Profiling singularities of geometric PDE

这个项目关注的是非线性抛物型偏微分方程（PDE）和系统所满足的几何对象的几何自然量，如曲率的演变。这些偏微分方程用于程序中，以使给定的几何形状朝着在适当的意义上是“最优”或“规范”的几何形状发展，从而可以进行分类。但是，因为这些PDE一般发展奇异性，这些奇异行为的分类是必要的，这些程序的成功完成。分析（有限或无限时间）奇点的形成是这个项目的统一目标。一个关键的方法是使用匹配渐近，一种技术，可以提供最精确的描述的一组点上的解决方案成为奇异的，以及该解决方案的行为在时空附近的奇异性。本文的主要目的是：（1）消除平均曲率流（MCF）和Ricci流（RF）的渐近奇异性分析中的对称性假设，从而证明某些奇异性轮廓在严格意义上是“普适的”;（2）构造和分析一个具有对称性的奇异性轮廓;（3）构造和分析一个具有对称性的奇异性轮廓;（4）构造和分析一个具有对称性的奇异性轮廓;（5）构造和分析一个具有对称性的奇异性轮廓。（非通用）II型RF奇点，其形成比自然抛物线速率更慢，因此具有更快的曲率爆破;（3）构造了余维2的RF奇异性，并研究了它们的渐近性、一般性和稳定性;（4）构造和研究了复杂曲面RF局部奇异性形成的新例子（和高维的复流形），并应用于这些维度的奇异模型的分类;（5）研究稳定性（6）证明了几何偏微分方程的奇异轮廓连续依赖于它们的初始数据，并应用于拓扑学;以及（7）研究在设计为提供时间无穷大处的渐近性的不同收敛方案下的无限时间RF奇异性模型的形成和稳定性，沿着其他几何信息，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，在曲率流中出现的PDE与模拟热传播、页岩和薄膜中石油的运动、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发展出各种临界或奇异行为。 这些模型的效用需要对这些行为有精确的数学理解。该项目将进一步发展数学技术，特别是匹配渐近展开，这将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.