The $L^1$-contraction principle in optimal transport

TL;DR

This paper develops a new $L^1$-contraction principle for diffusion problems using the JKO scheme, enabling analysis of complex inhomogeneous parabolic equations via optimal transport methods.

Contribution

It introduces a novel $L^1$-contraction principle that applies to irregular energy densities, broadening the applicability of optimal transport schemes in nonlinear diffusion equations.

Findings

Established a new $L^1$-contraction principle for the density variable.

Proved the principle relies only on the existence of an optimal transport map and convexity.

Enabled analysis of a wider class of inhomogeneous parabolic equations.

Abstract

In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.