Diffusion Approximation and Entropy based Moment Closure for Kinetic Equations

Diffusion Approximation and Entropy-based Moment Closure for Kinetic Equations Jean-Franc¸ois Coulombel1, Franc¸ois Golse2, Thierry Goudon1 1 CNRS & Universite Lille 1, Laboratoire Paul Painleve, UMR CNRS 8524 Cite scientifique, 59655 VILLENEUVE D'ASCQ Cedex, France 2 Laboratoire Jacques-Louis Lions, UMR CNRS 7598 Universite Paris 7, 175 rue du Chevaleret, 75013 PARIS, France E-mails: , , April 5, 2005 Abstract It is a well-known fact that, in small mean free path regimes, kinetic equations can lead to diffusion equations. Besides, kinetic equations can be approached by a closed system of moments equations. In this paper, we are interested in a special closure based on an entropy minimization principle, as introduced earlier by Levermore. We investigate the behavior of the resulting nonlinear hyperbolic system in the diffusive scaling. We first establish various fundamental facts on this system, then we show that the hyperbolic system admits global smooth solutions, and is consistent with the diffusion limit. Similar features are also discussed for a simpler limited flux equation. AMS subject classification: 82C40, 76N15, 35L65, 35Q99 Keywords: Diffusion Approximation, Hyperbolic Systems, Relaxation, Global Smooth Solu- tions, Nonlinear Parabolic Equations 1 Introduction This quite long Introduction is organized as follows.

• collision operator


• solution f?


• intermediate regimes


• smooth solution


• diffusion coefficient


• system admits global


• following claim


• ?1 term