1997 Regularity criterion of weak solutions to the Navier-Stokes equations
Hideo Kozono, Hermann Sohr
Adv. Differential Equations 2(4): 535-554 (1997). DOI: 10.57262/ade/1366741147

Abstract

In a domain $\Omega \subset \mathbb{R}^n $, consider a weak solution $u$ of the Navier-Stokes equations in the class $u \in L^{\infty}(0, T; L^n(\Omega))$. If $\limsup_{t\to t_*-0}\|u(t)\|_n^n -\|u(t_*)\|_n^n$ is small at each point of $t_*\in(0, T)$, then $u$ is regular on $\bar{\Omega}\times (0, T)$. As an application, we give a precise characterization of the singular time; i.e., we show that if a solution $u$ of the Navier-Stokes equations is initially smooth and loses its regularity at some later time $T_* < T$, then either $\limsup_{t\to T_*-0}\|u(t)\|_{L^n(\Omega)} =+ \infty$, or $u(t)$ oscillates in $L^n(\Omega)$ around the weak limit w-$\lim_{t\to T_*-0}u(t)$ with sufficiently large amplitude. Furthermore, we prove that every weak solution $u$ of bounded variation on $(0, T)$ with values in $L^n(\Omega)$ becomes regular.

Citation

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Hideo Kozono. Hermann Sohr. "Regularity criterion of weak solutions to the Navier-Stokes equations." Adv. Differential Equations 2 (4) 535 - 554, 1997. https://doi.org/10.57262/ade/1366741147

Information

Published: 1997
First available in Project Euclid: 23 April 2013

zbMATH: 1023.35523
MathSciNet: MR1441855
Digital Object Identifier: 10.57262/ade/1366741147

Subjects:
Primary: 35Q30
Secondary: 35D10 , 76D05

Rights: Copyright © 1997 Khayyam Publishing, Inc.

Vol.2 • No. 4 • 1997
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