Local and global solutions to the O(3)-sigma model with the Maxwell and the Chern-Simons gauges in R1+1

Abstract In this paper we study the well-posedness of the ( 1 + 1 ) -dimensional Maxwell-Chern-Simons gauged O ( 3 ) -sigma model with the Lorenz gauge condition. First, we provide the global well-posedness of a solution in a high regular space. We also prove the existence of a global solution with the finite total energy. Moreover, we consider local well-posedness below the energy regularity by exploiting null form and wave-Sobolev space.

[1]  Hyungjin Huh.  Cauchy problems of the gauged sigma model , 2005 .

[2]  Christopher D. Sogge,et al.  Lectures on Nonlinear Wave Equations , 2005 .

[3]  Hyungjin Huh.  Remarks on Chern–Simons–Dirac Equations in One Space Dimension , 2014 .

[4]  J. Ginibre,et al.  The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge , 1981 .

[5]  Taejin Lee,et al.  BPS domain wall solutions in self-dual Chern-Simons-Higgs systems , 1996, hep-th/9612183.

[6]  Jongmin Han,et al.  Bubbling solutions for the Chern–Simons gauged $$O(3)$$O(3) sigma model on a torus , 2015 .

[7]  Jongmin Han,et al.  Condensate solutions of the self-dual O(3) Maxwell–Chern–Simons–Higgs equations with symmetric vacua , 2019, Calculus of Variations and Partial Differential Equations.

[8]  Hyungjin Huh.  Global energy solutions of Chern–Simons–Higgs equations in one space dimension , 2014 .

[9]  C. Isham,et al.  RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS) 5470 The global existence in the Cauchy problem of the Maxwell-Chern- Simons-Higgs system , 2002 .

[10]  Hyungjin Huh,et al.  Global solutions to space–time monopole equations in one space dimension , 2015 .

[11]  Hyungjin Huh,et al.  Local and global solutions of Chern-Simons gauged O(3) sigma equations in one space dimension , 2016 .

[12]  B. Schroers.  The spectrum of Bogomol'nyi solitons in gauged linear sigma models , 1996, hep-th/9603101.

[13]  Hyungjin Huh,et al.  Reduction of Chern-Simons-Schrödinger Systems in One Space Dimension , 2013, J. Appl. Math..

[14]  V. Moncrief.  Global existence of Maxwell–Klein–Gordon fields in (2+1)‐dimensional space‐time , 1980 .