×

Asymptotic stability of harmonic maps under the Schrödinger flow. (English) Zbl 1170.35091

The paper aims to report results concerning the presence or absence of the dynamical collapse (blowup in a finite time) of finite-energy two-dimensional vortex solutions to the Landau-Lifshitz equation, which is fundamental equation governing the dynamics of local magnetization \({\mathbf u}(x,y,t)\) in ferromagnetic media: \[ \frac{\partial{\mathbf u}}{\partial t}= {\mathbf u}\times\Delta {\mathbf u}, \] where \(\Delta\) is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge \(m>0\), are looked for as \({\mathbf u}=e^{im\theta}{\mathbf v}(r)\), where \(r,\theta\) are the polar coordinates in the plane. The vortex solution decays at \(r\to\infty\), essentially, as \(r^{-m}\). First, the work produces a proof of theorems stating the local well-posedness and orbital stability of solutions close to the vortices, but only up to the moment of possible blowup (collapse) of the solutions.
The main result of the work is a theorem which states the absence of the collapse in solutions close to the vortices with \(m\geq 4\). This limitation is imposed by the necessity of a sufficiently quick decay of the unperturbed solution at \(r\to\infty\). The situation for the vortices with \(1\leq m\leq 3\), and for the zero-vorticity states, with \(m=0\), remains unknown. The proofs are based on the decomposition of the solution into the unperturbed part and dispersive perturbations, to which the so-called Strichartz estimates, following from the linearized version of the underlying equation, are applied.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I. Bejenaru, On Schrödinger maps , Amer. J. Math. 130 (2008), 1033–1065. · Zbl 1159.35065
[2] 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. · Zbl 1030.35024
[3] -, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay , Indiana Univ. Math. J. 53 (2004), 1665–1680. · Zbl 1084.35014
[4] 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. · Zbl 0765.53026
[5] N.-H. Chang, J. Shatah, and K. Uhlenbeck, Schrödinger maps , Comm. Pure Appl. Math. 53 (2000), 590–602. · Zbl 1028.35134
[6] M. Christ and A. Kiselev, Maximal functions associated to filtrations , J. Funct. Anal. 179 (2001), 409–425. · Zbl 0974.47025
[7] 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. · Zbl 1013.58007
[8] W. Ding and Y. Wang, Schrödinger flow of maps into symplectic manifolds , Sci. China Ser. A 41 (1998), 746–755. · Zbl 0918.53017
[9] -, Local Schrödinger flow into Kähler manifolds , Sci. China Ser. A 44 (2001), 1446–1464. · Zbl 1019.53032
[10] M. Grillakis and V. Stefanopoulos, Lagrangian formulation, energy estimates, and the Schrödinger map problem , Comm. Partial Differential Equations 27 (2002), 1845–1877. · Zbl 1021.35103
[11] S. Gustafson, K. Kang, and T.-P. Tsai, Schrödinger flow near harmonic maps , Comm. Pure Appl. Math. 60 (2007), 463–499. · Zbl 1144.53085
[12] A. D. Ionescu and C. E. Kenig, Low-regularity Schrödinger maps , Differential Integral Equations 19 (2006), 1271–1300. · Zbl 1212.35449
[13] 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. · Zbl 1387.35139
[14] T. Kato, Wave operators and similarity for some non-selfadjoint operators , Math. Ann. 162 (1965/1966), 258–279. · Zbl 0139.31203
[15] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Magnetic solitons , Phys. Rep. 194 (1990), 117–238.
[16] H. Mcgahagan, An approximation scheme for Schrödinger maps , Comm. Partial Differential Equations 32 (2007), 375–400. · Zbl 1122.35138
[17] 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. ; Mathematical Reviews (MathSciNet): · Zbl 1028.58018
[18] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness , Academic Press, New York, 1975. · Zbl 0308.47002
[19] -, Methods of Modern Mathematical Physics, IV: Analysis of Operators , Academic Press, New York, 1978. · Zbl 0401.47001
[20] I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials , Invent. Math. 155 (2004), 451–513. · Zbl 1063.35035
[21] I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical \(O(3)\) \(\sigma\)-model , preprint,\arxivmath/0605023v2[math.AP]
[22] 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. · Zbl 0614.35087
[23] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation , Comm. Partial Differential Equations 25 (2000), 1471–1485. · Zbl 0966.35027
[24] 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. · Zbl 1110.37056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.