July  2016, 36(7): 3857-3909. doi: 10.3934/dcds.2016.36.3857

Stability of stationary wave maps from a curved background to a sphere

1. 

University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor, MI 48109, United States

Received  March 2015 Revised  November 2015 Published  March 2016

We study time and space equivariant wave maps from $M\times\mathbb{R}\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of $SO(2)$ by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.
Citation: Sohrab Shahshahani. Stability of stationary wave maps from a curved background to a sphere. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3857-3909. doi: 10.3934/dcds.2016.36.3857
References:
[1]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical non-linear wave equations, Amer. J. Math., 121 (1999), 131-175.

[2]

M. Berger, A Panoramic View of Riemannian Geometry, Springer, 2003. doi: 10.1007/978-3-642-18245-7.

[3]

C. I. Cârstea, A Construction of blow up solutions for co-rotational wave maps, Comm. Math. Phys., 300 (2010), 487-528. doi: 10.1007/s00220-010-1118-4.

[4]

R. Côte, Instability of nonconstant harmonic maps for the $(1+2)$-dimensional equivariant wave map system, Int. Math. Res. Not., 57 (2005), 3525-3549. doi: 10.1155/IMRN.2005.3525.

[5]

D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091. doi: 10.1002/cpa.3160460705.

[6]

D. Christodoulou and A. S. Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps, Duke Math. J., 71 (1993), 31-69. doi: 10.1215/S0012-7094-93-07103-7.

[7]

T. Cazenave, J. Shatah and A. S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Comm. Math. Phys., 68 (1998), 315-349.

[8]

R. Côte, C. E. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system, Ann. Inst. H. Poincaré Phys. Théor., 284 (2008), 203-225. doi: 10.1007/s00220-008-0604-4.

[9]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., 1976.

[10]

F. G. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge University Press, 1975.

[11]

M. Keel andT. Tao, Local and global well-posedness for wave maps on $\mathbbR^{1+1}$ for rough data, Internat. Math. Res. Notices, (1998), 1117-1156. doi: 10.1155/S107379289800066X.

[12]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrodinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[13]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta. Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[14]

S. Klainerman and I. Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, Internat. Math. Res. Notices, 13 (2001), 655-677. doi: 10.1155/S1073792801000344.

[15]

S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential Integral Equations, 10 (1997), 1019-1030.

[16]

S. Klainerman and S. Selberg, Remarks on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations, 22 (1997), 901-918. doi: 10.1080/03605309708821288.

[17]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295. doi: 10.1142/S0219199702000634.

[18]

W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1, Walter de Gruyter and Co., Berlin, 1995.

[19]

J. Krieger, Null-form estimates and nonlinear waves, Adv. Differential Equations, 8 (2003), 1193-1236.

[20]

J. Krieger, Global Regularity of Wave Maps from $\mathbbR^{3+1}$ to Surfaces, Comm. Math. Phys., 238 (2003), 333-366. doi: 10.1007/s00220-003-0836-2.

[21]

J. Krieger, Global Regularity of Wave Maps from $\mathbbR^{2+1}$ to $\mathbbH^2$ Small Energy, Com. Pure Appl. Phys., 250 (2004), 507-580. doi: 10.1007/s00220-004-1088-5.

[22]

J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, European Mathematical Society Publishing House, 2012. doi: 10.4171/106.

[23]

J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., 171 (2008), 543-615. doi: 10.1007/s00222-007-0089-3.

[24]

L. Lemaire, Applications harmoniques de surfaces riemanniennes, Jour. Diff. Geom., 13 (1978), 51-78.

[25]

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

[26]

A. Lawrie and W. Schlag, Scattering for wave maps exterior to a ball, Advances in Mathematics, 232 (2013), 57-97. doi: 10.1016/j.aim.2012.09.005.

[27]

J. Nahas, Scattering of wave maps from $\mathbbR^{2+1}$ to general targets, Calc. Var. Partial Differ. Equ., 46 (2013), 427-437. doi: 10.1007/s00526-011-0489-5.

[28]

A. Nahmod, A. Stefanov and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom., 11 (2003), 49-83. doi: 10.4310/CAG.2003.v11.n1.a4.

[29]

P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Pub. Math. de L'IHES, 115 (2012), 1-122. doi: 10.1007/s10240-011-0037-z.

[30]

I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical $O(3) \sigma$-model, Ann. Math., 172 (2010), 187-242. doi: 10.4007/annals.2010.172.187.

[31]

J. Shatah, Weak Solutions and Development of Singularities of the $SU(2)$ $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469. doi: 10.1002/cpa.3160410405.

