The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.

中文翻译：

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具有多项式衰减摄动的非截止玻耳兹曼方程

当初始数据是整体Maxwellian的小扰动并在速度变量中代数衰减时，考虑不使用角度截止的Boltzmann方程。我们在微扰和强角奇异性的摄动框架中获得了适定性理论。证明中的三个主要因素是矩传播，线性算子的谱隙以及当初始数据位于具有负索引的Sobolev空间中时线性算子的正则化作用。精心设计的伪微分算子在捕获正则化效应中起着核心作用。此外，