The wave map problem. Small data critical regularity
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 965, p. 365-384

The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is (2+1), the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory. As a consequence, in this dimension the result is an important step in establishing global regularity at all energies, conjectured when the target manifold is negatively curved. The work advanced our understanding of the critical equations and already has been a catalyst for the new results for general target manifolds and other equations (Maxwell-Klein-Gordon, Yang-Mills).

Cet exposé présente le problème de l’application d’onde en se concentrant sur le travail révolutionnaire de T. Tao qui a montré que l’application d’onde, un analogue lorentzien dynamique d’une application harmonique, de l’espace de Minkowski dans la sphère avec une donnée initiale et une petite norme de Sobolev critique existe globalement en temps et reste lisse. Quand la dimension de l’espace de Minkowski est (2+1), la norme critique coïncide avec l’énergie, la seule quantité clairement conservée dans cette théorie lagrangienne. Comme conséquence, dans cette dimension, le résultat est une étape importante dans la preuve de la régularité globale pour n’importe quelle énergie, qui a été conjecturée quand la variété d’arrivée est de courbure négative. Ce travail a fait avancer notre compréhension des équations critiques et a déjà servi de catalyseur pour de nouveaux résultats pour des variétés d’arrivée plus générales et d’autres équations (Maxwell-Klein-Gordon, Yang-Mills).

Classification:  35L05,  35Q99
Keywords: wave map, critical regularity, renormalization
@incollection{SB_2005-2006__48__365_0,
     author = {Rodnianski, Igor},
     title = {The wave map problem. Small data critical regularity},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:965},
     pages = {365-384},
     zbl = {1194.35004},
     mrnumber = {2359050},
     language = {en},
     url = {http://www.numdam.org/item/SB_2005-2006__48__365_0}
}
Rodnianski, Igor. The wave map problem. Small data critical regularity, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 965, pp. 365-384. http://www.numdam.org/item/SB_2005-2006__48__365_0/

[1] A. A. Belavin & A. M. Polyakov - “Metastable states of two-dimensional isotropic ferromagnets”, JETP Lett. 22 (1975), p. 245-247, Russian.

[2] J. Bourgain - “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations”, Geom. Funct. Anal. 3 (1993), no. 2, p. 107-156. | MR 1209299 | Zbl 0787.35097

[3] -, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation”, Geom. Funct. Anal. 3 (1993), no. 3, p. 209-262. | MR 1215780 | Zbl 0787.35098 | Zbl 0787.35097

[4] T. Cazenave, J. Shatah & A. S. Tahvildar-Zadeh - “Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields”, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 3, p. 315-349. | Numdam | MR 1622539 | Zbl 0918.58074

[5] S.-Y. A. Chang, L. Wang & P. C. Yang - “Regularity of harmonic maps”, Comm. Pure Appl. Math. 52 (1999), no. 9, p. 1099-1111. | MR 1692152 | Zbl 1044.58019

[6] Y. Choquet-Bruhat - “Future complete U(1) symmetric Einsteinian spacetimes, the unpolarized case”, in The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, p. 251-298. | MR 2098918 | Zbl 1064.83005

[7] D. Christodoulou & A. S. Tahvildar-Zadeh - “On the regularity of spherically symmetric wave maps”, Comm. Pure Appl. Math. 46 (1993), no. 7, p. 1041-1091. | MR 1223662 | Zbl 0744.58071

[8] E. Fradkin - Field theories of condensed matter systems, Frontiers in Physics, vol. 82, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, 1991. | MR 1257400 | Zbl 0984.82504

[9] C. H. Gu - “On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space”, Comm. Pure Appl. Math. 33 (1980), no. 6, p. 727-737. | MR 596432 | Zbl 0475.58005

[10] F. Hélein - “Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne”, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, p. 591-596. | MR 1101039 | Zbl 0728.35015

[11] M. Keel & T. Tao - “Local and global well-posedness of wave maps on 𝐑 1+1 for rough data”, Internat. Math. Res. Notices 21 (1998), p. 1117-1156. | Article | MR 1663216 | Zbl 0999.58013

[12] S. Klainerman & M. Machedon - “Space-time estimates for null forms and the local existence theorem”, Comm. Pure Appl. Math. 46 (1993), no. 9, p. 1221-1268. | MR 1231427 | Zbl 0803.35095

[13] -, “Smoothing estimates for null forms and applications”, Duke Math. J. 81 (1995), no. 1, p. 99-133, a celebration of John F. Nash, Jr. | MR 1381973 | Zbl 0909.35094

