On the ill posedness of Force-Free Electrodynamics in Euler Potentials

Abstract

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 non continuous in that norm with respect to the initial data. We furthermore prove that this problem is also ill-posed in the Leray-Ohya sense.

arXiv comment13 pages, no figures, accepted for publication in Phys. Rev. D

References in corpus (7)

• On the Properties of Time-Dependent, Force-Free, Degenerate Electrodynamics
• Spacetime approach to force-free magnetospheres
• Magnetic field evolution in simulations with Euler potentials
• Strongly hyperbolic systems in General Relativity
• Nonlinear electrodynamics as a symmetric hyperbolic system
• Covariant Hyperbolization of Force-free Electrodynamics
• Hyperbolicity of Force-Free Electrodynamics

Cited by in corpus (5)

• On the final fate of compact boson star mergers
• Dark matter from primordial black holes would hold charge
• Force-free Electrodynamics and Foliations in an arbitrary Spacetime
• A necessary condition for strong hyperbolicity of general first order systems
• Ill-posedness of the mean-field dynamo equations with a linear electromotive force