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 RdRd. Here we call uu a Leray weak solution if uu is a weak solution of finite energy, i.e. u∈L∞((0,T);L2)∩L2((0,T);H.1). It is known that if a Leray weak solution uu belongs to equation(0.1)L21−r((0,T);Ldr)for some 0≤r≤1, then uu 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 uu in terms of pressure under the condition equation(0.2)p∈L22−r((0,T);X.r(Rd)d), where X.r(Rd) is the multiplier space (a definition is given in the text) for 0≤r≤10≤r≤1. Since this space X.r is wider than Ldr, 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 R3R3, Proc. Amer. Math. Soc. 134 (2006) 149–156].