Non-Uniqueness of Bubbling for Wave Maps
Ars Inveniendi Analytica (2022)
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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.
@article{10_15781_kz11_np83,
author = {Max Engelstein and Dana Mendelson},
title = {Non-Uniqueness of {Bubbling} for {Wave} {Maps}},
journal = {Ars Inveniendi Analytica},
publisher = {mathdoc},
year = {2022},
doi = {10.15781/kz11-np83},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.15781/kz11-np83/}
}
Max Engelstein; Dana Mendelson. Non-Uniqueness of Bubbling for Wave Maps. Ars Inveniendi Analytica (2022). doi: 10.15781/kz11-np83
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