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Some Dynamical Questions in Hamiltonian Partial Differential Equations

哈密顿偏微分方程中的一些动力学问题

基本信息

批准号：
2007457
负责人：
Zhiwu Lin
金额：
$ 24.2万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007457&HistoricalAwards=false
关键词：
Some Dynamical Questions Hamiltonian Partial

项目摘要

For many physical phenomena dissipation of energy effects are very small and ignoring them could provide a good approximation. Such phenomena include large scale water waves in the ocean, the plasmas in space and controlled fusion devices, dynamics of stars and galaxies in astrophysics. The dynamical behavior of mathematical models of these phenomena is complex and includes formation of coherent structures, such as tsunamis in the ocean, rotating stars (e.g. Sun) and the observed galaxy structures, to name a few. One of the objectives of this project is to contribute to the understanding of the formation and persistence of such coherent structures observed in experiments and in nature. Another topic is to understand and control the instability in many applications. One such example is to control the instability of plasmas in fusion devices to achieve the goal of controlled nuclear fusion for energy production. Another example is the instability of radially rotating neutron stars, which is related to the detection of gravitational waves. Methods of mathematical analysis are the primary tools employed in this investigation. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies. The project will incorporate research of students from graduate through postdoctoral levels thus providing for a vigorous mentoring program.Many conservative models have Hamiltonian structures. One goal of this proposal is to find an instability index formula for Hamiltonian PDEs with an indefinite energy functional, including gravity water waves, ion acoustic wave equations and Vlasov models for collisionless plasmas. The second goal is to find stability criteria (particularly turning point principles) for several stellar models including rotating and magnetic stars, neutron stars and relativistic galaxies. The third goal is to understand the long-time dynamics near linearly stable Bernstein–Greene–Kruskal (BGK) waves for the one-dimensional Vlasov-Poisson equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对于许多物理现象，能量耗散的影响非常小，忽略它们可以提供很好的近似值。这些现象包括海洋中的大规模水波、太空中的等离子体和受控聚变装置、天体物理学中恒星和星系的动力学。这些现象的数学模型的动力学行为非常复杂，包括相干结构的形成，例如海洋中的海啸、旋转的恒星（例如太阳）和观察到的星系结构等等。该项目的目标之一是有助于理解在实验和自然界中观察到的这种相干结构的形成和持久性。另一个主题是理解和控制许多应用程序中的不稳定性。其中一个例子就是控制聚变装置中等离子体的不稳定性，以实现受控核聚变产生能源的目标。另一个例子是径向旋转的中子星的不稳定性，这与引力波的探测有关。数学分析方法是本次调查中使用的主要工具。严格的数学使得进行稳定的数值计算和更好地理解数值和实验研究中发现的现象成为可能。该项目将纳入从研究生到博士后水平的学生研究，从而提供强有力的指导计划。许多保守模型都具有哈密顿结构。该提案的目标之一是找到具有不定能量泛函的哈密顿偏微分方程的不稳定指数公式，包括重力水波、离子声波方程和无碰撞等离子体的 Vlasov 模型。第二个目标是找到几种恒星模型的稳定性标准（特别是转折点原理），包括旋转恒星和磁星、中子星和相对论星系。第三个目标是了解一维 Vlasov-Poisson 方程的接近线性稳定的 Bernstein-Greene-Kruskal (BGK) 波的长期动态。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。