Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on \(\mathbb R^2\). (English) Zbl 1205.35294

The paper considers the Landau-Lifshitz (LL) equation which governs the evolution of the local magnetization in the 2D space. The equation also includes a dissipative term. It is known that various configurations of the 3D magnetization vector with a fixed length in the 2D space, i.e., the map of the 2D space into the 2D sphere, may be classified by an integer index \(m\) (“degree”). The energy of each subclass of the solutions, pertaining to a particular value of \(n\), is limited from below by the minimum value \(4\pi m\). In previous works, it was demonstrated that the solutions of the 2D LL equation with energies taken close enough to the \(m\)-dependent minimum are stable for \(m\geq 4\). In the present work, the same statement is proved for \(m=3\). The analysis is also extended for more particular (not most generic) types of the solutions pertaining to \(m=2\), which is much harder for the consideration, as in this case, it is difficult to decompose the general solution into an essential part and dispersing perturbations. In the present work, this is done by means of a specially devised nonorthogonal decomposition. It is concluded that the solutions for \(m=2\) may demonstrate different types of the dynamical behavior: asymptotic stability, collapse (the formation of a singularity), and persistent oscillations.


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
82D40 Statistical mechanics of magnetic materials
35B45 A priori estimates in context of PDEs
35B35 Stability in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
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[1] Angenent, S., Hulshof, J.: Singularities at t = in equivariant harmonic map flow. Contemp. Math. 367, Geometric evolution equations, Providence, RI: Amer. Math. Soc., 2005, pp. 1–15 · Zbl 1075.53057
[2] Bejenaru, I., Ionescu, A., Kenig, C., Tataru, D.: Global Schrödinger maps in dimensions d 2: small data in the critical Sobolev spaces. http://arxiv.org/abs/0807.0265v1 [math.AP], 2008 · Zbl 1233.35112
[3] Bergh J., Löfström J.: Interpolation spaces. Springer-Verlag, Berlin-Heidelberg-New York (1976) · Zbl 0344.46071
[4] Burq N., Planchon F., Stalker J., Tahvildar-Zadeh S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Ind. U. Math. J 53(6), 519–549 (2004) · Zbl 1030.35024
[5] Chang K.-C., Ding W.Y., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Diff. Geom. 36(2), 507–515 (1992) · Zbl 0765.53026
[6] Chang N.-H., Shatah J., Uhlenbeck K.: Schrödinger maps. Comm. Pure Appl. Math. 53(5), 590–602 (2000) · Zbl 1028.35134
[7] Germain P., Shatah J., Zeng C.: Self-similar solutions for the Schrödinger map equation. Math. Z. 264(3), 697–707 (2010) · Zbl 1186.35200
[8] Grotowski J.F., Shatah J.: Geometric evolution equations in critical dimensions. Calc. Var. Part. Diff. Eqs. 30(4), 499–512 (2007) · Zbl 1127.58003
[9] Guan, M., Gustafson, S., Kang, K., Tsai, T.-P.: Global Questions for Map Evolution Equations. CRM Proc. Lec. Notes 44, Providence, RI: Amer. Math. Soc., 2008, pp. 61–73
[10] Guan M., Gustafson S., Tsai T.-P.: Global existence and blow-up for harmonic map heat flow. J. Diff. Eq. 246, 1–20 (2009) · Zbl 1177.35104
[11] Gustafson S., Kang K., Tsai T.-P.: Schrödinger flow near harmonic maps. Comm. Pure Appl. Math. 60(4), 463–499 (2007) · Zbl 1144.53085
[12] Gustafson S., Kang K., Tsai T.-P.: Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Math. J. 145(3), 537–583 (2008) · Zbl 1170.35091
[13] Kosevich A., Ivanov B., Kovalev A.: Magnetic Solitons. Phys. Rep. 194, 117–238 (1990)
[14] Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) · Zbl 0922.35028
[15] Machihara S., Nakanishi K., Ozawa T.: Nonrelativistic limit in the energy space for the nonlinear Klein-Gordon equations. Math. Ann. 322(3), 603–621 (2002) · Zbl 0991.35080
[16] Montgomery-Smith S.J.: Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math. J. 91(2), 393–408 (1998) · Zbl 0955.35012
[17] O’Neil R.: Convolution operators and L(p,q) spaces. Duke Math. J. 30, 129–142 (1963) · Zbl 0178.47701
[18] Poláčik P., Yanagida E.: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003) · Zbl 1055.35055
[19] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985) · Zbl 0595.58013
[20] Tao T.: Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm. PDE 25(7–8), 1471–1485 (2000) · Zbl 0966.35027
[21] Topping P.M.: Rigidity in the harmonic map heat flow. J. Diff. Geom. 45, 593–610 (1997) · Zbl 0955.58013
[22] Topping P.M.: Winding behaviour of finite-time singularities of the harmonic map heat flow. Math. Z. 247, 279–302 (2004) · Zbl 1067.53055
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