Local regularity conditions on initial data for local energy solutions of the Navier-Stokes equations

TL;DR

This paper establishes new conditions on initial data that guarantee regularity of local energy solutions to the Navier-Stokes equations within certain space-time regions, refining previous theoretical results.

Contribution

It introduces weighted $L^2$ norm conditions on initial data that ensure local energy solutions are regular in specific space-time regions, extending prior theorems.

Findings

Weighted $L^2$ norm finiteness implies regularity in certain regions

Refines and generalizes classical regularity theorems

Provides conditions for initial data leading to regular solutions

Abstract

We study the regular sets of local energy solutions to the Navier-Stokes equations in terms of conditions on the initial data. It is shown that if a weighted $L^2$ norm of the initial data is finite, then all local energy solutions are regular in a region confined by space-time hypersurfaces determined by the weight. This result refines and generalizes Theorems C and D of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982) and our recent paper [15] as well.