Advertisement

Annales Henri Poincaré

, Volume 13, Issue 1, pp 103–144 | Cite as

On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

  • Roland Donninger
  • Birgit Schörkhuber
  • Peter C. Aichelburg
Article

Abstract

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f 0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f 0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f 0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than \({\frac{1}{2}}\). The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

Keywords

Cauchy Problem Minkowski Space Unstable Mode Global Regularity Regular Singular Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 65. Academic Press (a subsidiary of Harcourt Brace Jovanovich, Publishers), New York (1975)Google Scholar
  2. 2.
    Bizoń P.: An unusual eigenvalue problem. Acta Phys. Polon. B 36(1), 5–15 (2005)MathSciNetADSzbMATHGoogle Scholar
  3. 3.
    Bizoń P., Chmaj T., Rostworowski A., Zajac S.: Late-time tails of wave maps coupled to gravity. Class Quantum Gravity 26(22), 225015 (2009)ADSCrossRefGoogle Scholar
  4. 4.
    Bizoń P., Chmaj T., Tabor Z.: Dispersion and collapse of wave maps. Nonlinearity 13(4), 1411–1423 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Carstea, C.: A construction of blow up solutions for co-rotational wave maps. Preprint arXiv:0908.1201v1 (2009)Google Scholar
  6. 6.
    Cazenave T., Shatah J., Shadi T.-Z.A.: 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)zbMATHGoogle Scholar
  7. 7.
    Christodoulou D., Shadi T.-Z.A.: On the asymptotic behavior of spherically symmetric wave maps. Duke Math. J. 71(1), 31–69 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Christodoulou D., Shadi T.-Z.A.: On the regularity of spherically symmetric wave maps. Commun. Pure Appl. Math. 46(7), 1041–1091 (1993)zbMATHCrossRefGoogle Scholar
  9. 9.
    Clément G., Fabbri A.: The cosmological gravitating σ model: solitons and black holes. Class Quantum Gravity 17(13), 2537–2545 (2000)ADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Donninger, R.: Perturbation analysis of self-similar solutions of the SU(2) sigma-model on Minkowski spacetime. Master thesis, University of Vienna (2006)Google Scholar
  11. 11.
    Donninger, R.: On stable self-similar blow up for equivariant wave maps. Commun. Pure Appl. Math. 64(8) (2011)Google Scholar
  12. 12.
    Donninger R., Aichelburg P.C.: On the mode stability of a self-similar wave map. J. Math. Phys. 49(4), 043515–043519 (2008)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Donninger R., Aichelburg P.C.: Spectral properties and linear stability of self-similar wave maps. J. Hyperbolic Differ. Equ. 6(2), 359–370 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Donninger R., Aichelburg P.C.: A note on the eigenvalues for equivariant maps of the SU(2) sigma-model. Appl. Math. Comp. Sci. 1(1), 73–82 (2010)zbMATHGoogle Scholar
  15. 15.
    Engel K.-J., Nagel R.: One-parameter semigroups for linear evolution equations. In: Brendle, S., Campiti, M., Hahn, T., Metafune, G., Nickel, G., Pallara, D., Perazzoli, C., Rhandi, A., Romanelli, S., Schnaubelt, R. (eds) Graduate Texts in Mathematics, vol. 194, Springer, New York (2000)Google Scholar
  16. 16.
    Freire A., Müller S., Struwe M.: Weak convergence of wave maps from (1 + 2)-dimensional Minkowski space to Riemannian manifolds. Invent. Math. 130(3), 589–617 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Gell-Mann M., Lévy M.: The axial vector current in beta decay. Nuovo Cimento 16(10), 705–726 (1960)zbMATHGoogle Scholar
  18. 18.
    Helffer, B., Sjöstrand, J.: From resolvent bounds to semigroup bounds. Preprint arXiv:1001.4171 (2010)Google Scholar
  19. 19.
    Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin. Reprint of the 1980 edition (1995)Google Scholar
  20. 20.
    Keel M., Tao T.: Local and global well-posedness of wave maps on R 1+1 for rough data. Int. Math. Res. Not. 21, 1117–1156 (1998)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Klainerman S., Rodnianski I.: On the global regularity of wave maps in the critical Sobolev norm. Int. Math. Res. Not. 13, 655–677 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Klainerman S., Selberg S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2), 223–295 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Krieger, J.: Global regularity and singularity development for wave maps. In: Surveys in differential geometry. Geometric flows, vol. XII, pp. 167–201. International Press, Somerville (2008)Google Scholar
  24. 24.
    Krieger, J., Schlag, W.: Concentration compactness for critical wave maps. Preprint arXiv:0908.2474v1 (2009)Google Scholar
