The L 2 Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

Given a probability space ( X , p ) and a bounded domain R in R d equipped with the Lebesgue measure 1 . I (normalized so that 10 I = I ), it is shown (under additional technical assumptions on X and Q) that for every vector-valued function u E L p ( X , p; R d ) there is a unique "polar factorization" u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from ( X , p ) into ( Q , I . I), provided that u is nondegenerate, in the sense that p ( u -' ( E ) ) = 0 for each Lebesgue negligible subset E of Rd.Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified.The Monge-Amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate "Monge-Kantorovich" problem.