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Stabilizing Phenomenon for Incompressible Fluids

不可压缩流体的稳定现象

基本信息

批准号：
2104682
负责人：
Jiahong Wu
金额：
$ 26.7万
依托单位：
Oklahoma State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104682&HistoricalAwards=false
关键词：
Stabilizing Phenomenon Incompressible Fluids

项目摘要

This project seeks to understand an important universal stabilizing phenomenon concerning several incompressible fluids. The magnetic field stabilizes and damps electrically conducting fluids, a phenomenon observed in many physical experiments and numerical simulations. The temperature tames and stabilizes buoyancy driven fluids and helps the formation of stable structures in turbulent thermal convection. These are just two outstanding examples of a universal stabilizing phenomenon. The goal of this project is to fully comprehend this remarkable phenomenon and establish these observations as mathematically rigorous stability results on the models governing the dynamics of these flows. Understanding the proposed stability problems will help gain insight into many astronomical and meteorological phenomena such as Northern lights, solar flares, and severe weather events. The PIs will integrate the proposed research with the training of graduate students. In addition, the PIs will host events to get more K-16 students interested in these mathematical topics.This project focuses on the stability and large-time dynamics of the magnetohydrodynamic (MHD) equations near a background magnetic field and on the Boussinesq equations near the hydrostatic equilibrium. Mathematically these are extremely difficult problems. The fluid velocity in these MHD and Boussinesq systems is governed by the Euler or the Euler-like equations. The corresponding vorticity gradients could potentially grow rather rapidly in time. This makes the proposed stability problems appear to be impossible. The goal here is to solve these difficult problems by creating new strategies and innovative approaches. In particular, the PIs will reverse some of the classical approaches to the global regularity and stability problems on the MHD and the Boussinesq equations and treat the bad terms such as the Lorentz force and the buoyancy force in these models as good terms and exploit the smoothing and stabilizing effect of the magnetic field and the temperature. In addition, extensive numerical simulations will be performed to complement the theoretical studies.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.

这个项目旨在了解一个重要的普遍稳定现象，涉及几个不可压缩流体。磁场稳定和阻尼导电流体，这是许多物理实验和数值模拟中观察到的现象。温度驯服和稳定浮力驱动的流体，并有助于在湍流热对流中形成稳定的结构。这只是普遍稳定现象的两个突出例子。该项目的目标是充分理解这一显着的现象，并建立这些观测作为数学上严格的稳定性结果的模型，这些流动的动力学。了解所提出的稳定性问题将有助于深入了解许多天文和气象现象，如北方光，太阳耀斑和恶劣的天气事件。PI将把拟议的研究与研究生的培训结合起来。此外，PI将举办活动，让更多的K-16学生对这些数学主题感兴趣。这个项目的重点是在背景磁场附近的磁流体动力学（MHD）方程的稳定性和大时间动力学，以及在流体静力平衡附近的Boussinesq方程。在数学上，这些都是非常困难的问题。这些MHD和Boussinesq系统中的流体速度由欧拉或类欧拉方程控制。相应的涡度梯度可能会随着时间的推移而相当迅速地增长。这使得所提出的稳定性问题似乎是不可能的。我们的目标是通过制定新的战略和创新方法来解决这些难题。特别是，PI将扭转一些经典的方法来解决MHD和Boussinesq方程的全局正则性和稳定性问题，并将这些模型中的洛仑兹力和浮力等不良项视为良好项，并利用磁场和温度的平滑和稳定效果。此外，将进行广泛的数值模拟，以补充理论研究。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。