Regularity of the D-module associated to a symmetric pair
Yves Laurent
约化李代数上的不变本征分布是Kashiwara-Hotta已证明正则的完整D-模的解。我们在这里解决了一个猜想的关口说，在更一般的情况下对称对，相应的模块仍然经常。
The invariant eigendistributions on a reductive Lie algebra are solutions of a holonomic D-module which has been proved to be regular by Kashiwara-Hotta. We solve here a conjecture of Sekiguchi saying that in the more general case of symmetric pairs, the corresponding module are still regular.
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Boundary regularity for non-local operators with symmetric kernels and vanishing horizon
本文研究了具有奇异对称核且依赖于到域边界距离的非局部算子，证明了解的最优 Hölder 边界正则性及更高边界正则性。
A holonomic D-module on a complex analytic manifoldadmits always a b-function along any submanifold. If the module is regular, itadmits also a regular b-function, that is a b-function with a condition on the order of the lower terms of the equation.There is a weaker condition of regularity: regularity alonga submanifold. We prove that a module which is regular alonga submanifold admits a regular b-function along this submanifold.
A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations
本文证明了具有轴对称性的Navier-斯托克斯方程适当弱解的局部正则性的一个充分条件。我们的条件是一般局部正则性理论中所谓的$L_{3,\infty}$ -情形的轴对称模拟。
In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.
We establish partial regularity of BD-minima for variational integrals of linear growth which depend on the symmetric gradients and satisfy a weak ellipticity condition. Since there is no Korn Inequality in the $L^{1}$-Setup, the result does not follow from the respective result on BV due to Anzellotti and Giaquinta.
Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data
通过对Ginibre，Soffer和Velo \cite{gsv}关于能量临界方程的一个参数进行修正，在三维空间中建立了具有球对称初值的能量-超临界波动方程$\Box u = u^5 \log(2+u^2)$的整体正则性.这个例子表明，临界正则性参数可以渗透到超临界状态非常轻微。
We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, hav…
Remarks on Interior Regularity Criterion for an Axially Symmetric Suitable Weak Solution to the Navier Stokes Equations
我们证明了，如果v是Navier-Stokes方程的轴对称适当弱解（在L。卡法雷利河科恩公司Nirenberg）使得v的径向分量具有更高的正则性（即满足加权Serrin-Prodi型条件），则v的所有分量都是正则的。
Interior regularity of space derivatives to an evolutionary, symmetric $φ$-Laplacian
我们考虑Orlicz增长推广到进化$p$ -Laplacian和进化对称$p$ -Laplacian。我们得到了这些系统的局部弱解的空间二阶Caccioppoli估计。即使对于$p$ -情况，结果也是新的。
Maximal Sobolev regularity of the stress tensor for the symmetric gradient p-Laplace system
本文研究了对称p-拉普拉斯系统解的二次可微性，确立了非线性应力张量在局部平方可积右端项下的最大索波列夫正则性。
Regularity for solutions of non local, non symmetric equations
研究了完全非线性积分微分方程解关于非对称核的正则性。更准确地说，我们假设我们的操作是椭圆形的积分微分线性算子，其中的对称部分的内核有一个固定的同质性$σ$和反对称部分有严格较小的同质性$τ$的家庭。我们证明了一个弱ABP估计和$C^{1,α}$正则性。我们的估计保持一致，因为我们采取$σ\to 2$和$τ\to 1$，使这扩展了椭圆型微分方程的依赖于梯度的正则性理论。
Regularity of a D-Module along a submanifold
我们研究如何正则沿一个微分或微微分系统的子流形可以传播从一个家庭的子流形到另一个。第一个结果是沿拉格朗日叶理正则的微微分系统是正则的。然而，当限制到一个固定的子流形相应的结果是真的，只有在一个条件下的特征品种。
Regularity of Solution to Axis-symmetric Navier-Stokes Equations with a Slightly Supercritical Condition
考虑三维不可压Navier-斯托克斯方程的轴对称弱解。如果解满足一个稍微超临界的假设，我们将证明v是正则的。这推广了Chen-Strain-Tsai-Yau、Koch-Nadirashvili-Seregin-Sverak和Lei-Zhang的结果，其中证明了临界假设下的可积性。
In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $(p,δ)$-structure, for all $p>1$ and for all $δ>0$. In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting…
Global regularity for nonlinear systems with symmetric gradients
本文研究了依赖于梯度对称部分的偏微分方程非线性系统在 Dirichlet 边界条件下的全局正则性，涵盖了 $p \in (1,\infty)$ 的完整范围及 $p>2$ 的退化情形，并通过证明 Orlicz 增长系统的正则性提出了统一方法。
Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a t…
A single-component regularity criterion and Inviscid limit of axially symmetric MHD-Boussinesq system
该研究针对无粘轴对称MHD-Boussinesq系统给出仅含速度水平旋度分量的临界BKM型爆破准则，同时分析粘性系统的无粘极限并得到粘性系数趋于零时的收敛速率。
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.
Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data
本文研究了三维轴对称Navier-斯托克斯方程中的涡旋问题。本文证明了三维Navier-斯托克斯方程在大的各向异性初值下的整体正则性。此外，我们还得到了解在某些$L^p$范数下关于其初始值的整体上界。我们的结果还揭示了由于角速度场和角涡量场之间的相互作用，解的一些有趣的动力学增长行为。
On regularity for nonhomogeneous parabolic systems with a skew-symmetric part in BMO
本文针对系数含BMO斜对称部分的非齐次抛物型系统，证明了弱解梯度的改进Caccioppoli不等式与逆Hölder不等式，并允许无界系数。
The regularity and Neumann problem for non-symmetric elliptic operators
我们在平面上Lipschitz图上的区域上，根据Dirichlet数据或Neumann数据的切向导数边界处的L^p范数，建立了二阶椭圆算子解的梯度的非切向极大函数的最优L^p界，该椭圆算子为散度型，可能是非对称的，具有与垂直变量无关的有界可测系数。
Regularity Criterion to the axially symmetric Navier-Stokes Equations
轴对称Navier-斯托克斯方程的光滑解服从如下的最大值原理：$\|ru_θ(r,z,t)\|_{L^\infty}\leq\|ru_θ(r,z,0)\|_{L^\infty}.$我们首先证明了当$\|ru_θ(r,z,0)\|_{L^\infty}$或$ \|ru_θ(r,z,t)\|_{L^\infty(r\leq r_0)}$与一定的无量纲初值相比很小时，解的整体正则性。这一结果改进了Zhen Lei和Qi S的结果。张\cite{1}。作为推论，我们还证明了在假设$|ru_θ(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\ 0<r\leqδ_0\in(0,1/2).$
In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.
Comparability and regularity estimates for symmetric nonlocal Dirichlet forms
本文的目的是研究非局部Dirichlet型的可比性。我们提供了充分条件的内核的局部和全局的可比性。作为应用，我们证明了积分微分方程解在Hölder空间中的先验估计.这些解决方案的帮助下定义的对称非局部狄利克雷形式。
Nonlinear Schroedinger equations with radially symmetric data of critical regularity
研究了一类具有临界正则性的径向对称纯幂非线性Schroedinger方程小解的整体存在性。在径向对称条件下，我们将注意力集中在非线性项的幂比伪共形项的幂稍小且初始数据属于尺度不变的齐次索伯列夫空间的情形.尽管初值是负阶可微的，但只要初值具有小范数，非线性Schroedinger方程就有整体解。在证明这一结果的关键成分是一个有效的使用全球加权平滑估计特定的径向对称的解决方案。
Remarks on Regularity Criteria for Axially Symmetric Weak Solutions to the Navier-Stokes Equations, II
在轴对称数据的假设下，我们研究了Navier-斯托克斯方程解在整个三维空间中的条件正则性。证明了当速度的径向分量的正部分满足加权Serrin条件，且角分量满足一定条件时，解是正则的。
We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuin…
Microlocal study of ind-sheaves I: micro-support and regularity
定义了ind层的微支撑和正则性概念，并利用切触变换证明了它们的不变性。我们将所得结果应用于D-模的温带全纯解的ind层。我们证明了这样的ind层的微支撑是相应的D-模的特征簇，并且如果D-模是正则完整的，则ind层是正则的。
Wave equations and symmetric first-order systems in case of low regularity
我们分析了二阶波动方程和一阶对称双曲系统的柯西问题之间的过渡算法的情况下，系数以及数据是非光滑的，甚至允许正则性低于标准条件，保证适定性。将方程重写为系统所涉及的典型运算既不是经典定义的，也不能一致地扩展到分布理论设置。然而，采用非线性理论的广义函数在Colombeau的意义上，我们到达明确的声明转移问题的可解性和唯一性从波动方程对称双曲系统，反之亦然。最后，我们说明了如何转移方法允许得出新的结论，唯一的可解性的波动方程的柯西问题的非光滑系数。
Global geometry of T2 symmetric spacetimes with weak regularity
定义了一类具有T2对称性的弱正则时空，并研究了它们的整体几何结构。我们建立了弱正则性的Einstein真空方程的初值问题，并通过对称轨道面积R的水平集建立了整体叶理的存在性，使得每个叶理都可以看作是一个初始超曲面.除了明确已知的平坦卡斯纳时空外，R取所有正值。我们的弱正则性假设只要求R的梯度是连续的，而度量系数属于Sobolev空间H1（或具有更少的正则性）。
Regularity of Solution of the Schrödinger Equation on Symmetric Space
本文研究非紧致型秩对称空间上分数阶薛定谔方程解的逐点收敛性，证明了当初始数据属于 \\( H^s \\) 且 \\( s > 1/2 \\) 时，解在时间趋近于零时几乎处处收敛于初始数据，推广了 Sjölin 的结果。
