Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills Problems

Raphaël, Pierre; Rodnianski, Igor

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 22, 12 p.

This note summarizes the results obtained in [30]. 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 equivariant critical $SO\left(4\right)$ Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

[1] Atiyah, M.; Drinfield, V. G; Hitchin, N..; Manin, Y. I. Construction of instantons. Phys. Lett. A65 (1978), 185-187. | MR | Zbl

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

[3] Belavin A.A., Polyakov A.M., Schwarz, A.S, Tyupkin Y.S, Pseudoparticle solutions of the Yang-Mills equation, Phus. Lett B59, 85 (1975). | MR

[4] Bizon, P.; Chmaj, T.; Tabor, Z., Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere. Nonlinearity 14 (2001), no. 5, 1041–1053. | MR | Zbl

[5] Bizon, P.; Ovchinnikov, Y. N.; Sigal, I. M., Collapse of an instanton. Nonlinearity 17 (2004), no. 4, 1179–1191. | MR | Zbl

[6] Bogomol’nyi, E.B., The stability of classical solutions. Soviet J. Nuclear Phys. 24 (1976), no. 4, 449–454 (Russian). | MR

[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. I.H.P., section A 68 (1998), no. 3, 315–349. | EuDML | Numdam | MR | Zbl

[8] Christodoulou, D.; Tahvildar-Zadeh, A. S., On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math. 46 (1993), no. 7, 1041–1091. | MR | Zbl

[9] Côte, R., Instability of nonconstant harmonic maps for the $(1+2)$-dimensional equivariant wave map system. Int. Math. Res. Not. 2005, no. 57, 3525–3549. | MR | Zbl

[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. Comm. Math. Phys. 284 (2008), no. 1, 203–225 | MR

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

[12] Isenberg J.; Liebling, S.L., Singularity Formation in 2+1 Wave Maps. J. Math. Phys. 43 (2002), 678–683. | MR | Zbl

[13] Klainerman, S., Machedon, M., On the regularity properties of a model problem related to wave maps. Duke Math. J. 87 (1997), no. 3, 553–589 | MR | Zbl

[14] Klainerman S., Selberg, Z., Remark on the optimal regularity for equations of wave maps type. Comm. Partial Differential Equations 22 (1997), no. 5-6, 901–918. | MR | Zbl

[15] Kavian, O.; Weissler, F. B., Finite energy self-similar solutions of a nonlinear wave equation. Comm. Partial Differential Equations 15 (1990), no. 10, 1381–1420. l | MR | Zbl

[16] Kenig, C.E.; Merle, F., Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201 (2008), no. 2, 147–212. | MR

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

[18] Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. | MR | Zbl

[19] Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for the critical Yang-Mills problem, Adv. Math. 221 (2009), no. 5, 1445–1521. | MR

[20] Lemou, M.; Mehats, F.; Raphaël, P., Stable self similar blow up solutions to the relativistic gravitational Vlasov-Poisson system, J. Amer. Math. Soc. 21 (2008), no. 4, 1019–1063. | MR

[21] Manton, N.; Sutcliffe, P. Topological solitons. Cambridge University Press, 2004. | MR | Zbl

[22] Martel, Y.; Merle, F., Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation. J. Amer. Math. Soc. 15 (2002), no. 3, 617–664. | MR | Zbl

[23] Merle, F.; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math. 161 (2005), no. 1, 157–222. | MR

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

[25] Merle, F.; Raphaël, P., On universality of blow up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). | MR | Zbl

[26] Merle, F.; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37–90. | MR | Zbl

[27] Merle, F.; Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), no. 3, 675–704. | MR | Zbl

[28] Perelman, G., On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2 (2001), no. 4, 605–673. | MR | Zbl

[29] Raphaël, P., Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), no. 3, 577–609. | MR | Zbl

[30] Raphaël, P.; Rodnianski, R., Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills, submitted.

[31] Rodnianski, I., Sterbenz, J., On the formation of singularities in the critical $O\left(3\right)$ $\sigma$-model, to appear Ann. Math. | MR

[32] Shatah, J., Weak solutions and development of singularities of the $\mathrm{SU}\left(2\right)$ $\sigma$-model. Comm. Pure Appl. Math. 41 (1988), no. 4, 459–469 | MR | Zbl

[33] Shatah, J.; Tahvildar-Zadeh, A. S., On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. | MR | Zbl

[34] Sterbenz J., Tataru, D., Energy dispersed arge data wave maps in $2+1$ dimensions, preprint, arXiv:0906.3384.

[35] Sterbenz J., Tataru, D., Regularity of Wave-Maps in dimension $2+1$, preprint, arXiv:0907.3148.

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

[37] Tao, T., Global regularity of wave maps. II. Small energy in two dimensions. Comm. Math. Phys. 224 (2001), no. 2, 443–544. | MR | Zbl

[38] Tao, T., Geometric renormalization of large energy wave maps.

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

[40] Tataru, D., On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37–77. | MR | Zbl

[41] Ward, R. Slowly moving lumps in the $ℂP1$ model in $(2+1)$ dimensions, Phys. Lett. B158 (1985), 424–428. | MR

[42] Witten, E., Some exact multipseudoparticle solutions of the classical Yang-Mills theory. Phys. Rev. Lett. 38 (1977) 121–124.