Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect

TL;DR

This paper investigates the asymptotic stability of shock profiles in a Burgers equation with singular fast diffusion and boundary effects, revealing convergence to a shifted shock profile influenced by boundary layer dynamics.

Contribution

It introduces new weighted energy estimate techniques to handle the singular diffusivity and boundary layer effects, establishing stability results for the shock profiles.

Findings

Long-time convergence to shifted shock profiles

Boundary layer dynamics influence the shock shift

New energy estimate methods for singular diffusion problems

Abstract

In this paper, we study the asymptotic stability of viscous shock profile for the Burgers equation $u_t +f(u)_x = (\frac{u_{x}}{u^{1-m}})_x$ on the half-space $(0,+\infty)$, subject to the boundary conditions $u|_{x=0}=u_->0$ and $u|_{x=+\infty}=0$. Here, the parameter $\frac{1}{2}<m<1$ measures the strength of fast diffusion. A key challenge arises from the pronounced singularity in the diffusivity $\left(\frac{u_x}{u^{1-m}} \right)_x$ at $u=0$ and the boundary layer. We demonstrate that the long-time behavior of $u$ converges to a shifted shock profile $U(x-st-d(t))$, where $d(t)$ is governed by the boundary layer dynamics at $x=0$ and driven by the initial data $u(x,0)$. To overcome the singularity from fast diffusion compounded by the bad effect of boundary layer for wave stability, some new techniques for weighted energy estimates are introduced artfully.