Nonlinear stability of shock profiles to Burgers' equation with critical fast diffusion and singularity

TL;DR

This paper develops a novel framework to analyze the nonlinear stability of shock profiles in Burgers' equation with critical fast diffusion and singularities, combining weighted energy methods and numerical simulations.

Contribution

It introduces the first analytical framework for stability analysis of Burgers' equation with critical fast diffusion and singularities, addressing technical challenges with new weight functions.

Findings

Proved nonlinear stability of shock profiles with singularities.

Developed weighted energy method to handle singular behavior.

Numerical simulations confirmed theoretical stability results.

Abstract

In this paper we propose the first framework to study Burgers' equation featuring critical fast diffusion in form of $u_t+f(u)_x = (\ln u)_{xx}$. The solution possesses a strong singularity when $u=0$ hence bringing technical challenges. The main purpose of this paper is to investigate the asymptotic stability of viscous shocks, particularly those with shock profiles vanishing at the far field $x=+\infty$. To overcome the singularity, we introduce some weight functions and show the nonlinear stability of shock profiles through the weighted energy method. Numerical simulations are also carried out in different cases of fast diffusion with singularity, which illustrate and confirm our theoretical results.