Regularity of a inverse problem for generic parabolic equations
Nikolai Dokuchaev
研究一类最简单的抛物型方程的反边值问题，其中齐次Cauchy条件在初始时刻是不适定的。对于一类广泛的边值输入，建立了解的一些正则性。
The paper studies some inverse boundary value problem for simplest parabolic equations such that the homogenuous Cauchy condition is ill posed at initial time. Some regularity of the solution is established for a wide class of boundary value inputs.
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In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
On maximal parabolic regularity for non-autonomous parabolic operators
本文考虑与半双线性型相关的线性非齐次非自治抛物问题，其时间依赖性是不连续的。我们证明了对于这些问题，极大抛物正则性的性质可以外推到时间可积指数$r\neq 2$。这使我们能够证明非常一般的几何设置中不连续非自治二阶散度型算子的最大抛物型$L^r$ -正则性，并证明相关拟线性方程的存在性结果。
Regularity for Fully Nonlinear P-Laplacian Parabolic Systems: the Degenerate Case
研究一类强耦合退化抛物方程组有界弱解的Hölder正则性.
Regularity of the obstacle problem for the parabolic biharmonic equation
研究抛物型双调和方程障碍问题解的正则性。我们通过一个隐式时间离散化的问题进行分析，我们证明了一些规律性的解决方案。
Regularity results for fully nonlinear parabolic integro-differential operators
本文考虑具有对称核的完全非线性抛物型积分微分方程的正则性理论。我们能够找到Alexandrov-Backelman-Pucci估计的抛物线版本，其中0<σ<2。通过获得解的水平集的衰减估计，我们证明了解的Harnack不等式、Hölder正则性和C^{1，α}-正则性。
We obtain up to a flat boundary regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.
Parabolic Obstacle Problems. Quasi-convexity and Regularity
在广泛的一类所谓的抛物型障碍问题，它是如何提高最佳的正则性的解决方案，并因此如何获得相应的自由边界的时空正则性。
Maximal regularity for parabolic partial differential equations on manifolds with cylindrical ends
利用拟微分参数和预解式的R-有界性概念，给出了柱端流形上线性抛物型发展方程的极大正则性的一个简短的证明.
This note presents a regularity result with proof for an initial-boundary value problem of a linear parabolic system involving curl of the unknown vector field, subjected to the boundary condition of prescribing the tangential component of the solution.
Regularity results for nonlocal parabolic equations
我们调查最近的正则性结果抛物型方程涉及非局部算子，如分数拉普拉斯算子。我们推广了Felsinger-Kassmann（2013）的结果，得到了核关于勒贝格测度不绝对连续的非局部算子的正则性估计。
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
Further time regularity for fully non-linear parabolic equations
我们建立了完全非线性抛物型方程的解的时间导数的Hölder估计，它不一定有$C^{2,α}$估计。
A counterexample to $C^{2,1}$ regularity for parabolic fully nonlinear equations
我们解决了奇异抛物问题的自相似可解性，并表明抛物型完全非线性方程的解决方案预计不会是$C^{2,1}$。
Higher Hölder regularity for nonlocal parabolic equations with irregular kernels
本文研究了具有不规则核系数的非局部抛物型方程，在对非齐次项施加适当高阶可积性且对核系数作最小正则性假设的条件下，建立了该方程的高阶Hölder正则性。
Eventual regularity for the parabolic minimal surface equation
我们证明了抛物型极小曲面方程具有最终正则化效应，即在（严格正的）有限时间后，解变得光滑。
Regularity of Solutions of the Parabolic Normalized p-Laplace Equation
研究了抛物型归一化p-Laplace方程。证明了黏性解在Sobolev意义下具有局部属于$L^2$的时间导数。
A singular integral approach to the maximal L^p regularity of parabolic equations
本文描述了一种简单而基本的方法来求抛物型方程的极大L^p正则性，它只使用Volterra型奇异积分的概念.不需要解析半群、R-有界性或H^ \infty -谓词演算的知识。
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
Maximal parabolic regularity for divergence operators including mixed boundary conditions
证明了散度型椭圆二阶算子$A$在分布空间上满足极大抛物正则性，即使其下的区域是高度非光滑的，$A$的系数也是不连续的，并且$A$是用混合边界条件补的.应用于非光滑数据的拟线性抛物型方程。
Optimal regularity for the variable coefficients parabolic Signorini problem
