Helge Dietert, Jonas Hirsch
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.
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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
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。
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.
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.
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.
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。
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds just in terms of the supremum norm for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness and interior regularity of solutions for the Dirichlet problem with continuous boundary data.
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。
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.
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)>…
This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which satisfy mild assumptions concerning (a) the existence of cut-off functions, (b) a local ultracontractivity hypothesis, and (c) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous…
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。
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…
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…
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…
We prove a $C^{1,α}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.
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…
Given any solution $u$ of the Euler equations which is assumed to have some regularity in space - in terms of Besov norms, natural in this context - we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressure $p$ is twice as regular as $u$. This generalizes a recent result by Isett [16] (see also Colombo and De Rosa [8]), which covers the case of Hölder spaces.
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.
Higher-order boundary layers and regularity for Stokes systems over rough boundaries
本文研究了周期振荡 John 域中 Stokes 系统的大尺度边界正则性，通过构造任意阶边界层校正子，推广了已有的一阶和二阶结果，并证明了任意阶的大尺度正则性估计及 Liouville 定理。
Optimal regularity for degenerate Kolmogorov equations with rough coefficients
针对满足抛物型Hörmander条件、系数时间可测空间Hölder连续的退化方程，通过广义强解概念证明基本解存在性、最优Hölder正则性及高斯估计，为Langevin型扩散相关倒向Kolmogorov方程研究提供关键支撑。
We establish $C^{σ+α}$ interior estimates for concave nonlocal fully nonlinear equations of order $σ\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^α(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(σ)$, then $u$ belongs to $C^{σ+α}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^α$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs fo…
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。
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…
We prove space and time regularity for solutions of fully nonlinear parabolic integro-differential equations with rough kernels. We consider parabolic equations $u_t = \I u$, where $\I$ is translation invariant and elliptic with respect to the class $\mathcal L_0(σ)$ of Caffarelli and Silvestre, $σ\in(0,2)$ being the order of $\I$. We prove that if $u$ is a viscosity solution in $B_1 \times (-1,0]$ which is merely bounded in $\R^n \times (-1,0]$, then $u$ is $C^β$ in space and $C^{β/σ}$ in time in $\overline{B_{1/2}} \times [-1/2,0]$, for all $β< \min\{σ, 1+α\}$, where $α>0$. Our proof combine…
Higher-order regularity for a structurally damped plate equation on rough domains
本文研究了粗糙域上结构阻尼板方程的高阶正则性，通过加权Sobolev空间和最大正则性性质，处理非齐次边界条件下的适定性问题。
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。
The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lamé system with only bounded and measurable coefficients. Second, we derive a maximal $L^1$ regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal $L^1$ regularity for the Lamé system, from which it follows that the Lipschitz seminorm of the solutions to the Lamé system is globally $L^1$-in-time integrable. As an application, we use a Lagrangian approach to prove a…
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…
Hölder and Sobolev regularity of optimal transportation potentials with rough measures
本文研究粗糙测度间最优传输的 Kantorovich 势函数，在球上与 Lebesgue 测度可比的条件下，证明其至依赖 δ 的临界尺度具有 Hölder 与 Sobolev 正则性。
In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the …
We obtain $L^p(L^q)$ maximal regularity estimates for time dependent second order elliptic operators in divergence form with rough dependencies in the spatial variables.
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result u…
Regularity estimates for the non-cutoff soft potential Boltzmann equation with typical rough and slowly decaying data
本文针对非截止软势玻尔兹曼方程，构造典型粗糙且缓慢衰减的初始数据，证明弱解仅获得有限Sobolev平滑效应且Leibniz法则对碰撞算子高阶导数失效，从而揭示多项式衰减数据阻碍瞬时无穷光滑猜想成立的关键障碍。
We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t θ- Δθ+ b \cdot \nabla θ= 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $\frac{2}{q} + \frac{n}{p} < 2$, or $L^p_x L^q_t$, where $\frac{3}{q} + \frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of drifts with `bounded total speed' is a borderl…
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function $A=A(ξ,X,Y,t):\R^m\times\R^m\times\R^m\times\R\to\R^m$ is assumed to be continuous with respect to $ξ$, and measurable with respect to $X,Y$ and $t$. $A=A(ξ,X,Y,t)$ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and H{ö}lder continuity with quantitative estimates. In addition we e…
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.
In this paper, we obtain sharp Strichartz estimates for solutions of the wave equation $\square_\ggϕ=0$ where $\gg$ is a rough Lorentzian metric on a 4 dimensional space-time $\MM$. This is the last step of the proof of the bounded $L^2$ curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation…
Space Regularity of Evolution Equations Driven by Rough Paths
针对由 Hölder 粗糙路径驱动、含非磨光算子的线性发展方程，证明其 Cauchy 问题存在唯一 mild 解、具有即时正则化效应，并由积分解建立 Itô 公式。
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…
In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class.
We present a divergence free vector field in the Sobolev space $H^1$ such that the flow associated to the field does not belong to any Sobolev space. The vector field is deterministic but constructed as the realization of a random field combining simple elements. This construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well-posed flow which has nevertheless only very weak regularity.
In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrödinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schrödinger part with a careful iteration of the rough wave part in lower order Sobolev norms.
We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators.
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…
Trace theory for parabolic boundary value problems with rough boundary conditions
本文提出了一种适用于处理边界处导数发散的抛物型边值问题的迹理论，证明了热方程在粗糙边界数据下的适定性。
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