Nonlinear evolution equations on singular manifolds

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，以获得对奇性对发展过程的影响的精确理解。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动。前两个方程的研究传统上是在欧氏空间的区域上进行的，即光滑流形上的流。然而，最近他们对奇异对象的分析兴趣上升了。我们将集中在有边界和无边界的锥形奇点流形上，以及有边的流形上。我们感兴趣的是解的短时间和长时间的存在性，它们在奇点集附近的正则性和渐近性，以及它们的长时间行为。奇点分析在过去的30年里得到了迅速的发展。虽然最初的分析主要集中在线性椭圆问题和指数理论中的应用，但在过去的15年里，奇异空间上的抛物线和双曲型问题的工具已经发展起来。除了我们自己的贡献之外，我们还建立在Mazzeo及其合作者Bahuaud和Vertman以及Shao的重要工作的基础上。该理论的线性部分的主要工具是圆锥和边缘退化算子的伪微积分。这里，基本概念已经存在，但必须开发新的部件来分析非线性。我们需要对带边界流形上的锥拉普拉斯算子和边缘拉普拉斯算子找到合适的闭扩张，确定其预解子的结构，建立极大的$L p-正则性；此外，还需要更好地理解基空间和扩张的区域之间的实内插空间.在接下来的步骤中，我们将利用极大的$L^p-正则性技巧研究上述问题的短时间解的存在唯一性和正则性.正如我们以前的工作所表明的，奇性效应和解在奇点集附近的渐近性质在这一点上应该已经可见。下一个任务将是建立长时间解的存在性及其渐近性。我们将通过将拟线性方程的Hölder估计等经典技巧推广到奇异环境，并将它们与极大值$L^p$正则理论相结合来实现这一点。总之，我们希望对奇点附近的解的行为有一个清晰的认识，特别是我们希望展示在奇点附近的局部几何如何在短时间和长时间内决定演化的正则性和渐近性。

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.