${\mathcal{R}}$-bounded operator families arising from a compressible fluid model of Korteweg type with surface tension in the half-space

Let ${\mathbb{R}}^{N}_{+}$ and ${\mathbb{R}}^{N}_{0}$ be the upper half-space and its boundary for $N\geq 2$, respectively; namely,

where $x^{\prime}=(x_{1},\ldots,x_{N-1})$. In this paper, we consider the following resolvent problem arising from a compressible fluid model of Korteweg type in the half-space with taking the surface tension in account.

where $\lambda$ is the resolvent parameter varying in the sectorial region

for $0<\epsilon<\pi/2$ and $\lambda_{0}\geq 0$; $\rho=\rho(x)$, $x=(x_{1},\ldots,x_{N})$ and ${\mathbf{u}}={\mathbf{u}}(x)=(u_{1}(x),\ldots,u_{N}(x))^{\mathsf{T}}$^***${\mathbf{A}}^{\mathsf{T}}$denotes the transpose of ${\mathbf{A}}$. are unknown density field and velocity field, respectively; $h=h(x)$ is an unknown function on ${\mathbb{R}}^{N}_{0}$ obtained by linearization of the kinematic equation, $\Delta^{\prime}h=\sum^{N-1}_{j=1}\partial_{j}^{2}h$; ${\mathbf{n}}=(0,\ldots,0,-1)^{\mathsf{T}}$ is the unit outer normal to ${\mathbb{R}}^{N}_{0}$; $\rho_{*},\kappa,\sigma>0$ and $\gamma_{*}\in{\mathbb{R}}$ are constants; the right members $d=d(x)$, ${\mathbf{f}}={\mathbf{f}}(x)=(f_{1}(x),\ldots,f_{N}(x))^{\mathsf{T}}$, ${\mathbf{g}}={\mathbf{g}}(x)=(g_{1}(x),\ldots,g_{N}(x))^{\mathsf{T}}$, $k=k(x)$, and $\zeta=\zeta(x)$ are given functions. The viscous stress tensor ${\mathbf{S}}({\mathbf{u}})$ is given by

where $\mu>0$ and $\nu>0$ are the viscosity coefficients, ${\mathbf{D}}({\mathbf{u}})$ denotes the deformation tensor whose $(j,k)$ components are $D_{jk}({\mathbf{u}})=\partial_{j}u_{k}+\partial_{k}u_{j}$ with $\partial_{j}=\partial/\partial x_{j}$. For any vector of functions ${\mathbf{v}}=(v_{1},\ldots,v_{N})^{\mathsf{T}}$, we set ${\rm div}\,{\mathbf{v}}=\sum_{j=1}^{N}\partial_{j}v_{j}$ with $\partial_{j}=\partial/\partial x_{j}$, and also for any $N\times N$ matrix field ${\mathbf{L}}$ with $(j,k)^{\rm th}$ components $L_{jk}$, the quantity ${\rm Div}\,{\mathbf{L}}$ is an $N$-vector with $j^{\rm th}$ component $\sum_{k=1}^{N}\partial_{k}L_{jk}$; ${\mathbf{I}}=(\delta_{ij})_{1\leq i,j\leq N}$ is the $N\times N$ identity matrix.

A diffuse interface model for liquid-vapor two-phase flows was introduced by Korteweg [15] based on Van der Waals’s idea [31] and derived rigorously by Dunn and Serrin [5]. There are many results for the following whole space problem:

where $\partial_{t}=\partial/\partial t$, $t$ is the time variable. Here ${\mathbf{K}}(\rho)$ is the Korteweg stress tensor given by

where $\kappa$ is the capillary coefficient; $\nabla\rho\otimes\nabla\rho$ denotes an $N\times N$ matrix with $(j,k)^{\rm th}$ component $(\partial_{j}\rho)(\partial_{k}\rho)$ for $\nabla\rho=(\partial_{1}\rho,\dots,\partial_{N}\rho)^{\mathsf{T}}$; $P(\rho)$ is the pressure field satisfying a $C^{\infty}$ function defined on $\rho\geq 0$; $\rho_{0}$ and ${\mathbf{u}}_{0}$ are given initial data.

We refer to mathematical results on system (2). Bresch, Desjardins, and Lin [1] proved the existence of global weak solution, and then Haspot improved their result in [7]. Hattori and Li [8, 9] first showed the local and global unique existence of the strong solution in Sobolev space. Hou, Peng, and Zhu [10] improved the results [8, 9] when the total energy is small. Danchin and Desjardins [3] proved the local and global existence and uniqueness of the solution in critical Besov space. Chikami and Kobayashi [2] improved the result [3]. In particular, in the case $P^{\prime}(\rho_{*})=0$ for some constant $\rho_{*}$, they proved the global estimates under an additional low frequency assumption to control a pressure term. For the whole space problem in the case $P^{\prime}(\rho_{*})=0$, we also refer to Kobayashi and Tsuda [13]; Kobayashi and the second author [14]. Large time behavior of solutions was established by Wang and Tan [30], Tan and Wang [27], Tan, Wang, and Xu [28], and Tan and Zhang [29] in $L_{2}$-setting; the second author and Shibata [20] in $L_{p}$-$L_{q}$ setting. For the optimality of decay estimates in the $L^{p}$ critical framework, we refer to Kawashima, Shibata, and Xu [11].

