On a~boundary value problem for the time-dependent Stokes system with general boundary conditions

I. Sh. Mogilevskii

Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 37-66

Résumé

Solvability in Sobolev spaces $W_q^{2l,l}$ is proved and properties of solutions are investigated for the following initial boundary value problem: \begin{gather*} \frac{\partial\bar{\mathbf u}}{\partial t}=\nabla^2\bar{\mathbf v}+\nabla p=\bar{\mathbf f},\qquad\nabla\cdot\bar{\mathbf v}=\rho\quad\text{in}\quad Q_T=\Omega\times(0,T),\\ \bar{\mathbf v}|_{t=0}=\bar v^0,\qquad B\biggl(x,t,\frac\partial{\partial x},\frac\partial{\partial t}\biggr)(\bar{\mathbf v},p)\Bigr|_{x\in\partial\Omega}=\bar{\mathbf\Phi}, \end{gather*} where $\Omega$ is a bounded domain in $\mathbf R^3$ with smooth boundary, and $B$ is a matrix differential operator. It is proved that under particular conditions imposed on the data of the problem and boundary operator $B$ there exists a solution $\bar{\mathbf v}\in W_q^{2l,l}(Q_T)$, $\nabla\rho\in W_q^{2l-2,l-1}(Q_T)$. The question of necessity of these conditions is investigated. Bibliography: 18 titles.