Compressible-incompressible two-phase flows with phase transition: model problem

TL;DR

This paper develops a mathematical model for two-phase flows with phase transition and surface tension, proving the existence of solutions and addressing regularity issues by using Navier-Stokes-Korteweg equations.

Contribution

It introduces a model combining Navier-Stokes and Navier-Stokes-Korteweg equations for two-phase flows with phase transition, proving solution existence and handling regularity challenges.

Findings

Proved existence of $ ext{R}$-bounded solution operators.

Addressed regularity loss in phase density using Navier-Stokes-Korteweg equations.

Established a mathematical framework for sharp interface two-phase flow models.

Abstract

We study the compressible and incompressible two-phase flows separated by a sharp interface with a phase transition and a surface tension. In particular, we consider the problem in $\mathbb{R}^N$, and the Navier-Stokes-Korteweg equations is used in the upper domain and the Navier-Stokes equations is used in the lower domain. We prove the existence of $\mathcal{R}$-bounded solution operator families for a resolvent problem arising from its model problem. According to Shibata \cite{GS2014}, the regularity of $\rho_+$ is $W^1_q$ in space, but to solve the kinetic equation: $\mathbf{u}_\Gamma\cdot\mathbf{n}_t = [[\rho\mathbf{u}]]\cdot\mathbf{n}_t /[[\rho]]$ on $\Gamma_t$ we need $W^{2-1/q}_q$ regularity of $\rho_+$ on $\Gamma_t$, which means the regularity loss. Since the regularity of $\rho_+$ dominated by the Navier-Stokes-Korteweg equations is $W^3_q$ in space, we eliminate the problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.

Equations

$$[[f]](x_0,t)=\underset{\begin{array}{c}x\to x_0\\ x\in {\Omega }_{t-}\end{array}}{\lim }f(x,t)-\underset{\begin{array}{c}x\to x_0\\ x\in {\Omega }_{t+}\end{array}}{\lim }f(x,t),$$

$$\bullet$$

$$\{\begin{array}{rl}& {\partial }_t\rho +\mathrm{div}(\rho \mathbf{u})=0,\\ & {\partial }_t(\rho \mathbf{u})+\mathrm{div}(\rho \mathbf{u}\otimes \mathbf{u})-\mathrm{div}\mathbf{T}=\rho \mathbf{f},\\ & {\partial }_t(\frac{\rho }{2}|\mathbf{u}{|}^2+\rho e)+\mathrm{div}((\frac{\rho }{2}|\mathbf{u}{|}^2+\rho e)\mathbf{u})-\mathrm{div}(\mathbf{Tu}-\mathbf{q})=\rho \mathbf{f}\cdot \mathbf{u}+\rho r,\end{array}$$

$$\displaystyle\left\{\begin{aligned} &[[\mathbf{u}]]=\j[[1/\rho]]\mathbf{n}_{t},\quad\j[[\mathbf{u}]]-[[\mathbf{T}]]\mathbf{n}_{t}=-\sigma H_{\Gamma_{t}}\mathbf{n}_{t},\\ &[[\theta]]=0,\quad\j\theta[[\eta]]-[[d\nabla\theta]]\cdot\mathbf{n}_{t}=0,\\ &\cfrac{\j^{2}}{2}\Big{[}\Big{[}\cfrac{1}{\rho^{2}}\Big{]}\Big{]}+[[\psi]]-\Big{[}\Big{[}\cfrac{1}{\rho}(\mathbf{Tn}_{t}\cdot\mathbf{n}_{t})\Big{]}\Big{]}=0\\ &V_{t}=\mathbf{u}_{\Gamma}\cdot\mathbf{n}_{t}=\cfrac{[[\rho\mathbf{u}]]\cdot\mathbf{n}_{t}}{[[\rho]]},\\ &(\nabla\rho_{+})\cdot\mathbf{n}_{t}=0.\end{aligned}\right.$$

$${\Omega }_{t\pm }=$$

$${\Gamma }_t=$$

$${\rho }_{-}={\rho }_{\ast -},{\mathbf{T}}_{-}={\mu }_{-}\mathbf{D}({\mathbf{u}}_{-})-{\pi }_{-}\mathbf{I},{\mathbf{q}}_{-}=-d_{-}\nabla {\theta }_{-},$$

$$\displaystyle\begin{split}\mathbf{S}_{+}&=\mu_{+}\mathbf{D}(\mathbf{u}_{+})+(\nu_{+}-\mu_{+}){\rm div}\,\mathbf{u}_{+}\mathbf{I}-\pi_{+},\\ \mathbf{K}_{+}&=\Bigl{(}\frac{\kappa_{+}+\rho_{+}\kappa^{\prime}_{+}}{2}|\nabla\rho_{+}|^{2}+\kappa_{+}\rho_{+}\Delta\rho_{+}\Bigr{)}\mathbf{I}-\kappa_{+}\nabla\rho_{+}\otimes\nabla\rho_{+}.\end{split}$$

$${\mathbf{q}}_{+}=-d_{+}\nabla {\theta }_{+}+({\kappa }_{+}{\rho }_{+}\mathrm{div}{\mathbf{u}}_{+})\nabla {\rho }_{+}$$

$${\mathbb{R}}_{\pm }^N=\{x=(x_1,\dots ,x_N)\in {\mathbb{R}}^N|\pm x_N>0\},{\mathbb{R}}_0^N=\{x\in {\mathbb{R}}^N|x_N=0\},{\dot{\mathbb{R}}}^N={\mathbb{R}}_{+}^N\cup {\mathbb{R}}_{-}^N.$$

$$f{|}_{\pm }(x_0)=\underset{\begin{array}{c}x\to x_0\\ x\in {\mathbb{R}}_{\pm }^N\end{array}}{\lim }f(x),$$

$${\partial }_t{\rho }_{+}+{\rho }_{\ast +}\mathrm{div}{\mathbf{u}}_{+}$$

$${\rho }_{\ast +}{\partial }_t{\mathbf{u}}_{+}-{\mu }_{\ast +}\Delta {\mathbf{u}}_{+}-{\nu }_{\ast +}\nabla \mathrm{div}{\mathbf{u}}_{+}$$

$$-{\rho }_{\ast +}{\kappa }_{\ast +}\nabla \Delta {\rho }_{+}$$

$$\mathrm{div}{\mathbf{u}}_{-}=f_3$$

$${\rho }_{\ast -}{\partial }_t{\mathbf{u}}_{-}-{\mu }_{\ast -}\Delta {\mathbf{u}}_{-}+\nabla {\pi }_{-}$$

$$\displaystyle\partial_{t}h-\Big{(}\frac{\rho_{*-}}{\rho_{*-}-\rho_{*+}}u_{N-}|_{-}(x_{0})-\frac{\rho_{*+}}{\rho_{*-}-\rho_{*+}}u_{N+}|_{+}(x_{0})\Big{)}$$

$${\mu }_{\ast -}{D}_{mN}({\mathbf{u}}_{-}){|}_{-}(x_0)-{\mu }_{\ast +}{D}_{mN}({\mathbf{u}}_{+}){|}_{+}(x_0)$$

$$\{{\mu }_{\ast -}{D}_{NN}({\mathbf{u}}_{-})-{\pi }_{-}\}{|}_{-}(x_0)-\{{\mu }_{\ast +}{D}_{NN}({\mathbf{u}}_{+})$$

$$+({\nu }_{\ast +}-{\mu }_{\ast +})\mathrm{div}{\mathbf{u}}_{+}+{\rho }_{\ast +}{\kappa }_{\ast +}\Delta {\rho }_{+}\}{|}_{+}(x_0)-\sigma {\Delta }^{\prime }h$$

$$\frac{1}{{\rho }_{\ast -}}\{{\mu }_{\ast -}{D}_{NN}({\mathbf{u}}_{-})-{\pi }_{-}\}{|}_{-}(x_0)-\frac{1}{{\rho }_{\ast +}}\{{\mu }_{\ast +}{D}_{NN}({\mathbf{u}}_{+})$$

$$+({\nu }_{\ast +}-{\mu }_{\ast +})\mathrm{div}{\mathbf{u}}_{+}+{\rho }_{\ast +}{\kappa }_{\ast +}\Delta {\rho }_{+}\}{|}_{+}(x_0)$$

$$u_{m-}{|}_{-}(x_0)-u_{m+}{|}_{+}(x_0)$$

$${\partial }_N{\rho }_{+}{|}_{+}(x_0)$$

$${\mathbf{u}}_{\pm }{|}_{t=0}$$

$${\rho }_{+}{|}_{t=0}$$

$$h{|}_{t=0}$$

$${\rho }_{\ast +}{\kappa }_{v\ast +}{\partial }_t{\theta }_{+}-d_{\ast +}\Delta {\theta }_{+}={\tilde{f}}_{+}$$

$${\rho }_{\ast -}{\kappa }_{v\ast -}{\partial }_t{\theta }_{-}-d_{\ast -}\Delta {\theta }_{-}={\tilde{f}}_{-}$$

$${\theta }_{-}{|}_{-}(x_0)-{\theta }_{+}{|}_{+}(x_0)=0,d_{\ast -}{\partial }_N{\theta }_{-}{|}_{-}(x_0)-d_{\ast +}{\partial }_N{\theta }_{+}{|}_{+}(x_0)=\tilde{g},$$

$${\theta }_{\pm }{|}_{t=0}={\theta }_{0\pm }\mathrm{in}{\mathbb{R}}_{\pm }^N$$

$$\displaystyle\partial_{t}h-\Big{(}\frac{\rho_{*-}}{\rho_{*-}-\rho_{*+}}u_{N-}|_{-}(x_{0})-\frac{\rho_{*+}}{\rho_{*-}-\rho_{*+}}u_{N+}|_{+}(x_{0})\Big{)}$$

$${\mu }_{\ast -}{D}_{mN}({\mathbf{u}}_{-}){|}_{-}(x_0)-{\mu }_{\ast +}{D}_{mN}({\mathbf{u}}_{+}){|}_{+}(x_0)$$

Full text

1. Introduction

This paper deals with compressible-incompressible two-phase flows separated by a sharp interface. In particular, we consider the phase transition at the interface. Our problem is formulated as follows: Let $\Omega_{t+}$ and $\Omega_{t-}$ be two time dependent domains, and $\Gamma_{t}$ be the common boundary of $\Omega_{t+}$ and $\Omega_{t-}$. We assume that $\Omega_{t+}\cap\Omega_{t-}=\emptyset$ and $\Omega_{t+}\cup\Gamma_{t}\cup\Omega_{t-}=\mathbb{R}^{N}$, where $\mathbb{R}^{N}$ denotes the $N$-dimensional Euclidean space. Furthermore, we assume that $\Omega_{t+}$ and $\Omega_{t-}$ are occupied by a compressible viscous fluid and an incompressible viscous fluid, respectively. For example, $\Omega_{t-}$ is corresponding to an ocean of infinite extent without bottom, $\Omega_{t+}$ the atmosphere, and $\Gamma_{t}$ the surface of the ocean. Let $\mathbf{n}_{t}$ be the unit outer normal to $\Gamma_{t}$ pointed from $\Omega_{t+}$ to $\Omega_{t-}$. For any $x_{0}\in\Gamma_{t}$ and function $f$ defined on $\Omega_{t+}\cup\Omega_{t-}$, we set

