On Local Regularity of Distributional Solutions to the Navier--Stokes Equations

TL;DR

This paper proves that distributional solutions to the Navier-Stokes equations satisfying the Prodi-Serrin condition are regular in space, even if they are not in the Leray-Hopf class, sharpening previous regularity criteria.

Contribution

It establishes a new regularity result for distributional solutions under the Prodi-Serrin condition without requiring Leray-Hopf class membership.

Findings

Distributional solutions satisfying Prodi-Serrin are spatially regular.

Regularity holds without Leray-Hopf class assumptions.

Sharpness of the regularity criterion is demonstrated.

Abstract

We provide a sharp result that guarantees that a distributional solution satisfying the Prodi-Serrin condition is regular in the spatial variables. The solution does not need to belong to the (local) Leray-Hopf class.