Very weak solutions of the Stokes problem in a convex polygon

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 $L^2(\mathrm{\Omega }{)}^2$ for the Stokes problem in a domain $\mathrm{\Omega },$ when $\mathrm{\Omega }$ 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 $L^2(\mathrm{\partial }\mathrm{\Omega }{)}^2\mathrm{.}$ The results are also extended to the linear evolution Stokes problem.