The wave map problem. Small data critical regularity
[Le problème de l'application d'onde. Régularité critique des petites données initiales]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 965, pp. 365-384.

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).

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).

Classification : 35L05, 35Q99
Keywords: wave map, critical regularity, renormalization
Mot clés : application d'onde, régularité critique, renormalisation
@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},
     series = {Ast\'erisque},
     note = {talk:965},
     pages = {365--384},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     mrnumber = {2359050},
     zbl = {1194.35004},
     language = {en},
     url = {http://archive.numdam.org/item/SB_2005-2006__48__365_0/}
}
TY  - CHAP
AU  - Rodnianski, Igor
TI  - The wave map problem. Small data critical regularity
BT  - Séminaire Bourbaki : volume 2005/2006, exposés 952-966
AU  - Collectif
T3  - Astérisque
N1  - talk:965
PY  - 2007
SP  - 365
EP  - 384
IS  - 311
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/SB_2005-2006__48__365_0/
LA  - en
ID  - SB_2005-2006__48__365_0
ER  - 
%0 Book Section
%A Rodnianski, Igor
%T The wave map problem. Small data critical regularity
%B Séminaire Bourbaki : volume 2005/2006, exposés 952-966
%A Collectif
%S Astérisque
%Z talk:965
%D 2007
%P 365-384
%N 311
%I Société mathématique de France
%U http://archive.numdam.org/item/SB_2005-2006__48__365_0/
%G en
%F SB_2005-2006__48__365_0
Rodnianski, Igor. The wave map problem. Small data critical regularity, dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 965, pp. 365-384. http://archive.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. | EuDML | MR | Zbl

[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 | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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. | DOI | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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