In this article, we consider the Schrödinger flow of maps from two dimensional hyperbolic space $ {{\mathbb{H}}}^2 $ to sphere $ {{\mathbb{S}}}^2 $. First, we prove the local existence and uniqueness of Schrödinger flow for initial data $ u_0\in\mathbf{H}^3 $ using an approximation scheme and parallel transport introduced by McGahagan [
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[1] | J.-P. Anker and V. Pierfelice, Nonlinear Schrödinger equation on real hyperbolic spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1853-1869. doi: 10.1016/j.anihpc.2009.01.009. |
[2] | V. Banica, The Nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), 1643-1677. doi: 10.1080/03605300600854332. |
[3] | I. Bejenaru, Global results for Schrödinger maps in dimensions $n\geq3$, Comm. Partial Differential Equations, 33 (2008), 451-477. doi: 10.1080/03605300801895225. |
[4] | I. Bejenaru, A. D. Ionescu and C. E. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions $d\geq 4$, Adv. Math., 215 (2007), 263-291. doi: 10.1016/j.aim.2007.04.009. |
[5] | I. Bejenaru, A. D. Ionescu, C. E. Kenig and D. Tataru, Global Schrödinger maps in dimensions $d\geq 2$: Small data in the critical Sobolev spaces, Ann. of Math., 173 (2011), 1443-1506. doi: 10.4007/annals.2011.173.3.5. |
[6] | I. Bejenaru, A. D. Ionescu, C. E. Kenig and D. Tataru, Equivariant Schrödinger maps in two spatial dimensions, Duke Math. J., 162 (2013), 1967-2025. doi: 10.1215/00127094-2293611. |
[7] | I. Bejenaru, A. D. Ionescu, C. E. Kenig and D. Tataru, Equivariant Schrödinger maps in two spatial dimensions: The $\Bbb{H}^2$ target, Kyoto J. Math., 56 (2016), 283-323. doi: 10.1215/21562261-3478889. |
[8] | I. Bejenaru and D. Tataru, Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions, Mem. Amer. Math. Soc., 228 (2014), no. 1069. |
[9] | D. Borthwick and J. L. Marzuola, Dispersive estimates for scalar and matrix Schrödinger operators on $\Bbb{H}^{n+1}$, Math. Phys. Anal. Geom., 18 (2015), 26 pp. doi: 10.1007/s11040-015-9191-8. |
[10] | P. D'Ancona and Q. Zhang, Global existence of small equivariant wave maps on rotationally symmetric manifolds, Int. Math. Res. Not. IMRN, (2016), no. 4,978–1025. doi: 10.1093/imrn/rnv152. |
[11] | W. Ding and Y. Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A, 41 (1998), 746-755. doi: 10.1007/BF02901957. |
[12] | W. Ding and Y. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464. doi: 10.1007/BF02877074. |
[13] | S. Gustafson, K. Kang and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math., 60 (2007), 463-499. doi: 10.1002/cpa.20143. |
[14] | S. Gustafson, K. Kang and T.-P. Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Math. J., 145 (2008), 537-583. doi: 10.1215/00127094-2008-058. |
[15] | S. Gustafson, K. Nakanishi and T.-P. Tsai, Asymtotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on $\Bbb {R}^2$, Comm. Math. Phys., 300 (2010), 205-242. doi: 10.1007/s00220-010-1116-6. |
[16] | A. D. Ionescu and C. E. Kenig, Low-regularity Schrödinger maps. Ⅱ. Global well-posedness in dimensions $d\geq 3$, Comm. Math. Phys., 271 (2007), 523-559. doi: 10.1007/s00220-006-0180-4. |
[17] | A. D. Ionescu and G. Staffilani, Semilinear Schrödinger flows on hyperbolic spaces: Scattering $H^1$, Math. Ann., 345 (2009), 133-158. doi: 10.1007/s00208-009-0344-6. |
[18] | H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Vol. 45, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014. |
[19] | A. Lawrie, J. Lührmann, S.-J. Oh and S. Shahshahani, Asymptotic stability of harmonic maps on the hyperbolic plane under the Schrödinger maps evolution, preprint, arXiv: 1909.06899. |
[20] | A. Lawrie, S.-J. Oh and S. Shahshahani, Stability of stationary equivariant wave maps from the hyperbolic plane, Amer. J. Math., 139 (2017), 1085-1147. doi: 10.1353/ajm.2017.0028. |
[21] | A. Lawrie, S.-J. Oh and S. Shahshahani, Gap eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space, J. Funct. Anal., 271 (2016), 3111-3161. doi: 10.1016/j.jfa.2016.08.019. |
[22] | A. Lawrie, S.-J. Oh and S. Shahshahani, Equivariant wave maps on the hyperbolic plane with large energy, Math. Res. Lett., 24 (2017), 449-479. doi: 10.4310/MRL.2017.v24.n2.a10. |
[23] | A. Lawrie, S.-J. Oh and S. Shahshahani, The Cauchy problem for wave maps on hyperbolic space in dimensions $d\geq 4$, Int. Math. Res. Not. IMRN, (2018), no. 7, 1954–2051. doi: 10.1093/imrn/rnw272. |
[24] | M. Lemm and V. Markovic, Heat flows on hyperbolic spaces, J. Differential Geom., 108, (2018), 495–529. doi: 10.4310/jdg/1519959624. |
[25] | Z. Li, Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces, preprint, arXiv: 1707.01362. |
[26] | Z. Li, Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces, emph{Differential Integral Equations}, 32, (2019), 369–408. |
[27] | Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case, preprint, arXiv: 1811.10924. |
[28] | Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions, preprint, arXiv: 1903.05551. |
[29] | Z. Li, X. Ma and L. Zhao, Asymptotic stability of harmonic maps between 2D hyperbolic spaces under the wave map equation. Ⅱ. Small energy case, Dyn. Partial Differ. Equ., 15 (2018), 283-336. doi: 10.4310/DPDE.2018.v15.n4.a3. |
[30] | P. Li and L.-F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math., 105 (1991), 1-46. doi: 10.1007/BF01232256. |
[31] | Z. Li and L. Zhao, Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces, Discrete Contin. Dyn. Syst., 39 (2019), 607-638. doi: 10.3934/dcds.2019025. |
[32] | H. McGahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations, 32 (2007), 375-400. doi: 10.1080/03605300600856758. |
[33] | F. Merle, P. Raphaël and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem, Invent. Math., 193 (2013), 249-365. doi: 10.1007/s00222-012-0427-y. |
[34] | G. Perelman, Blow up dynamics for equivariant critical Schrödinger maps, Comm. Math. Phys., 330 (2014), 69-105. doi: 10.1007/s00220-014-1916-1. |
[35] | C. Song and Y. Wang, Uniqueness of Schrödinger flow on manifolds, Comm. Anal. Geom., 26 (2018), 217-235. doi: 10.4310/CAG.2018.v26.n1.a5. |
[36] | 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. doi: 10.1007/BF01220998. |
[37] | T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. |