \nabla \varrho_+ |^2
+ \kappa_+ \varrho_+ \Delta \varrho_+ \bigg) {\mathbf{I}}- \kappa_+
\nabla \varrho_+ \otimes \nabla \varrho_+, \\
{\mathbf{T}}_- = & \mu_- {\mathbf{D}}({\mathbf{v}}_-)- {\mathfrak{p}}_- {\mathbf{I}}.\end{aligned}$$ In the system, $\varrho_\pm$ are the densities, ${\mathbf{v}}_\pm$ the velocity fields, ${\mathfrak{p}}_\pm$ the pressure fields, $\psi_\pm$ the Helmholtz free energy functions, and $\rho_{*+}$ is some positive constant, $H_{\Gamma_t}$ the $(N-1)$-times mean curvature of $\Gamma_t$, $\sigma$ a positive constant describing the coefficient of the surface tension, $V_t$ the velocity of evolution of $\Gamma_t$ with respect to ${\mathbf{n}}_t$, ${\mathbf{v}}_{\Gamma_t}$ the interfacial velocity, ${\mathbf{n}}_t$ the unit outer normal to $\Gamma_t$ pointed from $\Omega_{t+}$ to $\Omega_{t-}$, and $\omega_+$ the unit outer normal to $\Gamma_{\mathrm{I}}$. In addition, for $x_0 \in \Gamma_t$ the quantity of $f$ across $\Gamma_t$ is defined by ${\llbracket}f{\rrbracket}(x_0, t) = f (x, t)|_- - f (x, t)|_+$, where the function $f (x, t)$ is defined on $x \in \Omega_{t+} \cup \Omega_{t-}$ and $f (x, t)|_\pm$ $(x \in \Omega_{t\pm})$ denote $$\begin{aligned}
f (x, t)|_\pm = \lim_{\substack{x \to x_0 \\ x \in \Omega_{t \pm}}} f (x, t).\end{aligned}$$

Pr[ü]{}ss *et al*. [@PS2012; @PSW2014; @PSSS2012; @PS2016; @PSSW2016] and Shimizu and Yagi [@SY2015; @SY2017] studied the thermodynamically correct model of the incompressible and incompressible two-phase flows with phase transitions. In particular, Pr[ü]{}ss, Shimizu, and Wilke [@PSW2014] proved the stability of the equilibria of the problem. On the other had, the compressible and incompressible two-phase flows with phase transitions was studied by Shibata [@Shi2016]. However, his result includes the derivative loss in the nonlinear term with respect to $\varrho_+$ due to the kinetic equation: ${\mathbf{v}}_{\Gamma_t} \cdot {\mathbf{n}}_t = {\llbracket}\varrho {\mathbf{v}}{\rrbracket}\cdot {\mathbf{n}}_t / {\llbracket}\varrho {\rrbracket}$. Namely, we can not prove the local solvability of the problem based on his result. To overcome this difficulty, the new model using the Navier-Stokes-Korteweg equations was proposed by the author [@Wat2018]. The new model is an extension of the *Navier-Stokes-Fourier equations*, and the physical consistency was discussed in the previous paper [@Wat2018], see also [@Wat2019]. The local solvability of the problem was also showed in the previous paper [@Wat2019]. For further historical review, the readers may consult the introduction in [@Wat2018; @Wat2019].

The difference between our model and the Navier-Stokes-Fourier equations is that we add the Korteweg tensor and interface condition $\nabla \varrho_+ \cdot \omega|_+ = 0$. The condition $\nabla \varrho_+ \cdot \omega|_+ = 0$ guarantees the *generalized Gibbs-Thomson law* and *Stefan law* and also means that the *interstitial working* $(\kappa_+ \varrho_+ {\mathrm{div}\,}{\mathbf{v}}_+)\nabla \varrho_+$ does not work in the normal direction of the interface. Notice that our model is thermodynamically consistent model, see Watanabe [@Wat2018 Section 2].

The goal of this paper is to show the global existence of unique strong solution to the free boundary problem – in bounded regions. To this end, we first transform the free boundary problem to a fixed boundary system. Let $B_R^\pm$ be the domains transformed from $\Omega_{t \pm}$ by the *Hanzawa transform* we explain below. In this paper, we consider the following two types of problem:

1.  The case: $\Gamma_{\mathrm{I}}= \emptyset$ and $\Gamma_{\mathrm{II}}= S_{R + R'}$,

2.  The case: $\Gamma_{\mathrm{I}}= S_{R + R'}$ and $\Gamma_{\mathrm{II}}= \emptyset$.

Here, $S_{R + R'}$ is defined by $S_{R + R'} := \{ x \in {\mathbb{R}}^N \mid |x| = R + R' \}$. We call the problem (I) *Type I* and the problem (II) *Type II*. Notice that we do not consider the boundary condition if the boundary is an empty set. Before explaining the fixed boundary system, we introduce some assumptions.

Let $B_R^+$ and $B_R^-$ be domains occupied by the compressible and incompressible fluid, respectively. In addition, let $S_R$ be hypersurface, which separate $\Omega$ into $B_R^+$ and $B_R^-$. Define $S_R := \{ x \in {\mathbb{R}}^N \mid |x| = R \}$ and $B_R := \{x \in {\mathbb{R}}^N \mid |x| < R \}$. We then assume that

1.  $B_R^\pm$ are either $B_R$ or $\Omega \backslash (B_R \cup S_R)$.

2.  \[A2\]$B_R^+ = B_R$ and $B_R^- = \Omega \backslash (B_R \cup S_R)$ for Type I; $B_R^- = B_R$ and $B_R^+ = \Omega \backslash (B_R \cup S_R)$ for Type II.

3.  \[A3\]The barycenter of $B_R^-$ is origin, that is, $$\begin{aligned}
    \int_{B_R^-} x {\,\mathrm{d}x}= 0.\end{aligned}$$

4.  $\Gamma$ is a normal perturbation of $S_R$, that is, $$\begin{aligned}
    \Gamma = \{x = y + h_0(y) (y / |y|) \mid y \in S_R \} = \{x = (1 + R^{-1} h_0(y)) y \mid y \in S_R \},\end{aligned}$$ where $h_0$ is some given small function defined on $S_R$.

5.  There exist positive constants $\pi_{*\pm}$ such that $$\begin{gathered}
    \psi_- (\rho_{*-}) - \Big(\psi_+ (\rho_{*+}, 0) + {\partial}_{\rho_+} \psi_+ (\rho_{*+}, 0) \varrho_+ |_+\Big) = \frac{\pi_{*-}}{\rho_{*-}} - \frac{\pi_{*+}}{\rho_{*+}}, \\
    \pi_{*-} - \pi_{*+} = \frac{\sigma(N - 1)}{R},\end{gathered}$$ which denotes the *modified Gibbs-Thomson condition* and the *Young-Laplace law*, respectively.

6.  \[A6\]The coefficients $\mu_+$, $\nu_+$, and $\kappa_+$ satisfy $$\begin{aligned}
    \bigg(\frac{\mu_+ + \nu_+}{2\rho_{*+}^2\kappa_+}\bigg)^2 \neq \frac{1}{\rho_{*+} \kappa_+},
    \quad \rho_{*+}^3 \kappa_+ \neq \mu_+ \nu_+, \quad
    \nu_+ > \frac{N -2}{N} \mu_+.\end{aligned}$$

Let the free boundary $\Gamma_t$ be given by $$\Gamma_t = \{x = y + h(y, t) (y / |y|) + \xi(t) \mid y \in S_R \} = \{x = (1 + R^{-1} h(y, t)) y + \xi (t) \mid y \in S_R \},$$ where $h(y, t)$ is an unknown height function with $h(y, 0) = h_0(y)$ for $y\in S_R$ and $\xi(t)$ is the barycenter of the domain $\Omega_{t-}$ defined by $$\begin{aligned}
\xi(t) = \frac{1}{|B_R^-|} \int_{\Omega_{t-}} x {\,\mathrm{d}x},\end{aligned}$$ which is also unknown function. Here, $|D|$ denotes the Lebesgue measure of a Lebesgue measure set $D \subset {\mathbb{R}}^N$. To solve the eigenvalue problem for $\Delta_{S_R}$ we have to consider the barycenter of the domain of the incompressible fluid, see Shibata [@Shi2018]. Note that by the assumption (A\[A3\]) we have $\xi(0) = 0$. In addition, from the Reynolds transport theorem we see that $$\begin{aligned}
\xi'(t) = \frac{1}{|B_R^-|} \int_{\Omega_{t-}} {\mathbf{v}}_- (x, t) {\,\mathrm{d}x}.\end{aligned}$$

For given function $h(y,t)$, let $H_h (y, t)$ be a solution to the following Dirichlet problem: $$\begin{aligned}
(1 - \Delta) H_h = 0 \quad \text{in $B_R$}, \quad H_h = h \quad \text{on $S_R$}.\end{aligned}$$ We then introduce the Hanzawa transform centered at $\xi(t)$ defined by $$\begin{aligned}
x= {\mathbf{H}}_{h, \xi}(y, t) := (1 + R^{-1} H_h (y, t)) y + \xi(t) \quad \text{for $y \in B_R$}.\end{aligned}$$ Furthermore, let $H_{h_0}(y)$ be a solution to the Dirichlet problem: $$\begin{aligned}
(1 - \Delta) H_{h_0} = 0 \quad \text{in $B_R$}, \quad H_{h_0} = h_0 \quad \text{on $S_R$}.\end{aligned}$$ From this viewpoint, we set ${\mathbf{H}}_{h_0, \xi}(y) := (1 + R^{-1} H_{h_0}(y)) y$ for $y \in B_R$ because $\xi(0) = 0$. In the sequel, we may assume that $$\begin{aligned}
\|H_h (\cdot, t)\|_{W^1_\infty(B_R)} \leq \varepsilon_0 \quad \text{for any $t \in [0,T)$}.\end{aligned}$$ Taking $\varepsilon_0$ suitably small such that $\varepsilon_0 \in (0, 1/4)$, the Hanzawa transform is a bijection mapping from $B_R$ onto $\Omega_t$, where we have set $$\begin{aligned}
\Omega_{t\pm} = \{x = y \pm (R^{-1} H_h(y, t) y + \xi(t) ) \mid y \in B^\pm_R \}.\end{aligned}$$ In fact, from $|H_h (y, t) - H_h (z, t)| \leq \| \nabla H_h (\cdot, t) \|_{L_\infty (B^\pm_R)} |y - z|$ we obtain $$\begin{aligned}
|{\mathbf{H}}_{h, \xi}(y, t) - {\mathbf{H}}_{h, \xi}(z, t)| \geq (1 - 2 \varepsilon_0) |y -z| \geq \frac{1}{2}|y - z|\end{aligned}$$ for any $y, z \in B^\pm_R$. Hence, in the following, we will choose $\varepsilon_0$ so small that our main results hold with some sufficient conditions.

Using the Hanzawa transform, which we explain above, we consider the fixed-boundary problem instead of the free boundary problem. To this end, we now introduce some symbols. We first set $$\begin{aligned}
\rho_{0+} = \varrho_{0+} \circ {\mathbf{H}}_{h_0, \xi}, \quad
{\mathbf{u}}_{0\pm} = {\mathbf{v}}_{0\pm} \circ {\mathbf{H}}_{h_0, \xi},\end{aligned}$$ where $\varrho_{0+}$ and ${\mathbf{v}}_{0\pm}$ are initial data . For functions $\varrho_+$, ${\mathbf{v}}_\pm$, and ${\mathfrak{p}}_-$ satisfying Eqs. and , we set $$\begin{aligned}
\rho_+ (y,t) = \varrho_+ \circ {\mathbf{H}}_{h, \xi}, \quad {\mathbf{u}}_\pm (y,t) = {\mathbf{v}}_\pm \circ {\mathbf{H}}_{h, \xi}, \quad \pi_- (y,t) = {\mathfrak{p}}_- \circ {\mathbf{H}}_{h,\xi} - \frac{\sigma(N - 1)}{R}.\end{aligned}$$ Then, by the Hanzawa transform, Eqs. , , , and are transformed to the following fixed boundary problem: $$\begin{aligned}
\label{eq-1.13}
\left\{\begin{aligned}
{\partial}_t \rho_+ + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_+ & = f_\rho (\rho_+, {\mathbf{u}}_+, h) &  & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_- = f_d ({\mathbf{u}}_-, h) & = \rho_{*-} {\mathrm{div}\,}{\mathbf{F}}_d ({\mathbf{u}}_-, h) & & \text{ in $B_R^- \times (0, T)$}, \\
\rho_{*+} {\partial}_t {\mathbf{u}}_+ - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}& \\
+ (- \gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} & = {\mathbf{f}}_+(\rho_+, {\mathbf{u}}_+, h) & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_- - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - \pi_- {\mathbf{I}}\} & = {\mathbf{f}}_- ({\mathbf{u}}_-, h) & &\text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t h - \frac{\langle \rho_{*-} {\mathbf{u}}_-, \omega \rangle|_- - \langle\rho_{*+} {\mathbf{u}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R \times (0, T)$}, \\
\Pi_0 (\mu_- {\mathbf{D}}({\mathbf{u}}_-) \omega)|_- - \Pi_0 (\mu_+ {\mathbf{D}}({\mathbf{u}}_+) \omega)|_+ & = g (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R \times (0, T)$}, \\
\{\langle {\mathbf{D}}({\mathbf{u}}_-) \omega, \omega \rangle - \pi_-\}|_- & \\
- \{\langle \mu_+ {\mathbf{D}}({\mathbf{u}}_+) \omega, \omega \rangle + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ & \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+\}|_+ - \sigma {\mathcal{B}}h & = f_a (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R\times(0, T)$}, \\
\frac{1}{\rho_{*-}} \langle \{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - \pi_- {\mathbf{I}}\} \omega, \omega \rangle \Big|_- & \\
- \Big\{\frac{1}{\rho_{*+}} \langle\{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}& \\
+ (- \gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} \omega, \omega \rangle \Big\} \Big|_+ & = f_b (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)
& & \text{ on $S_R \times (0, T)$}, \\
\Pi_0 {\mathbf{u}}_- |_- - \Pi_0 {\mathbf{u}}_+ |_+ & = {\mathbf{h}}({\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R\times(0,T)$}, \\
\langle \nabla \rho_+, \omega \rangle |_+ & = 0 & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{u}}_+ = 0, \qquad \langle \nabla \rho_+, \omega_+ \rangle & = 0 & &\text{ on $\Gamma_{\mathrm{I}}\times (0, T)$}, \\
{\mathbf{u}}_- & = 0 & & \text{ on $\Gamma_{\mathrm{II}}\times (0, T)$}, \\
(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) |_{t = 0} & = (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) & &\text{ on $I_R$}
\end{aligned}\right.    \end{aligned}$$ for $T \in (0, \infty]$, where $\gamma_{*+} = \pi_+'(\rho_{*+}) > 0$, $\omega = y / |y| \in S_1$, $I_R = B_R^+ \times B_R^+ \times B_R^- \times S_R$, ${\mathcal{B}}= \langle \Delta_{S_R} \omega, \omega \rangle + \Delta_{S_R}$, and $\Delta_{S_R}$ denotes the *Laplace-Beltrami operator* on $S_R$. The boundary $S^\pm_R$ exists and its condition holds for type $i$ ($i = {\mathrm{I}}, {\mathrm{II}}$), respectively.

