On Symmetry Breaking for the Navier–Stokes Equations

Inspired by an open question by Chemin, Zhang 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.