Hydrodynamic limits of a Vlasov-Fokker-Planck equation for granular media
The authors consider the two types of scaling of the Vlasov equation for the phase space density \(f\) with Fokker-Planck term [VFP eq.]. Using the solutions of the zeroth order equations in these scalings, resp., they linearize the mean-field operator which appears in the VFP eq. It is shown that the resulting linearized operators satisfy the Fredholm alternative. This allows to perform the Hilbert or Chapman-Enskog expansions for the solutions of VFP eq. in these scalings. The zeroth and the first order terms of these expansions give the corrections to the conservation law for the macroscopic density and the bulk velocity which are defined by the integrals of the phase space density \(f\).
Hilbert expansion, Chapman-Enskog expansion
scientific article; zbMATH DE number 5004818