Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath

Abstract

ABSTRACT. We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient α ∈ (0, 1)) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which α is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linearised collision operator.

Convergence to self-similarity for ballistic annihilation dynamics

“…Our approach is global in essence. This contrasts with the approach of [23] (see also [11]) where local stability estimates (in which exponential convergence is proven for small perturbations of the equilibrium) are first established and then suitable entropy estimates are used as a tool to pass from local to global stability. Here, even if we fully exploit the spectral properties of the linearized operator and the decay of the associated semigroup, our approach does not rely at all on the study of close-to-equilibrium solutions to (1.14).…”

Inelastic Boltzmann equation driven by a particle thermal bath

“…As a consequence, they obtained the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. Furthermore, their strategy inspired several works in the kinetic theory of granular gases like [1,2,6,7,10]. Inspired by the work of Gualdani et.…”

“…The strategy in this paper consists in combining the main ideas adopted in [22] with the arguments given by [6] and [10]. To develop a Cauchy theory for the equation ( 1), we first study the linearized problem around the equilibrium.…”

On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials

“…Interactions among the particles themselves are neglected, and thus the equation is linear. Various versions of the linear Boltzmann equation are used to model phenomena such as neutron scattering [28,29], radiative transfer [1] and cometary flows [16] (we refer to [13, Chapter XXI] for a detailed presentation of the mathematical theory of linear collisional kinetic equations), and appears in some non-linear models as a background interaction term [4,10,17]. On the other hand, a technical motivation for our results is that inequalities relating the logarithmic entropy to its production rate are interesting by themselves, and are helpful in the study of non-linear models involving a linear Boltzmann term.…”

“…Roughly speaking, the spectral gap properties of both equations (linear and linearised) are now understood in a variety of spaces. The difference in our present approach is that it is based on functional inequalities for the logarithmic entropy, which have their own interest and are more robust when applied to models with mixed linear and non-linear terms [4,10].…”