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On the spatially homogeneous Boltzmann equation. (English) Zbl 0946.35075
This interesting paper deals with the spatially homogeneous Boltzmann equation \[ \partial f/\partial t(t,v)= Q(f,f)(t,v),\;(t,v)\in (0,+ \infty) \times \mathbb{R}^3,\;f(0,v)= f_0(v),\;v\in\mathbb{R}^3, \] where \(f(t,v)\) is a nonnegative function, which describes the time evolution of the distribution of particles, which move with velocity \(v\), and \(Q(f,f)(t,v)\) is the collision operator. The main result is that for any initial data in \(L^1_2(\mathbb{R}^3)\) with finite mass and energy, there exists a unique solution \(f\in C([0, +\infty)\); \(L^1_2 (\mathbb{R}^3))\) for which the same two quantities are conserved. Here \(L^1_2 (\mathbb{R}^3)\) denotes the space of all functions \(f\) such that \(\|f\|_{1,s}= \int_{\mathbb{R}^3} f(v)(1+|v|^2)dv\) is bounded. Another interesting statement is that any solution satisfying certain bounds on moments of order \(s<2\) must necessarily have bounded energy. A time discretization of the Boltzmann equation with \(f_0\in L^1_s (\mathbb{R}^3)\) \((s\geq 2)\) is considered as well. Some estimates of the rate of convergence for the explicit and implicit Euler schemes are given. Two auxiliary results are of independent interest: a sharpened form of the Povzner inequality, and a regularity result for an iterated gain term.

MSC:
35Q35 PDEs in connection with fluid mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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