Abstract
LetH(f/M)=∫flog(f/M)dv be the relative entropy off and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation byD(f)=∫Q(f,f) logf dv. An example is presented which shows that |D(f)/H(f/M)| can be arbitrarily small. This example is a sequence of isotropic functions, and the estimates are very explicitly given by a simple formula forD which holds for such functions. The paper also gives a simplified proof of the so-called Povzner inequality, which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision. That inequality is then used to prove that ∫f(v) |v|s dt<C(t), wheref is the solution of the spatially homogeneous Boltzmann equation. HereC(t) is an explicitly given function dependings and the mass, energy, and entropy of the initial data.
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Wennberg, B. Entropy dissipation and moment production for the Boltzmann equation. J Stat Phys 86, 1053–1066 (1997). https://doi.org/10.1007/BF02183613
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DOI: https://doi.org/10.1007/BF02183613