The role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations

Abstract: We study the role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations. By introducing the notion of dissipative solutions, due to Duchon & Robert, we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach gives a new enlightenment of the role of the pressure in this theory in connection to Serrin's local regularity criterion.