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Zbl 1214.35043
Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations.
(English)
[J] Q. Appl. Math. 69, No. 1, 147-155 (2011). ISSN 0033-569X; ISSN 1552-4485/e

Summary: We establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in $\Bbb R^d$. It is known that if a Leray weak solution $u$ belongs to $$L^{\frac{2}{1-r}} \Big((0,T);L^{\frac dr}\Big) \quad\text{for some }\ 0\leq r\leq 1,$$ then $u$ is regular. It is proved that if the pressure $p$ associated to a Leray weak solution $u$ belongs to $$L^{\frac{2}{1-r}} \Big((0,T);\dot{\cal M}_{2,\frac dr}\big(\Bbb R^d\big)^d\Big), \tag *$$ where $\dot{\cal M}_{2,\frac dr}(\Bbb R^d)^d)$ is the critical Morrey-Campanato space (a definition is given in the text) for $0<r<1$, then the weak solution is actually regular. Since this space $\dot{\cal M}_{2,\frac dr}$ is wider than $L^{\frac dr}$ and $\dot X_r$, the above regularity criterion $(*)$ is an improvement of Zhou's result.
MSC 2000:
*35Q30 Stokes and Navier-Stokes equations
76D05 Navier-Stokes equations (fluid dynamics)
76D03 Existence, uniqueness, and regularity theory
35B65 Smoothness of solutions of PDE

Keywords: Navier-Stokes equation; Serrin-type regularity criterion; Leray weak solutions

Cited in: Zbl 1231.35146

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