Entropy dissipation and moment production for the Boltzmann equation

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.