Construction of infinite time bubble tower solutions to critical wave maps equation

TL;DR

This paper constructs infinite time bubble tower solutions for the critical wave maps equation, demonstrating the existence of multi-bubble configurations with arbitrary numbers of bubbles and specific asymptotic behaviors.

Contribution

It introduces a novel method using modulation analysis and a Morawetz-type functional to construct multi-bubble solutions with arbitrary bubble counts for the critical wave maps.

Findings

Existence of multi-bubble solutions with arbitrary bubbles

Solutions exhibit global behavior in one time direction

Asymptotic decomposition into concentric bubbles with alternating signs

Abstract

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers $k\geq3$ and $J\geq1$, we construct a solution that is global in one time direction, has $k$-corotational symmetry, and asymptotically decomposes into $J$-many concentric bubbles of alternating signs with asymptotically vanishing radiation. The scales of each bubble are of order $t^{-\alpha_{j}}$ with $\alpha_{j}=(\frac{k}{k-2})^{j-1}-1$. This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that $k\geq3$, global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.