397 research outputs found
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation
The aim of the present paper is twofold:(1) We carry on with developing an
abstract method for deriving decay estimates on the semigroup associated to
non-symmetric operators in Banach spaces as introduced in [10]. We extend the
method so as to consider the shrinkage of the functional space. Roughly
speaking, we consider a class of operators writing as a dissipative part plus a
mild perturbation, and we prove that if the associated semigroup satisfies a
decay estimate in some reference space then it satisfies the same decay
estimate in another-smaller or larger-Banach space under the condition that a
certain iterate of the "mild perturba- tion" part of the operator combined with
the dissipative part of the semigroup maps the larger space to the smaller
space in a bounded way. The cornerstone of our approach is a factorization
argument, reminiscent of the Dyson series.(2) We apply this method to the
kinetic Fokker-Planck equation when the spatial domain is either the torus with
periodic boundary conditions, or the whole space with a confinement potential.
We then obtain spectral gap es- timates for the associated semigroup for
various metrics, including Lebesgue norms, negative Sobolev norms, and the
Monge-Kantorovich-Wasserstein distance W\_1.Comment: Some typos corrected, proof of Lemma 4.7 only sketched to shorten the
paper, 41 page
Relaxation in time elapsed neuron network models in the weak connectivity regime
In order to describe the firing activity of a homogenous assembly of neurons,
we consider time elapsed models, which give mathematical descriptions of the
probability density of neurons structured by the distribution of times elapsed
since the last discharge. Under general assumption on the firing rate and the
delay distribution, we prove the uniqueness of the steady state and its
nonlinear exponential stability in the weak connectivity regime. The result
generalizes some similar results obtained in [10] in the case without delay.
Our approach uses the spectral analysis theory for semigroups in Banach spaces
developed recently by the first author and collaborators
Cauchy problem for the Boltzmann-BGK model near a global Maxwellian
In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK
model for a general class of collision frequencies. We prove that the
Boltzmann-BGK model linearized around a global Maxwellian admits a unique
global smooth solution if the initial perturbation is sufficiently small in a
high order energy norm. We also establish an asymptotic decay estimate and
uniform -stability for nonlinear perturbations.Comment: 26 page
Towards an -theorem for granular gases
The -theorem, originally derived at the level of Boltzmann non-linear
kinetic equation for a dilute gas undergoing elastic collisions, strongly
constrains the velocity distribution of the gas to evolve irreversibly towards
equilibrium. As such, the theorem could not be generalized to account for
dissipative systems: the conservative nature of collisions is an essential
ingredient in the standard derivation. For a dissipative gas of grains, we
construct here a simple functional related to the original ,
that can be qualified as a Lyapunov functional. It is positive, and results
backed by three independent simulation approaches (a deterministic spectral
method, the stochastic Direct Simulation Monte Carlo technique, and Molecular
Dynamics) indicate that it is also non-increasing. Both driven and unforced
cases are investigated
Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section
This paper focuses on the study of existence and uniqueness of distributional
and classical solutions to the Cauchy Boltzmann problem for the soft potential
case assuming integrability of the angular part of the collision
kernel (Grad cut-off assumption). For this purpose we revisit the
Kaniel--Shinbrot iteration technique to present an elementary proof of
existence and uniqueness results that includes large data near a local
Maxwellian regime with possibly infinite initial mass. We study the propagation
of regularity using a recent estimate for the positive collision operator given
in [3], by E. Carneiro and the authors, that permits to study such propagation
without additional conditions on the collision kernel. Finally, an
-stability result (with ) is presented assuming the
aforementioned condition.Comment: 19 page
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