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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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An L2 to L∞ Framework for the Landau Equation
Consider the Landau equation with Coulomb potential in a periodic box. We develop a new L2 to L∞ framework to construct global unique solutions near Maxwellian with small L∞ norm. The first step is to establish global L2 estimates with strong velocity weight and time decay, under the assumption of L∞ bound, which is further controlled by such L2 estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv :1505.04608 (2015)). The second step is to employ estimates in Sp spaces to control velocity derivatives to ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur in arXiv :1506.01908 (2015), Imbert and Mouhot in arXiv :1505.04608 (2015)).
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- Award ID(s):
- 1810868
- PAR ID:
- 10157275
- Date Published:
- Journal Name:
- Peking mathematical journal
- ISSN:
- 2096-6075
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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