Stability of Two-dimensional Viscous Incompressible Flows Under Three-dimensional Perturbations and Inviscid Symmetry Breaking

TL;DR

This paper investigates the stability and symmetry preservation of weak solutions in 3D incompressible flows, demonstrating that viscous flows maintain initial symmetries and exploring the potential for symmetry breaking in inviscid flows.

Contribution

It proves symmetry preservation for viscous flows and discusses conditions under which symmetry breaking can or cannot occur in inviscid flows.

Findings

Viscous flows preserve initial symmetry under perturbations.

Symmetry breaking is possible in inviscid flows due to recent constructions.

Certain initial data and solution restrictions prevent symmetry breaking.

Abstract

In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flows are stable under general three-dimensional perturbations, globally in time. We work in three different contexts: two-and-a-half-dimensional, helical and axi-symmetric flows. In the inviscid case, we observe that, as a consequence of recent work by De Lellis and Sz\'ekelyhidi, there are genuinely three-dimensional weak solutions of the Euler equations with two-dimensional initial data. We also present two partial results where restrictions on the set of initial data, and on the set of admissible solutions rule out spontaneous symmetry breaking; one is due to P.-L. Lions and the other is a consequence of our viscous stability result.