论文标题
Boltzmann方程式带有截止的Rutherford散射横截面附近的Maxwellian
Boltzmann equation with cutoff Rutherford scattering cross section near Maxwellian
论文作者
论文摘要
著名的卢瑟福差速器横截面，用$dΩ/dσ$表示，对应于与库仑电势的两体相互作用。它导致动量转移（或传输横截面）的对数分歧，该动量传递由$ \ int _ {{{\ Mathbb s}^2}（1- \cosθ）\ frac {dΩ} {dσ} {dσ}dσ\ sim \ sim \ sim \ int_0^phim \ sim \ int_0^pin_-int_0^pintious。 $$这里$θ$是散射事件中的偏差角。由于筛选效果，从物理上可以假设$θ_{\ min} $是最小角度的数量级，仍然可以将散射视为库仑散射。在偏离角度上的$θ\geqθ_ {\ min} $下，L. D. Landau在\ cite {landau1936transport}中得出了一个新方程，用于弱相互作用的气体，该气体现在称为fokker-planck-planck-planck-landau或landau或landau或landau或landau或landau或landau或landau或landau或landau。在目前的工作中，我们建立了一个统一的框架，以证明Landau在\ cite {landau1936transport}中的正式推导和在近距离平衡状态中提出的所谓的Landau近似近似问题。正是（i）。我们证明了玻尔兹曼方程的全球良好性，其截止卢瑟福横截面可能是相对速度和偏差角的最奇异内核。 （ii）。我们证明了以对数准确性对Boltzmann和Landau方程的解决方案之间的全球时间误差估计，这与著名的库仑对数一致。这些结果证明的关键成分包括对线性化玻尔兹曼碰撞算子的完整施加性估计，均匀的光谱间隙估计值和一种新颖的线性质量法方法。
The well-known Rutherford differential cross section, denoted by $ dΩ/dσ$, corresponds to a two body interaction with Coulomb potential. It leads to the logarithmically divergence of the momentum transfer (or the transport cross section) which is described by $$\int_{{\mathbb S}^2} (1-\cosθ) \frac{dΩ}{dσ} dσ\sim \int_0^π θ^{-1}dθ. $$ Here $θ$ is the deviation angle in the scattering event. Due to screening effect, physically one can assume that $θ_{\min}$ is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff $θ\geq θ_{\min}$ on the deviation angle, L. D. Landau derived a new equation in \cite{landau1936transport} for the weakly interacting gas which is now referred to as the Fokker-Planck-Landau or Landau equation. In the present work, we establish a unified framework to justify Landau's formal derivation in \cite{landau1936transport} and the so-called Landau approximation problem proposed in \cite{alexandre2004landau} in the close-to-equilibrium regime. Precisely, (i). we prove global well-posedness of the Boltzmann equation with cutoff Rutherford cross section which is perhaps the most singular kernel both in relative velocity and deviation angle. (ii). we prove a global-in-time error estimate between solutions to Boltzmann and Landau equations with logarithm accuracy, which is consistent with the famous Coulomb logarithm. Key ingredients into the proofs of these results include a complete coercivity estimate of the linearized Boltzmann collision operator, a uniform spectral gap estimate and a novel linear-quasilinear method.
