Stable self-similar blowup in the supercritical heat flow of harmonic maps

TL;DR

This paper demonstrates the nonlinear asymptotic stability of a specific self-similar blowup solution in the supercritical heat flow of harmonic maps, introducing a new systematic approach that bypasses traditional complex techniques.

Contribution

It presents a novel, constructive method for stability analysis of self-similar blowup in parabolic equations, avoiding traditional delicate tools and focusing on spectral analysis.

Findings

Proves stability of a particular self-similar shrinker in harmonic map heat flow.

Develops a new approach that simplifies stability analysis by spectral methods.

Provides a framework applicable to other parabolic evolution equations.

Abstract

We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators.