Analysis and PDE Seminar: On the Cauchy problem for the Hall-magnetohydrodynamics equations
Seminar | September 16 | 4:10-5 p.m. | 939 Evans Hall
Sung-Jin Oh, UC Berkeley
In this talk, I will describe a recent series of work with I.-J. Jeong on the Cauchy problem for the Hall-MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field). Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character.