1,225 research outputs found
A double large deviation principle for monge-ampere gravitation
Monge-Ampere gravitation is a nonlinear modification of classical Newtonian
gravitation, when the Monge-Ampere equation substitutes for the Poisson
equation. We establish, through two applications of the large deviation
principle, that the MA gravitation for a finite number of particles can be
reduced, through a double application of the large deviation principle, to the
simplest possible stochastic model: a collection of independent Brownian
motions with vanishing noise
The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem
We consider the Euler equations of incompressible fluids and attempt to solve
the initial value problem with the help of a concave maximization problem.We
show that this problem, which shares a similar structure with the optimal
transport problemwith quadratic cost, in its "Benamou-Brenier"
formulation,always admits a relaxed solution that can be interpretedin terms of
of the Euler equations in the sense of convex integration
theory.Moreover, any smooth solution of the Euler equations can be recovered
from this maximization problem, at least for short times
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
Geodesics in the space of measure-preserving maps and plans
We study Brenier's variational models for incompressible Euler equations.
These models give rise to a relaxation of the Arnold distance in the space of
measure-preserving maps and, more generally, measure-preserving plans. We
analyze the properties of the relaxed distance, we show a close link between
the Lagrangian and the Eulerian model, and we derive necessary and sufficient
optimality conditions for minimizers. These conditions take into account a
modified Lagrangian induced by the pressure field. Moreover, adapting some
ideas of Shnirelman, we show that, even for non-deterministic final conditions,
generalized flows can be approximated in energy by flows associated to
measure-preserving maps
Remarks on the Minimizing Geodesic Problem in Inviscid Incompressible Fluid Mechanics
We consider minimizing geodesics along the group of volume preserving
maps of a given 3-dimensional domain . The corresponding curves
describe the motion of an ideal incompressible fluid inside and are
(formally) solutions of the Euler equations. It is known that there is a unique
possible pressure gradient for these curves whenever their end points are
fixed. In addition, this pressure field has a limited but unconditional
(internal) regularity. The present paper completes these results by showing: 1)
the uniqueness property can be viewed as an infinite dimensional phenomenon
(related to the possibility of relaxing the corresponding minimization problem
by convex optimization), which is false for finite dimensional configuration
spaces such as O(3) for the motion of rigid bodies; 2) the unconditional
partial regularity is necessarily limited
Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics
There are well-established connections between combinatorial optimization,
optimal transport theory and Hydrodynamics, through the linear assignment
problem in combinatorics, the Monge-Kantorovich problem in optimal transport
theory and the model of inviscid, potential, pressure-less fluids in
Hydrodynamics. Here, we consider the more challenging quadratic assignment
problem (which is NP, while the linear assignment problem is just P) and find,
in some particular case, a correspondence with the problem of finding
stationary solutions of Euler's equations for incompressible fluids. For that
purpose, we introduce and analyze a suitable "gradient flow" equation.
Combining some ideas of P.-L. Lions (for the Euler equations) and
Ambrosio-Gigli-Savar\'e (for the heat equation), we provide for the initial
value problem a concept of generalized "dissipative" solutions which always
exist globally in time and are unique whenever theyare smooth
The Monge-Amp\`ere-Kantorovich approach to reconstruction in cosmology
Motion of a continuous fluid can be decomposed into an "incompressible"
rearrangement, which preserves the volume of each infinitesimal fluid element,
and a gradient map that transfers fluid elements in a way unaffected by any
pressure or elasticity (the polar decomposition of Y. Brenier). The Euler
equation describes a system whose kinematics is dominated by the incompressible
rearrangement. The opposite limit, in which the incompressible component is
negligible, corresponds to the Zel'dovich approximation, a model of motion of
self-gravitating fluid in cosmology.
We present a method of approximate reconstruction of the large-scale proper
motions of matter in the Universe from the present-day mass density field. The
method is based on recovering the corresponding gradient transfer map. We
discuss its algorithmics, tests of the method against mock cosmological
catalogues, and its application to observational data, which result in tight
constraints on the mean mass density Omega_m and age of the Universe.Comment: 6 pages, 2 figures; based on an invited lecture at the conference
"Euler's Equations: 250 Years On" (see http://www.obs-nice.fr/etc7/EE250/);
to be published in a special issue of Physica D containing the proceedings of
that conferenc
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