Nonlinear evolution equations on singular manifolds

奇异流形上的非线性演化方程

• 批准号: 329717144
• 负责人: Professor Dr. Elmar Schrohe
• 金额: --
• 依托单位: Institut für Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/329717144?language=en
• 关键词: Nonlinear; evolution; equations; singular; manifolds

The aim of this project is to study quasilinear parabolic evolution equations on singular spaces in order to obtain a precise understanding of the influence of the singularity on the evolution process. Prototypes of problems we are interested in are the Cahn-Hilliard equation, the porous medium equation and geometric flows. The former two equations have been studied traditionally in domains in Euclidean space, the flows on smooth manifolds. Recently, however, interest in their analysis on singular objects has risen.We will focus on manifolds with conical singularities both with and without boundary, and on manifolds with edges. We are interested in the short and long time existence of solutions, their regularity and asymptotics near the singular set and their long time behavior.Singular analysis has seen a rapid development during the past 30 years. While initially, the analysis mainly focused on linear elliptic problems and applications in index theory, over the past 15 years, tools for parabolic and hyperbolic problems on singular spaces have been developed. Apart from our own contributions we build on important work by Mazzeo and collaborators, Bahuaud and Vertman, and Shao.The principal tools for the linear part of the theory are the pseudodifferential calculi for conically and edge degenerate operators. Here, the basic concepts exist, but new parts will have to be developed for the analysis of the nonlinearities. One has to find suitable closed extensions for the cone Laplacian on manifolds with boundary and for the edge Laplacian, determine the structure of their resolvents and establish maximal $L^p$-regularity; moreover one needs to gain a better understanding of the real interpolation spaces between the base space and the domain of the extensions.In a subsequent step we shall study existence, uniqueness and regularity of short times solutions to the above problems via maximal $L^p$-regularity techniques. As our previous work indicates, singularity effects and asymptotic properties of the solutions near the singular set should already be visible at this point. The next task will be to establish the existence of long time solutions and their asymptotics. We will do this by extending classical techniques like Hölder estimates for quasilinear equations to the singular setting and combining them with maximal $L^p$-regularity theory. Altogether, we hope to obtain a clear view of the behavior of the solutions close to the singularity, in particular we expect to show how the local geometry near the singular set determines the regularity and the asymptotics of the evolution both for short and long times.

本项目的目的是研究奇异空间上的拟线性抛物型发展方程，以便精确地理解奇异性对发展过程的影响。我们感兴趣的问题的原型是Cahn-Hilliard方程，多孔介质方程和几何流。前两个方程传统上是在欧氏空间的区域上，即光滑流形上的流动中研究的。然而，最近，在奇异物体上的研究兴趣已经上升，我们将集中在有边界和无边界的锥奇点流形，以及有边流形。奇异性分析的研究主要集中在解的短时和长时存在性、解在奇异集附近的正则性和渐近性以及解的长时行为等方面。虽然最初的分析主要集中在线性椭圆问题和指数理论的应用，但在过去的15年里，奇异空间上的抛物和双曲问题的工具已经开发出来。 除了我们自己的贡献，我们建立在重要的工作Mazzeo和合作者，Bahuaud和Vertman，和Shao.的主要工具的线性部分的理论是pseudodiomatic微积分的圆锥和边缘退化运营商。在这里，基本的概念是存在的，但新的部分将不得不开发的非线性分析。对于有边流形上的锥Laplacian和边Laplacian，我们需要找到合适的闭扩张，确定它们的预解式的结构，并建立极大L^p-正则性;此外，人们需要更好地理解基空间和扩张域之间的真实的插值空间。在随后的步骤中，我们将研究存在性，利用极大L^p$-正则性技巧得到了上述问题的短时解的唯一性和正则性。正如我们以前的工作表明，奇异性效应和渐近性质的解决方案附近的奇异集应该已经可见在这一点上。下一个任务是建立长时间解的存在性及其渐近性。我们将通过扩展经典技术，如拟线性方程的Hölder估计到奇异设置，并将它们与最大L^p$-正则性理论相结合来做到这一点。总之，我们希望得到一个清晰的视图的行为的解决方案接近奇点，特别是我们希望展示如何在奇异集附近的局部几何确定的规律性和渐近的演化，无论是短期和长期的。