The diffusion approximation of the Boltzmann equation of Maxwellian molecules
The Landau equation describes the evolution of the one-particle density of a weakly coupled gas resp. a plasma. In contrast to the Boltzmann equation it has been studied rigorously only to a small extent. In this paper the Landau equation is derived as the diffusion limit of the Boltzmann equation. In this limit the collision rate increases and the change of velocities decreases in a diffusive scaling. This goes back to the original idea of Landau. It is proved that the corresponding semigroups converge in a suitable sense. The proof uses jump type Markov processes associated with the Boltzmann equation and a diffusion process associated with the Landau equation by showing their corresponding convergence. In two final chapters the time evolution of moments and trend to equilbrium is treated and a one-dimensional analogous model is studied.
scientific article; zbMATH DE number 3852184
scientific article