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June  2012, 5(3): 605-629. doi: 10.3934/dcdss.2012.5.605

On a p-curl system arising in electromagnetism

Fernando Miranda 1, , José-Francisco Rodrigues 2, and  Lisa Santos 1,

1. 

Department of Mathematics/CMAT, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal, Portugal
2. 

CMAF/Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa

Received  September 2010 Revised  June 2011 Published  October 2011

We prove existence of solution of a $p$-curl type evolutionary system arising in electromagnetism with a power nonlinearity of order $p$, $1 < p < \infty$, assuming natural tangential boundary conditions. We consider also the asymptotic behaviour in the power obtaining, when $p$ tends to infinity, a variational inequality with a curl constraint. We also discuss the existence, uniqueness and continuous dependence on the data of the solutions to general variational inequalities with curl constraints dependent on time, as well as the asymptotic stabilization in time towards the stationary solution with and without constraint.
Keywords: Electromagnetic problems, variational inequalities, variational methods, superconductivity models..
Mathematics Subject Classification: Primary: 35K87, 78M30; Secondary: 49J4.
Citation: Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 605-629. doi: 10.3934/dcdss.2012.5.605
References:
[1]

C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields, C. R. Math. Acad. Sci. Paris, 349 (2011), 529-534.  Google Scholar

[2]

A. Bermúdez, R. Muñoz-Sola and F. Pena, A nonlinear partial differential system arising in thermoelectricity, European J. Appl. Math., 16 (2005), 683-712.  Google Scholar

[3]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,'' Electromagnetism, Academic Press, Inc., San Diego, CA, 1998.  Google Scholar

[4]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 3, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990.  Google Scholar

[5]

A. Haraux, "Nonlinear Evolution Equations--Global Behavior of Solutions,'' Lecture Notes in Mathematics, 841, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[6]

L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,'' Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[7]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

F. Miranda, J.-F. Rodrigues and L. Santos, A class of stationary nonlinear Maxwell systems, Math. Models Methods Appl. Sci., 19 (2009), 1883-1905. doi: 10.1142/S0218202509003966.  Google Scholar

[9]

D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997), 131-192. doi: 10.1007/BF02649132.  Google Scholar

[10]

M. Mitrea, Boundary value problems for Dirac operators and Maxwell's equations in non-smooth domains, Math. Methods Appl. Sci., 25 (2002), 1355-1369. doi: 10.1002/mma.375.  Google Scholar

[11]

L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math., 7 (1996), 237-247.  Google Scholar

[12]

L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition, Port. Math., 48 (1991), 441-468.  Google Scholar

[13]

L. Santos, Variational problems with non-constant gradient constraints, Port. Math. (N.S.), 59 (2002), 205-248.  Google Scholar

[14]

C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992.  Google Scholar

[15]

J. Simon, Quelques propriétés de solutions d'équations et d'inéquations d'évolution paraboliques non linéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 585-609.  Google Scholar

[16]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar

[17]

S. Sobolev, "Applications of Functional Analysis in Mathematical Physics,'' Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963.  Google Scholar

[18]

W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.  Google Scholar

[19]

H.-M. Yin, On a nonlinear Maxwell's system in quasi-stationary electromagnetic fields, Math. Models Methods Appl. Sci., 14 (2004), 1521-1539. doi: 10.1142/S0218202504003787.  Google Scholar

[20]

H.-M. Yin, B. Li, and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Contin. Dyn. Syst., 8 (2002), 781-794. doi: 10.3934/dcds.2002.8.781.  Google Scholar

[21]

S. Zheng, "Nonlinear Evolution Equations,'' Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

show all references

References:
[1]

C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields, C. R. Math. Acad. Sci. Paris, 349 (2011), 529-534.  Google Scholar

[2]

A. Bermúdez, R. Muñoz-Sola and F. Pena, A nonlinear partial differential system arising in thermoelectricity, European J. Appl. Math., 16 (2005), 683-712.  Google Scholar

[3]

A. Bossavit, "Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements,'' Electromagnetism, Academic Press, Inc., San Diego, CA, 1998.  Google Scholar

[4]

R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology,'' Vol. 3, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990.  Google Scholar

[5]

A. Haraux, "Nonlinear Evolution Equations--Global Behavior of Solutions,'' Lecture Notes in Mathematics, 841, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[6]

L. Landau and E. Lifshitz, "Electrodynamics of Continuous Media,'' Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960.  Google Scholar

[7]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[8]

F. Miranda, J.-F. Rodrigues and L. Santos, A class of stationary nonlinear Maxwell systems, Math. Models Methods Appl. Sci., 19 (2009), 1883-1905. doi: 10.1142/S0218202509003966.  Google Scholar

[9]

D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $R^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl., 3 (1997), 131-192. doi: 10.1007/BF02649132.  Google Scholar

[10]

M. Mitrea, Boundary value problems for Dirac operators and Maxwell's equations in non-smooth domains, Math. Methods Appl. Sci., 25 (2002), 1355-1369. doi: 10.1002/mma.375.  Google Scholar

[11]

L. Prigozhin, On the Bean critical-state model in superconductivity, European J. Appl. Math., 7 (1996), 237-247.  Google Scholar

[12]

L. Santos, A diffusion problem with gradient constraint and evolutive Dirichlet condition, Port. Math., 48 (1991), 441-468.  Google Scholar

[13]

L. Santos, Variational problems with non-constant gradient constraints, Port. Math. (N.S.), 59 (2002), 205-248.  Google Scholar

[14]

C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992.  Google Scholar

[15]

J. Simon, Quelques propriétés de solutions d'équations et d'inéquations d'évolution paraboliques non linéaires, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 585-609.  Google Scholar

[16]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.  Google Scholar

[17]

S. Sobolev, "Applications of Functional Analysis in Mathematical Physics,'' Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963.  Google Scholar

[18]

W. von Wahl, Estimating $\nabla u$ by div $u$ and curl $u$, Math. Methods Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.  Google Scholar

[19]

H.-M. Yin, On a nonlinear Maxwell's system in quasi-stationary electromagnetic fields, Math. Models Methods Appl. Sci., 14 (2004), 1521-1539. doi: 10.1142/S0218202504003787.  Google Scholar

[20]

H.-M. Yin, B. Li, and J. Zou, A degenerate evolution system modeling Bean's critical-state type-II superconductors, Discrete Contin. Dyn. Syst., 8 (2002), 781-794. doi: 10.3934/dcds.2002.8.781.  Google Scholar

[21]

S. Zheng, "Nonlinear Evolution Equations,'' Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

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