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Renormalization and blow up for charge one equivariant critical wave maps

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Abstract

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→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, λ(t)=t -1-ν, and \(\mathcal{R}(t,r)\) is a radiative error with local energy going to zero as t→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.

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Authors and Affiliations

  1. Dept. of Mathematics, Science Center, Harvard University, Cambridge, MA, 1 Oxford Street, 02138, USA

    J. Krieger

  2. Department of Mathematics, The University of Chicago, Chicago, IL, 5734 South University Avenue, 60637, USA

    W. Schlag

  3. Department of Mathematics, The University of California at Berkeley, Berkeley, CA, Evans Hall, 94720, USA

    D. Tataru

Authors
  1. J. Krieger
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  2. W. Schlag
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  3. D. Tataru
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Correspondence to D. Tataru.

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Mathematics Subject Classification (1991)

35L05, 35Q75, 35P25

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

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  • DOI: https://doi.org/10.1007/s00222-007-0089-3

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