Hydrodynamic structure of the augmented Born-Infeld equations

The Born-Infeld system is a nonlinear version of Maxwell's equations. We first show that, by using the energy density and the Poynting vector as additional unknown variables, the BI system can be augmented as a 10x10 system of hyperbolic conservation laws. The resulting augmented system has some similarity with magnetohydrodynamics (MHD) equations and enjoys remarkable properties (existence of a convex entropy, Galilean invariance, full linear degeneracy). In addition, the propagation speeds and the characteristic fields can be computed in a very easy way, in contrast with the original BI equations. Then, we investigate several limit regimes of the augmented BI equations, by using a relative-entropy method going back to Dafermos, and recover the Maxwell equations for low fields, some pressureless MHD equations for high fields, and pressureless gas equations for very high fields.