Skip to main content
SpringerLink
Log in

Cauchy Problem for the Fermion Field Equation Coupled With the Chern–Simons Gauge

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study initial value problems of the Chern–Simons–Dirac equations. With the Lorentz gauge condition they are formulated in the second-order hyperbolic equations. Under the Coulomb gauge condition Dirac equation is coupled with the elliptic equations which show some smoothing properties of the gauge field. With the temporal gauge condition divergence-curl decomposition and elliptic estimates will be used.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bergé L., de Bouard A., Saut J.C. (1995) Blowing up time-dependent solutions of the planar Chern–Simons gauged nonlinear Schrödinger equation. Nonlinearity 8(2): 235–253

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Bournaveas N. (1996) Local existence for the Maxwell–Dirac equations in three space dimension. Commun. Partial Differ. Equ. 21(5–6): 693–720

    Article  MATH  MathSciNet  Google Scholar 

  3. Bournaveas N. (2001) Low regularity solutions of the Dirac Klein–Gordon equations in two space dimensions. Commun. Partial Differ. Equ. 26, 1345–1366

    Article  MATH  MathSciNet  Google Scholar 

  4. Chadam J.M., Glassey R.T. (1976) On the Maxwell–Dirac equations with zero magnetic field and their solution in two space dimensions. J. Math. Anal. Appl. 53, 495–507

    Article  MATH  MathSciNet  Google Scholar 

  5. Chae D., Choe K. (2002) Global existence in the Cauchy problem of the relativistic Chern–Simons-Higgs theory. Nonlinearity 15(3): 747–758

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Cho Y.M., Kim J.W., Park D.H. (1992) Fermionic vortex solutions in Chern–Simons electrodynamics. Phys. Rev. D 45, 3802–3806

    Article  ADS  MathSciNet  Google Scholar 

  7. Deser S., Jackiw R., Templeton S. (1982) Three dimensional massive gauge theories. Phys. Rev. Lett. 48, 975–978

    Article  ADS  Google Scholar 

  8. Eardley D.M., Moncrief V. (1982) The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. Commun. Math. Phys. 83, 171–191

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Flato, M., Simon, J., Taflin, E.: Asymptotic completeness, global existence and the infrared problem for the Maxwell–Dirac equations. Mem. Am. Math. Soc. 127(606), (1997)

  10. Foschi D., Klainerman S. (2000) Bilinear space-time estimates for homogeneous wave equations. Ann. Sci. École Norm. Sup.(4) 33(2): 211–274

    MATH  MathSciNet  Google Scholar 

  11. Georgiev V. (1991) Small amplitude solutions of the Maxwell–Dirac equations. Indiana Univ. Math. J. 40(3): 845–883

    Article  MATH  MathSciNet  Google Scholar 

  12. Glassey R.T., Strauss W.A. (1979) Conservation laws for the Maxwell–Dirac and Klein–Gordon–Dirac equations. J. Math. Phys. 20, 454–458

    Article  ADS  MathSciNet  Google Scholar 

  13. Hong J., Kim Y., Pac P.Y. (1990) Multivortex solutions of the abelian Chern–Simons-Higgs. Phys. Rev. Lett. 64, 2230–2233

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Huh H. (2005) Low regularity solutions of the Chern–Simons-Higgs equations. Nonlinearity 18, 1–9

    Article  MathSciNet  Google Scholar 

  15. Jackiw R., Pi S.Y. (1990) Soliton solutions to the gauged nonlinear Schrödinger equation on the plane Phys. Rev. Lett. 64, 2969–2972

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Kato T., Ponce G. (1988) Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math. 41, 891–907

    Article  MATH  MathSciNet  Google Scholar 

  17. Klainerman, S., Machedon, M.: On the regularity properties of the wave equation. In: Flato, M., Kerner, R., Lichnerowicz, A. (eds.) Physics on Manifolds, Kluwer Academic (1994)

  18. Klainerman S., Machedon M. (1994) On the Maxwell–Klein–Gordon equation with finite energy. Duke Math. J. 74(1):19–44

    Article  MATH  MathSciNet  Google Scholar 

  19. Klainerman S., Selberg S. (2002) Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2): 223–295

    Article  MATH  MathSciNet  Google Scholar 

  20. Li S., Bhaduri R.K. (1991) Planar solitons of the gauged Dirac equation. Phys. Rev. D 43, 3573–3574

    Article  ADS  Google Scholar 

  21. Lindblad H., Sogge C. (1995) On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2): 503–539

    Article  MathSciNet  Google Scholar 

  22. Ponce G., Sideris T. (1993) Local regularity of nonlinear wave equations in three space dimensions. Commun. Partial Differ. Equ. 18, 169–177

    Article  MATH  MathSciNet  Google Scholar 

  23. Psarelli M. (2005) Maxwell–Dirac equations in four-dimensional Minkowski space. Commun. Partial Differ. Equ. 30(1–3): 97–119

    Article  MATH  MathSciNet  Google Scholar 

  24. Tao T. (2001) Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123(5): 839–908

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Hyungjin Huh

Authors
  1. Hyungjin Huh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyungjin Huh.

Additional information

JSPS Research Follow supported by JSPS Grant-in-Aid

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huh, H. Cauchy Problem for the Fermion Field Equation Coupled With the Chern–Simons Gauge. Lett Math Phys 79, 75–94 (2007). https://doi.org/10.1007/s11005-006-0118-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-006-0118-y

Mathematics Subject Classification (2000)

Keywords

Search

Navigation