Optimal regularity in time and space for the porous medium equation

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. CONTENTS1. Introduction 1 2. Preliminaries, Notation and Function Spaces 5 3. Optimality of Estimates via Scaling 9 4. Averaging Lemma Approach 11 5. Application to Porous Medium Equations 26 Appendix A. Kinetic Solutions 31 Appendix B. Fourier Multipliers 33

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