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 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.

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

“…We would like to emphasize that this is a robust approach that uses no structure other than the spectral stability of the self-similar profile f 0 which is established in [3]. As a consequence, our method provides a universal framework for studying self-similar blowup in general parabolic evolution equations.…”

“…• The blowup profile f 0 is constructed in the companion paper [3] by a novel computerassisted (but rigorous) method. It is not known in closed form.…”

“…More precisely, we consider the semigroup e sL on L 2 σ (R 5 ) generated by the self-adjoint operator L. From [3] we know that L has precisely one nonnegative eigenvalue λ = 1 with eigenfunction ψ 1 . As usual, this instability is related to the freedom in choosing the parameter T in the similarity coordinates.…”

“…with f 0 constructed in [3]. Our goal is to study the evolution of small initial perturbations of u * T 0 .…”

“…By [3], f 0 is odd 1 and thus, V 0 ∈ C ∞ (R 5 ), see [45]. Now we define a differential operator L byL…”