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International Mathematics Research Papers (2006) Vol. 2006 : article ID 52976, 109 pages, doi:10.1155/IMRP/2006/52976
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Global stability for charged-scalar fields on Minkowski space

Hans Lindblad and Jacob Sterbenz

Department of Mathematics, University of California San Diego, La Jolla, CA 92093-0112, USA E-mail address: lindblad{at}math.ucsd.edu; E-mail address: jsterben{at}math.ucsd.edu

We prove that the charged-scalar-field equations are globally stable on (3+1)-dimensional Minkowski space for small initial data in certain gauge covariant weighted Sobolev spaces. These spaces can be chosen to be almost scale invariant with respect to the homogeneity of the equations, and our result is valid for initial data with nonzero charge that is also nonstationary at space-like infinity. The method of proof is a tensor-geometric approach which is based on a sharp family of weighted bilinear L2 space-time estimates.



References

  1. Choquet-Bruhat Y., Christodoulou D. Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in 3 + 1 dimensions. Annales Scientifiques de l'École Normale Supérieure. Quatrième Série (1981) 14(4):481–506. (1982).
  2. Christodoulou D. Global solutions of nonlinear hyperbolic equations for small initial data. Communications on Pure and Applied Mathematics (1986) 39(2):267–282.[CrossRef][Web of Science]
  3. Christodoulou D., Klainerman S. Asymptotic properties of linear field equations in Minkowski space. Communications on Pure and Applied Mathematics (1990) 43(2):137–199.[CrossRef][Web of Science]
  4. Christodoulou D., Klainerman S. The Global Nonlinear Stability of the Minkowski Space. In: Princeton Mathematical Series (1993) 41. New Jersey: Princeton University Press. x+514.
  5. Eardley D. M., Moncrief V. The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties. Communications in Mathematical Physics (1982) 83(2):171–191.[CrossRef][Web of Science]
  6. Eardley D. M., Moncrief V. The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof. Communications in Mathematical Physics (1982) 83(2):193–212.[CrossRef][Web of Science]
  7. Flato M., Simon J. C. H., Taflin E. Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations. Memoirs of the American Mathematical Society (1997) 127(606):x+311.
  8. Klainerman S. Uniform decay estimates and the Lorentz invariance of the classical wave equation. Communications on Pure and Applied Mathematics (1985) 38(3):321–332.[CrossRef][Web of Science]
  9. Klainerman S., Machedon M. On the Maxwell-Klein-Gordon equation with finite energy. Duke Mathematical Journal (1994) 74(1):19–44.[CrossRef][Web of Science]
  10. Klainerman S., Machedon M. Finite energy solutions of the Yang-Mills equations in R3 + 1. Annals of Mathematics. Second Series (1995) 142(1):39–119.
  11. Lieb E. H., Loss M. Analysis. In: Graduate Studies in Mathematics (2001) 14, 2nd. Rhode Island: American Mathematical Society. xxii+346.
  12. Lindblad H., Rodnianski I. Global existence for the Einstein vacuum equations in wave coordinates. Communications in Mathematical Physics (2005) 256(1):43–110.[CrossRef][Web of Science]
  13. Morawetz C. S. The limiting amplitude principle. Communications on Pure and Applied Mathematics (1962) 15:349–361.[CrossRef][Web of Science]
  14. Psarelli M. Asymptotic behavior of the solutions of Maxwell-Klein-Gordon field equations in 4-dimensional Minkowski space. Communications in Partial Differential Equations (1999) 24(1-2):223–272.[CrossRef][Web of Science]
  15. Shu W.-T. Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space. Communications in Mathematical Physics (1991) 140(3):449–480.[CrossRef][Web of Science]
  16. Shu W.-T. Global existence of Maxwell-Higgs fields. In: Pitman Res. Notes Math. Ser. (1992) 253. Nonlinear Hyperbolic Equations and Field Theory, 1990: Lake Como. Harlow: Longman Scientific & Technical. 214–227.
  17. Sideris T. C. Global existence of harmonic maps in Minkowski space. Communications on Pure and Applied Mathematics (1989) 42(1):1–13.[CrossRef][Web of Science]
  18. Stein E. M. Singular Integrals and Differentiability Properties of Functions. In: Princeton Mathematical Series, no. 30 (1970) New Jersey: Princeton University Press. xiv+290.

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This Article
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