We establish existence of finite energy weak solutions to the kinetic FokkerPlanck 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
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 longoutstanding 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 lowregularity 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 more »
 Award ID(s):
 1810868
 Publication Date:
 NSFPAR ID:
 10157273
 Journal Name:
 Archive for Rational Mechanics and Analysis
 Volume:
 236
 Issue:
 3
 Page Range or eLocationID:
 1389–1454
 Sponsoring Org:
 National Science Foundation
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