Merle, Frank; Raphaël, Pierre; Rodnianski, Igor Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. (Dynamique explosive de solutions régulières équivariantes de l’application de Schrödinger map.) (English. Abridged French version) Zbl 1213.35139 C. R., Math., Acad. Sci. Paris 349, No. 5-6, 279-283 (2011). Summary: We consider the energy critical Schrödinger map \(\partial_tu=u\wedge\Delta u\) to the 2-sphere for equivariant initial data of homotopy number \(k=1\). We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map \(Q_1\) in the scale invariant norm \(\dot H^1\) which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy\[ u(t,x)- e^{\Theta^*R}Q_1\bigg(\frac{x}{\lambda(t)}\bigg)\to u^* \quad\text{in }\dot H^1\text{ as }t\to T, \]where \(\Theta^*\in\mathbb R\), \(u^*\in\dot H^1\), \(R\) is a rotation and the concentration rate is given for some \(\kappa(u)>0\) by\[ \lambda(t)= \kappa(u) \frac{T-t}{|\log(T-t)|^2} \big(1+o(1)\big) \quad\text{as }t\to T. \] Cited in 20 Documents MSC: 35B44 Blow-up in context of PDEs 35K59 Quasilinear parabolic equations 35K15 Initial value problems for second-order parabolic equations Keywords:singularity formation; concentration of universal bubble PDFBibTeX XMLCite \textit{F. Merle} et al., C. R., Math., Acad. Sci. Paris 349, No. 5--6, 279--283 (2011; Zbl 1213.35139) Full Text: DOI arXiv References: [1] Bejenaru, I.; Tataru, D., Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions · Zbl 1303.58009 [2] I. Bejenaru, A. Ionescu, C. Kenig, D. Tataru, Global Schrödinger maps, Annals of Math., in press.; I. Bejenaru, A. Ionescu, C. Kenig, D. Tataru, Global Schrödinger maps, Annals of Math., in press. · Zbl 1233.35112 [3] Van den Bergh, J.; Hulshof, J.; King, J., Formal asymptotics of bubbling in the harmonic map heat flow, SIAM J. Appl. Math., 63, 1682-1717 (2003) · Zbl 1037.35023 [4] Chang, N.-H.; Shatah, J.; Uhlenbeck, K., Schrödinger maps, Comm. Pure Appl. Math., 53, 5 (2000) [5] Grotowski, J.; Shatah, J., Geometric evolution equations in critical dimensions, Calc. Var. Partial Differential Equations, 30, 4, 499-512 (2007) · Zbl 1127.58003 [6] Gustafson, S.; Kang, K.; Tsai, T.-P., Schrödinger flow near harmonic maps, Comm. Pure Appl. Math., 60, 4, 463-499 (2007) · Zbl 1144.53085 [7] Gustafson, S.; Kang, K.; Tsai, T.-P., Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Math. J., 145, 3, 537-583 (2008) · Zbl 1170.35091 [8] Gustafson, S.; Nakanishi, K.; Tsai, T.-P., Asymptotic stability, concentration and oscillations in harmonic map heat flow, Landau Lifschitz and Schrödinger maps on \(R^2\), Comm. Math. Phys., 300, 1, 205-242 (2010) · Zbl 1205.35294 [9] 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 [10] Merle, F.; Raphaël, P., Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal., 13, 591-642 (2003) · Zbl 1061.35135 [11] Merle, F.; Raphaël, P., On universality of blow up profile for \(L^2\) critical nonlinear Schrödinger equation, Invent. Math., 156, 565-672 (2004) · Zbl 1067.35110 [12] Merle, F.; Raphaël, P., Profiles and quantization of the blow up mass for critical non linear Schrödinger equation, Comm. Math. Phys., 253, 3, 675-704 (2004) · Zbl 1062.35137 [13] Merle, F.; Raphaël, P., Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Annals of Math., 161, 1, 157-222 (2005) · Zbl 1185.35263 [14] Merle, F.; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19, 1, 37-90 (2006) · Zbl 1075.35077 [15] F. Merle, P. Raphaël, I. Rodnianski, Blow up for the energy critical corotational Schrödinger map, preprint 2011.; F. Merle, P. Raphaël, I. Rodnianski, Blow up for the energy critical corotational Schrödinger map, preprint 2011. [16] Qing, J., On singularities of the heat flow for harmonic maps from surfaces into spheres, Comm. Anal. Geom., 3, 297-315 (1995) · Zbl 0868.58021 [17] Qing, J.; Tian, G., Bubbling of the heat flows for harmonic maps from surface, Comm. Pure Appl. Math., 50, 295-310 (1997) · Zbl 0879.58017 [18] Raphaël, P., Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. Ann., 331, 577-609 (2005) · Zbl 1082.35143 [19] P. Raphaël, I. Rodnianski, Stable blow up dynamics for the critical corotational Wave map and equivariant Yang-Mills problems, Publi. I.H.E.S., in press.; P. Raphaël, I. Rodnianski, Stable blow up dynamics for the critical corotational Wave map and equivariant Yang-Mills problems, Publi. I.H.E.S., in press. · Zbl 1284.35358 [20] Raphaël, P.; Szeftel, J., Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., 24, 471-546 (2011) · Zbl 1218.35226 [21] Struwe, M., On the evolution of harmonic mappings of Riemannian surfaces, Comm. Math. Helv., 60, 558-581 (1985) · Zbl 0595.58013 [22] Topping, P. M., Winding behaviour of finite-time singularities of the harmonic map heat flow, Math. Z., 247 (2004) · Zbl 1067.53055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.