Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect
Xiaowen Li, Ming Mei
TL;DR
This work analyzes the asymptotic stabilization of viscous shock profiles for a Burgers type equation with singular fast diffusion on the half-line, where the boundary layer at $x=0$ interacts with a degenerating diffusivity as $u\to0$. The authors reformulate the problem around a moving shock via a perturbation $\phi$ and a time-dependent shift $d(t)$, and then develop a tailored weighted energy method that accommodates the degeneracy and boundary effects. They establish a sequence of a priori estimates up to third order derivatives in carefully chosen weighted spaces, which enable local solutions to be extended globally. The main result proves the existence and uniqueness of a global solution that converges to a shifted shock profile $U(x-st-d_\infty)$ with $d(t)\to d_\infty$ and $d'(t)\in L^1(0,\infty)$, providing a rigorous description of boundary layer driven convergence in the fast diffusion regime. The techniques extend stability analysis to porous medium type diffusion with boundary layers and may inform analysis of related nonlinear degenerate 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.