Regarding boundary value problems, the system (2) was studied in a domain $\Omega$ with the boundary condition:

Kotschote [16] proved the existence and uniqueness of strong solutions in both bounded and exterior domains locally in time in the $L_{p}$-setting. He also considered a non-isothermal case for Newtonian and Non-Newtonian fluids in [17, 18]. Moreover, he proved the global existence and exponential decay estimates of strong solutions in a bounded domain for small initial data in [19]. Saito [21] proved the existence of ${\mathcal{R}}$-bounded solution operator families for the large resolvent parameter, and then he obtained the maximal $L_{p}$-$L_{q}$ regularity for the linearized problem in uniform $C^{3}$ domains and a generation of continuous analytic semigroup associated with the linearized problem. Concerning the resolvent estimates with small resolvent parameter, Kobayashi, the second author, and Saito [12] proved the resolvent estimate for $\lambda\in\overline{{\mathbb{C}}_{+}}=\{z\in{\mathbb{C}}\mid\operatorname{Re}z\geq 0\}$ in a bounded domain under the condition that the pressure $P(\rho)$ satisfies not only $P^{\prime}(\rho_{*})\geq 0$ but also $P^{\prime}(\rho_{*})<0$, and then they obtained a global solvability. Moreover, they proved the resolvent estimate for $\lambda\in\overline{{\mathbb{C}}_{+}}$ with $|\lambda|\geq\delta$ for any $\delta>0$ in an exterior domain if $P^{\prime}(\rho_{*})\geq 0$.

On the other hand, we are interested in a free boundary value problem with the surface tension; namely, the domain $\Omega$ and the boundary condition (3) are replaced by a time-dependent domain $\Omega_{t}$ and

where ${\mathbf{n}}_{t}$ is the unit outer normal; $\sigma$ is the coefficient of the surface tension; $H(\Gamma_{t})$ is the $N-1$-fold mean curvature on $\Gamma_{t}$; $\Gamma_{0}$ is the boundary of the given initial domain $\Omega_{0}$. The third equation of (4) is called the kinematic equation, where $V_{\Gamma_{t}}$ is the velocity of the evolution of free surface $\Gamma_{t}$ in the direction of ${\mathbf{n}}_{t}$.

Since $\Omega_{t}$ is unknown, we transform $\Omega_{t}$ to the fixed domain $\Omega_{0}$ by the so-called Hanzawa transform [6], and then the system of the linearized equations in $\Omega$ is given by the following forms:

where $\Delta_{\Gamma_{0}}$ is the Laplace-Beltrami operator on $\Gamma_{0}$; the right member $(d,{\mathbf{f}},{\mathbf{g}},k,\zeta)$, initial data $(\rho_{0},{\mathbf{u}}_{0})$, and ${\mathbf{B}}$ are given functions. Since ${\mathbf{B}}$ can be treated by the perturbation method, we consider the linearized problem (5) excluding ${\mathbf{B}}$, then we have the resolvent problem (1) by the Laplace transform if $\Omega_{0}={\mathbb{R}}^{N}_{+}$ and $\Gamma_{0}={\mathbb{R}}^{N}_{0}$.

Concerning a free boundary value problem, Saito [22] considered the resolvent problem arising from a free boundary value problem without surface tension in ${\mathbb{R}}^{N}_{+}$; namely, $\sigma=0$ in (4). He constructed ${\mathcal{R}}$-bounded operator families satisfying the resolvent problems in ${\mathbb{R}}^{N}$ and ${\mathbb{R}}^{N}_{+}$.

In this paper, we discuss the existence of the ${\mathcal{R}}$-bounded operator families for the resolvent problem (1). Once we obtain ${\mathcal{R}}$-boundedness for the solution operator families, we can consider the maximal $L_{p}$-$L_{q}$ regularity for the linearized problem by the Weis’s operator valued Fourier multiplier theorem [32], which is the key estimate when we consider the local solvability for the nonlinear problem in the maximal $L_{p}$-$L_{q}$ regularity class. Here we introduce the definition of ${\mathcal{R}}$-boundedness of operator families.

Concerning ${\mathcal{R}}$-boundedness, we introduce the following lemma proved by [4, Proposition 3.4].

