Continuous in time bubble decomposition for the harmonic map heat flow

Abstract

We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.

arXiv comment36 pages

References in corpus (7)

• Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow
• Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2
• Metastability of the proximal point algorithm with multi-parameters
• Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class
• Quantized slow blow up dynamics for the corotational energy critical harmonic heat flow
• A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm 1$ harmonic maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$
• Trichotomy dynamics of the 1-equivariant harmonic map flow