July  2020, 40(7): 4379-4425. doi: 10.3934/dcds.2020184

Equivariant Schrödinger map flow on two dimensional hyperbolic space

1. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

2. 

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

3. 

Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, AMSS, and School of Mathematical Sciences, UCAS, Beijing 100190, China

* Corresponding author: Lifeng Zhao

Received  August 2019 Revised  December 2019 Published  April 2020

Fund Project: J. Huang and L. Zhao are partially supported by the NSFC Grant No. 11771415. Y. Wang is partially supported by the NSFC GRant No. 11731001

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 [32]. Second, using the Coulomb gauge, we reduce the study of the equivariant Schrödinger flow to that of a system of coupled Schrödinger equations with potentials. Then we prove the global existence of equivariant Schrödinger flow for small initial data $ u_0\in\mathbf{H}^1 $ by Strichartz estimates and perturbation method.

Citation: Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184
References:
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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.  Google Scholar

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I. BejenaruA. 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.  Google Scholar

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I. BejenaruA. D. IonescuC. 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.  Google Scholar

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I. BejenaruA. D. IonescuC. E. Kenig and D. Tataru, Equivariant Schrödinger maps in two spatial dimensions, Duke Math. J., 162 (2013), 1967-2025.  doi: 10.1215/00127094-2293611.  Google Scholar

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I. BejenaruA. D. IonescuC. 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.  Google Scholar

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I. Bejenaru and D. Tataru, Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions, Mem. Amer. Math. Soc., 228 (2014), no. 1069.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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W. Ding and Y. Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A, 41 (1998), 746-755.  doi: 10.1007/BF02901957.  Google Scholar

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W. Ding and Y. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.  doi: 10.1007/BF02877074.  Google Scholar

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S. GustafsonK. Kang and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math., 60 (2007), 463-499.  doi: 10.1002/cpa.20143.  Google Scholar

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S. GustafsonK. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Vol. 45, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014.  Google Scholar

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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. Google Scholar

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A. LawrieS.-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.  Google Scholar

[21]

A. LawrieS.-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.  Google Scholar

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A. LawrieS.-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.  Google Scholar

[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.  Google Scholar

[24]

M. Lemm and V. Markovic, Heat flows on hyperbolic spaces, J. Differential Geom., 108, (2018), 495–529. doi: 10.4310/jdg/1519959624.  Google Scholar

[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. Google Scholar

[26]

Z. Li, Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces, emph{Differential Integral Equations}, 32, (2019), 369–408.  Google Scholar

[27]

Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case, preprint, arXiv: 1811.10924. Google Scholar

[28]

Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions, preprint, arXiv: 1903.05551. Google Scholar

[29]

Z. LiX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. McGahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations, 32 (2007), 375-400.  doi: 10.1080/03605300600856758.  Google Scholar

[33]

F. MerleP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

P.-L. SulemC. 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.  Google Scholar

[37]

T. TaoM. 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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

V. Banica, The Nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), 1643-1677.  doi: 10.1080/03605300600854332.  Google Scholar

[3]

I. Bejenaru, Global results for Schrödinger maps in dimensions $n\geq3$, Comm. Partial Differential Equations, 33 (2008), 451-477.  doi: 10.1080/03605300801895225.  Google Scholar

[4]

I. BejenaruA. 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.  Google Scholar

[5]

I. BejenaruA. D. IonescuC. 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.  Google Scholar

[6]

I. BejenaruA. D. IonescuC. E. Kenig and D. Tataru, Equivariant Schrödinger maps in two spatial dimensions, Duke Math. J., 162 (2013), 1967-2025.  doi: 10.1215/00127094-2293611.  Google Scholar

[7]

I. BejenaruA. D. IonescuC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Ding and Y. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A, 44 (2001), 1446-1464.  doi: 10.1007/BF02877074.  Google Scholar

[13]

S. GustafsonK. Kang and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math., 60 (2007), 463-499.  doi: 10.1002/cpa.20143.  Google Scholar

[14]

S. GustafsonK. 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.  Google Scholar

[15]

S. GustafsonK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

H. Koch, D. Tataru and M. Visan, Dispersive Equations and Nonlinear Waves, Vol. 45, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014.  Google Scholar

[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. Google Scholar

[20]

A. LawrieS.-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.  Google Scholar

[21]

A. LawrieS.-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.  Google Scholar

[22]

A. LawrieS.-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.  Google Scholar

[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.  Google Scholar

[24]

M. Lemm and V. Markovic, Heat flows on hyperbolic spaces, J. Differential Geom., 108, (2018), 495–529. doi: 10.4310/jdg/1519959624.  Google Scholar

[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. Google Scholar

[26]

Z. Li, Endpoint Strichartz estimates for magnetic wave equations on two dimensional hyperbolic spaces, emph{Differential Integral Equations}, 32, (2019), 369–408.  Google Scholar

[27]

Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case, preprint, arXiv: 1811.10924. Google Scholar

[28]

Z. Li, Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions, preprint, arXiv: 1903.05551. Google Scholar

[29]

Z. LiX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

H. McGahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations, 32 (2007), 375-400.  doi: 10.1080/03605300600856758.  Google Scholar

[33]

F. MerleP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

P.-L. SulemC. 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.  Google Scholar

[37]

T. TaoM. 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.  Google Scholar

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