American Institute of Mathematical Sciences

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December  2019, 39(12): 6979-6993. doi: 10.3934/dcds.2019240

The method of energy channels for nonlinear wave equations

Carlos E. Kenig ,

Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL, 60637-1514, USA

Received  October 2018 Revised  October 2018 Published  June 2019

Fund Project: The author is supported in part by NSF Grants DMS–1265249, DMS–1463746 and DMS–1800082.

This is a survey of some recent results on the asymptotic behavior of solutions to critical nonlinear wave equations.

Keywords: Nonlinear wave equations, asymptotic behavior, energy channels, traveling waves, soliton resolution.
Mathematics Subject Classification: Primary: 35L70; Secondary: 35L15.
Citation: Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240
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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