  25. 25.
    Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Krieger J.: Global regularity of wave maps from R 3+1 to surfaces. Commun. Math. Phys. 238(1–2), 333–366 (2003)MathSciNetADSzbMATHGoogle Scholar
  27. 27.
    Krieger J.: Global regularity of wave maps from R 1+2 to H 2. Small energy. Commun. Math. Phys. 250(3), 507–580 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Lechner C., Husa S., Aichelburg P.C.: Su(2) cosmological solitons. Phys. Rev. D 62(4), 044047 (2000)ADSCrossRefGoogle Scholar
  29. 29.
    Liebling S.L., Hirschmann E.W., Isenberg J.: Critical phenomena in nonlinear sigma models. J. Math. Phys. 41(8), 5691–5700 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Miller, P.D.: Applied asymptotic analysis. Graduate Studies in Mathematics, vol. 75. American Mathematical Society, Providence (2006)Google Scholar
  31. 31.
    Misner C.W.: Harmonic maps as models for physical theories. Phys. Rev. D (3) 18(12), 4510–4524 (1978)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Nahmod, A.: On global existence of wave maps with critical regularity. In: Surveys in differential geometry, vol. VIII (Boston, MA, 2002), pp. 307–335. International Press, Somerville (2003)Google Scholar
  33. 33.
    Nahmod A., Stefanov A., Uhlenbeck K.: On the well-posedness of the wave map problem in high dimensions. Commun. Anal. Geom. 11(1), 49–83 (2003)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Rodnianski, I., Raphaël, P.: Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills problems. Preprint arXiv:0911.0692v1 (2009)Google Scholar
  35. 35.
    Rodnianski, I., Sterbenz, J.: On the Formation of Singularities in the Critical O(3) Sigma-Model. Preprint arXiv:math/0605023v3 (2006)Google Scholar
  36. 36.
    Shatah J.: Weak solutions and development of singularities of the SU(2) σ-model. Commun. Pure Appl. Math. 41(4), 459–469 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Shatah, J., Struwe, M.: Geometric wave equations. In: Courant Lecture Notes in Mathematics, vol. 2. New York University Courant Institute of Mathematical Sciences, New York (1998)Google Scholar
  38. 38.
    Shatah J., Struwe M.: The Cauchy problem for wave maps. Int. Math. Res. Not. 11, 555–571 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Shatah J., Shadi Tahvildar-Zadeh A.: On the Cauchy problem for equivariant wave maps. Commun. Pure Appl. Math. 47(5), 719–754 (1994)zbMATHCrossRefGoogle Scholar
  40. 40.
    Sideris T.C.: Global existence of harmonic maps in Minkowski space. Commun. Pure Appl. Math. 42(1), 1–13 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Struwe M.: Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality. Commun. Pure Appl. Math. 52(9), 1179–1188 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Struwe M.: Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to the sphere. Math. Z. 242(3), 407–414 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Struwe M.: Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to general targets. Calc. Var. Partial Differ. Equ. 16(4), 431–437 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Tao, T.: Global regularity of wave maps III–VII. Preprints (2008–2009)Google Scholar
  46. 46.
    Tao T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Int. Math. Res. Not. 6, 299–328 (2001)Google Scholar
  47. 47.
    Tao T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001)ADSzbMATHCrossRefGoogle Scholar
  48. 48.
    Tataru, D., Sterbenz, J.: Energy dispersed large data wave maps in 2+1 dimensions. Preprint arXiv:0906.3384 (2009)Google Scholar
  49. 49.
    Tataru, D., Sterbenz, J.: Regularity of wave-maps in dimension 2+1. Preprint arXiv:0907.3148 (2009)Google Scholar
  50. 50.
    Tataru D.: Local and global results for wave maps. I. Commun. Partial Differ. Equ. 23(9–10), 1781–1793 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Tataru D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Tataru D.: Rough solutions for the wave maps equation. Am. J. Math. 127(2), 293–377 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Turok N., Spergel D.: Global texture and the microwave background. Phys. Rev. Lett. 64(23), 2736–2739 (1990)ADSCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Roland Donninger
    • 1
  • Birgit Schörkhuber
    • 2
  • Peter C. Aichelburg
    • 3
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Faculty of Mathematics and Geoinformation, Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria
  3. 3.Fakultät für Physik, Gravitational PhysicsUniversität WienWienAustria

Personalised recommendations