Mathematical Analysis of Magnetohydrodynamic Flows with Hall Effect

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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.