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Global large smooth solutions for 3-D Hall-magnetohydrodynamics
Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting
Emmanuel Hebey, Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France |
We investigate the system consisting of the the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting in the context of closed $ 3 $-dimensional manifolds. We prove existence of solutions up to the gauge, and compactness of the system both in the subcritical and in the critical case.
References:
[1] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
T. Aubin,
Espaces de Sobolev sur les variétés riemanniennes, Bull. Sc. Math., 100 (1976), 149-173.
|
[3] |
V. Benci and D. Fortunato,
Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Commun. Math. Phys., 295 (2010), 639-668.
doi: 10.1007/s00220-010-0985-z. |
[4] |
F. Bopp,
Eine lineare Theorie des Elektrons, Ann. Phys., 38 (1940), 345-384.
doi: 10.1002/andp.19404300504. |
[5] |
S. Brendle,
Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.
doi: 10.1090/S0894-0347-07-00575-9. |
[6] |
S. Brendle and F. C. Marques,
Blow-up phenomena for the Yamabe equation Ⅱ, J. Differential Geom., 81 (2009), 225-250.
doi: 10.4310/jdg/1231856261. |
[7] |
___, Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics, 20 (2011), 29–47. |
[8] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[9] |
P. d'Avenia, J. Medreski and P. Pomponio, Vortex ground states for Klein-Gordon-Maxwell-Proca type systems, J. Math. Phys., 58 (2017), 041503, 19 pp.
doi: 10.1063/1.4982038. |
[10] |
P. d'Avenia and G. Siciliano,
Nonlinear Schrödinger equation in thje Bopp-Podolsky electrodynamics: solutions in the electrostatic case, J. Differential Equations, 267 (2019), 1025-1065.
doi: 10.1016/j.jde.2019.02.001. |
[11] |
J. Dodziuk,
Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Diff. Geom., 16 (1981), 63-73.
doi: 10.4310/jdg/1214435988. |
[12] |
O. Druet,
From one bubble to several bubbles: The low-dimensional case, J. Differential Geom., 63 (2003), 399-473.
doi: 10.4310/jdg/1090426771. |
[13] |
___, Compactness for Yamabe metrics in low dimensions, Internat. Math. Res. Notices,
23 (2004), 1143–1191.
doi: 10.1155/S1073792804133278. |
[14] |
O. Druet and E. Hebey,
Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Analysis and PDEs, 2 (2009), 305-359.
doi: 10.2140/apde.2009.2.305. |
[15] |
___, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact
Riemannian manifolds, Math. Z., 263 (2009), 33–67.
doi: 10.1007/s00209-008-0409-3. |
[16] |
___., Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in
fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831–869.
doi: 10.1142/S0219199710004007. |
[17] |
O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, Princeton University Press, vol. 45, 2004.
doi: 10.1007/BF01158557. |
[18] |
O. Druet, E. Hebey and J. Vétois,
Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059.
doi: 10.1016/j.jfa.2009.07.004. |
[19] |
___, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds. Ⅱ,
J. reine angew. Math., 713 (2016), 149–179. |
[20] |
P. Esposito, A. Pistoia and J. Vétois,
The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.
doi: 10.1007/s00208-013-0971-9. |
[21] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Part. Diff. Eq., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[22] |
G. Gilbarg and N. S. Trüdinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin–New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[23] |
A. S. Goldhaber and M. M. Nieto,
Terrestrial and Extraterrestrial limits on the photon mass, Rev. Mod. Phys., 43 (1971), 277-296.
|
[24] |
___, Photon and Graviton mass limits, Rev. Mod. Phys., 82 (2010), 939–979. |
[25] |
E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2014.
doi: 10.4171/134. |
[26] |
E. Hebey and P. D. Thizy,
Stationary Kirchhoff systems in closed $3$-dimensional manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2085-2114.
doi: 10.1007/s00526-015-0858-6. |
[27] |
___, Klein-Gordon-Maxwell-Proca type systems in the electro-magneto-static case, J.
Part. Diff. Eq., 31 (2018), 119–58. |
[28] |
E. Hebey and M. Vaugon,
The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279.
