Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
MS system; global solvability; energy space; Lorenz gauge
MS system; global solvability; energy space; Lorenz gauge
Summary:
We investigate the Cauchy problem of the one dimensional Maxwell-Schrödinger (MS) system under the Lorenz gauge condition. Different from the classical case, we consider the electromagnetic and electrostatic potentials which are growing at space infinity. More precisely, the electrostatic potential is allowed to grow linearly, while for the electromagnetic potential the growth is sublinear. Based on the energy estimates and the gauge transformation, we prove the global existence and the uniqueness of the weak solutions to this system.
References:
[1] Antonelli, P., Marcati, P., Scandone, R.: Global well-posedness for the non-linear Maxwell-Schrödinger system. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 23 (2022), 1293-1324. DOI 10.2422/2036-2145.202010_033 | MR 4497745 | Zbl 1498.35412
[2] Antonelli, P., Michelangeli, A., Scandone, R.: Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials. Z. Angew. Math. Phys. 69 (2018), Article ID 46, 30 pages. DOI 10.1007/s00033-018-0938-5 | MR 3778934 | Zbl 1392.35278
[3] Aoki, K., Guimard, D., Nishioka, M., Nomura, M., Iwamoto, S., Arakawa, Y.: Coupling of quantum-dot light emission with a three-dimensional photonic-crystal nanocavity. Nature Photonics 2 (2008), 688-692. DOI 10.1038/nphoton.2008.202
[4] Ballentine, L. E.: Quantum Mechanics: A Modern Development. World Scientific, Singapore (2014). DOI 10.1142/3142 | MR 1629320 | Zbl 0997.81501
[5] Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50 (2012), 492-521. DOI 10.1137/11083080 | MR 2914273 | Zbl 1246.35188
[6] Bejenaru, I., Tataru, D.: Global wellposedness in the energy space for the Maxwell-Schrödinger system. Commun. Math. Phys. 288 (2009), 145-198. DOI 10.1007/s00220-009-0765-9 | MR 2491621 | Zbl 1171.81006
[7] Carles, R.: Nonlinear Schrödinger equations with repulsive harmonic potential and applications. SIAM J. Math. Anal. 35 (2003), 823-843. DOI 10.1137/S0036141002416936 | MR 2049023 | Zbl 1054.35090
[8] Colin, M., Watanabe, T.: Cauchy problem for the nonlinear Schrödinger equation coupled with the Maxwell equation. Ann. Henri Lebesgue 3 (2020), 67-85. DOI 10.5802/ahl.27 | MR 4060851 | Zbl 1483.35203
[9] Greiner, W., Reinhardt, J.: Quantum Electrodynamics. Springer, Berlin (2003). DOI 10.1007/978-3-662-05246-4 | MR 1987453 | Zbl 1092.81066
[10] Guo, Y., Nakamitsu, K., Strauss, W.: Global finite-energy solutions of the Maxwell-Schrödinger system. Commun. Math. Phys. 170 (1995), 181-196. DOI 10.1007/BF02099444 | MR 1331696 | Zbl 0830.35131
[11] Hayashi, N., Ozawa, T.: Remarks on nonlinear Schrödinger equations in one space dimension. Differ. Integral Equ. 7 (1994), 453-461. DOI 10.57262/die/1369330439 | MR 1255899 | Zbl 0803.35137
[12] Komech, A. I.: On quantum jumps and attractors of the Maxwell-Schrödinger equations. Ann. Math. Qué. 46 (2022), 139-159. DOI 10.1007/s40316-021-00179-1 | MR 4396073 | Zbl 1498.35456
[13] Liu, Y., Wada, T.: Long range scattering for the Maxwell-Schrödinger system in the Lorenz gauge without any restriction on the size of data. J. Differ. Equations 269 (2020), 2798-2852. DOI 10.1016/j.jde.2020.02.013 | MR 4097235 | Zbl 1434.35214
[14] Lourenço-Martins, H.: A brief introduction to nano-optics with fast electrons. Plasmon Coupling Physics Advances in Imaging and Electron Physics 222. Elsevier, Amsterdam (2022), 1-82. DOI 10.1016/bs.aiep.2022.05.001
