论文标题
泰勒分散和相混合在整个空间的非切割玻尔兹曼方程中
Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space
论文作者
论文摘要
在本文中，我们描述了在弱碰撞极限（即，无限的knudsen number $ 1/ν\ to \ infty $）中，在整个空间的全球Maxwellian背景附近具有柔软电位的非切割玻尔兹曼方程的长期行为。具体而言，我们证明，对于足够小的初始数据（与Knudsen数字无关），该解决方案显示了由运输运算符$ v \ cdot \ nabla_x $的相混合/分散效应引起的几种动态，以及与单数碰撞算子的相互作用。对于$ x $ -wavenumbers $ k $，带有$ | k | \ggν$，人们看到了增强的耗散效果，其中特征衰减的时间尺度加速到$ o（1/ν^{\ frac {1} {1} {1} {1+2S}} {1+2S}}}} | (0,1]$ is the singularity of the kernel ($s=1$ being the Landau collision operator, which is also included in our analysis); for $|k|\ll ν$, one sees Taylor dispersion, wherein the decay is accelerated to $O(ν/|k|^2)$. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $ρ$，这些界限意味着$（t \ nabla_x）^βρ$在$ l^\ infty_x $中的几乎均匀的$ν$衰减，这是由于landau抑制和分散衰减而导致的。
In this paper, we describe the long-time behavior of the non-cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (i.e. infinite Knudsen number $1/ν\to \infty$). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator $v \cdot \nabla_x$ and its interplay with the singular collision operator. For $x$-wavenumbers $k$ with $|k|\ggν$, one sees an enhanced dissipation effect wherein the characteristic decay time-scale is accelerated to $O(1/ν^{\frac{1}{1+2s}} |k|^{\frac{2s}{1+2s}})$, where $s \in (0,1]$ is the singularity of the kernel ($s=1$ being the Landau collision operator, which is also included in our analysis); for $|k|\ll ν$, one sees Taylor dispersion, wherein the decay is accelerated to $O(ν/|k|^2)$. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $ρ$, these bounds imply almost-uniform-in-$ν$ decay of $(t\nabla_x)^βρ$ in $L^\infty_x$ due to Landau damping and dispersive decay.
