Landau damping
Clément Mouhot, Cédric Villani
在本文中，我们提出了最近的工作hal-00376547/arXiv：0904.2760的主要结果，这是第一次在非线性背景下建立朗道阻尼。
In this note we present the main results from the recent work hal-00376547/arXiv:0904.2760, which for the first time establish Landau damping in a nonlinear context.
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On Landau damping
在朗道阻尼理论中，如何超越线性化研究一直是一个长期存在的问题。本文建立了解析正则的指数朗道阻尼。阻尼现象被重新解释在动力学和空间变量之间的规律性转移，而不是能量交换方面;相混合是驱动机制。分析涉及新的家庭的分析规范，测量规律性的自由输运方程的解决方案相比，新的功能不等式，控制非线性回波，尖锐的散射估计，和牛顿近似方案。我们的结果适用于任何潜在的不超过库仑或牛顿相互作用奇异的极限情况下，包括特定的技术努力。作为一个副结果，非线性Vlasov方程的齐次平衡点的稳定性建立在尖锐的假设。我们指出了与KAM理论的强类比，并讨论了物理意义。
In this paper, we consider the nonlinear Vlasov-Poisson equations in a weakly collisional regime and study the linear Boltzmann collision operator. We prove that Landau damping still occurs in this case.
We show that the linearized Vlasov-Poisson equations around self-similar non-homogeneous states near zero contain the full plasma echo mechanism, yielding Gevrey 3 as a critical stability class. Moreover, here Landau damping may persist despite blow-up: We construct a critical Gevrey regularity class in which the force field converges in $L^2$. Thus, on the one hand, the physical phenomenon of Landau damping holds. On the other hand, the density diverges to infinity in Sobolev regularity. Hence, ``strong damping'' cannot hold.
Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions however approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when Vlasov equation is reduced to the evolution equations for density and kinetic energy…
We prove Landau damping for the collisionless Vlasov equation with a class of $L^1$ interaction potentials (including the physical case of screened Coulomb interactions) on $\mathbb R^3_x \times \mathbb R^3_v$ for localized disturbances of an infinite, homogeneous background. Unlike the confined case $\mathbb T^3_x \times \mathbb R_v^3$, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from zero, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an add…
Landau damping in Sobolev spaces for the Vlasov-HMF model
我们考虑Vlasov-HMF（Hamiltonian Mean-Field）模型。我们认为解决方案在一个小的Sobolev附近的空间均匀的状态满足线性化的稳定性标准（彭罗斯标准）。我们证明了这些解对一个修正态的散射行为，这意味着非线性朗道阻尼效应的多项式阻尼率。
Landau damping: paraproducts and Gevrey regularity
我们给出了一个新的、更简单的证明，证明了T^d上的非线性朗道阻尼满足Gevrey-1/s正则性（s > 1/3），这与Mouhot和Villani在朗道阻尼的原始证明[Acta Mathematica 2011]中的形式分析所预测的正则性要求相匹配。我们的证明以一种新颖的方式结合了Landau阻尼的原始证明和2D Euler中无粘阻尼的证明[arXiv：1306.5028]。在2D欧拉的工作中，我们使用paraproduct分解和控制正则性损失，以取代牛顿迭代方案中采用的原始证明。我们执行的时间响应估计从原来的证明，以控制等离子体回波，并将它们耦合到能量估计的分布函数的风格的工作在2D欧拉。
In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. W…
The Kuramoto model is a mean-field model for the synchronisation behaviour of oscillators, which exhibits Landau damping. In a recent work, the nonlinear stability of a class of spatially inhomogeneous stationary states was shown under the assumption of analytic regularity. This paper proves the nonlinear Landau damping under the assumption of Sobolev regularity. The weaker regularity required the construction of a different more robust bootstrap argument, which focuses on the nonlinear Volterra equation of the order parameter.
Formalization of Landau damping in the Vlasov--Poisson equations in Lean
本文在 Lean 4 中形式化了 Mouhot 和 Villani 关于非线性朗道阻尼的定理，覆盖所有 Gevrey 指标 $s>1/3$ 的小背景情况。
In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter $\eps$ in front of the collision operator which will tend to $0$. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter $\eps$ as it goes to $0$.
