Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

TL;DR

This paper demonstrates a stable finite-time blow-up regime for a specific harmonic heat flow from  into a surface in , providing sharp asymptotics for singularity formation and extending previous studies on related geometric flows.

Contribution

It introduces a stable blow-up regime for the 1-corotational energy critical harmonic heat flow, with precise asymptotics and initial data close to the ground state.

Findings

Stable finite-time blow-up regime established.

Sharp asymptotics for singularity formation provided.

Initial data can be smooth, localized, and near the ground state.

Abstract

We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the semilinear parabolic problem $$\partial_t u -\pa^2_{r} u-\frac{\pa_r u}{r} + \frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$. The corresponding initial data can be chosen smooth, well localized and arbitrarily close to the ground state harmonic map in the energy critical topology. We give sharp asymptotics on the corresponding singularity formation which occurs through the concentration of a universal bubble of energy at the speed predicted in [Van den Bergh, J.; Hulshof, J.; King, J., Formal asymptotics of bubbling in the harmonic map heat flow, SIAM J. Appl. Math. vol 63, o5. pp 1682-1717]. Our approach lies in the continuation of the study of the 1-equivariant energy critical wave map and Schr\"odinger map with $\Bbb S^2$ target in [Rapha\"el, P.; Rodnianksi, I., Stable blow up dynamics for the critical corotational wave maps and equivariant Yang Mills problems, to appear in Prep. Math. IHES.], [Merle, F.; Rapha\"el, P.; Rodnianski, I., Blow up dynamics for smooth solutions to the energy critical Schr\"odinger map, preprint 2011.].