\(1) Setting $\Psi = R^{-1} H_h (y, t) y + \xi(t)$, the nonlinear terms can be seen in the previous paper [@Wat2019 Section 3] so that we omit the representation of the nonlinear terms. Since we apply not the Gibbs-Thomson condition but *modified* Gibbs-Thomson condition, the eighth equation in is different from the previous paper [@Wat2019 Section 3]. Namely, the lower order term ${\partial}_\rho \psi_+(\rho_{*+}, 0) \rho_+$ does not appear in the right-hand side of the eighth equation in .\
(2) Generally, the assumption $\gamma_{*+} > 0$ does not hold because the Helmholtz free energy $\psi_+$ is not monotone with respect ot density $\varrho_+$ in the Navier-Stokes-Korteweg flows. We, however, consider near the equibalium that $\gamma_{*+} > 0$ holds in this paper.

Before stating our main results, we first explain *compatibility conditions* for $h_0$. The assumptions (A\[A2\]) and (A\[A3\]) imply the following conditions for $h_0$: for Type I [$$\begin{gathered}
\begin{aligned}
\frac{\{(R + R')^N - R^N\} \omega^N}{N} & = \int_{B_R^-} {\,\mathrm{d}x}\\
& = \frac{(R + R')^N \omega^N}{N} - \int_{B_R^+} {\,\mathrm{d}x}\\
& = \frac{(R + R')^N \omega^N}{N} - \int_{|\omega| = 1} \int^{R + h_0 (R \omega)}_{0} r^{N-1} \,\mathrm{d}r \,\mathrm{d}\omega \\
& = \frac{(R + R')^N \omega^N}{N} - \int_{|\omega| = 1} \frac{(R + h_0 (R \omega))^N}{N} \,\mathrm{d}\omega,
\end{aligned} \\
\begin{aligned}
0 = \int_{B_R^-} x_i {\,\mathrm{d}x}& = \int_{|\omega| = 1} \int^{R + R'}_{R + h_0 (R \omega)} \omega_i r^N \,\mathrm{d}r \,\mathrm{d}\omega \\
& = \int_{|\omega| = 1} \frac{(R + R')^{N + 1} - (R + h_0 (R \omega))^{N + 1}}{N + 1} \omega_i \,\mathrm{d}\omega
\end{aligned}\end{gathered}$$]{} where $i$ denote $i = 1, \dots, N$; for Type II $$\begin{gathered}
\frac{R^N \omega^N}{N} = \int_{B_R^-} {\,\mathrm{d}x}=\int_{|\omega| = 1} \int^{R + h_0 (R \omega)}_{0} r^{N-1} \,\mathrm{d}r \,\mathrm{d}\omega
= \int_{|\omega| = 1} \frac{(R + h_0 (R \omega))^N}{N} \,\mathrm{d}\omega, \\
0 = \int_{B_R^-} x_i {\,\mathrm{d}x}= \int_{|\omega| = 1} \int_{0}^{R + h_0 (R \omega)} \omega_i r^N \,\mathrm{d}r \,\mathrm{d}\omega
= \int_{|\omega| = 1} \frac{(R + h_0 (R \omega))^{N + 1}}{N + 1} \omega_i \,\mathrm{d}\omega \quad (i = 1, \dots, N).\end{gathered}$$ Since $\int_{S_1} \,\mathrm{d}\omega = \omega_N$ and $\int_{S_1} \omega_i \,\mathrm{d}\omega = 0$, the compatibility conditions for $h_0$ is as follows: $$\begin{split}
\sum_{k = 1}^{N} \begin{pmatrix} N \\ k \end{pmatrix} \int_{S_R}(R^{-1} h_0 (y))^k \,\mathrm{d}\tau = 0, \\
\sum_{k = 1}^{N + 1} \begin{pmatrix} N + 1 \\ k \end{pmatrix} \int_{S_R} y_i (R^{-1} h_0(y))^k \,\mathrm{d}\tau = 0,
\end{split}$$ where $\mathrm{d}\tau$ denotes the area element of $S_R$ and we have set $$\begin{aligned}
\begin{pmatrix} N \\ k \end{pmatrix} = \frac{N!}{k! (N - k)!}, \quad
\begin{pmatrix} N + 1 \\ k \end{pmatrix} = \frac{N!}{k! (N + 1 - k)!}, \quad 
m! = 1 \cdot 2 \cdots \cdot m \enskip (m \geq 1), \quad 0! = 1.\end{aligned}$$ Note that the compatibility conditions for $h_0$ have no difference between type I and type II.

We know that ${\mathcal{B}}x_i = 0$ on $S_R$ for $i = 1, \dots, N$. In fact, it is wildly known that $\Delta$ is denoted by $$\begin{aligned}
\Delta = \frac{{\partial}^2}{{\partial}R^2} + (N - 1) \frac{1}{R} \frac{{\partial}}{{\partial}R} + \frac{1}{R^2} \Delta_{S_R},\end{aligned}$$ where $x_i = R \omega_i$ for $i = 1, \dots, N$. We then see that $$\begin{aligned}
0 = \Delta x = \bigg( \frac{{\partial}^2}{{\partial}R^2} + (N - 1) \frac{1}{R} \frac{{\partial}}{{\partial}R} + \frac{1}{R^2} \Delta_{S_R} \bigg) R \omega_i
= \frac{1}{R^2} \bigg((N - 1) + \Delta_{S_R} \bigg) \omega_i = {\mathcal{B}}\omega_i.\end{aligned}$$ From this view point, we set $\varphi_1 = |S_R|^{-1}$ and introduce the functions $\varphi_i$ for $i = 2, \dots, N + 1$, which are some linear combinations of $x_1, \dots, x_N$ such that $(\varphi_i, \varphi_j)_{S_R} = 0$ for $i, j = 1, \dots, N + 1$. Since we have $(\varphi_1, \varphi_i)_{S_R} = 0$ for $i = 2, \dots, N + 1$, the sequence $\{\varphi_j\}_{j = 1}^{N + 1}$ is the orthogonal subspace of $\{\psi \mid {\mathcal{B}}\psi = 0 \text{ on $S_R$} \}\cup {\mathbb{C}}$ with respect to the inner product in $L_2(S_R)$.

For simplicity of notation, we introduce boundary operators ${\mathbf{B}}_{\mathrm{I}}$ and ${\mathbf{B}}_{\mathrm{II}}$ such that $$\begin{aligned}
{\mathbf{B}}_{\mathrm{I}}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h)
& =
\begin{pmatrix}
\Pi_0 (\mu_- {\mathbf{D}}({\mathbf{u}}_-) \omega)|_- - \Pi_0 (\mu_+ {\mathbf{D}}({\mathbf{u}}_+) \omega)|_+ \\
B_1 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
B_2 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
\Pi_0 {\mathbf{u}}_- |_- - \Pi_0 {\mathbf{u}}_+ |_+ \\
\langle \nabla \rho_+, \omega \rangle|_+ \\
{\mathbf{u}}_-|_{S^-_R}
\end{pmatrix}, \\
{\mathbf{B}}_{\mathrm{II}}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h)
&=
\begin{pmatrix}
\Pi_0 (\mu_- {\mathbf{D}}({\mathbf{u}}_-) \omega)|_- - \Pi_0 (\mu_+ {\mathbf{D}}({\mathbf{u}}_+) \omega)|_+ \\
B_1 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
B_2 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
\Pi_0 {\mathbf{u}}_-|_- - \Pi_0 {\mathbf{u}}_+ |_+ \\
\langle \nabla \rho_+, \omega \rangle|_+ \\
{\mathbf{u}}_+|_{S^+_R} \\
\langle \nabla \rho_+, \omega_+ \rangle|_{S^+_R}
\end{pmatrix},\end{aligned}$$ where we have set $$\begin{aligned}
& B_1 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
& = \{\langle {\mathbf{D}}({\mathbf{u}}_-) \omega, \omega \rangle - \pi_-\}|_-
- \{\langle \mu_+ {\mathbf{D}}({\mathbf{u}}_+) \omega, \omega \rangle + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ + (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+\}|_+ \\
& \quad - \sigma {\mathcal{B}}h, \\
& B_2 (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \\
& = \frac{1}{\rho_{*-}} \langle \{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - \pi_- {\mathbf{I}}\} \omega, \omega \rangle \Big|_- \\
& \quad - \Big \{\frac{1}{\rho_{*+}} \langle \{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} \omega , \omega \rangle \Big\}\Big|_+.\end{aligned}$$ We then can write the Eq. as $$\begin{aligned}
\label{eq-1.15}
\left\{\begin{aligned}
{\partial}_t \rho_+ + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_+ & = f_\rho (\rho_+, {\mathbf{u}}_+, h) &  & \text{ in $B_R^+ \times (0, \infty)$,} \\
\rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_- = f_d ({\mathbf{u}}_-, h) & = \rho_{*-} {\mathrm{div}\,}{\mathbf{F}}_d ({\mathbf{u}}_-, h) & & \text{ in $B_R^- \times (0, \infty)$}, \\
\rho_{*+} {\partial}_t {\mathbf{u}}_+ - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} & = {\mathbf{f}}_+ (\rho_+, {\mathbf{u}}_+, h) & & \text{ in $B_R^+ \times (0, \infty)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_- - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - \pi_- {\mathbf{I}}\} & = {\mathbf{f}}_- ({\mathbf{u}}_-, h) & & \text{ in $B_R^- \times (0, \infty)$}, \\
{\partial}_t h - \frac{\langle \rho_{*-} {\mathbf{u}}_-, \omega \rangle|_- - \langle\rho_{*+} {\mathbf{u}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R \times (0, \infty)$}, \\
{\mathbf{B}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) & = {\mathbf{G}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S^i_R \times (0, \infty)$}, \\
(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)|_{t = 0} & = (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) & & \text{ on $I_R$},
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$, where we have set $$\begin{gathered}
{\mathbf{G}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h)
= \begin{cases}
{}^\top\! (g, f_a, f_b, {\mathbf{h}}, 0, 0, 0) & \text{for $i = {\mathrm{I}}$}, \\
{}^\top\! (g, f_a, f_b, {\mathbf{h}}, 0, 0) & \text{for $i = {\mathrm{II}}$},
\end{cases} \\
S^i_R
=\begin{cases}
(S_R)^5 \times (\Gamma_{\mathrm{II}})^2 & \text{for $i = {\mathrm{I}}$}, \\
(S_R)^5 \times \Gamma_{\mathrm{I}}& \text{for $i = {\mathrm{II}}$}.
\end{cases}\end{gathered}$$ In addition, let $$\begin{gathered}
S_{p, q} (a, b) =
\Set{(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) |
\begin{array}{l}
\rho_+ \in L_p((a, b), W^3_q(B_R^+)) \cap W^1_p((a, b), W^1_q(B_R^+)), \\
{\mathbf{u}}_\pm \in L_p((a, b), W^2_q(B_R^\pm)^N) \cap W^1_p((a, b), L_q(B_R^\pm)^N), \\
\pi_- \in L_p((a, b), W^1_q(B_R^-) + \hat{W}^1_{q, 0}(B_R^-)), \\
h \in L_p((a, b), W^{3 - 1 / q}_q(S_R)) \cap W^1_p((a, b), W^{2 - 1 / q}_q(S_R))
\end{array}},\end{gathered}$$ which is the solution class of Eq. . Here and in the following, $W^1_q(B_R^-) + \hat{W}^1_{q, 0}(B_R^-)$ is defined by $$\begin{aligned}
W^1_q(B_R^-) + \hat W^1_{q, 0}(B_R^-)
=\{p_1 + p_2 \mid p_1 \in W^1_q(B_R^-),\enskip p_2 \in \hat W^1_{q, 0}(B_R^-) \},\end{aligned}$$ where we have set $$\begin{aligned}
\hat{W}^1_{q, 0} (B_R^-) = \{\theta \in L_{q, \mathrm{loc}} (B_R^-) \mid \nabla \theta \in L_q (B_R^-)^N, \enskip \theta|_{S_R} = 0\}\end{aligned}$$ for $q \in (1, \infty)$.

