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The method of energy channels for nonlinear wave equations
Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, 60637-1514, USA |
This is a survey of some recent results on the asymptotic behavior of solutions to critical nonlinear wave equations.
References:
[1] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[2] |
M. Borghese, R. Jenkins and K. T.-R. McLaughlin,
Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 887-920.
doi: 10.1016/j.anihpc.2017.08.006. |
[3] |
G. Chen, Y. Liu and J. Wei, Nondegeneracy of harmonic maps from $ \mathbb R^{2}$ to $ \mathbb S^{2}$, preprint, arXiv: 1806.04131. |
[4] |
D. Christodoulou and A. 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. |
[5] |
D. Christodoulou and A. Tahvildar-Zadeh,
On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091.
doi: 10.1002/cpa.3160460705. |
[6] |
R. Côte,
On the soliton resolution for equivariant wave maps to the sphere, Comm. Pure. Appl. Math., 68 (2015), 1946-2004.
doi: 10.1002/cpa.21545. |
[7] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Characterization of large energy solutions of the equivariant wave map problem: Ⅰ, Amer. J. Math., 137 (2015), 139-207.
doi: 10.1353/ajm.2015.0002. |
[8] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Characterization of large energy solutions of the equivariant wave map problem: Ⅱ, Amer. J. Math., 137 (2015), 209-250.
doi: 10.1353/ajm.2015.0003. |
[9] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Profiles for the radial focusing $4d$ energy–critical wave equation, Comm. Math. Phys., 357 (2018), 943-1008.
doi: 10.1007/s00220-017-3043-2. |
[10] |
R. Côte, C. Kenig and W. Schlag,
Energy partition for the linear radial wave equation, Math. Ann., 358 (2014), 573-607.
doi: 10.1007/s00208-013-0970-x. |
[11] |
T. Duyckaerts, H. Jia, C. Kenig and F. Merle,
Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal., 27 (2017), 798-862.
doi: 10.1007/s00039-017-0418-7. |
[12] |
T. Duyckaerts, H. Jia, C. Kenig and F. Merle, Universality of blow–up profile small blow–up solutions to the energy critical wave map equation, preprint, arXiv: 1612.04927, to appear in IMRN.
doi: 10.1093/imrn/rnx073. |
[13] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow–up profile for small radial type Ⅱ blow–up solutions of the energy–critical wave equation, J. Eur. Math. Soc., 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[14] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow–up profile for small type Ⅱ blow–up solutions of the energy–critical wave equation: The nonradial case, J. Eur. Math. Soc., 14 (2012), 1389-1454.
doi: 10.4171/JEMS/336. |
[15] |
T. Duyckaerts, C. Kenig and F. Merle,
Profiles of bounded radial solutions of the focusing, energy–critical wave equation, Geom. Funct. Anal., 22 (2012), 639-689.
doi: 10.1007/s00039-012-0174-7. |
[16] |
T. Duyckaerts, C. Kenig and F. Merle,
Classification of radial solutions of the focusing, energy–critical wave equation, Cambridge Journ. of Math., 1 (2013), 75-144.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[17] |
T. Duyckaerts, C. Kenig and F. Merle,
Profiles for bounded solutions of dispersive equations, with applications to energy–critical wave and Schrödinger equations, Commun. Pure Appl. Anal., 14 (2015), 1275-1326.
doi: 10.3934/cpaa.2015.14.1275. |
[18] |
T. Duyckaerts, C. Kenig and F. Merle,
Solutions of the focusing nonradial critical wave equation with the compactness property, Ann. Sc. Norm. Super. Pisa Cl. Sci., 15 (2016), 731-808.
|
[19] |
T. Duyckaerts, C. Kenig and F. Merle, Scattering profile for global solutions of the energy–critical wave equation, preprint, arXiv: 1601.01871, to appear in J. Eur. Math. Soc.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[20] |
W. Eckhaus, The long–time behaviour for perturbed wave–equations and related problems, in Trends in Applications of Pure Mathematics to Mechanics (Bad Honnef, 1985) (eds. E. Kröner and K. Kirchgässner), Springer, 1986,168–194.
doi: 10.1007/BFb0016391. |
[21] |
W. Eckhaus and P. Schuur,
The emergence of solitons of the Korteweg–de Vries equation from arbitrary initial conditions, Math. Methods Appl. Sci., 5 (1983), 97-116.