[14] S. Klainerman & I. Rodnianski - “On the global regularity of wave maps in the critical Sobolev norm”, Internat. Math. Res. Notices 13 (2001), p. 655-677. | Article | MR 1843256 | Zbl 0985.58009

[15] S. Klainerman & S. Selberg - “Remark on the optimal regularity for equations of wave maps type”, Comm. Partial Differential Equations 22 (1997), no. 5-6, p. 901-918. | MR 1452172 | Zbl 0884.35102

[16] J. Krieger - “Global regularity of wave maps from 𝐑 3+1 to surfaces”, Comm. Math. Phys. 238 (2003), no. 1-2, p. 333-366. | MR 1990880 | Zbl 1046.58010

[17] -, “Global regularity of wave maps from 𝐑 2+1 to H 2 . Small energy”, Comm. Math. Phys. 250 (2004), no. 3, p. 507-580. | MR 2094472 | Zbl 1099.58010

[18] -, “Stability of spherically symmetric wave maps”, Mem. Amer. Math. Soc. 181 (2006). | MR 2214492 | Zbl 1387.35392

[19] O. Ladyzhenskaya & V. Shubov - “Unique solvability of the Cauchy problem for the equations of the two dimensional chiral fields, taking values in complete Riemann manifolds”, J. Soviet Math. 25 (1984), p. 855-864. | Article | Zbl 0531.58017

[20] N. Manton & P. Sutcliffe - Topological solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2004. | Article | MR 2068924 | Zbl 1100.37044

[21] A. Nahmod, A. Stefanov & K. Uhlenbeck - “On the well-posedness of the wave map problem in high dimensions”, Comm. Anal. Geom. 11 (2003), no. 1, p. 49-83. | MR 2016196 | Zbl 1085.58022

[22] G. Ponce & T. C. Sideris - “Local regularity of nonlinear wave equations in three space dimensions”, Comm. Partial Differential Equations 18 (1993), no. 1-2, p. 169-177. | MR 1211729 | Zbl 0803.35096

[23] I. Rodnianski & J. Sterbenz - “On the Formation of Singularities in the Critical O(3) σ-Model”, preprint http://arxiv.org/abs/math/0605023. | MR 2680419 | Zbl 1213.35392

[24] J. Shatah - “Weak solutions and development of singularities of the SU (2) σ-model”, Comm. Pure Appl. Math. 41 (1988), no. 4, p. 459-469. | MR 933231 | Zbl 0686.35081

[25] J. Shatah & M. Struwe - “The Cauchy problem for wave maps”, Int. Math. Res. Not. 11 (2002), p. 555-571. | Article | MR 1890048 | Zbl 1024.58014

[26] J. Shatah & A. S. Tahvildar-Zadeh - “On the Cauchy problem for equivariant wave maps”, Comm. Pure Appl. Math. 47 (1994), no. 5, p. 719-754. | MR 1278351 | Zbl 0811.58059

[27] T. C. Sideris - “Global existence of harmonic maps in Minkowski space”, Comm. Pure Appl. Math. 42 (1989), no. 1, p. 1-13. | MR 973742 | Zbl 0685.58016

[28] M. Struwe - “Equivariant wave maps in two space dimensions”, Comm. Pure Appl. Math. 56 (2003), no. 7, p. 815-823, Dedicated to the memory of Jürgen K. Moser. | MR 1990477 | Zbl 1033.53019

[29] -, “Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets”, Calc. Var. Partial Differential Equations 16 (2003), no. 4, p. 431-437. | MR 1971037 | Zbl 1039.58033

[30] T. Tao - “Global regularity of wave maps I. Small critical Sobolev norm in high dimension”, Internat. Math. Res. Notices 6 (2001), p. 299-328. | MR 1820329 | Zbl 0983.35080

[31] -, “Global regularity of wave maps II. Small energy in two dimensions”, Comm. Math. Phys. 224 (2001), no. 2, p. 443-544. | MR 1869874 | Zbl 1020.35046

[32] D. Tataru - “Local and global results for wave maps. I”, Comm. Partial Differential Equations 23 (1998), no. 9-10, p. 1781-1793. | MR 1641721 | Zbl 0914.35083

[33] -, “On global existence and scattering for the wave maps equation”, Amer. J. Math. 123 (2001), no. 1, p. 37-77. | MR 1827277 | Zbl 0979.35100

[34] -, “The wave maps equation”, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, p. 185-204 (electronic). | MR 2043751 | Zbl 1065.35199

[35] -, “Rough solutions for the wave maps equation”, Amer. J. Math. 127 (2005), no. 2, p. 293-377. | MR 2130618 | Zbl 1330.58021 | Zbl pre02164530