Asymptotic stability and bifurcation of time-periodic solutions for the viscous Burgers' equation

Abstract

We consider the Dirichlet boundary value problem for the viscous Burgers' equation with a time periodic force on a one dimensional finite interval. Under the boundedness assumption on the external force, we prove the existence of the time-periodic solution by using the Galerkin method and Schaefer's fixed point theorem. Furthermore, we show that this time-periodic solution is unique and time-asymptotically stable in the H1 sense under an additional smallness condition on the external force. It is naturally expected that when the amplitude of the external force increases and crosses a certain critical value, the time-periodic solution is no longer asymptotically stable. In the last part of the article, to support our theory, numerical experiments are carried out to investigate the exchange of stabilities of the time-periodic solutions when the amplitude of the force crosses the first critical value. We numerically find this critical value at which the stable solutions turn into the unstable ones.