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 $\mathbb {R}^{d}$. It is known that if a Leray weak solution $u$ belongs to \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 $u$ is regular. It is proved that if the pressure $p$ associated to a Leray weak solution $u$ belongs to \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 $\overset {.}{\mathcal {M}}_{2,\frac {d}{r}}\left ( \mathbb {R}^{d}\right )$ is the critical Morrey-Campanato space (a definition is given in the text) for $0<r<1$, then the weak solution is actually regular. Since this space $\overset {.}{\mathcal {M}}_{2,\frac {d}{r}}$ is wider than $L^{\frac {d}{r}}$ and $\overset {.}{X}_{r}$, the above regularity criterion (0.2) is an improvement of Zhou’s result.