93 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
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 discrete one-sided Lipschitz condition for convex scalar conservation laws
Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed
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