On a resolvent estimate of the interface problem for the Stokes system in a bounded domain
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 following: For \(1< p< \infty\) and \(0< \varepsilon< \pi/2\) there exist a \(\sigma>0\) such that for every \(\lambda\in \Sigma_\varepsilon\cup \{\lambda\in \mathbb{C}\mid |\lambda|< \varepsilon\}\), \(f_\pm\in L_p(\Omega_\pm)^n\) and \(h_\pm\in W_p^1 (\Omega_\pm)^n\), the above boundary problem admits an unique solution \((u,\pi)\in W_p^1(\Omega)^n\times W_p(\Omega)\) with \(u_\pm\in W_p^2(\Omega_\pm)^m\) satisfying an estimation given in the paper. For the proof of this assertion, another theorem is fundamental and the entire paper is devoted to these two theorem demonstrations. The paper has a marked mathematical character and there are used many mathematical procedures for the demonstrations.
scientific article; zbMATH DE number 1957269