On some related non homogeneous 3D Boltzmann models in the non cutoff case

We study some issues concerning the existence of weak solutions for two Boltzmann like equations: a Modified Boltzmann model and the Boltzmann Dirac model. The analysis of the collision operators rests on suitable decompositions. These are also provided for the Generalised Boltzmann operator. This study is performed without assuming Grad's angular cutoff hypothesis on the cross sections.