The Cauchy Problem for Boltzmann Bi-linear Systems: The Mixing of Monatomic and Polyatomic Gases

Journal of Statistical Physics, vol. 191

Abstract

Abstract From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate forp-binomial forms producing sharper estimates for thek-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a $$2^+$$ 2+ moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

Authors 3

1. Ricardo J. Alonso corresponding

   Texas A&M University · Texas A&M University at Qatar

   Affiliation as printed

   Texas A &M, Division of Arts and Sciences, Education City, Doha, Qatar

2. Milana Pavić‐Čolić Aachen Applied and Computational Mathematics

   RWTH Aachen University · University of Novi Sad

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062, Aachen, Germany

   Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000, Novi Sad, Serbia

3. Irene M. Gamba

   The University of Texas at Austin

   Affiliation as printed

   Department of Mathematics and Oden Institute of Computational Engineering and Sciences, University of Texas at Austin, 201 E. 24th Street, 1 University Station (C0200), POB 4.102, Austin, TX, 78712-1229, USA