BRENIER , Yann

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate...

We show that Kruzhkov's theory of entropy solutions to multidimensional scalar conservation laws can be entirely recast in L2 and fits into the general theory of maximal monotone operators in Hilbert spaces. Our approach is based on a combination of level-set, kinetic and transport-collapse approximations, in the spirit of previous works by...

Gaspard Monge a étudié un problème très concret — déplacer au mieux un tas de sable —, en lui appliquant une méthode rigoureuse. Aujourd'hui, on parle de « recherche opérationelle » pour désigner ce genre de méthodes.

We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves...

We show that bounded families of global classical relativistic strings that can be written as graphs are relatively compact in C0 topology, but their accumulation points include many non relativistic strings. We also provide an alternative formulation of these relativistic strings and characterize their ``semi-relativistic'' completion.

We address the early universe reconstruction (EUR) problem (as considered by Frisch and coauthors), and the related Zeldovich approximate model. By substituting the fully nonlinear Monge-Ampere equation for the linear Poisson equation to model gravitation, we introduce a modified mathematical model, for which the Zeldovich approximation becomes...

The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is unrevealed, robust existence and uniqueness results can be unexpectedly obtained for very general data. Of course,...

We derive classical particle, string, and membrane motion equations from a rigorous asymptotoc analysis of the Born-Infeld nonlinear electromagnetic theory...

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation which generates a set of...

We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Eu-ler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich-Bures metric space, which is a matricial analogue...

International audience

International audience

In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its...