[TABLE]

which is the jump quantity of $f$ across $\Gamma_{t}$. Let $\dot{\Omega}_{t}=\Omega_{t+}\cup\Omega_{t-}$, and for any function $f$ defined on $\dot{\Omega}_{t}$, we write $f_{\pm}=f|_{\Omega_{t\pm}}$. In the following, we use the following symbols:

[TABLE]

Here and in the following, ${}^{\top}\!M$ denotes the transposed $M$. And then, our problem is:

[TABLE]

where $\Omega_{t\pm}$ and $\Gamma_{t}$ are given by

[TABLE]

respectively, with unknown function $h(x^{\prime},t)$. Above, $H_{\Gamma_{t}}$ is the $N-1$ times mean curvature of $\Gamma_{t}$, $\sigma$ a positive constant describing the coefficient of the surface tension, and $V_{t}$ the velocity of evolution of $\Gamma_{t}$ with respect to $\mathbf{n}_{t}$. Furthermore, $\partial_{t}=\partial/\partial t$, $\partial_{i}=\partial/\partial x_{i}$, for any matrix field $\mathbf{K}$ with $(i,j)^{\rm th}$ component $K_{ij}$, the quantity ${\rm div}\,\mathbf{K}$ is the $N$-vector with $i^{\rm th}$ component $\sum_{j=1}^{N}\partial_{j}K_{ij}$, and for any $N$ vectors or $N$ vector fields, $\mathbf{a}={}^{\top}(a_{1},\ldots,a_{N})$, $\mathbf{b}={}^{\top}(b_{1},\ldots,b_{N})$, we set $\mathbf{a}\cdot\mathbf{b}=\sum_{j=1}^{N}a_{j}b_{j}$, ${\rm div}\,\mathbf{a}=\sum_{j=1}^{N}\partial_{j}a_{j}$, and $\mathbf{a}\otimes\mathbf{b}$ denotes the $N\times N$ matrix with $(i,j)^{\rm th}$ component $a_{i}b_{j}$. The interface condition (1.2) is explained in more detail in Sect. 2 below. To describe the motion of incompressible viscous flow occupying $\Omega_{t-}$ we use the usual Navier-Stokes equations, and so we set

[TABLE]

where $\rho_{*-}$ is a positive constant describing the mass density of the reference body $\Omega_{t-}$, $\mu_{-}$ is the viscosity coefficient, $\mathbf{D}(\mathbf{u})=(1/2)(\nabla\mathbf{u}+{}^{\top}\nabla\mathbf{u})$, is the deformation tensor with $(i,j)^{\rm th}$ element $D_{ij}(\mathbf{u})=\partial_{i}u_{j}+\partial_{j}u_{i}$ for $\partial_{i}=\partial/\partial x_{i}$ and $\mathbf{u}={}^{\top}(u_{1},\ldots,u_{N})$, and $d$ is the coefficient of the heat flux. In particular, the equation of mass conservation: $\partial_{t}\rho_{-}+{\rm div}\,(\rho_{-}\mathbf{u}_{-})=0$ leads to ${\rm div}\,\mathbf{u}_{-}=0$ in $\Omega_{t-}$.

On the other hand, to describe the motion of a compressible viscous fluid occupying $\Omega_{t+}$, we adopt the Navier-Stokes-Korteweg tensor of the following form: $\mathbf{T}_{+}=\mathbf{S}_{+}+\mathbf{K}_{+}-\pi_{+}\mathbf{I}$ with

[TABLE]

Here, $\mathbf{K}_{+}$ is called the Korteweg tensor (cf. Dunn and Serrin [9] and Kotschote [17]). According to Dunn and Serrin [9], in view of the second law of thermodynamics the energy flux includes not only a classical contribution corresponding to the Fourier law but also a nonclassical contribution, which we now call the interstitial working. In this sense, the energy flux $\mathbf{q}_{+}$ is given by

[TABLE]

when we use the Navier-Stokes-Korteweg equations, where $d_{+}$ is the coefficient of the heat flux.

We assume that $\mu_{+}=\mu_{+}(\rho_{+},\theta_{+})$, $\nu_{+}=\nu_{+}(\rho_{+},\theta_{+})$, $\kappa_{+}=\kappa_{+}(\rho_{+},\theta_{+})$, $e_{+}=e_{+}(\rho_{+},\theta_{+})$, $d_{+}=d_{+}(\rho_{+},\theta_{+})$ are positive $C^{\infty}$ functions with respect to $(\rho_{+},\theta_{+})\in(0,\infty)\times(0,\infty)$, and $\psi_{+}(\rho_{+},\theta_{+})$ and $\eta_{+}(\rho_{+},\theta_{+})$ are real valued $C^{\infty}$ functions with respect to $(\rho_{+},\theta_{+})\in(0,\infty)\times(0,\infty)$, while $\mu_{-}=\mu_{-}(\theta_{-})$, $e_{-}=e_{-}(\theta_{-})$, $d_{-}=d_{-}(\theta_{-})$ are positive $C^{\infty}$ functions with respect to $\theta_{-}\in(0,\infty)$, and $\psi_{-}(\theta_{-})$ and $\eta_{-}(\theta_{-})$ are real valued $C^{\infty}$ functions with respect to $\theta_{-}\in(0,\infty)$. Moreover, we assume that $\partial e_{+}/\partial\theta_{+}>0$, $e_{-}^{\prime}>0$, and $\pi_{+}$ is given by $\pi_{+}=P_{+}(\rho_{+},\theta_{+})$, where $P_{+}$ is some $C^{\infty}$ function with respect to $(\rho_{+},\theta_{+})\in(0,\infty)\times(0,\infty)$.

We now explain why the Navier-Stokes-Korteweg equations is used in $\Omega_{t+}$ to describe the motion of the compressible viscous fluid. Shibata [28] used the Navier-Stokes-Fourier equations for $\Omega_{t+}$, that is $\mathbf{T}_{+}=\mathbf{S}_{+}$ and $\mathbf{q}_{+}=-d_{+}\nabla\theta_{+}$, to formulate the compressible-incompressible two-phase flows separated by a sharp interface with the phase transition. He proved the existence of $\mathcal{R}$ bounded solution operators for the model problem that derives the maximal $L_{p}\mathchar 45\relax L_{q}$ regularity of solutions to the linearized equations automatically with the help of Weis’s operator valued Fourier multiplier theorem [35]. According to Shibata [13], the regularity of $\rho_{+}$ is $W^{1}_{q}$ in space, but to solve the kinetic equation: $\mathbf{u}_{\Gamma}\cdot\mathbf{n}_{t}=[[\rho\mathbf{u}]]\cdot\mathbf{n}_{t}/[[\rho]]$ on $\Gamma_{t}$ we need $W^{2-1/q}_{q}$ regularity of $\rho_{+}$ on $\Gamma_{t}$, which means the regularity loss. On the other hand, the regularity of $\rho_{+}$ dominated by the Navier-Stokes-Korteweg equations is $W^{3}_{q}$ in $\Omega_{t+}$ (cf. Kotschote [16, 17] and Saito [25]), which is enough to solve the kinetic equation. In addition, quite recently Gorban and Karlin [12] proved that the Navier-Stokes-Korteweg equations is implied by the Boltzmann equation that describes the statistical behavior of a gas. In this sense, to use the Navier-Stokes-Korteweg equations to describe the motion of compressible viscous fluid flow is meaningful. Furthermore, we would like to add some comments about the Navier-Stokes-Korteweg equations. More than one hundred years ago, Korteweg [15] derived the Navier-Stokes-Korteweg equations to describe the two phase problem with diffused interface like liquid and vapor flows with phase transition, which was based on the gradient theory for the interface developed by van der Waals [33]. In 1985, Dunn and Serrin [9] studied the Navier-Stokes-Korteweg equations with the second law of thermodynamics. As equations describing the two-phase flows with diffused interface, we also know the Navier-Stokes-Allen-Chan equations and the Navier-Stokes-Chan-Hilliard equations (cf. [2]), but they can be reduced to the Navier-Stokes-Korteweg equations, which is quite recently proved by Freisthüler and Kotchote [11]. Thus, our formulation (LABEL:10000) and (1.2) includes the following situation: The ocean and atmosphere are separated by a sharp interface and on this interface the phase transition occurs. In addition, the atmosphere part is two-phase flows with diffused interface like the mixture of gas and ice. Thus, we totally treat three phase problem, and liquid and gas-solid are separated by a sharp interface with phase transition and gas-solid part has diffused interface with phase transition.

Finally, let us mention related results about the initial-boundary value problem for the Navier-Stokes-Korteweg equations. In 2008 and 2010, Kotschote [16, 17] proved the existence and uniqueness of local strong solutions for an isothermal and non-isothermal model of capillary compressible fluids derived by Dunn and Serrin [9]. Recently, Tsuda [32] studied the existence and stability of time periodic solution to the Navier-Stokes-Korteweg equations in $\mathbb{R}^{3}$, and Saito [25] proved the existence of $\mathcal{R}$ bounded solution operators for the model problem of the Navier-Stokes-Korteweg equations with free boundary conditions.

The two phase problem has been studied by Abels [1], Denisova [3, 4], Denisova and Solonnikov [6, 7], Giga and Takahashi [14], Maryani and Saito [19], Nouri and Poupaund [20], Prüss et al. [21, 22, 23], Shibata and Shimizu [30], etc.. Although these works dealt with the two phase problem for the incompressible-incompressible case, as far as the author knows, the compressible-incompressible case is few. The compressible-incompressible case was studied by Denisova [5], Denisova and Solonnikov [8], Kubo, Shibata, and Soga [18], and Shibata [28]. In particular, Denisova [5] and Denisova and Solonnikov [8] studied the compressible-incompressible case in the $L_{2}$ Sobolev-Sobodetskii space. Denisova [5] proved the energy inequality without surface tension. Denisova and Solonnikov [8] proved the global-in-time solvability without surface tension under the assumption that the data are small. On the other hand, Kubo, Shibata, and Soga [18] and Shibata [28] studied the compressible-incompressible case in the $L_{p}$ in time and $L_{q}$ in space frame work. Kubo, Shibata, and Soga [18] proved the existence of $\mathcal{R}$-bounded solution operators to the corresponding generalized resolvent problem without surface tension and without phase transition and Shibata [28] prove it with surface tension and phase transition. However, the work in [18, 28] included the problem about the regularity of density, which we mentioned above. In this paper, we eliminate this problem by using the Navier-Stokes-Korteweg equations instead of the compressible Navier-Stokes equations.

