INTERPLAY OF ELASTIC AND INELASTIC SCATTERING OPERATORS IN EXTENDED KINETIC MODELS AND THEIR HYDRODYNAMIC LIMITS: REFERENCE MANUAL

ABSTRACT

In this paper we survey various hydrodynamic/diffusive limits which can occur in the linear kinetic equation describing elastic and inelastic collisions when the involved operators act on different time-scales. The first part of the review is devoted to the explanation to non-experts how to use the compressed Chapman-Enskog procedure to analyse the asymptotic behaviour of a kinetic equation when some physical parameters are small with respect to the others. In the second part it is shown that all the formal steps of the procedure can be mathematically justified in a natural way leading to a rigorous asymptotic theory.

Keywords:

• Inelastic scattering
• Lorentz gas
• Asymptotic analysis
• Abstract Cauchy problems
• AMS Subject Classification:
• 82C40
• 47D06
• 35Q35

ACKNOWLEDGMENTS

A significant part of this paper was prepared when one of the authors (J. Banasiak) visited Dipartimento di Matematica Applicata “G. Sansone” at the Università di Firenze. The support received for this visit from the National Group for Mathematical Physics of the Istituto Nazionale di Alta Matematica (INdAM-GNFM) is highly appreciated.

The work of J. Banasiak was also partly supported by National Reseach Foundation of South Africa.

This work was also partly supported by the Italian Ministery of University (MURST National Project “Problemi matematici delle teorie cinetiche”), by the CNR Special Project “Metodi Matematici in Fluidodinamica Molecolare,” and by the European TMR Network “Asymptotic Methods in Kinetic Theory.”

The authors would like to express their sincere thanks to all the colleagues who, on various occasions, contributed to this paper through stimulating discussions, suggestions and criticism.