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January/February 2017 On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations
Sri Maryani, Hirokazu Saito
Differential Integral Equations 30(1/2): 1-52 (January/February 2017). DOI: 10.57262/die/1484881218


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 $L_p{\theta}xt{-}L_q$ 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. The essential assumption of this paper is the unique solvability of a weak elliptic transmission problem for $\mathbf{f}\in L_q(\Omega )^N$, that is, it is assumed that the unique existence of solutions ${\theta}\in\mathcal{W}_q^1(\Omega )$ to the variational problem: $ (\rho^{-1}\nabla{\theta},\nabla{\varphi})_{\dot\Omega }=(\mathbf{f},\nabla{\varphi})_{\Omega } $ for any ${\varphi}\in\mathcal{W}_{q'}^1(\Omega )$ with $1 < q < \infty$ and $q'=q/(q-1)$, where $\rho$ is defined by $\rho=\rho_+$ ($x\in\Omega _+$), $\rho=\rho_-$ ($x\in\Omega _-$) and $\mathcal{W}_q^1(\Omega )$ is a suitable Banach space endowed with norm $\|\cdot\|_{\mathcal{W}_q^1(\Omega )}:=\|\nabla\cdot\|_{L_q(\Omega )}$. Our assumption covers e.g. the following domains as $\Omega $: ${\mathbf{R}^N}$, $\mathbf{R}_\pm^N$, perturbed $\mathbf{R}_\pm^N$, layers, perturbed layers, and bounded domains, where $\mathbf{R}_+^N$ and $\mathbf{R}_-^N$ are the open upper and lower half spaces, respectively.


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Sri Maryani. Hirokazu Saito. "On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations." Differential Integral Equations 30 (1/2) 1 - 52, January/February 2017.


Published: January/February 2017
First available in Project Euclid: 20 January 2017

zbMATH: 06738540
MathSciNet: MR3599794
Digital Object Identifier: 10.57262/die/1484881218

Primary: 35Q30 , 76D05

Rights: Copyright © 2017 Khayyam Publishing, Inc.


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Vol.30 • No. 1/2 • January/February 2017
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