# Asymptotic behaviour of solutions to fractional diffusion-convection equations

## TL;DR

This paper investigates the long-term behavior of solutions to a fractional diffusion-convection equation, showing they asymptotically resemble the entropy solution of the convection component, using a-priori estimates and Oleinik inequalities.

## Contribution

It establishes the asymptotic equivalence of solutions to the entropy solution of the convection part for fractional diffusion-convection equations.

## Key findings

- Solutions tend to the entropy solution of the convection equation over time
- Oleinik type inequalities are crucial in the analysis
- Provides a rigorous framework for asymptotic analysis of fractional PDEs

## Abstract

We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.