doi: 10.1002/mma.1670050108. |
[22] |
R. Grinis,
Quantization of time–like energy for wave maps into spheres, Comm. Math. Phys., 352 (2017), 641-702.
doi: 10.1007/s00220-016-2766-9. |
[23] |
M. Hillairet and P. Raphael,
Smooth type Ⅱ blow–up solutions to the four–dimensional energy critical wave equation, Anal. PDE, 5 (2012), 777-829.
doi: 10.2140/apde.2012.5.777. |
[24] |
J. Jendrej,
Construction of type Ⅱ blow–up solutions for the energy–critical wave equation in dimension 5, J. Funct. Anal., 272 (2017), 866-917.
doi: 10.1016/j.jfa.2016.10.019. |
[25] |
H. Jia and C. Kenig,
Asymptotic decomposition for semilinear wave and equivariant wave map equations, Amer. J. Math., 139 (2017), 1521-1603.
doi: 10.1353/ajm.2017.0039. |
[26] |
C. E. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations, in Proceedings of the International Congress of Mathematicians Volume 1, Hindustani Book Agency, 2010,326–338. |
[27] |
C. E. Kenig,
Critical non–linear dispersive equations: Global existence, scattering, blow–up and universal profiles, Japanese Journal of Mathematics, 6 (2011), 121-141.
doi: 10.1007/s11537-011-1108-0. |
[28] |
C. Kenig, A. Lawrie, B. Liu and W. Schlag,
Channels of energy for the linear radial wave equation, Adv. Math., 285 (2015), 877-936.
doi: 10.1016/j.aim.2015.08.014. |
[29] |
C. E. Kenig and F. Merle,
Global well–posedness, scattering and blow–up for the energy–critical, focusing, non–linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 646-675.
doi: 10.1007/s00222-006-0011-4. |
[30] |
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. |
[31] |
C. E. Kenig and F. Merle,
Scattering for $H^{1/2}$–bounded solutions to the cubic, defocusing non–linear Schrödinger equation in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962.
doi: 10.1090/S0002-9947-09-04722-9. |
[32] |
S. Klainerman and S. Selberg,
Remark on the optimal regularity for equations of wave maps type, C.P.D.E., 22 (1997), 901-918.
doi: 10.1080/03605309708821288. |
[33] |
S. Klainerman and S. Selberg,
Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.
doi: 10.1142/S0219199702000634. |
[34] |
S. Klainerman and M. Machedon,
Smoothing estimates for null forms and applications, Duke Math. J., 81 (1995), 99-133.
doi: 10.1215/S0012-7094-95-08109-5. |
[35] |
S. Klainerman and M. Machedon,
On the optimal local regularity for gauge field theories, Diff. and Integral Eq., 10 (1997), 1019-1030.
|
[36] |
S. Klainerman and M. Machedon,
On the algebraic properties of the $H^{n/2, 1/2}$ spaces, IMRN, 15 (1998), 765-774.
doi: 10.1155/S1073792898000464. |
[37] |
J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, 1$^{st}$ edition, European Mathematical Society, Zürich, 2012.
doi: 10.4171/106. |
[38] |
J. Krieger, W. Schlag and D. Tataru,
Slow blow–up solutions for the $H^1(\mathbb R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53.
doi: 10.1215/00127094-2009-005. |
[39] |
J. Krieger, W. Schlag and D. Tataru,
Renormalization and blow–up for charge one equivariant wave maps, Invent. Math., 171 (2008), 543-615.
doi: 10.1007/s00222-007-0089-3. |
[40] |
P. Raphaël and I. Rodnianski,
Stable blow–up dynamics for the critical co–rotational wave maps and equivariant Yang–Mills problems, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 1-122.
doi: 10.1007/s10240-011-0037-z. |
[41] |
I. Rodnianski and J. Sterbenz,
On the formation of singularities in the critical $O(3)$–model, Ann. of Math., 172 (2010), 187-242.
doi: 10.4007/annals.2010.172.187. |
[42] |
C. Rodriguez,
Profiles for the focusing energy critical wave equation in odd dimensions, Adv. Differential Equ., 21 (2016), 505-570.
