Universality of blow up profile for small blow up solutions to the energy critical wave map equation

Abstract

In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that for a wave map that has energy just above the degree one harmonic maps and that blows up in finite time, the solution asymptotically de-couples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called quantization of energy at blow up. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps.

50 pages

References in corpus (3)

• Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces

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• Sharp universal rate for stable blow-up of corotational wave maps
• Non-Uniqueness of Bubbling for Wave Maps