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Inventiones mathematicae

, Volume 213, Issue 3, pp 1249–1325 | Cite as

Two-bubble dynamics for threshold solutions to the wave maps equation

  • Jacek Jendrej
  • Andrew Lawrie
Article

Abstract

We consider the energy-critical wave maps equation \(\mathbb {R}^{1+2} \rightarrow \mathbb {S}^2\) in the equivariant case, with equivariance degree \(k \ge 2\). It is known that initial data of energy \(< 8\pi k\) and topological degree zero leads to global solutions that scatter in both time directions. We consider the threshold case of energy \(8 \pi k \). We prove that the solution is defined for all time and either scatters in both time directions, or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction. In the latter case, we describe the asymptotic behavior of the scales of the two harmonic maps. The proof combines the classical concentration-compactness techniques of Kenig–Merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation.

Notes

Acknowledgements

J. Jendrej was supported by the ERC Grant 291214 BLOWDISOL and by the NSF Grant DMS-1463746. This work was completed during his postdoc at the University of Chicago. A. Lawrie was supported by NSF Grant DMS-1700127. We would like to thank Raphaël Côte for many helpful discussions. And lastly, we would like to thank the anonymous referees for their careful reading of an earlier version of the manuscript and for suggesting substantial improvements.

References

  1. 1.
    Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S.: Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential. J. Funct. Anal. 203(2), 519–549 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burq, N., Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer, New York (1982)Google Scholar
  5. 5.
    Christodoulou, D., Tahvildar-Zadeh, A.S.: On the asymptotic behavior of spherically symmetric wave maps. Duke Math. J. 71(1), 31–69 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christodoulou, D., Tahvildar-Zadeh, A.S.: On the regularity of spherically symmetric wave maps. Commun. Pure Appl. Math. 46(7), 1041–1091 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Côte, R.: Instability of nonconstant harmonic maps for the \((1+2)\)-dimensional equivariant wave map system. Int. Math. Res. Not. 57, 3525–3549 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Côte, R.: On the soliton resolution for equivariant wave maps to the sphere. Commun. Pure Appl. Math. 68(11), 1946–2004 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Côte, R., Kenig, C., Lawrie, A., Schlag, W.: Characterization of large energy solutions of the equivariant wave map problem: I. Am. J. Math. 137(1), 139–207 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Côte, R., Kenig, C., Lawrie, A., Schlag, W.: Characterization of large energy solutions of the equivariant wave map problem: II. Am. J. Math. 137(1), 209–250 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Côte, R., Kenig, C.E., Schlag, W.: Energy partition for the linear radial wave equation. Math. Ann. 358(3–4), 573–607 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duyckaerts, T., Jia, H., Kenig, C., Merle, F.: Universality of blow up profile for small blow up solutions to the energy critical wave map equation (2016). arXiv:1612.04927
  13. 13.
    Duyckaerts, T., Jia, H., Kenig, C.E., Merle, F.: Soliton resolution along a sequence of times for the focusing energy critical wave equation. Geom. Funct. Anal. 27(4), 798–862 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Duyckaerts, T., Kenig, C., Merle, F.: Universality of the blow-up profile for small radial type II blow-up solutions of the energy critical wave equation. J. Eur. Math. Soc. (JEMS) 13(3), 533–599 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duyckaerts, T., Kenig, C., Merle, F.: Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal. 22(3), 639–698 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Duyckaerts, T., Kenig, C., Merle, F.: Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case. J. Eur. Math. Soc. (JEMS) 14(5), 1389–1454 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Duyckaerts, T., Kenig, C., Merle, F.: Classification of radial solutions of the focusing, energy critical wave equation. Camb. J. Math. 1(1), 75–144 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Duyckaerts, T., Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. 2008, rpn2002 (2008). https://doi.org/10.1093/imrp/rpn002
  19. 19.
    Duyckaerts, T., Merle, F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18(6), 1787–1840 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eells, J., Wood, J.C.: Restrictions on harmonic maps of surfaces. Topology 15(3), 263–266 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grinis, R.: Quantization of time-like energy for wave maps into spheres. Commun. Math. Phys. 352(2), 641–702 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jendrej, J.: Construction of two-bubble solutions for energy-critical wave equations. Am. J. Math. (to appear). arXiv:1602.06524
  23. 23.
    Jendrej, J.: Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation. Ann. Sci. Norm. Super. Pisa Cl. Sci. (to appear). arXiv:1510.03965
