Very weak solutions of the stationary Stokes equations on exterior domains

Abstract

We study the nonhomogeneous Dirichlet problem for the stationary Stokes equations on exterior smooth domains $\Omega$ in $\mathbb R^n , n \ge 3$. Our main result is the existence and uniqueness of very weak solutions in the Lorentz space $L^{p,q}(\Omega )^n$, where $(p,q)$ satisfies either $(p,q)=(n/(n-2),\infty)$ or $n/(n-2) < p < \infty , 1 \le q \le \infty$. This is deduced by a duality argument from our new solvability results on strong solutions in homogeneous Sobolev-Lorentz spaces. Homogeneous Sobolev-Lorentz spaces are also studied in quite details: particularly, we establish basic interpolation and density results, which are not only essential to our results for the Stokes equations but also themselves of independent interest.