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 and Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184
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.

[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. 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.

[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.

[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.

[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.

[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. GustafsonK. 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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

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.

[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. 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.

[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.

[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.

[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.

[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. GustafsonK. 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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

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