Our goal is to prove the local well-posedness and for this purpose, the key step is to prove the maximal $L_{p}$-$L_{q}$ regularity of the model problem. Let

[TABLE]

and for any $x_{0}\in\mathbb{R}^{N}_{0}$, we define $f|_{\pm}(x_{0})$ by

[TABLE]

Let $\rho_{*\pm}$ and $\theta_{*\pm}$ be the mass density and the absolute temperature of the reference domain: $\Omega_{t\pm}|_{t=0}$, all of which are positive constants. Transforming $\dot{\Omega}_{t}$ and $\Gamma_{t}$ to $\dot{\mathbb{R}}^{N}$ and $\mathbb{R}^{N}_{0}$, respectively, and linearizing the problem at $\rho_{*\pm}$ and $\theta_{*\pm}$, we have following two model problems. One is the following system:

[TABLE]

subject to the interface condition: for $x_{0}\in\mathbb{R}^{N}_{0}$ and $t\in(0,T)$

[TABLE]

and initial condition:

[TABLE]

where $m$ ranges from $1$ to $N-1$ and we have set $\mathbf{u}_{\pm}=(u_{1\pm},\dots,u_{N\pm})$, $\mu_{*+}=\mu_{+}(\rho_{*+},\theta_{*+})$, $\kappa_{*+}=\kappa_{+}(\rho_{*+},\theta_{*+})$, $\nu_{*+}=\nu_{+}(\rho_{*+},\theta_{*+})$, $\mu_{*-}=\mu_{-}(\theta_{*-})$, $\Delta^{\prime}h=\sum_{j=1}^{N-1}\partial_{j}^{2}h$.

The other is the heat equations:

[TABLE]

subject to the interface condition: for $x_{0}\in\mathbb{R}^{N}_{0}$ and $t\in(0,T)$

[TABLE]

and the initial condition:

[TABLE]

where we have set $d_{*+}=d(\rho_{*+},\theta_{*})$, $\kappa_{v*+}=(\partial e_{+}/\partial\theta_{+})(\rho_{*+},\theta_{*})$, $d_{*-}=d_{-}(\theta_{*})$, and $\kappa_{v*-}=e^{\prime}_{-}(\theta_{*})$. Here, the right-hand sides of (1.6), (1.7), (1.9), and (1.10) are nonlinear terms.

We note that the interface condition (1.7) can be rewritten as follows: for $x_{0}\in\mathbb{R}^{N}_{0}$ and $t\in(0,T)$

[TABLE]

with

[TABLE]

As in Shibata [26, 29], the maximal $L_{p}$-$L_{q}$ regularity and the generation of $C^{0}$ analytic semigroup follow automatically from the existence of $\mathcal{R}$ bounded solution operator families of the corresponding generalized resolvent problem. Hence, in this paper we concentrate on the existence of $\mathcal{R}$-bounded solution operator families for the resolvent problem arising from model problem with the interface condition:

[TABLE]

which is corresponding to the time dependent problem (1.6), (1.7), and (1.8). Here, $H(x,t)$ is an extension of $h(x^{\prime},t)$ such that $H=h$ on $\mathbb{R}^{N}_{0}$. In this paper, we do not consider (1.6), (1.7), and (1.8) anymore, and so we use the same symbols in the right-hand side of (1.12) as used in (1.6) and (1.7) below.

In order to state our main results precisely we introduce function spaces and some more symbols which will be used throughout the paper. For any scalar field $\theta$ we set $\nabla\theta=(\partial_{1}\theta,\ldots,\partial_{N}\theta)$, and for any $N$-vector field $\mathbf{u}={}^{\top}(u_{1},\ldots,u_{N})$, $\nabla\mathbf{u}$ is the $N\times N$ matrix with $(i,j)^{\rm th}$ component $\partial_{i}u_{j}$. For any domain $D$ in $\mathbb{R}^{N}$, integer $m$, and $1\leq q\leq\infty$, $L_{q}(D)$ and $W^{m}_{q}(D)$ denote the usual Lebesgue space and Sobolev space of functions defined on $D$ with norms: $\|\cdot\|_{L_{q}(D)}$ and $\|\cdot\|_{W^{m}_{q}(D)}$, respectively. We set $W^{0}_{q}(D)=L_{q}(D)$. The $\hat{W}^{1}_{q}(D)$ is a homogeneous space defined by $\hat{W}^{1}_{q}(D)=\{f\in L_{q,{\rm loc}}(D)\ |\ \nabla f\in L_{q}(D)\}$. For any Banach space $X$, interval $I$, integer $m$, and $1\leq p\leq\infty$, $L_{p}(I,X)$ and $W^{m}_{p}(I,X)$ denote the usual Lebesgue space and Sobolev space of the $X$-valued functions defined on $I$ with norms: $\|\cdot\|_{L_{p}(I,X)}$ and $\|\cdot\|_{W^{m}_{p}(I,X)}$, respectively. For any Banach space $X$, $X^{N}$ denote the $N$-product space of $X$, that is $X^{N}=\{f=(f_{1},\dots,f_{N})\ |\ f_{i}\in X\ (i=1,\dots,N)\}$. The norm of $X^{N}$ is also denoted by $\|\cdot\|_{X}$ for simplicity and $\|f\|_{X}=\sum_{j=1}^{N}\|f_{j}\|_{X}$ for $f=(f_{1},\dots,f_{N})\in X^{N}$. For any two Banach spaces $X$ and $Y$, $\mathcal{L}(X,Y)$ denotes the space of all bounded linear operators from $X$ to $Y$, and $\mathcal{L}(X)$ is the abbreviation of $\mathcal{L}(X,X)$. Let $U$ be a subset of $\mathbb{C}$. Then ${\rm Anal}\,(U,\mathcal{L}(X,Y))$ denotes the set of all $\mathcal{L}(X,Y)$-valued analytic functions defined on $U$. Throughout in this paper, the letter $C$ denotes generic constants and $C_{\alpha,\beta,\gamma,\dots}$ means that the constant depends on the quantities $\alpha,\beta,\gamma,\dots$. The values of constants $C$ and $C_{\alpha,\beta,\gamma,\dots}$ may change from line to line.

Before we state the main theorem, we first introduce the definition of $\mathcal{R}$-boundedness.

Definition 1.1**.**

($\mathcal{R}$-boundedness) Let $X$ and $Y$ be Banach spaces. A set of operators $\mathcal{T}\subset\mathcal{L}(X,Y)$ is called $\mathcal{R}$-bounded, if there is a constant $0<C<\infty$ and $1\leq p<\infty$ such that, for all $T_{1},\dots,T_{m}\in\mathcal{T}$ and $x_{1},\dots,x_{m}\in X$ with $m\in\mathbb{N}$, we have

[TABLE]

where $r_{n}(t)={\rm sign}\sin(2^{n}\pi t)$ are the Rademacher functions on $[0,1]$. The smallest such $C$ is called $\mathcal{R}$-bound of $\mathcal{T}$ on $\mathcal{L}(X,Y)$, which is denoted by $\mathcal{R}_{\mathcal{L}(X,Y)}(\mathcal{T})$.

Let $\eta_{*}$ be a constant given by

[TABLE]

and let $\varepsilon_{*}\in[0,\pi/2)$ be some angle that is given precisely in Lemma 6.1 below. In this paper, we assume $\eta_{*}\neq 0$ and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. We discuss these conditions in more detail in Remark 5.2 below (cf. Saito [25, Remark 3.3]).

The following theorem is a main theorem in this paper.

Theorem 1.2**.**

Let $1<q<\infty$ and $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Set

[TABLE]

Then, there exist a positive constant $\lambda_{0}$ and operator families $\mathcal{A}_{\pm}(\lambda)$, $\mathcal{B}_{+}(\lambda)$, $\mathcal{P}_{-}(\lambda)$, and $\mathcal{H}(\lambda)$ with

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$ and $\mathbf{F}=(f_{1},\mathbf{f}_{2},f_{3},\mathbf{f}_{4},\mathbf{f}_{5},\mathbf{g},\mathbf{h},k,d)\in X_{q}$, $\mathbf{u}_{\pm}=\mathcal{A}_{\pm}(\lambda)\widetilde{\mathbf{F}}$, $\rho_{+}=\mathcal{B}_{+}(\lambda)\widetilde{\mathbf{F}}$, $\pi_{-}=\mathcal{P}_{-}(\lambda)\widetilde{\mathbf{F}}$, and $H=\mathcal{H}(\lambda)\widetilde{\mathbf{F}}$ are unique solutions of problem (1.12). Furthermore, for $s=0,1$, we have

[TABLE]

with some positive constant $c_{0}$. Here, $G_{\lambda}^{1}\mathcal{A}_{\pm}(\lambda)=(\lambda\mathcal{A}_{\pm}(\lambda),\lambda^{1/2}\nabla\mathcal{A}_{\pm}(\lambda),\nabla^{2}\mathcal{A}_{\pm}(\lambda))$, $G_{\lambda}^{2}\mathcal{B}_{-}(\lambda)=(\lambda\mathcal{B}_{-}(\lambda),\lambda^{1/2}\nabla^{2}\mathcal{B}_{-}(\lambda),\nabla^{3}\mathcal{B}_{-}(\lambda))$, $G_{\lambda}^{3}\mathcal{H}(\lambda)=(\lambda\mathcal{H}(\lambda),\nabla\mathcal{H})$, and

[TABLE]

Remark 1.3**.**

(1) The uniqueness of solutions of problem (1.12) follows from the existence of solutions for a dual problem in a similar way to Shibata and Shimizu [31, Sect. 3], so that we omit its proof.

(2) It is easy to show the existence of $\mathcal{R}$-bounded solution operator families for the resolvent problem arising from (1.9), (1.10), and (1.11). In fact, when we employ the similar argumentation to that in the proof of Theorem 1.2 given in the sequel. Hence, we do not consider problem (1.9), (1.10), and (1.11) in this paper.

(3) We can show the maximal $L_{p}\mathchar 45\relax L_{q}$ regularity theorem for (1.6), (1.7), (1.8), (1.9), (1.10), and (1.11) due to the same theory as in Shibata [28] with the help of the $\mathcal{R}$-bounded solution operator and the operator valued Fourier multiplier theorem of Weis [35].

This paper is organized as follows. In Sect. 2, according to the argument due to Prüss et al. [21] (cf. Prüss and Simonett [24] and Shibata [27]) we explain the interface condition (1.2) in more detail from the point of conservation of mass, conservation of momentum, conservation of energy and increment of entropy and we show the complete model. In Sect. 3, we introduce some results of half spaces. From Sect.. 4 to Sect. 6, we consider the problem without the surface tension. In Sect. 4, by the partial Fourier transform, we have ordinary differential equations with respect to $x_{N}$. Then, we solve them and apply the inverse partial Fourier transform to its solution in order to obtain exact solution formulas to the resolvent problem. In Sect. 5, we introduce some technical lemmas and give some estimates for the multipliers appearing in the solution formula. In Sect. 6, we analyze the Lopatinski determinant appearing in the solution formula. Finally in Sect. 7, we prove the main theorem for the $\mathcal{R}$-bounded solution operator families.