Then the main result of this paper is the following.

\[th-1.3\] Let $2 < p < \infty$ and $N < q < \infty$ with $2/p + 1/q < 1$. Then there exists a positive constant $\varepsilon_* \in (0, 1)$ such that for any initial data $\rho_{0+} \in B^{3 - 2/p}_{q, p}(B_R^+)$, ${\mathbf{u}}_{0+} \in B^{2(1 - 1/p)}_{q, p} (B_R^+)$, ${\mathbf{u}}_{0-} \in B^{2(1- 1/p)}_{q, p}(B_R^+)$, $h_0 \in B^{3 - 1/p - 1/q}_{q, p}(S_R)$ satisfying the smallness condition: $$\begin{aligned}
\|\rho_{0+}\|_{B^{3-2/p}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0+}\|_{B^{2(1-1/p)}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0-}\|_{B^{2(1-1/p)}_{q,p}(B_R^{-})}
+\|h_0\|_{B^{3-1/p-1/q}_{q,p}(S_R)} \leq \varepsilon_*\end{aligned}$$ and the compatibility conditions: $$\begin{aligned}
{\mathrm{div}\,}{\mathbf{u}}_{0-} = f_d({\mathbf{u}}_{0-}, h_0) = {\mathrm{div}\,}{\mathbf{F}}_d ({\mathbf{u}}_{0-}, h_0) \quad \text{ in $B_R^-$}\end{aligned}$$ and $$\begin{aligned}
\left\{\begin{aligned}
\Pi_0(\mu_- {\mathbf{D}}({\mathbf{u}}_{0-}) \omega)|_- - \Pi_0(\mu_+ {\mathbf{D}}({\mathbf{u}}_{0+}) \omega)|_+ & = g (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) &\quad &\text{ on $S_R$}, \\
\Pi_0 {\mathbf{u}}_{0-}|_- - \Pi_0 {\mathbf{u}}_{0+}|_+ & = {\mathbf{h}}(\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) &\quad &\text{ on $S_R$}, \\
\langle \nabla \rho_{0+}, \omega \rangle|_+ & = 0 &\quad &\text{ on $S_R$}, \\
\langle \nabla \rho_{0+}, \omega_+ \rangle = 0, \quad
{\mathbf{u}}_{0+} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{I}}$}, \\
{\mathbf{u}}_{0-} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{II}}$}.
\end{aligned}\right.\end{aligned}$$ Furthermore, suppose that

1.  Assumption \[Ass-1.1\] holds.

2.  The pressure field $\pi_+ (\rho_+)$ is $C^2$ function defined on $\rho_{*+}/3 \leq \rho_+\leq 3\rho_{*+}$ such that $0\leq \pi'_{+}(\rho_+)\leq \pi^{*}$ with some positive constant $\pi^{*}$ for any $\rho_{*+}/3 \leq \rho_+\leq 3\rho_{*+}$.

3.  The Helmholtz free energy $\psi_+(\rho_+,|\nabla\rho_+|^2)$ is $C^2$ function defined on $(\rho_{*+}/3, 3\rho_{*+})\times(0,\infty)$ such that $0\leq \psi'_{+}(\rho_+,|\nabla\rho_+|^2)\leq \psi^{*}$ with some positive constant $\psi^{*}$ for any $\rho_{*+}/3 \leq \psi_{+}\leq 3\rho_{*+}$.

Then Eq. admit a unique solution $(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h) \in S_{p, q} (0, \infty)$ satisfying the estimate: $$\begin{aligned}
&\|e^{-\gamma t}{\partial}_t \rho_+\|_{L_p((0, \infty),W^1_q(B_R^{+}))}
+\|e^{-\gamma t}\rho_+\|_{L_p((0, \infty),W^3_q(B_R^{+}))} 
+\|e^{-\gamma t}{\partial}_t {\mathbf{u}}_+\|_{L_p((0, \infty),L_q(B_R^{+}))} \\
& +\|e^{-\gamma t}{\mathbf{u}}_+\|_{L_p((0, \infty),W^2_q(B_R^{+}))}
+\|e^{-\gamma t}{\partial}_t {\mathbf{u}}_-\|_{L_p((0, \infty),L_q(B_R^{-}))}
+\|e^{-\gamma t}{\mathbf{u}}_-\|_{L_p((0, \infty),W^2_q(B_R^{-}))} \\
& +\|e^{-\gamma t} \nabla \pi_-\|_{L_p((0, \infty),L_q(B_R^-))}
+ \|e^{-\gamma t}{\partial}_t h \|_{L_p((0, \infty),W^{2-1/p}_q(S_R))}
+\|e^{-\gamma t} h \|_{L_p((0, \infty),W^{3-1/p}_q(S_R))} \\
& \leq C \varepsilon_*\end{aligned}$$ for some positive constants $C$ and $\gamma$ independent of $\varepsilon_*$.

\(1) Taking $\varepsilon_0$ such that $0 < \varepsilon_0 < 1/4$, the Hanzawa transform ${\mathbf{H}}_{h, \xi}$ is a bijection mapping from $B_R^\pm$ onto $\Omega_{t \pm}$. Hence, there exists a unique solution $(\varrho_+, {\mathbf{v}}_+, {\mathbf{v}}_-, {\mathfrak{p}}_-, \Omega_{t+}, \Omega_{t-})$ satisfying – for any $t \in (0, \infty)$.\
(2) Theorem \[th-1.3\] holds for all compact boundary $\Gamma_{\mathrm{II}}$, although we assume that $\Gamma_{\mathrm{II}}$ is a sphere.

Preliminaries
=============

This section is devoted to the presentation of several definitions and theorem needed later on. We denote the sets of all complex numbers, natural numbers, and real numbers by ${\mathbb{C}}$, ${\mathbb{N}}$, and ${\mathbb{R}}$, respectively. We set ${\partial}_t={\partial}/{\partial}t$ and ${\partial}_i={\partial}/{\partial}x_i$ $(i=1,\dots,N)$. For multi-index $\alpha=(\alpha_1,\dots,\alpha_N)\in{\mathbb{N}}\cup \{ 0 \}$, we write ${\partial}^\alpha_x f={\partial}^{\alpha_1}_1\cdots{\partial}^{\alpha_N}_N f$.

For any scalar field $\theta$, we set $\nabla\theta=({\partial}_1\theta,\dots,{\partial}_N\theta)$. For any $N$-vectors ${\mathbf{a}}= {}^\top\!(a_1, \dots, a_N)$, $\nabla{\mathbf{a}}$ is the $N\times N$ matrix whose $(i,j)^\mathrm{th}$ component is ${\partial}_i a_j$, ${\mathbf{D}}({\mathbf{a}})$ is the deformation tensor whose $(j,k)^\mathrm{th}$ components are $D_{jk}({\mathbf{a}})={\partial}_j a_k + {\partial}_k a_j$, and $\nabla\cdot{\mathbf{a}}= {\mathrm{div}\,}{\mathbf{a}}= \sum_{j=1}^{N} {\partial}_j a_j$ is the divergence of ${\mathbf{a}}$. For any $N$-vectors ${\mathbf{a}}$ and ${\mathbf{b}}$, we set $\langle{\mathbf{a}},{\mathbf{b}}\rangle={\mathbf{a}}\cdot{\mathbf{b}}=\sum_{j=1}^{N}{\mathbf{a}}_j{\mathbf{b}}_j$. Let ${\mathbf{I}}$ be the $N\times N$ identity matrix. For any $N\times N$ matrix ${\mathbf{A}}$ with $(i,j)^\mathrm{th}$ components $A_{ij}$, the quantity ${\mathrm{Div}\,}{\mathbf{A}}$ is the $N$-vectors with $i^\mathrm{th}$ component of $\sum_{j=1}^N {\partial}_j A_{ij}$.

For any domain $D \subset {\mathbb{R}}^N$, let $B^s_{q,p}(D)$, $L_q(D)$, and $W^m_q(D)$ be the inhomogeneous Besov spaces, Lebesgue spaces, and Sobolev spaces on $D$, respectively, and we define their norms by $\|\cdot\|_{B^s_{q,p}(D)}$, $\|\cdot\|_{L_q(D)}$, and $\|\cdot\|_{W^m_q(D)}$, respectively. We denote the standard Lebesgue spaces and Sobolev spaces of $X$-valued functions defined on $(a, b) \subset {\mathbb{R}}$ as $L_p((a, b), X)$ and $W^m_p((a, b), X)$ for $1 < p < \infty$, respectively, and their norm are denoted by $\|\cdot\|_{L_p((a, b), X)}$ and $\|\cdot\|_{W^m_p((a, b), X)}$, respectively. For a Banach space $X$ the $X$-valued Bessel potential spaces of order $1/2$ are defined by $$\begin{aligned}
H^{1/2}_p({\mathbb{R}},X)
=&\{f\in L_p({\mathbb{R}},X)\mid \|f\|_{H^{1/2}_p({\mathbb{R}},X)}<\infty \},\\
\|f\|_{H^{1/2}_p({\mathbb{R}},X)}=&\|{\mathcal{F}}^{-1}[(1+|\xi|^2)^{-1/4}{\mathcal{F}}[f]]\|_{L_p({\mathbb{R}},X)},\end{aligned}$$ where ${\mathcal{F}}$ and ${\mathcal{F}}^{-1}$ are the Fourier transform and the Fourier inverse transform given by $$\begin{aligned}
{\mathcal{F}}[f](\xi)=\int_{{\mathbb{R}}^N}e^{-ix\cdot\xi}f(x){\,\mathrm{d}x}, \quad
{\mathcal{F}}^{-1}[g](x)=\frac{1}{(2\pi)^N}\int_{{\mathbb{R}}^N}
e^{i\xi\cdot x}g(\xi)\,\mathrm{d}\xi.\end{aligned}$$ For simplicity of notation, in this paper we use the following symbols: $$\begin{aligned}
H^{1,1/2}_{q,p}({\mathbb{R}}, D)=&L_p({\mathbb{R}},W^1_q(D))\cap H^{1/2}_p({\mathbb{R}},L_q(D)), \\
W^{2,1}_{q,p}({\mathbb{R}}, D)=&L_p({\mathbb{R}},W^2_q(D))\cap W^1_p({\mathbb{R}},L_q(D)), \end{aligned}$$ for $1 < p,q < \infty$ and a domain $D \subset {\mathbb{R}}^N$.

For a Banach space $X$ with norm $\|\cdot\|_X$, let $X^d=\{(f_1,\dots,f_d) \mid f_i\in X\enskip (i=1,\dots,d) \}$, and write the norm of $X^d$ as $\|\cdot\|_X$ defined by $\|f\|_X=\sum_{j=1}^{d}\|f_j\|_X$ for $f=(f_1,\dots,f_d)\in X^d$. We set $({\mathbf{f}},{\mathbf{g}})_D=\int_D {\mathbf{f}}\cdot{\mathbf{g}}{\,\mathrm{d}x}$ and $({\mathbf{f}},{\mathbf{g}})_{{\partial}D}=\int_{{\partial}D} {\mathbf{f}}\cdot{\mathbf{g}}{\,\mathrm{d}s}$, where ${\,\mathrm{d}s}$ denotes the surface element on ${\partial}D$. Let $C^\infty_0$ be the set of all $C^\infty$ functions with compact support. The letter $C$ denotes a constant and $C_{a,b,c,\dots}$ denotes the constant depending on $a$, $b$, $c$, ..., and the value of $C$ and $C_{a,b,c,\dots}$ may change from line to line.

We now recall the result of the local solvability of problem on the time interval $(0, T)$, where $T \in (0, \infty)$. For more details we refer the reader to Watanabe [@Wat2019 Theorem 2.5].

\[th-2.1\] Let $2 < p < \infty$ and $N < q < \infty$ with $2/p + 1/q < 1$. In addition, let $T > 0$ and assume that there exists a positive constant $\varepsilon_T$ depending on $T$ such that if the initial data satisfy the smallness condition: $$\begin{aligned}
\|\rho_{0+}\|_{B^{3-2/p}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0+}\|_{B^{2(1-1/p)}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0-}\|_{B^{2(1-1/p)}_{q,p}(B_R^{-})}
+\|h_0\|_{B^{3-1/p-1/q}_{q,p}(S_R)} \leq \varepsilon_T.\end{aligned}$$ Furthermore, suppose that

1.  Assumption \[Ass-1.1\] holds.

2.  The pressure field $\pi_+ (\rho_+)$ is $C^2$ function defined on $\rho_{*+}/3 \leq \rho_+\leq 3\rho_{*+}$ such that $0\leq \pi'_{+}(\rho_+)\leq \pi^{*}$ with some positive constant $\pi^{*}$ for any $\rho_{*+}/3 \leq \rho_+\leq 3\rho_{*+}$.

