Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow

Abstract

We prove the existence of a (spectrally) stable self-similar blow-up solution $f_0$ to the heat flow for corotational harmonic maps from ${\mathbb{R}}^3$ to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear stability of $f_0$. At the heart of our analysis lies a new existence result of a monotone self-similar solution $f_0$. Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of $f_0$, leading to the stability result. A key ingredient is the use of interval arithmetic: a rigorous computer-assisted method for estimating functions. It is easy to verify our results by robust numerics but the purpose of the present paper is to provide mathematically rigorous proofs.

arXiv comment26 pages, a Mathematica notebook is included, typos fixed