SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3093-3113, January 2020.
In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of the Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudodifferential calculus, and, in general, for a specific class of Weyl quantization, the $L^2$ dissipation implies $H^k_n$ dissipation. This kind of estimate is also known as Gårding's inequality.

中文翻译：

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$ H ^ k_n $上的耗散和半群：具有软势的非截止线性化玻尔兹曼算子

SIAM数学分析杂志，第52卷，第3期，第3093-3113页，2020
年1月。在本文中，我们发现具有软势的非截断Boltzmann方程的线性化碰撞算子$ L $生成一个强连续半群。 $ H ^ k_n $，以及$ k，n \ in \ mathbb {R} $。在没有角度截止的玻尔兹曼方程的理论中，加权的Sobolev空间起着基本作用。该证明基于伪微积分，通常，对于特定类别的Weyl量化，$ L ^ 2 $耗散意味着$ H ^ k_n $耗散。这种估计也称为戈丁不等式。