Jessica Guerand, Clément Mouhot
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.
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Quantitative regularity for parabolic De Giorgi classes
本文针对抛物 De Giorgi 类，用 De Giorgi 方法给出函数 Hölder 连续性的定量证明，核心是中间值引理的定量版本，且论文自包含。
De Giorgi's regularity theory for elliptic, parabolic and kinetic equations
本书系统梳理了椭圆、抛物线和动能方程的正则性理论，从经典椭圆问题到最新动能方程发展，涵盖德·吉奥吉和纳什等人的核心方法与贡献。
Quantitative De Giorgi methods in kinetic theory for non-local operators
本文基于轨迹和能量估计中的非局部项，定量推导了动理学积分微分方程的 Harnack 不等式，蕴含 Hölder 连续性，并恢复了非截断情形下非齐次 Boltzmann 方程的已有结果。
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。
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 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…
De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions
将De Giorgi-Nash-Moser理论推广到一类非局部次椭圆方程，通过弱解的速度非局部尾在漂移变量上的p-可和性假设，证明了局部L^2-L^∞估计和强Harnack不等式。
Formalization of De Giorgi--Nash--Moser Theory in Lean
本文在勒恩中正式化了均匀椭圆散度型方程的内部德吉奥-纳什-莫泽理论，包括弱次解的局部有界性、正弱超解的弱赫尔曼不等式、正弱解的莫泽赫尔曼不等式及内部霍尔德正则性，是现代PDE理论重要定理的首个机器验证正式化。
This paper gives the first affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the $L^\infty$-setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ with a polynomial decay tail, the Boltzmann equation has a global solution in the weighted $L^2\cap L^\infty$-space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemm…
We obtain local Holder continuity estimates up to the boundary for a kinetic Fokker-Planck equation with rough coefficients, with the prescribed influx boundary condition. Our result extends some recent developments that incorporate De Giorgi methods to kinetic Fokker-Planck equations. We also obtain higher order asymptotic estimates near the incoming part of the boundary. In particular, when the equation has a zero boundary conditions and no source term, we prove that the solution vanishes at infinite order on the incoming part of the boundary.
Kinetic Fokker--Planck Equations with Drifts in a Supercritical Range
该研究针对各向异性混合勒贝格空间中带粗糙无散漂移项的动力学福克-普朗克方程，结合能量抵消、各向异性正则化估计与De Giorgi迭代，建立弱次解局部有界性并证明超临界区域柯西问题整体适定性。
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…
We consider a Frenkel-Kontorova system of harmonic oscillators in a two-dimensional Euclidean lattice and we obtain a quantitative estimate on the angular function of the equilibria. The proof relies on a PDE method related to a classical conjecture by E. De Giorgi, also in view of an elegant technique based on complex variables that was introduced by A. Farina. In the discrete setting, a careful analysis of the reminders is needed to exploit this type of methodologies inspired by continuum models.
Quantitative analysis and its applications for Keller-Segel type systems
该研究运用De Giorgi迭代获得Keller-Segel型系统解的精细上界估计，并将其用于弱解大时间行为、模型正则性及多类趋化流体系统的渐近稳定性与Hölder连续性分析。
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。
On a kinetic Poincaré inequality and beyond
本文利用沿动力学向量场的轨迹方法证明动力学 Poincaré 不等式，改进已有结果，并推广至含 $k$ 步的 Kolmogorov 方程等更一般亚椭圆方程。
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 …
An entropy-level method for quantitative time-asymptotic in the Fokker-Planck model for fermions -- appearance of non saturation
本文针对费米子的Fermi-Dirac-Fokker-Planck方程，结合稳定性、熵方法和De Giorgi水平集迭代，证明了任意冷初始数据的定量时间渐近热化收敛。
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…
Coarse-grained ellipticity and De Giorgi-Nash-Moser theory
本文基于粗粒度椭圆性假设，证明了退化和奇异系数场的弱解局部有界性和Harnack不等式，扩展了经典结果的应用范围。
We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^ε\}_{ε> 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $ε$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and lar…
Harnack estimates for the nonlocal Trudinger equation
研究非局部Trudinger方程的逐点正则性理论，通过改编精细德乔治 - 莫泽机制，在最优尾条件下建立定量上界及抛物型哈拿克不等式，克服非局部双非线性框架技术难题。
We consider general linear kinetic equations combining transport and a linear collision on the kinetic variable with a spatial weight that can vanish on part of the domain. The considered transport operators include external potential forces and boundary conditions, e.g. specular, diffusive and Maxwell conditions. The considered collision operators include the linear relaxation (scattering) and the Fokker-Planck operators and the boundary conditions include specular, diffusive and Maxwell conditions. We prove quantitative estimates of exponential stabilisation (spectral gap) under a geometric …
Critical trajectories in kinetic geometry
该研究在动力学几何中构造了满足相切、连接性等性质的临界轨迹，基于牛顿定律通过力的叠加假设实现，提供近似指数映射并用于推导动力学Sobolev不等式、建立Kolmogorov方程超解估计及给出弱Harnack不等式的替代证明。
On the Cahn-Hilliard equation with kinetic rate dependent dynamic boundary condition and non-smooth potential: separation property and long-time behavior
研究带有动力学速率依赖动态边界条件和非光滑势的 Cahn-Hilliard 方程，通过 De Giorgi 迭代证明二维瞬时严格分离与三维最终严格分离性质，并据此获得解的高阶正则性及长时间收敛性。
We consider the kinetic Fokker-Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improve the rate of convergence for the Fitzhugh-Nagumo equation from non-quantitative to quantitative explicit rate.
