doi: 10.3934/dcds.2019240

The method of energy channels for nonlinear wave equations

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.

Citation: Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

M. BorgheseR. 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. Google Scholar

[3]

G. Chen, Y. Liu and J. Wei, Nondegeneracy of harmonic maps from $ \mathbb R^{2}$ to $ \mathbb S^{2}$, preprint, arXiv: 1806.04131.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. CôteC. KenigA. 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. Google Scholar

[8]

R. CôteC. KenigA. 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. Google Scholar

[9]

R. CôteC. KenigA. 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. Google Scholar

[10]

R. CôteC. 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. Google Scholar

[11]

T. DuyckaertsH. JiaC. 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. Google Scholar

[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. Google Scholar

[13]

T. DuyckaertsC. 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. Google Scholar

[14]

T. DuyckaertsC. 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. Google Scholar

[15]

T. DuyckaertsC. 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. Google Scholar

[16]

T. DuyckaertsC. 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. Google Scholar

[17]

T. DuyckaertsC. 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. Google Scholar

[18]

T. DuyckaertsC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

C. KenigA. LawrieB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, 1$^{st}$ edition, European Mathematical Society, Zürich, 2012. doi: 10.4171/106. Google Scholar

[38]

J. KriegerW. 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. Google Scholar

[39]

J. KriegerW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

C. Rodriguez, Profiles for the focusing energy critical wave equation in odd dimensions, Adv. Differential Equ., 21 (2016), 505-570. Google Scholar

[43]

P. C. Schuur, Asymptotic Analysis of Soliton Problems, 1$^{st}$ edition, Springer–Verlag, Berlin, 1986. doi: 10.1007/BFb0073054. Google Scholar

[44]

S. Selberg, Multilinear Space–Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. Google Scholar

[45]

J. Shatah and M. Struwe, Geometric Wave Equations, 1$^{st}$ edition, American Mathematical Society, New York, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

T. Tao, Global regularity of wave maps Ⅲ. Large energy from $ \mathbb R^{1+2}$ to hyperbolic spaces, preprint, arXiv: 0805.4666.Google Scholar

[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. Google Scholar

[53]

D. Tataru, Rough solutions for the wave maps equation, Am. J. Math., 127 (2005), 293-337. doi: 10.1353/ajm.2005.0014. Google Scholar

[54]

P. M. Topping, Rigidity in the harmonic map heat flow, J. Diff. Geom., 45 (1997), 593-610. doi: 10.4310/jdg/1214459844. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

M. BorgheseR. 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. Google Scholar

[3]

G. Chen, Y. Liu and J. Wei, Nondegeneracy of harmonic maps from $ \mathbb R^{2}$ to $ \mathbb S^{2}$, preprint, arXiv: 1806.04131.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. CôteC. KenigA. 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. Google Scholar

[8]

R. CôteC. KenigA. 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. Google Scholar

[9]

R. CôteC. KenigA. 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. Google Scholar

[10]

R. CôteC. 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. Google Scholar

[11]

T. DuyckaertsH. JiaC. 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. Google Scholar

[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. Google Scholar

[13]

T. DuyckaertsC. 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. Google Scholar

[14]

T. DuyckaertsC. 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. Google Scholar

[15]

T. DuyckaertsC. 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. Google Scholar

[16]

T. DuyckaertsC. 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. Google Scholar

[17]

T. DuyckaertsC. 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. Google Scholar

[18]

T. DuyckaertsC. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

C. KenigA. LawrieB. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, 1$^{st}$ edition, European Mathematical Society, Zürich, 2012. doi: 10.4171/106. Google Scholar

[38]

J. KriegerW. 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. Google Scholar

[39]

J. KriegerW. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

C. Rodriguez, Profiles for the focusing energy critical wave equation in odd dimensions, Adv. Differential Equ., 21 (2016), 505-570. Google Scholar

[43]

P. C. Schuur, Asymptotic Analysis of Soliton Problems, 1$^{st}$ edition, Springer–Verlag, Berlin, 1986. doi: 10.1007/BFb0073054. Google Scholar

[44]

S. Selberg, Multilinear Space–Time Estimates and Applications to Local Existence Theory for Nonlinear Wave Equations, Ph.D. Thesis, Princeton University, 1999. Google Scholar

[45]

J. Shatah and M. Struwe, Geometric Wave Equations, 1$^{st}$ edition, American Mathematical Society, New York, 1998. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

T. Tao, Global regularity of wave maps Ⅲ. Large energy from $ \mathbb R^{1+2}$ to hyperbolic spaces, preprint, arXiv: 0805.4666.Google Scholar

[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. Google Scholar

[53]

D. Tataru, Rough solutions for the wave maps equation, Am. J. Math., 127 (2005), 293-337. doi: 10.1353/ajm.2005.0014. Google Scholar

[54]

P. M. Topping, Rigidity in the harmonic map heat flow, J. Diff. Geom., 45 (1997), 593-610. doi: 10.4310/jdg/1214459844. Google Scholar

[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. Google Scholar

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