Mathematical Analysis of Magnetohydrodynamic Flows with Hall Effect

霍尔效应磁流体动力流的数学分析

基本信息

批准号：
2009422
负责人：
Mimi Dai
金额：
$ 20万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009422&HistoricalAwards=false
关键词：
Mathematical Analysis Magnetohydrodynamic Flows Hall

项目摘要

Magnetic reconnection is a fundamental process in highly conducting plasmas that involves rapid changes in the configuration of the magnetic flux lines, with the conversion of magnetic to kinetic energy. Solar flares, violent events with significant impact to telecommunications and the electric grid, may involve magnetic reconnection in a large scale. Magnetic reconnection is inherently a multi-scale process and causes difficulties in laboratory experimental study, satellite observations, and computational simulations. During magnetic reconnection, the magnetic force can create thin localized region wherein the elevated voltage difference generates intense electric currents and dissipation - the Hall effect. The Hall magnetohydrodynamics (Hall MHD) model has recently received increasing attention because of its improvements in predicting the fast-changing nature of magnetic reconnection compared to other models. Nevertheless, the mathematical theory of this model is far from being complete. This project will further advance the fundamental theoretical understanding of the Hall MHD model by addressing the issues such as existence and uniqueness of solutions, anomalous dissipation, and long-time behavior of the solutions. The project will provide opportunities for students to participate in the research. The Hall MHD model couples the Navier-Stokes equations and Maxwell's equations. The Hall MHD system is mathematically challenging due to the usual convective nonlinearities and the additional source term given by the Hall effect. The project includes investigations on the conservation or anomalous dissipation of energy and magnetic helicity for weak solutions, uniqueness, or lack thereof for weak solutions, weak-strong type of uniqueness, and asymptotic uniqueness in energy space within the framework of determining wavenumber.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.

磁重新连接是高度传导等离子体的基本过程，涉及磁通线的构型快速变化，并将磁性转换为动能。太阳耀斑，对电信和电网产生重大影响的暴力事件，可能涉及大规模的磁重新连接。磁重新连接本质上是一个多尺度过程，在实验室实验研究，卫星观测和计算模拟中造成困难。在磁重新连接期间，磁力可以产生薄的局部区域，在其中高架电压差会产生强烈的电流和耗散 - 霍尔效应。 Hall磁水动力学（HALL MHD）模型最近受到了越来越多的关注，因为与其他模型相比，它在预测磁重新连接的快速变化性质方面有所改善。然而，该模型的数学理论远非完整。该项目将通过解决解决方案的存在和唯一性，异常耗散以及解决方案的长期行为等问题，进一步推进对HALL MHD模型的基本理论理解。该项目将为学生提供参与研究的机会。 Hall MHD模型将Navier-Stokes方程式和Maxwell的方程式融合在一起。由于通常的对流非线性和Hall效应给出的附加源术语，Hall MHD系统在数学上具有挑战性。该项目包括有关弱解决方案，独特性或缺乏弱解决方案，弱的独特性类型以及能量独特性的弱点的弱点和磁性螺旋性的调查的研究标准。