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

Biernat, Paweł; Donninger, Roland; Schörkhuber, Birgit

doi:10.1007/s00526-017-1256-z

Cited by 8 publications

(15 citation statements)

References 53 publications

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“…where Q corresponds to the unique (up to scaling) harmonic map in this class, and L ∈ Z + , with L = 1 providing the generic blow-up rate. See also [14] for a related recent result, and [5,6,20,21] concerning the breakdown of solutions in higher (supercritical) dimensions.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

Gustafson,

Roxanas

2017

Preprint

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We study m-corotational solutions to the Harmonic Map Heat Flow from R 2 to S 2 . We first consider maps of zero topological degree, with initial energy below the threshold given by twice the energy of the harmonic map solutions. For m ≥ 2, we establish the smooth global existence and decay of such solutions via the concentration-compactness approach of Kenig-Merle, recovering classical results of Struwe by this alternate method. The proof relies on a profile decomposition, and the energy dissipation relation. We then consider maps of degree m and initial energy above the harmonic map threshold energy, but below three times this energy. For m ≥ 4, we establish the smooth global existence of such solutions, and their decay to a harmonic map (stability), extending results of Gustafson-Nakanishi-Tsai to higher energies. The proof rests on a stability-type argument used to rule out finitetime bubbling.

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Section: Introduction and Resultsmentioning

confidence: 99%

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

Gustafson,

Roxanas

2017

Preprint

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“…Our estimates (1.3) on f 0 lead to very precise bounds on V 0 and in turn allow us to prove the following stability result. This result is an indispensable ingredient in the proof of the nonlinear asymptotic stability of f 0 in the companion paper [4].…”

Section: Introductionmentioning

confidence: 90%

“…In addition to the existence result we have the following technical proposition to describe some qualitative properties of f 0 needed in the proof of nonlinear stability in [4].…”

Section: Introductionmentioning

confidence: 99%

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

Biernat

Donninger

2018

Nonlinearity

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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.

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“…The harmonic map heat flow forms also singularity in finite time, and the self-similar nature of the singularity appears only when 3 ≤ d ≤ 6, and for d ≥ 7 self-similar blowup solutions don't exist [6]. For 3 ≤ d ≤ 6, the existence of the self-similar solutions is known [13] and the stability has been proved only in the case d = 3 as in [1]. When d = 7 the blowup is not self-similar and the speed λ has a log correction [15], it turns out that the non-self-similar regime is stable when d = 7.…”

Section: Introductionmentioning

confidence: 99%

Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

Bensouilah¹,

Duong²,

Ghoul³

2022

Preprint

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In this paper, we consider the Yang-Mills heat flow on R d × SO(d) with d ≥ 11. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to :We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for d ≥ 11 and prove that the asymptotic of the solution is of the formwhere Q is the ground state with boundary conditions Q(0) = −1, Q ′ (0) = 0 and the blowup speedIn particular, when ℓ = 1, this asymptotic is stable whereas for ℓ ≥ 2 it becomes stable on a space of codimension ℓ − 1. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.

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

Cited by 8 publications

References 53 publications

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

Global solutions for the critical, higher-degree corotational harmonic map heat flow to $\mathbb{S}^2$

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

Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

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