The Landau Equation with the Specular Reflection Boundary Condition

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.