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Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates
Archive for Rational Mechanics and Analysis ( IF 2.6 ) Pub Date : 2021-02-16 , DOI: 10.1007/s00205-021-01613-y
Thomas Chen , Ryan Denlinger , Nataša Pavlović

We provide a new analysis of the Boltzmann equation with a constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is \(L^2_{x,v}\); we prove the global well-posedness and a version of scattering, assuming that the data \(f_0\) is sufficiently smooth and localized, and the \(L^2_{x,v}\) norm of \(f_0\) is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision “gain” term in Boltzmann’s equation, combined with a novel application of the Kaniel–Shinbrot iteration.



中文翻译:

基于双线性时空估计的玻尔兹曼方程的小数据全局适定性

我们提供了在两个空间维中具有恒定碰撞核的Boltzmann方程的新分析。缩放比例严格的Lebesgue空间为\(L ^ 2_ {x,v} \) ; 我们证明了全球适定性和散射的一个版本,假设数据\(F_0 \)足够光滑,并且局部的,以及\(L ^ 2_ {X,V} \)的范数\(F_0 \)是足够小。该证明依赖于对Boltzmann方程中的碰撞“增益”项的新的缩放临界双线性时空估计,并结合了Kaniel–Shinbrot迭代的新颖应用。

更新日期:2021-02-16
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