On Classical Solution in Finite Time of BGK-Poisson’s Equations

DOI: 10.5176/2251-1911_CMCGS14.14

Authors: Slim Ben Rejeb

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, entropy inequality. However contrary to Boltzmann model which has a quadratic form, the BGK model, presents a heavy nonlinearity who 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 (see [1]). In this paper we are interested to the uniqueness of such solution.

Keywords: Kinetic equations, BGK model, Maxellian, Boltzmann’s equations, Plasma’s physics, Schauder’s fixed point, Poisson’s equation, Fluid equations