Boltzmann's kernel and the spatially homogeneous Boltzmann equation (Q2778007)

The author presents a comprehensive review and classification of the properties of various spatially homogeneous (nonlinear) Boltzmann collision operators in the context of their applicability to the problems of existence, uniqueness, smoothness, and large time behavior of solutions. NEWLINENEWLINEFor the inverse power potentials, Boltzmann's cross section kernel has a singularity that makes it difficult (although not impossible) to write the Boltzmann operator in a traditional way as the difference between the gain and loss terms. After H.Grad, the above problem was remedied by introduction of the so called angular cutoff in Boltzmann's cross section kernel. Thus, the author considers both the cutoff and non cutoff Boltzmann collision kernels, and includes a series of results depending on the softness (or hardness) of the potential.NEWLINENEWLINEIn addition, the author presents various results for an interesting variant of the Boltzmann collision kernel, the Fokker-Planck-Landau form. While the angular cutoff in the Boltzmann collision kernel removes most of the so called grazing (long range) collisions, the Fokker-Planck-Landau kernel is obtained by retaining (in the limit) only the grazing collisions in the cross section kernel of the Boltzmann operator. In the case of the Coulomb potential the resulting equation is known as the Landau equation for weakly coupled gases and is closely related the Balescu-Lenard equation. NEWLINENEWLINEThe comprehensive approach of the author in presenting the results for the spatially homogeneous Boltzmann equation should be helpful in search for similar results in various spatially inhomogeneous problems of the kinetic theory of gases and fluids.