
Previous Article
Equicontinuous sweeping processes
 DCDSB Home
 This Issue

Next Article
On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited
OhmHall conduction in hysteresisfree ferromagnetic processes
1.  Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento)  Italia, Italy 
This model leads to the formulation of an initialvalue problem for a doublynonlinear parabolichyperbolic system in the whole $R^3$. Existence of a weak solution is proved, via approximation by timediscretization, derivation of a priori estimates, and passage to the limit. This final step rests upon a timedependent extension of the Murat and Tartar divcurl lemma, and on compactness by strict convexity.
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 SemiGroupes de Contractions dans les Espaces de Hilbert," NorthHolland, Amsterdam 1973. 
[6] 
H. Briane and G. W. Milton, Homogenization of the threedimensional Hall effect and change of sign of the Hall coefficient, Arch. Ration. Mech. Anal., 193 (2009), 715736. doi: 10.1007/s002050080200y. 
[7] 
H. Briane and G. W. Milton, An antisymmetric effective Hall matrix, SIAM J. Appl. Math., 70 (2010), 18101820. 
[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 GauthierVillars, 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), 489507. 
[17] 
F. Murat and L. Tartar, Hconvergence, in "Topics in the Mathematical Modelling of Composite Materials" (eds. A. Cherkaev and R. Kohn), Birkhäuser, Boston (1997), 2144. 
[18] 
M. Núñez, Formation of singularities in Hall magnetohydrodynamics, J. Fluid Mech., 634 (2009), 499507. 
[19] 
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. 
[20] 
V. Solonnikov and G. Mulone, On the solvability of some initialboundary value problems of magnetofluidmechanics with Hall and ionslip effects, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 117132. 
[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), 439466. 
[23] 
A. Visintin, Study of the eddycurrent problem taking account of Hall's effect, Appl. Anal., 19 (1985), 217226. 
[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), 138. 
[27] 
A. Visintin, Introduction to Stefantype problems, in "Handbook of Differential Equations: Evolutionary Differential Equations vol. IV" (eds. C. Dafermos and M. Pokorny), NorthHolland, Amsterdam (2008), chap. 8, 377484. 
[28] 
A. Visintin, Electromagnetic processes in doublynonlinear composites, Communications in P.D.E.s, 33 (2008), 808841. 
[29] 
A. Visintin, Scaletransformations 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. 
show all references
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 SemiGroupes de Contractions dans les Espaces de Hilbert," NorthHolland, Amsterdam 1973. 
[6] 
H. Briane and G. W. Milton, Homogenization of the threedimensional Hall effect and change of sign of the Hall coefficient, Arch. Ration. Mech. Anal., 193 (2009), 715736. doi: 10.1007/s002050080200y. 
[7] 
H. Briane and G. W. Milton, An antisymmetric effective Hall matrix, SIAM J. Appl. Math., 70 (2010), 18101820. 
[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 GauthierVillars, 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), 489507. 
[17] 
F. Murat and L. Tartar, Hconvergence, in "Topics in the Mathematical Modelling of Composite Materials" (eds. A. Cherkaev and R. Kohn), Birkhäuser, Boston (1997), 2144. 
[18] 
M. Núñez, Formation of singularities in Hall magnetohydrodynamics, J. Fluid Mech., 634 (2009), 499507. 
[19] 
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. 
[20] 
V. Solonnikov and G. Mulone, On the solvability of some initialboundary value problems of magnetofluidmechanics with Hall and ionslip effects, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 6 (1995), 117132. 
[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), 439466. 
[23] 
A. Visintin, Study of the eddycurrent problem taking account of Hall's effect, Appl. Anal., 19 (1985), 217226. 
[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), 138. 
[27] 
A. Visintin, Introduction to Stefantype problems, in "Handbook of Differential Equations: Evolutionary Differential Equations vol. IV" (eds. C. Dafermos and M. Pokorny), NorthHolland, Amsterdam (2008), chap. 8, 377484. 
[28] 
A. Visintin, Electromagnetic processes in doublynonlinear composites, Communications in P.D.E.s, 33 (2008), 808841. 
[29] 
A. Visintin, Scaletransformations 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. 
[1] 
Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure and Applied Analysis, 2013, 12 (5) : 22292266. doi: 10.3934/cpaa.2013.12.2229 
[2] 
J. J. Morgan, HongMing Yin. On Maxwell's system with a thermal effect. Discrete and Continuous Dynamical Systems  B, 2001, 1 (4) : 485494. 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 and Heterogeneous Media, 2021, 16 (2) : 283315. doi: 10.3934/nhm.2021007 
[4] 
Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022021 
[5] 
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 and Imaging, 2016, 10 (4) : 869898. doi: 10.3934/ipi.2016025 
[6] 
Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure and Applied Analysis, 2015, 14 (4) : 13571376. doi: 10.3934/cpaa.2015.14.1357 
[7] 
Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order HamiltonJacobiBellman equations. Approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 39333964. doi: 10.3934/dcds.2015.35.3933 
[8] 
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) : 10011014. doi: 10.3934/proc.2011.2011.1001 
[9] 
Alain Miranville. Asymptotic behavior of the conserved Caginalp phasefield system based on the MaxwellCattaneo law. Communications on Pure and Applied Analysis, 2014, 13 (5) : 19711987. doi: 10.3934/cpaa.2014.13.1971 
[10] 
Ahmad Makki, Alain Miranville, Georges Sadaka. On the nonconserved Caginalp phasefield system based on the MaxwellCattaneo law with two temperatures and logarithmic potentials. Discrete and Continuous Dynamical Systems  B, 2019, 24 (3) : 13411365. doi: 10.3934/dcdsb.2019019 
[11] 
Jincheng Gao, ZhengAn Yao. Global existence and optimal decay rates of solutions for compressible HallMHD equations. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 30773106. doi: 10.3934/dcds.2016.36.3077 
[12] 
Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Wellposedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307340. doi: 10.3934/ipi.2013.7.307 
[13] 
Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete and Continuous Dynamical Systems  S, 2015, 8 (4) : 769772. doi: 10.3934/dcdss.2015.8.769 
[14] 
Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 59295949. doi: 10.3934/dcds.2016060 
[15] 
Guangying Lv, Hongjun Gao, Jinlong Wei, JiangLun Wu. The effect of noise intensity on parabolic equations. Discrete and Continuous Dynamical Systems  B, 2020, 25 (5) : 17151728. doi: 10.3934/dcdsb.2019248 
[16] 
W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure and Applied Analysis, 2005, 4 (2) : 431444. doi: 10.3934/cpaa.2005.4.431 
[17] 
Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations and Control Theory, 2019, 8 (2) : 343357. doi: 10.3934/eect.2019018 
[18] 
PierreDamien Thizy. KleinGordonMaxwell equations in high dimensions. Communications on Pure and Applied Analysis, 2015, 14 (3) : 10971125. doi: 10.3934/cpaa.2015.14.1097 
[19] 
Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete and Continuous Dynamical Systems  B, 2005, 5 (3) : 631658. doi: 10.3934/dcdsb.2005.5.631 
[20] 
Percy D. Makita. Nonradial solutions for the KleinGordonMaxwell equations. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 22712283. doi: 10.3934/dcds.2012.32.2271 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]