3.  The Helmholtz free energy $\psi_+(\rho_+,|\nabla\rho_+|^2)$ is $C^2$ function defined on $(\rho_{*+}/3, 3\rho_{*+})\times(0,\infty)$ such that $0\leq \psi'_{+}(\rho_+,|\nabla\rho_+|^2)\leq \psi^{*}$ with some positive constant $\psi^{*}$ for any $\rho_{*+}/3 \leq \psi_{+}\leq 3\rho_{*+}$.

4.  The initial data $(\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) \in I_R$ satisfies the compatibility conditions: $$\begin{aligned}
    {\mathrm{div}\,}{\mathbf{u}}_{0-} = f_d({\mathbf{u}}_{0-}, h_0) = {\mathrm{div}\,}{\mathbf{F}}_d ({\mathbf{u}}_{0-}, h_0) \quad \text{ in $B_R^-$}\end{aligned}$$ and $$\begin{aligned}
    \left\{\begin{aligned}
    \Pi_0(\mu_- {\mathbf{D}}({\mathbf{u}}_{0-}) \omega)|_- - \Pi_0(\mu_+ {\mathbf{D}}({\mathbf{u}}_{0+}) \omega)|_+ & = g (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) &\quad &\text{ on $S_R$}, \\
    \Pi_0 {\mathbf{u}}_{0-}|_- - \Pi_0 {\mathbf{u}}_{0+}|_+ & = {\mathbf{h}}(\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) &\quad &\text{ on $S_R$}, \\
    \langle \nabla \rho_{0+}, \omega \rangle|_+ & = 0 &\quad &\text{ on $S_R$}, \\
    \langle \nabla \rho_{0+}, \omega_+ \rangle = 0, \quad
    {\mathbf{u}}_{0+} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{I}}$}, \\
    {\mathbf{u}}_{0-} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{II}}$}.
    \end{aligned}\right.\end{aligned}$$

Then there exists a unique solution $(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h)$ of on $[0,T]$ with $(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h) \in S_{p, q} (0, T)$ satisfying the estimate: $$\begin{aligned}
{\mathbb{I}}_{p,q}(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h,0;(0,T))\leq \varepsilon_T.\end{aligned}$$ Here and in the sequel, for $\delta\in [0,\infty)$ and $0\leq a<b\leq \infty$ we set $$\begin{aligned}
&{\mathbb{I}}_{p,q}(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h,\delta;(a,b))\\
&:=\|e^{-\delta t}{\partial}_t \rho_+\|_{L_p((a,b),W^1_q(B_R^{+}))}
+\|e^{-\delta t}\rho_+\|_{L_p((a,b),W^3_q(B_R^{+}))}
+\|e^{-\delta t}\rho_+\|_{L_\infty((a,b),B^{3-2/p}_{q,p}(B_R^{+}))}\\
&\quad + \|e^{-\delta t}{\partial}_t {\mathbf{u}}_+\|_{L_p((a,b),L_q(B_R^{+}))}
+\|e^{-\delta t}{\mathbf{u}}_+\|_{L_p((a,b),W^2_q(B_R^{+}))}
+\|e^{-\delta t}{\mathbf{u}}_+\|_{L_\infty((a,b),B^{2(1-1/p)}_{q,p}(B_R^{+}))}\\
&\quad +\|e^{-\delta t}{\partial}_t {\mathbf{u}}_-\|_{L_p((a,b),L_q(B_R^{-}))}
+\|e^{-\delta t}{\mathbf{u}}_-\|_{L_p((a,b),W^2_q(B_R^{-}))}
+\|e^{-\delta t}{\mathbf{u}}_-\|_{L_\infty((a,b),B^{2(1-1/p)}_{q,p}(B_R^{-}))}\\
&\quad+\|e^{-\delta t} \nabla \pi_-\|_{L_p((a,b),L_q(B_R^-))}
+ \|e^{-\delta t}{\partial}_t h \|_{L_p((a,b),W^{2-1/p}_q(S_R))}
+\|e^{-\delta t} h \|_{L_p((a,b),W^{3-1/p}_q(S_R))}\\
&\quad+\|e^{-\delta t} h \|_{L_\infty((a,b),B^{3-1/p-1/q}_{q,p}(S_R))}.\end{aligned}$$

Exponential stability of solutions to the linearized problem
============================================================

In this section, we prove the exponential stability of solutions to the following linearized system: $$\begin{aligned}
\label{eq-4.1}
\left\{\begin{aligned}
{\partial}_t \rho_+ + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_+
& = f_\rho &  & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_- = f_d & = \rho_{*-} {\mathrm{div}\,}{\mathbf{F}}_d
& & \text{ in $B_R^- \times (0, T)$}, \\
\rho_{*+} {\partial}_t {\mathbf{u}}_+ - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\}
& = {\mathbf{f}}_+ & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_- - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - \pi_- {\mathbf{I}}\}
& = {\mathbf{f}}_- & & \text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t h - \frac{\langle \rho_{*-} {\mathbf{u}}_-, \omega \rangle|_- -\langle \rho_{*+} {\mathbf{u}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{B}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) & = {\mathbf{G}}_i & & \text{ on $S^i_R \times (0, T)$}, \\
(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)|_{t = 0} & = (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0)
& & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$ and $T > 0$, where $f_\rho$, $f_d$, ${\mathbf{F}}_d$, ${\mathbf{f}}_\pm$, $d$, and ${\mathbf{G}}_i$ $(i = {\mathrm{I}}, {\mathrm{II}})$ are given functions defined on $t \in {\mathbb{R}}$.

The following is the main theorem of this section.

\[th-4.1\] Let $1 < p, q < \infty$ with $2/p + 1/q \neq 1$ and $2/p + 1/q \neq 2$. Furthermore, let $\{\varphi_j \}_{j = 1}^{N + 1}$ be orthogonal basis of $\{\psi \mid {\mathcal{B}}\psi = 0\enskip \text{on $S_R$} \} \cup {\mathbb{C}}$. Then there exists a constant $\gamma > 0$ such that the following assertion holds: Assume that

1.  Assumption \[Ass-1.1\] holds.

2.  The initial data $(\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) \in I_R$ satisfies the compatibility conditions: $$\begin{aligned}
    \label{com-1}
    {\mathrm{div}\,}{\mathbf{u}}_{0-} = f_d|_{t = 0} = {\mathrm{div}\,}{\mathbf{F}}_d|_{t = 0} \quad \text{ in $B_R^-$}.\end{aligned}$$ In addition, we assume that the compatibility conditions: $$\begin{aligned}
    \label{com-2}
    \left\{\begin{aligned}
    \Pi_0 (\mu_- {\mathbf{D}}({\mathbf{u}}_{0-}) \omega)|_- - \Pi_0 (\mu_+ {\mathbf{D}}({\mathbf{u}}_{0+}) \omega)|_+ & = g|_{t = 0} &\quad &\text{ on $S_R$}, \\
    \Pi_0 {\mathbf{u}}_{0-}|_- - \Pi_0 {\mathbf{u}}_{0+}|_+ & = {\mathbf{h}}|_{t = 0} &\quad &\text{ on $S_R$}, \\
    \langle \nabla \rho_{0+}, \omega \rangle|_+ & = 0 &\quad &\text{ on $S_R$}, \\
    \langle \nabla \rho_{0+}, \omega_+ \rangle = 0, \quad
    {\mathbf{u}}_{0+} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{I}}$}, \\
    {\mathbf{u}}_{0-} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{II}}$}
    \end{aligned}\right.\end{aligned}$$ hold when $2/p + 1/q < 1$, while we assume that the compatibility conditions: $$\begin{aligned}
    \label{com-3}
    \left\{\begin{aligned}
    \langle\nabla\rho_{0+}, \omega_+ \rangle = 0, \quad
    {\mathbf{u}}_{0+} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{I}}$}, \\
    {\mathbf{u}}_{0-} & = 0 &\quad &\text{ on $\Gamma_{\mathrm{II}}$}
    \end{aligned}\right.\end{aligned}$$ hold when $1 < 2/p + 1/q < 2$.

In addition, let $f_\rho$, $f_d$, ${\mathbf{F}}_d$, ${\mathbf{f}}_+$, ${\mathbf{f}}_-$, $d$, $g$, $f_a$, $f_b$, and ${\mathbf{h}}$ be given functions in the right-hand side of  with $$\begin{gathered}
e^{-\gamma t} f_\rho \in L_p ({\mathbb{R}},W^1_q(B_R^+)), \quad
e^{-\gamma t} {\mathbf{f}}_+ \in L_p ({\mathbb{R}},L_q(B_R^+)),\quad
e^{-\gamma t} {\mathbf{f}}_- \in L_p ({\mathbb{R}}, J_q(B_R^-) ), \\
e^{-\gamma t} {\mathbf{F}}_d \in W^1_p ({\mathbb{R}},L_q(B_R^-)^N),\quad
e^{-\gamma t} f_d \in L_p ({\mathbb{R}},W^1_q(B_R^-)^N) \cap H^{1/2}_p ({\mathbb{R}}, L_q(B_R^-)), \\
e^{-\gamma t} d \in L_p ({\mathbb{R}}, W^{2-1/q}_q(S_R)),\enskip
e^{-\gamma t} (g, f_a, f_b) \in H^{1,1/2}_{q,p} ({\mathbb{R}},\dot{B_R}),\enskip
e^{-\gamma t} {\mathbf{h}}\in W^{2,1}_{q,p}({\mathbb{R}}, \dot{B_R}),\end{gathered}$$ where $\dot{B_R} = B_R^+ \cup B_R^-$. Then there exists a unique solution $(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h)$ of with $(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \in S_{p, q}(0, \infty)$ satisfying the estimate: $$\begin{aligned}
{\mathbb{I}}_{p,q}(\rho_+,{\mathbf{u}}_+,{\mathbf{u}}_-,\pi_-,h,\gamma;(0,T))
& \leq C \bigg\{{\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma) \\
& \quad + \sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h(\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p} \bigg\}\end{aligned}$$ for some positive constant $C$ independent of $T$. Here and in the sequel, for $\delta\in [0,\infty)$ we set $$\begin{aligned}
& {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \delta) \\
& = \|\rho_{0+}\|_{B^{3 - 2/p}_{q, p}(B^+_R)} + \|{\mathbf{u}}_{0+}\|_{B^{2(1 - 1/p)}_{q, p}(B^+_R)} + \|{\mathbf{u}}_{0-}\|_{B^{2(1 - 1/p)}_{q, p}(B^-_R)} + \|h_0\|_{B^{3 - 1/p -1/q}_{q, p}(S_R)} \\
& \quad + \|e^{-\delta t} f_\rho\|_{L_p({\mathbb{R}}, W^1_q(B_R^+))}
+ \|e^{-\delta t} {\mathbf{f}}_+ \|_{L_p({\mathbb{R}}, L_q(B_R^+))} 
+ \|e^{-\delta t} {\mathbf{f}}_- \|_{L_p({\mathbb{R}}, L_q(B_R^-))} \\
&\quad + \|e^{-\delta t} f_d \|_{L_p({\mathbb{R}}, W^1_q(B_R^-))}
+ \|e^{-\delta t} f_d \|_{H^{1/2}_p({\mathbb{R}}, L_q(B_R^-))}
+ \|e^{-\delta t} {\partial}_t {\mathbf{F}}_d \|_{L_p({\mathbb{R}}, L_q(B_R^-))}\\
&\quad +\|e^{-\delta t} d \|_{L_p({\mathbb{R}}, W^{2-1/q}_q(S_R))}
+\|e^{-\delta t} (f_a, f_b)\|_{L_p({\mathbb{R}}, L_q(\dot{B_R}))}
+\|e^{-\delta t} (g, f_a, f_b, \nabla {\mathbf{h}})\|_{H^{1/2}_p({\mathbb{R}}, L_q(\dot{B_R}))}\\
&\quad+\|e^{-\delta t} (\nabla g, \nabla f_a, \nabla f_b, {\partial}_t{\mathbf{h}}, \nabla^2{\mathbf{h}}) \|_{L_p({\mathbb{R}}, L_q(\dot{B_R}))}.\end{aligned}$$

We first consider the solutions $\rho_+$, ${\mathbf{u}}_\pm$, $\pi_-$, and $h$ of Eq. of the form: $\rho_+ = \rho_{1+} + \rho_{2+}$, ${\mathbf{u}}_\pm = {\mathbf{u}}_{1\pm} + {\mathbf{u}}_{2\pm}$, $\pi_- = \pi_{1-} + \pi_{2-}$, and $h = h_1 + h_2$, where $\rho_{1+}$, ${\mathbf{u}}_{1+}$, ${\mathbf{u}}_{1-}$, $\pi_{1-}$, and $h_1$ satisfy the shifted equations: $$\begin{aligned}
\label{eq-4.2}
\left\{\begin{aligned}
{\partial}_t \rho_{1+} + \lambda_1 \rho_{1+} + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{1+}
& = f_\rho & & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_{1-} = f_d = \rho_{*-} {\mathrm{div}\,}& {\mathbf{F}}_d & & \text{ in $B_R^- \times (0, T)$},\\
\rho_{*+} {\partial}_t {\mathbf{u}}_{1+} + \lambda_1 {\mathbf{u}}_{1+} - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_{1+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_{1+} {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_{1+} {\mathbf{I}}\} & = {\mathbf{f}}_+ & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_{1-} + \lambda_1 {\mathbf{u}}_{1-} - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{1-}) - \pi_{1-} {\mathbf{I}}\} & = {\mathbf{f}}_- & &\text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t h_1 + \lambda_1 h_1 - \frac{\langle \rho_{*-} {\mathbf{u}}_{1-}, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_{1+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{B}}_i (\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1) & = {\mathbf{G}}_i & & \text{ on $S^i_R \times (0, T)$}, \\
(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, h_1)|_{t = 0} = (\rho_{0+}, {\mathbf{u}}_{0+}& , {\mathbf{u}}_{0-}, h_0) & & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$ and $T > 0$; while $\rho_{2+}$, ${\mathbf{u}}_{2+}$, ${\mathbf{u}}_{2-}$, $\pi_{2-}$, and $h_2$ satisfy the equations $$\begin{aligned}
\label{eq-4.3}
\left\{\begin{aligned}
{\partial}_t \rho_{2+} + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{2+} & = \lambda_1 \rho_{1+} & & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_{2-} & = 0 & & \text{ in $B_R^- \times (0, T)$}, \\
\rho_{*+} {\partial}_t {\mathbf{u}}_{2+} - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_{2+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_{2+} {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_{2+} {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_+ & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_{2-} - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{2-}) - \pi_{2-} {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1-} & & \text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t h_2 - \frac{\langle \rho_{*-} {\mathbf{u}}_{2-}, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_{2+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = \lambda_1 h_1 & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{B}}_i (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, \pi_{2-}, h_2) & = 0
& & \text{ on $S^i_R \times (0, T)$}, \\
(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)|_{t = 0} & = (0, 0, 0, 0) & &\text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$ and $T > 0$. For the shifted equation , we have the following theorem.

