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

Towards an HH-theorem for granular gases

The $H$-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 $\mathcal H$ related to the original $H$, 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 $S^{n-1}$ 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 $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.Comment: 19 page

Mechanical and electrochemical deterioration mechanisms in the tribocorrosion of Al alloys in NaCl and in NaNO3 solutions

Tribocorrosion of Al-Si-Cu-Mg alloys was investigated in 0.05 M NaCl and 0.1 M NaNO3 solutions under severe sliding and controlled electrochemical conditions. A simple galvanic coupling model was developed to analyze and quantitatively predict the evolution potential of the open circuit potential during tribocorrosion. According to this model and the obtained results, galvanic coupling was established in the NaNO3 solution within the wear track between passive and mechanically depassivated areas. In the NaCl solution, galvanic coupling was established between the whole depassivated wear track and the surrounding area. This difference was attributed to different mechanical properties of the passive surfaces.The research team was financially supported by the Portuguese Foundation for Science and Technology (FCT-Portugal), under a PhD scholarship (SFRH/BD/27911/2006). The authors thank also to Dr. Edith Ariza (University of Minho) and Pierre Mettraux (EPFL) for SEM analysis