8,278 research outputs found
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
Infinite energy solutions to the homogeneous Boltzmann equation
The goal of this work is to present an approach to the homogeneous Boltzmann
equation for Maxwellian molecules with a physical collision kernel which allows
us to construct unique solutions to the initial value problem in a space of
probability measures defined via the Fourier transform. In that space, the
second moment of a measure is not assumed to be finite, so infinite energy
solutions are not {\it a priori} excluded from our considerations. Moreover, we
study the large time asymptotics of solutions and, in a particular case, we
give an elementary proof of the asymptotic stability of self-similar solutions
obtained by A.V. Bobylev and C. Cercignani [J. Stat. Phys. {\bf 106} (2002),
1039--1071]
Well-posedness of the spatially homogeneous Landau equation for soft potentials
We consider the spatially homogeneous Landau equation of kinetic theory, and
provide a differential inequality for the Wasserstein distance with quadratic
cost between two solutions. We deduce some well-posedness results. The main
difficulty is that this equation presents a singularity for small relative
velocities. Our uniqueness result is the first one in the important case of
soft potentials. Furthermore, it is almost optimal for a class of moderately
soft potentials, that is for a moderate singularity. Indeed, in such a case,
our result applies for initial conditions with finite mass, energy, and
entropy. For the other moderatley soft potentials, we assume additionnally some
moment conditions on the initial data. For very soft potentials, we obtain only
a local (in time) well-posedness result, under some integrability conditions.
Our proof is probabilistic, and uses a stochastic version of the Landau
equation, in the spirit of Tanaka
A gradient flow approach to the Boltzmann equation
We show that the spatially homogeneous Boltzmann equation evolves as the
gradient flow of the entropy with respect to a suitable geometry on the space
of probability measures which takes the collision process into account. This
gradient flow structure allows to give a new proof for the convergence of Kac's
random walk to the homogeneous Boltzmann equation, exploiting the stability of
gradient flows.Comment: Presentation reworked and streamlined. Variational characterization
of the Boltzmann equation simplified using the action of curve without
referring to the associated distance function. Discussion of the distance
moved to appendix. Additional assumption missing in previous version on
moment bounds of order higher than 2 for Kac walk added in Thm 1.
Non-Equilibrium Steady States in Kac's Model Coupled to a Thermostat
This paper studies the existence, uniqueness and convergence to
non-equilibrium steady states in Kac's model with an external coupling. We work
in both Fourier distances and Wasserstein distances. Our methods work in the
case where the external coupling is not a Maxwellian equilibrium. This provides
an example of a non-equilibrium steady state. We also study the behaviour as
the number of particles goes to infinity and show quantitative estimates on the
convergence rate of the first marginal.Comment: 17 pages, no figure
Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem
We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann
equation for hard spheres, for a general form of collision rate which includes
in particular variable restitution coefficients depending on the kinetic energy
and the relative velocity as well as the sticky particles model. We prove
(local in time) non-concentration estimates in Orlicz spaces, from which we
deduce weak stability and existence theorem. Strong stability together with
uniqueness and instantaneous appearance of exponential moments are proved under
additional smoothness assumption on the initial datum, for a restricted class
of collision rates. Concerning the long-time behaviour, we give conditions for
the cooling process to occur or not in finite time.Comment: 45 page
Quantitative uniform in time chaos propagation for Boltzmann collision processes
This paper is devoted to the study of mean-field limit for systems of
indistinguables particles undergoing collision processes. As formulated by Kac
\cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1)
prove and quantify this property for Boltzmann collision processes with
unbounded collision rates (hard spheres or long-range interactions), (2) prove
and quantify this property \emph{uniformly in time}. This yields the first
chaos propagation result for the spatially homogeneous Boltzmann equation for
true (without cut-off) Maxwell molecules whose "Master equation" shares
similarities with the one of a L\'evy process and the first {\em quantitative}
chaos propagation result for the spatially homogeneous Boltzmann equation for
hard spheres (improvement of the %non-contructive convergence result of
Sznitman \cite{S1}). Moreover our chaos propagation results are the first
uniform in time ones for Boltzmann collision processes (to our knowledge),
which partly answers the important question raised by Kac of relating the
long-time behavior of a particle system with the one of its mean-field limit,
and we provide as a surprising application a new proof of the well-known result
of gaussian limit of rescaled marginals of uniform measure on the
-dimensional sphere as goes to infinity (more applications will be
provided in a forthcoming work). Our results are based on a new method which
reduces the question of chaos propagation to the one of proving a purely
functional estimate on some generator operators ({\em consistency estimate})
together with fine stability estimates on the flow of the limiting non-linear
equation ({\em stability estimates})
A gradient flow approach to linear Boltzmann equations
We introduce a gradient flow formulation of linear Boltzmann equations. Under
a diffusive scaling we derive a diffusion equation by using the machinery of
gradient flows
- …