Long time behavior for a nonlocal convection diffusion equation

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal viscous Burgers equation, demonstrating convergence to self-similar profiles under small initial data and deriving the first term of the asymptotic expansion.

Contribution

It establishes well-posedness and asymptotic behavior for a nonlocal Burgers equation, including the derivation of the first term in the asymptotic expansion.

Findings

Solutions behave as self-similar profiles for small initial data

Rescaling and compactness methods reveal the asymptotic structure

First term of the asymptotic expansion is explicitly characterized

Abstract

In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.