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, ��(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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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