East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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LOCAL EXISTENCE OF CHERN-SIMONS GAUGED O(3) SIGMA EQUATIONS

  • Zheng, Xueyan (Department of Mathematics, College of Science, Yanbian University)
  • Received : 2021.09.03
  • Accepted : 2021.09.27
  • Published : 2021.09.30
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Abstract

In this paper we study the Cauchy problem for the Chern-Simons gauged O(3) sigma model. We prove the local existence of solutions with low regularity initial data, observing null forms of the system and applying bilinear estimates for wave-Sobolev space Hs, b.

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References

  1. K. Choe, J. Han, C-S. Lin and T-C. Lin, Uniqueness and solution structure of nonlinear equations arising from the Chern-Simons gauged O(3) sigma models, J. Differential Equations 255 (2013), no. 8, 2136-2166. https://doi.org/10.1016/j.jde.2013.06.010
  2. P. D'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2 + 1 and 1 + 1 dimensions, Contemp. Math 526 (2010), 125-150. https://doi.org/10.1090/conm/526/10379
  3. P. K. Ghosh and S. K. Ghosh, Topological and Nontopological Solitons in a Gauged O(3) Sigma Model with Chern-Simons term, Phys. Lett. B. 366 (1996), no. 1-4, 199-204. https://doi.org/10.1016/0370-2693(95)01365-2
  4. V. Grigoryan and A. Nahmod, Almost critical well-posedness for nonlinear wave equations with Qµν null forms in 2D, Math. Res. Lett. 21 (2014), no. 2, 313-332. https://doi.org/10.4310/MRL.2014.v21.n2.a9
  5. C. H. Gu, On the Cauchy Problem for Harmonic Maps Defined on Two-Dimensional Minkowski Space, Commun. Pure Appl. Math 33 (1980), no. 6, 727-737. https://doi.org/10.1002/cpa.3160330604
  6. J. Han and H. Huh, Existence of solutions to the self-dual equations in the Maxwell gauged O(3) sigma model, J. Math. Anal. Appl. 386 (2012), no. 1, 61-74. https://doi.org/10.1016/j.jmaa.2011.07.046
  7. H. Huh, Global energy solutions of Chern-Simons-Higgs equations in one space dimension, J. Math. Anal. Appl. 420 (2014), no. 1, 781-791. https://doi.org/10.1016/j.jmaa.2014.06.013
  8. H. Huh and G. Jin, Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension, J. Math. Phys. 57 (2016), no. 8, 081511, 11 pp https://doi.org/10.1063/1.4960744
  9. K. Kimm, K. Lee and T. Lee, Anyonic Bogomol'nyi solitons in a gauged O(3) sigma model, Phys. Rev. D 53 (1996), no. 8, 4436-4440. https://doi.org/10.1103/physrevd.53.4436
  10. S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math. 4 (2002), no. 2, 223-295. https://doi.org/10.1142/S0219199702000634
  11. J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543-615. https://doi.org/10.1007/s00222-007-0089-3
  12. I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical O(3) σ-model, Ann. of Math. (2) 172 (2010), no. 1, 187-242. https://doi.org/10.4007/annals.2010.172.187
  13. J. Shatah, Weak solutions and developement of singularities in the SU(2) σ-model, Commun. Pure Appl. Math. 41 (1988), no. 4, 459-469. https://doi.org/10.1002/cpa.3160410405
  14. T. Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), no. 2, 443-544. https://doi.org/10.1007/PL00005588
  15. D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37-77. https://doi.org/10.1353/ajm.2001.0005
  16. D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math. 127 (2005), no. 2, 293-377. https://doi.org/10.1353/ajm.2005.0014