Two limit cases of Born-Infeld equations

We study two limit cases λ→∞ and λ→ 0 in Born-Infeld equations. Here the parameter λ > 0 is interpreted as the maximal electric field in the electromagnetic theory and the case λ = 0 corresponds to the string theory. Formal limits are governed by the classical Maxwell equations and pressureless magnetohydrodynamics system, respectively. For studying the limit λ → ∞, 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∞ in one space dimension, based on derived uniform estimates and techniques for linear Lagrangian systems.