[32]

J. Shatah and M. Struwe, Geometric Wave Equations, American Mathematical Society, Courant Lecture Notes 2, 1998.

[33]

J. Shatah and M. Struwe, The cauchy problem for wave maps, Int. Math. Res. Notices, 11 (2002), 555-571. doi: 10.1155/S1073792802109044.

[34]

J. Shatah and A. S. Tahvildar-Zadeh, On the stability of stationary wave maps, Commun. Math. Phys., 185 (1997), 231-256. doi: 10.1007/s002200050089.

[35]

J. Shatah and A. S. Tahvildar-Zadeh, Regularity of harmonic maps from the minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math., 45 (1992), 947-971. doi: 10.1002/cpa.3160450803.

[36]

R. M. Schoen and K. Uhlenbeck, Boundary regularity and the dirichlet problem for harmonic maps, J. Diff. Geom., 18 (1983), 253-268.

[37]

J. Sterbenz and D. Tataru, Regularity for wave maps in dimension 2+1, Comm. Math. Phys., 298 (2010), 231-264. doi: 10.1007/s00220-010-1062-3.

[38]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in 2+1 dimensions, Comm. Math. Phys., 298 (2010), 139-230. doi: 10.1007/s00220-010-1061-4.

[39]

M. Struwe, Equivariant wave maps in two space dimensions, Com. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[40]

M. Struwe, Radially symmetric wave maps from (1+2)-dimensional minkowski space to general targets, Calc. Var. Partial Differ. Equ., 16 (2003), 431-437. doi: 10.1007/s00526-002-0156-y.

[41]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Springer, 2008.

[42]

T. Tao, Global regularity of wave maps i. small critical sobolev norm in high dimension, Internat. Math. Res. Notices, 6 (2001), 299-328. doi: 10.1155/S1073792801000150.

[43]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544. doi: 10.1007/PL00005588.

[44]

T. Tao, Global regularity of wave maps III. Large energy from $\mathbbR^{1+2}$ to hyperbolic spaces, Preprint, 2008.

[45]

D. Tataru, Local and global results for wave maps, I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.

[46]

D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math., 123 (2001), 37-77. doi: 10.1353/ajm.2001.0005.

[47]

D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math., 127 (2005), 293-377. doi: 10.1353/ajm.2005.0014.

[48]

D. Tataru, The wave maps equation, Bull. Amer. Math. Soc., 41 (2004), 185-204. doi: 10.1090/S0273-0979-04-01005-5.

show all references

References:
[1]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical non-linear wave equations, Amer. J. Math., 121 (1999), 131-175.

[2]

M. Berger, A Panoramic View of Riemannian Geometry, Springer, 2003. doi: 10.1007/978-3-642-18245-7.

[3]

C. I. Cârstea, A Construction of blow up solutions for co-rotational wave maps, Comm. Math. Phys., 300 (2010), 487-528. doi: 10.1007/s00220-010-1118-4.

[4]

R. Côte, Instability of nonconstant harmonic maps for the $(1+2)$-dimensional equivariant wave map system, Int. Math. Res. Not., 57 (2005), 3525-3549. doi: 10.1155/IMRN.2005.3525.

[5]

D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091. doi: 10.1002/cpa.3160460705.

[6]

D. Christodoulou and A. S. Tahvildar-Zadeh, On the asymptotic behavior of spherically symmetric wave maps, Duke Math. J., 71 (1993), 31-69. doi: 10.1215/S0012-7094-93-07103-7.

[7]

T. Cazenave, J. Shatah and A. S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Comm. Math. Phys., 68 (1998), 315-349.

[8]

R. Côte, C. E. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system, Ann. Inst. H. Poincaré Phys. Théor., 284 (2008), 203-225. doi: 10.1007/s00220-008-0604-4.

[9]

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall Inc., 1976.

[10]

F. G. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge University Press, 1975.

[11]

M. Keel andT. Tao, Local and global well-posedness for wave maps on $\mathbbR^{1+1}$ for rough data, Internat. Math. Res. Notices, (1998), 1117-1156. doi: 10.1155/S107379289800066X.

[12]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrodinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[13]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta. Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[14]

S. Klainerman and I. Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, Internat. Math. Res. Notices, 13 (2001), 655-677. doi: 10.1155/S1073792801000344.

[15]

S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential Integral Equations, 10 (1997), 1019-1030.

[16]

S. Klainerman and S. Selberg, Remarks on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations, 22 (1997), 901-918. doi: 10.1080/03605309708821288.

[17]

S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295. doi: 10.1142/S0219199702000634.

[18]

W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1, Walter de Gruyter and Co., Berlin, 1995.

