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

TL;DR

This paper demonstrates that the initial value problem for Force-free Electrodynamics in Euler variables is ill-posed, showing discontinuity in evolution sequences and failure of well-posedness criteria.

Contribution

It proves the ill-posedness of Force-free Electrodynamics in Euler variables using Kreiss and Strang criteria, highlighting fundamental issues in the theory's mathematical formulation.

Findings

The initial value problem is not well posed in Sobolev norms.

The evolution sequence can become unbounded despite bounded initial data.

The problem is also ill-posed in the Leray-Ohya sense.

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.