The Łojasiewicz–Simon gradient inequality for open elastic curves

Dall’Acqua, Anna; Pozzi, Paola; Spener, Adrian

doi:10.1016/j.jde.2016.04.027

Cited by 39 publications

(32 citation statements)

References 17 publications

(27 reference statements)

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“…Note that A η is well defined by Lemma 3.4. It can be associated with the second variation of the energy E at η restricted to normal perturbations, see [6,Proposition 3.6]. In what follows we will use a general principle from analysis.…”

Section: Wwwmn-journalcommentioning

confidence: 99%

Short time existence for the elastic flow of clamped curves

Spener

2017

Mathematische Nachrichten

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We show well-posedness of the elastic flow of open curves with clamped boundary conditions. To show short time existence we prove that the linearised problem has the property of maximal L p -regularity and use the contraction principle to obtain the solution. Moreover, we show analyticity of the solution and its analytic dependency on the initial curve. With the developed methods we also prove long time existence of the flow if the initial curve is close to an elastica.1for some α > 0 and lim t→0 φ(t, ·) = φ 0 in C 2,α (I ).Furthermore the solution φ satisfies φ(t, ·)| ∂ I = ∂ x φ(t, ·)| ∂ I ≡ 0 for all t ∈ [0, T ] and depends analytically on the initial datum φ 0 .The proof of Theorem 1.2 will be given in Section 3. It consists of three steps: First we analyse the linearised equation and show that it has the property of maximal regularity, applying a general result from [7]. Afterwards, www.mn-journal.com

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Section: Wwwmn-journalcommentioning

confidence: 99%

Short time existence for the elastic flow of clamped curves

Spener

2017

Mathematische Nachrichten

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“…Additionally, from the global existence of the flow it is deduced that the family of curves subconverges after reparametrization by arc length to an equilibrium. Dall'Acqua, Pozzi, and Spener strengthened the result of Lin in [14] by showing that up to a time dependent reparametrization φ(t, ·) : I → I, t ∈ [0, ∞), the whole solution f (t, φ(t, ·)) converges to a critical point of the energy in L 2 for t → ∞, see [10].…”

Section: )mentioning

confidence: 69%

Short time existence for the curve diffusion flow with a contact angle

Abels

Butz

2019

Journal of Differential Equations

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We prove a blow-up criterion in terms of an L 2 -bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle α ∈ (0, π) with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

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“…[40] [ 40] [ 40,41] Open curves, clamped b.c. [52] [ 29] [ 18,41] Non compact curves [40] [ 40] Networks [15,22,23] [ 14,22] [ 14] We refer also to the two recent PhD theses [38,48]. The aim of this expository paper is to arrange (most of) this material in a unitary form, proving in full detail the results for the elastic flow of closed curves and underlying the differences with the other cases.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…In this part we compute the second variation of the elastic energy functional E μ . We are interested only in showing its structure and analyze some properties, instead of computing it explicitly (for the full formula of the second variation we refer to [18,47]). In fact, we will exploit the properties of the second variation only in the proof of the smooth convergence of the elastic flow of closed curves in Section 5.…”

Section: Second Variation Of the Elastic Energymentioning

confidence: 99%

“…The Lojasiewicz-Simon inequality we shall employ follows from the abstract results of [9] and the method is inspired by the strategy used in [10], where the authors study the Willmore flow of closed surfaces. We mention that another successful application of this strategy is contained in [18], where the authors study open curves with clamped boundary conditions. Finally, a recent application of this strategy to the flow of generalized elastic energies on Riemannian manifolds is contained in [47].…”

Section: Asymptotic Behaviormentioning

confidence: 99%

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A Survey of the Elastic Flow of Curves and Networks

2021

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We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in $${\mathbb {R}}^2$$ R 2 . In the last section of the paper we also discuss a list of open problems.

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The Łojasiewicz–Simon gradient inequality for open elastic curves

Cited by 39 publications

References 17 publications

Short time existence for the elastic flow of clamped curves

Short time existence for the elastic flow of clamped curves

Short time existence for the curve diffusion flow with a contact angle

A Survey of the Elastic Flow of Curves and Networks

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