Renormalization and blow up for charge one equivariant critical wave maps

By J. Krieger, Wilhelm Schlag, D. Tataru. 2006

249 citations recorded by the source corpus.

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S2 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79f0b9010c60bbf32459fd81787dc4863a2b0a2d

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Citations in the corpus

Nonradial type II blow up for the energy-supercritical semilinear heat equation (2016, background): “The possibility of various blow-up speeds is linked to the regularity of the solutions and this is why parabolic problems are more rigid, thanks to the regularizing effect, than dispersive problems, for which a wider range of concentration scenarios exists [20].”
Regularity of Wave-Maps in Dimension 2 + 1 (2009, background): “The results in [16] assert that blow-up with a rescaled Q profile can occur for initial data with energy arbitrarily close to E[Q].”
On collapsing ring blow up solutions to the mass supercritical NLS (2012, background): “Recall that no upper bound on the blow up rate is known in the mass critical case sc = 0, and arbitrary slow type II concentration(2) should be expected for the energy critical problem sc = 1 in the continuation of [17].”
Asymptotic decomposition for semilinear Wave and equivariant wave map equations (2015, methodology): “Indeed, singular solutions in the form of a shrinking soliton plus a residue term have been constructed for the energy critical wave equation in [28] for 3+ 1 dimensional case and in [20] for the 4+ 1 dimensional case, for the 2+1 equivariant dimensional wave map equation in [26, 30], and for the ra”
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation (2009, result): “We also refer to the works of Rodnianski and Sterbenz [RS], Krieger, Schlag and Tataru [ KST08 ] and Rapha¨el and Rodnianski [RR] for the construction of blow-up solutions.”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “More interesting is that for certain non-convex targets, for example when we take M = S(2), finite time blowup of smooth solutions is possible (see [13], [19]).”
Soliton resolution for equivariant wave maps to the sphere (2013, methodology): “Actual examples of wave maps blowing up in finite time were constructed by Rodnianski and Sterbenz [22] and Raphaël and Rodnianski [21] (as a perturbation of the self similar regime), and by Krieger, Schlag and Tataru [19] (with prescribed, polynomial blow up rate).”
Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, methodology): “[14] Krieger, J., Schlag, W. & Tataru, D., Renormalization and blow up for charge one equivariant critical wave maps.”
Blow up for the critical gKdV equation II: minimal mass dynamics (2012, background): “It is not clear to us whether a direct fixed point approach as in [3, 23, 24, 10, 11] is applicable here due to the poor localization in space of the minimal element.”
Concentration Compactness for Critical Wave Maps (2009, unknown): “Finally, Rodnianski, Sterbenz [38], as well as the authors together with Daniel Tataru [26] exhibited finite energy wave maps from R2+1 → S2 that blow up in finite time by suitable rescaling of harmonic maps.”
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “Thus, by passing to a subsequence, we may assume that φn[0] converges in Ḣ1 to a limit Φ∞, which has a maximal Cauchy development φ∞ on an open neighbourhood of 0.”
Blow up for the critical gKdV equation III: exotic regimes (2012, result): “[27] J. Krieger, W. Schlag and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps.”
Infinite-time blow-up for the 3-dimensional energy-critical heat equation (2017, background): “11, 12]. Construction of families of blow-up solutions for n= 3 has been achieved in [9, 19]. In particular, a continuum of blow-up and vanishing rates in inﬁnite time has been found in [9]. See also [18, 24] for connected results in related problems. In [35] Type-II sign changing, ﬁnite time blow-u”
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems (2009, background): “This gave rise to a plausible scenario of singularity formation in the corresponding time dependent equation which has been studied heuristically, numerically and very recently from a mathematical point of view, [5],[12],[18],[19],[31],[21] and references therein.”
Slow blow-up solutions for the $H^1({\mathbb R}^3)$ critical focusing semilinear wave equation (2009, methodology): “As this paper was written concurrently with our wave-map paper [ 10 ] it is only natural that there would be some similarities between this paper and [10].”