2. Derivation of interface conditions

In this section, assuming that the equation (LABEL:10000) holds in the bulk $\dot{\Omega}_{t}$, we derive interface conditions (1.2) under which balance of mass, balance of momentum, balance of energy, and entropy production hold. We follow the argument due to Prüss et al. in [21]. Our model is, however, different from Prüss et al. [21], and so we give a detailed explanation.

For our purpose we may assume that integration appearing below is finite. In this sense, our argument below is rather formal from the integrability point of view 11*1In order to make the discussion in this section rigorous, it is enough to assume that the domain is bounded and the outer boundary conditions are imposed like Prüss et al. [21]. In addition, we assume that there exists a smooth diffeomorphism $\phi_{t}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ such that

[TABLE]

for $t>0$. Let $\mathbf{v}=(\partial_{t}\phi_{t})(\phi_{t}^{-1}(x),t)$ and then, it follows from the Reynolds transport theorem that

[TABLE]

In particular, $\mathbf{u}_{\Gamma}\cdot\mathbf{n}_{t}=\mathbf{v}\cdot\mathbf{n}_{t}$ on $\Gamma_{t}$. We assume that

[TABLE]

hold in the bulk $\dot{\Omega}_{t}$. And then, we look for the interface conditions under which the following formulas hold:

[TABLE]

where $|\Gamma_{t}|$ is the Hausdorff measure of $\Gamma_{t}$. We know that

[TABLE]

Here and in the sequel, $d\tau$ denotes the surface element of $\Gamma_{t}$. **Balance of Mass:

**By (2.1), (2.2), and the divergence theorem of Gauss we have

[TABLE]

Thus, to obtain (2.5) it is sufficient to assume that

[TABLE]

and so we define $\j$ be

[TABLE]

which is called the phase flux, more precisely, the interfacial mass flux. Since $\mathbf{u}_{\Gamma}\cdot\mathbf{n}_{t}=\mathbf{u}_{+}\cdot\mathbf{n}_{t}-\j/\rho_{+}=\mathbf{u}_{-}\cdot\mathbf{n}_{t}-\j/\rho_{-}$, we have

[TABLE]

A phase transition takes place if $\j\not=0$.

Furthermore, by (2.10), we have

[TABLE]

which is the kinetic condition in the case that $\j\not=0$.

On the other hand, if $\j=0$, then we have $\mathbf{u}_{\pm}\cdot\mathbf{n}_{t}=\mathbf{u}_{\Gamma}\cdot\mathbf{n}_{t}$ on $\Gamma_{t}$. Thus, we have a usual kinetic condition: $V_{\Gamma_{t}}=\mathbf{u}\cdot\mathbf{n}_{t}$. If $\Gamma_{t}$ is defined by $F(x,t)=0$ locally, then $F(\phi_{t}(y),t)=0$ for $y\in\Gamma_{0}$. Thus, we have

[TABLE]

Since $\mathbf{n}_{t}$ is parallel to $\nabla F$, we have $(\nabla F)\cdot\mathbf{u}_{\Gamma}=(\nabla F)\cdot\mathbf{u}_{\pm}$ on $\Gamma_{t}$, and so we have

[TABLE]

This is a different representation formula of kinetic condition when the phase transition does not take place. Balance of Momentum:

We will prove that it follows from the balance of momentum that

[TABLE]

In fact, we write (2.3) componentwise as

[TABLE]

And then, by (2.1) we have

[TABLE]

By (2.10), we have $[[\rho u_{i}(\mathbf{u}_{\Gamma}-\mathbf{u})]]\cdot\mathbf{n}_{t}=-\j[[u_{i}]]$ on $\Gamma_{t}$, and so in order that (2.6) holds it is sufficient to assume that

[TABLE]

which leads to

[TABLE]

where $\mathbf{T}_{\Gamma_{t}}$ denotes surface stress and ${\rm div}\,_{\Gamma_{t}}$ denotes the surface divergence. When we consider surface tension on $\Gamma_{t}$, we have

[TABLE]

where $x$ is a position vector of $\Gamma_{t}$. Thus, we have the interface condition:

[TABLE]

Moreover, by (2.2) we have

[TABLE]

and so we can rewrite (2.3) as

[TABLE]

**Balance of Energy:

**We will prove that it follows from the balance of energy that

[TABLE]

In fact, by (2.1), (2.9) and (2.4), we have

[TABLE]

If we assume that

[TABLE]

then we have the balance of energy (2.7). By (2.10)

[TABLE]

Noting that $\mathbf{T}$ is a symmetric matrix, by (2.13) we have

[TABLE]

Putting these formulas together gives the following interface condition:

[TABLE]

By (2.2) and (2.3), we rewrite (2.4) as

[TABLE]

where we have set $\mathbf{T}:\nabla\mathbf{u}=\sum_{i,j=1}^{N}T_{ij}\partial_{i}u_{j}$. Putting this and (2.4) together gives

[TABLE]

Entropy Production:

We now introduce the fundamental thermodynamic relations which read

[TABLE]

The quantities $\kappa_{v}$ and $\ell$ are called heat capacity and latent heat, respectively. We assume that

[TABLE]

As constitutive laws in the phases, for the compressible viscous fluid part, $\Omega_{t+}$, we employ the Korteweg’s law for the stress tensor and the Dunn-Serrin law for the energy flux, while for the incompressible viscous fluid part, $\Omega_{t-}$, we employ the Newton’s law for the stress tensor and Fourier’s law for the energy flux. Namely, we assume that Constitutive Law in the Phases:

[TABLE]

Here, $\mathbf{D}(\mathbf{u})=(1/2)(\nabla\mathbf{u}+{}^{\top}\nabla\mathbf{u})$. To ensure non-negative entropy production in the bulk $\Omega_{t\pm}$, we assume that

[TABLE]

Assuming $\mathbf{q}_{+}=-d_{+}\nabla\theta_{+}+(\kappa_{+}(\rho_{+})\rho_{+}{\rm div}\,\mathbf{u}_{+})\nabla\rho_{+}$, Dunn and Serrin [9] proved that $\mathbf{K}_{+}$ should equal $(1/2)(\kappa_{+}+\rho_{+}\kappa_{+}^{\prime})|\nabla\rho_{+}|^{2}+\kappa_{+}\rho_{+}\Delta\rho_{+})\mathbf{I}-\kappa_{+}\nabla\rho_{+}\otimes\nabla\rho_{+}$ to ensure non-negative entropy production. But, in the following, assuming that $\mathbf{K}_{+}=(\alpha_{0}(\rho_{+})|\nabla\rho_{+}|^{2}+\alpha_{1}(\rho_{+})\Delta\rho_{+})\mathbf{I}+\beta(\rho_{+})\nabla\rho\otimes\rho$, we prove that $\mathbf{q}_{+}=-d_{+}\nabla\theta_{+}+(\kappa_{+}(\rho_{+})\rho_{+}{\rm div}\,\mathbf{u}_{+})\nabla\rho_{+}$, and $\alpha_{0}$, $\alpha_{1}$ and $\beta$ should be given as in (1.4) to ensure non-negative entropy production. Namely, our argument below is just opposite direction to Dunn and Serrin [9].

We also assume the following. Constitutive Law on the Interface $\Gamma_{t}$:

[TABLE]

Hence in our model, the temperature and the tangential part of velocity field are continuous across the interface, and the interstitial working does not take place in the normal direction of boundary $\Gamma_{t}$. In addition, the third boundary condition in (2.18) ensures the Fourier law and the generalized Gibbs-Thomson law, which we will explain below.

We first consider

[TABLE]

We assume that $e_{+}=e_{+}(\eta_{+},\rho_{+},\nabla_{+}\rho_{+})$. Let $z_{j}$ be a variable corresponding to $\partial_{j}\rho_{+}$. We then have

[TABLE]

By (2.2), we have

[TABLE]

Differentiating (2.19) by $x_{j}$, we have

[TABLE]

Inserting (2.19) and (2.20) and using the assumption: $\partial e_{+}/\partial\eta_{+}=\theta_{+}$, we have

[TABLE]

On the other hand, by (2.14), we have

[TABLE]

and so

[TABLE]

Using (2.2), we have $\partial_{t}(\rho\eta)+{\rm div}\,(\rho\eta\mathbf{u})=\rho(\partial_{t}\eta+\mathbf{u}\cdot\nabla\eta)$, and so by (2.1) we have

[TABLE]

From (2.21) we have

[TABLE]

We now assume that

[TABLE]

Since $\mathbf{D}(\mathbf{u})=(D_{ij}(\mathbf{u}))$ is symmetric, setting $\mathbf{u}={}^{\top}(u_{1},\ldots,u_{N})$, we have

[TABLE]

In the same manner, we have

[TABLE]

We then have

[TABLE]

Using chain rule and the fact that $(\nabla\alpha_{1})\cdot(\nabla\rho_{+})=\alpha^{\prime}_{1}|\nabla\rho_{+}|^{2}$, we have

[TABLE]

Inserting the formulas above into (2.23), we have

[TABLE]

We now assume that

[TABLE]

to obtain

[TABLE]

In particular, we have

[TABLE]

with some function $e^{\prime}_{+}(\eta_{+},\rho_{+})$. Thus, we have

[TABLE]

and so we have

[TABLE]

which, combined with (2.24), leads to

[TABLE]

We now assume that $\mathbf{q}_{+}=-d_{+}\nabla\theta_{+}+\alpha_{1}{\rm div}\,\mathbf{u}_{+}(\nabla\rho_{+})$, that is we employ the Dunn-Serrin law for the energy flux. Furthermore, we assume that

[TABLE]

We then have

[TABLE]

If we choose

[TABLE]

then, the formulas in (2.27) hold. In particular, we have the Korteweg tensor given in (1.4).

Since $\mathbf{T}_{-}=\mathbf{S}_{-}$, assuming that $e_{-}=e_{-}(\rho_{-},\eta_{-})$, $\mathbf{q}_{-}=-d_{-}\nabla\theta_{-}$ and $\dfrac{\partial e_{-}}{\partial\eta_{-}}=\theta_{-}$, and $\dfrac{\partial e_{-}}{\partial\rho_{-}}\rho_{-}^{2}=\pi_{-}$, we also have

[TABLE]

Putting (2.22), (2.28), and (2.30) together and assuming that $[[\theta]]=0$, by the divergence theorem of Gauss we have

[TABLE]

Since $({\rm div}\,\mathbf{u})^{2}\leq N|\mathbf{D}(\mathbf{u})|^{2}$, if we assume that

[TABLE]

then we have

[TABLE]

because ${\rm div}\,\mathbf{u}_{-}=0$ in $\Omega_{t-}$. Since $[[\rho\eta(\mathbf{u}-\mathbf{u}_{\Gamma})]]\cdot\mathbf{n}=\j[[\eta]]$ as follows from (2.10), to obtain Entropy Production:

[TABLE]

it is sufficient to assume that

[TABLE]

which is called the Stefan law.

