An $\epsilon$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data

TL;DR

This paper establishes an epsilon-regularity criterion for 3D Navier-Stokes equations based on initial data, providing insights into the regularity and energy concentration of solutions near potential singularities.

Contribution

It introduces a new epsilon-regularity criterion linked to initial data and applies it to estimate regular sets and analyze energy concentration in Navier-Stokes flows.

Findings

Regular solutions are guaranteed under small initial data in a scaled local L^2 norm.

The regular set can be estimated using initial data in weighted L^2 spaces.

Energy concentration near blow-up times can be characterized using the criterion.

Abstract

We prove an $\epsilon$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular in a set enclosed by a paraboloid started from the origin. The result is applied to the estimate of the regular set for local energy solutions with initial data in weighted $L^2$ spaces. We also apply this result to studying energy concentration near a possible blow-up time and regularity of forward discretely self-similar solutions.