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 ${H_t} + \nabla \times \left [ {{{\left | {\nabla \times H} \right |}^{p - 2}}\nabla \times H} \right ] = F$. 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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G. Aronsson, L. C. Evans, and Y. Wu, Fast/slow diffusion and growing sandpiles, J. Differential Equations 131, 304–335 (1996)
A. Bossavit, Numerical modeling of superconductors in three dimensions, a model and a finite element method, IEEE Trans. Magn. 30, 3363–3366 (1994)
C. P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys. 36, 31–39 (1964)
Ph. Benilan and M. Crandall, The continuous dependence on $\phi$ of solutions of ${u_t} - \Delta \phi \left ( u \right ) = 0$, Indiana Univ. Math. J. 30, 161–177 (1981)
L. A. Caffarelli and A. Friedman, Asymptotic behavior of solutions of ${u_t} = \Delta {u^{m}}$ as $m \to \infty$, Indiana Univ. Math. J. 36, 711–728 (1987)
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993
E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana University Math. J. 32, 83–118 (1983)
L. C. Evans, M. Feldman, and R. F. Gariepy, Fast/slow diffusion and collapsing sandpiles, J. Differential Equations 37, 166–209 (1997)
L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999)
A. Friedman, Variational Principles and Free Boundary Problems, John Wiley and Sons, New York, 1982
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2nd ed., 1983
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, New York, 1960
O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, Amer. Math. Soc., Providence, RI, 1967
L. Prigozhin, Variational model of sandpile growth, European J. Appl. Math. 7, 225–235 (1996)
L. Prigozhin, Solution of thin film magnetization problems in type-II superconductivity, J. Comput. Phys. 144, 180–193 (1998)
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990
H. M. Yin, Regularity of solutions to Maxwell’s system in quasi-stationary electromagnetic fields and applications, Comm. Partial Differential Equations 22, 1029–1053 (1997)
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© Copyright 2001
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