Entropy dissipation and moment production for the Boltzmann equation

Let H(f\M) = integral f log(f/M) dv be the relative entropy of f and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation by D(f) = integral Q(f,f) log f 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 for D which holds ibr 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 integral f(v)) \v\(s) dt < C(t), where f is the solution of the spatially homogeneous Boltzmann equation. Here C(t) is an explicitly given function depending s and the mass, energy, and entropy of the initial data.