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Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting. (English) Zbl 1330.76124
Summary: 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 [R. J. Alonso, Indiana Univ. Math. J. 58, No. 3, 999–1022 (2009; Zbl 1168.76047)] 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 [M. P. Gualdani, S. Mischler and C. Mouhot, “Factorization for non-symmetric operators and exponential H-theorem”, Preprint, arXiv:1006.5523] 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 [S. Mischler and C. Mouhot, Discrete Contin. Dyn. Syst. 24, No. 1, 159–185 (2009; Zbl 1160.76042)] in a spatially homogeneous framework. Concerning the case of a non-constant coefficient, this kind of results is new and we use results on steady states of the linearized equation from R. J. Alonso and B. Lods [Commun. Math. Sci. 11, No. 4, 851–906 (2013; Zbl 1302.76156)].

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76T25 Granular flows
47H20 Semigroups of nonlinear operators
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D05 Statistical mechanics of gases
74E20 Granularity
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
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