Quantization of time-like energy for wave maps into spheres

Abstract

In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru, but more specifically into the unit Euclidean sphere, and study further the dynamics of the sequence of wave maps that are obtained by Sterbenz and Tataru after the final rescaling at a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space.

arXiv comment62 pages, some corrections and feedback implemented, Lemma 2.8 refined, comments welcome

References in corpus (11)

• Global regularity of wave maps II. Small energy in two dimensions
• Renormalization and blow up for charge one equivariant critical wave maps
• Regularity of Wave-Maps in dimension 2+1
• Energy dispersed large data wave maps in 2+1 dimensions
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Global regularity of wave maps VII. Control of delocalised or dispersed solutions
• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces
• Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions
• Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class
• Soliton resolution along a sequence of times with dispersive error for type II singular solutions to focusing energy critical wave equation
• Asymptotic decomposition for semilinear wave and equivariant wave map equations

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• Two-bubble dynamics for threshold solutions to the wave maps equation
• Soliton resolution for critical co-rotational wave maps and radial cubic wave equation
• Universality of blow up profile for small blow up solutions to the energy critical wave map equation
• Optimal blowup stability for three-dimensional wave maps
• Sharp universal rate for stable blow-up of corotational wave maps