Non-Uniqueness of Bubbling for Wave Maps

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.

arXiv commentPublished version with the journal's style file. 30 pages. 2 figures

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