  The Born-Infeld (BI) model is a nonlinear correction of Maxwell's equations.
By adding the energy and Poynting vector as additional variables, it can be
augmented as a 10$\times$10 system of hyperbolic conservation laws, called the
augmented BI (ABI) equations. The author found that, through a quadratic change
of the time variable, the ABI system gives a simple energy dissipation model
that combines Darcy's law and magnetohydrodynamics (MHD). Using the concept of
"relative entropy" (or "modulated energy"), borrowed from the theory of
hyperbolic systems of conservation laws, we introduce a notion of generalized
solutions, that we call dissipative solutions. For given initial conditions,
the set of generalized solutions is not empty, convex, and compact. Smooth
solutions to the dissipative system are always unique in this setting.
