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

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 is an element of L(infinity) ((0, T) ; L(2)) boolean AND L(2) ((0, T) ; (H) over dot(1)). (0.1) It is known that if a Leray weak solution u belongs to L(2/1-r) ((0, T) ; L(d/r)) for some 0 <= r <= 1, (0.2) 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 p is an element of L(2/2-r) ((0, T) ; (X) over dot(r) (R(d))(d)), where (X) over dot(r) (R d) is the multiplier space (a definition is given in the text) for 0 <= r <= 1. Since this space (X) over dot(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]. Published by Elsevier Ltd