Advertisement

Local well-posedness of Yang–Mills equations in Lorenz gauge below the energy norm

  • Achenef TesfahunEmail author
Article

Abstract

We prove that the Yang–Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential A and the associated curvature \({F \,\,{\rm in}\,\, H^s \times H^{s-1} \,\,{\rm and}\,\, H^r \times H^{r-1} \,\,{\rm for}\,\, s = (\frac{6}{7}+,-\frac{1}{14}+)}\), respectively. This extends a recent result by Selberg and the present author on the local well-posedness of YM-LG for finite energy data (specifically, for (s, r) =  (1−, 0)). We also prove unconditional uniqueness of the energy class solution, that is, uniqueness in the classical space C([−T, T]; X 0), where X 0 is the energy data space. The key ingredient in the proof is the fact that most bilinear terms in YM-LG contain null structure some of which uncovered in the present paper.

Mathematics Subject Classification

35Q40 35L70 

References

  1. 1.
    D’Ancona P., Foschi D., Selberg S.: Atlas of products for wave-Sobolev spaces on \({\mathbb{R}^{1+3}}\). Trans. Am. Math. Soc. 364(1), 31–63 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D’Ancona P., Foschi D., Selberg S.: Null structure and almost optimal local well-posedness of the Maxwell-Dirac system. Am. J. Math. 132(3), 771–839 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eardley D.M., Moncrief V.: The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties. Comm. Math. Phys. 83(2), 171–191 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eardley D.M., Moncrief V.: The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. II. Completion of proof. Comm. Math. Phys. 83(2), 193–212 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Keel M.: Global existence for critical power Yang–Mills–Higgs equations in \({\mathbf{R}^{3+1}}\). Comm. Partial Differ. Equ. 22(7–8), 1161–1225 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Klainerman S., Machedon M.: Finite energy solutions of the Yang–Mills equations in \({\mathbf{R}^{{3+1}}}\). Ann. Math. (2) 142(1), 39–119 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Klainerman S., Tataru D.: On the optimal local regularity for Yang–Mills equations in \({\mathbf{R}^{4+1}}\). J. Am. Math. Soc. 12(1), 93–116 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Krieger J., Schlag W., Tataru D.: Renormalization and blow up for the critical Yang–Mills problem. Adv. Math. 221(5), 1445–1521 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lindblad H.: Counterexamples to local existence for semi-linear wave equations. Am. J. Math. 118(1), 1–16 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Oh, S.J.: Gauge choice for the Yang–Mills equations using the Yang–Mills heat flow and local well-posedness in \({{H}^{1}}\), preprint (2012). arXiv:1210.1558
  11. 11.
    Oh, S.J.: Finite energy global well-posedness of the Yang–Mills equations on \({\mathbb{R}^{1+3}}\): an approach using the Yang–Mills heat flow, preprint (2012). arXiv:1210.1557
  12. 12.
    Rodnianski I., Tao T.: Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions. Comm. Math. Phys. 251(2), 377–426 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ponce G., Sideris T.C.: Local regularity of nonlinear wave equations in three space dimensions. Comm. Partial Differ. Equ. 18(1–2), 169–177 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Irving S.: The Cauchy problem for the Yang–Mills equations. J. Funct. Anal. 33(2), 175–194 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Selberg S., Tesfahun A.: Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge. Comm. Partial Differ. Equ. 35(6), 1029–1057 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Selberg S., Tesfahun A.: Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete Contin. Dyn. Syst. 33(6), 2531–2546 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Selberg, S.; Tesfahun, A.: Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge. J. Eur. Math. Soc. http://arxiv.org/pdf/1309.1977 (to appear)
  18. 18.
    Selberg, S.: Anisotropic bilinear L 2 estimates related to the 3D wave equation. Int. Math. Res. Not. IMRN (2008). Art. ID rnn 107, 63. MR 2439535 (2010i:35216)Google Scholar
  19. 19.
    Tesfahun, A.: Finite energy local well-posedness for the Yang–Mills–Higgs equations in Lorenz Gauge. Int. Math. Res. Not. IMRN. (2014). doi: 10.1093/imrn/rnu087
  20. 20.
    Sterbenz J.: Global regularity and scattering for general non-linear wave equations. II. (4+1) dimensional Yang–Mills equations in the Lorentz gauge. Am. J. Math. 129(3), 611–664 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tao T.: Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J. Differ. Equ. 189(2), 366–382 (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Zhou Y.: Uniqueness of generalized solutions to nonlinear wave equations. Am. J. Math. 122(5), 939–965 (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

Personalised recommendations