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

M.H. Duong

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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 G-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: Gamma-convergence; porous medium equation; variational methods; Wasserstein gradient flow
Original languageEnglish
Pages (from-to)85-106
JournalAsymptotic Analysis
Volume91
Issue number1-2
DOIs
Publication statusPublished - 2015

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