THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

TLDR

The idea of using Riemannian calculus to derive contraction properties for partial differential equations was developed in conversations with Robert McCann in 1997 and later with Cedric Villani in 1998. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. The porous medium equation has a gradient flow structure that is both physically and mathematically natural. Keywords: AMS codes 35K55, 76S05, 58B20, 58F39; acknowledgments to advisor Stephan Luckhaus, coauthor David Kinderlehrer, and the Max–Planck Institute for Mathematics in the Sciences.

Abstract

Abstract We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. Keywords: AMS code: 35K55; 76S05; 58B20; 58F39 ACKNOWLEDGMENTS The author's PhD advisor, Stephan Luckhaus, always stressed the importance of underlying gradient flow structures. David Kinderlehrer is coauthor on an earlier paper on the relationship between diffusion equations and the Wasserstein metric, this paper owes to many conversations with him. The idea of using Riemannian calculus to derive contraction properties for partial differential equations developed in conversations with Robert McCann in 1997 and later with Cedric Villani in 1998. The author gave a lecture series on the results of this paper in 1998 at the Max–Planck Institute for Mathematics in the Sciences in Leipzig, and wishes to thank the Institute for this opportunity.