On symmetry breaking for the Navier-Stokes equations

TL;DR

This paper investigates symmetry breaking and preservation in the 3D Navier-Stokes equations, demonstrating conditions under which symmetry is broken or maintained, with implications for regularity and flow behavior.

Contribution

It provides new results on symmetry breaking, including norm inflation phenomena and explicit solutions that damp to Kolmogorov flow, advancing understanding of Navier-Stokes regularity.

Findings

Proves third component norm inflation from zero third component

Establishes symmetry breaking despite favorable pressure gradients

Constructs explicit solutions that inviscidly damp to Kolmogorov flow

Abstract

Inspired by an open question by Chemin and Zhang about the regularity of the 3D Navier-Stokes equations with one initially small component, we investigate symmetry breaking and symmetry preservation. Our results fall in three classes. First we prove strong symmetry breaking. Specifically, we demonstrate third component norm inflation (3rdNI) and Isotropic Norm Inflation (INI) starting from zero third component. Second we prove symmetry breaking for initially zero third component, even in the presence of a favorable initial pressure gradient. Third we study certain symmetry preserving solutions with a shear flow structure. Specifically, we give applications to the inviscid limit and exhibit explicit solutions that inviscidly damp to the Kolmogorov flow.