A weak formulation of the Boltzmann equation based on the Fourier transform

In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral. Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.