Hamiltonian Methods for Dispersive Fluids and Plasmas

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来模拟。该项目涉及这些方程的几个方面，包括定性和定量研究其渐近行为和各种平衡的稳定性（稳定平衡对应于可以遇到/观察到的对象）。该项目的一个基本主题是开发强大的方法来利用分散的稳定机制（事实上，由于所有物体都在无限空间中不断移动，随着时间的推移，在任何给定时间的任何给定点的高度集中变得越来越不可能）。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，这会引起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然是未知的。 该项目还将通过培训研究生、开发课程以及提供合作机会，为培养下一代科学家做出贡献。该项目解决了动力学和流体方程的各种稳定性和渐进问题，使用分散机制来获得解的长期控制。对于动力学方程，一个目标是理解具有点电荷部分的解的长时间行为，使用（和发展）渐近作用角方法。第一个例子是狄拉克质量的稳定性，在排斥或（暂时）吸引的情况下。第二个问题是研究大数据的情况和相对论情况的扩展。解决的另一个问题涉及均匀平衡，目的是更好地理解朗道阻尼（在整个空间）。PI将考虑胖尾平衡（泊松平衡）的模型，并将通过将电场分解为具有更快衰减的静电贡献和耗散缓慢的振荡分量来证明稳定性。使用规范形式技术，PI将证明缓慢但振荡分量的长期贡献仍然在控制之下，并将仅基于密度的勒贝格范数的自举参数关闭。还将继续进行其他归纳。最后，该项目还考虑了旋转引起稳定衰减机制的情况，从不可压缩三维欧拉刚体运动矢量场的全局稳定性开始。这反过来又减少了一个问题的小数据的全球存在的准线性色散问题，这将是解决使用的方法源于时空共振方法和extensions.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.