Geodesic
Div-curl lemma revisited: Applications in electromagnetism
Kybernetika, Tome 46 (2010) no. 2, pp. 328-340 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Two new time-dependent versions of div-curl results in a bounded domain $\Omega\subset\mathbb{R}^3$ are presented. We study a limit of the product ${\boldmath v}_k{\boldmath w}_k$, where the sequences ${\boldmath v}_k$ and ${\boldmath w}_k$ belong to $\L_{2}(\Omega)$. In Theorem 2.1 we assume that $\nabla\times{\boldmath v}_k$ is bounded in the $L_p$-norm and $\nabla\cdot{\boldmath w}_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla\times{\boldmath w}_k$ is bounded in the $L_p$-norm and $\nabla\cdot{\boldmath w}_k$ is controlled in the $L_r$-norm. The time derivative of ${\boldmath w}_k$ is bounded in both cases in the norm of $\H^{-1}(\Omega)$. The convergence (in the sense of distributions) of ${\boldmath v}_k{\boldmath w}_k$ to the product ${\boldmath v}{\boldmath w}$ of weak limits of ${\boldmath v}_k$ and ${\boldmath w}_k$ is shown.
Two new time-dependent versions of div-curl results in a bounded domain $\Omega\subset\mathbb{R}^3$ are presented. We study a limit of the product ${\boldmath v}_k{\boldmath w}_k$, where the sequences ${\boldmath v}_k$ and ${\boldmath w}_k$ belong to $\L_{2}(\Omega)$. In Theorem 2.1 we assume that $\nabla\times{\boldmath v}_k$ is bounded in the $L_p$-norm and $\nabla\cdot{\boldmath w}_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla\times{\boldmath w}_k$ is bounded in the $L_p$-norm and $\nabla\cdot{\boldmath w}_k$ is controlled in the $L_r$-norm. The time derivative of ${\boldmath w}_k$ is bounded in both cases in the norm of $\H^{-1}(\Omega)$. The convergence (in the sense of distributions) of ${\boldmath v}_k{\boldmath w}_k$ to the product ${\boldmath v}{\boldmath w}$ of weak limits of ${\boldmath v}_k$ and ${\boldmath w}_k$ is shown.
Classification : 35B05, 65J10, 65M99, 78A25
Keywords: compensated compactness; convergence; vector fields 
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Slodička, Marián; Buša, Ján Jr. Div-curl lemma revisited: Applications in electromagnetism. Kybernetika, Tome 46 (2010) no. 2, pp. 328-340. http://geodesic.mathdoc.fr/item/KYB_2010_46_2_a7/

[1] Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three dimensional nonsmooth domains. Math. Methods Appl. Sci. 21 (1998), 823–864. | DOI | MR

[2] Anderson, P. W., Kim, Y. B.: Hard superconductivity: Theory of the motion of Abrikosov flux lines. Rev. Mod. Phys. 36 (1964), 39–43. | DOI

[3] Bean, C. P.: Magnetization of high-field superconductors. Rev. Mod. Phys. 36 (1964), 31–39. | DOI

[4] Beasley, M. R., Labusch, R., Webb, W. W: Flux creep in type-II superconductors. Phys. Rev. 181 (1969), 682–700. | DOI

[5] Bossavit, A.: Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements. (Electromagnetism, Vol. XVIII.) Academic Press, Orlando 1998. | MR | Zbl

[6] Cessenat, M.: Mathematical methods in electromagnetism. Linear theory and applications. (Series on Advances in Mathematics for Applied Sciences, Vol. 41.) World Scientific Publishers, Singapore 1996. | MR | Zbl

[7] Chapman, S. J.: A hierarchy of models for type-II superconductors. SIAM Rev. 42 (2000), 4, 555–598. | DOI | MR | Zbl

[8] Costabel, M.: A remark on the regularity of Maxwell’s equations on Lipschitz domain. Math. Methods Appl. Sci. 12 (1990), 365–368. | DOI | MR

[9] Evans, L. C.: Partial Differential Equations. (Graduate Studies in Mathematics, Vol. 19.) American Mathematical Society, Providence, RI 1998. | MR

[10] Evans, L. C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. (Conference Board of the Mathematical Sciences, Vol. 74. Regional Conference Series in Mathematics.) American Mathematical Society, Providence 1990. | MR | Zbl

[11] Fabrizio, M., Morro, A.: Electromagnetism of Continuous Media. (Mathematical Modelling and Applications.) Oxford University Press, Oxford 2003. | MR | Zbl

[12] Gasser, I., Marcati, P.: On a generalization of the div-curl lemma. Osaka J. Math. 45 (2008), 211–214. | MR | Zbl

[13] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. (Grundlehren der Mathematischen Wissenschaften, Vol. 224.) Springer, Berlin 1977. | DOI | MR | Zbl

[14] Jost, J.: Partial Differential Equations. (Graduate Texts in Mathematics Vol. 214 .) Springer, New York xxxx. | MR | Zbl

[15] Kozono, H., Yanagisawa, T.: Global div-curl lemma on bounded domains in ${}^3$. J. Funct. Anal. 256 (2009), 11, 3847–3859. | DOI | MR

[16] Kufner, A., John, O., Fučík, S.: Function Spaces. (Monograpfs and Textbooks on Mechanics of Solids and Fluids.) Noordhoff International Publishing, Leyden 1977. | MR

[17] London, F.: Superfluids. Vol. I.: Macroscopic Theory of Superconductivity. New York: John Wiley & Sons, Inc. London: Chapman & Hall, Ltd., New York 1950. | Zbl

[18] London, F.: Superfluids. Vol. II. Macroscopic Theory of Superfluid Helium. John Wiley & Sons, Inc., New York 1954. | Zbl

[19] Mayergoyz, I. D.: Nonlinear Diffusion of Electromagnetic Fields with Applications to Eddy Currents and Surerconductivity. Academic Press, San Diego 1998.

[20] Monk, P.: Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford 2003. | MR | Zbl

[21] Murat, F.: Compacite par compensation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV Ser. 5 (1978), 489–507. | MR | Zbl

[22] Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. | MR

[23] Nečas, J.: Introduction to the Theory of Nonlinear Elliptic Equations. John Wiley & Sons Ltd., New York 1986. | MR

[24] Prigozhin, L.: The bean model in superconductivity: Variational formulation and numerical solution. J. Comput. Phys. 129 (1996), 1, 190–200. | DOI | MR | Zbl

[25] Prigozhin, L.: On the bean critical-state model in superconductivity. Eur. J. Appl. Math. 7 (1996), 3, 237–247. | DOI | MR | Zbl

[26] Slodička, M.: A time discretization scheme for a nonlinear degenerate eddy current model for ferromagnetic materials. IMA J. Numer. Anal. 26 (2006), 1, 173–187. | DOI | MR

[27] Slodička, M.: Nonlinear diffusion in type-II superconductors. J. Comput. Appl. Math. 216 (2008), 2, 568–576. | DOI | MR

[28] Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot–Watt Symp., Vol. 4, Edinburgh 1979, Res. Notes Math. 39 (1979), pp. 136–212. | MR | Zbl

[29] Vajnberg, M. M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. John Wiley & Sons, New York 1973. | Zbl

[30] Weber, C.: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980), 12–25. | DOI | MR | Zbl