Vol. 13, No. 8, 2020

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Optimal regularity in time and space for the porous medium equation

Benjamin Gess, Jonas Sauer and Eitan Tadmor

Vol. 13 (2020), No. 8, 2441–2480
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.

Keywords
porous medium equation, entropy solutions, kinetic formulation, velocity averaging, regularity results
Mathematical Subject Classification 2010
Primary: 35K59, 35B65, 35D30, 76S05
Milestones
Received: 19 April 2019
Revised: 30 July 2019
Accepted: 26 September 2019
Published: 28 December 2020
Authors
Benjamin Gess
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Fakultät für Mathematik
Universität Bielefeld
Bielefeld
Germany
Jonas Sauer
Max Planck Institute for Mathematics in the Sciences
Leipzig
Germany
Eitan Tadmor
Department of Mathematics, Institute for Physical Science and Technology
University of Maryland
College Park, MD
United States