Landau 方程 (1936) 在具有物理边界条件的一般有界域中的存在性和稳定性是一个长期悬而未决的开放问题。这项工作证明了具有库仑势的朗道方程在具有镜面反射边界条件的一般光滑有界域中的全局稳定性，用于麦克斯韦平衡态的初始扰动。这项工作的亮点还来自对初始分布所做的低正则性假设。这项工作概括了最近在周期框中的 Landau 方程的全局稳定性结果（Kim 等人在 Peking Math J，2020 年）。我们的方法包括福克-普朗克方程的适定性理论的推广（Hwang 等人 SIAM J Math Anal 50(2):2194-2232, 2018; Hwang 等人 Arch Ration Mech Anal 214(1):183 –233, 2014) 和将边界值问题扩展到整个空间问题，以及使用 De Giorgi-Nash-Moser 理论的最新扩展来处理动力学 Fokker-Planck 方程 (Golse 等人。Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) 和 Morrey 估计值 (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) 以进一步控制速度导数，从而确保唯一性。我们的方法为初始边界值问题的 Landau 理论中的碰撞碰撞提供了新的理解。1996）进一步控制速度导数，确保唯一性。我们的方法为初始边界值问题的 Landau 理论中的碰撞碰撞提供了新的理解。1996）进一步控制速度导数，确保唯一性。我们的方法为初始边界值问题的 Landau 理论中的碰撞碰撞提供了新的理解。
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.
