Stabilizing Phenomenon for Incompressible Fluids

这个项目试图理解关于几种不可压缩流体的一种重要的普遍稳定现象。磁场稳定并抑制导电流体，这是在许多物理实验和数值模拟中观察到的现象。温度驯服和稳定了浮力驱动的流体，并有助于在湍流热对流中形成稳定的结构。这只是普遍企稳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，并将这些观测结果确立为控制这些流动动态的模型的严格数学稳定性结果。了解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，如北极光、太阳耀斑和恶劣天气事件。PIS将把拟议的研究与研究生的培训结合起来。此外，PIS将举办活动，让更多的K-16年级的学生对这些数学主题感兴趣。本项目重点关注背景磁场附近磁流体动力学(MHD)方程的稳定性和大时间动力学，以及流体静力平衡附近的Boussinesq方程。从数学上讲，这些都是极其困难的问题。这些MHD和Boussinesq系统中的流体速度受欧拉方程或类似欧拉方程的支配。相应的涡度梯度可能会在时间上增长得相当快。这使得提出的稳定性问题似乎是不可能的。这里的目标是通过创造新的战略和创新的方法来解决这些难题。特别是，PI将逆转MHD和Boussinesq方程中关于整体正则性和稳定性问题的一些经典方法，并将这些模型中的洛伦兹力和浮力等坏项视为好项，并利用磁场和温度的平滑和稳定作用。此外，将进行广泛的数值模拟来补充理论研究。这一奖项反映了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 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.