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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 |
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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