doi: 10.1215/S0012-7094-95-07906-X. |
[29] |
___, Meilleures constantes dans le théorème d'inclusion de Sobolev, Ann. Inst. H.
Poincaré. Anal. Non Linéaire, 13 (1996), 57–93.
doi: 10.1016/S0294-1449(16)30097-X. |
[30] |
E. Hebey and J. Wei,
Schrödinger-Poisson systems in the $3$-sphere, Calc. Var. Partial Dif- ferential Equations, 47 (2013), 25-54.
doi: 10.1007/s00526-012-0509-0. |
[31] |
M. Khuri, F. C. Marques and R. Schoen,
A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.
doi: 10.4310/jdg/1228400630. |
[32] |
Y. Y. Li and L. Zhang,
A Harnack type inequality for the Yamabe equations in low dimensions, Calc. Var. PDE, 20 (2004), 133-151.
doi: 10.1007/s00526-003-0230-0. |
[33] |
___, Compactness of solutions to the Yamabe problem Ⅱ, Calc. Var. PDE, 24 (2005),
185–237. |
[34] |
___, Compactness of solutions to the Yamabe problem Ⅲ, J. Funct. Anal., 245 (2007),
438–474.
doi: 10.1016/j.jfa.2006.11.010. |
[35] |
Y. Y. Li and M. Zhu,
Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
[36] |
J. Luo, G. T. Gillies and L. C. Tu,
The mass of the photon, Rep. Prog. Phys., 68 (2005), 77-130.
|
[37] |
F. C. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.
doi: 10.4310/jdg/1143651772. |
[38] |
B. Podolsky,
A Generalized Electrodynamics, hys. Rev., 62 (1942), 68-71.
|
[39] |
H. Ruegg and M. Ruiz-Altaba,
The Stueckelberg field, Int. J. Mod. Phys. A, 19 (2004), 3265-3347.
doi: 10.1142/S0217751X04019755. |
[40] |
R. M. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
doi: 10.4310/jdg/1214439291. |
[41] |
___, Lecture Notes from Courses at Stanford, written by D.Pollack, preprint, 1988. |
[42] |
___, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math., Springer-Verlag, Berlin, 1365 (1989), 120–154.
doi: 10.1007/BFb0089180. |
[43] |
___, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A Symposium in Honor of Manfredo do Carmo, Proc. Int. Conf. (Rio de Janeiro, 1988)., Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 311–320. |
[44] |
___, A report on some recent progress on nonlinear problems in geometry, Surveys in Differential Geometry (Cambridge, 1990), Suppl. J. Diff. Geom., Lehigh University, Pennsylvania, 1 (1991), 201–241. |
[45] |
R. M. Schoen and S. T. Yau,
On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
doi: 10.1007/BF01940959. |
[46] |
P. D. Thizy,
Non-resonant states for Schrödinger-Poisson critical systems in high dimensions, Arch. Math., 104 (2015), 485-490.
doi: 10.1007/s00013-015-0763-4. |
[47] |
___, Schrödinger-Poisson systems in 4-dimensional closed manifolds, Discrete Contin.
Dyn. Syst.-Series A, 36 (2016), 2257–2284.
doi: 10.3934/dcds.2016.36.2257. |
[48] |
___, Blow-up for Schrödinger-Poisson critical systems in dimensions $4$ and $5$, Calc. Var. Partial Differential Equations, 55 (2016), Art. 20, 21 pp.
doi: 10.1007/s00526-016-0959-x. |
[49] |
___, Phase-stability for Schrödinger-Poisson critical systems in closed 5-manifolds, Int.
Math. Res. Not. IMRN, 20 (2016), 6245–6292.
doi: 10.1093/imrn/rnv344. |
[50] |
___, Unstable phases for the critical Schrödinger-Poisson system in dimension 4, Differential Integral Equations, 30 (2017), 825–832. |
[51] |
N. S. Trüdinger,
Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.
|
[52] |
E. Witten,
A new proof of the positive energy theorem, Comm. Math. Phys., 80 (1981), 381-402.
doi: 10.1007/BF01208277. |
show all references
References:
[1] |
A. Ambrosetti and P. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
T. Aubin,
Espaces de Sobolev sur les variétés riemanniennes, Bull. Sc. Math., 100 (1976), 149-173.
