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Well-posedness of the Maxwell equations with nonlinear Ohm law

Jens A. Griepentrog, Joachim Naumann

TL;DR

This work proves well-posedness for the Maxwell equations with a nonlinear Ohm law under perfect-conductor boundary conditions in a bounded domain. It develops a robust L^2 framework and shows that any weak solution admits a time-continuous representative and obeys a fundamental energy equality, enabling stability analysis. The key contributions include the construction of t-continuous representatives via Steklov averages, the derivation of an energy balance, and the existence of solutions by Faedo-Galerkin approximations, together with a Minty-type argument to identify the nonlinear current with its constitutive law under monotonicity. The results provide a rigorous basis for unique, energy-controlled evolution in nonlinear electromagnetic media and lay the groundwork for further numerical and analytical investigations in complex materials.

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).

Well-posedness of the Maxwell equations with nonlinear Ohm law

TL;DR

This work proves well-posedness for the Maxwell equations with a nonlinear Ohm law under perfect-conductor boundary conditions in a bounded domain. It develops a robust L^2 framework and shows that any weak solution admits a time-continuous representative and obeys a fundamental energy equality, enabling stability analysis. The key contributions include the construction of t-continuous representatives via Steklov averages, the derivation of an energy balance, and the existence of solutions by Faedo-Galerkin approximations, together with a Minty-type argument to identify the nonlinear current with its constitutive law under monotonicity. The results provide a rigorous basis for unique, energy-controlled evolution in nonlinear electromagnetic media and lay the groundwork for further numerical and analytical investigations in complex materials.

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).
Paper Structure (6 sections, 13 theorems, 193 equations)

This paper contains 6 sections, 13 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Weak solutions
  3. Existence of $t$-continuous representatives in the equivalence classes $e,h$
  4. Energy equality. Well-posedness
  5. Existence of weak solutions via the Faedo-Galerkin method
  6. On the solvability of the Cauchy problem for an ordinary differential equation

Key Result

Theorem 2.1

Let hypotheses complement--initial-values be satisfied and let Then the statements (i) and (ii) are equivalent:

Theorems & Definitions (40)

  • Example 1
  • Example 2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition
  • Remark 2.7
  • ...and 30 more