|
[43] |
P. C. Schuur, Asymptotic Analysis of Soliton Problems, 1$^{st}$ edition, Springer–Verlag, Berlin, 1986.
doi: 10.1007/BFb0073054. |
[44] |
S. Selberg, Multilinear Space–Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. |
[45] |
J. Shatah and M. Struwe, Geometric Wave Equations, 1$^{st}$ edition, American Mathematical Society, New York, 1998. |
[46] |
J. Shatah and A. S. Tahvildar–Zadeh,
On the stability of stationary wave maps, Comm. Math. Phys., 185 (1997), 231-256.
doi: 10.1007/s002200050089. |
[47] |
J. Shatah and A. S. Tahvildar-Zadeh,
On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.
doi: 10.1002/cpa.3160470507. |
[48] |
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. |
[49] |
J. Sterbenz and D. Tataru,
Regularity of wave maps in dimensions $2+1$, Comm. Math. Phys., 298 (2010), 231-264.
doi: 10.1007/s00220-010-1062-3. |
[50] |
T. Tao,
Global regularity of wave maps Ⅱ. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544.
doi: 10.1007/PL00005588. |
[51] |
T. Tao, Global regularity of wave maps Ⅲ. Large energy from $ \mathbb R^{1+2}$ to hyperbolic spaces, preprint, arXiv: 0805.4666. |
[52] |
D. Tataru,
On global existence and scattering for the wave maps equation, Am. J. Math., 123 (2001), 37-77.
doi: 10.1353/ajm.2001.0005. |
[53] |
D. Tataru,
Rough solutions for the wave maps equation, Am. J. Math., 127 (2005), 293-337.
doi: 10.1353/ajm.2005.0014. |
[54] |
P. M. Topping,
Rigidity in the harmonic map heat flow, J. Diff. Geom., 45 (1997), 593-610.
doi: 10.4310/jdg/1214459844. |
[55] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two–dimensional self–focusing and one dimensional self–modulation of waves in nonlinear media, Z. Eksper. Teoret. Fiz., 61 (1971), 118-134.
|
show all references
References:
[1] |
H. Bahouri and P. Gérard,
High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[2] |
M. Borghese, R. Jenkins and K. T.-R. McLaughlin,
Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 887-920.
doi: 10.1016/j.anihpc.2017.08.006. |
[3] |
G. Chen, Y. Liu and J. Wei, Nondegeneracy of harmonic maps from $ \mathbb R^{2}$ to $ \mathbb S^{2}$, preprint, arXiv: 1806.04131. |
[4] |
D. Christodoulou and A. 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. |
[5] |
D. Christodoulou and A. Tahvildar-Zadeh,
On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46 (1993), 1041-1091.
doi: 10.1002/cpa.3160460705. |
[6] |
R. Côte,
On the soliton resolution for equivariant wave maps to the sphere, Comm. Pure. Appl. Math., 68 (2015), 1946-2004.
doi: 10.1002/cpa.21545. |
[7] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Characterization of large energy solutions of the equivariant wave map problem: Ⅰ, Amer. J. Math., 137 (2015), 139-207.
doi: 10.1353/ajm.2015.0002. |
[8] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Characterization of large energy solutions of the equivariant wave map problem: Ⅱ, Amer. J. Math., 137 (2015), 209-250.
doi: 10.1353/ajm.2015.0003. |
[9] |
R. Côte, C. Kenig, A. Lawrie and W. Schlag,
Profiles for the radial focusing $4d$ energy–critical wave equation, Comm. Math. Phys., 357 (2018), 943-1008.
doi: 10.1007/s00220-017-3043-2. |
[10] |
R. Côte, C. Kenig and W. Schlag,
Energy partition for the linear radial wave equation, Math. Ann., 358 (2014), 573-607.
doi: 10.1007/s00208-013-0970-x. |
[11] |
T. Duyckaerts, H. Jia, C. Kenig and F. Merle,
Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal., 27 (2017), 798-862.
doi: 10.1007/s00039-017-0418-7. |
[12] |
T. Duyckaerts, H. Jia, C. Kenig and F. Merle, Universality of blow–up profile small blow–up solutions to the energy critical wave map equation, preprint, arXiv: 1612.04927, to appear in IMRN.
doi: 10.1093/imrn/rnx073. |
[13] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow–up profile for small radial type Ⅱ blow–up solutions of the energy–critical wave equation, J. Eur. Math. Soc., 13 (2011), 533-599.
