On uniqueness and continuity for the quasi-linear, bianisotropic Maxwell equations, using an entropy condition

The quasi-linear Maxwell equations describing electromagnetic wave propagation
in nonlinear media permit several weak solutions, which may be discontinuous
(shock waves). It is often conjectured that the solutions are unique
if they satisfy an additional entropy condition. The entropy condition states
that the energy contained in the electromagnetic fields is irreversibly dissipated
to other energy forms, which are not described by the Maxwell equations.
We use the method employed by Kruˇzkov to scalar conservation laws
to analyze the implications of this additional condition in the electromagnetic
case, i.e., systems of equations in three dimensions. It is shown that if a
certain term can be ignored, the solutions are unique.