*Submitted Paper*

**Inserted:** 7 feb 2017

**Year:** 2017

**Abstract:**

We discuss $L^p$ integrability estimates for the solution $u$ of the advection-diffusion equation $\partial_t u + div (bu) = \Delta u$, 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.

**Keywords:**
parabolic equations, Advection-diffusion equations, integrability estimates and their optimality, Duhamel formula, self-similar solutions