On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions
The modified Stokes equations \(\lambda u -\)\(\Delta u +\)\(\nabla p =f\), \(\text{div}\, u=0\), \(x\in \Omega\)\(\subset \mathbb{R}^n\), are considered. Two boundary-value problems are studied. The first describes leak jointly with friction and reads \(| \sigma_n | \leq g\), \(\sigma_n u_n +\)\(g| u_n| =0\), \(x\in\Gamma\subset\partial\Omega\), where \(\sigma_n\) and \(u_n\) are the normal components of the stress vector and the velocity vector. It is assumed that \(u=0\) on \(\partial\Omega \backslash\Gamma\) if \(\Gamma\neq\partial\Omega\). The second boundary-value problem is of the slip-friction type and reads \(| \sigma_{\tau} | \leq g\), \(\sigma_{\tau} u_{\tau} +\)\(g| u_{\tau}| =0\), \(x\in\Gamma\subset\partial\Omega\), where \(\sigma_{\tau}\) and \(u_{\tau} \) denote tangential components of the stress vector and the velocity vector. If \(\Gamma\neq\partial\Omega\) when \(u=0\) on \(\partial\Omega \backslash\Gamma\). Given \(\lambda(\geq 0)\), \(f(x,t)\) and \(g(x,t)\), unique existence theorems are proved in the class \(u\in\) \(H^2\), \(p\in\) \(H^1\).
Stokes equations, leak-friction and slip-friction boundary conditions, existence of strong solutions
scientific article; zbMATH DE number 2114251
scientific article