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Global solvability for a singular nonlinear Maxwell's equations

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  • 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.

    Mathematics Subject Classification: 35Q60.


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