We now derive the generalized Gibbs-Thomson law:

[TABLE]

provided that $\j\not=0$. Let $\tau_{i}$ ($i=1,\ldots,N-1$) be the tangent vectors of $\Gamma_{t}$. We then write

[TABLE]

Using the orthogonality of $\{\tau_{1},\ldots,\tau_{N-1},\mathbf{n}_{t}\}$, we have

[TABLE]

Since $[[\mathbf{u}-<\mathbf{u},\mathbf{n}_{t}>\mathbf{n}_{t}]]=0$ as follows from (2.18), we have

[TABLE]

and so by (2.10) we have

[TABLE]

Inserting this formula into the second formula in (2.14), using the relation: $\psi=e-\theta\eta$, and recalling the formulas: $\mathbf{q}_{+}=-d_{+}\nabla\theta_{+}+(\kappa_{+}\rho_{+}{\rm div}\,\mathbf{u}_{+})\nabla\rho_{+}$ and $\mathbf{q}_{-}=-d_{-}\nabla\theta_{-}$, by (2.18) and (2.33) we have

[TABLE]

We write the last term as

[TABLE]

By (2.10), we have

[TABLE]

On the other hand, by (2.18), (2.13), and (2.36), we have

[TABLE]

Summing up, we have obtained

[TABLE]

which, combined with (2.35), leads to (2.34). Notice that if $\j=0$, we do not have (2.34).

Finally, using $\theta$ we rewrite the first equation in (2.14). Since

[TABLE]

by (2.28) and (2.30), we have

[TABLE]

By (2.15),

[TABLE]

and so, by (2.2) we have

[TABLE]

which, combined with (2.37) leads to

[TABLE]

Summing up, we have obtained the following complete model. **The Complete Model:

**In the bulk:

[TABLE]

On the interface $\Gamma_{t}$:

[TABLE]

3. Results of half spaces

In this section, we introduce some results of half spaces. To prove Theorem 1.2, we consider the following systems:

[TABLE]

The existence of $\mathcal{R}$-bounded solution operators of (3.1) and (3) are proved by Saito [25] and Shibata [26], respectively. In fact, we know the following two lemmas.

Lemma 3.1**.**

([25])* Let $1<q<\infty$, $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Set*

[TABLE]

Then, there exists a positive constant $\lambda_{0}$ and operator families $\mathcal{A}^{1}_{+}(\lambda)$ and $\mathcal{B}^{1}_{+}(\lambda)$ with

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$ and $\mathbf{F}_{+}=(f_{1},\mathbf{f}_{2},\widetilde{\mathbf{g}},k)\in Y_{q+}$, $\mathbf{u}_{+}=\mathcal{A}^{1}_{+}(\lambda)\widetilde{\mathbf{F}}_{+}$ and $\rho_{+}=\mathcal{B}^{1}_{+}(\lambda)\widetilde{\mathbf{F}}_{+}$ are unique solutions of problem (3.1). Furthermore, for $s=0,1$, we have

[TABLE]

with some positive constant $c_{0}$. Here, $\widetilde{\mathbf{F}}_{+}=(f_{1},\mathbf{f}_{2},\lambda^{1/2}\widetilde{\mathbf{g}},\nabla\widetilde{\mathbf{g}},\lambda k,\lambda^{1/2}\nabla k,\nabla^{2}k)$.

Lemma 3.2**.**

([26])* Let $1<q<\infty$, $0<\varepsilon<\pi/2$. Set*

[TABLE]

Then, there exists operator families $\mathcal{A}^{1}_{-}(\lambda)$ and $\mathcal{P}^{1}_{-}(\lambda)$ with

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon}$ and $\mathbf{F}_{-}=(f_{3},\mathbf{f}_{4},\mathbf{f}_{5},g_{N})\in Y_{q-}$, $\mathbf{u}_{-}=\mathcal{A}^{1}_{-}(\lambda)\widetilde{\mathbf{F}}_{-}$ and $\pi_{-}=\mathcal{P}^{1}_{-}(\lambda)\widetilde{\mathbf{F}}_{-}$ are unique solutions of problem (3). Furthermore, for $s=0,1$, we have

[TABLE]

with some positive constant $c_{0}$. Here, $\widetilde{\mathbf{F}}_{-}=(\lambda^{1/2}f_{3},\nabla f_{3},\lambda\mathbf{f}_{4},\mathbf{f}_{5},\lambda^{1/2}g_{N},\nabla g_{N})$.

Thus, it is sufficient to consider the problem (1.12) with $f_{1}=0$, $\mathbf{f}_{2}=0$, $f_{3}=0$, $\mathbf{f}_{5}=0$, and $g_{k}=0$ ($k=1,\dots,N+1$). Finally, we consider one more auxiliary problem:

[TABLE]

From Sect. 4 to Sect. 6, we prove the following theorem.

Theorem 3.3**.**

Let $1<q<\infty$ and $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Set

[TABLE]

Then, there exist a positive constant $\lambda_{0}$ and operator families $\mathcal{A}_{\pm}(\lambda)$, $\mathcal{B}_{+}(\lambda)$, and $\mathcal{P}_{-}(\lambda)$ with

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$ and $\mathbf{h}\in Z_{q}$, $\mathbf{u}_{\pm}=\mathcal{A}^{2}_{\pm}(\lambda)\widetilde{\mathbf{H}}$, $\rho_{+}=\mathcal{B}^{2}_{+}(\lambda)\widetilde{\mathbf{H}}$, and $\pi_{-}=\mathcal{P}^{2}_{-}(\lambda)\widetilde{\mathbf{H}}$ are solutions of problem (3.2). Furthermore, for $s=0,1$, we have

[TABLE]

with some positive constant $c_{0}$. Here, $\widetilde{\mathbf{H}}=(\lambda\mathbf{h},\lambda^{1/2}\nabla\mathbf{h},\nabla^{2}\mathbf{h})$.

4. Solution formulas without surface tension

In this section, we consider the following equations:

[TABLE]

where we have added $\sigma_{\pm}\Delta^{\prime}H$ with $\sigma_{\pm}=\rho_{*\pm}\sigma/(\rho_{*-}-\rho_{*+})$ to (3.2) for the latter use. Let $\widehat{v}=\mathcal{F}_{x^{\prime}}[v](\xi^{\prime},x_{N})$ denote the partial Fourier transform with respect to the tangential variable $x^{\prime}=(x_{1},\dots,x_{N-1})$ with $\xi^{\prime}=(\xi_{1},\dots,\xi_{N-1})$ defined by $\mathcal{F}_{x^{\prime}}[v](\xi^{\prime},x_{N})=\int_{\mathbb{R}^{N-1}}e^{-i\xi^{\prime}\cdot\xi^{\prime}}v(x^{\prime},x_{N})dx^{\prime}$. Applying the partial Fourier transforms to (4.1) yields ordinary differential equations with respect to $x_{N}\neq 0$:

[TABLE]

subject to the interface condition:

[TABLE]

Here and in the sequel, $j$ runs from $1$ to $N-1$. According to Saito [25], from (4.2), (4.3), and (4.4), we obtain

[TABLE]

with

[TABLE]

The roots of $P_{\lambda}(t)=0$ are $t=\pm\sqrt{|\xi^{\prime}|^{2}+s_{i}\lambda}\ (i=1,2)$ and $s_{i}\ (i=1,2)$ are the root of the following equation:

[TABLE]

Here, $t_{i}$ are defined by $t_{i}=\sqrt{|\xi^{\prime}|^{2}+s_{i}\lambda}$, whose detail will be discussed in Sect. 5. As seen in Sect. 5, we have three roots $B_{+}$, $t_{1}$, and $t_{2}$ with positive real parts different from each other.

On the other hand, according to Shibata [28], from (4.5), (4.6), and (4.7), we obtain

[TABLE]

with

[TABLE]

In view of (4.13), (4.15), and (4.16), we look for solutions $\widehat{u}_{J\pm}$ and $\widehat{\pi}_{-}$ of the forms:

[TABLE]

Using $A$ and $B_{\pm}$, we rewrite (4.3), (4.4), (4.6), and (4.7) as follows:

[TABLE]

To state our solution formulas of equations: (4.2) - (4.12), we introduce some classes of multipliers.

Definition 4.1**.**

Let $0<\varepsilon<\pi/2,\ \lambda_{0}\geq 0$, and let $s$ be a real number. Set

[TABLE]

Let $m(\lambda,\xi^{\prime})$ be a function defined on $\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ which is infinitely times differentiable with respect to $\tau$ and $\xi^{\prime}$ when $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$.

(1) If there exists a real number $s$ such that for any multi-index $\alpha^{\prime}=(\alpha_{1},\dots,\allowbreak\alpha_{N-1})\in\mathbb{N}_{0}^{N-1}$ and $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ there hold the estimates:

[TABLE]

for some constant $C_{\alpha^{\prime}}$ depending on $s$, $\alpha^{\prime}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$. Then, $m(\lambda,\xi^{\prime})$ is called a multiplier of order $s$ with type 1.

(2) If there exists a real number $s$ such that for any multi-index $\alpha^{\prime}=(\alpha_{1},\dots,\allowbreak\alpha_{N-1})\in\mathbb{N}_{0}^{N-1}$ and $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ there hold the estimates:

[TABLE]

for some constant $C_{\alpha^{\prime}}$ depending on $s$, $\alpha^{\prime}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$. Then, $m(\lambda,\xi^{\prime})$ is called a multiplier of order $s$ with type 2.

In what follows, we denote the set of multipliers defined on $\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ of order $s$ with type $l\ (l=1,2)$ by $\mathbb{M}_{s,l,\varepsilon,\lambda_{0}}$.

Obviously, $\mathbb{M}_{s,l,\varepsilon,\lambda_{0}}$ are the vector spaces on $\mathbb{C}$. Furthermore, by the fact $|\lambda^{1/2}+A|^{-|\alpha^{\prime}|}\leq A^{-|\alpha^{\prime}|}$ and the Leibniz rule, we have the following lemma immediately.

Lemma 4.2**.**

Let $s_{1},s_{2}\in\mathbb{R}$, $0<\varepsilon<\pi/2$, $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$.

(1) Given $m_{i}\in\mathbb{M}_{s_{i},1,\varepsilon,\lambda_{0}}\ (i=1,2)$, we have $m_{1}m_{2}\in\mathbb{M}_{s_{1}+s_{2},1,\varepsilon,\lambda_{0}}$.

