# Monge-Kantorovich distance for PDEs: the coupling method

## TL;DR

This paper reviews the use of the coupling method to analyze the decay of Monge-Kantorovich distances between solutions of various PDEs, highlighting its effectiveness and limitations across different equations.

## Contribution

It introduces a unified coupling approach for PDEs to study solution distances and discusses alternative dual methods, expanding the analytical toolkit for PDE analysis.

## Key findings

- Coupling method effectively shows decay of solution distances for several PDEs.
- The method's limitations are identified for certain nonlinear PDEs like porous media.
- Alternative dual Monge-Kantorovich approaches are proposed for challenging cases.

## Abstract

We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.