Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
Publications Mathématiques de l'IHÉS, Tome 115 (2012), pp. 1-122.

We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

DOI : https://doi.org/10.1007/s10240-011-0037-z
PUBLISHER-ID : s10240-011-0037-z
@article{PMIHES_2012__115__1_0,
     author = {Rapha\"el, Pierre and Rodnianski, Igor},
     title = {Stable blow up dynamics for the critical co-rotational wave maps and equivariant {Yang-Mills} problems},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--122},
     publisher = {Springer-Verlag},
     volume = {115},
     year = {2012},
     doi = {10.1007/s10240-011-0037-z},
     zbl = {1284.35358},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-011-0037-z/}
}
TY  - JOUR
AU  - Raphaël, Pierre
AU  - Rodnianski, Igor
TI  - Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
JO  - Publications Mathématiques de l'IHÉS
PY  - 2012
DA  - 2012///
SP  - 1
EP  - 122
VL  - 115
PB  - Springer-Verlag
UR  - http://archive.numdam.org/articles/10.1007/s10240-011-0037-z/
UR  - https://zbmath.org/?q=an%3A1284.35358
UR  - https://doi.org/10.1007/s10240-011-0037-z
DO  - 10.1007/s10240-011-0037-z
LA  - en
ID  - PMIHES_2012__115__1_0
ER  - 
Raphaël, Pierre; Rodnianski, Igor. Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems. Publications Mathématiques de l'IHÉS, Tome 115 (2012), pp. 1-122. doi : 10.1007/s10240-011-0037-z. http://archive.numdam.org/articles/10.1007/s10240-011-0037-z/

[1.] Atiyah, M.; Drinfield, V. G.; Hitchin, N.; Manin, Y. I. Construction of instantons, Phys. Lett. A, Volume 65 (1978), pp. 185-187 | Article | MR 598562 | Zbl 0424.14004

[2.] Belavin, A. A.; Polyakov, A. M. Metastable states of two-dimensional isotropic ferromagnets, JETP Lett., Volume 22 (1975), pp. 245-247 (Russian)

[3.] Belavin, A. A.; Polyakov, A. M.; Schwarz, A. S.; Tyupkin, Y. S. Pseudoparticle solutions of the Yang-Mills equation, Phys. Lett. B, Volume 59 (1975), p. 85 | Article | MR 434183

[4.] Bizon, P.; Chmaj, T.; Tabor, Z. Formation of singularities for equivariant (2+1)-dimensional wave maps into the 2-sphere, Nonlinearity, Volume 14 (2001), pp. 1041-1053 | Article | MR 1862811 | Zbl 0988.35010

[5.] Bizon, P.; Ovchinnikov, Y. N.; Sigal, I. M. Collapse of an instanton, Nonlinearity, Volume 17 (2004), pp. 1179-1191 | Article | MR 2069700 | Zbl 1059.35081

[6.] Bogomol’nyi, E. B. The stability of classical solutions, Sov. J. Nucl. Phys., Volume 24 (1976), pp. 449-454 (Russian) | MR 443719

[7.] Cazenave, T.; Shatah, J.; Tahvildar-Zadeh, S. Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. Henri Poincaré, A, Volume 68 (1998), pp. 315-349 | MR 1622539 | Zbl 0918.58074

[8.] Christodoulou, D.; Tahvildar-Zadeh, A. S. On the regularity of spherically symmetric wave maps, Commun. Pure Appl. Math., Volume 46 (1993), pp. 1041-1091 | Article | MR 1223662 | Zbl 0744.58071

[9.] Côte, R. Instability of nonconstant harmonic maps for the (1+2)-dimensional equivariant wave map system, Int. Math. Res. Not., Volume 2005 (2005), pp. 3525-3549 | Article | Zbl 1101.35055

[10.] Côte, 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, Commun. Math. Phys., Volume 284 (2008), pp. 203-225 | Article | Zbl 1170.35064

[11.] Donaldson, S. K.; Kronheimer, P. B. Geometry of Four-Manifolds, Clarendon Press, Oxford, 1990 | Zbl 0820.57002

[12.] Eells, J.; Lemaire, L. Two Reports on Harmonic Maps, World Scientific Publishing Co., Inc., River Edge, 1995 | Article | Zbl 0836.58012

[13.] Hélein, F. Régularité des applications faiblement harmoniques entre une surface et une sphére, C. R. Acad. Sci. Paris Sér. I Math., Volume 311 (1990), pp. 519-524 | Zbl 0728.35014

[14.] Isenberg, J.; Liebling, S. L. Singularity formation in 2+1 wave maps, J. Math. Phys., Volume 43 (2002), pp. 678-683 | Article | MR 1872523 | Zbl 1052.58032

[15.] Kavian, O.; Weissler, F. B. Finite energy self-similar solutions of a nonlinear wave equation, Commun. Partial Differ. Equ., Volume 15 (1990), pp. 1381-1420 | Article | MR 1077471 | Zbl 0726.35085

[16.] Kenig, C. E.; Merle, F. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., Volume 201 (2008), pp. 147-212 | Article | MR 2461508 | Zbl 1183.35202

[17.] Klainerman, S.; Machedon, M. On the regularity properties of a model problem related to wave maps, Duke Math. J., Volume 87 (1997), pp. 553-589 | Article | MR 1446618 | Zbl 0878.35075

[18.] Klainerman, S.; Selberg, Z. Remark on the optimal regularity for equations of wave maps type, Commun. Partial Differ. Equ., Volume 22 (1997), pp. 901-918 | Article | MR 1452172 | Zbl 0884.35102

[19.] J. Krieger and W. Schlag, Concentration compactness for critical wave maps, preprint, arXiv:0908.2474.

