Duke Mathematical Journal

Asymptotic stability of harmonic maps under the Schrödinger flow

Stephen Gustafson, Kyungkeun Kang, and Tai-Peng Tsai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For Schrödinger maps from R2×R+ to the 2-sphere S2, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space--dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map

Article information

Duke Math. J., Volume 145, Number 3 (2008), 537-583.

First available in Project Euclid: 15 December 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions


Gustafson, Stephen; Kang, Kyungkeun; Tsai, Tai-Peng. Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Math. J. 145 (2008), no. 3, 537--583. doi:10.1215/00127094-2008-058. https://projecteuclid.org/euclid.dmj/1229349904

Export citation


  • I. Bejenaru, On Schrödinger maps, Amer. J. Math. 130 (2008), 1033--1065.
  • N. Burq, F. Planchon, J. G. Stalker, and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), 519--549.
  • —, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J. 53 (2004), 1665--1680.
  • K.-C. Chang, W. Y. Ding, and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), 507--515.
  • N.-H. Chang, J. Shatah, and K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), 590--602.
  • M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409--425.
  • W. Ding, ``On the Schrödinger flows'' in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 283--291.
  • W. Ding and Y. Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A 41 (1998), 746--755.
  • —, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A 44 (2001), 1446--1464.
  • M. Grillakis and V. Stefanopoulos, Lagrangian formulation, energy estimates, and the Schrödinger map problem, Comm. Partial Differential Equations 27 (2002), 1845--1877.
  • S. Gustafson, K. Kang, and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math. 60 (2007), 463--499.
  • A. D. Ionescu and C. E. Kenig, Low-regularity Schrödinger maps, Differential Integral Equations 19 (2006), 1271--1300.
  • J. Kato and H. Koch, Uniqueness of the modified Schrödinger map in $H^3/4 + \epsilon(\R^2)$, Comm. Partial Differential Equations 32 (2007), 415--429.
  • T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/1966), 258--279.
  • A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Magnetic solitons, Phys. Rep. 194 (1990), 117--238.
  • H. Mcgahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations 32 (2007), 375--400.
  • A. Nahmod, A. Stefanov, and K. Uhlenbeck, On Schrödinger maps, Comm. Pure Appl. Math. 56 (2003), 114--151.; Erratum, Comm. Pure Appl. Math. 57 (2004), 833--839. \!;
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
  • —, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978.
  • I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), 451--513.
  • I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical $O(3)$ $\sigma$-model, preprint,\arxivmath/0605023v2[math.AP]
  • P.-L. Sulem, C. Sulem, and C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), 431--454.
  • T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), 1471--1485.
  • C.-L. Terng and K. Uhlenbeck, ``Schrödinger flows on Grassmannians'' in Integrable Systems, Geometry, and Topology, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, 2006, 235--256.