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

TL;DR

This paper establishes the equivalence of two classes of weak solutions to the 3D Navier-Stokes equations with distributional pressure and introduces a new $ ext{epsilon}$-regularity criterion, leading to short-time regularity results.

Contribution

It proves the equivalence of dissipative and local suitable weak solutions allowing distributional pressure and develops a relaxed $ ext{epsilon}$-regularity criterion using a local Leray projection.

Findings

Equivalence of dissipative and local suitable weak solutions.

A relaxed $ ext{epsilon}$-regularity criterion for dissipative solutions.

Short-time regularity results on bounded domains.

Abstract

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 $\varepsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\varepsilon$-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.