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

TL;DR

This paper rigorously proves the existence and spectral stability of a self-similar blow-up solution to the supercritical corotational harmonic map heat flow, confirming a spectral gap conjecture and establishing a foundation for nonlinear stability analysis.

Contribution

It introduces a new rigorous existence proof of a monotone self-similar solution and demonstrates its spectral stability using interval arithmetic, advancing understanding of blow-up solutions.

Findings

Existence of a spectrally stable self-similar blow-up solution

Verification of the spectral gap conjecture

Application of computer-assisted interval arithmetic for rigorous proofs

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.