A new regularity criterion for steady Navier-Stokes equations

Abstract

We show that every weak solution $\mathbf{u}$ of the steady Navier--Stokes equations in a bounded domain $\Omega \subseteq \mathbb{R}^N$, $N\ge 5$, satisfying additionally $\mathbf{u}\in L^q(\Omega )$, where $ q\ge 4$ and $q > N/2$ (for the Dirichlet problem) or $ q\ge 4$ and $q > N/4$ (for the space periodic problem), is regular.