13,100 research outputs found
Asymptotic symmetries of three dimensional gravity and the membrane paradigm
The asymptotic symmetry group of three-dimensional (anti) de Sitter space is
the two dimensional conformal group with central charge . Usually
the asymptotic charge algebra is derived using the symplectic structure of the
bulk Einstein equations. Here, we derive the asymptotic charge algebra by a
different route. First, we formulate the dynamics of the boundary as a
1+1-dimensional dynamical system. Then we realize the boundary equations of
motion as a Hamiltonian system on the dual Lie algebra, , of
the two-dimensional conformal group. Finally, we use the Lie-Poisson bracket on
to compute the asymptotic charge algebra. This streamlines the
derivation of the asymptotic charge algebra because the Lie-Poisson bracket on
the boundary is significantly simpler than the symplectic structure derived
from the bulk Einstein equations. It also clarifies the analogy between the
infinite dimensional symmetries of gravity and fluid dynamics.Comment: 15 page
Uniform convergence to equilibrium for granular media
We study the long time asymptotics of a nonlinear, nonlocal equation used in
the modelling of granular media. We prove a uniform exponential convergence to
equilibrium for degenerately convex and non convex interaction or confinement
potentials, improving in particular results by J. A. Carrillo, R. J. McCann and
C. Villani. The method is based on studying the dissipation of the Wasserstein
distance between a solution and the steady state
Existence of Compactly Supported Global Minimisers for the Interaction Energy
The existence of compactly supported global minimisers for continuum models
of particles interacting through a potential is shown under almost optimal
hypotheses. The main assumption on the potential is that it is catastrophic, or
not H-stable, which is the complementary assumption to that in classical
results on thermodynamic limits in statistical mechanics. The proof is based on
a uniform control on the local mass around each point of the support of a
global minimiser, together with an estimate on the size of the "gaps" it may
have. The class of potentials for which we prove existence of global minimisers
includes power-law potentials and, for some range of parameters, Morse
potentials, widely used in applications. We also show that the support of local
minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun
Numerical Study of a Particle Method for Gradient Flows
We study the numerical behaviour of a particle method for gradient flows
involving linear and nonlinear diffusion. This method relies on the
discretisation of the energy via non-overlapping balls centred at the
particles. The resulting scheme preserves the gradient flow structure at the
particle level, and enables us to obtain a gradient descent formulation after
time discretisation. We give several simulations to illustrate the validity of
this method, as well as a detailed study of one-dimensional
aggregation-diffusion equations.Comment: 27 pages, 21 figure
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity
We consider a one dimensional transport model with nonlocal velocity given by
the Hilbert transform and develop a global well-posedness theory of probability
measure solutions. Both the viscous and non-viscous cases are analyzed. Both in
original and in self-similar variables, we express the corresponding equations
as gradient flows with respect to a free energy functional including a singular
logarithmic interaction potential. Existence, uniqueness, self-similar
asymptotic behavior and inviscid limit of solutions are obtained in the space
of probability measures with finite second
moments, without any smallness condition. Our results are based on the abstract
gradient flow theory developed in \cite{Ambrosio}. An important byproduct of
our results is that there is a unique, up to invariance and translations,
global in time self-similar solution with initial data in
, which was already obtained in
\textrm{\cite{Deslippe,Biler-Karch}} by different methods. Moreover, this
self-similar solution attracts all the dynamics in self-similar variables. The
crucial monotonicity property of the transport between measures in one
dimension allows to show that the singular logarithmic potential energy is
displacement convex. We also extend the results to gradient flow equations with
negative power-law locally integrable interaction potentials
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