From concentration to quantitative regularity: a short survey of recent developments for the Navier-Stokes equations

TL;DR

This survey reviews recent progress on the regularity and potential singularity formation in 3D Navier-Stokes solutions, highlighting new quantitative estimates and connections to nonlinear dispersive equations.

Contribution

It summarizes recent developments in understanding norm concentration, blow-up rates, and breaking criticality barriers for Navier-Stokes equations, inspired by dispersive PDE techniques.

Findings

Advances in quantifying norm blow-up rates.

Insights into concentration phenomena near singularities.

Progress in breaking the criticality barrier.

Abstract

In this short survey paper, we focus on some new developments in the study of the regularity or potential singularity formation for solutions of the 3D Navier-Stokes equations. Some of the motivating questions are: Are certain norms accumulating/concentrating on small scales near potential blow-up times? At what speed do certain scale-invariant norms blow-up? Can one prove explicit quantitative regularity estimates? Can one break the criticality barrier, even slightly? We emphasize that these questions are closely linked together. Many recent advances for the Navier-Stokes equations are directly inspired by results and methods from the field of nonlinear dispersive equations.