Semigroup generation propertiesof streaming operators with noncontractive boundary conditions

We present c₀-semigroup generation results for the free streaming operator with abstractboundary conditions. We recall some known results on the matter and establish a general theorem (already announced in [1]). We motivate our study with many examples and show that our result applies to the physical cases of Maxwell boundary conditions in the kinetic theory of gases, as well as to the nonlocal boundary conditions involved in transport-like equations from population dynamics.     

Dissipative Linear Boltzmann Equation for Hard Spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard-spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non-zero temperature lower than the background one. Moreover thanks to the H-Theorem we prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.     

Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach II. The Flexural Model

This paper is the sequel of Part I, in which the limiting displacement field of a thin shell when its thickness approaches zero is identified as the solution of a two‐dimensional nonlinear membrane shell model. When the geometry of the middle surface of the shell and the boundary conditions allow non‐zero “inextensional displacements”, the previous membrane limit model is not relevant. In this case, we show how to “update” the assumptions on the applied forces acting on the shell so that a limiting model can be derived by an asymptotic analysis. Furthermore, we identify this limit as the two‐dimensional nonlinear flexural shell model. (Accepted January 13, 1997)     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

On linear kinetic equations involving unbounded cross‐sections

This paper deals with the spectral properties of singular neutron transport equations involving unbounded collision frequencies and unbounded collision operators. Generation results are given as well as estimates of the essential type. Copyright © 2004 John Wiley & Sons, Ltd.     

A generation theorem for kinetic equations with non-contractive boundary operatorsUn résultat de génération pour des équations cinétiques soumises à des conditions frontières non contractives

In this Note, we present some c₀-semigroup generation results in L_p-spaces for the advection operator submitted to non-contractive boundary conditions covering in particular the classical Maxwell-type boundary conditions. To cite this article: B. Lods, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 655–660.     

Long-Time Behavior of Nonautonomous Fokker-Planck Equations and Cooling of Granular Gases

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases by showing that the Homogenous Cooling State attracts any solution at an algebraic rate. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 778–789, June, 2005.     

Boltzmann Model for Viscoelastic Particles: Asymptotic Behavior,Pointwise Lower Bounds and Regularity

We investigate the long-time behavior of a system of viscoelastic particles modeled with the homogeneous Boltzmann equation. We prove the existence of a universal Maxwellian intermediate asymptotic state with explicit rate of convergence towards it. Exponential lower pointwise bounds and propagation of regularity are also studied. These results can be seen as a generalization of several classical results holding for the pseudo-Maxwellian and constant normal restitution models.