Existence of weak solutions for fast diffusion equation with a divergence type of drift term

This paper investigates fast diffusion equations with a divergence type of drift term. We establish the existence of nonnegative $L^q$-weak solutions which satisfies energy estimates or even further with speed estimates in Wasserstein spaces under assuming certain classes for the drift that are differ from those in porous medium equations. When the drift has divergence-free structure, we construct weak solutions under weaker conditions on the drift. Moreover, the same techniques partially extend to porous medium equations. As an application, we also discuss a viscous Boussinesq system of fast diffusion type.