On the local regularity of suitable weak solutions to the generalized Navier–Stokes equations

Abstract

We consider the Navier–Stokes equations in three spatial dimensions and present a new proof of the Caffarelli–Kohn–Nirenberg theorem, based on a generalized notion of a local suitable weak solution, involving the local pressure. By estimating the integrals involving the pressure in terms of velocity, the pressure term is cancelled in the local decay estimates. In particular, our proof shows that the Caffarelli–Kohn–Nirenberg theorem holds for any open set \(\Omega \) without any restriction on the size and the regularity of the boundary. In addition, the method forms a basis for proving partial regularity results to other fluid models such as non-Newtonian models or models with heat conduction.