On the Cauchy problem for the Hall magnetohydrodynamics

Sungjin Oh
University of California Berkeley 
Wednesday, September 30, 2020 - 3:35pm to 4:35pm

 In this talk, I will describe a recent series of work with I.-J. Jeong on the incompressible 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 the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. Our ill posedness mechanism is sharp, in that it remains true under fractional dissipation of any subcritical order. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bi-characteristic flow associated with the appropriate principal symbol.Join Zoom Meetinghttps://umn.zoom.us/j/91765959125?pwd=RjRkSW9PNi9HNzVOb2lwb0tkWVVoZz09Meeting ID: 917 6595 9125Passcode: VinH16