Remark on a 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 \mathbb {R}^{d} . It is known that if a Leray weak solution u u belongs to L 2 1 − r ( ( 0 , T ) ; L d r )  \ \ for some \ \  0 ≤ r ≤ 1 , \begin{equation} L^{\frac {2}{1-r}}\left ( \left ( 0,T\right ) ;L^{\frac {d}{r}}\right ) \text { \ \ for some \ \ }0\leq r\leq 1, \end{equation} then u u is regular. It is proved that if the pressure p p associated to a Leray weak solution u u belongs to L 2 2 − r ( ( 0 , T ) ; M . 2 , d r ( R d ) d ) , \begin{equation} L^{\frac {2}{2-r}}\left ( \left ( 0,T\right ) ;\overset {.}{\mathcal {M}}_{2,\frac { d}{r}}\left ( \mathbb {R}^{d}\right ) ^{d}\right ) , \end{equation} where M . 2 , d r ( R d ) \overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 > r > 1 0>r>1 , then the weak solution is actually regular. Since this space M . 2 , d r \overset {.}{\mathcal {M}}_{2,\frac {d}{r}} is wider than L d r L^{\frac {d}{r}} and X . r \overset {.}{X}_{r} , the above regularity criterion (0.2) is an improvement of Zhou’s result.