Mathematical Studies of Magnetohydrodynamics with Hall Effect

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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.