[15] Nakamitsu, K., Tsutsumi, M.: The Cauchy problem for the coupled Maxwell-Schrödinger equations. J. Math. Phys. 27 (1986), 211-216. DOI 10.1063/1.527363 | MR 0816434 | Zbl 0606.35015
[16] Nakamura, M., Wada, T.: Local well-posedness for the Maxwell-Schrödinger equation. Math. Ann. 332 (2005), 565-604. DOI 10.1007/s00208-005-0637-3 | MR 2181763 | Zbl 1075.35065
[17] Nakamura, M., Wada, T.: Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations. Commun. Math. Phys. 276 (2007), 315-339. DOI 10.1007/s00220-007-0337-9 | MR 2346392 | Zbl 1134.81020
[18] Oh, Y.-G.: Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials. J. Differ. Equations 81 (1989), 255-274. DOI 10.1016/0022-0396(89)90123-X | MR 1016082 | Zbl 0703.35158
[19] Scandone, R.: Global solutions to the nonlinear Maxwell-Schrödinger system. Harmonic Analysis and Partial Differential Equations Trends in Mathematics. Birkhäuser, Cham (2022), 91-96. DOI 10.1007/978-3-031-24311-0_6 | MR 4696593
[20] Shi, Q.: Global boundedness for the nonlinear Klein-Gordon-Schrödinger system with power nonlinearity. Differ. Integral Equ. 36 (2023), 837-858. DOI 10.57262/die036-0910-837 | MR 4597865 | Zbl 07729567
[21] Shi, Q., Jia, Y., Cao, J.: Spatially singular solutions for Klein-Gordon-Schrödinger system. Appl. Math. Lett. 131 (2022), Article ID 108038, 7 pages. DOI 10.1016/j.aml.2022.108038 | MR 4395960 | Zbl 1487.35193
[22] Shi, Q., Peng, C., Wang, Q.: Blowup results for the fractional Schrödinger equation without gauge invariance. Discrete Contin. Dyn. Syst., Ser. B 27 (2022), 6009-6022. DOI 10.3934/dcdsb.2021304 | MR 4470533 | Zbl 1496.35368
[23] Shukla, P. K., Eliasson, B.: Nonlinear aspects of quantum plasma physics. Phys. Usp. 53 (2010), 51-76. DOI 10.3367/UFNe.0180.201001b.0055
[24] Shukla, P. K., Eliasson, B.: Colloquium: Nonlinear collective interactions in quantum plasmas with degenerate electron fluids. Rev. Mod. Phys. 83 (2011), 885-906. DOI 10.1103/RevModPhys.83.885
[25] M. Sugawara, N. Hatori, M. Ishida, H. Ebe, Y. Arakawa, T. Akiyama, K. Otsubo, T. Yamamoto, Y. Nakata: Recent progress in self-assembled quantum-dot optical devices for optical telecommunication: Temperature-insensitive 10 Gb s$^{-1}$ directly modulated lasers and 40 Gb s$^{-1}$ signal-regenerative amplifiers. J. Phys. D, Appl. Phys. 38 (2005), 2126-2134. DOI 10.1088/0022-3727/38/13/008
[26] Tsutsumi, Y.: Global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations in three space dimensions. Commun. Math. Phys. 151 (1993), 543-576. DOI 10.1007/BF02097027 | MR 1207265 | Zbl 0766.35061
[27] Tsutsumi, Y.: Global existence and uniqueness of energy solutions for the Maxwell-Schrödinger equations in one space dimension. Hokkaido Math. J. 24 (1995), 617-639. DOI 10.14492/hokmj/1380892611 | MR 1357033 | Zbl 0840.35091
[28] Tsutsumi, M., Nakamitsu, K.: Global existence of solutions to the Cauchy problem for coupled Maxwell-Schrödinger equations in two space dimensions. Physical Mathematics and Nonlinear Partial Differential Equations Lecture Notes in Pure and Applied Mathematics 102. Marcel Dekker, New York (1985), 139-155. DOI 10.1201/9781003072683 | MR 0826831 | Zbl 0575.35061
[29] Wada, T.: Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger equations. J. Funct. Anal. 263 (2012), 1-24. DOI 10.1016/j.jfa.2012.04.010 | MR 2920838 | Zbl 1251.35095
[30] Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110 (1987), 415-426. DOI 10.1007/BF01212420 | MR 0891945 | Zbl 0638.35036
[31] Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56 (1991), 29-76. DOI 10.1007/BF02820459 | MR 1243098 | Zbl 0739.35083
Search
Partner of
