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Hamiltonian Methods for Dispersive Fluids and Plasmas

色散流体和等离子体的哈密顿方法

基本信息

批准号：
2154162
负责人：
Benoit Pausader
金额：
$ 40.12万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-15 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154162&HistoricalAwards=false
关键词：
Hamiltonian Methods Dispersive Fluids Plasmas

项目摘要

Physical phenomena related to galaxies, gases, and plasmas can be modeled by the partial differential equations describing fluid motions. This project addresses several aspects of such equations, including the qualitative and quantitative study of their asymptotic behavior and the stability of various equilibriums (stable equilibriums correspond to objects that can be encountered/observed). An underlying theme of the project is to develop robust methods to leverage the stabilizing mechanisms of dispersion (the fact that, since all objects move constantly in an infinite space, a high concentration at any given point at any given time becomes increasingly unlikely as time passes). Another central theme of the project is to study the effect of rotation on fluids. While it has been already recognized that this induces a stabilizing dispersive effect, the extent of this effect, its precise mathematical expression, and its consequences, remain largely unknown. This project will also contribute to preparing the next generation of scientists by training graduate students, developing courses, as well as collaborative opportunities. This project addresses various stability and asymptotic questions on kinetic and fluid equations, using the dispersive mechanism to obtain long-time control of the solutions. For kinetic equations, a goal is to understand the long-time behavior of solutions with a point charge part, using (and developing) the method of asymptotic action-angle. A first example is the stability of a Dirac mass, in the repulsive or (tentatively) the attractive case. A second question is to investigate the case of large data and the extension to the relativistic case. Another problem addressed concerns homogeneous equilibrium with an aim to better understand Landau damping (in the whole space). The PI will consider a model of fat-tail equilibria (the Poisson equilibrium) and will prove stability, by decomposing the electric field into an electrostatic contribution with faster decay and an oscillatory component that dissipates slowly. Using normal form techniques, the PI will prove that the long-time contribution of the slow but oscillatory component remains under control, and will close with a bootstrap argument based on Lebesgue norm of the density alone. Other generalizations will also be pursued. Finally, this project also considers situations where the rotation induces a stabilizing decaying mechanism, starting with global stability of the vector field of rigid motion for the incompressible 3d Euler. This in turn reduces to a problem of small data global existence for a quasilinear dispersive problem, which will be tackled using methods originating from the space-time resonance method and extensions.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.

与星系、气体和等离子体有关的物理现象可以用描述流体运动的偏微分方程式来模拟。这个项目涉及这种方程的几个方面，包括对它们的渐近行为和各种平衡(稳定平衡对应于可以遇到/观察到的对象)的稳定性的定性和定量研究。该项目的一个基本主题是开发强有力的方法来利用分散的稳定机制(由于所有物体都在无限的空间中不断运动，随着时间的推移，在任何给定时间的任何给定点高度集中的可能性越来越小)。该项目的另一个中心主题是研究旋转对流体的影响。虽然人们已经认识到，这会导致一种稳定的色散效应，但这种效应的程度、其精确的数学表达式及其后果在很大程度上仍不清楚。该项目还将通过培训研究生、开发课程以及合作机会，为培养下一代科学家做出贡献。这个项目解决了运动方程和流体方程的各种稳定性和渐近性问题，使用色散机制来获得解的长期控制。对于动力学方程，目标是利用(并发展)渐近作用角法来理解具有点电荷部分的解的长期行为。第一个例子是狄拉克质量的稳定性，在排斥或(暂定)吸引的情况下。第二个问题是研究大数据的情况以及相对论情况的扩展。另一个问题涉及齐次平衡，目的是更好地理解朗道衰减(在整个空间)。PI将考虑厚尾均衡(泊松均衡)的模型，并通过将电场分解为衰减较快的静电分量和消散较慢的振荡分量来证明稳定性。使用正规形技术，PI将证明缓慢但振荡分量的长期贡献仍在控制之下，并将以仅基于密度的勒贝格范数的自举论证结束。还将继续进行其他概括。最后，从不可压缩三维欧拉刚体运动矢量场的全局稳定性出发，考虑了旋转引起稳定衰减机制的情况。这反过来又归结为准线性色散问题的全球小数据存在问题，该问题将使用源于时空共振方法和扩展的方法来解决。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。