This paper establishes global weighted Calderón-Zygmund type regularity estimates for weak solutions of a class of generalized Stokes systems in divergence form. The focus of the paper is on the case that the coefficients in the divergence-form Stokes operator consist of symmetric and skew-symmetric parts, which are both discontinuous. Moreover, the skew-symmetric part is not required to be bounded and therefore it could be singular. Sufficient conditions on the coefficients are provided to ensure the global weighted $W^{1,p}$-regularity estimates for weak solutions of the systems. As a direct…
We study solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $Ω^c$, in $C^{1,α}$ domains~$Ω$. We show that solutions $u$ satisfy $u/d^γ\in C^{\varepsilon_\circ}\big(\overlineΩ\big)$, where $d$ is the distance to $\partialΩ$, and $γ=γ(L,ν)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ν$ to the boundary $\partialΩ$. On the other hand, we establish new integration by parts identities in half s…
A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations
令$v$为轴对称三维Navier-斯托克斯方程的Leray-Hopf解的速度。证明了当角速度$v_θ$满足一个积分条件时，$v$是正则的.该条件允许满足\[ |v_θ(x, t)| \le \frac{C}{r |\ln r|^{2+ε}}, \quad r<1/2, \]的函数，其中$r$是从$x$到轴的距离，$C$和$ε$是任何正常数。与临界先验界\[ |v_θ(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2, \]相比，我们的条件在最坏的情况下偏离了对数因子$|\ln r|^{2+ε}$。这是受到最近有趣的论文\cite{CFZ:1}的启发，其中H. Chen，中国粘蝇D. Y. Fang和T. Zhang建立了关于角速度的几乎临界正则性条件。先前的正则性条件偏离因子$r^{-1}$。证明是基于新的观察，即从不同的角度看，三维轴对称Navier-斯托克斯方程中的所有涡拉伸项都是临界的，而不是通常认为的超临界。
The purpose of this paper is to study the incompressible non-resistive MHD equations in $\mathbb{R}^3$. We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the velocity and magnetic vorticity vanish. In particular, the special axially symmetric initial data can be arbitrarily large and satisfy low regularity assumptions.
A symmetric low-regularity integrator for the nonlinear Schrödinger equation
本文提出一种用于 NLS 方程的对称低正则性积分器，证明其在环面和有界区域上的 $L^2$ 分数阶收敛，并通过数值实验验证其结构保持与误差常数优势。
Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion
本文考虑$\mathbb{R}^d$中一个对称稳定过程与$\mathbb{R}^+$中一个一维布朗运动的乘积.然后定义了一类关于该乘积过程的调和函数。证明了上半空间中的有界非负调和函数满足Harnack不等式，并证明了它们是局部Hölder连续的.我们还讨论了由$L^p(\mathbb{R}^d)$函数的$α$ -调和扩张得到的Littlewood-Paley函数的一个结果。
In this paper we prove the higher Sobolev regularity of minimisers for convex integral functionals evaluated on linear differential operators of order one. This intends to generalise the already existing theory for the cases of full and symmetric gradients to the entire class of $\mathbb{C}$-elliptic operators therein including the trace-free symmetric gradient for dimension $n \geq 3$.
