| Title:
|
Div-curl lemma revisited: Applications in electromagnetism (English) |
| Author:
|
Slodička, Marián |
| Author:
|
Buša, Ján Jr. |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
328-340 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
compensated compactness |
| Keyword:
|
convergence |
| Keyword:
|
vector fields |
| MSC:
|
35B05 |
| MSC:
|
65J10 |
| MSC:
|
65M99 |
| MSC:
|
78A25 |
| idZBL:
|
Zbl 1201.78007 |
| idMR:
|
MR2663604 |
| . |
| Date available:
|
2010-09-13T16:42:27Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140747 |
| . |
| Reference:
|
[1] Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three dimensional nonsmooth domains.Math. Methods Appl. Sci. 21 (1998), 823–864. MR 1626990, 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B |
| Reference:
|
[2] Anderson, P. W., Kim, Y. B.: Hard superconductivity: Theory of the motion of Abrikosov flux lines.Rev. Mod. Phys. 36 (1964), 39–43. 10.1103/RevModPhys.36.39 |
| Reference:
|
[3] Bean, C. P.: Magnetization of high-field superconductors.Rev. Mod. Phys. 36 (1964), 31–39. 10.1103/RevModPhys.36.31 |
| Reference:
|
[4] Beasley, M. R., Labusch, R., Webb, W. W: Flux creep in type-II superconductors.Phys. Rev. 181 (1969), 682–700. 10.1103/PhysRev.181.682 |
| Reference:
|
[5] Bossavit, A.: Computational Electromagnetism.Variational Formulations, Complementarity, Edge Elements. (Electromagnetism, Vol. XVIII.) Academic Press, Orlando 1998. Zbl 0945.78001, MR 1488417 |
| Reference:
|
[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. Zbl 0917.65099, MR 1409140 |
| Reference:
|
[7] Chapman, S. J.: A hierarchy of models for type-II superconductors.SIAM Rev. 42 (2000), 4, 555–598. Zbl 0967.82014, MR 1814048, 10.1137/S0036144599371913 |
| Reference:
|
[8] Costabel, M.: A remark on the regularity of Maxwell’s equations on Lipschitz domain.Math. Methods Appl. Sci. 12 (1990), 365–368. MR 1048563, 10.1002/mma.1670120406 |
| Reference:
|
[9] Evans, L. C.: Partial Differential Equations.(Graduate Studies in Mathematics, Vol. 19.) American Mathematical Society, Providence, RI 1998. MR 1625845 |
| Reference:
|
[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. Zbl 0698.35004, MR 1034481 |
| Reference:
|
[11] Fabrizio, M., Morro, A.: Electromagnetism of Continuous Media.(Mathematical Modelling and Applications.) Oxford University Press, Oxford 2003. Zbl 1027.78001, MR 1996323 |
| Reference:
|
[12] Gasser, I., Marcati, P.: On a generalization of the div-curl lemma.Osaka J. Math. 45 (2008), 211–214. Zbl 1139.35379, MR 2416657 |
| Reference:
|
[13] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order.(Grundlehren der Mathematischen Wissenschaften, Vol. 224.) Springer, Berlin 1977. Zbl 1042.35002, MR 0473443, 10.1007/978-3-642-96379-7 |
| Reference:
|
[14] Jost, J.: Partial Differential Equations.(Graduate Texts in Mathematics Vol. 214 .) Springer, New York xxxx. Zbl 1121.35001, MR 1919991 |
| Reference:
|
[15] Kozono, H., Yanagisawa, T.: Global div-curl lemma on bounded domains in ${}^3$.J. Funct. Anal. 256 (2009), 11, 3847–3859. MR 2514064, 10.1016/j.jfa.2009.01.010 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[18] London, F.: Superfluids.Vol. II. Macroscopic Theory of Superfluid Helium. John Wiley & Sons, Inc., New York 1954. Zbl 0058.23405 |
| Reference:
|
[19] Mayergoyz, I. D.: Nonlinear Diffusion of Electromagnetic Fields with Applications to Eddy Currents and Surerconductivity.Academic Press, San Diego 1998. |
| Reference:
|
[20] Monk, P.: Finite Element Methods for Maxwell’s Equations.Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford 2003. Zbl 1024.78009, MR 2059447 |
| Reference:
|
[21] Murat, F.: Compacite par compensation.Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV Ser. 5 (1978), 489–507. Zbl 0464.46034, MR 0506997 |
| Reference:
|
[22] Nečas, J.: Les méthodes directes en théorie des équations elliptiques.Academia, Prague 1967. MR 0227584 |
| Reference:
|
[23] Nečas, J.: Introduction to the Theory of Nonlinear Elliptic Equations.John Wiley & Sons Ltd., New York 1986. MR 0874752 |
| Reference:
|
[24] Prigozhin, L.: The bean model in superconductivity: Variational formulation and numerical solution.J. Comput. Phys. 129 (1996), 1, 190–200. Zbl 0866.65081, MR 1419742, 10.1006/jcph.1996.0243 |
| Reference:
|
[25] Prigozhin, L.: On the bean critical-state model in superconductivity.Eur. J. Appl. Math. 7 (1996), 3, 237–247. Zbl 0873.49007, MR 1401169, 10.1017/S0956792500002333 |
| Reference:
|
[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. MR 2193975, 10.1093/imanum/dri030 |
| Reference:
|
[27] Slodička, M.: Nonlinear diffusion in type-II superconductors.J. Comput. Appl. Math. 216 (2008), 2, 568–576. MR 2406658, 10.1016/j.cam.2006.03.055 |
| Reference:
|
[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. Zbl 0437.35004, MR 0584398 |
| Reference:
|
[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 |
| Reference:
|
[30] Weber, C.: A local compactness theorem for Maxwell’s equations.Math. Methods Appl. Sci. 2 (1980), 12–25. Zbl 0432.35032, MR 0561375, 10.1002/mma.1670020103 |
| . |
