## Duke Mathematical Journal

### Asymptotic stability of harmonic maps under the Schrödinger flow

#### Abstract

For Schrödinger maps from $\mathbb{R}^2\times\mathbb{R}^+$ to the $2$-sphere $\mathbb{S}^2$, 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

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

Dates
First available: 15 December 2008

http://projecteuclid.org/euclid.dmj/1229349904

Digital Object Identifier
doi:10.1215/00127094-2008-058

Mathematical Reviews number (MathSciNet)
MR2462113

#### Citation

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

#### References

• 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.