Scaling limit in dispersive equations

色散方程中的标度极限

• 批准号: 1142293
• 负责人: Benoit Pausader
• 金额: $ 13.72万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1142293&HistoricalAwards=false
• 关键词: Scaling; limit; dispersive; equations

This project aims at developing the study of three fundamental equations from physics. First, stability issues for the 2-fluids Euler-Maxwell equation, which is one of the fundamental equations in plasma physics, are considered. The goal is to prove that, under certain conditions, small perturbations of an equilibrium will not develop shocks, and that, actually, the plasma will get back to equilibrium, even in the absence of dissipation. Such a result would be of great physical and mathematical importance as it is known to be false for the compressible Euler equation in the absence of a self-consistent electromagnetic field. In a second part, a study is made of the Schrodinger equation on a curved background. In this case, many classical tools from the Euclidean theory break down and one expects the appearance of many new phenomena due to the influence of the geometry of the background. In particular, a study will be made of the effect of the growth of the volume on the global existence and global behavior of the solutions to the energy-critical equation. Finally, in a third part, ideas developed before are used to upgrade some known results about the homogeneous fourth-order equation to the more physical inhomogeneous equation. Understanding how a fluid can be stabilized by a self-consistent electromagnetic field in the absence of any friction represents a cross-disciplinary collaboration between pure mathematics, applied mathematics, and physics, with applications in fluid engineering. More specifically, proving that a plasma at rest is stable under small perturbations would be a major physical discovery and would most certainly greatly enhance our ways to control plasma. This is especially fitting since plasma stability is one of the main factor limiting performances in tokamaks. Finally, beyond industrial applications, plasma represents the state of more than 99% of the matter in the universe and any work providing better understanding of this state would be of great importance. Dispersive equations and equations on curved spaces (manifolds) provide a rich area of interaction between various branches of mathematics as well as between different sciences. The study of equations on curved spaces is of interest to geometers, analysts, and number theorists in mathematics, as well as to theoretical physicists working in quantum chaos and general relativity. Fourth-order equations naturally arise in many different branches of physics and mathematics, especially those linked with elasticity and are critical to understand phenomena as diverse as the movement (and possible oscillations and breakdown) of bridges or the structure of blood vessels and the interaction between their membranes and the biofluids.

该项目旨在发展对物理学中三个基本方程的研究。首先，考虑等离子体物理学基本方程之一的 2 流体欧拉-麦克斯韦方程的稳定性问题。目标是证明，在某些条件下，平衡的小扰动不会产生冲击，并且实际上，即使没有耗散，等离子体也会恢复平衡。这样的结果将具有重大的物理和数学重要性，因为众所周知，在缺乏自洽电磁场的情况下，可压缩欧拉方程是错误的。第二部分研究了弯曲背景下的薛定谔方程。在这种情况下，欧几里得理论中的许多经典工具都失效了，并且由于背景几何形状的影响，人们预计会出现许多新现象。特别是，将研究体积增长对能量临界方程解的全局存在性和全局行为的影响。最后，在第三部分中，使用之前提出的想法将齐次四阶方程的一些已知结果升级为更物理的非齐次方程。了解如何在没有任何摩擦的情况下通过自洽电磁场来稳定流体代表了纯数学、应用数学和物理学之间的跨学科合作以及在流体工程中的应用。更具体地说，证明静止的等离子体在小扰动下是稳定的将是一项重大的物理发现，并且肯定会极大地增强我们控制等离子体的方法。这是特别合适的，因为等离子体稳定性是限制托卡马克性能的主要因素之一。最后，除了工业应用之外，等离子体代表了宇宙中 99% 以上物质的状态，任何能够更好地理解这种状态的工作都将非常重要。色散方程和弯曲空间（流形）上的方程为数学各分支之间以及不同科学之间的相互作用提供了丰富的领域。弯曲空间方程的研究引起了数学领域的几何学家、分析师和数论学家以及研究量子混沌和广义相对论的理论物理学家的兴趣。四阶方程自然出现在物理和数学的许多不同分支中，特别是那些与弹性相关的分支，并且对于理解诸如桥梁的运动（以及可能的振荡和破坏）或血管的结构以及之间的相互作用等各种现象至关重要。它们的膜和生物体液。