Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

• Published in: Communications on pure and applied mathematics 2021-05, Vol.74 (5), p.932-1020
• Main Authors: Duan, Renjun, Liu, Shuangqian, Sakamoto, Shota, Strain, Robert M.
• Format: Article
• Language: English
• Subjects: Boltzmann transport equation; Boundary conditions; Boundary value problems; Coulomb potential; Distribution functions; Domains; Function space; Mathematical analysis; Operators (mathematics); Regularity; Sobolev space; Specular reflection; Toruses; Variables
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21920

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc.