Very weak solutions of the Stokes problem in a convex polygon

Abstract

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in this article, the existence and uniqueness of solutions in $L^2(\Omega)^2$ for the Stokes problem in a domain $\Omega,$ when $\Omega$ is a smooth domain or a convex polygon. We establish also a trace theorem and show that the trace of $u$ can be arbitrary in $L^2(\partial\Omega)^2.$ The results are also extended to the linear evolution Stokes problem.