Magnetohydrodynamic regime of the born-infeld electromagnetism

TL;DR

This paper explores a nonlinear extension of Maxwell's equations called the augmented Born-Infeld system, revealing a connection to magnetohydrodynamics and introducing a new class of dissipative solutions with unique smooth solutions.

Contribution

It introduces a novel augmented BI model that links nonlinear electromagnetism to MHD and develops a framework of dissipative solutions using relative entropy.

Findings

The augmented BI system simplifies to an energy dissipation model combining Darcy's law and MHD.

Dissipative solutions form a non-empty, convex, and compact set for given initial conditions.

Smooth solutions within this framework are always unique.

Abstract

The Born-Infeld (BI) model is a nonlinear correction of Maxwell's equations. By adding the energy and Poynting vector as additional variables, it can be augmented as a 10$\times$10 system of hyperbolic conservation laws, called the augmented BI (ABI) equations. The author found that, through a quadratic change of the time variable, the ABI system gives a simple energy dissipation model that combines Darcy's law and magnetohydrodynamics (MHD). Using the concept of "relative entropy" (or "modulated energy"), borrowed from the theory of hyperbolic systems of conservation laws, we introduce a notion of generalized solutions, that we call dissipative solutions. For given initial conditions, the set of generalized solutions is not empty, convex, and compact. Smooth solutions to the dissipative system are always unique in this setting.