Bertrand Lods

Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications

AbstractnWe investigate the properties of the collision operator Q 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 Q in an Hilbert space setting, generalizing results from T.xa0Carleman (Publications Scientifiques de l’Institut Mittag-Leffler, vol.xa02, 1957) to granular gases. In the same way, we obtain from this integral representation of Q+ that the semigroup in L1(ℝ3×ℝ3,dx⊗dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991).n

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

Abstract We prove that any subcritical solution to the Becker–Doring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin and Niethammer (2003) [17] . Our approach is based on a careful spectral analysis of the linearized Becker–Doring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted l 1 spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.

FREE COOLING AND HIGH-ENERGY TAILS OF GRANULAR GASES WITH VARIABLE RESTITUTION COEFFICIENT

We prove the so-called generalized Haffs 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.

Entropy dissipation estimates for the linear Boltzmann operator

Abstract We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignanis conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to [27] , [32] . For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.

Uniqueness in the Weakly Inelastic Regime of the Equilibrium State to the Boltzmann Equation Driven by a Particle Bath

We consider the spatially homogeneous Boltzmann equation for inelastic hard-spheres (with constant restitution coefficient

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

alpha in (0,1)

Spectral analysis of transport equations with bounce-back boundary conditions

) 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

On The Spectrum of Absorption Operators With Maxwell Boundary Conditions Arising in Population Dynamics and Transport Theory. A Unified Treatment

alpha in (alpha_0,1)

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

, with some constructive

Trend to equilibrium for the Becker–Döring equations : an analogue of Cercignani’s conjecture

alpha_0 in [0, 1)