Uniqueness and asymptotic stability of time-periodic solution for the fractal Burgers equation

TL;DR

This paper proves the existence, uniqueness, and asymptotic stability of time-periodic solutions for the fractal Burgers equation with periodic forcing, extending previous results to the fractional case using Galerkin and Fourier methods.

Contribution

It establishes the first rigorous proof of T-periodic solutions for the fractional Burgers equation and demonstrates their stability in an energy space.

Findings

Existence of T-periodic solutions for the linearized equation.

Uniqueness of the T-periodic solution for the nonlinear equation.

Asymptotic stability of the unique T-periodic solution.

Abstract

The paper is concerned with the time-periodic (T-periodic) problem of the fractal Burgers equation with a T-periodic force on the real line. Based on the Galerkin approximates and Fourier series (transform) methods, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution to the nonlinear equation are established by the contraction mapping argument. Furthermore, we show that the unique T-periodic solution is asymptotically stable. This analysis, which is carried out in energy space $ H^{1}(0,T;H^{\frac{\alpha}{2}}(R))\cap L^{2}(0,T;\dot{H}^{\alpha})$ with $1<\alpha<\frac{3}{2}$, extends the T-periodic viscid Burgers equation in \cite{5} to the T-periodic fractional case.