Time-dependent Stokes and Navier-Stokes problems with boundary conditions involving pressure, existence and regularity

摘要

This paper is concerned with the time-dependent Stokes and Navier-Stokes problems with nonstandard boundary conditions: the pressure is given on some part of the boundary. The stationary case was first studied by Begue, Conca, Murat and Pironneau and, next, their study were completed by Bernard, mainly about regularity. In this paper, the Stokes problem is studied by a method analogous to that of Temam for the standard problem, combined with regularity results of Bernard for the nonstandard stationary case. We obtain existence, uniqueness and regularity H-2. In addition, in two dimensions, a regularity W-2,W-r, r greater than or equal to 2, is proved. Next, for the nonstandard Navier-Stokes problem, we present some existence, uniqueness and regularity H-2 results. The proof of existence is based on a fixed point method. [References: 12]