On regularity criteria in conjunction with the pressure of the Navier-Stokes equations

We present new regularity criteria involving the integrability of the pressure for the Navier-Stokes equations in bounded domains with smooth boundaries. We prove that either if the pressure belongs to $L_{x,t}^{\gamma ,q}$ with $3/\gamma +2/q\le 2$ and $3/2<\gamma \le \mathrm{\infty }$ or if the gradient of the pressure belongs to $L_{x,t}^{\gamma ,q}$ with $3/\gamma +2/q\le 3$ and $1<\gamma \le \mathrm{\infty }$ then weak solutions are regular. Local regularity criteria in terms of pressure are also established near a flat boundary as well as in the interior for suitable weak solutions.