Wendong Wang, Liqun Zhang
We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\ptl_t u}= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\a}$ regularity of homogeneous ultraparabolic equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove the $C^α$ regularity for weak solutions to a class of ultraparabolic equation, with measurable coefficients. The results generalized our recent $C^α$ regularity results of Prandtl's system to high dimensional cases.
The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients $a_{ij} (z)$. Assume that the operator $L_0$ obtained by freezing the coefficients $a_{ij}(z)$ at any point ${z_0} \in {\mathbb{R}^{N + 1}}$ is hypoelliptic. We first establish a Caccioppoli type inequality by choosing a cutoff function, a Sobolev type inequality by prosperities of the fundamental solution to $L_0$, and a Poincaré type inequality with a new cutoff function. Then $L^p$ estimate for weak solutions is derived by using the reverse Hölder inequality on homogeneous spaces. Final…
Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are Hölder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincaré inequality satisfied by non-negative weak sub-solutions and Moser iteration.
We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.
We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies H{ö}lder continuity with quantitative estimates. The paper is self-contained.
In this paper we establish a constructive method in order to show global existence and regularity for a class of degenerate parabolic Cauchy problems which satisfy a weak Hoermander condition on a subset of the domain where the data are measurable and which have regular data on the complementary set of the domain. This result has practical incentives related to the computation of Greeks in reduced LIBOR market models, which are standard computable approximations of the HJM-description of interest rate markets. The method leads to a probabilistic scheme for the computation of the value function…
Ultradifferentiable regularity properties of a class of hypoelliptic operators
研究一类由霍尔曼德引入的亚椭圆微分算子的超可微正则性，扩展了前人相关结果。
We show a multi-anisotropic Gevrey regularity of solutions of hypoelliptic equations. This result is a precision of a classical result of Hörmander
We present a general approach to obtain a weak Harnack inequality for rough hypoellipitic equations, e.g. kinetic equations. The proof is constructive and does not study the commutator structure but rather compares the rough solution with a smooth problem for which the estimates are assumed.
This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a Hölder regularity version of the classical averaging lemmas.
Sharp boundary regularity properties for hypoelliptic kinetic equations
该研究针对动理学Fokker-Planck方程，在给定流入边界条件下建立了精确的边界次椭圆正则性结果，刻画了不同系数条件下解在各类边界的性态并得到相关估计。
Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation
针对带外部约束势的相对论空间非齐次Fokker-Planck方程，通过构造李雅普诺夫泛函，证明解在加权L²和索伯列夫空间中指数衰减至平衡且收敛速率显式，同时证明半群具亚椭圆正则性并获显式正则化速率。
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。
Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator
本文研究了双曲算子的各向异性最大 $L^p$-正则性估计，证明了解的存在性并给出了相关算子的弱估计。
In this paper, we study the Gevrey regularity of weak solution for a class of linear and quasilinear Fokker-Planck equations.
Perturbations of globally hypoelliptic pseudo-differential operators on $\mathbb{R}^n$
本文研究了全局亚椭圆伪微分算子在低阶扰动下的稳定性，证明其全局正则性得以保持，并探讨了其在标准Hörmander类中的不适用性。
In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonloca…
Remarks on hypoelliptic equations
本文探讨了标量方程的光滑亚椭圆性，通过实例和反例分析了系统及不同正则性水平下的亚椭圆性问题。
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.
Propagation of regularity for a class of systems of real vector fields on torus
本文研究环面上一类实向量场系统，通过证明解的正则性传播定理，刻画了系统的全局亚椭圆性、几乎亚椭圆性和可解性及相应平方和算子的性质。
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.
We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan.
We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to develop blow-up analysis to prove a partial regularity result for weak solutions of certain non-linear systems.
