Linearized quantum and relativistic Fokker-Plack-Landau equations

Abstract

After a recent work on spectral properties and dispersion relations of the linearized classical Fokker-Planck-Landau operator [8], we establish in this paper analogous results for two more realistic collision operators: The first one is the Fokker-Planck-Landau collision operator obtained by relativistic calculations of binary interactions, and the second is a collision operator (of Fokker-Planck-Landau type) derived from the Boltzmann operator in which quantum effects have been taken into account. We apply Sobolev-Poincare inequalities to establish the spectral gap of the linearized operators. Furthermore, the present study permits the precise knowledge of the behaviour of these linear Fokker-Planck-Landau operators including the transport part. Relations between the eigenvalues of these operators and the Fourier-space variable in a neighbourhood of 0 are then investigated. This study is a first natural step when one looks for solutions near equilibrium and their hydrodynamic limit for the full non-linear problem in all space in the spirit of several works [3, 6, 20, 2] on the non-linear Boltzmann equation.