Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes

TL;DR

This paper advances the understanding of 3D Navier-Stokes equations by extending weak solution theory to supercritical classes, providing new asymptotic expansions, stability results, and insights into solution regularity and separation rates.

Contribution

It extends weak solution theory for 3D Navier-Stokes to supercritical classes, including a priori bounds, stability under weak-star convergence, and regularity results.

Findings

Provides a local short-time asymptotic expansion in time.

Establishes an upper bound on solution separation rates.

Demonstrates higher-order time regularity away from singularities.

Abstract

This paper extends the weak solution theory for the 3D Navier-Stokes equations of Barker, Seregin and Sverak from a critical setting to a supercritical setting making sure to include a useful a priori energy bound as well as a statement about stability under weak-star convergence. Two applications of the a priori bound are then explored. The first provides a spatially local, short-time asymptotic expansion in the time variable starting at $t=0$ which, as a corollary, provides an upper bound on how fast hypothetical non-unique solutions to the Navier-Stokes equations can separate locally. The second establishes higher-order time regularity at a singular time and at spatial points positioned away from the singularity. This quantifies the degree to which the non-local nature of the pressure allows a far flung singularity to disrupt the time regularity at a regular point.