Linearized quantum and relativistic Fokker-Planck-Landau equations
نویسنده
چکیده
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 rst 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 eeects have been taken into account. We apply Sobolev-Poincar e inequalities to establish the spectral gap of the linearized operators. Furthermore, The present study permits the precise knowledge of the behavior 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 neighborhood of 0 are then investigated. This study is a rst natural step when one looks for solutions near equilibrium and their hy-drodynamic limit for the full nonlinear problem in all space in the spirit of several works 3] 6]]20] 2] on the nonlinear Boltzmann equation.
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