January  2019, 39(1): 607-638. doi: 10.3934/dcds.2019025

Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces

1. 

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 10019, China

2. 

Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences and Department of Mathematic, University of Science and Technology of China, Hefei 230026, Anhui, China

* Corresponding author: Ze Li

Received  May 2018 Published  October 2018

In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t, x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to∞$. The main idea is to construct Tao's caloric gauge in the case where nontrivial harmonic maps exist and use it to prove the convergence to harmonic maps. On one side, since in our case the stationary solutions are asymptotically stable under the heat flow, the caloric gauge of Tao provides a natural geometric linearization. On the other side, although there exist infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow initiated from $u(t, x)$ for any given $t>0$ converges to the same harmonic map as the heat flow initiated from $u(0, x)$. The two observations enable us to construct Tao's caloric gauge to reduce the convergence to harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding heat tension field. This idea also works for dispersive geometric flows, see our succeeding works on wave maps for instance.

Citation: Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025
References:
[1]

I. BejenaruA. Ionescu and C. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions $d≥4$, Adv. Math., 215 (2007), 263-291.  doi: 10.1016/j.aim.2007.04.009.  Google Scholar

[2]

I. BejenaruA. IonescuC. Kenig and D. Tataru, Global Schrödinger maps in dimensions $d≥2$: 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. IonescuC. 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. IonescuC. 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

[5]

W. O. Bray, Aspects of harmonic analysis on real hyperbolic space, In Fourier Analysis (Orono, ME, 1992), volume 157 of Lecture Notes in Pure and Appl. Math., Dekker, New York, (1994), 77–102.  Google Scholar

[6]

K. C. ChangW. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36 (1999), 507-515.  doi: 10.4310/jdg/1214448751.  Google Scholar

[7]

N. H. ChangJ. Shatah and K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math., 53 (2000), 590-602.  doi: 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.3.CO;2-I.  Google Scholar

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E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, Proc. Lond. Math. Soc., 3 (1988), 182-208.  doi: 10.1112/plms/s3-57.1.182.  Google Scholar

[10]

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

[11]

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.  doi: 10.2307/2373037.  Google Scholar

[12]

B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations, 1 (1993), 311-334.  doi: 10.1007/BF01191298.  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, Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $\Bbb R^2$, Comm. Math. Phys., 300 (2000), 205-242.  doi: 10.1007/s00220-010-1116-6.  Google Scholar

[16]

R. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Math., 471, Springer-Verlag, Berlin, 1975.  Google Scholar

[17]

R. Hardt and M. Wolf, Harmonic extensions of quasiconformal maps to hyperbolic space, Indian J. Math., 46 (1997), 155-163.  doi: 10.1512/iumj.1997.46.1351.  Google Scholar

[18]

P. Hartman, On homotopic harmonic maps, Canad. J. Math., 19 (1967), 637-687.  doi: 10.4153/CJM-1967-062-6.  Google Scholar

[19]

E. Hebey, Sobolev Spaces on Riemannian Manifolds, Volume 1635 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996. doi: 10.1007/BFb0092907.  Google Scholar

[20]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, New York-London, 1978.  Google Scholar

[21]

A. IonescuB. Pausader and G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.  doi: 10.2140/apde.2012.5.705.  Google Scholar

[22]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sovietunion, 8 (1935), 153-169.   Google Scholar

[23]

A. Lawrie, The Cauchy problem for wave maps on a curved background, Calc. Var. Partial Differential Equations, 45 (2012), 505-548.  doi: 10.1007/s00526-011-0469-9.  Google Scholar

[24]

A. LawrieS. J. Oh and S. Shahshahani, Profile decompositions for wave equations on hyperbolic space with applications, Math. Ann., 365 (2016), 707-803.  doi: 10.1007/s00208-015-1305-x.  Google Scholar

[25]

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

[26]

A. LawrieS. J. Oh and S. Shahshahani, The Cauchy problem for wave maps on hyperbolic space in dimensions $d≥4$, IMRN, (2018), 1954-2051.  doi: 10.1093/imrn/rnw272.  Google Scholar

