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June  2020, 40(6): 3909-3955. doi: 10.3934/dcds.2020036

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

* Corresponding author: Xing-Bin Pan

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

Received  February 2019 Published  October 2019

Fund Project: This work was partially supported by the National Natural Science Foundation of China grant no. 11671143, and 11431005

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.

Citation: Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
References:
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show all references

References:
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S. AgmonA. 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.

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C. AmroucheC. BernardiM. 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.

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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.

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F. BachingerU. 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.

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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.

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[14]

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S. J. Chapman, Superheating fields of type Ⅱ superconductors, SIAM J. Appl. Math., 55 (1995), 1233-1258.  doi: 10.1137/S0036139993254760.

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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]

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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.

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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.

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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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J. L. Lions, Quelques Méthodes de R\'rsolution des Problémes aux Limites Non Linéaires, (French) Dunod, Gauthier-Villars, Paris 1969.

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J. Mederski, Ground states of time-harmonic semilinear Maxwell equations in $\Bbb R^3$ with vanishing permittivity, Arch. Ration. Mech. Anal., 218 (2015), 825-861.  doi: 10.1007/s00205-015-0870-1.

[38]

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