Variational and operator methods for Maxwell-Stokes system

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

Variational and operator methods for Maxwell-Stokes system

Xing-Bin Pan ^,

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 70^th 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

Abstract
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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.

Keywords: Curl equation, Maxwell equations, Maxwell-Stokes system, magneto-static problem, modified de Rham lemma, variational method, reduction method, domain topology.

Mathematics Subject Classification: Primary: 35Q61; Secondary: 35A15, 35J20, 35J47, 35J50, 35J57, 35J61, 35J62, 35Q60, 47J30, 78A25.

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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[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.
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[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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