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The Cubic Dirac Equation: Small Initial Data in \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\))

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Abstract

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\)) is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein–Gordon equation in dimension n = 2, which is captured by constructing an adapted systems of coordinate frames.

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References

  1. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bejenaru I., Ionescu A.D., Kenig C.E., Tataru D.: Global Schrödinger maps in dimensions d ≥ 2: small data in the critical Sobolev spaces. Ann. Math. (2) 173((3), 1443–1506 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bournaveas,N., Candy, T.: Globalwell-posedness for the massless cubic dirac equation. arXiv:1407.0655

  4. Candy T.: Global existence for an L 2 critical nonlinear Dirac equation in one dimension. Adv. Differ. Equ. 16(7-8), 643–666 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Cazenave T., Vázquez L.: Existence of localized solutions for a classical nonlinear Dirac field. Commun. Math. Phys. 105(1), 35–47 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. D’Ancona P., Foschi D., Selberg S.: Local well-posedness below the charge norm for the irac–Klein–Gordon system in two space dimensions. J. Hyperbol. Differ. Equ. 4(2), 295–330 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Escobedo, M., Vega, L.: A semilinear Dirac equation in \({H^{s} {\rm (\mathbf {R}}^{3})}\) for s >1. SIAM J. Math. Anal. 28(2), 338–362 (1997)

  8. Finkelstein R., LeLevier R., Ruderman M.: Nonlinear spinor fields. Phys. Rev. 83(2), 326–332 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Hadac M., Herr S., Koch H.: Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(3), 917–941 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Krieger J.: Global regularity of wave maps from \({\mathbb{R}^{3+1}}\) to surfaces. Commun. Math. Phys. 238, 333–366 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Krieger J.: Global regularity of wave maps from \({\mathbb{R}^{2+1}}\) to \({\mathbb{H}^{2}}\). small energy. Commun. Math. Phys. 250, 507–580 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Krieger, J., Schlag, W.: Concentration compactness for critical wave maps. In: EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich (2012)

  13. Machihara S., Nakamura M., Nakanishi K., Ozawa T.: Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219(1), 1–20 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Machihara S., Nakanishi K., Ozawa T.: Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev. Mat. Iberoamericana 19(1), 179–194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Machihara S., Nakanishi K., Tsugawa K.: Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50(2), 403–451 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Merle F.: Existence of stationary states for nonlinear Dirac equations. J. Differ. Equ. 74(1), 50–68 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Montgomery-Smith S.J.: Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math. J. 91(2), 393–408 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nakanishi K., Schlag,W.: Invariantmanifolds and dispersive Hamiltonian evolution equations. In: Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich (2011)

  19. Hartmut P.: Local well-posedness for the nonlinear Dirac equation in two space dimensions. Commun. Pure Appl. Anal. 13(2), 673–685 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Hartmut P.: Corrigendum of ’Local well-posedness for the nonlinear Dirac equation in two space dimensions’. Commun. Pure Appl. Anal. 14(2), 737–742 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Soler M.: Classial, stable, nonlinear spinor fields with positive rest energy. Phys. Rev. D 1(10), 2766–2769 (1970)

    Article  ADS  Google Scholar 

  22. Sterbenz J., Tataru D.: Energy dispersed large data wave maps in 2 + 1 dimensions. Commun. Math. Phys. 298(1), 139–230 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Strauss W., Vázquez L.: Stability under dilations of nonlinear spinor fields. Phys. Rev. D 34(2), 641–643 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Tao T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Tao, T.: A counterexample to an endpoint bilinear Strichartz inequality. Electron. J. Differ. Equ. arXiv:math/0609849 (2006)

  26. Tataru D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093-0112, USA

    Ioan Bejenaru

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501, Bielefeld, Germany

    Sebastian Herr

Authors
  1. Ioan Bejenaru
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  2. Sebastian Herr
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Correspondence to Ioan Bejenaru.

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Communicated by W. Schlag

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Bejenaru, I., Herr, S. The Cubic Dirac Equation: Small Initial Data in \({{H^{\frac{1}{2}}} (\mathbb{R}^{2}}\)). Commun. Math. Phys. 343, 515–562 (2016). https://doi.org/10.1007/s00220-015-2508-4

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  • DOI: https://doi.org/10.1007/s00220-015-2508-4

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