On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations
On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations
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关于两相Stokes求解方程解算子族的$mathcal{R}$有界性
DOI：
10.57262/die/1484881218
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发表时间：
2016
影响因子：
1.4
通讯作者：
Hirokazu Saito
中科院分区：
文献类型：
--
作者：
S. Maryani;Hirokazu Saito
本文的目的是证明$\dot\Omega=\Omega_+\cup\Omega_-$中两相Stokes预解方程的$\mathcal{R}$-有界解算子族的存在性，其中$\Omega_\pm$是$N$维欧氏空间$\mathbf{R}^N$（$N\geq 2$，$N<r<\infty$）的一致$W_r^{2-1/r}$域.更精确地说，给定一个具有两个边界$\Gamma_\pm$的一致$W_r^{2-1/r}$域$\Omega$，满足$\Gamma_+\cap\Gamma_-=\emptyset$，我们假设某个超曲面$\Gamma$将$\Omega$分成两个子域，即，存在域$\Omega_\pm\subset\Omega$，使得$\Omega_+\cap\Omega_-=\emptyset$和$\Omega\setminus\Gamma=\Omega_+\cup\Omega_-$，其中$\Gamma\cap\Gamma_+=\emptyset$，$\Gamma\cap\Gamma_-=\emptyset$，$\Omega_\pm$的边界分别由两部分$\Gamma$和$\Gamma_\pm$组成。域$\Omega_\pm$填充有粘性的、不可压缩的和不混溶的流体，其密度为$\rho_\pm$，粘度为$\mu_\pm$。这里$\rho_\pm$是正常数，而$\mu_\pm=\mu_\pm（x）$是x\in\mathbf{R}^N$的函数。在边界$\Gamma$，$\Gamma_+$和$\Gamma_-$上，我们分别考虑了界面条件，自由边界条件和Dirichlet边界条件。利用$\mathcal{R}$-有界解算子族，我们还证明了与两相Stokes预解方程相关的一类含时问题的最大L_p\text{-}L_q$正则性以及解析半群的生成.这类问题出现在两个粘性的，不可压缩的，不相溶的流体与自由表面的运动的数学研究。
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.
一类非牛顿流体两相自由边界问题的强解
影响因子：
1.4
作者：
M. Hieber;H. Saito
通讯作者：
H. Saito
无限层广义Stokes求解问题的解算子族的R有界性
影响因子：
2.9
作者：
H. Saito
通讯作者：
H. Saito
具有表面张力的两相纳维斯托克斯方程解的定性行为
影响因子：
1.4
作者：
M. Köhne;J. Prüss;M. Wilke
通讯作者：
M. Wilke