Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

Raphaël, Pierre; Schweyer, Remi

doi:10.2140/apde.2014.7.1713

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Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow

Pierre Raphaël and Remi Schweyer

Vol. 7 (2014), No. 8, 1713–1805

DOI: 10.2140/apde.2014.7.1713

Abstract

We consider the energy-critical harmonic heat flow from $ℝ^{2}$ into a smooth compact revolution surface of $ℝ^{3}$ . For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem

\partial_{t} u - \partial_{r}^{2} u - \frac{\partial_{r} u}{r} + \frac{f (u)}{r^{2}} = 0

for a suitable class of functions $f$ . Given an integer $L \in ℕ^{*}$ , we exhibit a set of initial data arbitrarily close to the least energy harmonic map $Q$ in the energy-critical topology such that the corresponding solution blows up in finite time by concentrating its energy

\nabla u (t, r) - \nabla Q (\frac{r}{(t)}) \to u^{*} in L^{2}

at a speed given by the quantized rates

(t) = c (u_{0}) (1 + o (1)) \frac{{(T - t)}^{L}}{| log (T - t) |^{2 L ∕ (2 L - 1)}},

in accordance with the formal predictions of van den Berg et al. (2003). The case $L = 1$ corresponds to the stable regime exhibited in our previous work (CPAM, 2013), and the data for $L \geq 2$ leave on a manifold of codimension $L - 1$ in some weak sense. Our analysis is a continuation of work by Merle, Rodnianski, and the authors (in various combinations) and it further exhibits the mechanism for the existence of the excited slow blow-up rates and the associated instability of these threshold dynamics.

Keywords

blow-up heat flow

Mathematical Subject Classification 2010

Primary: 35K58

Milestones

Received: 10 January 2013

Accepted: 22 December 2013

Published: 5 February 2015

Authors

Pierre Raphaël
Laboratoire J. A. Dieudonné Université de Nice Sophia Antipolis Institut Universitaire de France 06000 Nice France

Remi Schweyer
Institut de Mathématiques de Toulouse Université Paul Sabatier 31000 Toulouse France

Vol. 7, No. 8, 2014