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，这些理论在工程，气象学，空气动力学等问题中无处不在。 这些物质的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，从车辆的尾流到大气层，到处都可以观察到湍流。 其次，该项目将研究冲击波，或在天体物理等离子体中观察到的密度和流速等属性的明显不连续性。最后，该项目将研究背景剪切附近流体和等离子体的稳定特性，这些特性可用于许多重要应用中，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指导机会，并组织会议和研讨会。 这个项目的第一部分将构建不可压缩的欧拉和纳维-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。 这个项目的第二部分将开始建立小振幅的动力学冲击波的解决方案，玻尔兹曼和朗道方程近似的行波解的可压缩Navier-Stokes方程。工具从可压缩流体的研究，流体动力学极限，和动力学理论将被开发，然后用于研究稀带电粒子的模型，如Vlasov-Maxwell-Boltzmann系统。 最后，这个项目将研究流体动力学和磁流体动力学的稳定性和控制。 在剪切流附近经常观察到稳定机制、混合和增强耗散，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。

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.