Mathematical Analysis of Magnetohydrodynamic Flows with Hall Effect

磁重联是高导电性等离子体中的一个基本过程，它涉及磁通线构型的快速变化，伴随着磁能到动能的转换。太阳耀斑是对电信和电网有重大影响的猛烈事件，可能涉及大规模的磁场重新连接。磁重联本质上是一个多尺度的过程，给实验室实验研究、卫星观测和计算模拟带来困难。在磁重联过程中，磁场力可以产生薄的局域区，其中升高的电压差产生强烈的电流和耗散-霍尔效应。霍尔磁流体力学(Hall MHD)模型最近受到越来越多的关注，因为与其他模型相比，它在预测磁重联的快速变化性质方面有所改进。然而，该模型的数学理论还远远不完善。这个项目将通过解决解的存在唯一性、反常耗散和解的长时间行为等问题，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。霍尔MHD模型耦合了纳维-斯托克斯方程和麦克斯韦方程。由于通常的对流非线性和由霍尔效应给出的附加源项，霍尔MHD系统在数学上是具有挑战性的。该项目包括在确定波数的框架内研究能量守恒或反常耗散以及弱解的磁螺旋性、弱解的唯一性或缺乏唯一性、弱-强型唯一性以及能量空间中的渐近唯一性。该奖项反映了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。

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.