New Challenges in Nonlinear PDEs.

非线性偏微分方程的新挑战。

• 批准号: 1201443
• 负责人: Andrea Nahmod
• 金额: $ 25万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201443&HistoricalAwards=false
• 关键词: New; Challenges; Nonlinear; PDEs.

This project deals with research at the interface of harmonic analysis, nonlinear partial differential equations, geometry, and probability. On the one hand, it is concerned with the study of dispersive nonlinear wave phenomena from a nondeterministic viewpoint. In the last two decades enormous progress has been made in settling questions on existence of solutions to dispersive equations, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena, where sophisticated tools from nonlinear Fourier analysis, geometry, and analytic number theory have played crucial roles. Yet there remain some fundamental obstacles. A natural approach to overcome them is to consider evolution equations from a nondeterministic point of view and to incorporate into the analysis tools from probability. Some of the issues to be investigated in this project are the role of randomization in the well-posedness theory, the almost sure (in the sense of probability) global-in-time existence of solutions, the existence and dynamical properties of associated Gibbs measures, and the behavior of statistical ensembles under gauge transformations. On the other hand, the principal investigator will study the existence and long-time dynamics of special types of solutions to certain hyperbolic (or nonelliptic) nonlinear Schrodinger equations and systems. The aim is to develop a rigorous mathematical analysis of models arising in connection with the theory of vortex filaments, ferromagnetism, and current work in nonlinear optics (e.g., examining the evolution of optical pulses in normally dispersive optical media), models that have attracted the attention of the physics community. Wave phenomena in physics such as light, sound, and gravity are mathematically modeled using partial differential equations. Nonlinear wave models arise in quantum mechanics, ferromagnetism, vibrating systems, semiconductors, and optical fibers. Dispersive equations model important wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves while conserving energy or mass. The best known dispersive equations are the nonlinear Schrodinger equations that govern the motion of quantum particles (e.g., electrons), the macroscopic dynamics of the Bose-Einstein condensate, and signals in fiber optics. This project focuses on the rigorous mathematical analysis of dispersive equations that arise naturally in physics and engineering. The synergy of Fourier analysis, probability, geometry, and analytic number theory provides a well-adapted and powerful toolbox to study the nonlinear effects that allow waves to interact and produce new, modified propagation patterns. The problems that the principal investigator will study are of particular interest in the study of long internal gravity waves in deep stratified fluids, the theory of vortex filaments and aerodynamics, and in current work on nonlinear fiber optics that is of fundamental importance in today's telecommunication systems and internet traffic. The ubiquitous role of mathematics is to lay the foundations through rigorous research for the best predictions, based on which the technological advances and engineering applications we enjoy every day, can be efficiently enabled. The training of students and junior researchers is an integral part of the project.

该项目涉及谐波分析，非线性部分微分方程，几何形状和概率的界面的研究。 一方面，它与非确定性观点的分散性非线性波现象有关。在过去的二十年中，在解决有关分散方程，长期行为和奇异性形成的解决方案的问题方面取得了巨大进展。这项工作的推力主要集中在波浪现象的确定性方面，在这些方面，非线性傅立叶分析，几何和分析数理论的复杂工具起着至关重要的作用。然而，仍然存在一些基本障碍。克服它们的一种自然方法是从非确定的角度考虑进化方程，并从概率中纳入分析工具中。该项目中要研究的一些问题是随机在适当的度理论中的作用，几乎确定（从概率意义上）全球及时存在解决方案，相关Gibbs测量的存在和动态特性以及量学转换下的统计集合的行为。另一方面，主要研究者将研究某些双曲线（或非椭圆形）非线性schrodinger方程和系统的特殊类型解决方案的存在和长期动力学。目的是对与涡流丝，铁磁性和非线性光学的当前工作有关的模型进行严格的数学分析（例如，检查正常分散光学介质中光脉冲的演变），这些模型吸引了物理社区的注意力。 使用部分微分方程对光，声音和重力等物理学（例如光，声音和重力）的波浪现象进行了建模。非线性波模型在量子力学，铁磁，振动系统，半导体和光纤中出现。分散方程模型自然界重要的波传播现象。他们的解决方案是随着时间的流逝，在节省能量或质量的同时，随着时间的流逝，波浪在时空中散布。最著名的分散方程是控制量子颗粒（例如电子），Bose-Einstein冷凝物的宏观动力学和光纤中信号的非线性Schrodinger方程。该项目着重于对物理和工程中自然出现的分散方程的严格数学分析。傅立叶分析，概率，几何学和分析数理论的协同作用提供了一个良好的适应性和强大的工具箱，以研究非线性效应，使波浪可以相互作用并产生新的，修改的传播模式。首席研究者将研究的问题在研究深层流体，涡流细丝和空气动力学的理论以及当前关于非线性光纤光学的工作中特别重要的研究在当今电视连接系统和互联网流量中至关重要的当前工作。数学的无处不在作用是通过严格的研究为最佳预测奠定基础，这是根据我们每天享受的技术进步和工程应用程序可以有效地实现的。学生和初级研究人员的培训是该项目不可或缺的一部分。