Skip to main content
SpringerLink
Log in

Global solutions of nonlinear wave equations with large data

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

In this paper, we give a criterion on the Cauchy data for the semilinear wave equations satisfying the null condition in \({{\mathbb R}}^+\times {\mathbb {R}}^{3}\) such that the data can be arbitrarily large, while the solution is still globally in time in the future.

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

References

  1. Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39(2), 267–282 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Christodoulou, D.: The Formation of Black Holes in General Relativity. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2009)

    Book  Google Scholar 

  3. Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space, volume 41 of Princeton Mathematical Series. Princeton University Press, Princeton (1993)

    Google Scholar 

  4. Dafermos, M., Rodnianski, I.: A new physical-space approach to decay for the wave equation with applications to black hole spacetimes. In: XVIth International Congress on Mathematical Physics, pp. 421–432. World Scientific Publishing, Hackensack (2010)

  5. Keel, M., Smith, H., Sogge, C.: Global existence for a quasilinear wave equation outside of star-shaped domains. J. Funct. Anal. 189(1), 155–226 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33(1), 43–101 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Klainerman, S.: Long time behaviour of solutions to nonlinear wave equations. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Warsaw, 1983), pp. 1209–1215, Warsaw, PWN (1984)

  8. Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear systems of partial differential equations in applied mathematics, Part 1 (1984), vol. 23 of Lectures in Applied Mathematics, pp. 293–326. American Mathematical Society, Providence (1986)

  9. Klainerman, S., Luk, J., Rodnianski, I.: A fully anisotropic mechanism for formation of trapped surfaces in vacuum. Invent. Math. 198(1), 1–26 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Klainerman, S., Rodnianski, I.: On the formation of trapped surfaces. Acta Math. 208(2), 211–333 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Klainerman, S., Sideris, T.: On almost global existence for nonrelativistic wave equations in 3D. Commun. Pure Appl. Math. 49(3), 307–321 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krieger, J., Schlag, W.: Concentration compactness for critical wave maps. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2012)

    Book  Google Scholar 

  13. 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  MathSciNet  MATH  Google Scholar 

  14. Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. (2) 171(3), 1401–1477 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luk, J.: The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes. J. Eur. Math. Soc. (JEMS) 15(5), 1629–1700 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luk, J., Rodnianski, I.: Local Propagation of Impulsive Gravitational Waves. arXiv:1209.1130 (2012)

  17. Luk, J., Rodnianski, I.: Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations. arXiv:1301.1072 (2013)

  18. Metcalfe, J., Nakamura, M., Sogge, C.D.: Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains. Forum Math. 17(1), 133–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Metcalfe, J., Sogge, C.: Global existence of null-form wave equations in exterior domains. Math. Z. 256(3), 521–549 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

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

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  23. Sideris, T., Tu, S.-Y.: Global existence for systems of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal. 33(2), 477–488 (2001). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sogge, C.: Global existence for nonlinear wave equations with multiple speeds. In: Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), vol. 320 Contemporary Mathematics, pp. 353–366. American Mathematical Society, Providence (2003)

  25. Sterbenz, J., Tataru, D.: Energy dispersed large data wave maps in \(2+1\) dimensions. Commun. Math. Phys. 298(1), 139–230 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sterbenz, J., Tataru, D.: Regularity of wave-maps in dimension \(2+1\). Commun. Math. Phys. 298(1), 231–264 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Tao, T.: Global regularity of wave maps VII. Control of delocalised or dispersed solutions arXiv:0908.0776 (2009)

  29. Wang, C., Yu, X.: Global existence of null-form wave equations on small asymptotically Euclidean manifolds. J. Funct. Anal. 266(9), 5676–5708 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, J., Yu, P.: A Large Data Regime for non-linear Wave Equations. arXiv:1210.2056 (2012)

  31. Wang, J., Yu, P.: Long time solutions for wave maps with large data. J. Hyperbolic Differ. Equ. 10(2), 371–414 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, S.: Global stability of large solutions to nonlinear wave equations. arXiv: 1205.4216 (2012)

  33. Yang, S.: Global solutions of nonlinear wave equations in time dependent inhomogeneous media. Arch. Ration. Mech. Anal. 209(2), 683–728 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, S.: On the quasilinear wave equations in time dependent inhomogeneous media. arXiv:1312.7264 (2013)

  35. Yu, P: Dynamical formation of black holes due to the condensation of matter field. arXiv:1105.5898 (2011)

  36. Yu, P.: Energy estimates and gravitational collapse. Commun. Math. Phys. 317(2), 273–316 (2013)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank Igor Rodnianski for his consistent encouragement on this problem. He also thanks Pin Yu for helpful discussions. The author thanks the anonymous referee for providing helpful suggestions on improving the manuscript.

Author information

Authors and Affiliations

  1. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Shiwu Yang

Authors
  1. Shiwu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiwu Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, S. Global solutions of nonlinear wave equations with large data. Sel. Math. New Ser. 21, 1405–1427 (2015). https://doi.org/10.1007/s00029-014-0176-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-014-0176-4

Keywords

Mathematics Subject Classification

Search

Navigation