## Abstract and Applied Analysis

### A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces

Xiang'ou Zhu

#### Abstract

We exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to ${L}^{2/(2-\mathrm{r)}}((0,T);ℳ({\stackrel{˙}{H}}^{r}({\mathbb{R}}^{3})\to {\stackrel{˙}{H}}^{-r}({\mathbb{R}}^{3})))$, where $ℳ({\stackrel{˙}{H}}^{r}({ℝ}^{3})\to {\stackrel{˙}{H}}^{-r}({ℝ}^{3}))$ is the multipliers between Sobolev spaces whose definition is given later for $0, then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 682436, 7 pages.

Dates
First available in Project Euclid: 4 April 2013

https://projecteuclid.org/euclid.aaa/1365099940

Digital Object Identifier
doi:10.1155/2012/682436

Mathematical Reviews number (MathSciNet)
MR2935143

Zentralblatt MATH identifier
1242.35188

#### Citation

Zhu, Xiang'ou. A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 682436, 7 pages. doi:10.1155/2012/682436. https://projecteuclid.org/euclid.aaa/1365099940

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