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The Landau Equation with the Specular Reflection Boundary Condition

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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.

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Notes

  1. Note that the trace estimate (37) is actually uniform in the iteration process generated by the contraction mapping \({\mathcal {T}}\), and therefore the applicability can be justified.

  2. The authors of this work used spherical-type coordinates to make the map almost globally defined; here we just prefer the standard coordinates for simplicity.

  3. We use the column vector convention in the matrix operation expressions.

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Acknowledgements

Yan Guo’s research is supported in part by NSF Grants DMS-1611695 and DMS-1810868. Hyung Ju Hwang’s research is supported by the Basic Science Research Program through the National Research Foundation of Korea NRF-2017R1E1A1A0 3070105 and NRF-2019R1A5A1028324. Jin Woo Jang’s research is supported by the Korean IBS project IBS-R003-D1.

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Authors and Affiliations

  1. Brown University, Providence, RI, 02912, USA

    Yan Guo & Zhimeng Ouyang

  2. Department of Mathematics, POSTECH, Pohang, 37673, Republic of Korea

    Hyung Ju Hwang

  3. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Republic of Korea

    Jin Woo Jang

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  1. Yan Guo
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Guo, Y., Hwang, H.J., Jang, J.W. et al. The Landau Equation with the Specular Reflection Boundary Condition. Arch Rational Mech Anal 236, 1389–1454 (2020). https://doi.org/10.1007/s00205-020-01496-5

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