Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary

Richard Nutt, Roland Schnaubelt

TL;DR

This work analyzes a quasilinear Maxwell system with a state-dependent absorbing boundary to establish global existence and exponential decay for small data under a nontrapping condition. The authors develop a refined trace theory for the non-autonomous linear Maxwell problem, including a collar-operator-based normal-trace control and a div-curl framework in negative Sobolev spaces, culminating in an observability-type inequality. A key regularity-boost result then enables a bootstrap argument to absorb high-order error terms and derive decay for the nonlinear problem, while an autonomous linear case yields time-uniform trace bounds and exponential stability. Collectively, the results extend decay theory for Maxwell systems with nonlinear boundary damping beyond star-shaped domains, using advanced trace, microlocal, and div-curl techniques.

Abstract

We study the quasilinear Maxwell system with a strictly positive, state dependent boundary conductivity. For small data we show that the solution exists for all times and decays exponentially to $0$. As in related literature we assume a nontrapping condition. Our approach relies on a new trace estimate for the corresponding non-autonomous linear problem, an observability-type estimate, and a detailed regularity analysis. The results are improved in the linear autonomous case, using properties of the Helmholtz decomposition in Sobolev spaces of (small) negative order.

Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary

TL;DR

This work analyzes a quasilinear Maxwell system with a state-dependent absorbing boundary to establish global existence and exponential decay for small data under a nontrapping condition. The authors develop a refined trace theory for the non-autonomous linear Maxwell problem, including a collar-operator-based normal-trace control and a div-curl framework in negative Sobolev spaces, culminating in an observability-type inequality. A key regularity-boost result then enables a bootstrap argument to absorb high-order error terms and derive decay for the nonlinear problem, while an autonomous linear case yields time-uniform trace bounds and exponential stability. Collectively, the results extend decay theory for Maxwell systems with nonlinear boundary damping beyond star-shaped domains, using advanced trace, microlocal, and div-curl techniques.

Abstract

We study the quasilinear Maxwell system with a strictly positive, state dependent boundary conductivity. For small data we show that the solution exists for all times and decays exponentially to . As in related literature we assume a nontrapping condition. Our approach relies on a new trace estimate for the corresponding non-autonomous linear problem, an observability-type estimate, and a detailed regularity analysis. The results are improved in the linear autonomous case, using properties of the Helmholtz decomposition in Sobolev spaces of (small) negative order.
Paper Structure (8 sections, 28 theorems, 158 equations)

This paper contains 8 sections, 28 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Notation, assumptions and auxiliary results
  3. Main result for the nonlinear problem
  4. Trace inequality
  5. Proof of Proposition \ref{['prop:observability']}
  6. The autonomous linear case
  7. Proof of Proposition \ref{['prop:regularity-boost']}
  8. Div-curl estimate

Key Result

Theorem 3.1

Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with $C^5$-boundary and connected complement. Furthermore, assume that the permittivity $\varepsilon$ and permeability $\mu$ satisfy assumption:cont1, assumption:posdef1, assumption:cont2, and eq:technicalCond. We require that the initial values Then there exist constants $M, \omega, r>0$ such that for $\left\lVert E^{(0)}\right\rVert^2_{H^3(\

Theorems & Definitions (48)

  • proof : Sketch of proof of \ref{['eq:commutator-estimates']}
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof : Proof of Theorem \ref{['thm:mainThm']}
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 38 more