Hölder regularity of the Boltzmann equation past an obstacle

Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in Cx,v0,12−$C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elast...

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Published inCommunications on pure and applied mathematics Vol. 77; no. 4; pp. 2331 - 2386
Main Authors Kim, Chanwoo, Lee, Donghyun
Format Journal Article
LanguageEnglish
Published New York John Wiley and Sons, Limited 01.04.2024
Subjects
Barriers
Boltzmann transport equation
Boundary conditions
Mathematical analysis
Regularity
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Summary:Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in Cx,v0,12−$C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].
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content type line 14
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.22167