In this paper, we prove the nonlinear asymptotic stability of the Penrose-stable equilibria among solutions of the $2d$ Vlasov-Poisson system with massless electrons.
On nonlinear Landau damping and Gevrey regularity
本文研究了环面上Vlasov-Poisson方程的非线性Landau阻尼问题，证明了在大于Gevrey 3的正则类中关于$ε$一致的非线性稳定性，并构造了展现非线性Landau阻尼的Sobolev正则初始数据族。
A new proof of nonlinear Landau damping for the 3D Vlasov-Poisson system near Poisson equilibrium
本文针对三维弗拉索夫-泊松系统近泊松平衡态的非线性朗道阻尼问题，基于已有基础工作，通过精确衰减估计与新颖分解技术提供简化证明，揭示扰动密度的类自由输运行为，增进对无约束环境中朗道阻尼的理解。
Nonlinear Landau damping for the two-species screened Vlasov-Poisson system with large initial distributions
研究两物种屏敝泊松系统在大初始分布下的非线性兰道阻尼，证明全局强解的存在唯一性及Penrose稳定平衡态的时间渐进行为，确定净电荷密度的最优衰减率t^{-d}。
For the Vlasov-Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei, {\it J. Stat. Phys.}, 92:301-323, 1998) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the deterministic case. The main idea is to generalize the deterministic contraction argument to more complicated function spaces to estimate derivatives in space, velocity and random variables. This result suggests that the random space regularity can p…
We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this "scattering problem", closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.
Landau damping and the long-time collisionless limit of the Vlasov-Poisson-Landau Equation
本文研究小碰撞频率下弗拉索夫-泊松-朗道方程的朗道阻尼与无碰撞极限，提出新方法证明解的一致收敛性，推广相关结果并提供近物理侧证明。
Linear Landau damping for a two-species Vlasov-Poisson system for electrons and ions
本文研究电子与离子两物种Vlasov-Poisson系统在Penrose稳定平衡态附近的线性Landau阻尼，推广了单物种结果，证明电场能量指数衰减。
On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem
本文证明了一维空间中Vlasov-Poisson方程的一大类周期解的存在性，这些解随t的无穷大指数衰减。朗道阻尼问题的线性化形式的指数衰减是众所周知的，并且在[4]中已经证明了Vlasov-Poisson系统的一类解的指数衰减，其行为渐近地表现为自由流解并且在速度空间中足够平坦。本文的结果扩大了可能的渐近极限类，用线性化问题的稳定性条件代替了文[4]中的胖条件。
In this paper, we give an elementary proof of the nonlinear Landau damping for the Vlasov-Poisson system near Penrose stable equilibria on the torus $\mathbb{T}^d \times \mathbb{R}^d$ that was first obtained by Mouhot and Villani in \cite{MV} for analytic data and subsequently extended by Bedrossian, Masmoudi, and Mouhot \cite{BMM} for Gevrey-$γ$ data, $γ\in(\frac13,1]$. Our proof relies on simple pointwise resolvent estimates and a standard nonlinear bootstrap analysis, using an ad-hoc family of analytic and Gevrey-$γ$ norms.
