Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

Abstract

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by $P\mathbf{u}$, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by $(I-P)\mathbf{u}$, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for $P\mathbf{u}$ and $(I-P)\mathbf{u}$. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.