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Bertrand Lods

Publications and source records attributed to Bertrand Lods.

50 records · Page 3Linked to original sources
arXiv2011

Uniqueness in the weakly inelastic regime of the equilibrium state of the inelastic Boltzmann equation driven by a particle bath

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient $α\in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove uniqueness of the stationary solution (with given mass) in the weakly inelastic regime; i.e., for any inelasticity parameter $α\in (α_0,1)$, with some constructive $α_0 \in [0, 1)$. Our analysis is based on a perturbative argument which uses the knowledge of the stationary solution in the elastic limit and quantitative estimates of the convergence of stationary solutions as the inelasticity parameter goes to 1. In order to achieve this we give an accurate spectral analysis of the associated linearized collision operator in the elastic limit. Several qualitative properties of this unique steady state $F_α$ are also derived; in particular, we prove that $F_α$ is bounded from above and from below by two explicit universal (i.e. independent of $α$) Maxwellian distributions.

math.AP↗
arXiv2011

Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime

We revisit our recent contribution (SIAM J. Math. Analysis, 42 (2010) 2499--2538) and give two simpler proofs of the so-called Haff's law for granular gases (with non-necessarily constant restitution coefficient). The first proof is based upon the use of entropy and asserts that Haff's law holds whenever the initial datum is of finite entropy. The second proof uses only the moments of the solutions and holds in some weakly inelasticity regime which has to be clearly defined whenever the restitution coefficient is non-constant

math.AP↗
arXiv2010

Free cooling and high-energy tails of granular gases with variable restitution coefficient

We prove the so-called generalized Haff's law yielding the optimal algebraic cooling rate of the temperature of a granular gas described by the homogeneous Boltzmann equation for inelastic interactions with non constant restitution coefficient. Our analysis is carried through a careful study of the infinite system of moments of the solution to the Boltzmann equation for granular gases and precise Lp estimates in the selfsimilar variables. In the process, we generalize several results on the Boltzmann collision operator obtained recently for homogeneous granular gases with constant restitution coefficient to a broader class of physical restitution coefficients that depend on the collision impact velocity. This generalization leads to the so-called L1-exponential tails theorem. for this model.

math.AP↗
arXiv2009

On perturbed substochastic semigroups in abstract state spaces

The object of this paper is twofold: In the first part, we unify and extend the recent developments on honesty theory of perturbed substochastic semigroups (on $L^{1}(μ)$-spaces or noncommutative $L^{1}$ spaces) to general state spaces; this allows us to capture for instance a honesty theory in preduals of abstract von Neumann algebras or subspaces of duals of abstract $C^{\ast }$-algebras. In the second part of the paper, we provide another honesty theory (a semigroup-perturbation approach) independent of the previous resolvent-perturbation approach and show the equivalence of the two approaches. This second viewpoint on honesty is new even in $L^{1}(μ)$ spaces. Several fine properties of Dyson-Phillips expansions are given and a classical generation theorem by T. Kato is revisited.

math.AP↗
arXiv2009

A new approach to transport equations associated to a regular field: trace results and well-posedness

We generalize known results on transport equations associated to a Lipschitz field $\mathbf{F}$ on some subspace of $\mathbb{R}^N$ endowed with some general space measure $μ$. We provide a new definition of both the transport operator and the trace measures over the incoming and outgoing parts of $\partial Ω$ generalizing known results from the literature. We also prove the well-posedness of some suitable boundary-value transport problems and describe in full generality the generator of the transport semigroup with no-incoming boundary conditions.

math.AP↗
arXiv2008

Spectral analysis of transport equations with bounce-back boundary conditions

We investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from M. Sbihi \cite{Sbihi}, we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all $L^p$-spaces with $1<p<\infty$.

math.AP↗
arXiv2008

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.

math.AP↗
arXiv2008

Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models

We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases.

math.AP↗
arXiv2007

Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in $L^1(\R \times \R,\d \x \otimes \d\v)$ associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.

math.AP↗
arXiv2004

Semigroup generation properties of streaming operators with non--contractive boundary conditions

We present $c_0$--semigroup generation results for the free streaming operator with abstract boundary conditions. We recall some known results on the matter and establish a general theorem. We motivate our study with a lot of 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 non--local boundary conditions involved in transport--like equations from population dynamics.

math.AP↗
arXiv2004

Long time behavior of non-autonomous Fokker-Planck equations and the 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 as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.

cond-mat.soft↗
arXiv2003

The 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, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.

cond-mat.stat-mech↗