Sobolev regularity of the symmetric gradient of solutions to a class of $ϕ$-Laplacian systems
本文研究了ϕ-Laplace系统解的对称梯度的Sobolev正则性，通过构造近似问题并利用均匀高阶可微性估计，证明了在特定Orlicz-Sobolev空间下解的正则性。
In this paper, we study the problem of shock reflection by a wedge, with the potential flow equation, which is a simplification of the Euler System. In the work of M. Feldman and G. Chen, the existence theory of shock reflection problems with the potential flow equation was established, when the wedge is symmetric w.r.t. the direction of the upstream flow. As a natural extension, we study non-symmetric cases, i.e. when the direction of the upstream flow forms a nonzero angle with the symmetry axis of the wedge. The main idea of investigating the existence of solutions to non-symmetric problems…
On Axially Symmetric Incompressible Magnetohydrodynamics in Three Dimensions
三维不可压缩磁流体动力学方程组的整体正则性问题是流体力学和偏微分方程理论中的一个长期的开放性问题。Navier-斯托克斯方程可以看作是具有恒定磁场的MHD的一个特例，其整体正则性问题被称为Clay千禧奖问题。本文证明了一类非平凡磁场下三维理想磁流体力学轴对称解的整体正则性。这些证明是基于磁流体的特殊结构，当然也适用于电阻性磁流体。我们的结果可能表明，磁流体力学比Navier-斯托克斯方程有着更丰富的研究内容。
Temporal regularity of symmetric stochastic $p$-Stokes systems
本文研究有界域上的对称随机 $p$-Stokes 系统，证明了随机压力在 Besov 尺度上具有几乎 $-1/2$ 阶时间导数，且强解的速度分量及非线性对称梯度在 Nikolskii 空间中具有 $1/2$ 阶时间导数。
Decay structure of two hyperbolic relaxation models with regularity-loss
研究了具有非对称松弛的线性对称双曲型方程组的两种衰减结构。精确地说，系统是$(p,q)$型的，如果所有特征值的实部允许一个上界$-c|ξ|^{2p}/(1+|ξ|^2)^{q}$，其中$c$是一般的正常数，$ξ$是频率变量，并且如果$p<q$，则系统具有正则性损失性质。众所周知，标准类型$(1,1)$可以通过经典的川岛-静田条件来保证。在\cite{UDK}中引入了一个新的结构条件，用于分析具有非对称弛豫的正则性损失型$(1,2)$系统.本文构造了两个更复杂的正则性损失模型，分别对应于$p=m-3$，$q=m-2$和$p=(3m-10)/2$，$q=2(m-3)$，其中$m$表示相维数。证明是基于精细的傅立叶能量方法以及一系列能量不等式的适当的线性组合。由于任意较高的维度，它是不明显的，以捕捉相对于退化组件的能量耗散率。因此，对于每一个模型，我们总是从低相维的情况出发，以理解一般情况下的基本耗散结构，同时，我们也给出了补偿对称矩阵$K$和反对称矩阵$S$的显式构造.
In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local regularity theory of weak axisymmetric solutions.
Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type
研究紧致型黎曼对称空间上色散方程的解的逐点几乎处处收敛性，通过调和分析与数论方法，确定了不同秩空间下初始数据的Sobolev正则性阈值，并利用新传递原理推广至其他方程。
Weakly regular T2 symmetric spacetimes. The future causal geometry of Gowdy spaces
当度量满足弱正则性条件，使得度量系数（在适当的坐标下）只在Sobolev空间H1中或具有更弱的正则性时，我们研究了T3上Gowdy时空的未来渐近行为.作者最近在更广泛的T2对称时空的背景下引入了这类时空，并建立了当时间函数被选为对称曲面的面积时，类空超曲面的整体叶理的存在性。在本文中，我们确定了这些时空的整体因果几何，特别是，建立弱正则Gowdy时空未来因果测地线完成。这一结果扩展了林斯特伦的一个定理，使度量具有足够高的正则性。我们强调，我们的能量衰减的证明是基于一个能量功能的启发Gowdy到恩斯特变换。为了建立测地完备性的财产，我们证明了一个更高的正则性有关的度量系数沿类时曲线，我们提供了一个新的分析的测地方程Gowdy时空，这不需要高阶正则性估计。即使当充分的正则性假设，我们的证明提供了一个替代和更短的证明的能量衰减和测地完备性的Gowdy时空。
The characteristic initial value problem for plane symmetric spacetimes with weak regularity
当物质由压力等于其质量-能量密度的无旋理想流体组成时，研究了弱正则平面对称时空的存在性及其整体因果结构.我们的理论包括一类弱正则时空，其度量系数具有平方可积的一阶导数，并且其曲率必须在分布的意义上理解。我们制定的特征初值问题的数据上的两个零超曲面沿两个平面相交。依靠纽曼-彭罗斯的形式主义和表达我们的弱正则性条件的纽曼-彭罗斯标量，我们到达一个完全的几何公式，其中，沿着每个初始超曲面，两个标量场描述入射辐射必须规定。为了分析这样一个时空的未来边界，并确定其全球因果结构，我们引入了一个规范，减少爱因斯坦方程的耦合系统的波动方程和常微分方程精心挑选的未知数。我们证明了，在弱正则类考虑和通用的初始数据，一个真正的时空奇异形式在有限的适当的时间。我们的公式足够稳健，因此曲率或物质变量中的传播不连续性不会阻止我们构建一个曲率一般在未来边界上爆炸的时空。早期的工作对这个问题的研究在这里被限制在充分正规和真空时空。
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