Oscillating Maxwellians: Exact solutions for the Boltzmann equation

Abstract

Abstract For two classes of non linear Boltzmann equations, when the particles are of the Maxwell type and external forces are present, we present explicit distributions solutions. First, for the homogeneous one dimensional Kac model for which momentum conservation has been dropped, we include linear velocity forces and uniform source terms. Second, for the homogeneous and inhomogeneous Boltzmann equations of dimension d > 1, for which both energy and momentum conservations hold, we include either spatially dependent forces or velocity forces plus uniform sources. For both classes there exist examples of distributions which relax towards oscillating Maxwellians and other towards absolute Maxwellians. We give the complete proofs for the Kac model and present results for the d > 1 dimensional models.