Communications in Mathematical Physics

, Volume 343, Issue 1, pp 299–310 | Cite as

On the Stability of Self-Similar Solutions to Nonlinear Wave Equations

  • Ovidiu Costin
  • Roland Donninger
  • Irfan Glogić
  • Min Huang


We consider an explicit self-similar solution to an energy-supercritical Yang-Mills equation and prove its mode stability. Based on earlier work by one of the authors, we obtain a fully rigorous proof of the nonlinear stability of the self-similar blowup profile. This is a large-data result for a supercritical wave equation. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Actor A.: Classical solutions of SU(2) Yang-Mills theories. Rev. Mod. Phys. 51, 461–525 (1979)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bizoń P., Chmaj T.: Convergence towards a self-similar solution for a nonlinear wave equation: a case study. Phys. Rev. D 72(4), 045013 (2005)ADSCrossRefGoogle Scholar
  3. 3.
    Bizoń P., Ovchinnikov NY., Sigal I.M.: Collapse of an instanton. Nonlinearity 17(4), 1179–1191 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bizoń, P., Tabor, Z.: On blowup of Yang-Mills fields. Phys. Rev. D(3), 64(12), 121701, 4, (2001)Google Scholar
  5. 5.
    Bizoń P.: Formation of singularities in Yang-Mills equations. Acta Phys. Polon. B 33(7), 1893–1922 (2002)ADSMathSciNetMATHGoogle Scholar
  6. 6.
    Buslaev V.I., Buslaeva S.F.: Poincaré’s theorem on difference equations. Mat. Zametki 78(6), 943–947 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cazenave T., Shatah J., Tahvildar-Zadeh A.S.: 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)MathSciNetMATHGoogle Scholar
  8. 8.
    Costin O., Huang M., Schlag W.: On the spectral properties of L ± in three dimensions. Nonlinearity 25(1), 125–164 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Costin, O., Donninger, R., Xia, X.: A proof for the mode stability of a self-similar wave map. Preprint arXiv: arXiv:1411.2947, (2014)
  11. 11.
    Côte R., Kenig Carlos E., Merle F.: Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system. Comm. Math. Phys. 284(1), 203–225 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Donninger R.: On stable self–similar blow up for equivariant wave maps. Comm. Pure Appl. Math. 64(8), 1029–1164 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Donninger R.: Stable self-similar blowup in energy supercritical Yang-Mills theory. Math. Z. 278(3–4), 1005–1032 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Donninger R., Schörkhuber B.: Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Differ. Equ. 9(1), 63–87 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Donninger R., Schörkhuber B.: Stable blow up dynamics for energy supercritical wave equations. Trans. Am. Math. Soc. 366(4), 2167–2189 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Donninger R., Schörkhuber B., Aichelburg P.: On stable self-similar blow up for equivariant wave maps: the linearized problem. Ann. Henri Poincaré 13, 103–144 (2012). doi:10.1007/s00023-011-0125-0
  17. 17.
    Dumitraşcu O.: Equivariant solutions of the Yang-Mills equations. Stud. Cerc. Mat. 34(4), 329–333 (1982)MathSciNetMATHGoogle Scholar
  18. 18.
    Eardley Douglas M., Moncrief V.: The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties. Comm. Math. Phys. 83(2), 171–191 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Eardley Douglas M., Moncrief V.: The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof. Comm. Math. Phys. 83(2), 193–212 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Elaydi S.: An introduction to difference equations. Undergraduate Texts in Mathematics. third edition. Springer, New York (2005)Google Scholar
  21. 21.
    Gundlach, C., Martín-García, J.: Critical phenomena in gravitational collapse. Living Rev. Relat. 10(5), (2007)Google Scholar
  22. 22.
    Klainerman S., Machedon M.: Finite energy solutions of the Yang-Mills equations in R 3+1. Ann. Math. 2(142(1)), 39–119 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Klainerman S., Tataru D.: On the optimal local regularity for Yang-Mills equations in R 4+1. J. Am. Math. Soc. 12(1), 93–116 (1999)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Krieger J., Schlag W., Tataru D.: Renormalization and blow up for the critical Yang-Mills problem. Adv. Math. 221(5), 1445–1521 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    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)MathSciNetMATHGoogle Scholar
  26. 26.
    Phillips G.M.: Interpolation and approximation by polynomials. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York (2003)CrossRefGoogle Scholar
  27. 27.
    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., 1–122 (2012)Google Scholar
  28. 28.
    Stefanov A.: Global regularity for Yang-Mills fields in R 1+5. J. Hyperbolic Differ. Equ. 7(3), 433–470 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hooft, G.: 50 years of Yang-Mills theory. World Scientific Publishing Co. Pte. Ltd., Hackensack (2005)Google Scholar
  31. 31.
    Titchmarsh, E.C.: The theory of functions. Oxford University Press, Oxford (1958) (Reprint of the second (1939) edition) Google Scholar
  32. 32.
    Wall H.S.: Polynomials whose zeros have negative real parts. Am. Math. Monthly 52, 308–322 (1945)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ovidiu Costin
    • 1
  • Roland Donninger
    • 2
  • Irfan Glogić
    • 1
  • Min Huang
    • 3
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Mathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  3. 3.Department of MathematicsCity University of Hong KongKowloonHong Kong

Personalised recommendations