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A note on the regularity criterion in terms of pressure for the Navier–Stokes equations

✍ Scribed by Samia Benbernou

Publisher
Elsevier Science
Year
2009
Tongue
English
Weight
372 KB
Volume
22
Category
Article
ISSN
0893-9659

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✦ Synopsis

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R d . Here we call u a Leray weak solution if u is a weak solution of finite energy, i.e.

It is known that if a Leray weak solution u belongs to

then u is regular (see [J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal. 9 (1962) 187-195]). We succeed in proving the regularity of the Leray weak solution u in terms of pressure under the condition

where .

X r R d is the multiplier space (a definition is given in the text) for 0 ≤ r ≤ 1.

Since this space

.

X r is wider than L d r , the above regularity criterion (0.2) is an improvement on Zhou's result [Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in R 3 , Proc. Amer. Math. Soc. 134 (2006) 149-156].

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