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 γ,q with 3/γ+2/q≤2 and 3/2<γ≤∞ or if the gradient of the pressure belongs to L x,t γ,q with 3/γ+2/q≤2 and 1<γ≤∞, 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.