Regularity of Navier-Stokes flows with bounds for the pressure

This study derives regularity criteria for solutions of the Navier-Stokes equations. Let Ω(t)≔{x:|u(x,t)|>cuLr(R3)}, for some r≥3 and constant c independent of t, with measure |Ω|. It is shown that if p+PL3∕2(Ω) becomes sufficiently small as |Ω| decreases, then uL(r+6)∕3(R3) decays and regularity is secured. Here p is the physical pressure and P is a pressure moderator of relatively broad forms. The implications of the results are discussed and regularity criteria in terms of bounds for |p+P| within Ω are deduced.