Dynamics of Bubbling Wave Maps with Prescribed Radiation

TL;DR

This paper constructs and classifies blow-up solutions for energy critical wave maps into the two-sphere, focusing on prescribed radiation and the dynamics of bubbling phenomena.

Contribution

It introduces a method to construct blow-up solutions with prescribed radiation and provides a sharp classification of their blow-up rates.

Findings

Successfully constructed blow-up solutions with prescribed radiation.

Provided a sharp classification of the blow-up rates.

Enhanced understanding of energy concentration and radiation in wave maps.

Abstract

We study energy critical one-equivariant wave maps taking values in the two-sphere. It is known that any finite energy wave map that develops a singularity does so by concentrating the energy of (possibly) several copies of the ground state harmonic map at the origin. If only a single bubble of energy is concentrated, the solution decomposes into a dynamically rescaled harmonic map plus a term that accounts for the energy that radiates away from the singularity. In this paper, we construct blow up solutions by prescribing the radiative component of the map. In addition, we give a sharp classification of the dynamical blow up rate for every solution with this prescribed radiation.