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

TL;DR

This paper establishes the existence of weak solutions for fast diffusion equations with divergence-type drift terms, extending previous results and applying to related porous medium and Boussinesq systems.

Contribution

It introduces new existence results for weak solutions under integrability conditions, especially relaxing these conditions for divergence-free drifts, applicable to various related equations.

Findings

Existence of non-negative weak solutions under certain integrability conditions.

Relaxation of integrability conditions for divergence-free drift terms.

Application to viscous Boussinesq systems of fast diffusion type.

Abstract

We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term. Furthermore, in the case that the drift term has a divergence-free structure, it turns out that its integrability conditions can be relaxed, which is also applicable to porous medium equations, thereby improving previous results. As an application, the existence of weak solutions is also discussed for a viscous Boussinesq system of the fast diffusion type.