Nonlinear asymptotic stability of non-self-similar rarefaction wave for two-dimensional viscous Burgers equation

TL;DR

This paper proves the nonlinear asymptotic stability of a non-self-similar rarefaction wave for the 2D viscous Burgers equation, including decay rates, even with large initial perturbations and wave strength.

Contribution

It establishes the time-asymptotic stability of non-self-similar rarefaction waves in 2D viscous Burgers equations, a novel result for large initial perturbations.

Findings

Proved the stability of non-self-similar rarefaction waves in 2D viscous Burgers equation.

Derived decay rates for solutions approaching the rarefaction wave.

Demonstrated stability even with large initial perturbations and wave strength.

Abstract

We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+uu_x+uu_y=\Delta u$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states, seperated by a curve $y=\varphi(x)$, and prove that the above 2D non-self-similar rarefaction wave is time-asymptotically stable. Furthermore, we also get the decay rate. Both the rarefaction wave strength and the initial perturbation can be large.