|
[3] |
V. Benci and D. Fortunato,
Spinning $Q$-balls for the Klein-Gordon-Maxwell equations, Commun. Math. Phys., 295 (2010), 639-668.
doi: 10.1007/s00220-010-0985-z. |
[4] |
F. Bopp,
Eine lineare Theorie des Elektrons, Ann. Phys., 38 (1940), 345-384.
doi: 10.1002/andp.19404300504. |
[5] |
S. Brendle,
Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.
doi: 10.1090/S0894-0347-07-00575-9. |
[6] |
S. Brendle and F. C. Marques,
Blow-up phenomena for the Yamabe equation Ⅱ, J. Differential Geom., 81 (2009), 225-250.
doi: 10.4310/jdg/1231856261. |
[7] |
___, Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Advanced Lectures in Mathematics, 20 (2011), 29–47. |
[8] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear ellitpic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[9] |
P. d'Avenia, J. Medreski and P. Pomponio, Vortex ground states for Klein-Gordon-Maxwell-Proca type systems, J. Math. Phys., 58 (2017), 041503, 19 pp.
doi: 10.1063/1.4982038. |
[10] |
P. d'Avenia and G. Siciliano,
Nonlinear Schrödinger equation in thje Bopp-Podolsky electrodynamics: solutions in the electrostatic case, J. Differential Equations, 267 (2019), 1025-1065.
doi: 10.1016/j.jde.2019.02.001. |
[11] |
J. Dodziuk,
Sobolev spaces of differential forms and de Rham-Hodge isomorphism, J. Diff. Geom., 16 (1981), 63-73.
doi: 10.4310/jdg/1214435988. |
[12] |
O. Druet,
From one bubble to several bubbles: The low-dimensional case, J. Differential Geom., 63 (2003), 399-473.
doi: 10.4310/jdg/1090426771. |
[13] |
___, Compactness for Yamabe metrics in low dimensions, Internat. Math. Res. Notices,
23 (2004), 1143–1191.
doi: 10.1155/S1073792804133278. |
[14] |
O. Druet and E. Hebey,
Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Analysis and PDEs, 2 (2009), 305-359.
doi: 10.2140/apde.2009.2.305. |
[15] |
___, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact
Riemannian manifolds, Math. Z., 263 (2009), 33–67.
doi: 10.1007/s00209-008-0409-3. |
[16] |
___., Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in
fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831–869.
doi: 10.1142/S0219199710004007. |
[17] |
O. Druet, E. Hebey and F. Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes, Princeton University Press, vol. 45, 2004.
doi: 10.1007/BF01158557. |
[18] |
O. Druet, E. Hebey and J. Vétois,
Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian, J. Funct. Anal., 258 (2010), 999-1059.
doi: 10.1016/j.jfa.2009.07.004. |
[19] |
___, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds. Ⅱ,
J. reine angew. Math., 713 (2016), 149–179. |
[20] |
P. Esposito, A. Pistoia and J. Vétois,
The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.
doi: 10.1007/s00208-013-0971-9. |
[21] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Part. Diff. Eq., 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
[22] |
G. Gilbarg and N. S. Trüdinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer, Berlin–New York, 1983.
doi: 10.1007/978-3-642-61798-0. |
[23] |
A. S. Goldhaber and M. M. Nieto,
Terrestrial and Extraterrestrial limits on the photon mass, Rev. Mod. Phys., 43 (1971), 277-296.
|
[24] |
___, Photon and Graviton mass limits, Rev. Mod. Phys., 82 (2010), 939–979. |
[25] |
E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2014.
doi: 10.4171/134. |
[26] |
E. Hebey and P. D. Thizy,
Stationary Kirchhoff systems in closed $3$-dimensional manifolds, Calc. Var. Partial Differential Equations, 54 (2015), 2085-2114.
doi: 10.1007/s00526-015-0858-6. |
[27] |
___, Klein-Gordon-Maxwell-Proca type systems in the electro-magneto-static case, J.
Part. Diff. Eq., 31 (2018), 119–58. |
[28] |
E. Hebey and M. Vaugon,
The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79 (1995), 235-279.
