Profiling singularities of geometric PDE

这个项目涉及非线性抛物型偏微分方程组(PDE)和由几何对象(如曲率)演化而成的几何对象所满足的系统。这些偏微分方程用在程序中，以使给定的几何形状演变成在适当意义上“最佳”或“规范”的几何形状，因此可以进行分类。但是因为这些偏微分方程会产生奇点，所以对这些奇异行为进行分类是成功完成这些程序所必需的。分析(有限或无限时间)奇点的形成是这个项目的统一目标。一种关键的方法是使用匹配的渐近性，这是一种技术，可以提供关于解成为奇点的点集的最精确描述，以及解在该奇点的时空邻域中的行为。该方案的主要目的包括：(1)去掉平均曲率流(MCF)和Ricci流(RF)渐近奇点分析中的对称性假设，从而证明某些奇点轮廓在严格意义上是“普适的”；(2)构造和分析(非一般的)II型RF奇点，这种奇点的形成速度比自然抛物线速度慢，因此具有更快的曲率爆破；(3)构造余维-2 RF奇点，并研究它们的渐近性、一般性和稳定性；(4)构造和研究了复杂曲面(和高维复杂流形)的RF局部奇性形成的新实例，并将其应用于这些维奇性模型的分类；(5)研究了低维RF保持的乘积结构和相关曲率条件的稳定性(正确理解)；(6)证明了几何PDE的奇异轮廓连续地依赖于它们的初始数据，并应用于拓扑学；以及(7)研究在不同收敛格式下无限时间RF奇性模型的形成和稳定性。几何偏微分方程(PDE)理论与非线性双曲型和色散型方程有着惊人的相似之处。此外，曲率流动中产生的偏微分方程组与模拟热传播、石油在页岩和薄膜中的运动、多孔介质中的燃烧以及等离子体物理中的某些效应的方程非常相似。在所有这些应用中，底层模型基本上是非线性的，这一特性导致相关的PDE发展各种关键或奇异行为。这些模型的实用性需要对这些行为有精确的数学理解。这个项目将进一步发展数学技术，特别是匹配的渐近展开，这应该有助于分析这些不同系统和应用中的奇点形成。

This project is concerned with nonlinear parabolic partial differential equations (PDE) and systems satisfied by geometric objects evolving by geometrically natural quantities such as curvature. These PDE are used in programs to evolve given geometries toward ones which are in suitable senses "optimal" or "canonical," and which are thus amenable to classification. But because these PDE generically develop singularities, a classification of those singular behaviors is necessary for the successful completion of those programs. Analyzing formation of (finite- or infinite-time) singularities is the unifying goal of this project. A key approach is the use of matched asymptotics, a technique that can provide the most precise description of the set of points on which a solution becomes singular, and of the behavior of the solution in a space-time neighborhood of that singularity. Major objectives of this proposal include: (1) removing symmetry hypotheses in asymptotic singularity analysis for mean curvature flow (MCF) and Ricci flow (RF), thereby proving that certain singularity profiles are "universal" in a rigorous sense; (2) constructing and analyzing (non-generic) Type-II RF singularities, which form more slowly than the natural parabolic rate and thus feature faster curvature blow-up; (3) constructing codimension-2 RF singularities and studying their asymptotics, genericity, and stability; (4) constructing and studying new examples of RF local singularity formation for complex surfaces (and complex manifolds of higher dimension) with applications to the classification of singularity models in those dimensions; (5) studying stability (properly understood) of product structures and related curvature conditions preserved by RF in low dimensions; (6) showing that singular profiles of geometric PDE depend continuously on their initial data, with applications to topology; and (7) studying formation and stability of infinite-time RF singularity models under distinct convergence schemes designed to provide asymptotics at temporal infinity, along with other geometric information.The theory of geometric partial differential equations (PDE) has surprising similarities with nonlinear hyperbolic and dispersive equations. Furthermore, the PDE that arise in curvature flows are remarkably similar to equations that model heat propagation, the movement of oil in shale and thin films, combustion in porous media, and certain effects in plasma physics. In all of these applications, the underlying models are fundamentally nonlinear, a property which causes the associated PDE to develop various critical or singular behaviors. The utility of these models requires a precise mathematical understanding of these behaviors. This project will further develop mathematical techniques, particularly matched asymptotic expansions, that should help the analysis of singularity formation in these varied systems and applications.