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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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