On classical solution in finite time of BGK-Poisson’s equations

Abstract

BGK model is a collision operator for the evolution of gases which satisfies several fundamental properties. Different collision operators for gas evolutions have been introduced earlier, but none of them could satisfy all the basic physical properties: conservation, positivity, correct exchange coefficients, and entropy inequality. However, contrary to Boltzmann model which has a quadratic form, the BGK model presents a heavy nonlinearity which explains the complexity of this analysis. The existence of solution to the periodic Boltzmann BGK model (Bhatnagar– Gross–Krook) coupled with Poisson’s equation has been proved by the same author. In this paper we are interested to the uniqueness of such solution.