Well-posedness of the Maxwell equations with nonlinear Ohm law

TL;DR

This paper proves an energy equality for weak solutions of Maxwell's equations with nonlinear Ohm law under perfect conductor boundary conditions, establishing well-posedness in an L^2 framework.

Contribution

It demonstrates the energy equality for weak solutions of Maxwell's equations with nonlinear Ohm law, advancing the mathematical understanding of their well-posedness.

Findings

Energy equality holds for all weak solutions

Weak solutions are well-defined in L^2 spaces

Results apply to nonlinear Ohm law in Maxwell equations

Abstract

This paper is concerned with weak solutions (e,h) in L^2 x L^2 of the Maxwell equations with nonlinear Ohm law and under perfect conductor boundary conditions. These solutions are defined in terms of integral identities with appropriate test functions. The main result of our paper is an energy equality that holds for any weak solution (e,h).