Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
compensated compactness; convergence; vector fields
compensated compactness; convergence; vector fields
Summary:
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.
References:
[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 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B | MR 1626990
[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 10.1103/RevModPhys.36.39
[3] Bean, C. P.: Magnetization of high-field superconductors. Rev. Mod. Phys. 36 (1964), 31–39. DOI 10.1103/RevModPhys.36.31
[4] Beasley, M. R., Labusch, R., Webb, W. W: Flux creep in type-II superconductors. Phys. Rev. 181 (1969), 682–700. DOI 10.1103/PhysRev.181.682
[5] Bossavit, A.: Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements. (Electromagnetism, Vol. XVIII.) Academic Press, Orlando 1998. MR 1488417 | Zbl 0945.78001
[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 1409140 | Zbl 0917.65099
[7] Chapman, S. J.: A hierarchy of models for type-II superconductors. SIAM Rev. 42 (2000), 4, 555–598. DOI 10.1137/S0036144599371913 | MR 1814048 | Zbl 0967.82014
[8] Costabel, M.: A remark on the regularity of Maxwell’s equations on Lipschitz domain. Math. Methods Appl. Sci. 12 (1990), 365–368. DOI 10.1002/mma.1670120406 | MR 1048563
[9] Evans, L. C.: Partial Differential Equations. (Graduate Studies in Mathematics, Vol. 19.) American Mathematical Society, Providence, RI 1998. MR 1625845
[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 1034481 | Zbl 0698.35004
[11] Fabrizio, M., Morro, A.: Electromagnetism of Continuous Media. (Mathematical Modelling and Applications.) Oxford University Press, Oxford 2003. MR 1996323 | Zbl 1027.78001
[12] Gasser, I., Marcati, P.: On a generalization of the div-curl lemma. Osaka J. Math. 45 (2008), 211–214. MR 2416657 | Zbl 1139.35379
[13] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. (Grundlehren der Mathematischen Wissenschaften, Vol. 224.) Springer, Berlin 1977. DOI 10.1007/978-3-642-96379-7 | MR 0473443 | Zbl 1042.35002
[14] Jost, J.: Partial Differential Equations. (Graduate Texts in Mathematics Vol. 214 .) Springer, New York xxxx. MR 1919991 | Zbl 1121.35001
[15] Kozono, H., Yanagisawa, T.: Global div-curl lemma on bounded domains in ${}^3$. J. Funct. Anal. 256 (2009), 11, 3847–3859. DOI 10.1016/j.jfa.2009.01.010 | MR 2514064
[16] Kufner, A., John, O., Fučík, S.: Function Spaces. (Monograpfs and Textbooks on Mechanics of Solids and Fluids.) Noordhoff International Publishing, Leyden 1977. MR 0482102
[17] London, F.: Superfluids. Vol. I.: Macroscopic Theory of Superconductivity. New York: John Wiley & Sons, Inc. London: Chapman & Hall, Ltd., New York 1950. Zbl 0058.23405
[18] London, F.: Superfluids. Vol. II. Macroscopic Theory of Superfluid Helium. John Wiley & Sons, Inc., New York 1954. Zbl 0058.23405
[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 2059447 | Zbl 1024.78009
[21] Murat, F.: Compacite par compensation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV Ser. 5 (1978), 489–507. MR 0506997 | Zbl 0464.46034
[22] Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Academia, Prague 1967. MR 0227584
[23] Nečas, J.: Introduction to the Theory of Nonlinear Elliptic Equations. John Wiley & Sons Ltd., New York 1986. MR 0874752
[24] Prigozhin, L.: The bean model in superconductivity: Variational formulation and numerical solution. J. Comput. Phys. 129 (1996), 1, 190–200. DOI 10.1006/jcph.1996.0243 | MR 1419742 | Zbl 0866.65081
[25] Prigozhin, L.: On the bean critical-state model in superconductivity. Eur. J. Appl. Math. 7 (1996), 3, 237–247. DOI 10.1017/S0956792500002333 | MR 1401169 | Zbl 0873.49007
[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 10.1093/imanum/dri030 | MR 2193975
[27] Slodička, M.: Nonlinear diffusion in type-II superconductors. J. Comput. Appl. Math. 216 (2008), 2, 568–576. DOI 10.1016/j.cam.2006.03.055 | MR 2406658
[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 0584398 | Zbl 0437.35004
[29] Vajnberg, M. M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. John Wiley & Sons, New York 1973. Zbl 0279.47022
[30] Weber, C.: A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980), 12–25. DOI 10.1002/mma.1670020103 | MR 0561375 | Zbl 0432.35032
Search
Partner of
