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

We first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak solutions. Then, an $\epsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\epsilon$-regularity criterion with a new approach using a local version of Leray projection operator. As an application of the approach, we obtain the short-time regularity result on a bounded domain for dissipative solutions.