Nonlinear partial differential equations arising in differential geometry

主要研究者：Simon Brendle，Princeton UniversityDMS-0245208拟议的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一般目标流形的调和映射。我们还计划研究非线性波动方程，使用调和分析的方法以及源自微分几何的向量场方法。此外，我们的目标是在共形几何Yamabe流的一般收敛结果。在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（如R。Ye），或者它们需要对初始能量有相当严格的限制（例如M。Struwe和H. Schwetlick）。它是已知的，在奇点之前不久的解决方案必须看起来像一个叠加的“峰值解决方案”，其渐近轮廓是明确已知的。对不同峰之间的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上，这会消耗能量。由于演化方程是为了降低能量而设计的，这表明不应该形成奇点。虽然结果是已知的初始能量小于两个峰值，一般情况下提供了一个更有趣的图片。在这个项目中要研究的问题主要是由微分几何的动机。然而，似乎令人惊讶的是，这些方程中的许多方程在应用科学中也发挥着重要作用。例如，到二维球的调和映射与宏观铁磁连续统的朗道-利夫希茨方程密切相关。此外，共形几何中的Yamabe流可以归结为多孔介质方程的快速扩散情形。在正标量曲率的情况下，反应项的效果与扩散项的效果相反，并且决定这些效果中哪一个将占上风是一个重要的问题。

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.