On a resolvent estimate of the interface problem for the Stokes system in a bounded domain
type: Item
instance of: scholarly article
title: On a resolvent estimate of the interface problem for the Stokes system in a bounded domain
zbMATH Open document ID: 1030.35134
DOI: 10.1016/S0022-0396(03)00023-8
author: Senjo Shimizu
author: Yoshihiro Shibata
published in: Journal of Differential Equations
publication date: 2003-07-30T00:00:00Z
review text: The authors' purpose is to give a resolvent estimate of the Stokes system of equations defined in a bounded domain, in the presence of an interface between two different viscous fluids in motion. The movement is defined in the domain \(\Omega\subset \mathbb{R}^n\) \((n\geq 2)\) with the boundary \(\partial\Omega\). \(\Omega\) contains a subdomain \(\Omega_+\) with the boundary \(\Gamma\), strictly contained in \(\Omega\); \(\Omega_+\subset \Omega\) and the distance between \(\Gamma\) and \(\partial\Omega\) is positive. The domain \(\Omega_+\) is occupied by a viscous fluid and \(\Omega_-= \Omega_- \Omega_+\), by another one; \(\Gamma\) is the interface. They formulate the Stokes resolvent boundary problem with interface in a bounded domain as follows: \(\lambda u_\pm- \operatorname {Div} T_\pm (u_\pm,\pi_\pm)= f_\pm\), \(\nabla u_\pm= 0\) in \(\Omega_\pm\), \(\nu T_+(u_+,\pi_+)- \nu T_-(u_-,\pi_-)= h_+-h_-\), \(u_+=u_-\) on \(\Gamma\), \(u_-=0\) on \(\partial\Omega\). The used notations are: \(u_\pm= (u_{\pm 1},\dots, u_{\pm n})\) -- unknown velocity vectors, \(\pi_\pm\) -- unknown pressures in \(\Omega_\pm\), \(f_\pm= (f_{\pm 1},\dots, f_{\pm n})\) -- given external force vectors, \(h_\pm= (h_{\pm 1},\dots, h_{\pm n})\) -- given interface force vectors, \(T_\pm(u_\pm,\pi_\pm)\) -- the stress vectors in \(\Omega_\pm\), \(T_{\pm jk} (u_\pm,\pi_\pm)= 2\mu_\pm D_{jk}(u_\pm)- \delta_{jk} \pi_\pm\), \(\mu_\pm\) -- the viscous coefficients, \(\nu\) -- the unit outer normal to \(\Gamma\), \(\lambda\) -- the resolvent parameter contained in the domain \(\Sigma_\varepsilon= \{\lambda\in \mathbb{C}\mid \lambda\neq 0\), \(|\arg \lambda|\leq \pi-\varepsilon\}\), \(0< \varepsilon< \pi/2\). The authors search estimates of velocities and pressure in \(L_p\) and they consider the paper as a first step in the problem of viscous motions with free boundary. Other authors, quoted in references, considered these motions in some restrictive cases. The main result of the paper is the foll…
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D07
zbMATH DE Number: 1957269
zbMATH Keywords: Stokes system
zbMATH Keywords: resolvent problem \(L_p\) estimate
zbMATH Keywords: interface condition
zbMATH Keywords: bounded domain
zbMATH Keywords: Stokes flows
zbMATH Keywords: free boundary problems
zbMATH Keywords: interface
zbMATH Keywords: resolvent estimates problems
zbMATH Keywords: weak solutions
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
cites work: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
cites work: Q4289774
cites work: Q3362264
cites work: Q4870253
cites work: Q4097611
cites work: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains
cites work: Q4292463
cites work: On Global Weak Solutions of the Nonstationary Two-Phase Stokes Flow
cites work: On a resolvent estimate for the Stokes system with Neumann boundary condition.
cites work: Q4839311
cites work: On the free boundary value problem for compressible viscous fluid motion
cites work: Two-phase free boundary problem for compressible viscous fluid motion
cites work: Q5531981
description: scientific article; zbMATH DE number 1957269