Abstract and Applied Analysis

Energy Solution to the Chern-Simons-Schrödinger Equations

Hyungjin Huh

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Abstract

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 ( 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 590653, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511764

Digital Object Identifier
doi:10.1155/2013/590653

Mathematical Reviews number (MathSciNet)
MR3035224

Zentralblatt MATH identifier
1276.35138

Citation

Huh, Hyungjin. Energy Solution to the Chern-Simons-Schrödinger Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 590653, 7 pages. doi:10.1155/2013/590653. https://projecteuclid.org/euclid.aaa/1393511764


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References

  • R. Jackiw and S.-Y. Pi, “Classical and quantal nonrelativistic Chern-Simons theory,” Physical Review D, vol. 42, no. 10, pp. 3500–3513, 1990.
  • R. Jackiw and S.-Y. Pi, “Self-dual Chern-Simons solitons,” Progress of Theoretical Physics. Supplement, no. 107, pp. 1–40, 1992.
  • G. Dunne, Self-Dual Chern-Simons Theories, Springer, Berlin, Germany, 1995.
  • P. A. Horvathy and P. Zhang, “Vortices in (abelian) Chern-Simons gauge theory,” Physics Reports, vol. 481, no. 5-6, pp. 83–142, 2009.
  • K. Nakamitsu and M. Tsutsumi, “The Cauchy problem for the coupled Maxwell-Schrödinger equations,” Journal of Mathematical Physics, vol. 27, no. 1, pp. 211–216, 1986.
  • L. Bergé, A. de Bouard, and J.-C. Saut, “Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation,” Nonlinearity, vol. 8, no. 2, pp. 235–253, 1995.
  • S. Demoulini, “Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 24, no. 2, pp. 207–225, 2007.
  • H. Huh, “Blow-up solutions of the Chern-Simons-Schrödinger equations,” Nonlinearity, vol. 22, no. 5, pp. 967–974, 2009.
  • J. Byeon, H. Huh, and J. Seok, “Standing waves of nonlinear Schrödinger equations with the gauge field,” Journal of Functional Analysis, vol. 263, no. 6, pp. 1575–1608, 2012.
  • H. Huh, “Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field,” Journal of Mathematical Physics, vol. 53, no. 6, p. 063702, 8, 2012.
  • S. Demoulini and D. Stuart, “Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schrödinger system,” Communications in Mathematical Physics, vol. 290, no. 2, pp. 597–632, 2009.
  • J. Kato, “Existence and uniqueness of the solution to the modified Schrödinger map,” Mathematical Research Letters, vol. 12, no. 2-3, pp. 171–186, 2005.
  • C. E. Kenig and A. R. Nahmod, “The Cauchy problem for the hyperbolic-elliptic Ishimori system and Schrödinger maps,” Nonlinearity, vol. 18, no. 5, pp. 1987–2009, 2005.
  • J. Kato and H. Koch, “Uniqueness of the modified Schrödinger map in ${H}^{3/4+\epsilon }({\mathbb{R}}^{2})$,” Communications in Partial Differential Equations, vol. 32, no. 1–3, pp. 415–429, 2007.
  • H. Brezis and J.-M. Coron, “Multiple solutions of $H$-systems and Rellich's conjecture,” Communications on Pure and Applied Mathematics, vol. 37, no. 2, pp. 149–187, 1984.
  • H. C. Wente, “An existence theorem for surfaces of constant mean curvature,” Journal of Mathematical Analysis and Applications, vol. 26, pp. 318–344, 1969.
  • A. Nahmod, A. Stefanov, and K. Uhlenbeck, “On Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 56, no. 1, pp. 114–151, 2003.
  • A. Nahmod, A. Stefanov, and K. Uhlenbeck, “Erratum: on Schrödinger maps,” Communications on Pure and Applied Mathematics, vol. 57, no. 6, pp. 833–839, 2004.
  • C. E. Kenig and K. D. Koenig, “On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations,” Mathematical Research Letters, vol. 10, no. 5-6, pp. 879–895, 2003.
  • H. Koch and N. Tzvetkov, “On the local well-posedness of the Benjamin-Ono equation in ${H}^{s}(\mathbb{R})$,” International Mathematics Research Notices, no. 26, pp. 1449–1464, 2003.
  • T. Ogawa, “A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 14, no. 9, pp. 765–769, 1990.
  • M. V. Vladimirov, “On the solvability of a mixed problem for a nonlinear equation of Schrödinger type,” Doklady Akademii Nauk SSSR, vol. 275, no. 4, pp. 780–783, 1984.