Skip to main content
SpringerLink
Log in

Stable self-similar blowup in energy supercritical Yang–Mills theory

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in \((1+5)\)-dimensional Yang–Mills theory. A certain self-similar solution \(W_0\) of this model is conjectured to act as an attractor for generic large data evolutions. Assuming mode stability of \(W_0\), we prove a weak version of this conjecture, namely that the self-similar solution \(W_0\) is (nonlinearly) stable. Phrased differently, we prove that mode stability of \(W_0\) implies its nonlinear stability. The fact that this statement is not vacuous follows from careful numerical work by Bizoń and Chmaj that verifies the mode stability of \(W_0\) beyond reasonable doubt.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In order to avoid notational clutter we usually omit the arguments and write \(\psi \) instead of \(\psi (t,r)\).

  2. Strictly speaking, the cases \(\lambda \in \{-1,0,1\}\) require special attention since for these values of \(\lambda \) there exist two possibilities: the nonsmooth solution involves a logarithmic term or all solutions are smooth at \(\rho =1\). In either case, however, we arrive at the same conclusion as for \(\lambda \notin \{-1,0,1\}\).

  3. There exist at least five nonequivalent notions of essential spectra for nonself-adjoint operators, see [18] for a detailed discussion. We stick to the definition given by Kato [21] as the set of all \(\lambda \) such that \(\lambda -\mathbf {L}\) fails to be semi-Fredholm.

  4. Recall that global existence in the variable \(\tau \) really means local existence for the original equation in the backward lightcone \(\mathcal C_T\).

References

  1. Actor, A.: Classical solutions of SU(2) Yang-Mills theories. Rev. Mod. Phys. 51, 461–525 (1979)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  3. Bizoń, P., Ovchinnikov, Y.N., Sigal, I.M.: Collapse of an instanton. Nonlinearity 17(4), 1179–1191 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bizoń, P.: Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere. Commun. Math. Phys. 215(1), 45–56 (2000)

    Article  MATH  Google Scholar 

  5. Bizoń, P.: Formation of singularities in Yang–Mills equations. Acta Phys. Polon. B 33(7), 1893–1922 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Bizoń, P., Maison, D., Wasserman, A.: Self-similar solutions of semilinear wave equations with a focusing nonlinearity. Nonlinearity 20(9), 2061–2074 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bulut, A.: Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation (2010, preprint). arXiv:1006.4168v1

  8. Bulut, A.: The defocusing energy-supercritical cubic nonlinear wave equation in dimension five (2011, preprint). arXiv:1112.0629v1

  9. 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)

    MathSciNet  MATH  Google Scholar 

  10. Côte, R., Kenig, C.E., Merle, F.: Scattering below critical energy for the radial 4D Yang–Mills equation and for the 2D corotational wave map system. Commun. Math. Phys. 284(1), 203–225 (2008)

    Article  MATH  Google Scholar 

  11. Donninger, R.: The radial wave operator in similarity coordinates. J. Math. Phys. 51(2), 023527, 10 (2010)

  12. Donninger, R.: On stable self-similar blow up for equivariant wave maps. Commun. Pure Appl. Math. 64(8), 1029–1164 (2011)

    Article  MathSciNet  Google Scholar 

  13. Donninger, R., Schörkhuber, B.: Stable self-similar blowup for energy subcritical wave equations (2012, preprint). arXiv:1201.4337

  14. 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

    Article  MATH  Google Scholar 

  15. Dumitraşcu, O.: Equivariant solutions of the Yang–Mills equations. Stud. Cerc. Mat. 34(4), 329–333 (1982)

    MathSciNet  MATH  Google Scholar 

  16. Eardley, D.M., Moncrief, V.: The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties. Commun. Math. Phys. 83(2), 171–191 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eardley, D.M., Moncrief, V.: The global existence of Yang–Mills–Higgs fields in \(4\)-dimensional Minkowski space. II. Completion of proof. Commun. Math. Phys. 83(2), 193–212 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Edmunds, D.E., Evans, W.D.: Spectral theory and differential operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications

  19. Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer, New York, (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

  20. Gundlach, C., Martín-García, J.M.: Critical phenomena in gravitational collapse. Living Rev. Relat. 10(5), 34 (2007)

  21. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition

  22. Keel, M.: Global existence for critical power Yang–Mills–Higgs equations in \({ R}^{3+1}\). Commun. Partial Differ. Equ. 22(7–8), 1161–1225 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Kenig, C.E., Merle, F.: Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications. Am. J. Math. 133(4), 1029–1065 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kenig, C.E., Merle, F.: Radial solutions to energy supercritical wave equations in odd dimensions. Disc. Cont. Dyn. Sys. A 4, 1365–1381 (2011)

    MathSciNet  Google Scholar 

  25. Killip, R., Visan, M.: The defocusing energy-supercritical nonlinear wave equation in three space dimensions. Trans. Am. Math. Soc. 363(7), 3893–3934 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Killip, R., Visan, M.: The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions. Proc. Am. Math. Soc. 139(5), 1805–1817 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Klainerman, S., Machedon, M.: Finite energy solutions of the Yang-Mills equations in \( {R}^{3+1}\). Ann. of Math. 142(1), 39–119 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Klainerman, S., Tataru, D.: On the optimal local regularity for Yang–Mills equations in \({\bf R}^{4+1}\). J. Am. Math. Soc. 12(1), 93–116 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for the critical Yang–Mills problem. Adv. Math. 221(5), 1445–1521 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Krieger, J., Sterbenz, J.: Global regularity for the Yang-Mills equations on high dimensional Minkowski space (2005, preprint). arXiv:math/0503382v1

  31. Linhart, J.M., Sadun, L.A.: Fast and slow blowup in the \(S^2\) \(\sigma \)-model and the (4+1)-dimensional Yang-Mills model. Nonlinearity 15(2), 219–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Merle, F., Zaag, H.: Determination of the blow-up rate for the semilinear wave equation. Am. J. Math. 125(5), 1147–1164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Merle, F., Zaag, H.: Determination of the blow-up rate for a critical semilinear wave equation. Math. Ann. 331(2), 395–416 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Merle, F., Zaag, H.: Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension. J. Funct. Anal. 253(1), 43–121 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Raphaël, P. Rodnianski, I.: Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang–Mills problems. (2009 preprint). arXiv:0911.0692

  36. Shatah, J.: Weak solutions and development of singularities of the SU(2) \(\sigma \)-model. Commun. Pure Appl. Math. 41(4), 459–469 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stefanov, A.: Global regularity for Yang-Mills fields in \({\bf R}^{1+5}\). J. Hyperbolic Differ. Equ. 7(3), 433–470 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  MATH  Google Scholar 

  39. ’t Hooft, G. (ed.): 50 years of Yang–Mills theory. World Scientific Publishing Co., Pte. Ltd., Hackensack (2005)

  40. Tao, T.: Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J. Differ. Equ. 189(2), 366–382 (2003)

    Article  MATH  Google Scholar 

  41. Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil 1. Mathematische Leitfäden. [Mathematical Textbooks]. B. G. Teubner, Stuttgart, (2000). Grundlagen. [Foundations]

Download references

Author information

Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne, MA B1 487, Station 8, 1015 , Lausanne, Switzerland

    Roland Donninger

Authors
  1. Roland Donninger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Donninger.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Donninger, R. Stable self-similar blowup in energy supercritical Yang–Mills theory. Math. Z. 278, 1005–1032 (2014). https://doi.org/10.1007/s00209-014-1344-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-014-1344-0

Keywords

Search

Navigation