  24. 24.
    Jendrej, J.: Construction of two-bubble solutions for the energy-critical NLS. Anal. PDE 10(8), 1923–1959 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Jia, H., Kenig, C.: Asymptotic decomposition for semilinear wave and equivariant wave map equations. Am. J. Math. 139(6), 1521–1603 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46(9), 1221–1268 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Int. Math. Res. Not. 1994(9), 383–389 (1994).  https://doi.org/10.1155/S1073792894000425 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Duke Math. J. 81(1), 99–133 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Klainerman, S., Machedon, M.: On the regularity properties of a model problem related to wave maps. Duke Math. J. 87(3), 553–589 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Klainerman, S., Selberg, S.: Remark on the optimal regularity for equations of wave maps type. Commun. Partial Differ. Equ. 22(5–6), 901–918 (1997)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Klainerman, S., Selberg, S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2), 223–295 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Krieger, J.: Global regularity of wave maps from \({ R}^{2+1}\) to \(H^2\). Small energy. Commun. Math. Phys. 250(3), 507–580 (2004)CrossRefzbMATHGoogle Scholar
  35. 35.
    Krieger, J.: On stability of type II blow up for the critical NLW on \(\mathbb{R}^3\) (2017). arXiv:1705.03907
  36. 36.
    Krieger, J., Nakanishi, K., Schlag, W.: Global dynamics away from the ground state for the energy-critical nonlinear wave equation. Am. J. Math. 135(4), 935–965 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Krieger, J., Nakanishi, K., Schlag, W.: Center-stable manifold of the ground state in the energy space for the critical wave equation. Math. Ann. 361(1–2), 1–50 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Krieger, J., Schlag, W.: Concentration Compactness for Critical Wave Maps. EMS Monographs. European Mathematical Society, Zürich (2012)CrossRefzbMATHGoogle Scholar
  39. 39.
    Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for charge one equivariant wave critical wave maps. Invent. Math. 171(3), 543–615 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lawrie, A., Oh, S.-J.: A refined threshold theorem for \((1+2)\)-dimensional wave maps into surfaces. Commun. Math. Phys. 342(3), 989–999 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Martel, Y., Merle, F.: Description of two soliton collision for the quartic gKdV equation. Ann. Math. (2) 174(2), 757–857 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Martel, Y., Merle, F.: Inelastic interaction of nearly equal solitons for the quartic gKdV equation. Invent. Math. 183(3), 563–648 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Planchon, F., Stalker, J.G., Tahvildar-Zadeh, A.S.: \(L^p\) estimates for the wave equation with the inverse-square potential. Discrete Cont. Dyn. Syst. 9(2), 427–442 (2003)zbMATHGoogle Scholar
  44. 44.
    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. 115, 1–122 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Raphaël, P., Szeftel, J.: Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous \({L}^2\)-critical NLS. J. Am. Math. Soc. 24(2), 471–546 (2011)CrossRefzbMATHGoogle Scholar
  46. 46.
    Rodnianski, I., Sterbenz, J.: On the formation of singularities in the critical \({O}(3)\) \(\sigma \)-model. Ann. Math. 172, 187–242 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1–24 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Shatah, J., Tahvildar-Zadeh, A.: Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds. Commun. Pure Appl. Math. 45(8), 947–971 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Shatah, J., Tahvildar-Zadeh, A.S.: On the Cauchy problem for equivariant wave maps. Commun. Pure Appl. Math. 47(5), 719–754 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Sterbenz, J., Tataru, D.: Energy dispersed large data wave maps in \(2+1\) dimensions. Commun. Math. Phys. 1, 139–230 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Sterbenz, J., Tataru, D.: Regularity of wave maps in \(2+1\) dimensions. Commun. Math. Phys. 1, 231–264 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Struwe, M.: Equivariant wave maps in two space dimensions. Commun. Pure Appl. Math. 56(7), 815–823 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Tao, T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Int. Math. Res. Not. 6, 299–328 (2001)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Tao, T.: Global regularity of wave maps II: small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Tao, T.: Global regularity of wave maps III–VII (2008–2009). arXiv:0805.4666, arXiv:0806.3592, arXiv:0808.0368, arXiv:0906.2833, arXiv:0908.0776
  56. 56.
    Tataru, D.: Local and global results for wave maps. I. Commun. Partial Differ. Equ. 23(9–10), 1781–1793 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Tataru, D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LAGA, UMR 7539CNRS and Université Paris 13VilletaneuseFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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