A drift-diffusion equation is like the heat equation with an extra first order term. In some cases, the Laplacian is replaced by a fractional Laplacian. There are several nonlinear models in a variety of contexts that fit into this scheme. In order to understand the solvability of the non linear models, it is essential to obtain a priori estimates on the smoothness of the solution for linear drift-diffusion equations when the first order term is given by a very irregular vector field times the gradient. We will analyze different smoothness estimates in different situations. It is particularly important to understand the consequences of assuming that the drift is divergence free, given their applications to models related to incompressible fluids.
