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The Landau Equation with the Specular Reflection Boundary Condition

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Title
The Landau Equation with the Specular Reflection Boundary Condition
Author(s)
Guo, Y; Hwang, HJ; JinWoo Jang; Ouyang, ZM
Publication Date
2020-06
Journal
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, v.236, no., pp.1389 - 1454
Publisher
SPRINGER
Abstract
The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker-Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194-2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183-233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253-295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332-354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem. © Springer-Verlag GmbH Germany, part of Springer Nature (2020)
URI
https://pr.ibs.re.kr/handle/8788114/7120
ISSN
0003-9527
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
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