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

Abstract

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 Lx,tγ,q with 3/γ + 2/q ≤ 2 and 3/2 < γ ≤ ∞ or if the gradient of the pressure belongs to Lx,tγ,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.