doi: 10.4171/JEMS/261. |
[14] |
T. Duyckaerts, C. Kenig and F. Merle,
Universality of blow–up profile for small type Ⅱ blow–up solutions of the energy–critical wave equation: The nonradial case, J. Eur. Math. Soc., 14 (2012), 1389-1454.
doi: 10.4171/JEMS/336. |
[15] |
T. Duyckaerts, C. Kenig and F. Merle,
Profiles of bounded radial solutions of the focusing, energy–critical wave equation, Geom. Funct. Anal., 22 (2012), 639-689.
doi: 10.1007/s00039-012-0174-7. |
[16] |
T. Duyckaerts, C. Kenig and F. Merle,
Classification of radial solutions of the focusing, energy–critical wave equation, Cambridge Journ. of Math., 1 (2013), 75-144.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[17] |
T. Duyckaerts, C. Kenig and F. Merle,
Profiles for bounded solutions of dispersive equations, with applications to energy–critical wave and Schrödinger equations, Commun. Pure Appl. Anal., 14 (2015), 1275-1326.
doi: 10.3934/cpaa.2015.14.1275. |
[18] |
T. Duyckaerts, C. Kenig and F. Merle,
Solutions of the focusing nonradial critical wave equation with the compactness property, Ann. Sc. Norm. Super. Pisa Cl. Sci., 15 (2016), 731-808.
|
[19] |
T. Duyckaerts, C. Kenig and F. Merle, Scattering profile for global solutions of the energy–critical wave equation, preprint, arXiv: 1601.01871, to appear in J. Eur. Math. Soc.
doi: 10.4310/CJM.2013.v1.n1.a3. |
[20] |
W. Eckhaus, The long–time behaviour for perturbed wave–equations and related problems, in Trends in Applications of Pure Mathematics to Mechanics (Bad Honnef, 1985) (eds. E. Kröner and K. Kirchgässner), Springer, 1986,168–194.
doi: 10.1007/BFb0016391. |
[21] |
W. Eckhaus and P. Schuur,
The emergence of solitons of the Korteweg–de Vries equation from arbitrary initial conditions, Math. Methods Appl. Sci., 5 (1983), 97-116.
doi: 10.1002/mma.1670050108. |
[22] |
R. Grinis,
Quantization of time–like energy for wave maps into spheres, Comm. Math. Phys., 352 (2017), 641-702.
doi: 10.1007/s00220-016-2766-9. |
[23] |
M. Hillairet and P. Raphael,
Smooth type Ⅱ blow–up solutions to the four–dimensional energy critical wave equation, Anal. PDE, 5 (2012), 777-829.
doi: 10.2140/apde.2012.5.777. |
[24] |
J. Jendrej,
Construction of type Ⅱ blow–up solutions for the energy–critical wave equation in dimension 5, J. Funct. Anal., 272 (2017), 866-917.
doi: 10.1016/j.jfa.2016.10.019. |
[25] |
H. Jia and C. Kenig,
Asymptotic decomposition for semilinear wave and equivariant wave map equations, Amer. J. Math., 139 (2017), 1521-1603.
doi: 10.1353/ajm.2017.0039. |
[26] |
C. E. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations, in Proceedings of the International Congress of Mathematicians Volume 1, Hindustani Book Agency, 2010,326–338. |
[27] |
C. E. Kenig,
Critical non–linear dispersive equations: Global existence, scattering, blow–up and universal profiles, Japanese Journal of Mathematics, 6 (2011), 121-141.
doi: 10.1007/s11537-011-1108-0. |
[28] |
C. Kenig, A. Lawrie, B. Liu and W. Schlag,
Channels of energy for the linear radial wave equation, Adv. Math., 285 (2015), 877-936.
doi: 10.1016/j.aim.2015.08.014. |
[29] |
C. E. Kenig and F. Merle,
Global well–posedness, scattering and blow–up for the energy–critical, focusing, non–linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 646-675.
doi: 10.1007/s00222-006-0011-4. |
[30] |
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. |
[31] |
C. E. Kenig and F. Merle,
Scattering for $H^{1/2}$–bounded solutions to the cubic, defocusing non–linear Schrödinger equation in 3 dimensions, Trans. Amer. Math. Soc., 362 (2010), 1937-1962.
