Global solvability for a singular nonlinear Maxwell's equations

W. Wei; H. M. Yin

doi:10.3934/cpaa.2005.4.431

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2005, Volume 4, Issue 2: 431-444. Doi: 10.3934/cpaa.2005.4.431

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Global solvability for a singular nonlinear Maxwell's equations

W. Wei^1, and
H. M. Yin^2,

1.
Department of Mathematics, Guizou University, Guiyang, Guizhou Province
2.
Department of Mathematics, Washington State University, Pullman, WA 99164

Received: May 31, 2004

Revised: August 31, 2004

Published: May 2005

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Abstract

In this paper we study a singular nonlinear evolution system:

$\frac{\partial}{\partial t}[\mu(x,|\mathbf H|)\mathbf H]+ \nabla\times [r(x,t) \nabla \times \mathbf H]=\mathbf F(x,t),$

where $\mathbf H$ represents the magnetic field in a quasi-stationary electromagnetic field and $\mu(x,|\mathbf H|)$ is the magnetic permeability in a conductive medium, which strongly depends on the strength of $\mathbf H$ such as $\mu(x,|\mathbf H|)=|\mathbf H|^b$ with $b>0$. We prove that under appropriate initial and boundary conditions the system has a global weak solution and the solution is also unique.

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Mathematics Subject Classification: 35Q60.

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