March  2013, 18(2): 551-563. doi: 10.3934/dcdsb.2013.18.551

Ohm-Hall conduction in hysteresis-free ferromagnetic processes

1. 

Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento) - Italia, Italy

Received  October 2011 Revised  January 2012 Published  November 2012

Electromagnetic processes in a ferromagnetic conductor (e.g., an electric transformer) are here described by coupling the Maxwell equations with nonlinear constitutive laws of the form $$ \vec B \in \mu_0\vec H + {\mathcal M}(x) \vec H/|\vec H|, \qquad \vec J = \sigma(x) \big( \vec E + \vec E_a(x,t) + h(x)\vec J \!\times\! \vec B \big). $$ Here $\vec E_a$ stands for an applied electromotive force; the saturation ${\mathcal M}(x)$, the conductivity $\sigma(x)$ and the Hall coefficient $h(x)$ are also prescribed. The first relation accounts for hysteresis-free ferromagnetism, the second one for the Ohm law and the Hall effect.
    This model leads to the formulation of an initial-value problem for a doubly-nonlinear parabolic-hyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by time-discretization, derivation of a priori estimates, and passage to the limit. This final step rests upon a time-dependent extension of the Murat and Tartar div-curl lemma, and on compactness by strict convexity.
Citation: Augusto Visintin. Ohm-Hall conduction in hysteresis-free ferromagnetic processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 551-563. doi: 10.3934/dcdsb.2013.18.551
References:
[1]

N. W. Ashcroft and N. D. Mermin, "Solid State Physics,", Holt, (1976). Google Scholar

[2]

V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010). Google Scholar

[3]

A. Bossavit, "Électromagnétisme en Vue de la Modélisation,", Springer, (1993). Google Scholar

[4]

A. Bossavit and J. C. Verité, "A Mixed Finite Element Boundary Integral Equation Method to Solve the Three Dimensional Eddy Current Problem,", COMPUMAG Congress, (1981). Google Scholar

[5]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973). Google Scholar

[6]

H. Briane and G. W. Milton, Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,, Arch. Ration. Mech. Anal., 193 (2009), 715. doi: 10.1007/s00205-008-0200-y. Google Scholar

[7]

H. Briane and G. W. Milton, An antisymmetric effective Hall matrix,, SIAM J. Appl. Math., 70 (2010), 1810. Google Scholar

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). Google Scholar

[9]

C. M. Elliott and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems,", Pitman, (1982). Google Scholar

[10]

I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles,", Dunod Gauthier-Villars, (1974). Google Scholar

[11]

D. S. Jones, "The Theory of Electromagnetism,", Pergamon Press, (1964). Google Scholar

[12]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989). Google Scholar

[13]

L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,", Pergamon Press, (1960). Google Scholar

[14]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003). Google Scholar

[15]

A. M. Meirmanov, "The Stefan Problem,", De Gruyter, (1992). Google Scholar

[16]

F. Murat, Compacité par compensation,, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489. Google Scholar

[17]

F. Murat and L. Tartar, H-convergence,, in, (1997), 21. Google Scholar

[18]

M. Núñez, Formation of singularities in Hall magnetohydrodynamics,, J. Fluid Mech., 634 (2009), 499. Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969). Google Scholar

[20]

V. Solonnikov and G. Mulone, On the solvability of some initial-boundary value problems of magnetofluidmechanics with Hall and ion-slip effects,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 117. Google Scholar

[21]

L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,", Springer Berlin; UMI, (2009). Google Scholar

[22]

A. Visintin, Strong convergence results related to strict convexity,, Communications in P.D.E.s, 9 (1984), 439. Google Scholar

[23]

A. Visintin, Study of the eddy-current problem taking account of Hall's effect,, Appl. Anal., 19 (1985), 217. Google Scholar

[24]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). Google Scholar

[25]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996). Google Scholar

[26]

A. Visintin, Maxwell's equations with vector hysteresis,, Archive Rat. Mech. Anal., 175 (2005), 1. Google Scholar

[27]

A. Visintin, Introduction to Stefan-type problems,, in, (2008), 377. Google Scholar

[28]

A. Visintin, Electromagnetic processes in doubly-nonlinear composites,, Communications in P.D.E.s, 33 (2008), 808. Google Scholar

[29]

A. Visintin, Scale-transformations and homogenization of maximal monotone relations, and applications,, (forthcoming), (). Google Scholar

[30]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators,", Springer, (1990). Google Scholar

show all references

References:
[1]

N. W. Ashcroft and N. D. Mermin, "Solid State Physics,", Holt, (1976). Google Scholar

[2]

V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces,", Springer, (2010). Google Scholar

[3]

A. Bossavit, "Électromagnétisme en Vue de la Modélisation,", Springer, (1993). Google Scholar

[4]

A. Bossavit and J. C. Verité, "A Mixed Finite Element Boundary Integral Equation Method to Solve the Three Dimensional Eddy Current Problem,", COMPUMAG Congress, (1981). Google Scholar

[5]

H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (1973). Google Scholar

[6]

H. Briane and G. W. Milton, Homogenization of the three-dimensional Hall effect and change of sign of the Hall coefficient,, Arch. Ration. Mech. Anal., 193 (2009), 715. doi: 10.1007/s00205-008-0200-y. Google Scholar

[7]

H. Briane and G. W. Milton, An antisymmetric effective Hall matrix,, SIAM J. Appl. Math., 70 (2010), 1810. Google Scholar

[8]

M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). Google Scholar

[9]

C. M. Elliott and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems,", Pitman, (1982). Google Scholar

[10]

I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles,", Dunod Gauthier-Villars, (1974). Google Scholar

[11]

D. S. Jones, "The Theory of Electromagnetism,", Pergamon Press, (1964). Google Scholar

[12]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis,", Springer, (1989). Google Scholar

[13]

L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,", Pergamon Press, (1960). Google Scholar

[14]

I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications,", Elsevier, (2003). Google Scholar

[15]

A. M. Meirmanov, "The Stefan Problem,", De Gruyter, (1992). Google Scholar

[16]

F. Murat, Compacité par compensation,, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489. Google Scholar

[17]

F. Murat and L. Tartar, H-convergence,, in, (1997), 21. Google Scholar

[18]

M. Núñez, Formation of singularities in Hall magnetohydrodynamics,, J. Fluid Mech., 634 (2009), 499. Google Scholar

[19]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1969). Google Scholar

[20]

V. Solonnikov and G. Mulone, On the solvability of some initial-boundary value problems of magnetofluidmechanics with Hall and ion-slip effects,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 117. Google Scholar

[21]

L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,", Springer Berlin; UMI, (2009). Google Scholar

[22]

A. Visintin, Strong convergence results related to strict convexity,, Communications in P.D.E.s, 9 (1984), 439. Google Scholar

[23]

A. Visintin, Study of the eddy-current problem taking account of Hall's effect,, Appl. Anal., 19 (1985), 217. Google Scholar

[24]

A. Visintin, "Differential Models of Hysteresis,", Springer, (1994). Google Scholar

[25]

A. Visintin, "Models of Phase Transitions,", Birkhäuser, (1996). Google Scholar

[26]

A. Visintin, Maxwell's equations with vector hysteresis,, Archive Rat. Mech. Anal., 175 (2005), 1. Google Scholar

[27]

A. Visintin, Introduction to Stefan-type problems,, in, (2008), 377. Google Scholar

[28]

A. Visintin, Electromagnetic processes in doubly-nonlinear composites,, Communications in P.D.E.s, 33 (2008), 808. Google Scholar

[29]

A. Visintin, Scale-transformations and homogenization of maximal monotone relations, and applications,, (forthcoming), (). Google Scholar

[30]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators,", Springer, (1990). Google Scholar

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