• Previous Article
    On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds
  • CPAA Home
  • This Issue
  • Next Article
    Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay
September  2015, 14(5): 1705-1741. doi: 10.3934/cpaa.2015.14.1705

Optimal polynomial blow up range for critical wave maps

1. 

Bâtiment des Mathématiques, EPFL, Station 8, CH-1015 Lausanne, Switzerland

Received  April 2014 Revised  March 2015 Published  June 2015

We prove that the critical Wave Maps equation with target $S^2$ and origin $R^{2+1}$ admits energy class blow up solutions of the form \begin{eqnarray} u(t, r) = Q(\lambda(t)r) + \varepsilon(t, r) \end{eqnarray} where $Q:R^2\rightarrow S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu>0$. This extends the work [14], where such solutions were constructed under the assumption $\nu>\frac{1}{2}$. In light of a result of Struwe [23], our result is optimal for polynomial blow up rates.
Citation: Can Gao, Joachim Krieger. Optimal polynomial blow up range for critical wave maps. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1705-1741. doi: 10.3934/cpaa.2015.14.1705
References:
[1]

P. d'Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system, Comm. Pure Appl. Math., 57 (2004), 357-383. doi: 10.1002/cpa.3043.

[2]

P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201. doi: 10.1088/0951-7715/17/6/009.

[3]

F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials, Math. Nachr., 279 (2006), 1041-1082. doi: 10.1002/mana.200510410.

[4]

R. Cote, C. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system, Communications in Mathematical Physics, 284 (2008), 203-225. doi: 10.1007/s00220-008-0604-4.

[5]

R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I, preprint. doi: 10.1353/ajm.2015.0002.

[6]

R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II, preprint. doi: 10.1353/ajm.2015.0003.

[7]

R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation, preprint, arXiv: 1201.3258v1. doi: 10.1007/s00208-013-0898-1.

[8]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation, J. Eur. Math. Soc., 13 (2011), 533-599. doi: 10.4171/JEMS/261.

[9]

T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case, preprint, arXiv:1003.0625, to appear in JEMS. doi: 10.4171/JEMS/336.

[10]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, preprint, arXiv:1201.4986, to appear in GAFA. doi: 10.1007/s00039-012-0174-7.

[11]

T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, preprint, arXiv:1204.0031. doi: 10.4310/CJM.2013.v1.n1.a3.

[12]

S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1995), 99-133. doi: 10.1215/S0012-7094-95-08109-5.

[13]

J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions, Journal de Mathematiques Pures et Appliquees, to appear. doi: 10.1016/j.matpur.2013.10.008.

[14]

J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., 171 (2008), 543-615. doi: 10.1007/s00222-007-0089-3.

[15]

J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53. doi: 10.1215/00127094-2009-005.

[16]

F. Merle, P. Raphael and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schröinger map problem, Invent. Math., 193 (2013), 249-365. doi: 10.1007/s00222-012-0427-y.

[17]

P. Raphael and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes tudes Sci., 115 (2012), 1-122. doi: 10.1007/s10240-011-0037-z.

[18]

I. Rodnianski and J. Sterbenz, On the Formation of Singularities in the Critical $O(3)$ Sigma-Model, Annals of Math., 172 (2010), 187-242. doi: 10.4007/annals.2010.172.187.

[19]

J. Shatah and S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754. doi: 10.1002/cpa.3160470507.

[20]

J. Sterbenz and D. Tataru, Regularity of Wave-Maps in dimension $2+1$, Preprint 2009. doi: 10.1007/s00220-010-1062-3.

[21]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Preprint 2009. doi: 10.1007/s00220-010-1061-4.

[22]

M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems, Second edition, Springer Verlag, New York, 1996. doi: 10.1007/978-3-662-03212-1.

[23]

M. Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[24]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544. doi: 10.1007/PL00005588.

[25]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices, 2001, 299-328. doi: 10.1155/S1073792801000150.

[26]

D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math., 127 (2005), 293-377.

show all references

References:
[1]

P. d'Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system, Comm. Pure Appl. Math., 57 (2004), 357-383. doi: 10.1002/cpa.3043.

[2]

P. Bizoń, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity, 17 (2004), 2187-2201. doi: 10.1088/0951-7715/17/6/009.

