Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

Article type: Research Article

Authors: Duong, Manh Hong

Affiliations: Department of Mathematics and Computer Sciences, Technische Universiteit Eindhoven, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected]

Abstract: In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional Kh, which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, Kh is asymptotically equivalent, in the sense of Γ-convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.

Keywords: porous medium equation, Gamma-convergence, Wasserstein gradient flow, variational methods

DOI: 10.3233/ASY-141272

Journal: Asymptotic Analysis, vol. 92, no. 1-2, pp. 85-106, 2015