On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions. (English) Zbl 0918.35136

Summary: We are interested in the influence of grazing collisions, with deflection angle near \(\pi/2\), in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-\(s\)th-power force laws, and we deal with general initial data with bounded mass, energy, and entropy.
First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, for \(s\geq 7/3\). Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value \(\pi/2\). For very soft interactions, \(2< s<7/3\), the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term.


35Q72 Other PDE from mechanics (MSC2000)
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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