Section: New Results

Optimal Transportation numerical methods for Fluid models

F-X. Vialard Q. Mérigot L. Nenna G. Carlier J-D. Benamou

Several new algorithms based on Optimal Transport have applied to Generalized Euler Geodesics and the Cauchy problem for the Euler equation. The methods rely on the generalized polar decomposition of Brenier, numerically implemented whether through semi-discrete optimal transport or through entropic regularization. It is robust enough to extract non-classical, multi-valued solutions of Euler’s equations predicted by Brenier and Schnirelman. The semi-discrete approach also leads to a numerical scheme able to approximate regular solutions to the Cauchy problem for Euler equations. See Luca Nenna Thesis and [15].

A new link between optimal transport and fluid dynamic was discovered in [42]. Since the work of Brenier, optimal transport is tightly linked with the incompressible Euler equation and can be seen as a nonlinear extension of the pressure. Recently, a new optimal transport model between unbalanced measures has been proposed by some of the members of Mokaplan. In [41], it is shown that the corresponding fluid dynamic equation is the Camassa-Holm equation, well known to model waves in shallow water and wave breaking. On the theoretical side, we prove that the solutions to the Camassa-Holm equation can be seen as particular solutions of the incompressible Euler equation. This work paves the way for the study of the generalized Camassa-Holm geodesics and numerical methods based on unbalanced optimal transport scaling algorithms to solve it.