Harmonic Analysis, Mathematical Physics, and Nonlinear PDE

调和分析、数学物理和非线性偏微分方程

• 批准号: 0653841
• 负责人: Wilhelm Schlag
• 金额: $ 28.8万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653841&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Mathematical; Physics; Nonlinear

Harmonic Analysis, Mathematical Physics and Nonlinear PDEAbstract of Proposed ResearchWilhelm Schlag This project is explore the long time behavior, or prove that singularities will form in finite time, of solutions of equations that arise in mathematical physics. The equations under consideration typically admit nonlinear bound states (solitons or instantons) and much research has recently been devoted to the perturbative analysis of such solutions. It is well known that solutions of nonlinear Schroedinger and wave equations of the focusing type may blow up in finite time (if the energy of the data is negative, for example). It turns out, however, that global solutions exist if the data belong to a submanifold of finite co-dimension (a "center-stable" manifold in the language of dynamical systems). We shall investigate whether there is a manifold that divides a region of blow-up from one of scattering. Our goal is to obtain a deeper understanding of blow-up phenomena. Recently, progress was made for the critical wave-map equation into the two-dimensional sphere with regard to blow-up. It can be shown in a very precise and quantitative way that blow-up for this equation occurs through the bubbling off of energy via a non-constant harmonic map. Moreover, it turns out that the blow-up rate can be prescribed a priori. Similar phenomena occur for the semi-linear energy critical focusing equation in three plus one dimensions. Currently we do not understand which classes of equations admit this kind of phenomenon.Much of the success of science and engineering lies with its effective use of mathematical tools, both in terms of modeling and for computational simulation. The nonlinear Schroedinger equation arises in various applications in optics where a bound state (soliton) for represents a particle, or beam, that travels for a long time without disintegrating. An important issue is to understand the stability or instability of these solitons. That is, whether they persist under small perturbations or not? The theoretical understanding of these issues is very difficult, and is requiring new insights into mathematical problems. This project will investigate these problems and develop methods that may be used by practicing scientists and engineers.

谐波分析，数学物理学和非线性pdabstract的拟议研究wilhelmschlag这个项目正在探索长期行为，或者证明在数学物理学中出现的方程式解决方案的解决方案将形成奇异性。所考虑的方程通常允许非线性结合状态（孤子或Instantons），最近许多研究致力于对此类溶液的扰动分析。众所周知，非线性Schroedinger的溶液和聚焦类型的波动方程可能会在有限的时间内爆炸（例如，如果数据的能量为阴性）。但是，事实证明，如果数据属于有限的共同维度（动态系统语言中的“中心稳定”歧管），则存在全局解决方案。我们将调查是否有一个歧管将炸毁区域与散射之一分开。我们的目标是获得对爆炸现象的更深入的了解。最近，关于爆炸的临界波映射方程式进入二维领域的进展。可以以非常精确和定量的方式显示，该方程式通过非恒定谐波映射通过能量冒泡而发生爆炸。此外，事实证明，可以先验规定爆炸率。半线性能量临界聚焦方程在三个加一个维度上也发生了类似的现象。目前，我们不了解哪些等式类别承认这种现象。科学和工程的成功在于其在建模和计算模拟方面的有效利用数学工具。非线性Schroedinger方程在光学元件中出现在各种应用中，其中一个绑定的状态（soliton）代表粒子或光束，该粒子或光束在不瓦解的情况下长时间传播。一个重要的问题是了解这些孤子的稳定性或不稳定。也就是说，它们是否在小扰动下持续存在？ 对这些问题的理论理解非常困难，并且需要对数学问题的新见解。该项目将调查这些问题，并开发从执业科学家和工程师使用的方法。