Two limit cases of Born-Infeld equations

We study two limit cases lambda ->infinity and lambda -> 0 in Born - Infeld equations. Here the parameter lambda > 0 is interpreted as the maximal electric field in the electromagnetic theory and the case lambda=0 corresponds to the string theory. Formal limits are governed by the classical Maxwell equations and pressureless magnetohydrodynamics system, respectively. For studying the limit lambda ->infinity, a new scaling is introduced. We give the relations between these limits and Brenier high and low field limits. Finally, using compensated compactness arguments, the limits are rigorously justified for global entropy solutions in L-infinity in one space dimension, based on derived uniform estimates and techniques for linear Lagrangian systems.