On The Asymptotic Behavior Of A Subcritical Convection-Diffusion Equation With Nonlocal Diffusion

TL;DR

This paper studies the long-term behavior of solutions to a subcritical convection-diffusion equation with nonlocal diffusion, establishing well-posedness, uniform estimates, and asymptotic decay properties.

Contribution

It introduces Oleinik-type estimates for nonlocal diffusion, proves existence of entropy solutions via viscous approximation, and analyzes their large-time asymptotics.

Findings

Established Oleinik-type estimates for nonlocal diffusion.

Proved well-posedness of entropy solutions with $L^1$ initial data.

Derived the first term in the asymptotic behavior of solutions.

Abstract

In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term $\mu u_{xx}$ and letting $\mu\to 0$. Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with $L^1$-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik's inequality, we obtain the first term in the asymptotic behavior of the nonnegative solutions. Finally, the large time behavior of changing sign solutions is proved using the classical flux-entropy method and estimates for the nonlocal operator.