Let $1 < p, q < \infty$ with $2/p + 1/q \neq 1$ and $2/p + 1/q \neq 2$. Then there exists a constant $\gamma_0 > 0$ and $\lambda_1 > 0$ such that the following assertion holds: Suppose that

1.  Assumption \[Ass-1.1\] holds.

2.  The initial data $(\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) \in I_R$ satisfies the compatibility condition . In addition, we assume that the compatibility conditions hold when $2/p + 1/q < 1$, while we assume that the compatibility conditions hold when $1 < 2/p + 1/q < 2$.

In addition, let $f_\rho$, $f_d$, ${\mathbf{F}}_d$, ${\mathbf{f}}_+$, ${\mathbf{f}}_-$, $d$, $g$, $f_a$, $f_b$, and ${\mathbf{h}}$ be given functions in the right-hand side of  with $$\begin{gathered}
e^{-\gamma t} f_\rho \in L_p ({\mathbb{R}},W^1_q(B_R^+)), \quad
e^{-\gamma t} {\mathbf{f}}_+ \in L_p ({\mathbb{R}},L_q(B_R^+)),\quad
e^{-\gamma t} {\mathbf{f}}_- \in L_p ({\mathbb{R}}, J_q(B_R^-) ), \\
e^{-\gamma t} {\mathbf{F}}_d \in W^1_p ({\mathbb{R}},L_q(B_R^-)^N),\quad
e^{-\gamma t} f_d \in L_p ({\mathbb{R}},W^1_q(B_R^-)^N) \cap H^{1/2}_p ({\mathbb{R}}, L_q(B_R^-)), \\
e^{-\gamma t} d \in L_p ({\mathbb{R}}, W^{2-1/q}_q(S_R)),\enskip
e^{-\gamma t} (g, f_a, f_b) \in H^{1,1/2}_{q,p} ({\mathbb{R}},\dot{B_R}),\enskip
e^{-\gamma t} {\mathbf{h}}\in W^{2,1}_{q,p}({\mathbb{R}}, \dot{B_R}).\end{gathered}$$ Then there exists a unique solution $(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1)$ of with $(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1) \in S_{p, q}(0, \infty)$ satisfying the estimate: $$\begin{aligned}
\label{4.7}
{\mathbb{I}}_{p,q}(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1, \gamma;(0,t))
& \leq C {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma)\end{aligned}$$ for any $t \in (0, T]$ and $\gamma \leq \gamma_0$ with some positive constant $C$ independent of $t$ and $T$.

Using the similar argument as in the proof of the previous paper [@Wat2019 Theorem 2.5], we see that there exists a unique solution $(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1)$ satisfying the required estimate.

To consider Eq. in semigroup framework, we have to eliminate the divergence equation and the pressure term $\pi_-$. Let $J_q (B_R^-)$ be the *solenoidal space* for $B_R^-$ defined by $$\begin{aligned}
J_q (B_R^-) = \{{\mathbf{f}}_- \in L_q (B_R^-)^N \mid ({\mathbf{f}}_-, \nabla \varphi)_{B_R^-} =0 \quad \text{for any $\varphi \in \hat{W}^1_{q', 0} (B_R^-)$} \},\end{aligned}$$ where we have set $$\begin{aligned}
\hat{W}^1_{q', 0} (B_R^-) = \{\varphi \in L_{q, \mathrm{loc}} (B_R^-) \mid \nabla \varphi \in L_q (B_R^-)^N, \quad \varphi|_{S_R} = 0\}.\end{aligned}$$ Since $C^\infty_0 (B^-_R)$ is dense in $W^1_{q, 0} (B_R^-) := \{\varphi \in W^1_q (B_R^-) \mid \varphi|_{S_R} = 0\}$, ${\mathbf{u}}_- \in J_q (B_R^-)$ holds if and only if ${\mathrm{div}\,}{\mathbf{u}}_- = 0$ in $B_R^-$. Let $\widetilde{\varphi} \in W^1_{q, 0} (B^-_R)$ be a solution to the following variational problem: $$\begin{aligned}
(\nabla \widetilde{\varphi}, \nabla \varphi)_{B_R^-} = ({\mathbf{u}}_{1-}, \nabla \varphi)_{B^-_R} \quad \text{for any $\varphi \in \hat{W}^1_{q', 0} (B^-_R) $}.\end{aligned}$$ Note that by the Poincaré inequality we see that $\widetilde{\varphi} \in \hat{W}^1_{q', 0} (B^-_R)$. In addition, let ${\mathbf{w}}_-$ be a function defined by ${\mathbf{w}}_- = {\mathbf{u}}_{1-} - \nabla \widetilde{\varphi}$, and then we see that ${\mathbf{w}}_- \in J_q (B_R^-)$. Since $B_R^-$ is a bounded domain, the weak Dirichlet-Neumann problem is uniquely solvable on $\hat{W}^1_{m}(B^-_R)$ $(m = q, q')$ and the following estimate holds: $$\begin{aligned}
\| \widetilde{\varphi} \|_{L_q (B_R^-)} \leq C \| {\mathbf{u}}_{1-} \|_{L_q (B_R^-)}.\end{aligned}$$ Hence, we obtain $$\begin{aligned}
\| {\mathbf{w}}_- \|_{L_q (B_R^-)} + \| \widetilde{\varphi} \|_{L_q (B_R^-)} \leq C \| {\mathbf{u}}_{1-} \|_{L_q (B_R^-)}.\end{aligned}$$ Using ${\mathbf{w}}_-$ and $\widetilde{\varphi}$, which we define above, we rewrite the fourth equation in as follows: $$\begin{aligned}
\rho_{*-} {\partial}_t {\mathbf{u}}_{2-} - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{2-}) - (\pi_{2-} + \lambda_1 \widetilde{\varphi}) {\mathbf{I}}\} = \lambda_1 {\mathbf{w}}_- \quad \text{ in $B_R^- \times (0, T)$}.\end{aligned}$$ In the following, we may assume that ${\mathbf{u}}_{1-}$ satisfy $$\begin{aligned}
{\mathbf{u}}_{1-} \in W^1_p ((0, T), J_q (B_R^-)) \cap L_p ((0, T), W^2_q (B_R^-)).\end{aligned}$$ We now define a functional $$\begin{aligned}
K ({\mathbf{u}}_-, h) \in W^1_q (B^-_R) + \hat{W}^1_{q, 0} (B_R^-)\end{aligned}$$ as a unique solution to the *weak Dirichlet-Neumann problem*: $$\begin{aligned}
(\nabla K ({\mathbf{u}}_-, h), \nabla \varphi)_{B_R^-} = (\mu_-{\mathrm{Div}\,}({\mathbf{D}}({\mathbf{u}}_-)) - \rho_{*-} \nabla {\mathrm{div}\,}{\mathbf{u}}_-, \nabla \varphi)_{B_R^-}\end{aligned}$$ for any $\varphi \in \hat{W}^1_{q', 0} (B_R^-)$ subject to $$\begin{aligned}
K ({\mathbf{u}}_-, h) = \mu_- \langle {\mathbf{D}}({\mathbf{u}}_-) \omega, \omega \rangle - \rho_{*-} {\mathrm{div}\,}{\mathbf{u}}_- - \frac{\rho_{*-} \sigma}{\rho_{*-} - \rho_{*+}} {\mathcal{B}}h \quad \text{on $S_R$}.\end{aligned}$$ We then consider the following initial value problem instead of : $$\begin{aligned}
\label{IVP}
\left\{\begin{aligned}
{\partial}_t \rho_{2+} + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{2+} & = \lambda_1 \rho_{1+} &  & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*+} {\partial}_t {\mathbf{u}}_{2+} - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_{2+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_{2+} {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_{2+} {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1+} & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t {\mathbf{u}}_{2-} - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{2-}) - K({\mathbf{u}}_{2-}, h_2) {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1-} & & \text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t h_2 - \frac{\langle \rho_{*-} {\mathbf{u}}_{2-}, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_{2+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = \lambda_1 h_1 & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{B}}_i (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) & = 0 & & \text{ on $S^i_R \times (0, T)$}, \\
(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)|_{t = 0} & = (0, 0, 0, 0) & & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$ and $T > 0$. Notice that the boundary condition: $B_1 (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, K({\mathbf{u}}_{2-}, h_2), h_2) = 0$ and $B_2 (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, K({\mathbf{u}}_{2-}, h_2), h_2)  = 0$ are equivalent to $$\begin{aligned}
\label{4.9}
\begin{aligned}
\langle \{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - K({\mathbf{u}}_{2-}, h_2) {\mathbf{I}}\} \omega, \omega \rangle \Big|_- - \frac{\rho_{*-} \sigma}{\rho_{*-} - \rho_{*+}} {\mathcal{B}}h & = 0, \\
\langle \{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_- {\mathbf{I}}+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} \omega, \omega \rangle \Big|_+ 
- \frac{\rho_{*+} \sigma}{\rho_{*-} - \rho_{*+}} {\mathcal{B}}h & = 0,
\end{aligned}\end{aligned}$$ see Shibata and Shimizu [@SS2011 Lemma 2.1] (cf. Solonnikov [@Sol2003 p.155]).

Defining the function spaces ${\mathcal{H}}_q$ and ${\mathcal{D}}_q$ by $$\begin{split}
{\mathcal{H}}_q & = \Set{(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) |
\begin{array}{l}
\rho_{2+} \in W^1_q (B_R^+), \enskip {\mathbf{u}}_{2+} \in L_q (B_R^+), \\
{\mathbf{u}}_{2-} \in J_q (B_R^-), \enskip h_2 \in W^{2 - 1 / q}_q (S_R)
\end{array}}, \\ 
{\mathcal{D}}_q & = \Set{(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) \in {\mathcal{H}}_q |
\begin{array}{l}
\rho_{2+} \in W^3_q(B_R^+), \enskip {\mathbf{u}}_{2\pm} \in W^2_q(B_R^\pm), \\ h_2 \in W^{3 - 1 / q}_q(S_R), \\
{\mathbf{B}}_i(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)|_{S_R^i} = 0 \enskip \text{for $i=1,2$}
\end{array}},
\end{split}$$ respectively, and the operator ${\mathcal{A}}_q (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)$ by $$\begin{aligned}
A_q (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)
=\begin{pmatrix}
-\rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{2+} \\
{\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_{2+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_{2+} {\mathbf{I}}+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_{2+} {\mathbf{I}}\} \\
{\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{2-}) - K ({\mathbf{u}}_{2-}, h_2) {\mathbf{I}}\} \\
\displaystyle
\frac{\langle \rho_{*-} {\mathbf{u}}_{2-}, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_{2+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
\end{pmatrix}\end{aligned}$$ for $(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) \in {\mathcal{D}}_q$. According to Watanabe [@Wat2019 Section 6], we know that the operator $A_q$ generates a continuous analytic semigroup $\{T (t)\}_{t \geq 0}$ on ${\mathcal{H}}_q$. We now show the uniqueness of solution to the homogeneous equation: $$\begin{aligned}
\label{eq-resol}
(\lambda - A_q) (\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) = (0, 0, 0, 0) \quad \text{in $B_R^+ \times B_R^+ \times B_R^- \times S_R$},\end{aligned}$$ which implies the exponential stability of $\{T (t)\}_{t \geq 0}$.

\[lem-4.2\] Let $1 < q < \infty$. Assume that $\mu_+$, $\nu_+$, and $\kappa_+$ satisfy the condition $(\mathrm{A}\ref{A6})$. Let $(\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) \in {\mathcal{D}}_q$ satisfy with $(h_\lambda, \varphi_j)_{S_R} = 0$ $(j = 1, \dots, N + 1)$. If $\lambda \in {\mathbb{C}}\backslash (- \infty, 0)$, then $(\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) = (0, 0, 0, 0)$.

