American Institute of Mathematical Sciences

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

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.  Google Scholar

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

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

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

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

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.  Google Scholar

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.   Google Scholar

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

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

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

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

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

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

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

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

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

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.  Google Scholar

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

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

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

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

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

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

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

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

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

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.  Google Scholar

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.  Google Scholar

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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