On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations

The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dot\Omega=\Omega_+\cup\Omega_-$, where $\Omega_\pm$ are uniform $W_r^{2-1/r}$ domains of $N$-dimensional Euclidean space $\mathbf{R}^N$ ($N\geq 2$, $N<r<\infty$). More precisely, given a uniform $W_r^{2-1/r}$ domain $\Omega$ with two boundaries $\Gamma_\pm$ satisfying $\Gamma_+\cap\Gamma_-=\emptyset$, we suppose that some hypersurface $\Gamma$ divides $\Omega$ into two sub-domains, that is, there exist domains $\Omega_\pm\subset\Omega$ such that $\Omega_+\cap\Omega_-=\emptyset$ and $\Omega\setminus\Gamma=\Omega_+\cup\Omega_-$, where $\Gamma\cap\Gamma_+=\emptyset$, $\Gamma\cap\Gamma_-=\emptyset$, and the boundaries of $\Omega_\pm$ consist of two parts $\Gamma$ and $\Gamma_\pm$, respectively. The domains $\Omega_\pm$ are filled with viscous, incompressible, and immiscible fluids with density $\rho_\pm$ and viscosity $\mu_\pm$, respectively. Here $\rho_\pm$ are positive constants, while $\mu_\pm=\mu_\pm(x)$ are functions of $x\in\mathbf{R}^N$. On the boundaries $\Gamma$, $\Gamma_+$, and $\Gamma_-$, we consider an interface condition, a free boundary condition, and the Dirichlet boundary condition, respectively. We also show, by using the $\mathcal{R}$-bounded solution operator families, some maximal $L_p\text{-}L_q$ regularity as well as generation of analytic semigroup for a time-dependent problem associated with the two-phase Stokes resolvent equations. This kind of problems arises in the mathematical study of the motion of two viscous, incompressible, and immiscible fluids with free surfaces.