We first consider the case where $2 \leq q < \infty$. Since $B_R^\pm$ and $S_R$ are bounded, we see that $(\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) \in {\mathcal{D}}_2$. By the first equation in , we have that $$\begin{aligned}
\label{4.12}
\frac{\gamma_{*+}}{\rho_{*+}}\lambda \|\rho_{\lambda+}\|^2_{L_2 (B_R^+)} + (\gamma_{*+} \rho_{\lambda+}, {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} & = 0, \\
\kappa_+ \lambda \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)} + (\rho_{*+} \kappa_+ \nabla \rho_{\lambda+}, \nabla {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} & = 0.\end{aligned}$$ Multiplying the second equation in with ${\mathbf{u}}_{\lambda+}$, the third equation in with ${\mathbf{u}}_{\lambda-}$, and the fourth equation in with ${\mathcal{B}}h_\lambda$ and using the divergence theorem of Gauss, we obtain that $$\begin{split}
0 & = \lambda \Big(\rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)} - ({\mathcal{B}}h_\lambda, h_\lambda)_{S_R} \Big) \\
& \quad + \frac{\mu_+}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda+})\|^2_{L_2 (B_R^+)} + (\nu_+ - \mu_+) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad - (\gamma_{*+}\rho_{\lambda+}, {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} 
+ (\rho_{*+} \kappa_+ \Delta \rho_{\lambda+}, {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} + \frac{\mu_-}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda-})\|^2_{L_2 (B_R^-)}\\
& = \lambda \Big(\rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)} - ({\mathcal{B}}h_\lambda, h_\lambda)_{S_R} \Big) \\
& \quad + \frac{\mu_+}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda+})\|^2_{L_2 (B_R^+)} + (\nu_+ - \mu_+) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad - (\gamma_{*+}\rho_{\lambda+}, {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} 
- (\rho_{*+} \kappa_+ \nabla \rho_{\lambda+}, {\mathrm{div}\,}{\mathbf{u}}_{\lambda+})_{B_R^+} + \frac{\mu_-}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda-})\|^2_{L_2 (B_R^-)},
\end{split}$$ which, combined with , we see that $$\label{4.15}
\begin{split}
0 & = \lambda \bigg(\frac{\gamma_{*+}}{\rho_{*+}} \|\rho_{\lambda+}\|^2_{L_2 (B_R^+)} + \kappa_+ \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad + \rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)} - ({\mathcal{B}}h_\lambda, h_\lambda)_{S_R} \bigg) \\
& \quad + \frac{\mu_+}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda+})\|^2_{L_2 (B_R^+)}
+ (\nu_+ - \mu_+) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \frac{\mu_-}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda-})\|^2_{L_2 (B_R^-)}.
\end{split}$$ To treat $({\mathcal{B}}h_\lambda, h_\lambda)$, we use the following lemma proved by Shibata [@Shi2018 Lemma 4.5].

\[lem-4.3\] Let $\dot{W}^2_2 (S_R)$ be the space defined by $$\begin{aligned}
\dot{W}^2_2 (S_R) = \{h \in W^2_2 (S_R) \mid (h, 1)_{S_R} = 0, \enskip (h, x_j)_{S_R} = 0 \enskip (j = 1, \dots, N) \}.\end{aligned}$$ Let $\{\lambda_j\}_{j = 1}^\infty$ is the set of all eigenvalues of the Laplace-Beltrami operator $\Delta_{S_R}$ on $S_R$, $E_j$ be the eigen-space corresponding to $\lambda_j$, and $d_j = \dim E_j$. Then, $$\begin{aligned}
- ({\mathcal{B}}h, h)_{S_R} = \sum_{i = 3}^{\infty} \sum_{j = 1}^{d_i} |a_{ij}|^2 \bigg(- \lambda_i- \frac{N-1}{R}\bigg) \|\varphi_{ij}\|^2_{L_2 (S_R)},\end{aligned}$$ where $\{\varphi_{ij}\}_{j=1}^{d_j}$ is the orthogonal basis of $E_i$ in $L_2 (S_R)$ and $a_{ij} = (h, \varphi_{ij})_{S_R}$. In particular, we have the following estimate: $$\begin{aligned}
- ({\mathcal{B}}h, h)_{S_R} \geq c_{\mathcal{B}}\|h\|^2_{L_2 (S_R)}\end{aligned}$$ provided that we take a positive constant $c_{\mathcal{B}}$ such that $c_{\mathcal{B}}\leq - \lambda_i- R^{-1} (N - 1) $.

When ${\mathrm{Im}\,}\lambda \neq 0$, taking the imaginary part of , we obtain $$\begin{aligned}
0 & = \frac{\gamma_{*+}}{\rho_{*+}} \|\rho_{\lambda+}\|^2_{L_2 (B_R^+)} + \kappa_+ \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)}
+ \rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)}
+ \sum_{i = 3}^{\infty} \sum_{j = 1}^{d_i} |a_{ij}|^2 \bigg(- \lambda_i- \frac{N-1}{R}\bigg) \|\varphi_{ij}\|^2_{L_2 (S_R)},\end{aligned}$$ which yields $(\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}) = (0, 0, 0)$ and $a_{ij} = \varphi_{ij} = 0$. Since $h$ is represented by $$\begin{aligned}
h_\lambda = \sum_{i = 3}^{\infty} \sum_{j = 1}^{d_i} a_{ij} \varphi_{ij},\end{aligned}$$ we have $h_\lambda = 0$.

We next consider the case where ${\mathrm{Re}\,}\lambda > 0$. By Lemma \[lem-4.3\] we have that $$\label{4.16}
\begin{split}
0 & \geq \lambda \bigg(\frac{\gamma_{*+}}{\rho_{*+}} \|\rho_{\lambda+}\|^2_{L_2 (B_R^+)} + \kappa_+  \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad + \rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)} + c_{\mathcal{B}}\|h\|^2_{L_2 (S_R)} \bigg) \\
& \quad + \frac{\mu_+}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda+})\|^2_{L_2 (B_R^+)}
+ (\nu_+ - \mu_+) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \frac{\mu_-}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda-})\|^2_{L_2 (B_R^-)} \\
& \geq \lambda \bigg(\frac{\gamma_{*+}}{\rho_{*+}} \|\rho_{\lambda+}\|^2_{L_2 (B_R^+)} + \kappa_+ \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)} \\
& \quad + \rho_{*+} \|{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*-} \|{\mathbf{u}}_{\lambda-}\|^2_{L_2 (B_R^+)} + c_{\mathcal{B}}\|h\|^2_{L_2 (S_R)} \bigg) \\
& \quad + \bigg(\nu_+ - \frac{N - 2}{N} \mu_+ \bigg) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)} + \frac{\mu_-}{2} \|{\mathbf{D}}({\mathbf{u}}_{\lambda-})\|^2_{L_2 (B_R^-)}
\end{split}$$ because $\|{\mathbf{D}}({\mathbf{u}}_{\lambda+})\|^2_{L_2 (B_R^+)} \geq (4/N) \|{\mathrm{div}\,}{\mathbf{u}}_{\lambda+}\|^2_{L_2 (B_R^+)}$. According to Assumption (A\[A6\]), we have $(\rho_{\lambda+}, \allowbreak {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) = (0, 0, 0, 0)$.

We finally deal with the case: $\lambda = 0$. From the first equation in , we see that ${\mathrm{div}\,}{\mathbf{u}}_{\lambda+} = 0$. Then, by the inequality , we have ${\mathbf{D}}({\mathbf{u}}_{\lambda+}) = 0$ and ${\mathbf{D}}({\mathbf{u}}_{\lambda-}) = 0$. We now consider type I. In this case, by the Dirichlet boundary condition ${\mathbf{u}}_{\lambda-}|_{S_R^-} = 0$ we have ${\mathbf{u}}_{\lambda-} = 0$ in $B^-_R$. Furthermore, from the fourth equation in with $\lambda = 0$, we have $\langle {\mathbf{u}}_{\lambda+}, \omega \rangle|_+ = 0$. Hence, the jump condition: $\Pi_0 {\mathbf{u}}_{\lambda-}|_- - \Pi_0 {\mathbf{u}}_{\lambda+}|_+ = 0$ implies ${\mathbf{u}}_{\lambda+}|_+ = 0$, and then ${\mathbf{u}}_{\lambda+} = 0$ in $B^+_R$. Notice that the similar argument enable us to obtain ${\mathbf{u}}_{\lambda\pm} = 0$ in $B^\pm_R$ for type II. Since ${\mathbf{u}}_{\lambda-} = 0$ in $B^-_R$, from we see that ${\mathcal{B}}h = 0$ on $S_R$, which, combined with Lemma \[lem-4.3\], implies that $h = 0$ on $S_R$. In addition, multiplying the second condition in with $\nabla \rho_{\lambda+}$ and using the fact that ${\mathbf{u}}_{\lambda+} = 0$ in $B_R^+$, $u_{\lambda+} = 0$, $h = 0$ on $S_R$, and the divergence theorem of Gauss, we have $$\begin{split}
0 & = - \gamma_{*+} (\rho_{\lambda+}, \Delta \rho_{\lambda+})_{B_R^+} + \rho_{*+} \kappa_+ \|\Delta \rho_{\lambda+}\|_{L_2 (B_R^+)}^2 \\
& = \gamma_{*+} \|\nabla \rho_{\lambda+}\|^2_{L_2 (B_R^+)} + \rho_{*+} \kappa_+ \|\Delta \rho_{\lambda+}\|_{L_2 (B_R^+)}^2.
\end{split}$$ Namely, we obtain $\nabla \rho_{\lambda+} = 0$, which implies $\rho_{\lambda+}$ is a constant in $B_R^+$. However, from the second condition in , $\rho_{\lambda+}$ should be 0 on $S_R$, that is, $\rho_{\lambda+} = 0$ in $B_R^+$. Summing up, we obtain $(\rho_{\lambda+}, {\mathbf{u}}_{\lambda+}, {\mathbf{u}}_{\lambda-}, h_\lambda) = (0, 0, 0, 0)$ when $\lambda = 0$.

Finally, we treat the case: $1 < q < 2$. Notice that $B_R^\pm$ are bounded and the resolvent set of $A_{q'}$ includes $\{\lambda \in {\mathbb{R}}\mid |\arg \lambda| \leq \pi - \varepsilon, \enskip |\lambda| \geq c \}$ for any $\varepsilon \in (\varepsilon_1, \pi / 2)$ with some $c$ and $\varepsilon_1$ depending on $q'$, and then from the standard homotopic argument (cf. Enomoto and Shibata [@ES2013 Sect. 7]) we see that there exists $(\lambda {\mathbf{I}}- A_{q'})^{-1}$ on ${\mathcal{H}}_{q'}$ for ${\mathrm{Im}\,}\lambda \neq 0$ or for ${\mathrm{Re}\,}\lambda \geq 0$, because we have already showed that the uniqueness of solution to for $2 \leq q' < \infty$. Hence, the uniqueness of solution to for $1 < q < 2$ follows from the invertibility of $\lambda {\mathbf{I}}- A_{q'}$. We then complete the proof.

By the standard semigroup theory, Lemma \[lem-4.2\] implies the following proposition (cf. Pazy [@P1983 Cahpter 4, Theorem 4.3]).

\[lem-4.4\] Let $1 < q < \infty$. Suppose that $\mu_+$, $\nu_+$, and $\kappa_+$ satisfy the condition $(\mathrm{A}\ref{A6})$. In addition, let $\{T(t)\}_{t \geq 0}$ be the analytic semigroup associated with . Then there exist positive constants $C$ and $ \gamma_1$ such that $$\begin{aligned}
\|T(t) (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)\|_{{\mathcal{H}}_q} \leq C e^{- \gamma_1 t} \|(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)\|_{{\mathcal{H}}_q}\end{aligned}$$ for any $t > 0$ and $(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) \in {\mathcal{D}}_q$ provided that $(h, \varphi_j)_{S_R} = 0$ for $(j = 1, \dots, N + 1)$.

