Optimality of integrability estimates for advection-diffusion equations

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation ${\mathrm{\partial }}_{t}u+\mathrm{d}\mathrm{i}\mathrm{v}(bu)=\mathrm{\Delta }u$, 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.