Viscous shock waves of Burgers equation with fast diffusion and singularity

Abstract

In this paper, we study the asymptotic stability of viscous shock waves for Burgers' equation with fast diffusion $u_t+f(u)_x=\mu (u^m)_{xx}$ on $\mathbb{R} \times (0, +\infty)$ when $0 u_+=0$, the equation with fast diffusion $(u^m)_{xx}=m\left(\frac{u_x}{u^{1-m}}\right)_x$ processes a strong singularity at $u_+=0$, which causes the stability study to be challenging. We observe that, there exist two different types of viscous shocks, one is the non-degenerate shock satisfying Lax's entropy condition with fast algebraic decay to the singular state $u_+=0$, which causes much strong singularity to the system in the form of $m\left(\frac{u_x}{u^{1-m}}\right)_x$, and the other is the degenerate viscous shock with slow algebraic decay to $u_+=0$, which makes less strong singularity to the system. In order to overcome the singularity at $u_+=0$, we technically use the weighted energy method and develop a new strategy where the weights related to the shock waves are carefully selected, while the chosen weights for the non-degenerate case are stronger than the degenerate case. Numerical simulations are also carried out in different cases to illustrate and validate our theoretical results. In particular, we numerically approximate the solution for different value of $0<m<1$, and find that the shapes of shock waves become steeper when the singularity $\left(\frac{u_x}{u^{1-m}}\right)_x$ is stronger as $m\rightarrow 0$, which indicates that the effect of singular fast diffusion on the solution is essential.