Global solutions to Maxwell equations in a ferromagnetic medium
type: Item
instance of: scholarly article
title: Global solutions to Maxwell equations in a ferromagnetic medium
zbMATH Open document ID: 0964.35155
DOI: 10.1007/PL00001007
author: Jean-Luc Joly
author: Guy Métivier
author: Jeffrey Rauch
published in: Annales Henri Poincaré
publication date: 2000-12-04T00:00:00Z
review text: The Cauchy problem for the Maxwell system with the separated magnetization \(M\) \[ \partial_tE-\text{curl} H=0,\quad \partial_tH+\text{curl} E=0,\quad \partial_tM=F(M,H), \] \[ \text{div} E=0,\quad \text{div} (H+M)=0 \] is studied. The nonlinear term \(F(M,H)\) corresponds to the Landau-Lifschitz model for the propagation of the electromagnetic field in a ferromagnetic medium. The aim of the paper is to prove the existence of global finite energy solutions. The distribution \((E,H,M)\) is a finite energy solution in \(\Omega_T=[0,T]\times \mathbb{R}^3\) if all its components belong to \(C^0([0,T]; L^2(\mathbb{R}^3))\) and \(M\) is in \(L^{\infty}([0,T]\times \mathbb{R}^3)\). The uniqueness and regularity properties of these solutions are studied also. Particularly, it is shown that if the initial data are smooth, then the solution of the Cauchy problem remains smooth for all time. By suitable assumptions on \(F\) it is proved that if \(E_0, {H_0}_{\perp}\) belong to \(H^{\mu}, \mu\in]0,1]\), then \(E\) and \(H_{\perp}\) belong to \(C^0([0,+\infty[; H^{\mu}(\mathbb{R}^3))\).
reviewed by: Nikolai Pleshchinskii
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 78A25
zbMATH DE Number: 1540531
zbMATH Keywords: Cauchy problem
zbMATH Keywords: finite energy solutions
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: pl00001007
OpenAlex ID: W2001703461
description: scientific article; zbMATH DE number 1540531