Stability of the Maxwell–Stefan System in the Diffusion Asymptotics of the Boltzmann Multi-species Equation

• Published in: Communications in mathematical physics Vol. 382; no. 1; pp. 381 - 440
• Main Authors: Bondesan, Andrea, Briant, Marc
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-02-2021; Springer Nature B.V
• Subjects: Asymptotic properties; Boltzmann transport equation; Classical and Quantum Gravitation; Complex Systems; Fluid dynamics; Fluid flow; Incompressible flow; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Toruses
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-021-03976-5

We investigate the diffusion asymptotics of the Boltzmann equation for gaseous mixtures, in the perturbative regime around a local Maxwellian vector whose fluid quantities solve a flux-incompressible Maxwell–Stefan system. Our framework is the torus and we consider hard-potential collision kernels with angular cutoff. As opposed to existing results about hydrodynamic limits in the mono-species case, the local Maxwellian we study here is not a local equilibrium of the mixture due to cross-interactions. By means of a hypocoercive formalism and introducing a suitable modified Sobolev norm, we build a Cauchy theory which is uniform with respect to the Knudsen number
ε
. In this way, we shall prove that the Maxwell–Stefan system is stable for the Boltzmann multi-species equation, ensuring a rigorous derivation in the vanishing limit
ε
→
0
.