Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting

Abstract

Abstract. In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard spheres. In the weak thermalization regime, i.e. when the diffusion parameter is sufficiently small, we prove existence of global solutions considering the close-to-equilibrium regime as well as the weakly inhomogeneous regime in the case of a constant restitution coefficient. It is the very first existence theorem of global solution in an inelastic "collision regime" (that is excluding [1] where an existence theorem is proven in a near to the vacuum regime). We also study the long-time behavior of these solutions and prove a convergence to equilibrium with an exponential rate. The basis of the proof is the study of the linearized equation. We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces and a result of exponential stability for the semigroup generated by the linearized operator. To do that, we develop a perturbative argument around the spatially inhomogeneous equation for elastic hard spheres and we take advantage of the recent paper [14] where this equation has been considered. We then link the linearized theory with the nonlinear one in order to handle the full non-linear problem thanks to new bilinear estimates on the collision operator that we establish. As far as the case of a constant coefficient is concerned, the present paper largely improves similar results obtained in [20]

Convergence to self-similarity for ballistic annihilation dynamics

“…This is an important contrast with respect to the results obtained so far in the context of granular gases [23,32] for which the rate of convergence to self-similarity is not continuous with respect to the elastic limit. To be more precise, in [23,32], if η ∈ (0, 1) denotes the inelasticity parameter, then the decay to the self-similar profile is O(e −c (1−η) t ) for some explicit c > 0. As a consequence, the elastic limit η → 1 yields no relaxation at all, whereas it is well-known that the solution to the elastic Boltzmann equation converges exponentially fast to equilibrium.…”

“…. , d+ 2) do not play any role in our subsequent analysis which is an important contrast with respect to the analysis performed in [23] and [32]. On this point, it is an interesting open question to determine the sign of the eigenvalues µ i α .…”

“…By virtue of the point 2, the rate of convergence to equilibrium for the rescaled equation is sharp in the sense that it allows to recover, in the limit α → 0, the decay to equilibrium for the Boltzmann equation in O(e −µ⋆t ). We already commented on this point in Remark 1.5 explaining the contrast with the analysis in [23,32]. Let us emphasize at this point that recovering the sharp decay rate is made possible again thanks to the conservative form of the rescaled equation and the method described in point 2 and in the previous paragraph.…”

“…Remark 1.8. The approach to prove the above result is reminiscent to the recent contributions [15,23,32] and consists in a perturbation argument around the elastic limit combined with some abstract enlargement and factorization arguments as developed in [15]. Remark 1.9.…”

Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

“…The stability in L 1 (R 3 x × R 3 v ) under the same assumptions was derived for instance in [33]. Finally the existence and convergence to equilibrium in T 3 x × R 3 v for a diffusively heated, weakly inhomogeneous granular gas was proved in [31].…”