We summarize several symbols and functional spaces used throughout the paper. Let ${\mathbb{N}}$, ${\mathbb{R}}$ and ${\mathbb{C}}$ denote the sets of all natural numbers, real numbers, and complex numbers, respectively. We use boldface letters, e.g. ${\mathbf{u}}$ to denote vector-valued functions.

For scalar function $f$ and $N$-vector functions ${\mathbf{g}}$, we set

where $\partial_{i}=\partial/\partial x_{i}$.

Let ${\mathbb{N}}_{0}={\mathbb{N}}\cup\{0\}$. For multi-index $\alpha^{\prime}=(\alpha_{1},\ldots,\alpha_{N-1})\in{\mathbb{N}}_{0}^{N-1}$ and scalar function $f=f(\xi_{1},\ldots,\xi_{N-1})$,

For complex valued functions $f=f(x)$ and $g=g(x)$; $N$-vector functions ${\mathbf{f}}=(f_{1}(x),\ldots,f_{N}(x))$ and ${\mathbf{g}}=(g_{1}(x),\ldots,g_{N}(x))$, the inner products $(f,g)_{{\mathbb{R}}^{N}_{+}}$, $(f,g)_{{\mathbb{R}}^{N}_{0}}$, $({\mathbf{f}},{\mathbf{g}})_{{\mathbb{R}}^{N}_{+}}$, and $({\mathbf{f}},{\mathbf{g}})_{{\mathbb{R}}^{N}_{0}}$ are defined by

where $\overline{g(x)}$ is the complex conjugate of $g(x)$.

For Banach spaces $X$ and $Y$, ${\mathcal{L}}(X,Y)$ denotes the set of all bounded linear operators from $X$ into $Y$, ${\mathcal{L}}(X)$ is the abbreviation of ${\mathcal{L}}(X,X)$, and $\rm{Hol}\,(U,{\mathcal{L}}(X,Y))$ denotes the set of all ${\mathcal{L}}(X,Y)$ valued holomorphic functions defined on a domain $U$ in ${\mathbb{C}}$.

For any $1<q<\infty$, $m\in{\mathbb{N}}$, $L_{q}({\mathbb{R}}^{N}_{+})$ and $H_{q}^{m}({\mathbb{R}}^{N}_{+})$ denote the usual Lebesgue space and Sobolev space; while $\|\cdot\|_{L_{q}({\mathbb{R}}^{N}_{+})}$, $\|\cdot\|_{H_{q}^{m}({\mathbb{R}}^{N}_{+})}$ denote their norms, respectively; $W^{m+s}_{q}({\mathbb{R}}^{N}_{0})=(H^{m}_{q}({\mathbb{R}}^{N}_{0}),H^{m+1}_{q}({\mathbb{R}}^{N}_{0}))_{s,q}$ for $m\in{\mathbb{N}}_{0}$ and $0<s<1$, where $(\cdot,\cdot)_{s,q}$ denotes the real interpolation functor; $C^{\infty}((a,b))$ denotes the set of all $C^{\infty}$ functions defined on $(a,b)$. The $d$-product space of $X$ is defined by $X^{d}=\{f=(f,\ldots,f_{d})\mid f_{i}\in X\,(i=1,\ldots,d)\}$, while its norm is denoted by $\|\cdot\|_{X}$ instead of $\|\cdot\|_{X^{d}}$ for the sake of simplicity.

For the Banach space X, we also denote the usual Lebesgue space and Sobolev space of $X$-valued functions defined on time interval $I$ by $L_{p}(I,X)$ and $H^{m}_{p}(I,X)$ with $m\in{\mathbb{N}}$; while $\|\cdot\|_{L_{p}(I,X)}$, $\|\cdot\|_{H_{p}^{m}(I,X)}$ denote their norms, respectively.

Let $\gamma\geq 1$. Set

Let ${\mathcal{L}}$ and ${\mathcal{L}}^{-1}$ denote the Laplace transform and the Laplace inverse transform, respectively, which are defined by

with $\lambda=\gamma+i\tau\in{\mathbb{C}}$. Given $s\geq 0$ and $X$-valued function $f(t)$, we set

The Bessel potential space of $X$-valued functions of order $s$ is defined by the following:

The letter $C$ denotes generic constants and the constant $C_{a,b,\ldots}$ depends on $a,b,\ldots$. The values of constants $C$ and $C_{a,b,\ldots}$ may change from line to line.

This paper is organized as follows. In the next section, we state the main theorem concerning the ${\mathcal{R}}$-bounded operator families arising from the free boundary value problem for the compressible fluid model of Korteweg type with surface tension. Section 3 and Section 4 prove the main theorem. As preliminaries, we study the reduced resolvent problem by using the result for the Navier-Stokes-Korteweg system without surface tension. Section 5 proves the existence of the ${\mathcal{R}}$-bounded operator families for the system containing lower-order derivatives. As an application of the main theorem, Section 6 proves the maximal $L_{p}$-$L_{q}$ regularity for the linearized equations.