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

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.

Figures (6)

• Figure 1: Case 1--the graph of $f(u)$
• Figure 2: Case 1--the solution $u(x,t)$ behaves like a monotone viscous shock wave despite being initially perturbed
• Figure 3: Case 2--graph of $f(u)$
• Figure 4: Case 2--the solution $u(x,t)$ behaves like a monotone viscous shock wave
• Figure 5: Case 3--the graph of $f(u)$

Theorems & Definitions (7)

• proof : Proof of Theorem \ref{['Existence of the shock profile']}
• proof
• proof
• proof
• proof
• proof : Proof of Proposition \ref{['proposition priori estimate']}
• proof : Proof of Theorem \ref{['phi stability']}