Nonlinear partial differential equations arising in differential geometry

微分几何中出现的非线性偏微分方程

• 批准号: 0245208
• 负责人: Simon Brendle
• 金额: $ 10.11万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245208&HistoricalAwards=false
• 关键词: Nonlinear; partial; differential; equations; arising

PI: Simon Brendle, Princeton UniversityDMS-0245208The proposed project is concerned with the use of analytical tools to understand the qualitative behavior of nonlinear partial differential equations arising in both differential geometry and mathematical physics. For example, we intend to study harmonic maps from higher dimensional Riemannian manifolds into general target manifolds. We also plan to study nonlinear wave equations, using methods from harmonic analysis as well as vector field methods originating in differential geometry. In addition, we aim for a general convergence result for the Yamabe flow in conformal geometry. All known results in this direction either assume the manifold to be locally conformally flat (such as the work of R. Ye), or they require a rather restrictive bound on the initial energy (such as the recent work of M. Struwe and H. Schwetlick). It is known that shortly before a singularity the solution must look like a superposition of "peak solutions", whose asymptotic profile is explicitly known. A careful analysis of the interaction between different peaks suggests that it costs energy to make the peak higher, i.e. to concentrate the energy on a smaller region. Since the evolution equation is designed to decrease the energy, this indicates that no singularities should form. While the result is known for initial energy less than two peaks, the general case offers a more interesting picture.The questions to be studied in this project are mainly motivated by differential geometry. However, it may seem surprising that many of these equations also play an important role in applied sciences. For example, harmonic maps into the two-dimensional sphere are closely related to the Landau-Lifschitz equation for a macroscopic ferromagnetic continuum. Moreover, the Yamabe flow in conformal geometry can be reduced to the fast diffusion case of the porous medium equation. In the case of positive scalar curvature, the effect of the reaction term is opposed to that of the diffusion term, and it is a non-trivial issue to decide which of these effects will prevail.

PI：Simon Brendle，普林斯顿大学DMS-0245208该项目涉及使用分析工具来理解微分几何和数学物理中出现的非线性偏微分方程的定性行为。例如，我们打算研究从高维黎曼流形到通用目标流形的调和映射。我们还计划使用调和分析方法以及源自微分几何的矢量场方法来研究非线性波动方程。此外，我们的目标是获得共形几何中 Yamabe 流的一般收敛结果。这个方向上的所有已知结果要么假设流形是局部共形平坦的（例如 R. Ye 的工作），要么需要对初始能量有相当严格的限制（例如 M. Struwe 和 H. Schwetlick 的最近工作）。众所周知，在奇点之前不久，解必须看起来像“峰值解”的叠加，其渐近轮廓是明确已知的。对不同峰之间相互作用的仔细分析表明，使峰更高（即将能量集中在较小的区域）需要消耗能量。由于演化方程旨在降低能量，这表明不应形成奇点。虽然已知结果的初始能量少于两个峰值，但一般情况提供了更有趣的情况。该项目要研究的问题主要是由微分几何激发的。然而，令人惊讶的是，其中许多方程在应用科学中也发挥着重要作用。例如，二维球体的调和映射与宏观铁磁连续体的朗道-利夫希茨方程密切相关。此外，共形几何中的山边流可以简化为多孔介质方程的快速扩散情况。在正标量曲率的情况下，反应项的影响与扩散项的影响相反，决定哪种影响占主导地位是一个重要的问题。