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Conditional Global Existence and Scattering for a Semi-Linear Skyrme Equation with Large Data

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Abstract

We study a generalization of energy super-critical wave maps due to Adkins and Nappi that can also be viewed as a simplified version of the Skyrme model. These are maps from 1 + 3 dimensional Minkowski space that take values in the 3-sphere, and it follows that every finite energy Adkins–Nappi wave map has a fixed topological degree which is an integer. Here we initiate the study of the large data dynamics for Adkins–Nappi wave maps by proving that there is no type II blow-up in the class of maps with topological degree zero. In particular, any degree zero map whose critical norm stays bounded must be global-in-time and scatter to zero as \({t \to \pm \infty}\).

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References

  1. Adkins G.S., Nappi C.R.: Stabilization of chiral solitons via vector mesons. Phys. Lett. B 137(3-4), 251–256 (1984)

    Article  ADS  Google Scholar 

  2. Bahouri P., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121, 131–175 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bizoń P., Chmaj T., Maliborski M.: Equivariant wave maps exterior to a ball. Nonlinearity 25(5), 1299–1309 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Bizoń, P., Chmaj, T., Rostworowski, A.: Asymptotic stability of the Skyrmion. Phys. Rev. D 75(12), 121702, 5 (2007)

  5. Bulut A.: Maximizers for the Strichartz inequalities for the wave equation. Differ. Integral Equ. 23(11–12), 1035–1072 (2010)

    MATH  MathSciNet  Google Scholar 

  6. Bulut, A.: The defocusing cubic nonlinear wave equation in the energy-supercritical regime. In: Recent Advances in Harmonic Analysis and Partial Differential Equations, vol. 581. Contemporary Mathematics, pp. 1–11. American Mathematical Society, Providence, RI (2012)

  7. Bulut A.: Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation. J. Funct. Anal. 263(6), 1609–1660 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Côte, R., Kenig, C., Lawrie, A., Schlag, W.: Characterization of large energy solutions of the equivariant wave map problem: I. Amer. J. Math, Preprint (to appear, 2012)

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Duyckaerts T., Kenig C., Merle F.: Classification of radial solutions of the focusing, energy critical wave equation. Cambridge J. Math. 1(1), 75–144 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Duyckaerts T., Kenig C., Merle F.: Scattering for radial, bounded solutions of focusing supercritical wave equations. Int. Math. Res. Notices 2014(1), 224–258 (2014)

    MATH  MathSciNet  Google Scholar 

  13. 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)

    Article  MATH  MathSciNet  Google Scholar 

  14. Geba D.-A., Nakanishi K., Rajeev S.G.: Global well-posedness and scattering for Skyrme wave maps. Commun. Pure Appl. Anal. 11(5), 1923–1933 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Geba, D.-A., Rajeev, S.G.: A continuity argument for a semilinear Skyrme model. Electron. J. Differ. Equ. 86, 9 (2010)

  16. Geba D.-A., Rajeev S.G.: Nonconcentration of energy for a semilinear Skyrme model. Ann. Phys. 325(12), 2697–2706 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Ionescu A.D., Pausader B., Staffilani G.: On the global well-posedness of energy-critical Schrödinger equations in curved spaces. Anal. PDE 5(4), 705–746 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 120(5), 955–980 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kenig C., Lawrie A., Schlag W.: Relaxation of wave maps exterior to a ball to harmonic maps for all data. Geom. Funct. Anal. 24(2), 610–647 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. 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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kenig C., Merle F.: Radial solutions to energy supercritical wave equations in odd dimensions. Discrete Contin. Dyn. Syst. 31(4), 1365–1381 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kenig C.E., Merle F.: Scattering for \({\dot{H}^{1/2}}\) bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc. 362(4), 1937–1962 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Lawrie A., Schlag W.: Scattering for wave maps exterior to a ball. Adv. Math. 232(1), 57–97 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Li, D.: Global global well-posedness of hedgehog solutions for the (3+1) Skyrme model. Preprint, (2012)

  29. Lindblad H., Sogge C.D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130(2), 357–426 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. McLeod J.B., Troy W.C.: The Skyrme model for nucleons under spherical symmetry. Proc. Roy. Soc. Edinburgh Sect. A 118(3–4), 271–288 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Shatah J., Shadi Tahvildar-Zadeh A.: On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47(5), 719–754 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Shen, R.: On the energy subcritical nonlinear wave equation with radial data for 3 < p < 5. Preprint, (2012)

  34. Skyrme T.H.R.: Selected Papers With Commentary of Tony Hilton Royle Skyrme, vol. 3. World Scientific, Singapore (1994)

    Google Scholar 

  35. Turok N., Spergel D.: Global texture and the microwave background. Phys. Rev. Lett. 64(23), 2736–2739 (1990)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of California, Berkeley 859 Evans Hall #3840, Berkeley, CA, 94720, USA

    Andrew Lawrie

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  1. Andrew Lawrie
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Correspondence to Andrew Lawrie.

Additional information

Communicated by W. Schlag

Support from the National Science Foundation, DMS-1302782 is gratefully acknowledged. The author would also like to thank Carlos Kenig and Wilhelm Schlag for many helpful conversations.

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Lawrie, A. Conditional Global Existence and Scattering for a Semi-Linear Skyrme Equation with Large Data. Commun. Math. Phys. 334, 1025–1081 (2015). https://doi.org/10.1007/s00220-014-2207-6

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  • DOI: https://doi.org/10.1007/s00220-014-2207-6

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