[3]

F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials, Math. Nachr., 279 (2006), 1041-1082. doi: 10.1002/mana.200510410.

[4]

R. Cote, C. Kenig and F. Merle, Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system, Communications in Mathematical Physics, 284 (2008), 203-225. doi: 10.1007/s00220-008-0604-4.

[5]

R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: I, preprint. doi: 10.1353/ajm.2015.0002.

[6]

R. Cote, C. Kenig, A. Lawrie and W. Schlag, Characterization of large energy solutions of the equivariant wave map problem: II, preprint. doi: 10.1353/ajm.2015.0003.

[7]

R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation, preprint, arXiv: 1201.3258v1. doi: 10.1007/s00208-013-0898-1.

[8]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation, J. Eur. Math. Soc., 13 (2011), 533-599. doi: 10.4171/JEMS/261.

[9]

T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation: the non-radial case, preprint, arXiv:1003.0625, to appear in JEMS. doi: 10.4171/JEMS/336.

[10]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation, preprint, arXiv:1201.4986, to appear in GAFA. doi: 10.1007/s00039-012-0174-7.

[11]

T. Duyckaerts, C. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, preprint, arXiv:1204.0031. doi: 10.4310/CJM.2013.v1.n1.a3.

[12]

S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications. A celebration of John F. Nash, Jr, Duke Math. J., 81 (1995), 99-133. doi: 10.1215/S0012-7094-95-08109-5.

[13]

J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions, Journal de Mathematiques Pures et Appliquees, to appear. doi: 10.1016/j.matpur.2013.10.008.

[14]

J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., 171 (2008), 543-615. doi: 10.1007/s00222-007-0089-3.

[15]

J. Krieger, W. Schlag and D. Tataru, Slow blow-up solutions for the $H^1(\R^3)$ critical focusing semilinear wave equation, Duke Math. J., 147 (2009), 1-53. doi: 10.1215/00127094-2009-005.

[16]

F. Merle, P. Raphael and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schröinger map problem, Invent. Math., 193 (2013), 249-365. doi: 10.1007/s00222-012-0427-y.

[17]

P. Raphael and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes tudes Sci., 115 (2012), 1-122. doi: 10.1007/s10240-011-0037-z.

[18]

I. Rodnianski and J. Sterbenz, On the Formation of Singularities in the Critical $O(3)$ Sigma-Model, Annals of Math., 172 (2010), 187-242. doi: 10.4007/annals.2010.172.187.

[19]

J. Shatah and S. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719-754. doi: 10.1002/cpa.3160470507.

[20]

J. Sterbenz and D. Tataru, Regularity of Wave-Maps in dimension $2+1$, Preprint 2009. doi: 10.1007/s00220-010-1062-3.

[21]

J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Preprint 2009. doi: 10.1007/s00220-010-1061-4.

[22]

M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems, Second edition, Springer Verlag, New York, 1996. doi: 10.1007/978-3-662-03212-1.

[23]

M. Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math., 56 (2003), 815-823. doi: 10.1002/cpa.10074.

[24]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys., 224 (2001), 443-544. doi: 10.1007/PL00005588.

[25]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices, 2001, 299-328. doi: 10.1155/S1073792801000150.

[26]

D. Tataru, Rough solutions for the wave maps equation, Amer. J. Math., 127 (2005), 293-377.

[1]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[2]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[3]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[4]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459

[5]

Joachim Krieger, Kenji Nakanishi, Wilhelm Schlag. Global dynamics of the nonradial energy-critical wave equation above the ground state energy. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2423-2450. doi: 10.3934/dcds.2013.33.2423

[6]

César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067

[7]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[8]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351

[9]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261

[10]

Yue Pang, Xingchang Wang, Furong Wu. Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4439-4463. doi: 10.3934/dcdss.2021115

[11]

George Osipenko. Linearization near a locally nonunique invariant manifold. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 189-205. doi: 10.3934/dcds.1997.3.189

[12]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735

[13]

Antonios Zagaris, Christophe Vandekerckhove, C. William Gear, Tasso J. Kaper, Ioannis G. Kevrekidis. Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2759-2803. doi: 10.3934/dcds.2012.32.2759

[14]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[15]

Boris Hasselblatt. Critical regularity of invariant foliations. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931

[16]

Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285

[17]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[18]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

[19]

Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure and Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921

[20]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]