(2) Given $l_{i}\in\mathbb{M}_{s_{i},i,\varepsilon,\lambda_{0}}\ (i=1,2)$, we have $l_{1}l_{2}\in\mathbb{M}_{s_{1}+s_{2},2,\varepsilon,\lambda_{0}}$.

(3) Given $n_{i}\in\mathbb{M}_{s_{i},2,\varepsilon,\lambda_{0}}\ (i=1,2)$, we have $n_{1}n_{2}\in\mathbb{M}_{s_{1}+s_{2},2,\varepsilon,\lambda_{0}}$.

Remark 4.3**.**

We easily see that $i\xi_{j}\in\mathbb{M}_{1,2,\varepsilon,0}$ ($j=1,\dots,N-1$), and $A\in\mathbb{M}_{1,2,\varepsilon,0}$. Especially, $i\xi_{j}/A\in\mathbb{M}_{0,2,\varepsilon,0}\ (j=1,\dots,N-1)$. In addition, $\mathbb{M}_{s,1,\varepsilon,\lambda}\subset\mathbb{M}_{s,2,\varepsilon,\lambda}$ for any $s\in\mathbb{R}$.

Then, we arrive at the following solution formulas for equations (4.2) - (4.12):

[TABLE]

with

[TABLE]

Here and in the following, $J$ runs from $1$ through $N$. Recall that $j$ and $m$ run from $1$ through $N-1$, respectively. Furthermore, we define $M_{0+}(x_{N})$, $M_{1+}(x_{N})$, $M_{2+}(x_{N})$, and $M_{-}(x_{N})$ as follows:

[TABLE]

From now on, we prove (4.26). On the other hand, we prove (4.27) in Sect.5. By (4.19), we obtain

[TABLE]

with

[TABLE]

Then, from (4.20) and (4.21), we have

[TABLE]

which furnishes that

[TABLE]

By (4.5), (4.22), and (4.23), we have

[TABLE]

Combining (4.31) and (4.32), we deduce $i\xi^{\prime}\cdot\beta^{\prime}_{-}+A\beta_{N-}=0$. Hence, by (4.31), (4.32), and (4.33), we observe

[TABLE]

Next, we consider the interface condition. From (4.2) and (4.10), we have

[TABLE]

Substituting (4.28) and (4.30) into (4.37) to obtain

[TABLE]

because $\lambda\mu_{*+}\neq 0$. In addition, by (4.28) and (4.38), it follows that

[TABLE]

Together with (4.11) and (4.39), this shows

[TABLE]

By (4.9), we have

[TABLE]

Combining with (4.35) and (4.36), this yields

[TABLE]

From (4.8), we have

[TABLE]

which implies

[TABLE]

Substituting (4.30), (4.34), (4.38), (4.39), (4.40), and (4.41) into (4.43), we obtain

[TABLE]

with

[TABLE]

By (4.2) and (4.12), we have

[TABLE]

Consequently, by (4.44) and (4.46), we have

[TABLE]

where

[TABLE]

By direct calculations, we have

[TABLE]

According to Saito [25], we have following formula:

[TABLE]

with

[TABLE]

Then, we rewrite (4.52) as follows:

[TABLE]

If $\det L\neq 0$, the inverse of $L$ exists and we see

[TABLE]

with

[TABLE]

In this section, we assume $\det L\neq 0$ and continue to obtain the solution formula. We shall prove $\det L\neq 0$ when $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ in Sect. 6. By (4.58) and (4.63), we obtain

[TABLE]

Setting

[TABLE]

with $m=1,\dots,N-1$, we have

[TABLE]

From (4.30), we have

[TABLE]

with

[TABLE]

Furthermore, combined with (4.29), (4.30), and (4.65), we have

[TABLE]

with

[TABLE]

By (4.38), we have

[TABLE]

with

[TABLE]

Substituting (4.65) into (4.40) to obtain

[TABLE]

Combining (4.41) and (4.67), we obtain

[TABLE]

with

[TABLE]

Substituting (4.67) and (4.68) into (4.35) and (4.36), we have

[TABLE]

respectively. Here we set

[TABLE]

for short. From (4.32) and (4.70), we have

[TABLE]

with

[TABLE]

Accordingly, by (4.11), (4.42), and (4.71), we obtain

[TABLE]

with

[TABLE]

This completes the proof of (4.26).

To prove Theorem 3.3, we consider problem (3.2), namely, problem (4.1) with $H=0$. First of all, we define our solution operators $\mathcal{A}_{Ji}^{3+}(\lambda)$ $(i=1,2,3,4)$, $\mathcal{A}_{Ji}^{3-}(\lambda)$ $(i=1,2,3)$, $\mathcal{B}_{+}^{3}(\lambda)$, and $\mathcal{P}_{-}^{3}(\lambda)$ of problem (4.26) such that

[TABLE]

with

[TABLE]

In order to prove Theorem 3.3, we introduce following lemma and corollary.

Lemma 4.4**.**

Let $1<q<\infty$, $\lambda_{0}\geq 0$, $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. For $m(\lambda,\xi^{\prime})\in\mathbb{M}_{0,2,\varepsilon,\lambda_{0}}$, $i=0,1,2,3$, and $j=1,2$, we define operators $J_{i}(\lambda)$, $K_{i}(\lambda)$, and $L_{j}(\lambda)$ by

[TABLE]

Then, for $i=0,1,2$, $j=1,2$, and $s=0,1$, the sets $\{(\tau\partial\tau)^{s}J_{i}(\lambda)\}$, $\{(\tau\partial\tau)^{s}K_{i}(\lambda)\}$, and $\{(\tau\partial\tau)^{s}L_{j}(\lambda)\}$ are $\mathcal{R}$-bounded families in $\mathcal{L}(L_{q}(\mathbb{R}^{N}_{+}))$, whose $\mathcal{R}$-bounds do not exceed some constant $C_{N,q,\lambda_{0},\varepsilon,\mu_{*+},\nu_{*+},\kappa_{*+}}$ depending essentially only on $N$, $q$, $\lambda_{0}$, $\varepsilon$, $\mu_{*+}$, $\nu_{*+}$, and $\kappa_{*+}$.

Proof.

First we consider $K_{0}(\lambda)$. Setting

[TABLE]

we have

[TABLE]

Employing the same argumentation due to Shibata and Shimizu [31, Lemma 5.4], it is sufficient to prove

[TABLE]

According to Saito [25, Lemma 4.8], we have

[TABLE]

Here and in the sequel, $b$ is a positive constant depending on $\varepsilon$, $\mu_{*+}$, $\nu_{*+}$, and $\kappa_{*+}$. On the other hand, $C$ is a positive constant depending on $\alpha^{\prime}$, $\lambda_{0}$, $\varepsilon$, $\mu_{*+}$, $\nu_{*+}$, and $\kappa_{*+}$. Then, by the Leibniz rule and the assumption we have

[TABLE]

which, combined with Theorem 3.6 in Shibata and Shimizu [31], furnishes that

[TABLE]

On the other hand, using (4.73) with $s=0$ and $\alpha^{\prime}=0$, for $s=0,1$, we have

[TABLE]

which, combined with (4.73), implies (4.72). Thus this completes the case $K_{0}(\lambda)$.

Next we consider $K_{1}(\lambda)$ and $K_{2}(\lambda)$. By the identities:

[TABLE]

we have for $j=1,2$

[TABLE]

Since $M_{j+}(x_{N})=-\mathfrak{r}_{j}(\lambda,\xi^{\prime})x_{N}\int_{0}^{1}e^{-\{\theta t_{j}+(1-\theta)B_{+}\}x_{N}}\,d\theta$ and $\mathfrak{r}_{j}(\lambda,\xi^{\prime})\in\mathbb{M}_{0,1,\varepsilon,0}$ by (5.6) below, we can prove the required properties in the same manner as the case $K_{0}(\lambda)$. In addition, we can prove the case $J_{i}(\lambda)$ ($i=0,1,2$) in the same manner as the case $K_{0}(\lambda)$, so that we may omit those proof.

Then, we consider $L_{1}(\lambda)$. If we set

[TABLE]

the operator $L_{1}(\lambda)$ is given by the formula:

[TABLE]

so that to prove that $L_{1}$ has the required properties it is sufficient to prove

[TABLE]

According to Saito [25, Lemma 4.5] and Shibata and Shimizu [31, Lemma 5.3], we have

[TABLE]

Then, by the Leibniz rule and the assumption we have

[TABLE]

which, combined with Theorem 3.6 in Shibata and Shimizu [31], furnishes that

[TABLE]

On the other hand, using (4.76) with $s=0$ and $\alpha^{\prime}=0$, for $s=0,1$, we have

[TABLE]

which, combined with (4.76), implies (4.75). Hence, this completes the case $L_{1}(\lambda)$. Furthermore, we can prove the case $L_{2}(\lambda)$ in the same manner as the case $L_{1}(\lambda)$, so that we omit its proof. ∎

Corollary 4.5**.**

Let $1<q<\infty$ and $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Set $N_{1}=N+1$ and $N_{2}=N^{2}+N+1$.

(1) Let $r=1,2$. For $k_{r,1}(\lambda,\xi^{\prime})\in\mathbb{M}_{r-3,2,\varepsilon,0}$ and $l_{r}(\lambda,\xi^{\prime})\in\mathbb{M}_{r-4,2,\varepsilon,0}$, we define operators $K_{0}^{r}(\lambda)$, $L_{j}^{r}(\lambda)$ ($j=1,2$) by

[TABLE]

for $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$, and $f\in W^{r}_{q}(\mathbb{R}^{N}_{+})$. Then, there exist operators $\widetilde{K}^{r}_{0}(\lambda)$, $\widetilde{L}_{j}^{r}(\lambda)$, with

[TABLE]

such that for any $g\in W^{1}_{q}(\mathbb{R}_{+}^{N})$ and any $h\in W^{2}_{q}(\mathbb{R}_{+}^{N})$

[TABLE]

Furthermore, for $s=0,1$, $j=1,2$

[TABLE]

with a positive constant $C=C_{N,q,\varepsilon,\lambda_{0},\mu_{*+},\nu_{*+},\kappa_{*+}}$.

(2) Let $i,r=1,2$. For $k_{r,2}(\lambda,\xi^{\prime})\in\mathbb{M}_{r-2,2,\varepsilon,0}$, we define operators $K_{i}^{r}(\lambda)$ by

[TABLE]

for $\lambda\in\Sigma_{\varepsilon,0}$, and $f\in W^{r}_{q}(\mathbb{R}^{N}_{+})$. Then, there exist operators $\widetilde{K}^{r}_{0}(\lambda)$, $\widetilde{L}_{j}^{r}(\lambda)$, with

[TABLE]

such that for any $g\in W^{1}_{q}(\mathbb{R}_{+}^{N})$ and any $h\in W^{2}_{q}(\mathbb{R}_{+}^{N})$

[TABLE]

Furthermore, for $s=0,1$,

[TABLE]

with a positive constant $C=C_{N,q,\varepsilon,\mu_{*+},\nu_{*+},\kappa_{*+}}$.

