On some related non homogeneous 3D Boltzmann models in the non cutoff case
The aim of this paper is to study the following three models: modified Boltzmann equation, generalized Boltzmann equation, and the Dirac Boltzmann model. The models are non-homogeneous, and the collision operator cannot be splitted into the usual gain and loss terms in view of the high singularity of the collision cross sections. The analysis is performed without assuming Grad's angular cutoff hypothesis on the cross sections. In the modified Boltzmann model the collision operator is regularized by dividing the zero moment of the solution, and the author proves existence results in the distribution sense. Then the existence of a weak solution is proved for the Dirac Boltzmann model. The final chapter of the paper deals with the collision operator (generalized Boltzmann operator), which is represented as an integral of usual Boltzmann collision term. In all the cases the author mollifies the collision kernel in the neighborhood of approximately equal velocities.
high singularity of the collision cross sections
scientific article; zbMATH DE number 1590229