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Regularity of wave-maps in dimension \(2+1\). (English) Zbl 1218.35057
There is an investigation on large data wave-maps in \(2+1\) dimensions. Questions of finite time regularity for wave-maps as well as scattering for large data wave-maps are considered. Similar results, but under a stronger energy dispersion assumption, have been proved by the authors in a previous article [J. Sterbenz, D. Tataru, Commun. Math. Phys. 298, No. 1, 139–230 (2010; Zbl 1218.35129)].

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35L05 Wave equation
35P25 Scattering theory for PDEs
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[1] Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121(1), 131–175 (1999) · Zbl 0919.35089
[2] Choquet-Bruhat, Y.: General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford: Oxford University Press, 2009 · Zbl 1157.83002
[3] Christodoulou D., Tahvildar-Zadeh A.S.: On the asymptotic behavior of spherically symmetric wave maps. Duke Math. J. 71(1), 31–69 (1993) · Zbl 0791.58105
[4] Christodoulou D., Tahvildar-Zadeh A.S.: On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math. 46(7), 1041–1091 (1993) · Zbl 0789.58072
[5] Cote, R., Kenig, C.E., Merle, F.: Scattering below critical energy for the radial 4d Yang-Mills equation and for the 2d corotational wave map system. http://arxiv.org/abs/0709.3222v1[math.AP] , 2007
[6] Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Universitext. Berlin: Springer-Verlag, Second edition, 1990 · Zbl 0716.53001
[7] Grillakis, M.G.: On the wave map problem. In: Nonlinear wave equations (Providence, RI, 1998), Volume 263 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2000, pp. 71–84 · Zbl 0966.35079
[8] Gromov M.L.: Isometric imbeddings and immersions. Dokl. Akad. Nauk SSSR 192, 1206–1209 (1970) · Zbl 0214.50404
[9] Günther, M.: Isometric embeddings of Riemannian manifolds. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Tokyo: Math. Soc. Japan, 1991, pp. 1137–1143 · Zbl 0745.53031
[10] Hélein, F.: Harmonic maps, conservation laws and moving frames. Volume 150 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, second edition, 2002 · Zbl 1010.58010
[11] Jost, J.: Riemannian geometry and geometric analysis. Universitext. Berlin: Springer-Verlag, 1995 · Zbl 0828.53002
[12] Kenig C.E., Merle F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008) · Zbl 1183.35202
[13] Klainerman S., Machedon M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46(9), 1221–1268 (1993) · Zbl 0803.35095
[14] Klainerman S., Machedon M.: Smoothing estimates for null forms and applications. Duke Math. J. 81(1), 99–133 (1996) · Zbl 0909.35094
[15] Krieger, J.: Global regularity and singularity development for wave maps. In: Surveys in differential geometry. Vol. XII. Geometric flows, Volume 12 of Surv. Differ. Geom., Somerville, MA: Int. Press, 2008, pp. 167–201 · Zbl 1166.53027
[16] Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008) · Zbl 1139.35021
[17] Krieger J.: Global regularity of wave maps from \({{\mathbb R}^ {2+1}}\) to H 2. Small energy. Commun. Math. Phys. 250(3), 507–580 (2004) · Zbl 1099.58010
[18] Krieger, J., Schlag, W.: Concentration compactness for critical wave-maps. http://arxiv.org/abs/0908.2474[math.AP] , 2009 · Zbl 1387.35006
[19] Lemaire L.: Applications harmoniques de surfaces riemanniennes. J. Diff. Geom. 13(1), 51–78 (1978) · Zbl 0388.58003
[20] Nash J.: The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63, 20–63 (1956) · Zbl 0070.38603
[21] Qing J.: Boundary regularity of weakly harmonic maps from surfaces. J. Funct. Anal. 114(2), 458–466 (1993) · Zbl 0785.53048
[22] Qing J., Tian G.: (1993) Bubbling of the heat flows for harmonic maps from surfaces. Comm. Pure Appl. Math. 50(4), 295–310 (1997) · Zbl 0879.58017
[23] Rodnianski, I., Sterbenz, J.: On the formation of singularities in the critical O(3) sigma-model. http://arxiv.org/abs/math/0605023v3[math.AP] , 2008
[24] Schoen, R., Yau, S.T.: Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology, II. Cambridge, MA: International Press, 1997 · Zbl 0886.53004
[25] Shatah, J., Struwe, M.: Geometric wave equations, Volume 2 of Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences, 1998 · Zbl 0993.35001
[26] Shatah J., Tahvildar-Zadeh A.S.: On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47(5), 719–754 (1994) · Zbl 0811.58059
[27] Sterbenz, J., Tataru, D.: Energy dispersed large data wave maps in 2 + 1 dimensions. http://arxiv.org/abs/0906.3384v1[math.AP] , 2009. doi: 10.1007/s00220-010-1061-4 · Zbl 1218.35129
[28] Struwe M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(4), 558–581 (1985) · Zbl 0595.58013
[29] Struwe M.: Equivariant wave maps in two space dimensions. Comm. Pure Appl. Math. 56(7), 815–823 (2003) · Zbl 1033.53019
[30] Struwe M.: Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to general targets. Calc. Var. Part. Diff. Eqs. 16(4), 431–437 (2003) · Zbl 1039.58033
[31] Tao, T.: Global regularity of wave maps III. Large energy from \({{\mathbb {R}}^ {2+1} }\) to hyperbolic spaces. http://arxiv.org/abs/0805.4666v3[math.AP] , 2009
[32] Tao, T.: Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class. http://arxiv.org/abs/0806.3592v2[math.AP] , 2009
[33] Tao, T.: Global regularity of wave maps V. Large data local well-posedness in the energy class. http://arxiv.org/abs/0808.0368v2[math.AP] , 2009
[34] Tao, T.: Global regulrity of wave maps VI. Minimal energy blowup solutions. http://arxiv.org/abs/0906.2833v2[math.AP] , 2009
[35] Tao, T.: An inverse theorem for the bilinear L 2 Strichartz estimate for the wave equation. http://arxiv.org/abs/0904.2880v1[math.NT] , 2009
[36] Tao T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001) · Zbl 1020.35046
[37] Tao, T.: Geometric renormalization of large energy wave maps. In: Journées ”Équations aux Dérivées Partielles”, pages Exp. No. XI, 32. École Polytech., Palaiseau, 2004
[38] Tao, T.: Nonlinear dispersive equations. Volume 106 of CBMS Regional Conference Series in Mathematics. Washington, DC: Conference Board of the Mathematical Sciences, 2006 · Zbl 1106.35001
[39] Tataru D.: On global existence and scattering for the wave maps equation. Amer. J. Math. 123(1), 37–77 (2001) · Zbl 0979.35100
[40] Tataru D.: The wave maps equation. Bull. Amer. Math. Soc. (N.S.) 41(2), 185–204 (2004) (electronic) · Zbl 1065.35199
[41] Tataru D.: Rough solutions for the wave maps equation. Amer. J. Math. 127(2), 293–377 (2005) · Zbl 1330.58021
[42] Topping, P.: An example of a nontrivial bubble tree in the harmonic map heat flow. In: Harmonic morphisms, harmonic maps, and related topics (Brest, 1997), Volume 413 of Chapman & Hall/CRC Res. Notes Math. Boca Raton, FL.: Chapman & Hall/CRC, 2000, pp. 185–191 · Zbl 0947.58016
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