We now prove Theorem \[th-4.1\]. Set $$\begin{aligned}
\widetilde{\rho}_{1+} = \rho_{1+},\quad \widetilde{{\mathbf{u}}}_{1\pm} = {\mathbf{u}}_{1\pm}, \quad \widetilde{h}_1 = h_1 + \sum_{j = 1}^{N + 1} (h_1 (\cdot, t), \varphi_j)_{S_R} \varphi_j,\end{aligned}$$ and then $(\widetilde{h}_1, \varphi_j)_{S_R} = 0$ for $j = 1, \dots, N+1$, where $\varphi_1 = |S_R|^{-1}$ and $\varphi_i$ $(i = 2, \dots, N + 1)$ are some linear combinations of $x_1, \dots, x_N$ such that $(\varphi_i, \varphi_j)_{S_R} = 0$ for $i, j = 1, \dots, N + 1$. We next define the function $\widetilde{\rho}_+$, $\widetilde{{\mathbf{u}}}_+$, $\widetilde{{\mathbf{u}}}_-$, and $\widetilde{h}$ by $$\begin{aligned}
(\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h}) (\cdot, s)
= \int_{0}^{s} T (s - r) (\lambda_1 \rho_{1+} (\cdot, r), \lambda_1 {\mathbf{u}}_{1+} (\cdot, r), \lambda_1 {\mathbf{u}}_{1-} (\cdot, r), \lambda_1 \widetilde{h}_1 (\cdot, r))\,\mathrm{d}r\end{aligned}$$ for $s \in (0, T)$. Then, by Duhamel principle, we see that $\widetilde{\rho}_+$, $\widetilde{{\mathbf{u}}}_+$, $\widetilde{{\mathbf{u}}}_-$, and $\widetilde{h}$ satisfy the following equations: $$\begin{aligned}
\label{4.18}
\left\{\begin{aligned}
{\partial}_t \rho_{2+} + \rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{2+} & = \lambda_1 \rho_{1+} &  & \text{ in $B_R^+ \times (0, T)$,} \\
\rho_{*+} {\partial}_t \widetilde{{\mathbf{u}}}_+ - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}(\widetilde{{\mathbf{u}}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}\widetilde{{\mathbf{u}}}_+ {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \widetilde{\rho}_+ {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1+} & & \text{ in $B_R^+ \times (0, T)$}, \\
\rho_{*-} {\partial}_t \widetilde{{\mathbf{u}}}_- - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}(\widetilde{{\mathbf{u}}}_-) - K(\widetilde{{\mathbf{u}}}_-, \widetilde{h}) {\mathbf{I}}\}
& = \lambda_1 {\mathbf{u}}_{1-} & & \text{ in $B_R^- \times (0, T)$}, \\
{\partial}_t \widetilde{h} - \frac{\langle \rho_{*-} \widetilde{{\mathbf{u}}}_-, \omega \rangle|_- - \langle \rho_{*+} \widetilde{{\mathbf{u}}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = \lambda_1 \widetilde{h}_1 & & \text{ on $S_R \times (0, T)$}, \\
{\mathbf{B}}_i (\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h}) & = 0 & & \text{ on $S^i_R \times (0, T)$}, \\
(\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h})|_{t = 0} & = (0, 0, 0, 0) & & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ By Theorem \[lem-4.4\] and the H[ö]{}lder inequality, we have $$\begin{aligned}
&\|(\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h}) (\cdot, s)\|_{{\mathcal{H}}_q} \\
& \leq C \int_{0}^{s} e^{-\gamma_1 (s - r)} \|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q} \,\mathrm{d}r \\
& \leq C (\gamma_1 p')^{-1/p'}
\bigg(\int_{0}^{s} e^{-\gamma_1 (s - r) p} \|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q}^p \,\mathrm{d}r \bigg)^{1/p} \\
& \leq C (\gamma_1 p')^{-1/p'} \bigg(\int_{0}^{s} e^{-\gamma_1 (s - r) p} \|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q}^p \,\mathrm{d}r \bigg)^{1/p}.\end{aligned}$$ We may assume that $0 < \gamma p < \gamma_1$ choosing $\gamma > 0$ small if necessary, and then we obtain $$\begin{aligned}
&\int_{0}^{t} (e^{\gamma s} \|(\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h}) (\cdot, s)\|_{{\mathcal{H}}_q} )^p {\,\mathrm{d}s}\\
& \leq C (\gamma_1 p')^{-p/p'} \int_{0}^{t} \bigg(\int_{0}^{s} e^{\gamma s p} e^{-\gamma_1 (s - r) p} \|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q}^p \,\mathrm{d}r \bigg) {\,\mathrm{d}s}\\
& \leq C (\gamma_1 p')^{-p/p'} \int_{0}^{t} \bigg(\int_{0}^{s} e^{- (\gamma_1 - \gamma p)(s -r)} (e^{\gamma r}\|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q})^p \,\mathrm{d}r \bigg) {\,\mathrm{d}s}\\
& \leq C (\gamma_1 p')^{-p/p'} \int_{0}^{t} (e^{\gamma r}\|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q})^p \bigg(\int_{r}^{t} e^{- (\gamma_1 - \gamma p)(s -r)} {\,\mathrm{d}s}\bigg)\,\mathrm{d}r \\
& \leq C (\gamma_1 p')^{-p/p'} (\gamma_1 - \gamma p)^{-1} \int_{0}^{t} (e^{\gamma r}\|(\widetilde{\rho}_{1+} (\cdot, r), \widetilde{{\mathbf{u}}}_{1+ }(\cdot, r), \widetilde{{\mathbf{u}}}_{1-} (\cdot, r), \widetilde{h}_1 (\cdot, r))\|_{{\mathcal{H}}_q})^p {\,\mathrm{d}s}.\end{aligned}$$ Combined with we arrive at $$\label{4.23}
\begin{split}
&\|e^{\gamma s} (\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, \widetilde{h}) (\cdot, s)\|_{L_p ((0, t), {\mathcal{H}}_q)} \\
&\leq C {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma)
\end{split}$$ for $t \in (0, T)$ and $i={\mathrm{I}}, {\mathrm{II}}$. Assume that $\widetilde{\rho}_{3+}$, $\widetilde{{\mathbf{u}}}_{3+}$, $\widetilde{{\mathbf{u}}}_{3-}$, and $\widetilde{h}_3$ satisfy the shifted equations: $$\begin{aligned}
\left\{\begin{aligned}
{\partial}_t \widetilde{\rho}_{3+} + \rho_{*+} {\mathrm{div}\,}\widetilde{{\mathbf{u}}}_{3+} & = f_{\rho,3} &  & \text{ in $B_R^+ \times (0, \infty)$,} \\
{\mathrm{div}\,}\widetilde{{\mathbf{u}}}_{3-} & = 0  &  & \text{ in $B_R^- \times (0, \infty)$,} \\
\rho_{*+} {\partial}_t \widetilde{{\mathbf{u}}}_{3+} + \lambda_1 \widetilde{{\mathbf{u}}}_{3+} - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}(\widetilde{{\mathbf{u}}}_{3+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}\widetilde{{\mathbf{u}}}_{3+} {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \widetilde{\rho}_{3+} {\mathbf{I}}\} & = {\mathbf{f}}_{3,+} & & \text{ in $B_R^+ \times (0, \infty)$}, \\
\rho_{*-} {\partial}_t \widetilde{{\mathbf{u}}}_{3-} + \lambda_1 \widetilde{{\mathbf{u}}}_{3-} - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}(\widetilde{{\mathbf{u}}}_{3-}) - K(\widetilde{{\mathbf{u}}}_{3-}, \widetilde{h}_3) {\mathbf{I}}\}
& = {\mathbf{f}}_{3,-} & & \text{ in $B_R^- \times (0, \infty)$}, \\
{\partial}_t \widetilde{h}_3 + \lambda_1 \widetilde{h}_3 - \frac{\langle \rho_{*-} \widetilde{{\mathbf{u}}}_{3-}, \omega \rangle|_- - \langle \rho_{*+} \widetilde{{\mathbf{u}}}_{3+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d_3 & & \text{ on $S_R \times (0, \infty)$}, \\
{\mathbf{B}}_i (\widetilde{\rho}_{3+}, \widetilde{{\mathbf{u}}}_{3+}, \widetilde{{\mathbf{u}}}_{3-}, \widetilde{h}_3) & = 0 & & \text{ on $S^i_R \times (0, \infty)$}, \\
(\widetilde{\rho}_{3+}, \widetilde{{\mathbf{u}}}_{3+}, \widetilde{{\mathbf{u}}}_{3-}, \widetilde{h}_3)|_{t = 0} = (0, & 0, 0, 0) & & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$ and $T > 0$, where we have set $$\begin{aligned}
f_{\rho, 3}
& = \begin{cases}
\lambda_1 \rho_{1+} + \lambda_1 \widetilde{\rho}_{3+} & \text{for $t \in (0, T)$,} \\
0 & \text{for $t \notin (0, T)$,}
\end{cases} \\
{\mathbf{f}}_{\pm, 3}
& = \begin{cases}
\lambda_1 {\mathbf{u}}_{1\pm} + \lambda_1 \widetilde{{\mathbf{u}}}_{3\pm} & \text{for $t \in (0, T)$,} \\
0 & \text{for $t \notin (0, T)$,}
\end{cases} \\
d_3
& = \begin{cases}
\lambda_1 h_1 + \lambda_1 \widetilde{h}_3 & \text{for $t \in (0, T)$,} \\
0 & \text{for $t \notin (0, T)$}.
\end{cases}\end{aligned}$$ From and , we have $$\begin{aligned}
&{\mathbb{I}}_{p,q} (\widetilde{\rho}_{3+}, \widetilde{{\mathbf{u}}}_{3+}, \widetilde{{\mathbf{u}}}_{3-}, K(\widetilde{{\mathbf{u}}}_{3-}, \widetilde{h}_3),\widetilde{h}_3; \gamma, (0,T)) \\
&\leq C {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma).\end{aligned}$$ Since $\widetilde{\rho}_+$, $\widetilde{{\mathbf{u}}}_\pm$, and $\widetilde{h}$ satisfy , we see that $\widetilde{\rho}_{3+} = \widetilde{\rho}_+$, $\widetilde{{\mathbf{u}}}_{3+} = \widetilde{{\mathbf{u}}}_\pm$, and $\widetilde{h}_3 = \widetilde{h}$ for $t \in (0, T)$ by the uniqueness, and then we have $$\label{4.25}
\begin{split}
&{\mathbb{I}}_{p,q} (\widetilde{\rho}_+, \widetilde{{\mathbf{u}}}_+, \widetilde{{\mathbf{u}}}_-, K(\widetilde{{\mathbf{u}}}_-, \widetilde{h}),\widetilde{h}; \gamma, (0,T)) \\
&\leq C {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma).
\end{split}$$ Set $$\begin{aligned}
\rho_{2+} = \widetilde{\rho}_+, \quad {\mathbf{u}}_{2\pm} = \widetilde{{\mathbf{u}}}_\pm, \quad h_2 = \widetilde{h} + \lambda_1 \sum_{j = 1}^{N + 1} \int_{0}^{t} (h_1(\cdot, s), \varphi_j)_{S_R} {\,\mathrm{d}s}\varphi_j.\end{aligned}$$ Note that $$\begin{aligned}
K({\mathbf{u}}_{2-}, h_2) = K(\widetilde{{\mathbf{u}}}_{2-}, \widetilde{h}) + \lambda_1 \sum_{j = 1}^{N + 1} \int_{0}^{t} (h_1(\cdot, s), \varphi_j)_{S_R} {\,\mathrm{d}s}K(0, \varphi_j).\end{aligned}$$ We have $K(0, \varphi_j) = 0$ for $j = 2, \dots, N + 1$ because ${\mathcal{B}}\varphi_j = 0$ on $S_R$ for $j = 2, \dots, N + 1$. In addition, we have $K(0, \varphi_1) = -\sigma (N - 1) R^{-2} \varphi_1$ because $\varphi_1 = |S_1|^{-1}$ is constant and ${\mathcal{B}}\varphi_1 = (N - 1) R^{-2} \varphi_1$ on $S_R$. Hence, by we see that $\rho_{2+}$, ${\mathbf{u}}_{2\pm}$, and $h_2$ satisfy Eq. . In addition, from and we have the estimate $$\label{4.26}
\begin{split}
&\|e^{\gamma s} {\partial}_s (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) \|_{L_p((0, t), W^1_q (B_R^+) \times L_q (B_R^+) \times L_q (B_R^-) \times W^{2 - 1/q}_q(S_R))} \\
& \leq C {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma).
\end{split}$$ To estimate $\|e^{\gamma s} {\partial}_s (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) \|_{L_p((0, t), W^3_q (B_R^+) \times W^2_q (B_R^+) \times W^2_q (B_R^-) \times W^{3 - 1/q}_q(S_R))}$ we need the following lemma.

\[lem-4.6\] Let $1 < q < \infty$. Furthermore, let $\rho_+$, ${\mathbf{u}}_+$, ${\mathbf{u}}_-$, and $h$ satisfy the following equations: $$\begin{aligned}
\label{4.27}
\left\{\begin{aligned}
\rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_+ & = f_\rho &  & \text{ in $B_R^+$,} \\
- {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_+ {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_+ {\mathbf{I}}\} & = {\mathbf{f}}_+ & & \text{ in $B_R^+$}, \\
- {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_-) - K({\mathbf{u}}_-, h) {\mathbf{I}}\}
& = {\mathbf{f}}_- & & \text{ in $B_R^-$}, \\
- \frac{\langle \rho_{*-} {\mathbf{u}}_-, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = d & & \text{ on $S_R$}, \\
{\mathbf{B}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & = 0 & & \text{ on $S^i_R$},
\end{aligned}\right.\end{aligned}$$ where $(f_\rho, {\mathbf{f}}_+, {\mathbf{f}}_-, d) \in {\mathcal{H}}_q$. Then the following estimate hold: $$\begin{aligned}
\| (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)\|_{{\mathcal{D}}_q} \leq C \bigg(\|(f_\rho, {\mathbf{f}}_+, {\mathbf{f}}_-, d)\|_{{\mathcal{H}}_q} + \sum_{j = 1}^{N + 1}|(h, \varphi_j)_{S_R}| \bigg)\end{aligned}$$ with some positive constant $C > 0$.

