# Optimality of integrability estimates for advection-diffusion equations

## TL;DR

This paper examines the conditions under which solutions to advection-diffusion equations are integrable in $L^p$ spaces, establishing the optimality of known exponent ranges through new examples.

## Contribution

It provides a rigorous proof of the optimality of classical integrability estimates for advection-diffusion equations using novel counterexamples.

## Key findings

- Classical $L^p$ estimates hold within certain exponent ranges.
- The paper proves these ranges are optimal with new examples.
- Results clarify the limits of integrability for solutions based on velocity field properties.

## Abstract

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation $\partial_t u + \mathrm{div} (bu) = \Delta u$, where the velocity field $b \in L^r_t L^q_x$. We first summarize some classical results proving such estimates for certain ranges of the exponents $r$ and $q$. Afterwards we prove the optimality of such ranges by means of new original examples.