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

On a p-curl system arising in electromagnetism

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.
Citation: Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete and 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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

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