Entropy dissipation and moment production for the Boltzmann equation |
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Authors: | Bernt Wennberg |
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Institution: | (1) Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden |
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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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Keywords: | Boltzmann equation entropy production Povzner inequality moments |
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