# Porous Medium Equation with A Drift: Free boundary Regularity

## TL;DR

This paper investigates the regularity of free boundaries in porous medium equations with drift, establishing $C^{1,eta}$ regularity under directional monotonicity, and introduces new non-degeneracy estimates.

## Contribution

It provides the first local non-degeneracy estimates for porous medium equations with drift, leading to $C^{1,eta}$ free boundary regularity under directional monotonicity.

## Key findings

- Proves $C^{1,eta}$ regularity of free boundaries with drift.
- Establishes new local non-degeneracy estimates for the equation.
- Extends regularity results to cases with drift, including zero drift.

## Abstract

We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show the $C^{1,\alpha}$ regularity of the free boundary, when the solution is directionally monotone in space variable in a local neighborhood. The main challenge lies in establishing a local non-degeneracy estimate (Theorem 1.3 and Proposition 1.5), which appears new even for the zero drift case.