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Mass and energy balance laws derived from high-field limits of thermostatted Boltzmann equations. (English) Zbl 1140.82033

Summary: We derive coupled mass and energy balance laws from a high-field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a high-field equilibrium obtained when balancing the thermostatted field drift operator with the elastic collision operator. To this aim, a hydrodynamic type scaling of the thermostatted Boltzmann equation is used, considering that the leading ‘collision operator’ actually consists of the combination of the thermostatted field operator and of the elastic collision operator. At leading order in the Knudsen number, the resulting model consist of coupled nonlinear first order partial differential equations. We investigate two cases. The first one is based on a one-dimensional BGK-type operator. The second one is three dimensional and concerns a Fokker-Planck collision operator. In both cases, we show that the resulting models are hyperbolic, thereby indicating that they might be appropriate for a use in physically realistic situations.

MSC:

82C70 Transport processes in time-dependent statistical mechanics
35F20 Nonlinear first-order PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82B40 Kinetic theory of gases in equilibrium statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
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