Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow. (27th June 2018)

• Record Type: Journal Article
• Authors: Biernat, Pawel; Donninger, Roland
• Is Part Of: Nonlinearity. Volume 31:Number 8(2018:Aug.)
• Page Start: 3543
• Page End: 3566
• Publication Date: 2018-06-27
• Subjects: harmonic map heat flow -- blowup -- self-similar solution -- spectral stability; 58E20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aabe4c
• Languages: English
• ISSNs: 0951-7715

Abstract

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 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 .