Advertisement

Annals of PDE

, 4:17 | Cite as

On the Division Problem for the Wave Maps Equation

  • Timothy Candy
  • Sebastian Herr
Article
  • 17 Downloads

Abstract

We consider Wave Maps into the sphere and give a new proof of small data global well-posedness and scattering in the critical Besov space, in any space dimension \(n \geqslant 2\). We use an adapted version of the atomic space \(U^2\) as the single building block for the iteration space. Our approach to the so-called division problem is modular as it systematically uses two ingredients: atomic bilinear (adjoint) Fourier restriction estimates and an algebra property of the iteration space, both of which can be adapted to other phase functions.

Keywords

Wave maps Division problem Bilinear Fourier restriction Atomic spaces 

Mathematics Subject Classification

35L15 35L52 

Notes

Acknowledgements

The authors thank Kenji Nakanishi and Daniel Tataru for sharing their part in the story of the division problem with us. Also, the authors thank Daniel Tataru for providing a preliminary version of [31]. In addition, we thank the referees for their helpful comments, in particular for the alternative strategy for the proof of Theorem 4.6 summarised in Remark 4.7. Financial support by the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is acknowledged.

References

  1. 1.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  2. 2.
    Bejenaru, I.: Optimal bilinear restriction estimates for general hypersurfaces and the role of the shape operator. Int. Math. Res. Not. IMRN 2017(23), 7109–7147 (2017)MathSciNetGoogle 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.
    Bejenaru, I., Herr, S.: The cubic Dirac equation: small initial data in \(H^{\frac{1}{2}}({\mathbb{R}}^2)\). Commun. Math. Phys. 343(2), 515–562 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Bejenaru, I., Ionescu, A.D., Kenig, C.E., Tataru, D.: Global Schrödinger maps in dimensions \(d\ge 2\): small data in the critical Sobolev spaces. Ann. Math. (2) 173(3), 1443–1506 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bournaveas, N., Candy, T.: Global well-posedness for the massless cubic Dirac equation. Int. Math. Res. Not. IMRN 2016(22), 6735–6828 (2016)MathSciNetGoogle Scholar
  7. 7.
    Candy, T.: Multi-scale bilinear restriction estimates for general phases. Preprint (2017). arXiv:1707.08944 [math.CA]
  8. 8.
    Candy, T., Herr, S.: Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system. Anal. PDE 11(5), 1171–1240 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Geba, D.-A., Grillakis, M.G.: An Introduction to the Theory of Wave Maps and Related Geometric Problems. World Scientific Publishing Co Pte. Ltd., Hackensack (2017)zbMATHGoogle Scholar
  10. 10.
    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)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33(1), 43–101 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Klainerman, S., Rodnianski, I.: On the global regularity of wave maps in the critical Sobolev norm. Int. Math. Res. Not. 2001(13), 655–677 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klainerman, S., Selberg, S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2), 223–295 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58(2), 217–284 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Koch, H., Tataru, D.: Conserved energies for the cubic nonlinear Schrödinger equation in one dimension. Duke Math. J. 167(17), 3207–3313 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Koch, H., Tataru, D., Visan, M.: Dispersive Equations and Nonlinear Waves: Generalized Korteweg–de Vries, Nonlinear Schrödinger, Wave and Schrödinger Maps, vol. 45. Springer, Basel (2014)zbMATHGoogle Scholar
  17. 17.
    Krieger, J.: Null-form estimates and nonlinear waves. Adv. Differ. Equ. 8(10), 1193–1236 (2003)ADSMathSciNetzbMATHGoogle Scholar
  18. 18.
    Krieger, J.: Global regularity of wave maps from \( {\bf R}^{2+1}\) to \(H^2\). Small energy. Commun. Math. Phys. 250(3), 507–580 (2004)ADSCrossRefGoogle Scholar
  19. 19.
    Krieger, J., Schlag, W.: Concentration Compactness for Critical Wave Maps. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2012)CrossRefGoogle Scholar
  20. 20.
    Krieger, J., Tataru, D.: Global well-posedness for the Yang–Mills equation in \(4+1\) dimensions. Small energy. Ann. Math. (2) 185(3), 831–893 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lee, S., Vargas, A.: Restriction estimates for some surfaces with vanishing curvatures. J. Funct. Anal. 258(9), 2884–2909 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sung-Jin, O., Tataru, D.: Global well-posedness and scattering of the \((4+1)\)-dimensional Maxwell–Klein–Gordon equation. Invent. Math. 205(3), 781–877 (2016)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Peetre, J.: New Thoughts on Besov spaces. Mathematics Department, Duke University, Durham, Duke University Mathematics Series, No. 1 (1976)Google Scholar
  24. 24.
    Pisier, Gilles, X., Quan H.: Random Series in the Real Interpolation Spaces Between the Spaces \(v_p\), Geometrical Aspects of Functional Analysis (1985/86), Lecture Notes in Mathematics, vol. 1267. Springer, Berlin, pp. 185–209 (1987)CrossRefGoogle Scholar
  25. 25.
    Shatah, J., Struwe, M.: Geometric Wave Equations, Courant Lecture Notes in Mathematics, vol. 2. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  26. 26.
    Sterbenz, J., Tataru, D.: Energy dispersed large data wave maps in \(2+1\) dimensions. Commun. Math. Phys. 298(1), 139–230 (2010)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Sterbenz, J., Tataru, D.: Regularity of wave-maps in dimension \(2+1\). Commun. Math. Phys. 298(1), 231–264 (2010)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Tao, T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238(2), 215–268 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tao, T.: Global regularity of wave maps II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Tao, T.: Nonlinear Dispersive Equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, Local and global analysis (2006)Google Scholar
  31. 31.
    Tataru, D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tataru, D.: The wave maps equation. Bull. Am. Math. Soc. (NS) 41(2), 185–204 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tataru, D.: Rough solutions for the wave maps equation. Am. J. Math. 127(2), 293–377 (2005)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wiener, N.: The quadratic variation of a function and its Fourier coefficients. J. Math. Phys. 3(2), 72–94 (1924)CrossRefGoogle Scholar
  35. 35.
    Wolff, T.: A sharp bilinear cone restriction estimate. Ann. Math. (2) 153(3), 661–698 (2001)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251–282 (1936)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

Personalised recommendations