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On the Division Problem for the Wave Maps Equation

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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.

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References

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. Bournaveas, N., Candy, T.: Global well-posedness for the massless cubic Dirac equation. Int. Math. Res. Not. IMRN 2016(22), 6735–6828 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Candy, T.: Multi-scale bilinear restriction estimates for general phases. Preprint (2017). arXiv:1707.08944 [math.CA]

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  11. Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33(1), 43–101 (1980)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Klainerman, S., Selberg, S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2), 223–295 (2002)

    Article  MathSciNet  Google Scholar 

  14. Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58(2), 217–284 (2005)

    Article  MathSciNet  Google Scholar 

  15. Koch, H., Tataru, D.: Conserved energies for the cubic nonlinear Schrödinger equation in one dimension. Duke Math. J. 167(17), 3207–3313 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  17. Krieger, J.: Null-form estimates and nonlinear waves. Adv. Differ. Equ. 8(10), 1193–1236 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  19. Krieger, J., Schlag, W.: Concentration Compactness for Critical Wave Maps. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2012)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Lee, S., Vargas, A.: Restriction estimates for some surfaces with vanishing curvatures. J. Funct. Anal. 258(9), 2884–2909 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  23. Peetre, J.: New Thoughts on Besov spaces. Mathematics Department, Duke University, Durham, Duke University Mathematics Series, No. 1 (1976)

  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)

  25. Shatah, J., Struwe, M.: Geometric Wave Equations, Courant Lecture Notes in Mathematics, vol. 2. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  26. 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  Google Scholar 

  27. Sterbenz, J., Tataru, D.: Regularity of wave-maps in dimension \(2+1\). Commun. Math. Phys. 298(1), 231–264 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  28. Tao, T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238(2), 215–268 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  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)

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

    Article  MathSciNet  Google Scholar 

  32. Tataru, D.: The wave maps equation. Bull. Am. Math. Soc. (NS) 41(2), 185–204 (2004)

    Article  MathSciNet  Google Scholar 

  33. Tataru, D.: Rough solutions for the wave maps equation. Am. J. Math. 127(2), 293–377 (2005)

    Article  MathSciNet  Google Scholar 

  34. Wiener, N.: The quadratic variation of a function and its Fourier coefficients. J. Math. Phys. 3(2), 72–94 (1924)

    Article  Google Scholar 

  35. Wolff, T.: A sharp bilinear cone restriction estimate. Ann. Math. (2) 153(3), 661–698 (2001)

    Article  MathSciNet  Google Scholar 

  36. Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251–282 (1936)

    Article  MathSciNet  Google Scholar 

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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.

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

    Timothy Candy & Sebastian Herr

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  1. Timothy Candy
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  2. Sebastian Herr
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Correspondence to Sebastian Herr.

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Candy, T., Herr, S. On the Division Problem for the Wave Maps Equation. Ann. PDE 4, 17 (2018). https://doi.org/10.1007/s40818-018-0054-z

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