Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

TL;DR

This paper proves the local asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws, specifically the fractal Burgers equation, providing explicit algebraic decay rates for perturbations.

Contribution

It establishes the stability of traveling waves in a Sobolev space setting and derives explicit decay rates, advancing understanding of nonlocal conservation laws.

Findings

Traveling wave solutions are locally asymptotically stable.

Explicit algebraic decay rates for perturbations are derived.

Stability proof uses a Lyapunov functional approach.

Abstract

We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decreasing profile has been established recently (in special cases). We show the local asymptotic stability of these traveling wave solutions in a Sobolev space setting by constructing a Lyapunov functional. Most importantly, we derive the algebraic-in-time decay of the norm of such perturbations with explicit algebraic-in-time decay rates.