On symmetry breaking for the Navier-Stokes equations

Tobias Barker, Christophe Prange, Jin Tan

Published 2023-02-06Version 1

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 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 introduce a new class of genuinely three-dimensional symmetry preserving global Navier-Stokes flows with a shear flow structure. With regard to that point we give applications to the inviscid limit and exhibit explicit solutions that inviscidly damp to the Kolmogorov flow.