Mathematical Studies of Magnetohydrodynamics with Hall Effect

霍尔效应磁流体动力学的数学研究

基本信息

批准号：
1815069
负责人：
Mimi Dai
金额：
$ 18.88万
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：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1815069&HistoricalAwards=false
关键词：
Mathematical Studies Magnetohydrodynamics Hall Effect

项目摘要

Magnetic reconnection is a fundamental process in highly conducting plasmas, which allows for 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 address fundamental mathematical questions for the Hall MHD model and provide theoretical insights for experiments and numerical simulations. The project will involve a graduate student in the research. The Hall MHD model is mathematically challenging due to the usual convective nonlinearities and the additional source term given by the Hall effect. The main objectives are: 1) Explore the largest possible space of well-posedness corresponding to the major scalings in the system. 2) Search the optimal Sobolev spaces in which the model is well-posed. 3) Examine ill-posedness of solutions in some physically relevant spaces. 4) Seek minimal conditions for weak solutions to satisfy energy identity. 5) Study the asymptotic behavior of solutions and stability of the steady state. The project will combine known tools from harmonic analysis and partial differential equations, and develop new methods to study these questions.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，它可以随着磁通量的转换为动能，从而可以快速变化磁通量线的构型。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。该项目将解决HALL MHD模型的基本数学问题，并为实验和数值模拟提供理论见解。该项目将涉及研究生。由于通常的对流非线性和Hall效应给出的其他源术语，MHD模型在数学上具有挑战性。主要目标是：1）探索与系统中主要量表相对应的最大拟合度的空间。 2）搜索模型所在的最佳Sobolev空间。 3）检查某些与物理相关的空间中解决方案的不良性。 4）寻求最小的条件以满足能源身份的弱解决方案。 5）研究解决方案的渐近行为和稳定性的稳定性。该项目将结合谐波分析和部分微分方程的已知工具，并开发新的方法来研究这些问题。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来提供支持。