[19]

J. Krieger, Null-form estimates and nonlinear waves, Adv. Differential Equations, 8 (2003), 1193-1236.

[20]

J. Krieger, Global Regularity of Wave Maps from $\mathbbR^{3+1}$ to Surfaces, Comm. Math. Phys., 238 (2003), 333-366. doi: 10.1007/s00220-003-0836-2.

[21]

J. Krieger, Global Regularity of Wave Maps from $\mathbbR^{2+1}$ to $\mathbbH^2$ Small Energy, Com. Pure Appl. Phys., 250 (2004), 507-580. doi: 10.1007/s00220-004-1088-5.

[22]

J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, European Mathematical Society Publishing House, 2012. doi: 10.4171/106.

[23]

J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., 171 (2008), 543-615. doi: 10.1007/s00222-007-0089-3.

[24]

L. Lemaire, Applications harmoniques de surfaces riemanniennes, Jour. Diff. Geom., 13 (1978), 51-78.

[25]

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

[26]

A. Lawrie and W. Schlag, Scattering for wave maps exterior to a ball, Advances in Mathematics, 232 (2013), 57-97. doi: 10.1016/j.aim.2012.09.005.

[27]

J. Nahas, Scattering of wave maps from $\mathbbR^{2+1}$ to general targets, Calc. Var. Partial Differ. Equ., 46 (2013), 427-437. doi: 10.1007/s00526-011-0489-5.

[28]

A. Nahmod, A. Stefanov and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom., 11 (2003), 49-83. doi: 10.4310/CAG.2003.v11.n1.a4.

[29]

P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Pub. Math. de L'IHES, 115 (2012), 1-122. doi: 10.1007/s10240-011-0037-z.

[30]

I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical $O(3) \sigma$-model, Ann. Math., 172 (2010), 187-242. doi: 10.4007/annals.2010.172.187.

[31]

J. Shatah, Weak Solutions and Development of Singularities of the $SU(2)$ $\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469. doi: 10.1002/cpa.3160410405.

[32]

J. Shatah and M. Struwe, Geometric Wave Equations, American Mathematical Society, Courant Lecture Notes 2, 1998.

[33]

J. Shatah and M. Struwe, The cauchy problem for wave maps, Int. Math. Res. Notices, 11 (2002), 555-571. doi: 10.1155/S1073792802109044.

[34]

J. Shatah and A. S. Tahvildar-Zadeh, On the stability of stationary wave maps, Commun. Math. Phys., 185 (1997), 231-256. doi: 10.1007/s002200050089.

[35]

J. Shatah and A. S. Tahvildar-Zadeh, Regularity of harmonic maps from the minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math., 45 (1992), 947-971. doi: 10.1002/cpa.3160450803.

[36]

R. M. Schoen and K. Uhlenbeck, Boundary regularity and the dirichlet problem for harmonic maps, J. Diff. Geom., 18 (1983), 253-268.

[37]

J. Sterbenz and D. Tataru, Regularity for wave maps in dimension 2+1, Comm. Math. Phys., 298 (2010), 231-264. doi: 10.1007/s00220-010-1062-3.

[38]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in 2+1 dimensions, Comm. Math. Phys., 298 (2010), 139-230. doi: 10.1007/s00220-010-1061-4.

[39]

M. Struwe, Equivariant wave maps in two space dimensions, Com. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[40]

M. Struwe, Radially symmetric wave maps from (1+2)-dimensional minkowski space to general targets, Calc. Var. Partial Differ. Equ., 16 (2003), 431-437. doi: 10.1007/s00526-002-0156-y.

[41]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Springer, 2008.

[42]

T. Tao, Global regularity of wave maps i. small critical sobolev norm in high dimension, Internat. Math. Res. Notices, 6 (2001), 299-328. doi: 10.1155/S1073792801000150.

[43]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544. doi: 10.1007/PL00005588.

[44]

T. Tao, Global regularity of wave maps III. Large energy from $\mathbbR^{1+2}$ to hyperbolic spaces, Preprint, 2008.

[45]

D. Tataru, Local and global results for wave maps, I, Comm. Partial Differential Equations, 23 (1998), 1781-1793. doi: 10.1080/03605309808821400.

[46]

D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math., 123 (2001), 37-77. doi: 10.1353/ajm.2001.0005.

[47]

D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math., 127 (2005), 293-377. doi: 10.1353/ajm.2005.0014.

[48]

D. Tataru, The wave maps equation, Bull. Amer. Math. Soc., 41 (2004), 185-204. doi: 10.1090/S0273-0979-04-01005-5.

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