Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On a $ p$-Laplacian type of evolution system and applications to the Bean model in the type-II superconductivity theory

Author: Hong-Ming Yin
Journal: Quart. Appl. Math. 59 (2001), 47-66
MSC: Primary 35K65; Secondary 35K55, 35K60, 82D55
DOI: https://doi.org/10.1090/qam/1811094
MathSciNet review: MR1811094
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Abstract: In this paper we study the Cauchy problem for a $ p$-Laplacian type of evolution system $ {H_t} + \nabla \times \left[ {{{\left\vert {\nabla \times H} \right\vert}^{p - 2}}\nabla \times H} \right] = F$. This system governs the evolution of a magnetic field H, where the displacement currently is neglected and the electrical resistivity is assumed to be some power of the current density. The existence, uniqueness, and regularity of solutions to the system are established. Furthermore, it is shown that the limit solution as the power $ p \to \infty $ solves the problem of Bean's model in the type-II superconductivity theory. The result provides us information about how the superconductor material under the external force becomes the normal conductor and vice versa. It also provides an effective method for finding numerical solutions to Bean's model.

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DOI: https://doi.org/10.1090/qam/1811094
Article copyright: © Copyright 2001 American Mathematical Society

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