Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition

Tom 81 / 2008

Reinhard Farwig, Hideo Kozono, Hermann Sohr Banach Center Publications 81 (2008), 175-184 MSC: Primary 35Q30; Secondary 76D05, 35B65. DOI: 10.4064/bc81-0-11


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 $(0,T)$.


  • Reinhard FarwigFachbereich Mathematik
    Technische Universität Darmstadt
    64289 Darmstadt, Germany
  • Hideo KozonoMathematical Institute
    Tôhoku University
    Sendai, 980-8578 Japan
  • Hermann SohrFakultät für Elektrotechnik
    Informatik und Mathematik
    Universität Paderborn
    33098 Paderborn, Germany

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