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Annals of PDE

, 5:10 | Cite as

Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

  • Cristian GavrusEmail author
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Abstract

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on \(\mathbb {R}^{1+d}\) \((d \ge 4)\) for data with small critical Sobolev norm. This extends to the general case \( m^2 > 0 \) the results of Krieger–Sterbenz–Tataru (\(d=4,5 \)) and Rodnianski–Tao (\( d \ge 6 \)), who considered the case \( m=0\). We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp \(L^{2}\) null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of \( \Box ^{-1} \) in conjunction with endpoint Strichartz estimates in Lorentz spaces.

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References

  1. 1.
    Bejenaru, I., Herr, S.: On global well-posedness and scattering for the massive Dirac–Klein–Gordon system. arXiv preprint arXiv:1409.1778 (2014)
  2. 2.
    Bejenaru, I., Herr, S.: The cubic Dirac equation: small initial data in \(H^{1/2}(\mathbb{R}^{2})\). Commun. Math. Phys. 335, 1–48 (2015)CrossRefGoogle Scholar
  3. 3.
    Bejenaru, I., Herr, S.: The cubic Dirac equation: small initial data in \(H^1(\mathbb{R}^3)\). Commun. Math. Phys. 335(1), 43–82 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    Cuccagna, S.: On the local existence for the Maxwell–Klein–Gordon system in \(\mathbb{R}^{3+1}\). Commun. Partial Differ. Equ. 24(5–6), 851–867 (1999)CrossRefGoogle Scholar
  5. 5.
    Eardley, D.M., Moncrief, V.: The global existence of Yang–Mills–Higgs fields in \(4\)-dimensional Minkowski space. Commun. Math. Phys. 83(2), 171–191 (1982)ADSCrossRefGoogle Scholar
  6. 6.
    Foschi, D., Klainerman, S.: Bilinear space-time estimates for homogeneous wave equations. Annales Scientifiques de l’Ecole Normale Superieure 33(2), 211–274 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gavrus, C., Oh, S.-J.: Global well-posedness of high dimensional Maxwell–Dirac for small critical data. arXiv preprint arXiv:1604.07900 (2016)
  8. 8.
    Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008)zbMATHGoogle Scholar
  9. 9.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, New York (2003)CrossRefGoogle Scholar
  10. 10.
    Keel, M., Roy, T., Tao, T.: Global well-posedness of the Maxwell–Klein–Gordon equation below the energy norm. Discrete Contin. Dyn. Syst 30(3), 573–621 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Keel, M., Tao, T.: Endpoint strichartz estimates. Am. J. Math. 120, 955–980 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46(9), 1221–1268 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klainerman, S., Machedon, M.: On the Maxwell–Klein–Gordon equation with finite energy. Duke Math. J. 74(1), 19–44 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klainerman, S., Machedon, M.: On the optimal local regularity for gauge field theories. Differ. Integral Equ. 10(6), 1019–1030 (1997)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Klainerman, S., Tataru, D.: On the optimal local regularity for the Yang–Mills equations in \( \mathbb{R}^{4+1} \). J. Am. Math. Soc. 12(1), 93–116 (1999)CrossRefGoogle Scholar
  16. 16.
    Krieger, J., Lührmann, J.: Concentration compactness for the critical Maxwell–Klein–Gordon equation. Ann. PDE 1(1), 1–208 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Krieger, J., Sterbenz, J.: Global regularity for the Yang–Mills equations on high dimensional Minkowski space. Mem. Am. Math. Soc. 223(1047), vi+99 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Krieger, J., Sterbenz, J., Tataru, D.: Global well-posedness for the Maxwell–Klein–Gordon equation in \(4+1\) dimensions: small energy. Duke Math. J. 164(6), 973–1040 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Krieger, J., Tataru, D.: Global well-posedness for the Yang–Mills equation in \(4+1\) dimensions. Small Energy. arXiv:1509.00751 (2015)
  20. 20.
    Machedon, M., Sterbenz, J.: Almost optimal local well-posedness for the \((3+1)\)-dimensional Maxwell–Klein–Gordon equations. J. Am. Math. Soc. 17(2), 297–359 (2004). (electronic)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nakanishi, K., Schlag, W.: Invariant Manifolds and Dispersive Hamiltonian Evolution Equations. European Mathematical Society (EMS), Zürich (2011)CrossRefGoogle Scholar
  22. 22.
    O-Neil, R.: Convolution operators and \({L}(p,q)\) spaces. Duke Math. J. 30(1), 129–142, 03 (1963)Google Scholar
  23. 23.
    Oh, S.-J.: Gauge choice for the Yang Mills equations using the Yang Mills heat flow and local well-posedness in \( {H}^1\). J. Hyperb. Differ. Equ. 11(01), 1–108 (2014)CrossRefGoogle Scholar
  24. 24.
    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. Duke Math. J. 164(9), 1669–1732, 06 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Oh, S.-J., Tataru, D.: Energy dispersed solutions for the (\(4+1\))-dimensional Maxwell–Klein–Gordon equation. Am. J. Math. 140, 1–82 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Oh, S.-J., Tataru, D.: Global well-posedness and scattering of the (\(4+1\))-dimensional Maxwell–Klein–Gordon equation. Invent. Math. 205, 781–877 (2016)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Oh, S.-J., Tataru, D.: Local well-posedness of the (\(4+1\))-dimensional Maxwell–Klein–Gordon equation at energy regularity. Ann. PDE. 2(2), (2016)Google Scholar
  28. 28.
    Pecher, H.: Low regularity local well-posedness for the Maxwell–Klein–Gordon equations in Lorenz gauge. Adv. Differ. Equ. 19(3/4), 359–386 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rodnianski, I., Tao, T.: Global regularity for the Maxwell–Klein–Gordon equation with small critical Sobolev Norm in high dimensions. Commun. Math. Phys. 251(2), 377–426 (2004)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Selberg, S.: Almost optimal local well-posedness of the Maxwell–Klein–Gordon equations in \(1 + 4\) dimensions. Commun. Partial Differ. Equ. 27(5–6), 1183–1227 (2002)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Selberg, S., Tesfahun, A.: Finite-energy global well-posedness of the Maxwell–Klein–Gordon system in Lorenz gauge. Commun. Partial Differ. Equ. 35(6), 1029–1057 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shatah, J., Struwe, M.: The Cauchy problem for wave maps. Int. Math. Res. Not. 11, 555–571 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tao, T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Tao, T.: Multilinear weighted convolution of \( {L}^{2}\) functions, and applications to nonlinear dispersive equations. Am. J. Math. 123(5), 839–908 (2001)CrossRefGoogle Scholar
  36. 36.
    Tataru, D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Tataru, D.: Rough solutions for the wave maps equation. Am. J. Math. 127(2), 293–377 (2005)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsThe Johns Hopkins UniversityBaltimoreUSA

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