Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1131.35060
Struwe, Michael
On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure. (English)
[J] J. Math. Fluid Mech. 9, No. 2, 235-242 (2007). ISSN 1422-6928; ISSN 1422-6952/e

The author proves a regularity result for a weak solution (in the sense of Leray-Schauder) of the Navier-Stokes system posed in the space-time region $ \Omega \times (0,T)$, where $T>0$ and $\Omega $ is a domain of $\Bbb R^{n}$, $ n\geq 3$. Homogeneous Dirichlet boundary conditions are imposed on the lateral boundary $\partial \Omega \times (0,T)$. Here, the author restricts the study to the cases where $\Omega $ is the whole $\Bbb R^{n}$ or in a spatially periodic situation. He extends a previous result by {\it L. C. Berselli} and {\it G. P. Galdi} [Proc. Am. Math. Soc. 130, No. 12, 3585--3595 (2002; Zbl 1075.35031)]. Assuming that $u\in L^{\infty }([0,T];L^{2}(\Bbb R^{n}))\cap L^{2}([0,T];H_{0}^{1}(\Bbb R^{n}))$ is a strong solution of the Navier-Stokes system and that the gradient of the pressure belongs to $L_{x,t}^{r,s}$, with $n/r+2/s\leq 3$, and $n/3<r<+\infty $ and $ 2/3<s<+\infty $, the author proves that $u$ is a smooth solution of the Navier-Stokes system. The proof of the regularity result is obtained using some properties of the spaces $L_{x,t}^{r,s}$, through Sobolev embeddings, and interpolation tools.
[Alain Brillard (Riedisheim)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
35B65 Smoothness of solutions of PDE
76D03 Existence, uniqueness, and regularity theory
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: Navier-Stokes equations; regularity; Sobolev embeddings; interpolation; weak solution

Citations: Zbl 1075.35031

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]