The Gevrey hypoellipticity for kinetic equations
Hua Chen, Weixi Li, Chao-Jiang Xu
本文研究了一类线性和半线性动力学方程的弱解的Gevrey正则性，这类方程是空间非齐次Boltzmann方程的无角截断的线性模型.
In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.
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Gevrey hypoellipticity for a class of kinetic equations
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Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators
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Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds
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Smooth and Gevrey Microlocal Hypoellipticity for a Class of Hypocomplex Tube Structures
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The Metivier inequality and ultradifferentiable hypoellipticity
本文推广Métivier关于线性微分算子亚椭圆性的刻画至Denjoy-Carleman超可微类，并讨论超函数解情形及应用。
Global hypoellipticity for a class of time-periodic operators on asymptotically Euclidean manifolds
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Fractional order kinetic equations and hypoellipticity
本文给出了具有分数阶扩散部分的动力学方程模型的亚椭圆估计的简单证明。证明是基于能量估计与F。Bouchut和B. Perthame以前的想法。
On local Gevrey regularity for Gevrey vectors of subelliptic sums of squares -- an elementary proof of a sharp Gevrey Kotake-Narasimhan theorem
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Averaging lemmas and hypoellipticity
本文使用交换子和基本解方法，为纯动力学输运方程建立平均化引理和亚椭圆估计，并在一般非齐次通量条件下推导出正则性性质，应用于平流-扩散方程及双曲守恒律的动力学形式。
The purpose of this paper is to extend the hypocoercivity results for the kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir \cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To remedy this issue, we shall modify the usual $H^k$ norm with certain well-chosen mixed terms and with suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time (c.f. \cite{Herau}), usually referred as Hérau's method, can be employ…
Concentrations in kinetic transport equations and hypoellipticity
利用对相空间的适当分解，我们建立了动力学输运方程解的改进的亚椭圆估计。我们的主要结果表明，满足适当迁移关系$$v\cdot\nabla_x f_λ= (1-Δ_x)^\fracβ{2}(1-Δ_v)^\fracα{2}g_λ$$的有界族$f_λ(x,v)\in L^p$的所有变元的相对紧性可以仅由它在$v$中的紧性来推断.当输运方程的右侧具有非常低的正则性时，例如由于外力场，该方法被引入作为$L^1$中缺乏已知的合适的平均引理的替代。在即将到来的工作中，作者提出了一个重要的应用，这种新的方法来研究流体动力学极限的玻尔兹曼方程与粗糙力场。
Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff
本文研究了无角截断的空间齐次Boltzmann方程的Gevrey正则性。证明了空间齐次Boltzmann方程Cauchy问题的$C^\infty$解的Gevrey正则性在Maxwell衰减下的传播.我们在这里使用的想法是基于Morimoto最近的论文的框架（见Morimoto：J. Oper（2010）1：139-159，DOI：10.1007/s11868-010-0008-z），但是我们扩展了满足$γ+ 2s \in (-1,1)$、$s\in (0,1/2)$的指数$γ$的范围，并且在这种情况下，我们考虑$Φ(v)=|v|^γ$形式的动力学因子，而不是Morimoto之前所做的$\la v \ra ^γ$。
The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are Hölder-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obta…
The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently. We prove an L $\infty$ bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads u…
On Linear Landau Damping for Relativistic Plasmas via Gevrey Regularity
通过对相对论性Vlasov-Poisson系统（环面上和$\mathbb{R}^3$上）在一个足够好的、空间均匀的动力学平衡点附近线性化的研究，我们考察了相对论性等离子体中的朗道阻尼现象.我们发现，指数衰减的空间傅立叶模式是不可能的适度对称性假设下。然而，通过假设平衡和初始数据是足够经常的函数的速度为一个给定的波矢量（特别是，他们表现出一种Gevrey规律性），我们表明，它是可能的模式与此波矢量衰减次指数，如果其幅度超过一定的临界尺寸。我们也给出了一个启发式的论点，为什么不应该期望这样的快速衰减模式与波矢量低于这个阈值。
We study the kinetic Fokker-Planck equation on the whole space with a confining potential. We show quantitative rates of exponential convergence to equilibrium in a well chosen Wasserstein-1 distance. We use the Wasserstein-1 version of Harris's theorem introduced by Hairer and Mattingly. We make use of similarities between hypocoercivity and hypoellipticity in order to use Malliavin calculus to see hypocoercivity for this equation on the level of the SDE.
Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift
研究了具有线性漂移项的亚强制Fokker-Planck方程在全空间上的定态和指数衰减的存在性。对于这类方程，我们首先建立了它的生成元的亚椭圆性和系统的禁闭等价于存在一个唯一的归一化稳态。这两个条件也意味着hypoverravity，即指数收敛的解决方案的平衡。由于标准的熵方法不适用于退化抛物方程，我们开发了一种新的修改后的熵方法（基于修改后的，非退化熵耗散类功能）证明这种指数衰减相对熵（对数到二次）-与尖锐的速度。此外，我们还计算了生成元的谱和特征空间以及高斯函数的流不变流形。接下来，我们将我们的方法推广到一类非二次势的动力学Fokker-Planck方程。最后，我们将这种新方法应用于非对称的，一致抛物型的线性漂移的Fokker-Planck方程。至少在2D中，这总是产生熵函数的尖锐指数包络。在这种情况下，我们甚至得到了一个尖锐的乘法常数的衰减估计的非对称半群。
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