Minimal Geodesics Along Volume-Preserving Maps, Through Semidiscrete Optimal Transport

Abstract

We introduce a numerical method for extracting minimal geodesics along the group of volume-preserving maps, equipped with the $L^2$ metric, which as observed by Arnold [Ann. Inst. Fourier (Grenoble), 16 (1966), pp. 319--361] solve the Euler equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier [Comm. Pure Appl. Math., 44 (1991), pp. 375--417], numerically implemented through semidiscrete optimal transport. It is robust enough to extract nonclassical, multivalued solutions of Euler's equations, for which the flow dimension---defined as the quantization dimension of Brenier's generalized flow---is higher than the domain dimension, a striking and unavoidable consequence of thismodel [A. I. Shnirelman, Geom. Funct. Anal., 4 (1994), pp. 586--620]. Our convergence results encompass this generalizedmodel, and our numerical experiments illustrate it for the first time in two space dimensions.