Renormalization and blow up for charge one equivariant critical wave maps. (English) Zbl 1139.35021

Summary: We prove the existence of equivariant finite time blow-up solutions for the wave map problem from \(\mathbb R^{2+1}\rightarrow S^{2}\) of the form \(u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)\) where \(u\) is the polar angle on the sphere, \(Q(r)=2\arctan r\) is the ground state harmonic map, \(\lambda (t)= t^{-1-\nu}\), and \(\mathcal{R}(t,r)\) is a radiative error with local energy going to zero as \(t \rightarrow 0\). The number \(\nu>\frac{1}{2}\) can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.


35B40 Asymptotic behavior of solutions to PDEs
35Q75 PDEs in connection with relativity and gravitational theory
35P25 Scattering theory for PDEs
35L70 Second-order nonlinear hyperbolic equations
Full Text: DOI arXiv


[1] Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Natl. Bureau Stand. Appl. Math. Ser., vol. 55. U.S. Government Printing Office, Washington, D.C. (1964) · Zbl 0171.38503
[2] Bizon, P., Tabor, Z.: Formation of singularities for equivariant 2+1-dimensional wave maps into the 2-sphere. Nonlinearity 14(5), 1041–1053 (2001) · Zbl 0988.35010
[3] Bizon, P., Ovchinnikov, Y.N., Sigal, I.M.: Collapse of an instanton. Nonlinearity 17(4), 1179–1191 (2004) · Zbl 1059.35081
[4] Cazenave, T., Shatah, J., Tahvildar-Zadeh, A.S.: Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang–Mills fields. Ann. Inst. Henri Poincaré, Phys. Théor. 68(3), 315–349 (1998) · Zbl 0918.58074
[5] Christodoulou, D., Tahvildar-Zadeh, A.S.: On the regularity of spherically symmetric wave maps. Commun. Pure Appl. Math. 46, 1041–1091 (1993) · Zbl 0789.58072
[6] Cote, R.: Instability of non-constant harmonic maps for the 2+1-dimensional equivariant wave map system. Int. Math. Res. Not. 2005(57), 3525–3549 (2005) · Zbl 1101.35055
[7] Dunford, N., Schwartz, J.: Linear Operators. Part II. Wiley Classics Library. John Wiley & Sons, Inc., New York (1988)
[8] Gesztesy, F., Zinchenko, M.: On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279, 1041–1082 (2006) · Zbl 1108.34063
[9] Isenberg, J., Liebling, S.: Singularity formation for 2+1 wave maps. J. Math. Phys. 43(1), 678–683 (2002) · Zbl 1052.58032
[10] Karageorgis, P., Strauss, W.A.: Instability of steady states for nonlinear wave and heat equations. Preprint (2006) · Zbl 1130.35015
[11] Krieger, J.: Global regularity of wave maps from R 2+1 to H 2. Small energy. Commun. Math. Phys. 250(3), 507–580 (2004) · Zbl 1099.58010
[12] Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19(4), 815–920 (2006) · Zbl 1281.35077
[13] Krieger, J., Schlag, W.: On the focusing critical semi-linear wave equation. Am. J. Math. 129(3), 843–913 (2007) · Zbl 1219.35144
[14] Krieger, J., Schlag, W.: Non-generic blow-up solutions for the critical focusing NLS in 1-d. Preprint (2005) · Zbl 1163.35035
[15] Krieger, J., Schlag, W., Tataru, D.: Slow blow-up solutions for the H 1 critical focusing semi-linear wave equation in \(\mathbb{R}\)3. Preprint (2007) · Zbl 1170.35066
[16] Merle, F., Raphael, P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13(3), 591–642 (2003) · Zbl 1061.35135
[17] Merle, F., Raphael, P.: On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004) · Zbl 1067.35110
[18] Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. (2) 161(1), 157–222 (2005) · Zbl 1185.35263
[19] Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19(1), 37–90 (2006) · Zbl 1075.35077
[20] Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001) · Zbl 1007.35087
[21] Rodnianski, I., Sterbenz, J.: On the formation of singularities in the critical O(3) {\(\sigma\)}-model. Preprint (2006) · Zbl 1213.35392
[22] Schlag, W.: Stable manifolds for an orbitally unstable NLS. To appear in Ann. Math. · Zbl 1180.35490
[23] Shatah, J.: Weak solutions and development of singularities of the SU(2) {\(\sigma\)}-model. Commun. Pure Appl. Math. 41(4), 459–469 (1988) · Zbl 0686.35081
[24] Shatah, J., Tahvildar-Zadeh, S.A.: Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds. Commun. Pure Appl. Math. 45(8), 947–971 (1992) · Zbl 0769.58015
[25] Shatah, J., Tahvildar-Zadeh, S.A.: On the Cauchy problem for equivariant wave maps. Commun. Pure Appl. Math. 47(5), 719–754 (1994) · Zbl 0811.58059
[26] Stein, E.: Harmonic Analysis. Princeton University Press (1993) · Zbl 0821.42001
[27] Struwe, M.: Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets. Calc. Var. Partial Differ. Equ. 16(4), 431–437 (2003) · Zbl 1039.58033
[28] Struwe, M.: Equivariant wave maps in two space dimensions. Commun. Pure Appl. Math. 56(7), 815–823 (2003) · Zbl 1033.53019
[29] Tao, T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001) · Zbl 1020.35046
[30] Tataru, D.: On global existence and scattering for the wave maps equation. Am. J. Math. 123(1), 37–77 (2001) · Zbl 0979.35100
[31] Tataru, D.: The wave maps equation. Bull. Am. Math. Soc. 41(2), 185–204 (2004) · Zbl 1065.35199
[32] Tataru, D.: Rough solutions for the wave maps equation. Am. J. Math 127(2), 293–377 (2005) · Zbl 1330.58021
[33] Watson, G.: A treatise on the theory of Bessel functions. Cambridge (1944) · Zbl 0063.08184
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