  The goal of the present paper is three-fold. First is to clarify the
connection between the dominant energy condition and hyperbolicity properties
of Lagrangian field theories. Second is to provide further analysis on the
breakdown of hyperbolicity for the Skyrme model, sharpening the results of
Crutchfield and Bell and comparing against a result of Gibbons, and provide a
local well-posedness result for the dynamical problem in the Skyrme model.
Third is to provide a short summary of the framework of regular hyperbolicity
of Christodoulou for the relativity community. In the process, a general
theorem about dominant energy conditions for Lagrangian theories of maps is
proved, as well as several results concerning hyperbolicity of those maps.
