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

(9 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

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

Contact Info

Product

Resources

About