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

Abstract

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. u ∈ L ∞ ( ( 0 , T ) ; L 2 ) ∩ L 2 ( ( 0 , T ) ; H . 1 ) . It is known that if a Leray weak solution u belongs to (0.1) L 2 1 − r ( ( 0 , T ) ; L d r ) for some 0 ≤ r ≤ 1 , 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 (0.2) p ∈ L 2 2 − r ( ( 0 , T ) ; X . r ( R d ) d ) , 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].