Skip to main content
SpringerLink
Log in

Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

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

Abstract

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation

$$\left\{ {\begin{array}{*{20}c} {( - i\gamma ^\mu \partial _\mu + M)\psi = 0} \\ {( - \square + m^2 )\varphi = g(t)\psi ^\dag \gamma ^0 \psi } \\ \end{array} } \right.$$

in two space dimensions with complex-valued initial data ψ(0,x)∈L2 (ℝ2;ℂ4), real valued ϕ(0, x) ∈ W1,2 (ℝ2) and ϕt (0, x) ∈ L2 (ℝ2) have regularity

Here ℊ 1loc (ℝ3) denotes the (local) Hardy space, andg(t) is assumed to be inC 1(ℝ) andg(0)=0. Consequently nonlinear terms φψ which appear in the classical coupled Dirac-Klein-Gordon system (with the modificationg=g(t)∈C 1 andg(0)=0) can then be defined in L ∞loc (ℝ2; L1 (ℝ2)). We hope these results will be useful in establishing the existence of weak solutions to the classical coupled Dirac-Klein-Gordon system in the framework of compensated compactness.

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. Bachelot, A.: Solutions globales des systèmes de Dirac-Klein-Gordon. Journées “Équations aux Derivées Partielles” (Saint Jean de Monts, 1987), Exp. No. XV, 10pp., École Polytech., Palaiseau, 1987

  2. Bachelot, A.: Global existence of large amplitude solutions for Dirac-Klein-Gordon systems in Minkowski space, nonlinear hyperbolic problems. (Bordeaux, 1988), Lect. Notes in Math., vol. 1402, Berlin, Heidelberg, New York: Springer-Verlag, 1989, pp. 99–113

    Google Scholar 

  3. Bachelot, A.: Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. H. Poincaré Phys. Théor.48, 387–422 (1988)

    Google Scholar 

  4. Bachelot, A.: Global solutions to nonlinear Dirac equations in Minkowski space, Integrable Systems and Applications. (Île d'Oléron, 1988), Lect. Notes in Phys., vol. 342, Berlin, Hedelberg, New York: Springer-Verlag, 1989, pp. 1–11

    Google Scholar 

  5. Bjorken, J.D., Drell, S.D.: Relativistic quantum mechanics and relativistic quantum fields. New York: McGraw-Hill, 1964, 1965.

    Google Scholar 

  6. Chadam, J.M.: Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension. J. Funct. Anal.13, 173–184 (1973)

    Google Scholar 

  7. Chadam, J.M., Glassey, R.T.: On certain global solutions of the Cauchy problem for the (classical) coupled Klein-Gordon-Dirac equations in one and three space dimensions. Arch. Ratl. Mech. Anal.54, 223–237 (1974)

    Google Scholar 

  8. Coifman, R.R., Lions, P.L., Meyer, Y., Semmes, S.: Compacité par compensation et espaces de Hardy. Preprint

  9. Evans, L.C.: Partial regularity for stationary harmonic maps into spheres. Preprint

  10. Fefferman, C.: Characterization of bounded mean oscillation. Bull. AMS77, 585–587 (1971)

    Google Scholar 

  11. Fefferman, C., Stein, E.:H p spaces of several variables. Acta Math.129, 137–193 (1972)

    Google Scholar 

  12. Hélein, F.: Regularité des applications faiblement harmoniques entre une surface et une sphere. CRAS, to appear

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

    Google Scholar 

  14. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J.46, 27–42 (1979)

    Google Scholar 

  15. Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math.38, 321–332 (1985)

    Google Scholar 

  16. Marshall, B., Strauss, W., Wainger, S.:L p-L q estimates for the Klein-Gordon equation. J. Math. Pures et Appl.59, 417–440 (1980)

    Google Scholar 

  17. Müller, S.: Higher integrability of determinants and weak convergence inL 1. Preprint

  18. Peral, J.C.:L p estimates for the wave equation. J. Funct. Anal.36, 114–145 (1980)

    Google Scholar 

  19. Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math.38, 685–696 (1985)

    Google Scholar 

  20. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press, 1970

    Google Scholar 

  21. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press, 1971

    Google Scholar 

  22. Tartar, L.: Compensated compactness and applications to partial differential equations. Edinburg: Heriot-Watt, vol. 4

  23. Torchinsky, A.: Real variable methods in harmonic analysis. New York: Academic Press, 1986

    Google Scholar 

  24. Wente, W.C.: The Dirichlet problem with a volume constraint. Manuscripta Math.11, 141–157 (1974)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute, New York University, 10012, New York, NY, USA

    Yuxi Zheng

Authors
  1. Yuxi Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Research at MSRI and IAS supported in part by NSF Grant DMS-8505550 and DMS-9100383

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, Y. Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations. Commun.Math. Phys. 151, 67–87 (1993). https://doi.org/10.1007/BF02096749

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096749

Keywords

Search

Navigation