Section: New Results
Generalized Solution of Euler
Minimal geodesics along volume preserving maps, through semidiscrete optimal transport
Q. Mérigot and J.M. Mirebeau introduced a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the ${\mathrm{L}}^{2}$ metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semidiscrete optimal transport. It is robust enough to extract nonclassical, multivalued solutions of Euler's equations, for which the dimension of the support of the flow is higher than the dimension of the domain, a striking and unavoidable consequence of this model. Our convergence results encompass this generalized model, and our numerical experiments illustrate it for the first time in two space dimensions (see Figure 14 ).