Let $\rho_{4+}$, ${\mathbf{u}}_{4+}$, ${\mathbf{u}}_{4-}$, and $h_4$ be functions such that $$\begin{aligned}
\rho_{4+} = \rho_+, \quad {\mathbf{u}}_{4\pm} = {\mathbf{u}}_\pm, \quad h_4 = h + \sum_{j = 1}^{N + 1} (h, \varphi_j)_{S_R} \varphi_j.\end{aligned}$$ Recall that $K(0, \varphi_j) = 0$ for $j = 2, \dots, N + 1$ and $K(0, \varphi_1) = -\sigma (N - 1) R^{-2} \varphi_1$, that is, ${\mathcal{B}}h_4 = {\mathcal{B}}h + (h, \varphi_j)_{S_R} (N - 1)R^{-2} |S_R|^{-1}$. We then see that $\rho_{4+}$, ${\mathbf{u}}_{4+}$, ${\mathbf{u}}_{4-}$, and $h_4$ satisfy . From Lemmas \[lem-4.3\] and \[lem-4.4\] we have the estimate $\|(\rho_{4+}, {\mathbf{u}}_{4+}, {\mathbf{u}}_{4-}, h_4)\|_{{\mathcal{D}}_q} \leq C \|(f_\rho, {\mathbf{f}}_+, {\mathbf{f}}_-, d)\|_{{\mathcal{H}}_q}$, and then by the estimate: $$\begin{aligned}
\|(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h)\|_{{\mathcal{D}}_q} \leq \|(\rho_{4+}, {\mathbf{u}}_{4+}, {\mathbf{u}}_{4-}, h_4)\|_{{\mathcal{D}}_q}
+ C \sum_{j = 1}^{N + 1}|(h, \varphi_j)_{S_R}|\end{aligned}$$ we obtain the required estimate.

By we see that $\rho_{2+}$, ${\mathbf{u}}_{2\pm}$, and $h_2$ satisfy the following elliptic equations: $$\begin{aligned}
\left\{\begin{aligned}
\rho_{*+} {\mathrm{div}\,}{\mathbf{u}}_{2+} & = \lambda_1 \rho_{1+} - {\partial}_t \rho_{2+}&  & \text{ in $B_R^+$,} \\
- {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}({\mathbf{u}}_{2+}) + (\nu_+ - \mu_+) {\mathrm{div}\,}{\mathbf{u}}_{2+} {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \rho_{2+} {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1+} - \rho_{*+} {\partial}_t {\mathbf{u}}_{2+} & & \text{ in $B_R^+$}, \\
- {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}({\mathbf{u}}_{2-}) - K({\mathbf{u}}_{2-}, h_2) {\mathbf{I}}\} & = \lambda_1 {\mathbf{u}}_{1-} - \rho_{*-} {\partial}_t {\mathbf{u}}_{2-} & & \text{ in $B_R^-$}, \\
- \frac{\langle \rho_{*-} {\mathbf{u}}_{2-}, \omega \rangle|_- - \langle \rho_{*+} {\mathbf{u}}_{2+}, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = \lambda_1 h_1 -{\partial}_t h_2 & & \text{ on $S_R$}, \\
{\mathbf{B}}_i (\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2) & = 0 & & \text{ on $S^i_R$}, \\
(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, h_2)|_{t = 0} & = (0, 0, 0, 0) & & \text{ on $I_R$}
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$. Applying Lemma \[lem-4.6\] and using and , we have $$\label{4.28}
\begin{split}
&\| e^{\gamma s} \rho_{2+} \|_{L_p ((0, t), W^3_q (B^+_R))} + \| e^{\gamma s} {\mathbf{u}}_{2+} \|_{L_p ((0, t), W^2_q (B^+_R))} + \| e^{\gamma s} {\mathbf{u}}_{2-} \|_{L_p ((0, t), W^2_q (B^-_R))} \\
& \quad + \| e^{\gamma s} \nabla \pi_{2-} \|_{L_p ((0, t), W^3_q (B^+_R))} + \| e^{\gamma s} h_2 \|_{L_p ((0, t), W^{3 - 1/q}_q (S_R))} \\
& \leq C \bigg\{{\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma) \\
& \quad + \sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h_2(\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p} \bigg\},
\end{split}$$ where we have set $\pi_{2-} = K ({\mathbf{u}}_{2-}, h_2)$. Recall that $\rho_+ = \rho_{1+} + \rho_{2+}$, ${\mathbf{u}}_\pm = {\mathbf{u}}_{1\pm} + {\mathbf{u}}_{2\pm}$, $\pi_- = \pi_{1-} + \pi_{2-}$, $h = h_1 + h_2$. We then see that $(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h)$ satisfy because $(\rho_{1+}, {\mathbf{u}}_{1+}, {\mathbf{u}}_{1-}, \pi_{1-}, h_1)$ and $(\rho_{2+}, {\mathbf{u}}_{2+}, {\mathbf{u}}_{2-}, \pi_{2-}, h_2)$ satisfy Eq. and , respectively. By we have $$\label{4.29}
\begin{split}
&\sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h_2 (\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p} \\
&\leq C {\mathbb{J}}_{p, q}^\gamma + \sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h (\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p},
\end{split}$$ where we write ${\mathbb{J}}_{p, q}^\gamma = {\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, f_\rho, f_d, {\mathbf{F}}_d, {\mathbf{f}}_+, {\mathbf{f}}_-, d, {\mathbf{G}}_i; \gamma)$ for short. Hence, we have the required estimate as in Theorem \[th-4.1\] from , , and .

Global solvability
==================

Finally, we prove Theorem \[th-1.3\]. To this end, it is enough to show that the inequality $$\begin{aligned}
\label{5.1}
{\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))
\leq C \Big({\mathcal{I}}_{p, q} + \big({\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))\big)^2\Big)\end{aligned}$$ holds, where ${\mathcal{I}}_{p, q}$ is defined by $$\begin{aligned}
{\mathcal{I}}_{p, q} :=
\|\rho_{0+}\|_{B^{3-2/p}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0+}\|_{B^{2(1-1/p)}_{q,p}(B_R^{+})}
+\|{\mathbf{u}}_{0-}\|_{B^{2(1-1/p)}_{q,p}(B_R^{-})}
+\|h_0\|_{B^{3-1/p-1/q}_{q,p}(S_R)}.\end{aligned}$$ Indeed, if we assume that assumptions as in Theorem \[th-2.1\] hold, by Theorem \[th-2.1\] we know that there exists a unique solution $(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h) \in S_{p, q}(0, T_0)$ to Eq. , where $T_0$ is some positive constant. Then, by using a standard bootstrap argument we can extend $T_0$ to $\infty$ under the assumption that the initial data are small. Namely, the inequality immediately implies Theorem \[th-1.3\], see also Shibata [@Shi2018 Section 6] for more details.

Let $\overline{f}_\rho$, $\overline{f}_d$, $\overline{{\mathbf{F}}}_d$, $\overline{{\mathbf{f}}}_\pm$, $\overline{d}$, $\overline{g}$, $\overline{f}_a$, $\overline{f}_b$, and $\overline{{\mathbf{h}}}$ be the suitable extensions defined on $t \in {\mathbb{R}}$ of $f_\rho$, $f_d$, ${\mathbf{F}}_d$, ${\mathbf{f}}_\pm$, $d$, $g$, $f_a$, $f_b$, and ${\mathbf{h}}$, respectively, see Watanabe [@Wat2019 Section 7.2] for some details. Set $$\begin{aligned}
\overline{{\mathbf{G}}}_i := 
\begin{cases}
{}^\top\!(\overline g, \overline f_a, \overline f_b , \overline {\mathbf{h}}, 0, 0) & \text{ for $i = {\mathrm{I}}$,} \\
{}^\top\!(\overline g, \overline f_a, \overline f_b , \overline {\mathbf{h}}, 0, 0, 0) & \text{ for $i = {\mathrm{II}}$}.
\end{cases}\end{aligned}$$ Furthermore, let $(\overline \rho_+, \overline {\mathbf{u}}_+, \overline {\mathbf{u}}_-, \overline \pi_-, \overline h) \in S_{p, q}(0,\infty)$ be a unique solution to the following equations: $$\begin{aligned}
\left\{\begin{aligned}
{\partial}_t \overline \rho_+ + \rho_{*+} {\mathrm{div}\,}\overline {\mathbf{u}}_+ & = \overline f_\rho (\rho_+, {\mathbf{u}}_+, H_h) &  & \text{ in $B_R^+ \times (0, \infty)$,} \\
\rho_{*-} {\mathrm{div}\,}\overline {\mathbf{u}}_- = \overline f_d ({\mathbf{u}}_-, H_h) & = \rho_{*-} {\mathrm{div}\,}\overline {\mathbf{F}}_d ({\mathbf{u}}_-, H_h) & & \text{ in $B_R^- \times (0, \infty)$}, \\
\rho_{*+} {\partial}_t \overline {\mathbf{u}}_+ - {\mathrm{Div}\,}\{\mu_+ {\mathbf{D}}(\overline {\mathbf{u}}_+) + (\nu_+ - \mu_+) {\mathrm{div}\,}\overline {\mathbf{u}}_+ {\mathbf{I}}& \\
+ (-\gamma_{*+} + \rho_{*+} \kappa_+ \Delta) \overline \rho_+ {\mathbf{I}}\} & = \overline {\mathbf{f}}_+ (\rho_+, {\mathbf{u}}_+, h) & & \text{ in $B_R^+ \times (0, \infty)$}, \\
\rho_{*-} {\partial}_t \overline {\mathbf{u}}_- - {\mathrm{Div}\,}\{\mu_- {\mathbf{D}}(\overline {\mathbf{u}}_-) - \overline \pi_- {\mathbf{I}}\} & = \overline {\mathbf{f}}_- ({\mathbf{u}}_-, H_h) & & \text{ in $B_R^- \times (0, \infty)$}, \\
{\partial}_t \overline h - \frac{\langle \rho_{*-} \overline {\mathbf{u}}_-, \omega \rangle|_- - \langle\rho_{*+} \overline {\mathbf{u}}_+, \omega \rangle|_+}{\rho_{*-} - \rho_{*+}}
& = \overline d (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, h) & & \text{ on $S_R \times (0, \infty)$}, \\
{\mathbf{B}}_i (\overline \rho_+, \overline {\mathbf{u}}_+, \overline {\mathbf{u}}_-, \overline \pi_-, \overline h) & = \overline {\mathbf{G}}_i (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, H_h) & & \text{ on $S^i_R \times (0, \infty)$}, \\
(\overline \rho_+, \overline {\mathbf{u}}_+, \overline {\mathbf{u}}_-, \overline h)|_{t = 0} & = (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0) & & \text{ on $I_R$},
\end{aligned}\right.\end{aligned}$$ for $i = {\mathrm{I}}, {\mathrm{II}}$. The uniqueness of the solutions implies that $\rho_+ = \overline \rho_+$, ${\mathbf{u}}_\pm = \overline {\mathbf{u}}_\pm$, $\pi_- = \overline \pi_-$, and $h = \overline h$ because the extensions $\overline{f}_\rho$, $\overline{f}_d$, $\overline{{\mathbf{F}}}_d$, $\overline{{\mathbf{f}}}_\pm$, $\overline{d}$, $\overline{g}$, $\overline{f}_a$, $\overline{f}_b$, and $\overline{{\mathbf{h}}}$ coincide with $f_\rho$, $f_d$, ${\mathbf{F}}_d$, ${\mathbf{f}}_\pm$, $d$, $g$, $f_a$, $f_b$, and ${\mathbf{h}}$ for $t \in (0, T)$, respectively. Hence, from Theorem \[th-4.1\] we have $$\label{5.2}
\begin{split}
{\mathbb{I}}_{p,q} (\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma; (0,T)) 
& \leq C \bigg\{{\mathbb{J}}_{p, q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, \overline f_\rho, \overline f_d, \overline {\mathbf{F}}_d, \overline {\mathbf{f}}_+, \overline {\mathbf{f}}_-, \overline d, \overline {\mathbf{G}}_i; \gamma) \\
& \quad + \sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h(\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p} \bigg\}.
\end{split}$$ Since we have already known the estimate of nonlinear terms due to Watanabe [@Wat2019 Section 7.2], we see that $$\begin{aligned}
&{\mathbb{J}}_{p,q} (\rho_{0+}, {\mathbf{u}}_{0+}, {\mathbf{u}}_{0-}, h_0, \overline f_\rho, \overline f_d, \overline {\mathbf{F}}_d, \overline {\mathbf{f}}_+, \overline {\mathbf{f}}_-, \overline d, \overline {\mathbf{G}}_i; \gamma) \\
& \leq C \Big({\mathcal{I}}_{p, q} + \big({\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))\big)^2\Big).\end{aligned}$$ We, therefore, have the estimate $$\begin{aligned}
{\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))
& \leq C \Big({\mathcal{I}}_{p, q} + \big({\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))\big)^2\Big) \\
& \quad + \sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h(\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p}\end{aligned}$$ by . According to Shibata [@Shi2018 Section 6, (134)], we have $$\begin{aligned}
\sum_{j = 1}^{N + 1} \bigg(\int_{0}^{T} (e^{\gamma s}|(h(\cdot, s), \varphi_j)_{S_R}|)^p {\,\mathrm{d}s}\bigg)^{1/p}
& \leq C \sup_{0 < s < T} \|h(\cdot, t)\|_{W^{2-1/q}_q (S_R)} \|h\|_{L_p ((0,T), L_q(S_R))} \\
& \leq C \big({\mathbb{I}}_{p,q}(\rho_+, {\mathbf{u}}_+, {\mathbf{u}}_-, \pi_-, h, \gamma;(0,T))\big)^2.\end{aligned}$$ Hence, we obtain , and then the proof of Theorem \[th-1.3\] is completed.

Acknowledgment {#acknowledgment .unnumbered}
==============

This work was partially supported by JSPS KAKENHI Grant Number 17H01097, Top Global University Project (Waseda University), and Waseda Research Institute for Science and Engineering, Grant-in-Aid for Young Scientists (Early Bird).

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