English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The aim of this paper is to show the existence of $\mathcal{R}$-boundedsolution 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 twosub-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 theboundaries 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, afree boundary condition, and the Dirichlet boundary condition, respectively. Wealso show, by using the $\mathcal{R}$-bounded solution operator families, somemaximal $L_p\text{-}L_q$ regularity as well as generation of analytic semigroupfor a time-dependent problem associated with the two-phase Stokes resolventequations. This kind of problems arises in the mathematical study of the motionof two viscous, incompressible, and immiscible fluids with free surfaces.

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