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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

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