On the well posedness of Robinson Trautman Maxwell solutions

TL;DR

This paper demonstrates that Robinson-Trautman-Maxwell equations lack well-posedness, meaning their solutions are highly sensitive to initial data and cannot reliably be used outside the analytic domain.

Contribution

It establishes that Robinson-Trautman-Maxwell equations do not form a well-posed initial value problem, highlighting limitations in their applicability for solving outside the analytic domain.

Findings

Solutions depend discontinuously on initial data.

The equations are not well-posed in any norm based on initial data.

They cannot be used for non-analytic initial conditions.

Abstract

We show that the so called Robinson-Trautman-Maxwell equations do not constitute a well posed initial value problem. That is, the dependence of the solution on the initial data is not continuous in any norm built out from the initial data and a finite number of its derivatives. Thus, they can not be used to solve for solutions outside the analytic domain.