## Advances in Differential Equations

- Adv. Differential Equations
- Volume 2, Number 4 (1997), 535-554.

### Regularity criterion of weak solutions to the Navier-Stokes equations

#### 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.

#### Article information

**Source**

Adv. Differential Equations Volume 2, Number 4 (1997), 535-554.

**Dates**

First available in Project Euclid: 23 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366741147

**Mathematical Reviews number (MathSciNet)**

MR1441855

**Zentralblatt MATH identifier**

1023.35523

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 35D10 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Kozono, Hideo; Sohr, Hermann. Regularity criterion of weak solutions to the Navier-Stokes equations. Adv. Differential Equations 2 (1997), no. 4, 535--554. https://projecteuclid.org/euclid.ade/1366741147