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 Kruzkov 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 cubic term can be ignored, the solutions are unique and depend continuously on given data.