[20.] Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., Volume 171 (2008), pp. 543-615 | Article | MR 2372807 | Zbl 1139.35021

[21.] Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for the critical Yang-Mills problem, Adv. Math., Volume 221 (2009), pp. 1445-1521 | Article | MR 2522426 | Zbl 1183.35203

[22.] Lemou, M.; Mehats, F.; Raphaël, P. Stable self similar blow up solutions to the relativistic gravitational Vlasov-Poisson system, J. Am. Math. Soc., Volume 21 (2008), pp. 1019-1063 | Article | Zbl 1206.82092

[23.] Manton, N.; Sutcliffe, P. Topological Solitons, Cambridge University Press, Cambridge, 2004 | Article | Zbl 1100.37044

[24.] Martel, Y.; Merle, F. Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equation, J. Am. Math. Soc., Volume 15 (2002), pp. 617-664 | Article | MR 1896235 | Zbl 0996.35064

[25.] Merle, F.; Raphaël, P. Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal., Volume 13 (2003), pp. 591-642 | Article | MR 1995801 | Zbl 1061.35135

[26.] Merle, F.; Raphaël, P. On universality of blow up profile for L 2 critical nonlinear Schrödinger equation, Invent. Math., Volume 156 (2004), pp. 565-672 | Article | MR 2061329 | Zbl 1067.35110

[27.] Merle, F.; Raphaël, P. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math., Volume 161 (2005), pp. 157-222 | Article | Zbl 1185.35263

[28.] Merle, F.; Raphaël, P. Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Commun. Math. Phys., Volume 253 (2005), pp. 675-704 | Article | Zbl 1062.35137

[29.] Merle, F.; Raphaël, P. Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Am. Math. Soc., Volume 19 (2006), pp. 37-90 | Article | Zbl 1075.35077

[30.] Morrey, C. B. The problem of Plateau on a Riemannian manifold, Ann. Math., Volume 49 (1948), pp. 807-851 | Article | MR 27137 | Zbl 0033.39601

[31.] Perelman, G. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, Volume 2 (2001), pp. 605-673 | Article | MR 1852922 | Zbl 1007.35087

[32.] Piette, B.; Zakrzewski, W. J. Shrinking of solitons in the (2+1)-dimensional S 2 sigma model, Nonlinearity, Volume 9 (1996), pp. 897-910 | Article | MR 1399478 | Zbl 0895.58030

[33.] Raphaël, P. Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann., Volume 331 (2005), pp. 577-609 | Article | MR 2122541 | Zbl 1082.35143

[34.] I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical O(3) σ-model, Ann. Math., to appear | MR 2680419 | Zbl 1213.35392

[35.] Shatah, J. Weak solutions and development of singularities of the SU(2) σ-model, Commun. Pure Appl. Math., Volume 41 (1988), pp. 459-469 | Article | MR 933231 | Zbl 0686.35081

[36.] Shatah, J.; Tahvildar-Zadeh, A. S. On the Cauchy problem for equivariant wave maps, Commun. Pure Appl. Math., Volume 47 (1994), pp. 719-754 | Article | MR 1278351 | Zbl 0811.58059

[37.] I. M. Sigal and Y. N. Ovchinnikov, On collapse of wave maps, preprint, arXiv:0909.3085. | MR 2831768 | Zbl 1232.35138

[38.] J. Sterbenz and D. Tataru, Energy dispersed arge data wave maps in 2+1 dimensions, preprint, arXiv:0906.3384. | MR 2657817

[39.] J. Sterbenz and D. Tataru, Regularity of wave-maps in dimension 2+1, preprint, arXiv:0907.3148. | MR 2657818 | Zbl 1218.35057

[40.] Struwe, M. Equivariant wave maps in two space dimensions. Dedicated to the memory of Jürgen K. Moser, Commun. Pure Appl. Math., Volume 56 (2003), pp. 815-823 | Article | MR 1990477 | Zbl 1033.53019

[41.] Tao, T. Global regularity of wave maps. II. Small energy in two dimensions, Commun. Math. Phys., Volume 224 (2001), pp. 443-544 | Article | MR 1869874 | Zbl 1020.35046

[42.] T. Tao, Geometric renormalization of large energy wave maps. | Zbl 1087.58019

[43.] T. Tao, Global regularity of wave maps III–VII, preprints, arXiv:0908.0776.

[44.] Tataru, D. On global existence and scattering for the wave maps equation, Am. J. Math., Volume 123 (2001), pp. 37-77 | Article | MR 1827277 | Zbl 0979.35100

[45.] Uhlenbeck, K. Removable singularities in Yang-Mills fields, Commun. Math. Phys., Volume 83 (1982), pp. 11-29 | Article | MR 648355 | Zbl 0491.58032

[46.] Ward, R. Slowly moving lumps in the C P1 model in (2+1) dimensions, Phys. Lett. B, Volume 158 (1985), pp. 424-428 | Article | MR 802039

[47.] Witten, E. Some exact multipseudoparticle solutions of the classical Yang-Mills theory, Phys. Rev. Lett., Volume 38 (1977), pp. 121-124 | Article

Cité par Sources :