doi: 10.1090/S0002-9947-09-04722-9. |
[32] |
S. Klainerman and S. Selberg,
Remark on the optimal regularity for equations of wave maps type, C.P.D.E., 22 (1997), 901-918.
doi: 10.1080/03605309708821288. |
[33] |
S. Klainerman and S. Selberg,
Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., 4 (2002), 223-295.
doi: 10.1142/S0219199702000634. |
[34] |
S. Klainerman and M. Machedon,
Smoothing estimates for null forms and applications, Duke Math. J., 81 (1995), 99-133.
doi: 10.1215/S0012-7094-95-08109-5. |
[35] |
S. Klainerman and M. Machedon,
On the optimal local regularity for gauge field theories, Diff. and Integral Eq., 10 (1997), 1019-1030.
|
[36] |
S. Klainerman and M. Machedon,
On the algebraic properties of the $H^{n/2, 1/2}$ spaces, IMRN, 15 (1998), 765-774.
doi: 10.1155/S1073792898000464. |
[37] |
J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, 1$^{st}$ edition, European Mathematical Society, Zürich, 2012.
doi: 10.4171/106. |
[38] |
J. Krieger, W. Schlag and D. Tataru,
Slow blow–up solutions for the $H^1(\mathbb R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53.
doi: 10.1215/00127094-2009-005. |
[39] |
J. Krieger, W. Schlag and D. Tataru,
Renormalization and blow–up for charge one equivariant wave maps, Invent. Math., 171 (2008), 543-615.
doi: 10.1007/s00222-007-0089-3. |
[40] |
P. Raphaël and I. Rodnianski,
Stable blow–up dynamics for the critical co–rotational wave maps and equivariant Yang–Mills problems, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 1-122.
doi: 10.1007/s10240-011-0037-z. |
[41] |
I. Rodnianski and J. Sterbenz,
On the formation of singularities in the critical $O(3)$–model, Ann. of Math., 172 (2010), 187-242.
doi: 10.4007/annals.2010.172.187. |
[42] |
C. Rodriguez,
Profiles for the focusing energy critical wave equation in odd dimensions, Adv. Differential Equ., 21 (2016), 505-570.
|
[43] |
P. C. Schuur, Asymptotic Analysis of Soliton Problems, 1$^{st}$ edition, Springer–Verlag, Berlin, 1986.
doi: 10.1007/BFb0073054. |
[44] |
S. Selberg, Multilinear Space–Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. |
[45] |
J. Shatah and M. Struwe, Geometric Wave Equations, 1$^{st}$ edition, American Mathematical Society, New York, 1998. |
[46] |
J. Shatah and A. S. Tahvildar–Zadeh,
On the stability of stationary wave maps, Comm. Math. Phys., 185 (1997), 231-256.
doi: 10.1007/s002200050089. |
[47] |
J. Shatah and A. S. Tahvildar-Zadeh,
On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754.
doi: 10.1002/cpa.3160470507. |
[48] |
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. |
[49] |
J. Sterbenz and D. Tataru,
Regularity of wave maps in dimensions $2+1$, Comm. Math. Phys., 298 (2010), 231-264.
doi: 10.1007/s00220-010-1062-3. |
[50] |
T. Tao,
Global regularity of wave maps Ⅱ. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544.
doi: 10.1007/PL00005588. |
[51] |
T. Tao, Global regularity of wave maps Ⅲ. Large energy from $ \mathbb R^{1+2}$ to hyperbolic spaces, preprint, arXiv: 0805.4666. |
[52] |
D. Tataru,
On global existence and scattering for the wave maps equation, Am. J. Math., 123 (2001), 37-77.
doi: 10.1353/ajm.2001.0005. |
[53] |
D. Tataru,
Rough solutions for the wave maps equation, Am. J. Math., 127 (2005), 293-337.
doi: 10.1353/ajm.2005.0014. |
[54] |
P. M. Topping,
Rigidity in the harmonic map heat flow, J. Diff. Geom., 45 (1997), 593-610.
doi: 10.4310/jdg/1214459844. |
[55] |
V. E. Zakharov and A. B. Shabat,
Exact theory of two–dimensional self–focusing and one dimensional self–modulation of waves in nonlinear media, Z. Eksper. Teoret. Fiz., 61 (1971), 118-134.
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