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We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the \begin{document}$$ S_p $$\end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces \begin{document}$$ S_p $$\end{document} and anisotropic Hölder spaces for such weak solutions. Such \begin{document}$$ S_p $$\end{document} regularity leads to the uniqueness of weak solutions.
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The Landau Equation with the Specular Reflection Boundary Condition.
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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- Award ID(s):
- 1810868
- PAR ID:
- 10157273
- Date Published:
- Journal Name:
- Archive for Rational Mechanics and Analysis
- Volume:
- 236
- Issue:
- 3
- Page Range / eLocation ID:
- 1389–1454
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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