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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[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.  doi: 10.1007/s00220-008-0604-4.  Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1155/S107379289800066X.  Google Scholar

[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.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[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.  doi: 10.1007/s11511-008-0031-6.  Google Scholar

[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.  doi: 10.1155/S1073792801000344.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1,, Walter de Gruyter and Co., (1995).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.4310/CAG.2003.v11.n1.a4.  Google Scholar

[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.  doi: 10.1007/s10240-011-0037-z.  Google Scholar

[30]

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

[31]

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

[32]

J. Shatah and M. Struwe, Geometric Wave Equations,, American Mathematical Society, (1998).   Google Scholar

[33]

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

[34]

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

[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.  doi: 10.1002/cpa.3160450803.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Fourth edition, (2008).   Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[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.  doi: 10.1007/s00220-008-0604-4.  Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1155/S107379289800066X.  Google Scholar

[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.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[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.  doi: 10.1007/s11511-008-0031-6.  Google Scholar

[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.  doi: 10.1155/S1073792801000344.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics 1,, Walter de Gruyter and Co., (1995).   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.  doi: 10.4310/CAG.2003.v11.n1.a4.  Google Scholar

[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.  doi: 10.1007/s10240-011-0037-z.  Google Scholar

[30]

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

[31]

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

[32]

J. Shatah and M. Struwe, Geometric Wave Equations,, American Mathematical Society, (1998).   Google Scholar

[33]

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

[34]

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

[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.  doi: 10.1002/cpa.3160450803.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,, Fourth edition, (2008).   Google Scholar

[42]

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

[43]

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

[44]

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

[45]

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

[46]

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

[47]

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

[48]

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

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