IUMJ

Title: Local in time regularity properties of the Navier-Stokes equations

Authors: R. Farwig, H. Kozono and H. Sohr

Issue: Volume 56 (2007), Issue 5, 2111-2132

Abstract:

Let $u$ be a weak solution of the Navier-Stokes equations in a smooth domain $\Omega \subseteq \mathbb{R}^3$ and a time interval $[0,T)$, $0 < T \leq \infty$, with initial value $u_0$, and vanishing external force. As is well known, global regularity of $u$ for general $u_0$ is an unsolved problem unless we pose additional assumptions on $u_0$ or on the solution $u$ itself such as Serrin's condition $\|u\|_{L^s(0,T;L^q(\Omega))} < \infty$ where $2/s + 3/q = 1$. In the present paper we prove several new local and global regularity properties by using assumptions beyond Serrin's condition, e.g. as follows: If $\Omega$ is bounded and the norm $\|u\|_{L^1(0,T;L^q(\Omega))}$, with Serrin's number $2/1 + 3/q$ strictly larger than $1$, is sufficiently small, then $u$ is regular in $(0,T)$. Further local regularity conditions for general smooth domains are based on energy quantities such as $\|u\|_{L^{\infty}(T_0,T_1;L^2(\Omega))}$ and $\|\nabla u\|_{L^2(T_0,T_1;L^2(\Omega))}$.