Geodesic
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.
Publié le :
DOI : 10.15781/kz11-np83 
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     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/}
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Max Engelstein; Dana Mendelson. Non-Uniqueness of Bubbling for Wave Maps. Ars Inveniendi Analytica (2022). doi : 10.15781/kz11-np83. http://geodesic.mathdoc.fr/articles/10.15781/kz11-np83/

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