[27]

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

[28]

M. Lemm and V. Markovic, Heat flows on hyperbolic spaces, J. Differential Geom., 108 (2018), 495–529, arXiv: 1506.04345. doi: 10.4310/jdg/1519959624.  Google Scholar

[29]

J. Li and X. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math., 226 (2011), 4456-4491.  doi: 10.1016/j.aim.2010.12.009.  Google Scholar

[30]

P. Li and L. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math., 105 (1991), 1-46.  doi: 10.1007/BF01232256.  Google Scholar

[31]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203.  Google Scholar

[32]

Z. Li, Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces, arXiv preprint. Google Scholar

[33]

Z. Li and L. Zhao, Asymptotic behaviors of Landau-Lifshitz flows from $\Bbb R^2$ to Kähler manifolds, Calc. Var. Partial Differential Equations, 56 (2017), Art. 96, 35 pp. doi: 10.1007/s00526-017-1182-0.  Google Scholar

[34]

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

[35]

C. Melcher, Existence of Partially Regular Solutions for Landau-Lifshitz Equations in $R^3$, Comm. Partial Differential Equations, 30 (2005), 567-587.  doi: 10.1081/PDE-200050122.  Google Scholar

[36]

F. MerleP. Raphael 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

[37]

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

[38]

I. RodnianskiY. Rubinstein and G. Staffilani, On the global well-posedness of the one-dimensional Schrödinger map flow, Anal. PDE, 2 (2009), 187-209.  doi: 10.2140/apde.2009.2.187.  Google Scholar

[39]

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

[40]

T. Tao, Geometric renormalization of large energy wave maps, Journees Equations Aux Derivees Partielles, (2014), 1-32.   Google Scholar

[41]

T. Tao, Global regularity of wave maps Ⅲ-Ⅶ, arXiv preprint, 2008-2009. Google Scholar

[42]

T. Tao, Global regularity of wave maps Ⅳ. Absence of stationary or self-similar solutions in the energy class, arXiv preprint. Google Scholar

[43]

Y. L. ZhouB. L. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Sci. China Math., Ser. A, 34 (1991), 257-266.   Google Scholar

show all references

References:
[1]

I. BejenaruA. Ionescu and C. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions $d≥4$, Adv. Math., 215 (2007), 263-291.  doi: 10.1016/j.aim.2007.04.009.  Google Scholar

[2]

I. BejenaruA. IonescuC. Kenig and D. Tataru, Global Schrödinger maps in dimensions $d≥2$: Small data in the critical Sobolev spaces, Ann. of Math., 173 (2011), 1443-1506.  doi: 10.4007/annals.2011.173.3.5.  Google Scholar

[3]

I. BejenaruA. IonescuC. 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

[4]

I. BejenaruA. IonescuC. 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

[5]

W. O. Bray, Aspects of harmonic analysis on real hyperbolic space, In Fourier Analysis (Orono, ME, 1992), volume 157 of Lecture Notes in Pure and Appl. Math., Dekker, New York, (1994), 77–102.  Google Scholar

[6]

K. C. ChangW. Y. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36 (1999), 507-515.  doi: 10.4310/jdg/1214448751.  Google Scholar

[7]

N. H. ChangJ. Shatah and K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math., 53 (2000), 590-602.  doi: 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.3.CO;2-I.  Google Scholar

[8]

M. CowlingS. Giulini and S. Meda, $L^p$-$L^ q$ estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces. I, Duke Math. J., 72 (1993), 109-150.  doi: 10.1215/S0012-7094-93-07206-7.  Google Scholar

[9]

E. B. Davies and N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, Proc. Lond. Math. Soc., 3 (1988), 182-208.  doi: 10.1112/plms/s3-57.1.182.  Google Scholar

[10]

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

[11]

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160.  doi: 10.2307/2373037.  Google Scholar

[12]

