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Cooling process for inelastic Boltzmann equations for hard spheres. I: The Cauchy problem. (English) Zbl 1135.82325
Summary: We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the long-time behaviour, we give conditions for the cooling process to occur or not in finite time.
For Part II see [S. Mischler and C. Mouhot, J. Stat. Phys. 124, No. 2-4, 703–746 (2006; Zbl 1135.82030)].

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35F25 Initial value problems for nonlinear first-order PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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