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

TL;DR

This paper establishes a universal blow-up profile for small energy wave maps into the sphere, showing that solutions near blow-up decompose into a regular part plus a traveling wave, demonstrating energy quantization at blow-up.

Contribution

Introduces a channel of energy argument for energy critical wave maps and proves energy quantization through decomposition into traveling waves at blow-up.

Findings

Proves a channel of energy inequality for small energy wave maps.

Shows solutions near blow-up decompose into regular part plus traveling wave.

Demonstrates energy concentration occurs only through traveling waves.

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.