Optimality of integrability estimates for advection-diffusion equations

We discuss L-p integrability estimates for the solution u of the advection-diffusion equation partial derivative(t)u+div(bu) =Delta u, where the velocity field b is an element of (LtLxq)-L-r. 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