Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition
Tom 81 / 2008
Streszczenie
Let $u$ be a weak solution of the Navier-Stokes equations in a smooth bounded domain $\Omega \subseteq \mathbb R^3$ and a time interval $[0,T)$, $0< T\leq \infty$, with initial value $u_0$, external force $f=\mathop{\rm div} F$, and viscosity $\nu>0$. As is well known, global regularity of $u$ for general $u_0$ and $f$ 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 local and global regularity properties by using assumptions beyond Serrin's condition e.g. as follows: If the norm $\| u\|_{L^r(0,T;L^q(\Omega))}$ and a certain norm of $F$ satisfy a $\nu$-dependent smallness condition, where Serrin's number ${2}/{r} + {3}/{q}>1$, or if $u$ satisfies a local leftward $L^s$-$L^q$-condition for every $t\in(0,T)$, then $u$ is regular in $(0,T)$.
