# 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 & Continuous Dynamical Systems - B, 2013, 18 (2) : 551-563. doi: 10.3934/dcdsb.2013.18.551
##### References:

show all references

##### References:
 [1] Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229 [2] J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485 [3] Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007 [4] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [5] Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357 [6] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [7] Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001 [8] Alain Miranville. Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1971-1987. doi: 10.3934/cpaa.2014.13.1971 [9] Ahmad Makki, Alain Miranville, Georges Sadaka. On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1341-1365. doi: 10.3934/dcdsb.2019019 [10] Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077 [11] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307 [12] Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769 [13] W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431 [14] Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations & Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018 [15] Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097 [16] Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631 [17] Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271 [18] Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257 [19] Yuri Kalinin, Volker Reitmann, Nayil Yumaguzin. Asymptotic behavior of Maxwell's equation in one-space dimension with thermal effect. Conference Publications, 2011, 2011 (Special) : 754-762. doi: 10.3934/proc.2011.2011.754 [20] Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060

2020 Impact Factor: 1.327