Mathematical treatment of the homogeneous Boltzmann equation for Maxwellian molecules in the presence of singular kernels

This paper proves the existence of weak solutions to the the spatially homogeneous Boltzmann equation for Maxwellian molecules, when the initial data are chosen from the space of all Borel probability measures on R^3 with finite second moments, and the (angular) collision kernel satisfies a very weak cutoff condition. For the equation at issue, the uniqueness of the solution corresponding to a specific initial datum has been recently established in Fournier and Guérin (J Stat Phys 131:749–781, 2008). Finally, conservation of momentum and energy is also proved for these weak solutions, without resorting to any boundedness of the entropy.