A survey of geometric constraints on the blowup of solutions of the Navier--Stokes equation

TL;DR

This survey reviews geometric regularity criteria for the Navier--Stokes equation that could prevent finite-time blowup, highlighting their physical implications and significance.

Contribution

It compiles and discusses various geometric constraints on Navier--Stokes solutions that influence blowup behavior, offering a comprehensive overview of current regularity criteria.

Findings

Geometric constraints can prevent finite-time blowup.

Regularity criteria have significant physical interpretations.

The survey consolidates key results in Navier--Stokes regularity theory.

Abstract

In this survey article, we will discuss some regularity criteria for the Navier--Stokes equation that provide geometric constraints on any possible finite-time blowup. We will also discuss the physical significance of such regularity criteria.