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# Communications in Mathematical Sciences

## Volume 5 (2007)

### Number 2

### Mass and energy balance laws derived from High-Field limits of thermostatted Boltzmann equations

Pages: 355 – 382

DOI: http://dx.doi.org/10.4310/CMS.2007.v5.n2.a7

#### Authors

#### Abstract

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.

#### Keywords

Boltzmann equation; Fokker-Planck equation; thermostat; hydrodynamic limit; High-Field limit; hyperbolic balance laws; mass and energy transfer

#### 2010 Mathematics Subject Classification

35J05, 76M28, 82B21, 82B40, 82C21, 82C40, 82C70