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.

One-dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line, which stick under collision with conservation of mass and momentum.

We consider the displacement of an electronic cloud generated by the local difference of charge with a uniform neutralizing background of non-moving ions. The equations are given by the Vlasov-Poisson system, with a coupling constant c = ( & ) 2 where r is the (constant) oscillation period of the electrons. In the so-called quasi-neutral regime, namely as 6 + 0, the current is expected to converge to a solution of the incompressible Euler equations, at least in the case of a vanishing initial temperature. This result is proved by adapting an argument used by P.-L. Lions [Li] to prove the convergence of the Leray solutions of the 3d Navier-Stokes equation to the so-called dissipative solutions of the Euler equations. For this purpose, the total energy of the system is modulated by a test-function. An alternative proof is given, based on the concept of measure-valued (mu) solutions introduced by DiPerna and Majda [DM] and already used by Brenier and Grenier [BG], [Gr2] for the asymptotic analysis of the Vlasov-Poisson system in the quasi-neutral regime. Through this analysis, a link is established between Lions' dissipative solutions and Diperna-Majda's mu solutions of the Euler equations. A second interesting asymptotic regime, still leading to the Euler equations, known

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Abstract. We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by R.DiPerna and A.Majda in their landmark paper [10], where in particular global existence to any L 2 initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open.We also show that DiPerna's measure-valued solutions to systems of conservation laws have the weak-strong uniqueness property.

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour clarifications in the text, additional material on the history of the Monge-Ampere equation, improved description of the auction algorithm, updated bibliography. Version 3: several misprints correcte

The link between the Euler equations of perfect incompressible flows and the Least Action Principle has been known for a long time [1]. Solutions can be considered as geodesic curves along the manifold of volume preserving mappings. Here the “shortest path problem” is investigated. Given two different volume preserving mappings at two different times, find, for the intermediate times, an incompressible flow map that minimizes the kinetic energy (or, more generally, the Action). In its classical formulation, this problem has been solved [7] only when the two different mappings are sufficiently close in some very strong sense. In this paper, a new framework is introduced, where generalized flows are defined, in the spirit of L. C. Young, as probability measures on the set of all possible trajectories in the physical space. Then the minimization problem is generalized as the “continuous linear programming” problem that is much easier to handle. The existence problem is completely solved in the case of the d d -dimensional torus. It is also shown that under natural restrictions a classical solution to the Euler equations is the unique optimal flow in the generalized framework. Finally, a link is established with the concept of measure-valued solutions to the Euler equations [6], and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic.