Abstract
Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv 2), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA 1 exp[−B 1(t)v 2] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.
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Bobylev, A.V. Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems. J Stat Phys 88, 1183–1214 (1997). https://doi.org/10.1007/BF02732431
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DOI: https://doi.org/10.1007/BF02732431