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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 & 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,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 357. doi: 10.1002/cpa.3043. Google Scholar

[2]

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

[3]

F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials,, \emph{Math. Nachr.}, 279 (2006), 1041. doi: 10.1002/mana.200510410. Google Scholar

[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,, \emph{Communications in Mathematical Physics}, 284 (2008), 203. doi: 10.1007/s00220-008-0604-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation,, preprint, (1201). doi: 10.1007/s00208-013-0898-1. Google Scholar

[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,, \emph{J. Eur. Math. Soc.}, 13 (2011), 533. doi: 10.4171/JEMS/261. Google Scholar

[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, (). doi: 10.4171/JEMS/336. Google Scholar

[10]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.1007/s00039-012-0174-7. Google Scholar

[11]

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

[12]

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

[13]

J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions,, \emph{Journal de Mathematiques Pures et Appliquees}, (). doi: 10.1016/j.matpur.2013.10.008. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems,, Second edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar

[23]

M. Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074. Google Scholar

[24]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions,, \emph{Comm. Math. Phys.}, 224 (2001), 443. doi: 10.1007/PL00005588. Google Scholar

[25]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension,, \emph{Internat. Math. Res. Notices}, (2001), 299. doi: 10.1155/S1073792801000150. Google Scholar

[26]

D. Tataru, Rough solutions for the wave maps equation,, \emph{Amer. J. Math.}, 127 (2005), 293. Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

F. Gesztesy and M. Zinchenko, On spectral theory for Schrodinger operators with strongly singular potentials,, \emph{Math. Nachr.}, 279 (2006), 1041. doi: 10.1002/mana.200510410. Google Scholar

[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,, \emph{Communications in Mathematical Physics}, 284 (2008), 203. doi: 10.1007/s00220-008-0604-4. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

R. Donninger and J. Krieger, Nonscattering solutions and blow up at infinity for the critical wave equation,, preprint, (1201). doi: 10.1007/s00208-013-0898-1. Google Scholar

[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,, \emph{J. Eur. Math. Soc.}, 13 (2011), 533. doi: 10.4171/JEMS/261. Google Scholar

[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, (). doi: 10.4171/JEMS/336. Google Scholar

[10]

T. Duyckaerts, C. Kenig and F. Merle, Profiles of bounded radial solutions of the focusing, energy-critical wave equation,, preprint, (). doi: 10.1007/s00039-012-0174-7. Google Scholar

[11]

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

[12]

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

[13]

J. Krieger and W. Schlag, Full range of blow up exponents for the quintic wave equation in three dimensions,, \emph{Journal de Mathematiques Pures et Appliquees}, (). doi: 10.1016/j.matpur.2013.10.008. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. Struwe, Variational methods, Applications to Nonlinear PDEs and Hamiltonian Systems,, Second edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar

[23]

M. Struwe, Equivariant wave maps in two space dimensions,, \emph{Comm. Pure Appl. Math.}, 56 (2003), 815. doi: 10.1002/cpa.10074. Google Scholar

[24]

T. Tao, Global regularity of wave maps II. Small energy in two dimensions,, \emph{Comm. Math. Phys.}, 224 (2001), 443. doi: 10.1007/PL00005588. Google Scholar

[25]

T. Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension,, \emph{Internat. Math. Res. Notices}, (2001), 299. doi: 10.1155/S1073792801000150. Google Scholar

[26]

D. Tataru, Rough solutions for the wave maps equation,, \emph{Amer. J. Math.}, 127 (2005), 293. Google Scholar

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