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Diffusive limit of a kinetic model for cometary flows. (English) Zbl 1181.82055
A kinetic equation with a relaxation time model for wave-particle collisions is considered, motivated by the analysis of cometary flows and cosmic-ray transport in astrophysical plasmas [L. L. Williams and J. R. Jokipii, Astrophys. J. 417, 725–734 (1993)]. Similarly to the BGK-model of gas dynamics, it involves a projection onto the set of equilibrium distributions isotropic around the velocity. This equation depends nonlinearly on the moments of the distribution function. In the two first sections of the paper, the authors provide a short and useful presentation of the relevant properties of this equation and on the mathematical results obtained about it since its proposal in 1996. In Section 3, the macroscopic limit is formally computed using diffusive and low Mach number scaling. The result is a generalization of the incompressible Navier-Stokes equations, where the momentum equations are coupled to a diffusive equation for an energy distribution function. In Section 4, an entropy inequality and some other properties of the limiting system are discussed, and a simplified version is obtained by a moment approximation around a local Maxwellian energy distribution. The resulting system can be related to a low Mach number model with constant viscosity of fluid mechanics, which already appeared in the literature [P. Embid, Commun. Partial Differ. Equ. 12, 1227–1283 (1987; Zbl 0632.76075)]. The authors discuss the difficulties that hinder to extend to general energy distributions the weak existence theory known for this equation [P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)]. Finally, in Section 5, for a linearized version around global equilibrium corresponding to Stokes flow for the divergence-free set of the velocity and a decoupled nonlocal equation for the energy distribution, an existence result for initial value problems is proved by using the entropy inequality derived in the paper.

MSC:
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82D05 Statistical mechanical studies of gases
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