Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions

Marc Briant

doi:10.3934/krm.2017014

Article Contents

2017, Volume 10, Issue 2: 329-371. Doi: 10.3934/krm.2017014

This issue Previous Article Next Article Dispersion relations in cold magnetized plasmas

Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions

Marc Briant

Sorbonne Universités, UPMC Univ. Paris 06/ CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received: June 30, 2015

Revised: June 30, 2016

Published: May 2017

The author was supported by the 150^th Anniversary Postdoctoral Mobility Grant of the London Mathematical Society and by the Division of Applied Mathematics of Brown University

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Abstract

We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo [21] in $L_{x,v}^\infty \left( {{{\left( {1 + |v|} \right)}^\beta }{e^{|v{|^2}/4}}} \right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.

Keywords:

Boltzmann equation,
perturbative theory,
exponential trend to equilibrium,
specular reflection boundary conditions,
Maxwellian diffusion boundary conditions.

Mathematics Subject Classification: Primary: 35Q20, 35B20; Secondary: 76P05.

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References

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