# Optimal regularity in time and space for the porous medium equation

## TL;DR

This paper establishes optimal regularity estimates in time and space for solutions to the porous medium equation within Sobolev spaces, including higher regularity for solution powers, aligning with linear case results.

## Contribution

It provides new optimal regularity estimates for the porous medium equation in Sobolev spaces, extending to higher powers of solutions.

## Key findings

- Regularity estimates are optimal and consistent with the linear case.
- Higher spatial regularity for powers of solutions is achieved.
- Scaling arguments support the optimality of these estimates.

## Abstract

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case.