Mild criticality breaking for the Navier-Stokes equations

TL;DR

This paper proves global regularity for Navier-Stokes solutions under bounded slightly supercritical quantities, partially confirming a conjecture by Tao, and introduces a novel method inspired by nonlinear Schrödinger equations.

Contribution

It establishes regularity results for Navier-Stokes equations under new supercritical bounds, advancing understanding of potential blow-up scenarios.

Findings

Solutions remain regular if certain supercritical norms are bounded.

Blow-up implies unboundedness of specific supercritical Orlicz norms.

Method transfers subcritical initial data information to large times.

Abstract

In this short paper we prove the global regularity of solutions to the Navier-Stokes equations under the assumption that slightly supercritical quantities are bounded. As a consequence, we prove that if a solution $u$ to the Navier-Stokes equations blows-up, then certain slightly supercritical Orlicz norms must become unbounded. This partially answers a conjecture recently made by Terence Tao. The proof relies on quantitative regularity estimates at the critical level and transfer of subcritical information on the initial data to arbitrarily large times. This method is inspired by a recent paper of Aynur Bulut, where similar results are proved for energy supercritical nonlinear Schr\"odinger equations.