zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 ${\bold u}(x,y,t)$ in ferromagnetic media: $$ \frac{\partial{\bold u}}{\partial t}= {\bold u}\times\Delta {\bold u}, $$ where $\Delta$ is the Laplacian (two-dimensional, in the present case). Vortex solutions, with integer topological charge $m>0$, are looked for as ${\bold u}=e^{im\theta}{\bold v}(r)$, 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:
35Q55NLS-like (nonlinear Schrödinger) equations
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B35Stability of solutions of PDE
WorldCat.org
Full Text: DOI arXiv
References:
[1] I. Bejenaru, On Schrödinger maps , Amer. J. Math. 130 (2008), 1033--1065. · Zbl 1159.35065 · doi:10.1353/ajm.0.0014
[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 · doi:10.1016/S0022-1236(03)00238-6
[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 · doi:10.1512/iumj.2004.53.2541
[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 · doi:10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R
[6] M. Christ and A. Kiselev, Maximal functions associated to filtrations , J. Funct. Anal. 179 (2001), 409--425. · Zbl 0974.47025 · doi:10.1006/jfan.2000.3687
[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 · doi:10.1007/BF02901957
[9] -, Local Schrödinger flow into Kähler manifolds , Sci. China Ser. A 44 (2001), 1446--1464. · Zbl 1019.53032 · doi:10.1007/BF02877074
[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 · doi:10.1081/PDE-120016130
[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 · doi:10.1002/cpa.20143
[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 05150098 · doi:10.1080/03605300600910332
[14] T. Kato, Wave operators and similarity for some non-selfadjoint operators , Math. Ann. 162 (1965/1966), 258--279. · Zbl 0139.31203 · doi:10.1007/BF01360915 · eudml:161339
[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 · doi:10.1080/03605300600856758
[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 · doi:10.1002/cpa.10054 · http://www.ams.org/mathscinet-getitem?mr=2038118
[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 · doi:10.1007/s00222-003-0325-4
[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 · doi:10.1007/BF01220998
[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 · doi:10.1080/03605300008821556
[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