Ill posedness of Force-Free Electrodynamics in Euler Potentials

REULA, OSCAR A.; RUBIO, MARCELO E.

Conferencia; IperPV2017, XVII Italian Meeting on Hyperbolic Equations; 2017

We prove that the initial value problem for force-free electrodynamics in Euler variables is not well posed. We establish this result by showing that a well-posedness criterion provided by Kreiss fails to hold for this theory, and using a theorem provided by Strang. To show the nature of the problem we display a particular bounded (in Sobolev norms) sequence of initial data for the force-free equations such that at any given time as close to zero as one wishes, the corresponding evolution sequence is not bounded. Thus, the force-free evolution is noncontinuous in that norm with respect to the initial data. We furthermore prove that this problem is also ill-posed in the Leray-Ohya sense.