The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space $\mathcal{W}$, suitable for the study of weak solutions to $\mathscr{L}u = f$, that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combi…
Hypoelliptic Regularization in the Obstacle Problem for the Kolmogorov Operator
本文研究Kolmogorov算子障碍问题，利用Bernstein技巧与Evans-Krylov理论思想将解在x变量上的正则性提升至$C^{0,1}_{t,x}$，并据此结合新的单调性公式与交换子估计首次证明了自由边界的正则性。
Weak and renormalized solutions to a hypoelliptic Mean Field Games system
本文证明了退化次椭圆平均场博弈系统弱解与重整化解的存在唯一性，利用De-Giorgi论证和动力学正则性，覆盖Lipschitz及二次增长Hamilton量。
A class of globally analytic hypoelliptic operators on compact Lie groups
本文研究紧李群上一类微分算子的全局解析亚椭圆性，在Hörmander有限型条件、闭子群不变性及横截方向椭圆性三个条件下证明了该性质，并指出这为研究主纤维丛上平方和的 regularity 奠定了基础。
Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators
该研究提出基于轨迹的系统方法，证明含任意多个Hörmander交换子的次椭圆方程弱非负下解的庞加莱不等式，进而推导弱哈纳克不等式与赫尔德正则性。
We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are Hölder continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions; second in proving that the gradient with respect to the velocity variable is $L^{2+ε}_{\mathrm{loc}}$; third, in the proof of an "hypoelliptic isoperimetric De Giorgi lemma." To get such a lemma, we develop a …
In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.
We present sufficient conditions to have global hypoellipticity for a class of Vekua-type operators defined on a compact Lie group. When the group has the property that every non-trivial representation is not self-dual we show that these sufficient conditions are also necessary. We also present results about the global solvability for this class of operators.
On the radius of analyticity and Gevrey regularity for the Boltzmann equation
本文研究了Boltzmann方程在硬势场中的分析半径和Gevrey正则性，通过宏微观分解和次椭圆估计获得短时和全局时间的正则性估计。
Consider the following Kolmogorov type hypoelliptic operator $$ \mathscr L_t:=\mbox{$\sum_{j=2}^n$}x_j\cdot\nabla_{x_{j-1}}+{\rm Tr} (a_t \cdot\nabla^2_{x_n}), $$ where $n\geq 2$, $x=(x_1,\cdots,x_n)\in(\mathbb R^d)^n =\mathbb R^{nd}$ and $a_t$ is a time-dependent constant symmetric $d\times d$-matrix that is uniformly elliptic and bounded.. Let $\{\mathcal T_{s,t}; t\geq s\}$ be the time-dependent semigroup associated with $\mathscr L_t$; that is, $\partial_s {\mathcal T}_{s, t} f = - {\mathscr L}_s {\mathcal T}_{s, t}f$. For any $p\in(1,\infty)$, we show that there is a constant $C=C(p,n,d)>…
We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{ö}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic…
For $p\in(1,\infty)$, let $u(t,x,v)$ and $f(t,x,v)$ be in $L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)$ and satisfy the following nonlocal kinetic Fokker-Plank equation on $\mathbb{R}^{1+2d}$ in the weak sense: $$ \partial_t u+v\cdot\nabla_x u=Δ^{α/{2}}_v u+f, $$ where $α\in(0,2)$ and $Δ^{α/{2}}_v$ is the usual fractional Laplacian applied to $v$-variable. We show that there is a constant $C=C(p,α,d)>0$ such that for any $f(t, x, v)\in L^p(\mathbb{R} \times \mathbb{R}^d \times \mathbb{R}^d)=L^p(\mathbb{R}^{1+2d})$, $$ \|Δ_x^{α/{(2(1+α))}}u\|_p+\|Δ_v^{α/{2}}u\|_p\leq C\|f\|_p, $$ wh…
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …
Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincaré inequality, and as a consequence, we deduce a version of the classical Moser iteration technique using in a crucial way the geometry of the equation. The point of this contribution is to emphasize that one can use the {\sl elliptic} version of the Moser argument at the price of the lack of uniformity, even in the {\sl parabolic } setting. This is nevertheless enough to deduce Hölder regularity of weak solutions. The proof is elementary and…
A global real analytic regularity theorem for a quasilinear sum of squares of vector fields of Hormander rank 2 is given. A related local result for a special case was proved recently by the second author and L. Zanghirati in a paper titled "Local Real Analyticity of Solutions for sums of squares of non-linear vector fields".
