A Hopf bifurcation in the planar Navier-Stokes equations

TL;DR

This paper demonstrates a Hopf bifurcation in the planar Navier-Stokes equations with specific boundary conditions, showing how stationary solutions transition to periodic solutions as viscosity varies, using a constructive, computer-assisted approach.

Contribution

It provides a rigorous, computer-assisted proof of a Hopf bifurcation in the Navier-Stokes equations under Navier boundary conditions, which is a novel analytical result.

Findings

Identification of a Hopf bifurcation point as viscosity varies.

Construction of periodic solutions branching from stationary solutions.

Use of computer-assisted estimates for rigorous proof.

Abstract

We consider the Navier-Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.