本文针对变系数抛物型Signorini问题，利用Carleman估计和epiperimetric不等式，证明了在特定系数和非齐次项条件下解及正则自由边界的最优正则性。
Regularity for solutions of non local parabolic equations II
证明了抛物型非局部方程$u_t-Iu=f$的边界正则性和紧性结果，其中算子$I$不一定是平移不变的.由于这一点和正则性结果的平移不变的情况下，我们得到了$C^{1,α}$内部估计空间的非平移不变运营商在一定的假设下的时间规则性的边界数据。
Further Time Regularity for Non-Local, Fully Non-Linear Parabolic Equations
在边界条件较弱的假设下，建立了非局部抛物型方程解的时间导数的Hölder估计.因此，我们能够扩展粗糙核的埃文斯-克雷洛夫估计抛物型方程。
Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem
我们给出了一个全面的治疗抛物Signorini问题的基础上推广的Almgren的单调性的频率。其中包括解的最优正则性证明、自由边界点的分类、正则集的正则性和奇异集的结构。
Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems
本文针对非光滑域上带混合边界条件的非自治拟线性抛物系统，建立了$L^p$-极大正则性，通过外推技术获得一致估计，仅需温和边界正则性及有界可测复系数，并允许时间非局部非线性，涵盖稳态不满足椭圆性的情形。
Regularity of Solutions to a Parabolic Free Boundary Problem with Variable Coefficients
本文的主要结果是证明了变系数抛物型自由边界问题的黏性解在具有Lipschitz自由边界且满足非退化条件的条件下是Lipschitz连续的.这是这个问题的最佳规律。
Regularity of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
考虑一维抛物型自由边界值问题，在自由边界上具有非局部（积分-微分）条件。得到了自由边界的$C^m$正则性结果.特别地，给出了自由边界无穷可微的一个充要条件。
Regularity of solutions to the parabolic fractional obstacle problem
在本文中，我们研究了一个抛物型版本的分数障碍问题，证明了几乎最优的正则性的解决方案。这个问题是由Menton提出的美式期权模型所激发的，该模型将股票价格动态中的不连续路径引入期权评估理论。
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.
Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations
该文研究一致退化抛物方程解的全局Hölder正则性，并在极限一致退化椭圆方程特征指数的附加假设下，分析时间趋于无穷时解的收敛性。
In this paper, we study the second order Sobolev regularity of solutions to the parabolic $p$-Laplace equation. For any $p$-parabolic function $u$, we show that $D(|Du|^{\frac{p-2+s}{2}}Du)$ exists as a function and belongs to $L^{2}_{\text{loc}}$ with $s>-1$ and $1<p<\infty$. The range of $s$ is sharp.
This paper deals with a class of nonlinear anisotropic parabolic equations with degenerate coercivity. Using the anisotropic Gagliardo-Nirenberg-type inequality, we prove some existence and regularity results for the solutions under the framework of anisotropic Sobolev spaces, which generalize the previous results of [10,18,23].
Generalized Morrey regularity for parabolic equations with discontinuity data
本文研究了抛物方程在不连续数据下的正则性问题，通过广义Morrey空间中的连续性估计，探讨了线性均匀抛物方程的全局解的正则性。
Singular p-Laplacian parabolic system in exterior domains: higher regularity of solutions and related properties of extinction and asymptotic behavior in time
我们考虑p-Laplacian抛物型方程组在外部区域上的IBVP。我们证明了p \in（2n/（n+2），2n/（n+1））的边界正则性，灭绝性和p= 2n/（n+1）的指数衰减性.
On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative
在连续函数空间和Hölder连续函数空间中，证明了带分数阶时间导数$\mathbb{D}_t^α$的混合线性Cauchy-狄利克雷问题的两个极大正则性结果.该衍生物的含义为Caputo，$α$的含义为$(0, 2)$。在$α= 1$的情况下，我们获得了混合抛物问题的最大正则性结果，已知在Artemtica文献。
On the regularity of the Non-dynamic Parabolic Fractional Obstacle Problem
在所谓的非动态抛物型分数阶障碍问题中，给出了如何获得解的空间导数的更高正则性和最佳正则性。进一步证明了在正抛物密度的自由边界点上，解的时间导数是Hölder连续的.最后，在正则自由边界点处，对任意分数$s\in(0,1)$，得到了相应自由边界的时空正则性。
Linear reaction-diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal Lp regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.
Hölder regularity for parabolic De Giorgi classes in metric measure spaces
给出了度量测度空间中抛物型De Giorgi类中函数的Hölder连续性的一个证明.我们假设测度是加倍的，以支持弱的$(1,p)$ -Poincaré不等式并满足环形衰减性质.