B. Guo and M. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations, 1 (1993), 311-334.  doi: 10.1007/BF01191298.  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, Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $\Bbb R^2$, Comm. Math. Phys., 300 (2000), 205-242.  doi: 10.1007/s00220-010-1116-6.  Google Scholar

[16]

R. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Math., 471, Springer-Verlag, Berlin, 1975.  Google Scholar

[17]

R. Hardt and M. Wolf, Harmonic extensions of quasiconformal maps to hyperbolic space, Indian J. Math., 46 (1997), 155-163.  doi: 10.1512/iumj.1997.46.1351.  Google Scholar

[18]

P. Hartman, On homotopic harmonic maps, Canad. J. Math., 19 (1967), 637-687.  doi: 10.4153/CJM-1967-062-6.  Google Scholar

[19]

E. Hebey, Sobolev Spaces on Riemannian Manifolds, Volume 1635 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996. doi: 10.1007/BFb0092907.  Google Scholar

[20]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, New York-London, 1978.  Google Scholar

[21]

A. IonescuB. Pausader and G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE, 5 (2012), 705-746.  doi: 10.2140/apde.2012.5.705.  Google Scholar

[22]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sovietunion, 8 (1935), 153-169.   Google Scholar

[23]

A. Lawrie, The Cauchy problem for wave maps on a curved background, Calc. Var. Partial Differential Equations, 45 (2012), 505-548.  doi: 10.1007/s00526-011-0469-9.  Google Scholar

[24]

A. LawrieS. J. Oh and S. Shahshahani, Profile decompositions for wave equations on hyperbolic space with applications, Math. Ann., 365 (2016), 707-803.  doi: 10.1007/s00208-015-1305-x.  Google Scholar

[25]

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

[26]

A. LawrieS. J. Oh and S. Shahshahani, The Cauchy problem for wave maps on hyperbolic space in dimensions $d≥4$, IMRN, (2018), 1954-2051.  doi: 10.1093/imrn/rnw272.  Google Scholar

[27]

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

[28]

M. Lemm and V. Markovic, Heat flows on hyperbolic spaces, J. Differential Geom., 108 (2018), 495–529, arXiv: 1506.04345. doi: 10.4310/jdg/1519959624.  Google Scholar

[29]

J. Li and X. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math., 226 (2011), 4456-4491.  doi: 10.1016/j.aim.2010.12.009.  Google Scholar

[30]

P. Li and L. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math., 105 (1991), 1-46.  doi: 10.1007/BF01232256.  Google Scholar

[31]

P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.  doi: 10.1007/BF02399203.  Google Scholar

[32]

Z. Li, Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces, arXiv preprint. Google Scholar

[33]

Z. Li and L. Zhao, Asymptotic behaviors of Landau-Lifshitz flows from $\Bbb R^2$ to Kähler manifolds, Calc. Var. Partial Differential Equations, 56 (2017), Art. 96, 35 pp. doi: 10.1007/s00526-017-1182-0.  Google Scholar

[34]

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

[35]

C. Melcher, Existence of Partially Regular Solutions for Landau-Lifshitz Equations in $R^3$, Comm. Partial Differential Equations, 30 (2005), 567-587.  doi: 10.1081/PDE-200050122.  Google Scholar

[36]

F. MerleP. Raphael 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

[37]

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

[38]

I. RodnianskiY. Rubinstein and G. Staffilani, On the global well-posedness of the one-dimensional Schrödinger map flow, Anal. PDE, 2 (2009), 187-209.  doi: 10.2140/apde.2009.2.187.  Google Scholar

[39]

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

[40]

T. Tao, Geometric renormalization of large energy wave maps, Journees Equations Aux Derivees Partielles, (2014), 1-32.   Google Scholar

[41]

T. Tao, Global regularity of wave maps Ⅲ-Ⅶ, arXiv preprint, 2008-2009. Google Scholar

[42]

T. Tao, Global regularity of wave maps Ⅳ. Absence of stationary or self-similar solutions in the energy class, arXiv preprint. Google Scholar

[43]

Y. L. ZhouB. L. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Sci. China Math., Ser. A, 34 (1991), 257-266.   Google Scholar

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