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 (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) 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 $L^\infty_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff 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 $L^\infty_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(\Omega)$ 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.

中文翻译：

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朗道方程和非截止玻尔兹曼方程的全局温和解

本文证明了包括库仑势的朗道方程和无角截止的玻尔兹曼方程的小幅度全局时间唯一温和解的存在。自从著名的著作 (Guo, 2002) 和 (Gressman-Strain-2011, AMUXY-2012) 在平滑 Sobolev 空间中构建经典解，特别是在空间变量中是规则的，它仍然是一个悬而未决的问题在 $L^\infty_{x,v}$ 框架中获得全局解，类似于 (Guo-2010) 中的框架，用于在一般有界域中具有截止的 Boltzmann 方程。一个主要的困难来自于非截止玻尔兹曼方程和朗道方程中传输算子和速度扩散型碰撞算子之间的相互作用；另一个主要困难是可能形成奇点以解决边值问题。在目前的工作中，我们引入了一个新的空间变量具有低规律性的函数空间，以在空间域是环面或具有边界的有限通道的情况下处理问题。对于后一种情况，考虑流入边界条件或镜面反射边界条件。函数空间的一个重要性质是分布函数的速度和时间的 $L^\infty_T L^2_v$ 范数在空间变量中的维纳代数 $A(\Omega)$ 中。除了在这些函数空间中构建全局解之外，我们还研究了硬势和软势解的长时间行为，