American Institute of Mathematical Sciences

Advanced
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 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.

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 & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020036
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, (Italian) Rend. Sem. Padova., 31 (1961), 308-340.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

G. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. B. Pan, Minimizing curl in a multiconnected domain, J. Math. Phys., 50 (2009), 033508, 10 pp. doi: 10.1063/1.3082373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

X. B. Pan, General magneto-static model, submitted, https://arXiv.org/pdf/1908.03882.pdf. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[67]

M. Wiegner, Schauder estimates for boundary layer potentials, Math. Methods Appl. Sci., 16 (1993), 877-894.  doi: 10.1002/mma.1670161204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, (Italian) Rend. Sem. Padova., 31 (1961), 308-340.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

G. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40] P. Monk, Finite Element Methods for Maxwell's Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. B. Pan, Minimizing curl in a multiconnected domain, J. Math. Phys., 50 (2009), 033508, 10 pp. doi: 10.1063/1.3082373.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

X. B. Pan, General magneto-static model, submitted, https://arXiv.org/pdf/1908.03882.pdf. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[67]

M. Wiegner, Schauder estimates for boundary layer potentials, Math. Methods Appl. Sci., 16 (1993), 877-894.  doi: 10.1002/mma.1670161204.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Jishan Fan, Fucai Li, Gen Nakamura. Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain. Kinetic & Related Models, 2016, 9 (3) : 443-453. doi: 10.3934/krm.2016002
[2]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631
[3]

Luigi Fontana, Steven G. Krantz and Marco M. Peloso. Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space. Electronic Research Announcements, 1995, 1: 103-107.

[4]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[5]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045
[6]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159
[7]

Gang Bao, Bin Hu, Peijun Li, Jue Wang. Analysis of time-domain Maxwell's equations in biperiodic structures. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 259-286. doi: 10.3934/dcdsb.2019181
[8]

Wenjing Song, Ganshan Yang. The regularization of solution for the coupled Navier-Stokes and Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2113-2127. doi: 10.3934/dcdss.2016087
[9]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[10]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[11]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423
[12]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135
[13]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
[14]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[15]

S. S. Krigman. Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain. Conference Publications, 2007, 2007 (Special) : 590-601. doi: 10.3934/proc.2007.2007.590
[16]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010
[17]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056
[18]

Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283
[19]

Xiaofeng Hou, Limei Zhu. Serrin-type blowup criterion for full compressible Navier-Stokes-Maxwell system with vacuum. Communications on Pure & Applied Analysis, 2016, 15 (1) : 161-183. doi: 10.3934/cpaa.2016.15.161
[20]

Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (19)
  • HTML views (62)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences