Optimality of integrability estimates for advection-diffusion equations

We discuss Lp integrability estimates for the solution u of the advection–diffusion equation ∂tu+div(bu)=Δu, where the velocity field b∈LrtLqx. 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