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 $\dotΩ=Ω_+\cupΩ_-$, where $Ω_\pm$ 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 $Ω$ with two boundaries $Γ_\pm$ satisfying $Γ_+\capΓ_-=\emptyset$, we suppose that some hypersurface $Γ$ divides $Ω$ into two sub-domains, that is, there exist domains $Ω_\pm\subsetΩ$ such that $Ω_+\capΩ_-=\emptyset$ and $Ω\setminusΓ=Ω_+\cupΩ_-$, where $Γ\capΓ_+=\emptyset$, $Γ\capΓ_-=\emptyset$, and the boundaries of $Ω_\pm$ consist of two parts $Γ$ and $Γ_\pm$, respectively. The domains $Ω_\pm$ are filled with viscous, incompressible, and immiscible fluids with density $ρ_\pm$ and viscosity $μ_\pm$, respectively. Here $ρ_\pm$ are positive constants, while $μ_\pm=μ_\pm(x)$ are functions of $x\in\mathbf{R}^N$. On the boundaries $Γ$, $Γ_+$, and $Γ_-$, 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.

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