arXiv subjects

Yann Brenier

Publications and source records attributed to Yann Brenier.

23 records · Page 2Linked to original sources

arXiv2008

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

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 version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology.

math.AP↗

arXiv2006

Non relativistic strings may be approximated by relativistic strings

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.

math.AP↗

arXiv2006

L2 formulation of multidimensional scalar conservation laws

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 Giga, Miyakawa, Osher, Tsai and the author.

math.AP↗

arXiv2005

A geometric approximation to the Euler equations: the Vlasov-Monge-Ampere system

This paper studies the Vlasov-Monge-Ampere system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampere equation substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.

math.AP↗

arXiv2003

Some geometric PDEs related to hydrodynamics and electrodynamics

We discuss several geometric PDEs and their relationship with Hydrodynamics and classical Electrodynamics. We start from the Euler equations of ideal incompressible fluids that, geometrically speaking, describe geodesics on groups of measure preserving maps with respect to the $L^2$ metric. Then, we introduce a geometric approximation of the Euler equation, which involves the Monge-Ampère equation and the Monge-Kantorovich optimal transportation theory. This equation can be interpreted as a fully nonlinear correction of the Vlasov-Poisson system that describes the motion of electrons in a uniform neutralizing background through Coulomb interactions. Finally we briefly discuss an equation for generalized extremal surfaces in the 5 dimensional Minkowski space, related to the Born-Infeld equations, from which the Vlasov-Maxwell system of classical Electrodynamics can be formally derived.

math.AP↗