Partial regularity for parabolic systems with VMO-coefficients
我们建立了抛物方程组向量值解的部分正则性，其中系数关于（x，t）可能是不连续的。更确切地说，我们假设一个VMO条件关于（x，t）和连续性关于u和证明Hölder连续性的解决方案以外的奇异集。
Regularity Results for a class of Semilinear Parabolic Degenerate Equations and Applications
本文考虑了一个可能强退化的抛物半线性问题，它可以应用于金融衍生品定价的微分模型。我们证明了所要求的正则性应用伊藤的公式，用于建立微分模型。
Everywhere regularity of certain types of parabolic systems
在本文中，我讨论的非线性抛物型方程组的标量扩散方程的推广。我表明，当潜在的是一个凸函数，只依赖于规范的解决方案，然后有界的弱解这些抛物型系统是到处持有人连续，从而无处不在光滑。证明了该结果的证明方法可以方便地用于简化Wiegner关于强耦合抛物方程组有界弱解处处正则性的证明。
Boundary regularity for a general nonlinear parabolic equation in non-divergence form
本文通过构造障碍函数族刻画了一般非线性抛物方程的正则边界点，推广了标准p-拉普拉斯方程，并证明q<2时单个障碍不充分。
We investigate global and local regularity of generalized solutions to parabolic initial-boundary value problem for Petrovskii system of second order differential equations. Results are formulated in terms of the belonging of right-hand sides of the problem to some generalized Sobolev spaces. We also obtain new sufficient conditions under which the generalized solution should be classical.
Regularity for solutions of non local parabolic equations
研究抛物型完全非线性非局部方程解的正则性。我们证明了在空间和时间上的Holder正则性以及对于平移不变方程的Holder正则性，并在不同的核假设下证明了空间导数的Holder正则性。证明依赖于一个弱抛物ABP的启发，最近的工作L。西尔维斯特和K. Tso和L.王.我们的结果保持一致的顺序的方程去2，使我们能够理解的非局部理论作为一个扩展的经典之一。
Local regularity for quasi-linear parabolic equations in non-divergence form
本文研究非散度型$p$ -Laplacian型非齐次退化奇异抛物型方程的黏性解。我们提供当地的Hölder和Lipschitz估计的解决方案。在退化的情况下，我们证明了梯度的Hölder正则性。我们的研究是基于替代的方法和平坦度估计的改进相结合。
In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are assumed on that part of the lateral boundary.
On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach
证明了对于双调和方程，典型抛物面顶点的最优正则性准则可以用1934年导出的热方程的Petrovskii型Osgood-Dini积分条件表示.讨论了第一Fourier系数法在其它偏微分方程中的推广。综述了椭圆和抛物型偏微分方程的边界点正则性。
In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{σ,σ/2}$-regularity for every $0<σ<1$, as well as $H^{β,β/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<β<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.
Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth
本文在Lipschitz区域上建立了拟线性散度型椭圆和抛物型方程弱解的整体正则性，并在低阶项上给出了控制增长条件.首项系数属于关于$x$具有小平均振荡的BMO函数类。
A note on Besov regularity for parabolic initial boundary value problems
本文考虑了具有Besov数据的Lipschitz区域上的柱上热方程的初边值问题。我们得到了一个正则性指数的解决方案，提高了非线性逼近方法收敛速率。
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.
We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $Ω\times \R^+$, where $Ω$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.
Maximal $L_q$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces
本文研究了变系数分数阶抛物型方程，在任意非负阶Bessel位势空间中建立了极大$L_q$-正则性，并应用于证明分数阶多孔介质方程与非局部Kirchhoff方程的高阶正则性和瞬时平滑性。
In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.
Regularity for obstacle problems to anisotropic parabolic equations
本文沿用Dibenedetto的内在标度法，在障碍仅满足Hölder连续性的条件下，证明了部分各向异性抛物方程障碍问题弱解的局部Hölder连续性。
We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.
We prove $C^{1, α}$ regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.
A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts
研究了具有奇异无散度漂移的非齐次抛物型方程弱解的加权Sobolev正则性。在漂移满足一定的正则性条件下，建立了弱解梯度的局部加权$L^p$ -估计。我们的结果改进了经典的边界情况下取代$L^\infty$ -假设的解决方案的解决方案在John-Nirenberg \textup{BMO}空间。所得结果也推广到具有小振荡椭圆对称系数的散度型抛物型方程，从而改进了许多已知结果。
On the regularity of entropy solutions to stochastic degenerate parabolic equations
本文研究了具有各向异性退化性和随机强迫的拟线性抛物型方程熵解的正则性，通过使用包含两个输运方程的动能公式并利用“抛物正则性”，在更弱的非退化条件下建立了时空正则性。
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