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建模。该项目涉及此类方程的几个方面，包括对其渐近行为和各种平衡稳定性的定性和定量研究（稳定平衡对应于可以遇到/观察到的对象）。该项目的一个潜在主题是开发健壮的方法来利用分散的稳定机制（事实上，由于所有物体在无限空间中不断移动，因此随着时间的推移，任何给定时间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。该项目还将通过培养研究生、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。本项目解决动力学和流体方程的各种稳定性和渐近问题，利用色散机制获得解的长期控制。对于动力学方程，目标是使用（并发展）渐近作用角方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。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将考虑一个肥尾平衡模型（泊松平衡），并将通过将电场分解为具有更快衰减的静电贡献和缓慢消散的振荡成分来证明稳定性。使用范式技术，PI将证明缓慢但振荡分量的长期贡献仍然在控制之下，并且将以基于密度的勒贝格范数的自引导参数结束。还将进行其他概括。最后，本项目还考虑了旋转诱导稳定衰减机制的情况，从不可压缩三维欧拉的刚性运动矢量场的全局稳定性开始。这反过来又简化为拟线性色散问题的小数据全局存在性问题，该问题将使用源自时空共振方法和扩展的方法来解决。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.