Quantum Landau Damping with Coulomb Repulsion in the Whole Space
本文研究空间均匀稳定平衡附近线性化Hartree方程解的长时间行为，探讨量子朗道阻尼，得到电场对普朗克常数一致的时间衰减性，以及量子密度向经典密度的一致时间收敛性。
In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces on $\mathbb T^n_x \times \mathbb R^n_v $ is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, …
Landau damping in mixed hyperbolic-kinetic systems and thick sprays
本文研究稠密喷雾模型中 Vlasov 方程与正压可压缩 Euler 方程耦合的朗道阻尼现象，证明声波因与近速粒子相互作用而被阻尼或放大，并揭示该机制在双曲守恒律与 Vlasov 方程耦合时自然出现。
Remarks on Landau damping
该研究针对经典Vlasov-Poisson系统的非线性朗道阻尼给出多维度注记，涵盖面向非专业者的解析性框架等内容，还在一类全新的Vlasov-Riesz系统中建立了非线性朗道阻尼，且该类系统包含锐解析空间中的临界Vlasov-Dirac-Benney系统。
Landau damping on expanding backgrounds
本文研究了牛顿宇宙学中膨胀对带电自相互作用等离子体渐进行为的影响，证明在特定膨胀因子下，初始数据满足强Gevrey类时，系统表现出非线性兰道阻尼。
We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $C^n$ ($n\geq 4$). A criterion for linear stability of the uniform stationary state is established which, for basic examples of frequency distributions, is equivalent to the standard condition on the coupling strength in the literature. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate $n$) for every trajectory issued …
Landau damping on the torus for the Vlasov-Poisson system with massless electrons
本文研究带无质量电子的Vlasov-Poisson系统在环面上的非线性Landau阻尼，证明了解析或Gevrey初值下密度和力场指数衰减，将相关结果推广到离子及任意解析非线性耦合情形。
We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (both on the torus and on $\mathbb{R}^3$) linearized around a sufficiently nice, spatially uniform kinetic equilibrium. We find that exponential decay of spatial Fourier modes is impossible under modest symmetry assumptions. However, by assuming the equilibrium and initial data are sufficiently regular functions of velocity for a given wavevector (in particular that they exhibit a kind of Gevrey regularity), we show that it is possible for the mode associated to this waveve…
Weak collision effect on nonlinear Landau damping for the Vlasov-Poisson-Fokker-Planck system
本文研究环面上Vlasov-Poisson-Fokker-Planck系统中弱碰撞对非线性Landau阻尼的影响，针对不同Gevrey指标建立了小初值的全局稳定性与增强耗散，并揭示了相位混合、增强耗散和等离子体回声的作用。
Phase mixing for the Hartree equation and Landau damping in the semiclassical limit
研究Hartree方程在平移不变稳态附近的渐近行为，通过相位混合估计证明密度衰减和散射，提供朗道阻尼的量子类比，并在半经典极限下一致成立。
Nonlinear Landau damping for the unconfined Vlasov-Poisson system in 2D
该论文证明了二维无约束Vlasov-Poisson系统在均匀平衡附近的非线性Landau阻尼，建立了全局存在性、电场衰减估计及散射结果，是首个此类结果。
In these short, rather informal, expository notes I review the current state of the field regarding the mathematics of Landau damping, based on lectures given at the CIRM Research School on Kinetic Theory, November 14--18, 2022. These notes are mainly on Vlasov-Poisson in $(x,v) \in \mathbb T^d \times \mathbb R^d$ however a brief discussion of the important case of $(x,v) \in \mathbb R^d \times \mathbb R^d$ is included at the end. The focus will be nonlinear and these notes include a proof of Landau damping on $(x,v) \in \mathbb T^d \times \mathbb R^d$ in the Vlasov--Poisson equations meant fo…
Nonlinear Landau damping and wave operators in sharp Gevrey spaces
本文在最优加权 Gevrey-3 空间中证明受限 Vlasov–Poisson 系统小扰动解的非线性 Landau 阻尼及相应非线性散射算子的存在性、单射性与 Lipschitz 估计，解决了两个公开问题。
A note on Landau damping of two-species Vlasov-Poisson system
本文借鉴 GNR 方法证明双物种 Vlasov-Poisson 系统的非线性 Landau 阻尼，处理有限电子质量与大离子质量情形，并改进 $G$-泛函为仅含一阶导数的 $L^{d+1}$ 范数。
We prove that in dimensions three and higher the Landau-Lifshitz- Gilbert equation with small initial data in the critical Besov space is globally wellposed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schrodinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.
Nonlinear Landau damping for the Vlasov-Poisson system in $\R^3$: the Poisson equilibrium
研究欧氏空间中弗拉索夫-泊松系统的泊松齐次平衡渐近稳定性，证明其小光滑局域扰动存在全局解且以多项式速率散射到线性解，阐明了该欧氏问题与周期情形朗道阻尼的关键差异。
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