Variational and operator methods for Maxwell-Stokes system

Xing-Bin Pan

doi:10.3934/dcds.2020036

Article Contents

2020, Volume 40, Issue 6: 3909-3955. Doi: 10.3934/dcds.2020036 open access

This issue Previous Article Convergence and structure theorems for order-preserving dynamical systems with mass conservation Next Article The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs

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
^* Corresponding author: Xing-Bin Pan

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

Received: January 31, 2019

Published: March 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

FullText(HTML)

Related Papers

Cited by

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.
[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.
[36]	J. L. Lions, Quelques Méthodes de R\'rsolution des Problémes aux Limites Non Linéaires, (French) Dunod, Gauthier-Villars, Paris 1969.
[37]	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]	A. Milani and R. Picard, Weak solution theory for Maxwell's equstions in the semistatic limit case, J. Math. Anal. Appl., 191 (1995), 77-100. doi: 10.1016/S0022-247X(85)71121-3.
[39]	D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the hodge laplacian and global boundary problems in non-smooth riemannian manifolds, Mem. Amer. Math. Soc., 150 (2001), x+120 pp. doi: 10.1090/memo/0713.
[40]	P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.
[41]	R. Monneau, Quasilinear elliptic system arising in a three-dimensional type I\!I superconductor for infinite $\kappa$, Nonlinear Anal., 52 (2003), 917-930. doi: 10.1016/S0362-546X(02)00142-6.
[42]	J. Nedelec, Éléments finis mixtes incompressibles pour l'équation de Stokes dans $\Bbb R^3$, (French) [Incompressible mixed finite elements for the Stokes equation in $\mathbb{R}\^{3}$], Numer. Math., 39 (1982), 97-112. doi: 10.1007/BF01399314.
[43]	X. B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys., 239 (2003), 343-382. doi: 10.1007/s00220-003-0875-8.
[44]	X. B. Pan, Nucleation of instability of meissner state of superconductors and related mathematical problems, Trends in Partial Differential Equations, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 10 (2010), 323–372.
[45]	X. B. Pan, Minimizing curl in a multiconnected domain, J. Math. Phys., 50 (2009), 033508, 10 pp. doi: 10.1063/1.3082373.
[46]	X. B. Pan, On a quasilinear system involving the operator Curl, Calc. Var. Partial Differential Equations, 36 (2009), 317-342. doi: 10.1007/s00526-009-0230-9.
[47]	X. B. Pan, Asymptotic behavior of solutions of a quasilinear system involving the operator curl, J. Math. Phys., 52 (2011), 023517, 34 pp. doi: 10.1063/1.3543618.
[48]	X. B. Pan, Regularity of weak solutions to nonlinear Maxwell systems, J. Math. Phys., 56 (2015), 071508, 24 pp. doi: 10.1063/1.4927427.
[49]	X. B. Pan, Existence and regularity of solutions to quasilinear systems of Maxwell type and Maxwell-Stokes type, Calc. Var. Partial Differential Equations, 55 (2016), art. 143, 43 pp. doi: 10.1007/s00526-016-1081-9.
[50]	X. B. Pan, Meissner states of type I\!I superconductors, J. Elliptic Parabol. Equ., 4 (2018), 441-523. doi: 10.1007/s41808-018-0027-0.
[51]	X. B. Pan, General magneto-static model, submitted, https://arXiv.org/pdf/1908.03882.pdf.
[52]	X. B. Pan and Y. W. Qi, Asymptotics of minimizers of variational problems involving curl functional, J. Math. Phys., 41 (2000), 5033-5063. doi: 10.1063/1.533391.
[53]	X. B. Pan and Z. B. Zhang, Existence, regularity and uniqueness of weak solutions to a quasilinear Maxwell system, Nonlinearity, 32 (9) (2019), 3342-3366.
[54]	R. Picard, On the boundary value problems of electro- and magnetostatics, Proc. Royal Soc. Edinburgh A, 92 (1982), 165-174. doi: 10.1017/S0308210500020023.
[55]	J. Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl., 88 (1982), 104-115. doi: 10.1016/0022-247X(82)90179-2.
[56]	G. Schwarz, Hodge Decomposition– A Method for Solving Boundary Value Problems, Lecture Notes in Math., Springer-Verlag, Berlin Heidelberg, 1995. doi: 10.1007/BFb0095978.
[57]	C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $l^q$-spaces for bounded and exterior domains, Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci., World Sci. Publ., River Edge, NJ, 11 (1992), 1–35. doi: 10.1142/9789814503594_0001.
[58]	J. Simon, Démonstration constructive d'un théoréme de G. de Rham, (French) [A constructive proof of a theorem of G. de Rham], C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 1167-1172.
[59]	H. Sohr, The Navier-Stokes Equations, An elementary functional analytic approach. [2013 reprint of the 2001 original] [MR1928881]. Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001.
[60]	M. Struwe, Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 34, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04194-9.
[61]	C. Stuart and H. S. Zhou, A variational problem related to self-trapping of an electromagnetic field, Math. Meth. Appl. Sci., 19 (1996), 1397-1407.
[62]	C. Stuart and H. S. Zhou, A constrained minimization problem and its application to guided cylindrical TM-modes in an anisotropic self-focusing dielectric, Calc. Var. Partial Differential Equations, 16 (2003), 335-373. doi: 10.1007/s005260100153.
[63]	X. H. Tang and D. D. Qin, Ground state solutions for semilinear time-harmonic Maxwell equations, J. Math. Phys., 57 (2016), 041505, 19 pp. doi: 10.1063/1.4947179.
[64]	R. Temam, Navier-Stokes Equations–Theory and Numerical Analysis, Theory and numerical analysis. Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. xiv+408 pp. doi: 10.1090/chel/343.
[65]	W. von Wahl, Estimating $\nabla{\bf u}$ by $ \rmdiv $ ${\bf u}$ and $ \rmcurl $ ${\bf u}$, Math. Meth. Appl. Sci., 15 (1992), 123-143. doi: 10.1002/mma.1670150206.
[66]	X. M. Wang, A remark on the characterization of the gradient of a distribution, Appl. Anal., 51 (1993), 35-40. doi: 10.1080/00036819308840202.
[67]	M. Wiegner, Schauder estimates for boundary layer potentials, Math. Methods Appl. Sci., 16 (1993), 877-894. doi: 10.1002/mma.1670161204.
[68]	T. T. Wu and C. N. Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Review D, 12 (1975), 3845-3857. doi: 10.1103/PhysRevD.12.3845.
[69]	X. F. Xiang, Radially symmetric solutions for a limiting form of the Ginzburg-Landau model, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 185-203. doi: 10.1017/S0308210511000825.
[70]	X. F. Xiang, On the shape of Meissner solutions to a limiting form of Ginzburg-Landau systems, Arch. Ration. Mech. Anal., 222 (2016), 1601-1640. doi: 10.1007/s00205-016-1029-4.
[71]	H. M. Yin, Optimal regularity of solutions to a degenerate elliptic system arising in electromagnetic fields, Comm. Pure Appl. Anal., 1 (2002), 127-134. doi: 10.3934/cpaa.2002.1.127.
[72]	E. Zeidler, Nonlinear Functional Analysis and Its Applicationss, I, Fixed-point theorems. Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.
[73]	X. Y. Zeng, Cylindrically symmetric ground state solutions for curl-curl equations with critical exponent, Z. Angew. Math. Phys., 68 (2017), art. 135, 12 pp. doi: 10.1007/s00033-017-0887-4.