132 research outputs found
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
Contractive metrics for scalar conservation laws
We consider nondecreasing entropy solutions to 1-d scalar conservation laws
and show that the spatial derivatives of such solutions satisfy a contraction
property with respect to the Wasserstein distance of any order. This result
extends the L^1-contraction property shown by Kruzkov
A new approach to quantitative propagation of chaos for drift, diffusion and jump processes
This paper is devoted the the study of the mean field limit for many-particle
systems undergoing jump, drift or diffusion processes, as well as combinations
of them. The main results are quantitative estimates on the decay of
fluctuations around the deterministic limit and of correlations between
particles, as the number of particles goes to infinity. To this end we
introduce a general functional framework which reduces this question to the one
of proving a purely functional estimate on some abstract generator operators
(consistency estimate) together with fine stability estimates on the flow of
the limiting nonlinear equation (stability estimates). Then we apply this
method to a Boltzmann collision jump process (for Maxwell molecules), to a
McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision
jump process with (stochastic) thermal bath. To our knowledge, our approach
yields the first such quantitative results for a combination of jump and
diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction
of a few typos, to appear In Probability Theory and Related Field
Structure preserving schemes for mean-field equations of collective behavior
In this paper we consider the development of numerical schemes for mean-field
equations describing the collective behavior of a large group of interacting
agents. The schemes are based on a generalization of the classical Chang-Cooper
approach and are capable to preserve the main structural properties of the
systems, namely nonnegativity of the solution, physical conservation laws,
entropy dissipation and stationary solutions. In particular, the methods here
derived are second order accurate in transient regimes whereas they can reach
arbitrary accuracy asymptotically for large times. Several examples are
reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic
Problem
An exact solution of the inelastic Boltzmann equation for the Couette flow with uniform heat flux
In the steady Couette flow of a granular gas the sign of the heat flux
gradient is governed by the competition between viscous heating and inelastic
cooling. We show from the Boltzmann equation for inelastic Maxwell particles
that a special class of states exists where the viscous heating and the
inelastic cooling exactly compensate each other at every point, resulting in a
uniform heat flux. In this state the (reduced) shear rate is enslaved to the
coefficient of restitution , so that the only free parameter is the
(reduced) thermal gradient . It turns out that the reduced moments of
order are polynomials of degree in , with coefficients that
are nonlinear functions of . In particular, the rheological properties
() are independent of and coincide exactly with those of the
simple shear flow. The heat flux () is linear in the thermal gradient
(generalized Fourier's law), but with an effective thermal conductivity
differing from the Navier--Stokes one. In addition, a heat flux component
parallel to the flow velocity and normal to the thermal gradient exists. The
theoretical predictions are validated by comparison with direct Monte Carlo
simulations for the same model.Comment: 16 pages, 4 figures,1 table; v2: minor change
Dimension dependent hypercontractivity for Gaussian kernels
We derive sharp, local and dimension dependent hypercontractive bounds on the
Markov kernel of a large class of diffusion semigroups. Unlike the dimension
free ones, they capture refined properties of Markov kernels, such as trace
estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and
a dimensional and refined (transportation) Talagrand inequality when applied to
the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck
semigroup driven by a non-diffusive L\'evy semigroup are also investigated.
Curvature-dimension criteria are the main tool in the analysis.Comment: 24 page
Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state
The collisional rates associated with the isotropic velocity moments
and
are exactly derived in the case of the
inelastic Maxwell model as functions of the exponent , the coefficient of
restitution , and the dimensionality . The results are applied to
the evolution of the moments in the homogeneous free cooling state. It is found
that, at a given value of , not only the isotropic moments of a degree
higher than a certain value diverge but also the anisotropic moments do. This
implies that, while the scaled distribution function has been proven in the
literature to converge to the isotropic self-similar solution in well-defined
mathematical terms, nonzero initial anisotropic moments do not decay with time.
On the other hand, our results show that the ratio between an anisotropic
moment and the isotropic moment of the same degree tends to zero.Comment: 7 pages, 2 figures; v2: clarification of some mathematical statements
and addition of 7 new references; v3: Published in "Special Issue: Isaac
Goldhirsch - A Pioneer of Granular Matter Theory
Tanaka Theorem for Inelastic Maxwell Models
We show that the Euclidean Wasserstein distance is contractive for inelastic
homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its
associated Kac-like caricature. This property is as a generalization of the
Tanaka theorem to inelastic interactions. Consequences are drawn on the
asymptotic behavior of solutions in terms only of the Euclidean Wasserstein
distance
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
We study a quasilinear parabolic Cauchy problem with a cumulative
distribution function on the real line as an initial condition. We call
'probabilistic solution' a weak solution which remains a cumulative
distribution function at all times. We prove the uniqueness of such a solution
and we deduce the existence from a propagation of chaos result on a system of
scalar diffusion processes, the interactions of which only depend on their
ranking. We then investigate the long time behaviour of the solution. Using a
probabilistic argument and under weak assumptions, we show that the flow of the
Wasserstein distance between two solutions is contractive. Under more stringent
conditions ensuring the regularity of the probabilistic solutions, we finally
derive an explicit formula for the time derivative of the flow and we deduce
the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations
(2013) http://dx.doi.org/10.1007/s40072-013-0014-
Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
This paper deals with a one--dimensional model for granular materials, which
boils down to an inelastic version of the Kac kinetic equation, with
inelasticity parameter . In particular, the paper provides bounds for
certain distances -- such as specific weighted --distances and the
Kolmogorov distance -- between the solution of that equation and the limit. It
is assumed that the even part of the initial datum (which determines the
asymptotic properties of the solution) belongs to the domain of normal
attraction of a symmetric stable distribution with characteristic exponent
\a=2/(1+p). With such initial data, it turns out that the limit exists and is
just the aforementioned stable distribution. A necessary condition for the
relaxation to equilibrium is also proved. Some bounds are obtained without
introducing any extra--condition. Sharper bounds, of an exponential type, are
exhibited in the presence of additional assumptions concerning either the
behaviour, near to the origin, of the initial characteristic function, or the
behaviour, at infinity, of the initial probability distribution function
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