We study an asymptotic limit of Vlasov type equation with nonlocal interaction forces where the friction terms are dominant. We provide a quantitative estimate of this large friction limit from the kinetic equation to a continuity type equation with a nonlocal velocity field, the so-called aggregation equation, by employing $2$-Wasserstein distance. By introducing an intermediate system, given by the pressureless Euler equations with nonlocal forces, we can quantify the error between the spatial densities of the kinetic equation and the pressureless Euler system by means of relative entropy ty…
Derivation of the bacterial run-and-tumble kinetic model : quantitative and strong convergence results
本文从包含内部细胞机制的综合模型出发，通过合适坐标与相对熵方法推导梯度依赖翻滚率的奔跑-翻滚动力学方程，并证明具有形式最优速率的定量强收敛。
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
本文研究混合各向异性与非局部p-Laplace算子及双重非线性抛物方程，建立存在唯一性、比较定理、Hardy不等式及正则性估计，并利用Picone恒等式与能量估计等方法。
Quantitative Schauder estimates for hypoelliptic equations
本文利用 Campanato 型方法为一类局部次椭圆算子与非局部动理学方程建立了涵盖散度及非散度形式的定量 Schauder 估计，并适用于非齐次 Landau 与无截断 Boltzmann 方程。
Quantitative Propagation of Chaos and Fluctuations for Kinetic McKean--Vlasov SDEs with Singular Interaction Kernels
研究一类含奇异相互作用核的退化动力学McKean-Vlasov SDEs相关粒子系统，通过动力学Krylov-Khasminskii估计等方法，证明混沌定量传播估计和中心极限定理，得到路径空间相对熵界及Berry-Esseen型界。
In this paper we consider a system of $N$ particles on the real line evolving according to Newton's law, interacting through a singular (repulsive) force deriving from the potential $\frac{|x|^{1-α}}{1-α}$ with $α\in (0,1/8)$ and Brownian force. Thanks to the entropy dissipation along the Liouville equation associated to this particles system and a control of the mechanic energy, we establish a quantitative estimate which enables to conclude to a convergence/consistency the particles system toward the limiting Vlasov-Fokker-Planck equation with singular force.
On the nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potential: well-posedness, regularity and asymptotic limits
研究带非局部动态边界条件与奇异势的非局部Cahn-Hilliard方程初边值问题，用Yosida逼近和Faedo-Galerkin方法证明整体弱解的存在唯一性，分析动力学速率趋于零或无穷的渐近极限及收敛率，还得到正则性传播与瞬时严格分离性质。
We establish a quantitfied overdamped limit for kinetic Vlasov-Fokker-Planck equations with nonlocal interaction forces. We provide explicit bounds on the error between solutions of that kinetic equation and the limiting equation, which is known under the names of aggregation-diffusion equation or McKean-Vlasov equation. Introducing an intermediate system via a coarse-graining map, we quantitatively estimate the error between the spatial densities of the Vlasov-Fokker-Planck equation and the intermediate system in the Wasserstein distance of order 2. We then derive an evolution-variational-lik…
For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce existence of classical solutions near equilibrium for the full non-linear models associated, with explicit regularity bounds, and we obtain explicit estimates on the rate of ex…
Quantitative approximation of stochastic kinetic equations: from discrete to continuum
本文研究带奇异系数的动力学型随机微分方程，证明低正则漂移下系统在强弱意义下良定义，且通用驯服EM格式均以1/2速率收敛。
This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights $ϕ_α(x) = |x|^{-α}$ for some $α> 0$. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure $p(ρ) = ρ^γ$, $γ= 1 + \frac2d$ from that kinetic equation.
We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behav…
On quantitative hypocoercivity estimates based on Harris-type theorems
本文综述了利用 Harris 型定理研究空间非均匀动力学方程收敛至稳态的定量结果，为应用该概率技术提供指导并探讨了其在气体理论与生物学中的扩展前景。
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.
We propose and study a fully discrete finite volume scheme for the Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L 2 relative entropy. Then, we adapt the arguments developed in [12] based the hypocoercivity method to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.
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