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Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow
Analysis of PDEs · Submitted 29 Oct 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | result 0.83 | 0.96 | 0.95 | asserted 1.00 |
| s1 | 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 $f_0$. | interpretation 0.95 | 0.85 | 0.15 | asserted 0.99 |
| s2 | At the heart of our analysis lies a new existence result of a monotone self-similar solution $f_0$. | contribution 0.84 | 0.92 | 0.09 | asserted 1.00 |
| s3 | Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of $f_0$, leading to the stability result. | contribution 0.59 | 0.66 | 0.05 | asserted 1.00 |
| s4 | A key ingredient is the use of interval arithmetic: a rigorous computer-assisted method for estimating functions. | method 0.99 | 0.77 | 0.02 | asserted 1.00 |
| s5 | It is easy to verify our results by robust numerics but the purpose of the present paper is to provide mathematically rigorous proofs. | method 0.29 | 0.54 | 0.03 | asserted 0.99 |
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The paper
Original 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.
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How the literature uses it
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See also [ref] for a related recent result, and [ref] concerning the breakdown of solutions in higher (supercritical) dimensions.
Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$ · Introduction and Results · 0.91
What it cites · 4 sentences without a confident label
Cites as background 1 paper
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In higher dimensions $d>7$, the blow-up takes a more complicated form as described in [ref] .
Type II blow-up mechanism for supercritical harmonic map heat flow · Introduction · 0.93
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