Turbulence, Shocks, and Stability in Fluids and Plasmas

这个项目将发展液体、气体和等离子体的数学理论，它们在工程、气象学、空气动力学等问题中普遍存在。物质的这些基本状态表现出广泛的行为，包括湍流和混沌行为、冲击波和稳定性。第一个目标是推进湍流的数学理论，从车辆尾迹到大气，到处都可以观察到湍流。其次，这个项目将研究冲击波，即在天体物理等离子体中观察到的明显的不连续性质，如密度和流速。最后，该项目将研究背景剪切附近的流体和等离子体的稳定特性，这可能会用于一些重要的应用，以控制流体或等离子体的行为。该项目还包括为研究生提供培训和辅导机会，以及组织会议和研讨会。这个项目的第一部分将构造不可压缩欧拉方程和纳维-斯托克斯方程的耗散解，以及其他流体和等离子体模型。间歇性、基于小波的迭代等将在分析中发挥关键作用。该项目的第二部分将首先建立Boltzmann方程和Landau方程的小振幅动力学激波解，它们近似于可压缩的Navier-Stokes方程的行波解。研究可压缩流体、流体动力学极限和动力学理论的工具将被开发出来，然后用于研究稀薄带电粒子的模型，如Vlasov-Maxwell-Boltzmann系统。最后，本项目将对流体动力和磁流体动力稳定性与控制进行研究。在剪切流附近经常观察到稳定机制、混合和增强的消散，并将以一种新的方式用于解决流体和等离子体的控制问题。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。

This project will develop the mathematical theory of liquids, gases, and plasmas, which are ubiquitous in problems from engineering, meteorology, aerodynamics, and more. These fundamental states of matter exhibit a wide range of behaviors, including turbulent and chaotic behavior, shock waves, and stability. The first goal is to advance the mathematical theory of turbulent fluids, which may be observed everywhere from the wakes of vehicles to the atmosphere. Second, this project will study shock waves, or the apparent discontinuities in properties such as density and flow velocity, which are observed in astrophysical plasmas and more. Finally, the project will investigate the stabilizing properties of fluids and plasmas near background shears, which may be used in a number of important applications to control the behavior of the fluid or plasma. This project also includes training and mentoring opportunities for graduate students and the organization of conferences and seminars. The first portion of this project will construct dissipative solutions of the incompressible Euler and Navier-Stokes equations, as well as other models of fluid and plasmas. Intermittency, wavelet-based iterations, and more will play a key role in the analysis. The second portion of this project will begin by building small-amplitude kinetic shock solutions to the Boltzmann and Landau equations which approximate traveling wave solutions of the compressible Navier-Stokes equations. Tools from the study of compressible fluids, the hydrodynamic limit, and kinetic theory will be developed and then used to investigate models of dilute charged particles, such as the Vlasov-Maxwell-Boltzmann system. Finally, this project will study hydrodynamic and magnetohydrodynamic stability and control. Stabilizing mechanisms, mixing, and enhanced dissipation are often observed in the vicinity of shear flows and will be used in a novel way to solve control problems for fluids and plasmas.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.