article

Title: Regularity for the Navier-Stokes equations with a solution in a Morrey space

Authors: Igor Kukavica

Issue: Volume 57 (2008), Issue 6, 2843-2860

Abstract:

$We consider conditional regularity of local weak solutions of the Navier-Stokes system. We prove that if a weak solution $(u,p) \in L_{\text{\upshape loc}}^{q} \times L_{\text{\upshape loc}}^{q/2}$ satisfies \begin{equation*} \sup_{(x,t) \in D}~\sup_{r \in (0,r_0)} \frac{1}{r^{(n+2)/q-1}} \left( \int_{t - r^2}^{t + r^2} \int_{|y - x|<r} |u(y,s)|^{q} \mathrm{d}y \mathrm{d}s \right)^{1/q} < \epsilon_{*} \end{equation*} where $D \subseteq \mathbb{R}^{n} \times \mathbb{R}$ is a domain, $q > 2$, and $\epsilon_{*}$ is a sufficiently small constant, then $u$ is regular in $D$.$