Schwartz regularity of differential operators on the cylinder
本文利用混合Fourier变换研究圆柱面上的一类微分算子，通过分析混合Fourier系数给出其Schwartz全局亚椭圆性和全局可解性的充要条件。
The main result of the paper is on the continuity of weak solutions of infinitely degenerate quasilinear second order equations. Namely, we show that every weak solution to a certain class of degenerate quasilinear equations is continuous. More precisely, we show that it is Hölder continuous with respect to a certain metric associated to the operator. One of the essential features of this metric is that the metric balls are non doubling with respect to Lebesgue measure. The proof of the continuity together with a recent result by Rios et al. completes the result on hypoellipticity of a class o…
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…
Higher order hypoelliptic damped wave equations on graded Lie groups with data from negative order Sobolev spaces
本文研究分次李群上带正Rockland算子的半线性次椭圆阻尼波动方程柯西问题，在负阶Sobolev空间框架下得到新临界指数，证明小初值全局存在性与次临界情形有限时间爆破及生命跨度上下界，结果在高阶微分算子与分层李群情形下也具新颖性。
We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{ö}lder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of H{ö}rmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent exten…
This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goals are to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked forward limit. As in the case where wavenumbers take values in R^{d+1}, we show that the t…
Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients
本文研究具有一般传输算子和粗糙系数的动力学 Fokker-Planck 方程弱解的局部有界性和 Hölder 连续性，通过结合椭圆 De Giorgi-Nash-Moser 理论与速度平均引理的方法，利用扩散与传输的混合效应及亚椭圆结构证明了该结果。
For a class of ordinary differential operators $P$ with polynomial coefficients, we give a necessary and sufficient condition for $P$ to be globally regular in $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply $u\in \cS(\R)$ (this can be regarded as a global version of the Schwartz' hypoellipticity notion). The condition involves the asymptotic behaviour, at infinity, of the roots $ξ=ξ_j(x)$ of the equation $p(x,ξ)=0$, where $p(x,ξ)$ is the (Weyl) symbol of $P$.
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 theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau--Weyl pseudodifferential operators". The link between standard Weyl calculus and Landau--Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfuncti…
We study the regularity properties of the second order linear operator in $\mathbb{R}^{N+1}$: \begin{equation*} \mathscr{L} u := \sum_{j,k= 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum_{j,k= 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} where $A = \left( a_{jk} \right)_{j,k= 1, \dots, m}, B= \left( b_{jk} \right)_{j,k= 1, \dots, N}$ are real valued matrices with constant coefficients, with $A$ symmetric and strictly positive. We prove that, if the operator $\mathscr{L}$ satisfies Hörmander's hypoellipticity condition, and $f$ is a Dini continuous function, then the second ord…
In the paper, for the Cauchy problem on the non-cutoff Boltzmann equation in torus, we establish the global-in-time Gevrey smoothness in velocity and space variables for a class of low-regularity mild solutions near Maxwellians with the Gevrey index depending only on the angular singularity. This together with [24] provides a self-contained well-posedness theory for both existence and regularity of global solutions for initial data of low regularity in the framework of perturbations. For the proof we treat in a subtle way the commutator between the regularization operators and the Boltzmann co…
In this paper we study the Cauchy problem for the wave equations for sums of squares of left invariant vector fields on compact Lie groups and also for hypoelliptic homogeneous left-invariant differential operators on graded Lie groups (the positive Rockland operators), when the time-dependent propagation speed satisfies a Log-Lipschitz condition. We prove the well-posedness in the associated Sobolev spaces exhibiting a finite loss of regularity with respect to the initial data, which is not true when the propagation speed is a ${\rm H\ddot{o}lder}$ function. We also indicate an extension to g…
We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide …
Global Well-posedness and Regularity of the Dynamical Prandtl Equation
研究动态Prandtl方程在单调性条件下的整体适定性和正则性，通过Kolmogorov方程基本解和Hörmander亚椭圆估计，证明了解的整体存在性和光滑性，解决了Oleinik和Samokhin提出的公开问题。
Introduction to quantitative De Giorgi methods
本课程回顾了椭圆和抛物情形下的经典 De Giorgi 方法，并重点介绍了基于轨迹构造的定量方法及其在动力学理论中亚椭圆方程的最新推广。
The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander's condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。