Construction of two-bubble solutions for energy-critical wave equations

TL;DR

This paper constructs special two-bubble solutions for certain energy-critical wave equations, where one bubble remains fixed while the other concentrates over time, providing new insights into the solution structure of these equations.

Contribution

It introduces the construction of pure two-bubbles with one fixed and one concentrating in energy-critical wave equations, including Yang-Mills and wave map models.

Findings

Existence of solutions with one bubble fixed and the other concentrating

Concentration speed varies: exponential or power law

Solutions exist globally with energy tending to zero error

Abstract

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k > 2. The concentrating speed of the second bubble is exponential for the first two models and a power function in the last case.