The Nonlinear Stability of the Trivial Solution to the Maxwell-Born-Infeld System

TL;DR

This paper proves the global stability and decay of small solutions to the Maxwell-Born-Infeld system in Minkowski space, using a gauge-free approach and null condition analysis.

Contribution

It establishes the existence, decay, and hyperbolicity of solutions to the nonlinear Maxwell-Born-Infeld system in a new gauge-free framework.

Findings

Global small-data solutions exist and decay at linear rates.

The MBI system is hyperbolic within the real-valued Lagrangian regime.

Null condition satisfied by the nonlinearities aids decay analysis.

Abstract

In this article, we use an electromagnetic gauge-free framework to establish the existence of small-data global solutions to the Maxwell-Born-Infeld (MBI) system on the Minkowski space background in 1 + 3 dimensions. Because the nonlinearities in the system satisfy a version of the null condition, we are also able to show that these solutions decay at exactly the same rates as solutions to the linear Maxwell-Maxwell system. In addition, we show that on any Lorentzian manifold, the MBI system is hyperbolic in the interior of the field-strength regime in which its Lagrangian is real-valued.