Proof.

We only prove the case $K_{0}^{1}(\lambda)$. The proof of other cases are similar to $K_{0}^{1}(\lambda)$, so that we may omit the detailed proof (cf. Saito [25, Corrolary 4.9] and Shibata and Shimizu [31, Lemma 5.6, Lemma 5.7]).

By the definition of $M_{0+}(x_{N})$, we have

[TABLE]

which, combined with (4.74), implies that

[TABLE]

Here, we use Volevich’s formula [34]:

[TABLE]

with $f(\xi^{\prime},x_{N}+y_{N})\widehat{g}(y_{N})\to 0$ as $y_{N}\to\infty$.

First, we estimate $\widetilde{K}^{1}_{0,n_{0}}(\lambda)$ ($n_{0}=1,2,3,4,7,8$). Here, we consider $\widetilde{K}^{1}_{0,8}(\lambda)$ only, because we can treat other terms similarly. Let $j,k,l=1,\dots,N-1$. By (4.77), $\widetilde{K}^{1}_{0,8}(\lambda)$ can be written as

[TABLE]

Then, by Lemma 4.2, the assumption for $k_{1,1}(\lambda,\xi^{\prime})$, and (5.6) below, we have

[TABLE]

Combining these properties with Lemma 4.4 furnishes for $s=0,1$

[TABLE]

with some positive constant $C=C_{N,q,\varepsilon,\lambda_{0},\mu_{*+},\nu_{*+},\kappa_{*+}}$. Analogously, we have

[TABLE]

for $n_{1}=1,2,3,4,7$.

Next, we estimate $\widetilde{K}^{1}_{0,n_{2}}(\lambda)$ ($n_{2}=5,6$). By Lemma 4.2, the assumption for $k_{1,1}(\lambda,\xi^{\prime})$, and (5.6) below, we have for $m=1,\dots,N-1$

[TABLE]

Accordingly, if we employ the same argument as in proving (4.78), we obtain

[TABLE]

for $s=0,1$ and $n_{2}=3,4$ with $C=C_{N,q,\varepsilon,\lambda_{0},\mu_{*+},\nu_{*+},\kappa_{*+}}$, which, combined with (4.78) and (4.79), complete the proof. ∎

Employing the argument in Shibata [28, Sect. 4], by (4.26) and (4.27), there exist operator families $\widetilde{\mathcal{A}}^{3+}_{Ji}(\lambda)$ $(i=1,2,3,4)$, $\widetilde{\mathcal{A}}^{3-}_{Jl}(\lambda)$ $(l=1,2,3)$, and $\widetilde{\mathcal{P}}^{3}_{-}(\lambda)$ such that

[TABLE]

respectively. Using (4.26), (4.27), and Corollary 4.5, there exists operator family $\widetilde{\mathcal{B}}^{3}_{+}(\lambda)$ such that

[TABLE]

Consequently, if we define operators $\mathcal{A}^{2}_{\pm}(\lambda)$, $\mathcal{B}_{+}^{2}(\lambda)$, and $\mathcal{P}_{-}^{2}(\lambda)$ by

[TABLE]

respectively, from (4.26) we have

[TABLE]

Combining Corollary 4.5 and the argument in Shibata [28, Sect. 4], we have (3.3), so that we have completed the proof of Theorem 3.3.

5. Analysis of multipliers

In this section, we estimate several multipliers. To this end, we start with the following wildly known estimate:

[TABLE]

for any $\lambda\in\Sigma_{\varepsilon}$ and positive numbers $\alpha$ and $\beta$.

First, we estimate $B_{\pm}^{s}$ and $(\mu_{*+}B_{+}+\mu_{*-}B_{-})$. For this purpose, we use the estimates:

[TABLE]

for any $(\lambda,\xi)\in\widetilde{\Sigma}_{\varepsilon,0}$ with some positive constant $c_{1}$ and $c_{2}$, which immediately follows from (5.1). Here and in the sequel, $c_{1}$ and $c_{2}$ denote some positive constants essentially depending on $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$. In particular, by (5.2) we have

[TABLE]

for any $(\lambda,\xi)\in\widetilde{\Sigma}_{\varepsilon,0}$. As shown in Enomoto and Shibata [10, Lemma 4.3], using (5.2), (5.3) and the Bell’s formula:

[TABLE]

with suitable coefficients $\Gamma^{\alpha^{\prime}}_{\alpha_{1},\dots,\alpha_{l}}$, where $f^{(l)}(t)=d^{l}f(t)/dt^{l}$, we see that

[TABLE]

Second, we estimate $(t_{i})^{s},\enskip t_{i}+B_{+}$, and $t_{i}B_{+}+A^{2}\enskip(i=1,2)$. As seen in Saito [25], we have the following lemma.

Lemma 5.1**.**

Let $i=1,2$. Then, the roots $s_{i}$ of (4.14) are given by

[TABLE]

with

[TABLE]

In addition, there exist positive constants $c_{1}$ and $c_{2}$ such that

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,0}$.

Remark 5.2**.**

We have in general the following situations concerning roots with positive real parts for the characteristic equation of (4.13):

(1) When $\eta_{*}<0$, it holds that $B_{+}\neq t_{1}$, $B_{+}\neq t_{2}$, and $t_{1}\neq t_{2}$.

(2) When $\eta_{*}=0$, there are two cases: $B_{+}\neq t_{1}$ and $t_{1}=t_{2}$; $B_{+}=t_{1}=t_{2}$.

(3) When $\eta_{*}>0$, there are three cases: $B_{+}\neq t_{1}$, $B_{+}\neq t_{2}$, and $t_{1}\neq t_{2}$; $B_{+}=t_{1}$ and $t_{1}\neq t_{2}$; $B_{+}=t_{2}$ and $t_{1}\neq t_{2}$.

We assume $\eta_{*}\neq 0$ and $\kappa_{*+}\neq\nu_{*+}\mu_{*+}$. Under these assumptions, we have the three roots with positive real parts different from each other. We consider, however, that our technique in this paper can be applied to the case of equal roots.

From the Bell’s formula and Lemma 5.1, for $i=1,2$, we have

[TABLE]

Then, by (4.45), (4.53), (4.63), (4.64), and Lemma 5.1, we have

[TABLE]

with $s_{1}-\rho_{*+}\mu_{*+}^{-1}\neq 0$. Hence, by Remark 4.3, (5.5), Lemma 5.1, (5.6), and Lemma 6.1 bellow, we have $Q_{Nm}^{+}\in\mathbb{M}_{0,2,\varepsilon,0}$. Analogously, we have $Q_{jm}^{+}\in\mathbb{M}_{0,2,\varepsilon,0}$ ($j=1,\dots,N-1$), which yields $Q_{Jm}^{+}\in\mathbb{M}_{0,2,\varepsilon,0}$ ($J=1,\dots,N$). Furthermore, employing the same argument as $Q_{Jm}^{+}$, we have other assertions in (4.27). Then, we finish the estimate of multipliers.

6. Analysis of Lopatinski determinant

Lemma 6.1**.**

Let $\varepsilon_{*}<\varepsilon<\pi/2$ and $\mathfrak{l}(\lambda,\xi^{\prime})$ be defined in (4.53). Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Then, there exists a positive constant $C$ such that

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,0}$. Here, positive constant $C_{\alpha^{\prime}}$ is depending on $\alpha^{\prime}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$.

In addition, we have

[TABLE]

for any multi-index $\alpha^{\prime}\in\mathbb{N}^{N-1}_{0}$ and $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,0}$, that is, $\mathfrak{l}(\lambda,\xi)\in\mathbb{M}_{-6,2,\varepsilon,0}$.

Proof.

We can prove (6.2) by using (6.1) with the Leibniz rule and the Bell’s formula (5.4) with $f(t)=t^{-1}$ and $g(\xi^{\prime})=\mathfrak{l}(\lambda,\xi^{\prime})$.

In order to prove (6.1), we consider the three cases: (1) $R_{1}|\lambda|^{1/2}\leq A$, (2) $R_{2}A\leq|\lambda|^{1/2}$, (3) $R_{2}^{-1}|\lambda|^{1/2}\leq A\leq|\lambda|^{1/2}$ for large $R_{1}\geq 1$ and $R_{2}\geq 1$.

First, we consider the case: $R_{1}|\lambda|^{1/2}\leq A$ with large $R_{1}\geq 1$. In this case, we set $\delta_{1}=\lambda^{1/2}/A$ and see that

[TABLE]

which imply that

[TABLE]

Here, by Lemma 5.1 we have $s_{i}-\rho_{*+}\mu_{*+}^{-1}\neq 0$. Then, we obtain

[TABLE]

Summing up, there exists a positive constant $C_{1}:=\omega_{1}/2$ such that

[TABLE]

Second, we consider the case $R_{2}A\leq|\lambda|^{1/2}$ for large $R_{2}$. In this case, we set $\delta_{2}=A/\lambda^{1/2}$ and see that

[TABLE]

which imply that

[TABLE]

Then, we obtain

[TABLE]

From Lemma 5.1, we obtain $\omega_{2}\neq 0$. Summing up, there exists a positive constant $C_{2}:=|\omega_{2}|/2$ such that

[TABLE]

Third, we consider the case $R_{2}^{-1}|\lambda|^{1/2}\leq A\leq R_{1}|\lambda|^{1/2}$. Here, we set

[TABLE]

If $(\lambda,\xi^{\prime})$ satisfies the condition: $R_{2}^{-1}|\lambda|^{1/2}\leq A\leq R_{1}|\lambda|^{1/2}$ and $\lambda\in\Sigma_{\varepsilon}$, then $(\widetilde{\lambda},\widetilde{A})\in D_{\varepsilon}(R_{1},R_{2})$. We also define $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})$ by replacing $A,B_{\pm}$ and $t_{i}\ (i=1,2)$ by $\widetilde{A},\widetilde{B}_{\pm}$ and $\widetilde{t_{i}}$, respectively. Then, we have

[TABLE]

We prove that $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})\neq 0$ provided that $(\widetilde{\lambda},\widetilde{A})\in D_{\varepsilon}(R_{1},R_{2})$ by contradiction. Suppose that $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})=0$, namely, $\det L=0$. In this case, in view of (4.51) we assume that there exist $\mathbf{u}_{\pm}(x_{N})=(u_{1\pm}(x_{N}),\dots,u_{N\pm}(x_{N}))\neq 0$, $\rho_{+}(x_{N})\neq 0$, and $\pi_{-}(x_{N})\neq 0$ satisfying (4.2) - (4.12) with $\widehat{d}(0)=0$, $\widehat{H}(0)=0$, $\widehat{h}_{m}(0)=0$, and $\rho_{+}\neq 0$, that is, $\mathbf{u}_{\pm}(x_{N})\neq 0$, $\rho_{+}(x_{N})\neq 0$, and $\pi_{-}(x_{N})\neq 0$ satisfy the following homogeneous equations: for $x_{N}\neq 0$ and $w_{+}=\sum_{j=1}^{N-1}i\xi_{j}u_{j+}(x_{N})+\partial_{N}u_{N+}(x_{N})$ ($x_{N}>0$),

