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Convergence and structure theorems for order-preserving dynamical systems with mass conservation
Variational and operator methods for Maxwell-Stokes system
School of Mathematical Sciences, East China Normal University and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, Shanghai 200062, China |
In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.
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show all references
References:
[1] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
doi: 10.1002/cpa.3160120405. |
[2] |
Y. Aharonov and D. Bohm,
Significance of electromagnetic potentials in the quantum theory, Phys. Rev., 115 (1959), 485-491.
doi: 10.1103/PhysRev.115.485. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
C. Amrouche, C. Bernardi, M. Dauge and V. Girault,
Vector potentials in three-dimensional non-smooth domains, Math. Meth. Appl. Sci., 21 (1998), 823-864.
doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. |
[5] |
C. Amrouche and V. Girault,
On the existence and regularity of the solution of Stokes problem in arbitrary dimension, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 171-175.
doi: 10.3792/pjaa.67.171. |
[6] |
C. Amrouche and N. Seloula,
$L^p$-teory for vector potentials and Sobolev's inequalities for vector fields. Application to the Stokes equations with presure boundary conditions, Math. Models Meth. Appl. Sci., 23 (2013), 37-92.
doi: 10.1142/S0218202512500455. |
[7] |
G. Auchmuty and J. C. Alexander,
$L^2$-well-posedness of $3D$ div-curl boundary value problems, Quart. Appl. Math., 63 (2005), 479-508.
doi: 10.1090/S0033-569X-05-00972-5. |
[8] |
F. Bachinger, U. Langer and J. Schöberl,
Numerical analysis of nonlinear multiharmonic eddy current problems, Numer. Math., 100 (2005), 593-616.
doi: 10.1007/s00211-005-0597-2. |
[9] |
T. Bartsch and J. Mederski,
Ground and bound state solutions of semilinear timeharmonic Maxwell equations in a bounded domain, Arch. Ration. Mech. Anal., 215 (2015), 283-306.
doi: 10.1007/s00205-014-0778-1. |
[10] |
P. Bates and X. B. Pan,
Nucleation of instability of the Meissner state of 3-dimensional superconductors, Comm. Math. Phys., 276 (2007), 571-610.
doi: 10.1007/s00220-007-0335-y. |
[11] |
V. Benci and D. Fortunato,
Towards a unified field theory for classical electrodynamics, Arch. Ration. Mech. Anal., 173 (2004), 379-414.
doi: 10.1007/s00205-004-0324-7. |
[12] |
J. Bolik and W. von Wahl,
Estimating $\nabla {\bf{u}}$ in terms of $ \rm div $ ${\bf u}$, $ \rm curl $ $ {\bf u}$, either $(\nu, {\bf u})$ or $\nu\times{\bf u}$ and the topology, Math. Meth. Appl. Sci., 20 (1997), 737-744.
doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9. |
[13] |
F. Boyer and P. Fabrie, Mathematical Tools for the Navier-Stokes Equations and Related Models, Applied. Math. Sci., vol. 183, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[14] |
A. Buffa, M. Costbel and D. Sheen,
On traces for $H({\rm {curl}}\; ,\Omega)$ in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845-867.
doi: 10.1016/S0022-247X(02)00455-9. |
[15] |
L. Cattabriga,
Su un problema al contorno relativo al sistema di equazioni di Stokes, (Italian) Rend. Sem. Padova., 31 (1961), 308-340.
|
[16] |
M. Cessenat, Mathematical Methods in Electromagnetism-Linear Theory and Applications, Series on Advances in Mathematics for Applied Sciences, 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/2938. |
[17] |
S. C. Chapman, Macroscopic Models of Superconductivity, Ph. D thesis, Oxford University, 1991. Google Scholar |
[18] |
S. J. Chapman,
Superheating fields of type Ⅱ superconductors, SIAM J. Appl. Math., 55 (1995), 1233-1258.
doi: 10.1137/S0036139993254760. |
[19] |
J. Chen and X. B. Pan,
Functionals with operator curl in an extended magnetostatic Born-Infeld model, SIAM J. Math. Anal., 45 (2013), 2253-2284.
doi: 10.1137/120891496. |
[20] |
J. Chen and X. B. Pan, An extended magnetostatic Born-Infeld model with a concave lower order term, J. Math. Phys., 54 (2013), 111501, 29 pp.
doi: 10.1063/1.4826995. |
[21] |
J. Chen and X. B. Pan,
Quasilinear degenerate systems with pperator curl, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 243-279.
doi: 10.1017/S0308210517000014. |
[22] |
M. Costabel,
A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci., 12 (1990), 365-368.
doi: 10.1002/mma.1670120406. |
[23] |
M. Costabel and M. Dauge,
Singularities of electromagnetic fields in polyhedral domains, Arch. Ration. Mech. Anal., 151 (2000), 221-276.
doi: 10.1007/s002050050197. |
[24] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Functional and variational methods. With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily. Translated from the French by Ian N. Sneddon, Springer-Verlag, Berlin, 1988.
doi: 10.1007/978-3-642-61566-5. |
[25] |
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3. Spectral theory and applications. With the collaboration of Michel Artola and Michel Cessenat. Translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. |
[26] |
G. de Rham, Differentiable Manifolds, Forms, Currents, Harmonic Forms, Translated from the French by F. R. Smith. With an introduction by S. S. Chern. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 266, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-61752-2. |
[27] |
G. Galdi, Introduction to the Mathematical Theory of the Navier-Stokes Equations-Steady State Problems, Second edition. Springer Monographs in Mathematics, Springer, New York, 2011.
doi: 10.1007/978-0-387-09620-9. |
[28] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[29] |
V. Girault and P. Raviart, Finite Element Methods for the Navier-Stokes Equations, Theory and algorithms. Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[30] |
X. Jiang and W. Zheng,
An efficient eddy current model for nonlinear Maxwell equations with laminated conductors, SIAM J. Appl. Math., 72 (2012), 1021-1040.
doi: 10.1137/110857477. |
[31] |
H. Kozono and T. Yanagisawa,
$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853-1920.
doi: 10.1512/iumj.2009.58.3605. |
[32] |
H. Kozono and T. Yanagisawa,
Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems, Manuscripta Math., 141 (2013), 637-662.
doi: 10.1007/s00229-012-0586-6. |
[33] |
L. Landau and E. Lifshitz, Electrodynamics of Continuous Media, Course of Theoretical Physics, Vol. 8. Translated from the Russian by J. B. Sykes and J. S. Bell, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass. 1960. |
[34] |
G. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8679. |
[35] |
G. Lieberman and X. B. Pan,
On a quasilinear system arising in the theory of superconductivity, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 397-407.
doi: 10.1017/S0308210509001395. |
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