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Mode Stability of Self-Similar Wave Maps in Higher Dimensions

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Abstract

We consider co-rotational wave maps from Minkowski space in d + 1 dimensions to the d-sphere. Recently, Bizoń and Biernat found explicit self-similar solutions for each dimension \({d\geq 4}\). We give a rigorous proof for the mode stability of these self-similar wave maps.

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References

  1. Bizoń P., Biernat P.: Generic self-similar blowup for equivariant wave maps and Yang–Mills fields in higher dimensions. Commun. Math. Phys. 338(3), 1443–1450 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bizoń Piotr: An unusual eigenvalue problem. Acta Phys. Polon. B. 36(1), 5–15 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Bizoń Piotr, Chmaj Tadeusz, Tabor Zbisław: Dispersion and collapse of wave maps. Nonlinearity 13(4), 1411–1423 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Buslaev V.I., Buslaeva S.F.: Poincaré’s theorem on difference equations. Mat. Zametki 78(6), 943–947 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cazenave Thierry, Shatah Jalal, Tahvildar-Zadeh A. Shadi: Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang–Mills fields. Ann. Inst. H. Poincaré Phys. Théor. 68(3), 315–349 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Costin O., Costin R.D., Lebowitz J.L., Rokhlenko A.: Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Commun. Math. Phys. 221(1), 1–26 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Costin O., Huang M., Schlag W.: On the spectral properties of \({L_\pm}\) in three dimensions. Nonlinearity 25(1), 125–164 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Costin O., Donninger R., Glogić I., Huang M.: On the stability of self-similar solutions to nonlinear wave equations. Commun. Math. Phys. 343(1), 299–310 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Costin O., Donninger R., Xia X.: A proof for the mode stability of a self-similar wave map. Nonlinearity 29(8), 2451 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Costin O., Huang M., Tanveer S.: Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of P I . Duke Math. J. 163(4), 665–704 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Costin O., Kim T.E., Tanveer S.: A quasi-solution approach to nonlinear problems—the case of the Blasius similarity solution. Fluid Dyn. Res. 46(3), 031419,19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Côte R., Kenig C.E., Lawrie A., Schlag W.: Characterization of large energy solutions of the equivariant wave map problem: I. Amer. J. Math. 137(1), 139–207 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Côte R., Kenig C.E., Lawrie A., Schlag W.: Characterization of large energy solutions of the equivariant wave map problem: II. Amer. J. Math. 137(1), 209–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Donninger Roland: On stable self-similar blowup for equivariant wave maps. Comm. Pure Appl. Math. 64(8), 1095–1147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Donninger R.: Stable self-similar blowup in energy supercritical Yang-Mills theory. Math. Z. 278(3-4), 1005–1032 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donninger Roland: Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation. arXiv:1509.02041 (2015) (Preprint)

  17. Donninger R., Schörkhuber Birgit: Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Differ. Equ. 9(1), 63–87 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Donninger R., Schörkhuber B.: Stable blow up dynamics for energy supercritical wave equations. Trans. Amer. Math. Soc. 366(4), 2167–2189 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Donninger R., Schörkhuber B.: On blowup in supercritical wave equations. Comm. Math. Phys. 346(3), 907–943 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Donninger R., Schörkhuber B., Aichelburg P.C.: On stable self-similar blow up for equivariant wave maps: the linearized problem. Ann. Henri Poincaré 13(1), 103–144 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Elaydi, S.: An Introduction to Difference Equations. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2005)

  22. Jaffé, G.: Zur theorie des wasserstoffmolekülions. Zeitschrift für Physik 87(7), 535– (1934)

  23. Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  25. Leaver E.W.: An analytic representation for the quasi-normal modes of kerr black holes. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 402(1823), 285–298 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  26. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, Charles W. (eds): NIST Handbook of Mathematical Functions. US Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge (2010)

  27. Phillips G.M.: Interpolation and Approximation by Polynomials. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 14. Springer, New York (2003)

    Book  Google Scholar 

  28. Raphaël, P., Rodnianski, I.: Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. Publ. Math. Inst. Hautes Études Sci., pp. 1–122 (2012)

  29. Rodnianski I., Sterbenz J.: On the formation of singularities in the critical \({{\rm O(3)}}\) \({\sigma}\)-model. Ann. Math. (2) 172(1), 187–242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shatah Jalal: Weak solutions and development of singularities of the \({{\rm SU(2)}}\) \({\sigma}\)-model. Comm. Pure Appl. Math. 41(4), 459–469 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sterbenz J., Tataru D.: Regularity of wave-maps in dimension \({{\rm 2+1}}\). Comm. Math. Phys. 298(1), 231–264 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Struwe, M.: Equivariant wave maps in two space dimensions. Comm. Pure Appl. Math., 56(7), 815–823 (2003) (Dedicated to the memory of Jürgen K. Moser)

  33. Titchmarsh, E.C.: The theory of functions. Oxford University Press, Oxford, (1958). Reprint of the second (1939) edition.

  34. Turok N., Spergel D.: Global texture and the microwave background. Phys. Rev. Lett. 64, 2736–2739 (1990)

    Article  ADS  Google Scholar 

  35. Wall H.S.: Polynomials whose zeros have negative real parts. Amer. Math. Mon. 52, 308–322 (1945)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH, 43220, USA

    Ovidiu Costin & Irfan Glogić

  2. Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut, Endenicher Allee 60, 53115, Bonn, Germany

    Roland Donninger

  3. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Roland Donninger

Authors
  1. Ovidiu Costin
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  2. Roland Donninger
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  3. Irfan Glogić
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Corresponding author

Correspondence to Irfan Glogić.

Additional information

Communicated by W. Schlag

O. C. and I. G. were partially supported by the NSF DMS grant 1515755. R. D. is supported by a Sofja Kovalevskaja Award Granted by the Alexander von Humboldt Foundation and the German Federal Ministry of Education and Research; partial support by the DFG, CRC 1060, is also gratefully acknowledged.

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Costin, O., Donninger, R. & Glogić, I. Mode Stability of Self-Similar Wave Maps in Higher Dimensions. Commun. Math. Phys. 351, 959–972 (2017). https://doi.org/10.1007/s00220-016-2776-7

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  • DOI: https://doi.org/10.1007/s00220-016-2776-7

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