Cercignani-Lampis boundary in the Boltzmann theory

Hongxu Chen

doi:10.3934/krm.2020019

Article Contents

2020, Volume 13, Issue 3: 549-597. Doi: 10.3934/krm.2020019

This issue Previous Article Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions Next Article The Vlasov-Maxwell-Boltzmann system near Maxwellians with strong background magnetic field

Cercignani-Lampis boundary in the Boltzmann theory

Hongxu Chen

Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, USA

Received: June 2019

Revised: December 2019

Published: March 2020

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Abstract

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an $ L^\infty $ estimate. In particular, for the $ L^\infty $ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Moreover, under some constraints on the wall temperature and the accommodation coefficients, we construct a unique steady solution of the Boltzmann equation.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Maxwell boundary condition with $ c = 1/2 $

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Figure 2. C-L boundary condition with $ r_\perp = r_\parallel = 1/2 $

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Figure 3. C-L boundary condition with $ r_\perp = r_\parallel = 1/10 $

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Figure 4. C-L boundary condition with $ r_\perp = r_\parallel = 1/30 $

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