Existence of weak solutions for Porous medium equation with a divergence type of drift term

TL;DR

This paper proves the existence of weak solutions for porous medium equations with divergence-type drift terms, expanding the class of admissible drifts and providing new regularity and uniqueness results using Wasserstein space methods.

Contribution

It introduces broader classes of divergence-type drift terms for which weak solutions exist, including divergence-free cases, and develops a splitting method for constructing solutions with regularity estimates.

Findings

Existence of $L^{q}$-weak solutions under new drift conditions.

Wider drift classes including divergence-free cases.

Improved regularity results for porous medium type Keller-Segel systems.

Abstract

We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on $q$ and $m$ caused by the nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, bypassing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of the crucial points in the construction is uniform H\"{o}lder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.