Taiwanese Journal of Mathematics

A CRITICAL CASE ON THE DIRAC-KLEIN-GORDON EQUATIONS IN ONE SPACE DIMENSION

Yung-Fu Fang and Hsiu-Chuan Huang

Full-text: Open access

Abstract

We establish local and global existence results for a critical case of Dirac-Klein-Gordon equations in one space dimension, employing a null form estimate, a bilinear estimate and a fixed point argument.

Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1045-1059.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574246

Digital Object Identifier
doi:10.11650/twjm/1500574246

Mathematical Reviews number (MathSciNet)
MR2431878

Zentralblatt MATH identifier
1172.35453

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations

Keywords
Dirac-Klein-Gordon equations null form estimates

Citation

Fang, Yung-Fu; Huang, Hsiu-Chuan. A CRITICAL CASE ON THE DIRAC-KLEIN-GORDON EQUATIONS IN ONE SPACE DIMENSION. Taiwanese J. Math. 12 (2008), no. 5, 1045--1059. doi:10.11650/twjm/1500574246. https://projecteuclid.org/euclid.twjm/1500574246


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References

  • [1.] J. Bourgain, Invariant Measures for NLS in Infinite Volume, Commun. Math. Phys, 210 (2000), 605-620.
  • [2.] A. Bachelot, Global existence of large amplitude solutions for Dirac-Klein-Gordon systems in Minkowski space, Lecture Notes in Math., 1402 (1989), 99-113, Springer, Berlin.
  • [3.] N. Bournaveas, A new proof of global existence for the Dirac-Klein-Gordon equations in one space dimension, J. Funct. Anal., 173 (2000), 203-213.
  • [4.] N. Bournaveas & D. Gibbeson, Low Regularity Global Solutions of the Dirac-Klein-Gordon Equations in One Space Dimension, Differential and Integral Equations, 19 (2006), 211-222.
  • [5.] J. Chadam, Global Solutions of the Cauchy Problem for the (Classical) Coupled Maxwell-Dirac Equations in one Space Dimension, J. Funct. Anal., 13 (1973), 173-184
  • [6.] J. Chadam & R. Glassey, On Certain Global Solutions of the Cauchy Problem for the (Classical) Coupled Klein-Gordon-Dirac equations in one and three Space Dimensions, Arch. Rational Mech. Anal., 54 (1974), 223-237.
  • [7.] Yung-fu Fang, Local Existence for Semilinear Wave Equations and Applications to Yang-Mills Equations, Ph.D dissertation, University of Maryland, 1996.
  • [8.] Yung-fu Fang, A Direct Proof of Global Existence for the Dirac-Klein-Gordon Equations in One Space Dimension, Taiwanese J. Math., 8 (2004), 33-41.
  • [9.] Yung-fu Fang, On the Dirac-Klein-Gordon Eqautions in One Space Dimension, Differential and Integral Equations, 17 (2004), 1321-1346.
  • [10.] Yung-fu Fang, Low Regularity Solutions for the Dirac-Klein Equations in One Space Dimension, Electronic J. Diff. Equations, 2004 (2004), 1-19.
  • [11.] Yung-fu Fang and Manoussos Grillakis, Existence and Uniqueness for Boussinesq type Equations on a Circle, Comm. PDE, 21 (1996), 1253-1277.
  • [12.] V. Georgiev, Small amplitude solutions of the Maxwell-Dirac equations, Indiana Univ. Math. J., 40 (1991), 845-883.
  • [13.] R. Glassey & W. Strauss, Conservation laws for the classical Maxwell-Dirac and Klein-Gordon-Dirac equations, J. Math. Phys., 20 (1979), 454-458.
  • [14.] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., XLVI (1993), 1221-1268.
  • [15.] Sergej Kuksin, Infinite-Dimensional Symplectic Capacities and a Squeezing Theorem for Hamiltonian PDE's, Commun. Math. Phys, 167 (1995), 531-552.
  • [16.] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, (1970).
  • [17.] Y. Zheng, Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations, Commun. Math. Phys., 151 (1993), 67-87.
  • [18.] Hsiu-Chuan Huang, Interative Methods for Dirac-Klein-Gordon Equations and Variational Methods for Ellrptic Equations, PhD Thesis, Cheng-Kung University, 2008.