Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff

We consider the 3-dimensional spatially homogeneous Boltzmann equation, which describes the evolution in time of the velocity distribution in a gas, where particles are assumed to undergo binary elastic collisions. We consider a cross section bounded in the relative velocity variable, without angular cutoff, but with a moderate angular singularity. We show that there exists at most one weak solution with finite mass and momentum. We use a Wasserstein distance. Although our result is far from applying to physical cross sections, it seems to be the first one which deals with cross sections without cutoff for non Maxwellian molecules.