[TABLE]

Set $(f,g)_{\pm}=\pm\int_{0}^{\infty}f(x_{N})\overline{g(x_{N})}dx_{N}$ and $\|f\|_{\pm}=(f,f)_{\pm}^{1/2}$. Multiplying the equations in (6.6) by $\overline{u_{J\pm}}$ and using integration by parts and interface conditions in (6.6), we have

[TABLE]

Here, we use the identity

[TABLE]

Taking the real part of (6.7) to obtain

[TABLE]

which, combined with the inequality:

[TABLE]

furnishes that

[TABLE]

When ${\rm Im}\,\lambda=0$, we have $\lambda>0$ because $\lambda\in\Sigma_{\varepsilon}$. However, this contradict to (6.8). Summing up, we have $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})\neq 0$ for $(\widetilde{\lambda},\widetilde{A})\in D(R_{1},R_{2})$, where

[TABLE]

Then, there exists a positive constant $C_{3}$ such that

[TABLE]

because $D(R_{1},R_{2})$ is compact. Since $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})$ is continuous in $\Sigma_{\varepsilon,\lambda_{0}}\times\mathbb{R}^{N-1}$ for $0<\varepsilon<\pi/2$ and $D_{\varepsilon}(R_{1},R_{2})$ is compact, $\mathfrak{l}(\widetilde{\lambda},\widetilde{\xi^{\prime}})$ is uniformly continuous in $D_{\varepsilon}(R_{1},R_{2})\times\mathbb{R}^{N-1}$ for $0<\varepsilon<\pi/2$. Then, there exist a constant $\varepsilon_{*}\in(0,\pi/2)$ such that

[TABLE]

which, combined with (6.5), furnishes that

[TABLE]

provided that $R_{2}^{-1}|\lambda|^{1/2}\leq A\leq R_{1}|\lambda|^{1/2}$ and $\lambda\in\Sigma_{\varepsilon}$.

Summing up, setting $C=\min(C_{1},C_{2},C_{3})$, by (6.3), (6.4), and (6.9), we have (6.1), which completes the proof of Lemma 6.1. ∎

7. Problem with surface tension and height function

In this final section, we consider the following problem:

[TABLE]

where $\sigma_{\pm}=\rho_{*\pm}\sigma/(\rho_{*-}-\rho_{*+})$, and prove the following theorem.

Theorem 7.1**.**

Let $1<q<\infty$ and $\varepsilon_{*}<\varepsilon<\pi/2$. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Then, there exist a positive constant $\lambda_{0}$ and operator families $\mathcal{A}^{4}_{\pm}(\lambda)$, $\mathcal{B}^{4}_{+}(\lambda)$, $\mathcal{P}^{4}_{-}(\lambda)$, and $\mathcal{H}(\lambda)$ with

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$ and $d\in W^{2}_{q}(\mathbb{R}^{N})$, $\mathbf{u}_{\pm}=\mathcal{A}^{4}_{\pm}(\lambda)d$, $\rho_{+}=\mathcal{B}^{4}_{+}(\lambda)d$, $\pi_{-}=\mathcal{P}^{4}_{-}(\lambda)d$, and $H=\mathcal{H}(\lambda)d$ are solutions of problem (7.1). Furthermore, for $s=0,1$, we have

[TABLE]

with some constant $c_{0}$.

Remark 7.2**.**

Combining Theorem 7.1 with Theorem 3.1, Theorem 3.2, and Theorem 3.3, we have Theorem 1.2 immediately.

As discussed in Sect. 4, applying the partial Fourier transform to (7.1), we have

[TABLE]

with the interface condition:

[TABLE]

and the resolvent equation for $H$:

[TABLE]

Our task is to represent $\widehat{H}$ in terms of $\widehat{d}(0)$, so that we look for solutions $\widehat{u}_{J\pm}$ and $\widehat{\pi}_{-}$ of the form (4.26) with $g_{j}=h_{j}=k=0$. In view of (4.17), (4.18), and (4.19), when $g_{j}=h_{j}=k=0$, we have

[TABLE]

Inserting these formulas into (7.2), we have

[TABLE]

with

[TABLE]

We now prove the following lemma.

Lemma 7.3**.**

Let $\varepsilon_{*}<\varepsilon<\pi/2$ and let $K_{H}$ be the function defined in (7.4). Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Then there exists a positive constant $\lambda_{0}$ depending on $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$ such that

[TABLE]

for any multi-index $\alpha^{\prime}\in\mathbb{N}^{N-1}_{0}$ and $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ with some constant $C_{\alpha^{\prime}}$ depending on $\alpha^{\prime}$, $\lambda_{0}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$.

Proof.

To prove (7.5) with $\alpha^{\prime}=0$ and $s=0$, first we consider the case where $R_{1}|\lambda|^{1/2}\leq A$ with large $R_{1}$. In the following, $\delta_{1}$ is the same small number as in the proof of Lemma. 6.1. In this case, we see that

[TABLE]

Then, we have

[TABLE]

Since $\lambda\in\Sigma_{\varepsilon}$, by (5.1) and (7.6) we have

[TABLE]

If we choose $\delta_{1}$ so small that $O(\delta_{1})\leq\sin(\varepsilon/2)/2$, we have

[TABLE]

provided that $R_{1}|\lambda|^{1/2}\leq A$ with large $R_{1}>0$ and $\lambda\in\Sigma_{\varepsilon}$.

Next we consider the case where $A\leq R_{1}|\lambda|^{1/2}$. Here and in the sequel, $C$ denotes a generic constant depending on $R_{1}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$. From (4.45), (4.53), (4.63), and (4.64), we have

[TABLE]

Then, by (4.66) and (4.69), we easily see

[TABLE]

which, combined with (7.4), furnishes that

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,0}$ provided that $A\leq R_{1}|\lambda|$. Thus we have $|\lambda+K_{H}|\geq|\lambda|^{1/2}(|\lambda|^{1/2}-C)$. Consequently, when we take $\lambda_{0}>0$ so large that $\lambda_{0}^{1/2}/2\geq C$, we have

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ provided that $A\leq R_{1}|\lambda|$. Since $A\leq R_{1}|\lambda|^{1/2}$, we have

[TABLE]

Choosing $R_{1}$ so large that $\lambda_{0}^{1/2}R_{1}^{-1}\leq\omega_{3}$, we obtain

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,0}$ provided that $A\leq R_{1}|\lambda|$. Accordingly, by $\Sigma_{\varepsilon,\lambda_{0}}\subset\Sigma_{\varepsilon}$, combining (7.7) and (7.8), we obtain

[TABLE]

for any $(\lambda,\xi^{\prime})\in\widetilde{\Sigma}_{\varepsilon,\lambda_{0}}$ with

[TABLE]

Finally, we prove (7.5) for any multi-index $\alpha^{\prime}\in\mathbb{N}^{N-1}_{0}$. By Lemma 4.2, (4.27), and Lemma 6.1, we have $K_{H}\in\mathbb{M}_{1,2,\varepsilon,0}$, so that by the Bell’s formula (5.4) with $f(t)=(\lambda+t)^{-1}$ and $g=K_{H}$, we have

[TABLE]

Analogously, we have

[TABLE]

Summing up, we have (7.5). ∎

From (7.4) and Lemma 7.3, we have

[TABLE]

so that we define $\widehat{H}(\xi^{\prime},x_{N})$ by $\widehat{H}(\xi^{\prime},x_{N})=e^{-(1+A^{2})^{1/2}x_{N}}(\lambda+K_{H})^{-1}\widehat{d}(\xi^{\prime},0)$. The following lemma was proved in Shibata [28].

Lemma 7.4**.**

Let $1<q<\infty,\enskip\varepsilon_{*}<\varepsilon<\pi/2$ and let $\lambda_{0}$ be the same constant as in Lemma 7.3. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Given that the operator $\widetilde{\mathcal{H}}(\lambda)$ is defined by

[TABLE]

for any $d\in W^{2}_{q}(\mathbb{R}^{N})$, $\widetilde{\mathcal{H}}(\lambda)\in{\rm Anal}\,(\Sigma_{\varepsilon,\lambda_{0}},\mathcal{L}(W^{2}_{q}(\mathbb{R}^{N}),W^{3}_{q}(\mathbb{R}^{N}_{+}))$, and

[TABLE]

with some constant $\gamma$ depending on $\lambda_{0}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$.

We extend $\widetilde{\mathcal{H}}(\lambda)d$ to $x_{N}<0$, namely, we define $\mathcal{H}(\lambda)$ by

[TABLE]

where $a_{j}$ are constants satisfying the equations: $\sum_{j=1}^{4}a_{j}(-j)^{k}=1$ for $k=0,1,2,3$. By Lemma 7.4, we have the following corollary of Lemma 7.4 immediately.

Corollary 7.5**.**

Let $1<q<\infty$, $\varepsilon_{*}<\varepsilon<\pi/2$ and let $\lambda_{0}$ be the same constant as in Lemma 7.3. Assume that $\rho_{*+}\neq\rho_{*-}$, $\eta_{*}\neq 0$, and $\kappa_{*+}\neq\mu_{*+}\nu_{*+}$. Then, there exists an operator family

[TABLE]

such that for any $\lambda\in\Sigma_{\varepsilon,\lambda_{0}}$ and $d\in W^{2}_{q}(\mathcal{R}^{N})$, $\mathcal{F}^{-1}_{\xi^{\prime}}[\widehat{d}(\xi^{\prime},0)/(\lambda+K_{H})](x^{\prime})=\mathcal{H}(\lambda)d|_{x_{N}=0}$ and

[TABLE]

with some constant $c_{0}$ depending on $\lambda_{0}$, $\varepsilon$, $\mu_{*\pm}$, $\nu_{*+}$, $\kappa_{*+}$, and $\rho_{*\pm}$.

Then, we construct solution operators of (7.1). Using (4.26) and (7.3), we have

[TABLE]

which yields

[TABLE]

Accordingly, if we the define operator $\mathcal{A}^{4}_{\pm}(\lambda)$ by

[TABLE]

by (4.27), Corollary 4.5, Lemma 7.3, and the argument in Shibata [28, Sect. 6], we have Theorem 7.1, which completes the proof of Theorem 1.2.

8. Acknowledgment

The author would like to thank Prof. Yoshihiro Shibata and Dr. Hirokazu Saito for stimulating discussions on the subject of this paper.