Advertisement

Communications in Mathematical Physics

, Volume 359, Issue 1, pp 375–426 | Cite as

Soliton Resolution for Equivariant Wave Maps on a Wormhole

  • Casey Rodriguez
Article

Abstract

We study finite energy \({\ell}\)–equivariant wave maps from the (1+3)–dimensional spacetime \({\mathbb{R} \times (\mathbb{R} \times \mathbb{S}^2) \rightarrow \mathbb{S}^3}\) where the metric on \({\mathbb{R} \times (\mathbb{R} \times \mathbb{S}^2)}\) is given by
$$ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d\varphi^2 \right ), \quad t,r \in \mathbb{R}, (\theta,\varphi) \in \mathbb{S}^2.$$
The constant time slices are each given by a Riemannian manifold with two asymptotically Euclidean ends at \({r = \pm \infty}\) that are connected by a 2–sphere at r =  0. The spacetime \({\mathbb{R} \times (\mathbb{R} \times \mathbb{S}^2)}\) has appeared in the general relativity literature as a prototype wormhole geometry (but is not expected to exist in nature). Each \({\ell}\)–equivariant finite energy wave map can be indexed by its topological degree n. For each \({\ell}\) and n, there exists a unique, linearly stable energy minimizing \({\ell}\)–equivariant harmonic map \({Q_{\ell,n} : \mathbb{R} \times \mathbb{S}^2 \rightarrow \mathbb{S}^3}\) of degree n. In this work, we prove the soliton resolution conjecture for this model. More precisely, we show that modulo a free radiation term every \({\ell}\)–equivariant wave map of degree n converges strongly to \({Q_{\ell,n}}\). This fully resolves a conjecture made by Bizon and Kahl. Previous work by the author proved this result for the corotational case \({\ell = 1}\) and established many preliminary results that are used in the current work.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bizon P., Chmaj T, Maliborski M.: Equivariant wave maps exterior to a ball. Nonlinearity 5(25), 1299–1309 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bizon P., Kahl M.: Wave maps on a wormhole. Phys. Rev. D 91, 065003 (2015)ADSCrossRefGoogle Scholar
  3. 3.
    Bulut A., Czubak M., Li D., Pavlovi N., Pavlovi N.: Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions. Comm. Partial Differ. Equ. 38(4), 575–607 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coulhon T., Russ E., Russ E.: Sobolev algebras on Lie groups and Riemannian manifolds. Am. J. Math. 123(2), 283–342 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    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 Physicss, pp. 421–432. World Sci. Publ., Hackensack (2010)Google Scholar
  7. 7.
    Duyckaerts T., Kenig C., Merle F.: Classification of radial solutions of the focusing, energy critical wave equation. Camb. J. Math. 1(1), 74–144 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Franklin, P., James, O., Thorne, K.S., von Tunzelmann, E.: Visualizing Interstellar’s wormhole. Am. J. Phys. 83 (2015). doi: 10.1119/1.4916949
  9. 9.
    Hidano K., Metcalfe J., Smith Hart F., Sogge Christopher D., Zhou Y.: On abstract Strichartz estimates and the Strauss conjecture for nontrapping obstacles. Trans. Am. Math. Soc. 362(5), 2789–2809 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kenig C.E., 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kenig C.E., 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
  12. 12.
    Kenig C.E., 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kenig C., Lawrie A., Liu B., Schlag W.: Channels of energy for the linear radial wave equation. Adv. Math. 285, 877–936 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kenig C., Lawrie A., Liu B., Schlag W.: Stable soliton resolution for exterior wave maps in all equivariance classes. Adv. Math. 285, 235–300 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lawrie A., Schlag W.: Scattering for wave maps exterior to a ball. Adv. Math. 232, 57–97 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McLeod J.B., Troy W.C.: The Skyrme model for nucleons under spherical symmetry. Proc. Roy. Soc. Edinb. Sect. A 118(3–4), 271–288 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Morris M.S., Thorne K. S.: Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys. 56(5), 395–412 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rodriguez C.: Profiles for the radial focusing energy-critical wave equation in odd dimensions. Adv. Differ. Equ. 21(5-6), 505–570 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rodriguez, C.: Soliton resolution for corotational wave maps on a wormhole. IMRN (2016). arXiv:1609.08477
  20. 20.
    Schlag, W.: Semilinear wave equations. In: ICM Proceedings (2014)Google Scholar
  21. 21.
    Shatah, J., Struwe, M.: Geometric wave equations. In: Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, viii+153 pp. (1998)Google Scholar
  22. 22.
    Yang, S.: On global behavior of solutions of the Maxwell–Klein–Gordon equations. (2015). arXiv:1511.00250
  23. 23.
    Yang, S.: Decay of solutions of Maxwell–Klein–Gordon equations with arbitrary Maxwell field. Anal. PDE 9(8), 1829–1902 (2016)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations