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

TL;DR

This paper explores how pressure influences partial regularity in weak solutions of Navier-Stokes equations, extending existing theories through the concept of dissipative solutions and connecting to Serrin's regularity criterion.

Contribution

It introduces dissipative solutions to generalize the Caffarelli-Kohn-Nirenberg theory and clarifies the pressure's role in local regularity criteria.

Findings

Generalization of partial regularity theory for Navier-Stokes

New insights into pressure's influence on solution regularity

Connection established between pressure and Serrin's criterion

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.