Non-Uniqueness of Bubbling for Wave Maps

TL;DR

This paper presents the first example of non-uniqueness in bubbling phenomena for wave maps, showing multiple distinct bubbles can form at the same point during blow-up, inspired by similar behavior in harmonic map heat flow.

Contribution

It constructs the first known example of non-uniqueness of bubbling in dispersive wave equations, revealing complex blow-up behavior similar to harmonic map heat flow.

Findings

Multiple distinct bubbles form at the same point during blow-up.

Non-uniqueness of bubbling occurs along different sequences of times.

The mechanism involves a winding behavior similar to harmonic map heat flow.

Abstract

We consider wave maps from $\mathbb R^{2+1}$ to a $C^\infty$-smooth Riemannian manifold, $\mathcal N$. Such maps can exhibit energy concentration, and at points of concentration, it is known that the map (suitably rescaled and translated) converges weakly to a harmonic map, known as a bubble. We give an example of a wave map which exhibits a type of non-uniqueness of bubbling. In particular, we exhibit a continuum of different bubbles at the origin, each of which arise as the weak limit along a different sequence of times approaching the blow-up time. This is the first known example of non-uniqueness of bubbling for dispersive equations. Our construction is inspired by the work of Peter Topping [Topping 2004], who demonstrated a similar phenomena can occur in the setting of harmonic map heat flow, and our mechanism of non-uniqueness is the same 'winding' behavior exhibited in that work.