Rarefaction waves in nonlocal convection-diffusion equations

TL;DR

This paper studies the long-term behavior of solutions to a nonlocal convection-diffusion equation with step-like initial data, demonstrating convergence to a rarefaction wave similar to solutions of the Burgers equation.

Contribution

It proves the convergence of solutions of a nonlocal convection-diffusion equation to a rarefaction wave, extending methods from fractal Burgers equation analysis.

Findings

Solutions converge to a rarefaction wave over time.

The convergence is towards the entropy solution of the Riemann problem.

Methods are inspired by fractal Burgers equation studies.

Abstract

We consider the "convection-diffussion" equation $u_t=J*u-u-uu_x,$ where $J$ is a probability density. We supplement this equation with step-like initial conditions and prove a convergence of corresponding solution towards a rarefaction wave, i.e. a unique entropy solution of the Riemann problem for the nonviscous Burgers equation. Methods and tools used in this paper are inspired by those used in [Karch, Miao and Xu, SIAM J. Math. Anal. {\bf 39} (2008), no. 5, 1536--1549.], where the fractal Burgers equation was studied.