On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques

In [J. D. Benamou and Y. Brenier, Numer. Math., 84 ( 2000), pp. 375 - 393], a computational fluid dynamic approach was introduced for computing the optimal map occurring in the Monge-Kantorovich problem. Though the described augmented Lagrangian method involves a Hilbertian framework, the discussion was purely formal. Taking advantage of the recent progress in optimal transport theory [ L. A. Caffarelli, Comm. Pure Appl. Math., 45 ( 1992), pp. 1141 - 1151], [ L. A. Caffarelli, Ann. of Math. ( 2), 144 ( 1996), pp. 453 - 496], [ D. Cordero-Erausquin, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), pp. 199 - 202], [ R. J. McCann, Geom. Funct. Anal., 11 ( 2001), pp. 589 - 608] and despite the lack of coercivity of the Hilbertian problem, we establish an existence result. Then, under a reasonable assumption of positivity for the density, we prove the existence of saddle-points for both Lagrangians defined in Benamou and Brenier, and finally prove the convergence of the numerical method.