On the regularity set and angular integrability for the Navier--Stokes equation

TL;DR

This paper studies how the angular integrability of initial data influences the size of the regular set for solutions to the Navier--Stokes equations, showing higher angular integrability leads to larger regular regions.

Contribution

It introduces a new analysis linking weighted Lebesgue spaces with angular integrability to the regularity of Navier--Stokes solutions, extending previous regularity results.

Findings

Higher angular integrability enlarges the regular set.

The regular set approaches the entire half-space as data's weighted norm tends to zero.

Regularity improves with increased angular integrability of initial data.

Abstract

We investigate the size of the regular set for suitable weak solutions of the Navier--Stokes equation, in the sense of Caffarelli--Kohn--Nirenberg. We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space $\{t>0\}$ in an appropriate limit. In particular, we obtain that if the $L^{2}$ norm with weight $|x|^{-\frac12}$ of the data tends to 0, the regular set invades $\{t>0\}$.