Optimality of integrability estimates for advection-diffusion equations

Abstract

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation ${\partial }_{t}u+\mathrm{div}(bu)=\hat{I}”u$, where the velocity field $b\in L_t^r{L}_x^q$. 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.

19 pages