非紧流形上投射结构的全局Gevrey次椭圆性

一类亚复管结构的平滑与Gevrey微局部亚椭圆性

Smooth and Gevrey Microlocal Hypoellipticity for a Class of Hypocomplex Tube Structures

本文利用部分F.B.I.变换，证明了与实解析局部可积管型结构相关的复向量场系统的平滑与Gevrey-$s$微局部亚椭圆性，并给出了波前集的微局部刻画。

Global analytic hypoellipticity and solvability of certain operators subject to group actions

On $T \times G$, where $T$ is a compact real-analytic manifold and $G$ is a compact Lie group, we consider differential operators $P$ which are invariant by left translations on $G$ and are elliptic in $T$. Under a mild technical condition, we prove that global hypoellipticity of $P$ implies its global analytic-hypoellipticity (actually Gevrey of any order $s \geq 1$). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.

非紧流形上involution系统全局拟解析性

Global hypoellipticity for involutive systems on non-compact manifolds

本文研究了非紧流形上involution系统算子的全局拟解析性，通过形式的算术性质和散射度量下的微局技术进行刻画。

Metivier不等式与超可微亚椭圆性

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。

渐近欧几里得流形上一类时间周期算子的全局亚椭圆性

Global hypoellipticity for a class of time-periodic operators on asymptotically Euclidean manifolds

该研究在渐近欧几里得流形上引入时间周期Gevrey-Sobolev-Kato空间并研究其表征，进而应用于一类时间周期演化方程的全局亚椭圆性问题分析。

On the sharp Gevrey regularity for a generalization of the Métivier operator

The sharp Gevrey hypoellipticity is provided for the following generalization of the Métivier operator, "Non-hypoellipticité analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. Métivier, \begin{align*} D_{x}^{2}+\left(x^{2n+1}D_{y}\right)^{2}+\left(x^{n}y^{m}D_{y}\right)^{2}, \end{align*} in $Ω$ open neighborhood of the origin in $\mathbb{R}^{2}$, where $n$ and $m$ are positive integers.

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Sums of Squares of Vector Fields

We consider an operator $ P $ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $ σ$ is not constant on $ \Char P $. Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for $ P $ consists of closed curves in a ring-shaped open set around the origin. We prove that then $ P $ is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for $ P $ is $ G^{k+1} $ and that this is sharp.

非紧流形上involution系统全局可解性研究

Global Solvability for Involutive Systems on Non-Compact Manifolds

本文研究了非紧流形上involution系统微分算子的全局可解性条件，并扩展了此前在紧流形上关于次椭圆性结果的结论。

Gevrey超分布中管复形的最高次整体可解性

Top-Degree Global Solvability for Tube Complexes in Gevrey Ultradistributions

本文研究非紧实解析流形与环面乘积上的管微分复形，证明其最高次算子在Roumieu Gevrey超分布中整体可解，无需亚椭圆或周期算术条件，与紧流形情形形成本质差异。

Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity $ν\to 0$), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Ge…

Global properties of vector fields on compact Lie groups in Komatsu classes. II. Normal forms

Let $G_1$ and $G_2$ be compact Lie groups, $X_1 \in \mathfrak{g}_1$, $X_2 \in \mathfrak{g}_2$ and consider the operator \begin{equation*} L_{aq} = X_1 + a(x_1)X_2 + q(x_1,x_2), \end{equation*} where $a$ and $q$ are ultradifferentiable functions in the sense of Komatsu, and $a$ is real-valued. We characterize completely the global hypoellipticity and the global solvability of $L_{aq}$ in the sense of Komatsu. For this, we present a conjugation between $L_{aq}$ and a constant-coefficient operator that preserves these global properties in Komatsu classes. We also present examples of globally hypo…

时间周期性Gelfand-Shilov空间中的微分复形

Differential Complexes in Time-Periodic Gelfand-Shilov Spaces

本文研究了时间周期性Gelfand-Shilov空间中由时间依赖实系数生成的微分复形的全局可解性与次椭圆性，提出了基于Diophantine条件的可解性刻画。

Global regularity and solvability of left-invariant differential systems on compact Lie groups

We are interested in global properties of systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and finite dimensionality of their cohomology spaces, when acting on various function spaces e.g. smooth, analytic and Gevrey. Extending the methods of Greenfield and Wallach (1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of results due to Greenfield (1972), Greenfield and Wallach (1972), as well as global versions of a result of Caetano and Cor…