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Energy solution to the Chern-Simons-Schrödinger equations. (English) Zbl 1276.35138

Summary: We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space \(H^1(\mathbb R^2)\). The Coulomb gauge provides elliptic features for gauge fields \(A_0, A_j\). The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente’s inequalities.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35Q40 PDEs in connection with quantum mechanics
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[1] Jackiw, R.; Pi, S.-Y., Classical and quantal nonrelativistic Chern-Simons theory, Physical Review D, 42, 10, 3500-3513, (1990)
[2] Jackiw, R.; Pi, S.-Y., Self-dual Chern-Simons solitons, Progress of Theoretical Physics. Supplement, 107, 1-40, (1992)
[3] Dunne, G., Self-Dual Chern-Simons Theories, (1995), Berlin, Germany: Springer, Berlin, Germany · Zbl 0834.58001
[4] Horvathy, P. A.; Zhang, P., Vortices in (abelian) Chern-Simons gauge theory, Physics Reports, 481, 5-6, 83-142, (2009)
[5] Nakamitsu, K.; Tsutsumi, M., The Cauchy problem for the coupled Maxwell-Schrödinger equations, Journal of Mathematical Physics, 27, 1, 211-216, (1986) · Zbl 0606.35015
[6] Bergé, L.; de Bouard, A.; Saut, J.-C., Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8, 2, 235-253, (1995) · Zbl 0822.35125
[7] Demoulini, S., Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 24, 2, 207-225, (2007) · Zbl 1166.35035
[8] Huh, H., Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22, 5, 967-974, (2009) · Zbl 1173.35313
[9] Byeon, J.; Huh, H.; Seok, J., Standing waves of nonlinear Schrödinger equations with the gauge field, Journal of Functional Analysis, 263, 6, 1575-1608, (2012) · Zbl 1248.35193
[10] Huh, H., Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, Journal of Mathematical Physics, 53, 6, 063702, 8, (2012) · Zbl 1276.81053
[11] Demoulini, S.; Stuart, D., Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schrödinger system, Communications in Mathematical Physics, 290, 2, 597-632, (2009) · Zbl 1185.81071
[12] Kato, J., Existence and uniqueness of the solution to the modified Schrödinger map, Mathematical Research Letters, 12, 2-3, 171-186, (2005) · Zbl 1082.35140
[13] Kenig, C. E.; Nahmod, A. R., The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps, Nonlinearity, 18, 5, 1987-2009, (2005) · Zbl 1213.35358
[14] Kato, J.; Koch, H., Uniqueness of the modified Schrödinger map in \(H^{3 / 4 + \epsilon}(\mathbb{R}^2),\) Communications in Partial Differential Equations, 32, 1–3, 415-429, (2007) · Zbl 1387.35139
[15] Brezis, H.; Coron, J.-M., Multiple solutions of \(H\)-systems and Rellich’s conjecture, Communications on Pure and Applied Mathematics, 37, 2, 149-187, (1984) · Zbl 0537.49022
[16] Wente, H. C., An existence theorem for surfaces of constant mean curvature, Journal of Mathematical Analysis and Applications, 26, 318-344, (1969) · Zbl 0181.11501
[17] Nahmod, A.; Stefanov, A.; Uhlenbeck, K., On Schrödinger maps, Communications on Pure and Applied Mathematics, 56, 1, 114-151, (2003) · Zbl 1028.58018
[18] Nahmod, A.; Stefanov, A.; Uhlenbeck, K., Erratum: on Schrödinger maps, Communications on Pure and Applied Mathematics, 57, 6, 833-839, (2004)
[19] Kenig, C. E.; Koenig, K. D., On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Mathematical Research Letters, 10, 5-6, 879-895, (2003) · Zbl 1044.35072
[20] Koch, H.; Tzvetkov, N., On the local well-posedness of the Benjamin-Ono equation in \(H^s(\mathbb{R}),\) International Mathematics Research Notices, 26, 1449-1464, (2003) · Zbl 1039.35106
[21] Ogawa, T., A proof of Trudinger’s inequality and its application to nonlinear Schrödinger equations, Nonlinear Analysis. Theory, Methods & Applications, 14, 9, 765-769, (1990) · Zbl 0715.35073
[22] Vladimirov, M. V., On the solvability of a mixed problem for a nonlinear equation of Schrödinger type, Doklady Akademii Nauk SSSR, 275, 4, 780-783, (1984)
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