On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

Germain, Pierre; Ghoul, Tej Eddine; Miura, Hideyuki

doi:10.1002/cpa.21716

Cited by 17 publications

(22 citation statements)

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“…The case d = 2 is of special interest since it is energy-critical and blowup takes place via shrinking of a soliton [32,33,44]. The unique continuation beyond blowup is investigated in [2,22]. Needless to say, there are similar results for closely related problems like the Yang-Mills heat flow or the nonlinear heat equation, see the discussion in [17] for a brief overview.…”

Section: Related Resultsmentioning

confidence: 99%

“…(1.1) is not time-reversible, there is another, independent class of self-similar solutions, so-called expanders, which take the form f ( r √ t−T 0 ). The latter have also attracted considerable interest, in particular in connection with the question of unique continuation beyond blowup [2,22,23], but they play no role in the present paper.…”

Section: (R ) Has a Unique Solution U H That Blows Up At T = T H And mentioning

confidence: 99%

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

2017

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We consider the heat flow of corotational harmonic maps from R 3 to the threesphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators. Mathematics Subject Classification

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Section: Related Resultsmentioning

confidence: 99%

Section: (R ) Has a Unique Solution U H That Blows Up At T = T H And mentioning

confidence: 99%

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

2017

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“…Also for any finite energy data u 0 2 H 1 .M I N /, the result of Chen-Struwe [3] yields a global, partially regular weak solution u of the harmonic map heat flow, which, however, as shown by Coron [4], may fail to be unique among partially regular weak solutions with finite energy. It is an open question whether the monotonicity formula or Feldman's [13] notion of stationary flows can give uniqueness; see Germain-Ghoul-Miura [16] for a recent study of these questions in the equivariant (corotational) setting and further references. The book by Lin-Wang [21] gives a comprehensive account of harmonic maps and their heat flows with a more thorough discussion of these issues.…”

Section: Higher Dimensionsmentioning

confidence: 99%

Normalized Harmonic Map Heat Flow

Struwe

2019

Comm Pure Appl Math

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For any closed Riemannian manifold N we propose the normalized harmonic map heat flow as a means to obtain nonconstant harmonic maps uW S m ! N , m 3.Finding nonconstant harmonic maps uW S m ! N R n for a closed target manifold N and m 3 is a prototype of a supercritical variational problem.El Soufi [12] showed that any nontrivial (sufficiently smooth) harmonic map uW S m ! N either achieves a strict maximum of the Dirichlet energywith respect to the action 3 ! u ı of the Möbius group of S m on u, or is constant in the direction defined by some a 2 S m . Thus, given a smooth map u 0 W S m ! N , one might hope to find a harmonic map homotopic to u 0 as a critical point of D of mountain-pass type, minimizing the quantity sup 2 D.uı / among maps u homotopic to u 0 . Even though this min-max procedure at first may look similar to the study of sweepouts in the work of Marques-Neves [22] on the Willmore conjecture, the fact that in our case the Möbius group acts from the right makes a decisive difference. Indeed, in our setting for any u each 2 induces a "variation of the independent variables" while keeping the image of u fixed, whereas in the setting of Marques-Neves the map u itself is deformed. A further complication arises from the fact that when applying the harmonic map heat flow to -type sweepouts of the form .u ı / 2 , singularities arise instaneously, possibly destroying any topological data. Moreover, since the heat flow in general does not commute with 2 , the family of such -type sweepouts is not invariant under the flow; that is, flowing a -type sweepout for any time t > 0 in general will fail to produce a -type sweepout.

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“…Then tools such as the maximum principle or the shooting method can be used to show the existence of solutions. We refer to [19,21,23,7,8,6,22] and the references therein for more details on such results for maps taking values in S d , with d ≥ 3. Recently, Deruelle and Lamm [17] have studied the Cauchy problem for the harmonic map heat flow with initial data m 0 : R N → S d , with N ≥ 3 and d ≥ 2, where m 0 is a Lipschitz 0-homogeneous function, homotopic to a constant, which implies the existence of expanders coming out of m 0 .…”

Section: The Landau-lifshitz-gilbert Equation: Self-similar Solutionsmentioning

confidence: 99%

Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

Gutiérrez

Laire

2015

Nonlinearity

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The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

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On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

Cited by 17 publications

References 50 publications

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

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

Normalized Harmonic Map Heat Flow

Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

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