doi: 10.1215/S0012-7094-95-07906-X. |
[29] |
___, Meilleures constantes dans le théorème d'inclusion de Sobolev, Ann. Inst. H.
Poincaré. Anal. Non Linéaire, 13 (1996), 57–93.
doi: 10.1016/S0294-1449(16)30097-X. |
[30] |
E. Hebey and J. Wei,
Schrödinger-Poisson systems in the $3$-sphere, Calc. Var. Partial Dif- ferential Equations, 47 (2013), 25-54.
doi: 10.1007/s00526-012-0509-0. |
[31] |
M. Khuri, F. C. Marques and R. Schoen,
A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.
doi: 10.4310/jdg/1228400630. |
[32] |
Y. Y. Li and L. Zhang,
A Harnack type inequality for the Yamabe equations in low dimensions, Calc. Var. PDE, 20 (2004), 133-151.
doi: 10.1007/s00526-003-0230-0. |
[33] |
___, Compactness of solutions to the Yamabe problem Ⅱ, Calc. Var. PDE, 24 (2005),
185–237. |
[34] |
___, Compactness of solutions to the Yamabe problem Ⅲ, J. Funct. Anal., 245 (2007),
438–474.
doi: 10.1016/j.jfa.2006.11.010. |
[35] |
Y. Y. Li and M. Zhu,
Yamabe type equations on three dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
[36] |
J. Luo, G. T. Gillies and L. C. Tu,
The mass of the photon, Rep. Prog. Phys., 68 (2005), 77-130.
|
[37] |
F. C. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.
doi: 10.4310/jdg/1143651772. |
[38] |
B. Podolsky,
A Generalized Electrodynamics, hys. Rev., 62 (1942), 68-71.
|
[39] |
H. Ruegg and M. Ruiz-Altaba,
The Stueckelberg field, Int. J. Mod. Phys. A, 19 (2004), 3265-3347.
doi: 10.1142/S0217751X04019755. |
[40] |
R. M. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
doi: 10.4310/jdg/1214439291. |
[41] |
___, Lecture Notes from Courses at Stanford, written by D.Pollack, preprint, 1988. |
[42] |
___, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math., Springer-Verlag, Berlin, 1365 (1989), 120–154.
doi: 10.1007/BFb0089180. |
[43] |
___, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A Symposium in Honor of Manfredo do Carmo, Proc. Int. Conf. (Rio de Janeiro, 1988)., Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, 52 (1991), 311–320. |
[44] |
___, A report on some recent progress on nonlinear problems in geometry, Surveys in Differential Geometry (Cambridge, 1990), Suppl. J. Diff. Geom., Lehigh University, Pennsylvania, 1 (1991), 201–241. |
[45] |
R. M. Schoen and S. T. Yau,
On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys., 65 (1979), 45-76.
doi: 10.1007/BF01940959. |
[46] |
P. D. Thizy,
Non-resonant states for Schrödinger-Poisson critical systems in high dimensions, Arch. Math., 104 (2015), 485-490.
doi: 10.1007/s00013-015-0763-4. |
[47] |
___, Schrödinger-Poisson systems in 4-dimensional closed manifolds, Discrete Contin.
Dyn. Syst.-Series A, 36 (2016), 2257–2284.
doi: 10.3934/dcds.2016.36.2257. |
[48] |
___, Blow-up for Schrödinger-Poisson critical systems in dimensions $4$ and $5$, Calc. Var. Partial Differential Equations, 55 (2016), Art. 20, 21 pp.
doi: 10.1007/s00526-016-0959-x. |
[49] |
___, Phase-stability for Schrödinger-Poisson critical systems in closed 5-manifolds, Int.
Math. Res. Not. IMRN, 20 (2016), 6245–6292.
doi: 10.1093/imrn/rnv344. |
[50] |
___, Unstable phases for the critical Schrödinger-Poisson system in dimension 4, Differential Integral Equations, 30 (2017), 825–832. |
[51] |
N. S. Trüdinger,
Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 265-274.
|
[52] |
E. Witten,
A new proof of the positive energy theorem, Comm. Math. Phys., 80 (1981), 381-402.
doi: 10.1007/BF01208277. |
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Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113 |
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