# American Institute of Mathematical Sciences

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 and 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, Rinehart and Winston, Philadelphia, 1976. [2] V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces," Springer, Berlin, 2010. [3] A. Bossavit, "Électromagnétisme en Vue de la Modélisation," Springer, Paris, 1993. [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, Chicago, 1981. [5] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam 1973. [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), 715-736. doi: 10.1007/s00205-008-0200-y. [7] H. Briane and G. W. Milton, An antisymmetric effective Hall matrix, SIAM J. Appl. Math., 70 (2010), 1810-1820. [8] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Heidelberg, 1996. [9] C. M. Elliott and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems," Pitman, Boston, 1982. [10] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles," Dunod Gauthier-Villars, Paris, 1974. [11] D. S. Jones, "The Theory of Electromagnetism," Pergamon Press, Oxford, 1964. [12] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis," Springer, Berlin, 1989. Russian edition: Nauka, Moscow, 1983. [13] L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media," Pergamon Press, Oxford, 1960. [14] I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications," Elsevier, Amsterdam, 2003. [15] A. M. Meirmanov, "The Stefan Problem," De Gruyter, Berlin 1992. (Russian edition: Nauka, Novosibirsk 1986). [16] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489-507. [17] F. Murat and L. Tartar, H-convergence, in "Topics in the Mathematical Modelling of Composite Materials" (eds. A. Cherkaev and R. Kohn), Birkhäuser, Boston (1997), 21-44. [18] M. Núñez, Formation of singularities in Hall magnetohydrodynamics, J. Fluid Mech., 634 (2009), 499-507. [19] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. [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), 117-132. [21] L. Tartar, "The General Theory of Homogenization. A Personalized Introduction," Springer Berlin; UMI, Bologna, 2009. [22] A. Visintin, Strong convergence results related to strict convexity, Communications in P.D.E.s, 9 (1984), 439-466. [23] A. Visintin, Study of the eddy-current problem taking account of Hall's effect, Appl. Anal., 19 (1985), 217-226. [24] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [25] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [26] A. Visintin, Maxwell's equations with vector hysteresis, Archive Rat. Mech. Anal., 175 (2005), 1-38. [27] A. Visintin, Introduction to Stefan-type problems, in "Handbook of Differential Equations: Evolutionary Differential Equations vol. IV" (eds. C. Dafermos and M. Pokorny), North-Holland, Amsterdam (2008), chap. 8, 377-484. [28] A. Visintin, Electromagnetic processes in doubly-nonlinear composites, Communications in P.D.E.s, 33 (2008), 808-841. [29] A. Visintin, Scale-transformations and homogenization of maximal monotone relations, and applications,, (forthcoming), (). [30] E. Zeidler, "Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators," Springer, New York, 1990.

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##### References:
 [1] N. W. Ashcroft and N. D. Mermin, "Solid State Physics," Holt, Rinehart and Winston, Philadelphia, 1976. [2] V. Barbu, "Nonlinear Differential Equations of Monotone Types in Banach Spaces," Springer, Berlin, 2010. [3] A. Bossavit, "Électromagnétisme en Vue de la Modélisation," Springer, Paris, 1993. [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, Chicago, 1981. [5] H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam 1973. [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), 715-736. doi: 10.1007/s00205-008-0200-y. [7] H. Briane and G. W. Milton, An antisymmetric effective Hall matrix, SIAM J. Appl. Math., 70 (2010), 1810-1820. [8] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions," Springer, Heidelberg, 1996. [9] C. M. Elliott and J. R. Ockendon, "Weak and Variational Methods for Moving Boundary Problems," Pitman, Boston, 1982. [10] I. Ekeland and R. Temam, "Analyse Convexe et Problèmes Variationnelles," Dunod Gauthier-Villars, Paris, 1974. [11] D. S. Jones, "The Theory of Electromagnetism," Pergamon Press, Oxford, 1964. [12] M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, "Systems with Hysteresis," Springer, Berlin, 1989. Russian edition: Nauka, Moscow, 1983. [13] L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media," Pergamon Press, Oxford, 1960. [14] I. D. Mayergoyz, "Mathematical Models of Hysteresis and Their Applications," Elsevier, Amsterdam, 2003. [15] A. M. Meirmanov, "The Stefan Problem," De Gruyter, Berlin 1992. (Russian edition: Nauka, Novosibirsk 1986). [16] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489-507. [17] F. Murat and L. Tartar, H-convergence, in "Topics in the Mathematical Modelling of Composite Materials" (eds. A. Cherkaev and R. Kohn), Birkhäuser, Boston (1997), 21-44. [18] M. Núñez, Formation of singularities in Hall magnetohydrodynamics, J. Fluid Mech., 634 (2009), 499-507. [19] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. [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), 117-132. [21] L. Tartar, "The General Theory of Homogenization. A Personalized Introduction," Springer Berlin; UMI, Bologna, 2009. [22] A. Visintin, Strong convergence results related to strict convexity, Communications in P.D.E.s, 9 (1984), 439-466. [23] A. Visintin, Study of the eddy-current problem taking account of Hall's effect, Appl. Anal., 19 (1985), 217-226. [24] A. Visintin, "Differential Models of Hysteresis," Springer, Berlin, 1994. [25] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [26] A. Visintin, Maxwell's equations with vector hysteresis, Archive Rat. Mech. Anal., 175 (2005), 1-38. [27] A. Visintin, Introduction to Stefan-type problems, in "Handbook of Differential Equations: Evolutionary Differential Equations vol. IV" (eds. C. Dafermos and M. Pokorny), North-Holland, Amsterdam (2008), chap. 8, 377-484. [28] A. Visintin, Electromagnetic processes in doubly-nonlinear composites, Communications in P.D.E.s, 33 (2008), 808-841. [29] A. Visintin, Scale-transformations and homogenization of maximal monotone relations, and applications,, (forthcoming), (). [30] E. Zeidler, "Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators," Springer, New York, 1990.
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