Bifurcations of viscous shear flows in a strip

TL;DR

This paper proves that viscous shear flows in a strip undergo a Hopf bifurcation near their stability limit, leading to periodic solutions in time and space under certain spectral conditions.

Contribution

It establishes the occurrence of Hopf bifurcations in viscous shear flows in a strip, extending understanding of flow stability and pattern formation.

Findings

Shear flows undergo Hopf bifurcation near the upper marginal stability curve.

Existence of time and space periodic solutions near the bifurcation point.

Results apply under spectral assumptions valid for convex or concave analytic flows.

Abstract

It is well-established that shear flows in a periodic strip are linearly unstable for the incompressible Navier Stokes equations provided the viscosity is small enough. In this article, under a natural spectral assumption which is satisfied for convex or concave analytic flows, we prove that shear flows undergo a Hopf bifurcation near their upper marginal stability curve. In particular, near this curve, there exist solutions which are periodic in $t$ and $x$.