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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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This content will become publicly available on March 1, 2026
Uniqueness of the Ginzburg–Rallis model: the p-adic case
We prove the uniqueness of the Ginzburg–Rallis models over p-adic local fields of characteristic zero, which completes the local uniqueness problem for the Ginzburg–Rallis models, starting from the work of Nien (Models of representations of general linear groups over p-adic fields, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-University of Minnesota, 2006) that proves the non-split case, and the work of Jiang et al. (Trans Am Math Soc 363(5): 2763–2802, 2011) that proves the general case over Archimedean local fields. Our proof extends the strategy of [16] to the p-adic case with the help of the refined structure of the wavefront sets of z-finite distributions as developed by Aizenbud et al. (Adv Math 285:1376–1414,2015).
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- Award ID(s):
- 2200890
- PAR ID:
- 10614799
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 2522-0160
- Page Range / eLocation ID:
- 1-46
- Subject(s) / Keyword(s):
- Representations of p-adic general linear groups, Ginzburg–Rallis model, Multiplicity one theorem, z-Finite distributions, Wavefront set
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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