A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

Abels, Helmut; Butz, Julia

doi:10.1137/19m1242914

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…It is worth to mention also results about the Helfrich flow [17,56], the elastic flow with constraints [25,43,44] and other fourth (or higher) order flows [1,2,36,37,57].…”

Section: Introductionmentioning

confidence: 99%

A survey of the elastic flow of curves and networks

Mantegazza,

Pluda,

Pozzetta

2020

Preprint

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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 R 2 . In the last section of the paper we also discuss a list of open problems.

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“…It is worth to mention also results about the Helfrich flow [17,56], the elastic flow with constraints [25,43,44] and other fourth (or higher) order flows [1,2,36,37,57].…”

Section: Introductionmentioning

confidence: 99%

A survey of the elastic flow of curves and networks

Mantegazza,

Pluda,

Pozzetta

2020

Preprint

View full text Add to dashboard Cite

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“…ds ≤ E μ (γ 0 ), k| 2 ds ≤ C(E μ (γ 0 )).These estimates allows us to conclude that ∂2 xγ , ∂3 xγ , ∂4 …”

mentioning

confidence: 65%

“…It is worth to mention also results about the Helfrich flow [17,56], the elastic flow with constraints [25,43,44] and other fourth (or higher) order flows [1,2,36,37,57].…”

Section: Introductionmentioning

confidence: 99%

A Survey of the Elastic Flow of Curves and Networks

2021

View full text Add to dashboard Cite

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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“…More precisely, we exploit a short time existence result Theorem 2.14 in [1], which guarantees that the curve diffusion flow starts for initial curves given by a certain sufficiently small height function of class W…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the following, ρ : d) serves as a height function such that Γ can be expressed via (2.2). The main result of [1] reads: Theorem 2.14 (Local Well-Posedness for Data Close to a Reference Curve) Let Φ * and η be given such that (2.1) and (2.3) are fulfilled, respectively. Let ρ 0 ∈ W 4(µ− 1 /2) 2 (I), with I = (0, 1) and µ ∈ 7 8 , 1 , fulfill the conditions…”

Section: A Local Wellposedness Results For the Curve Diffusion Flowmentioning

confidence: 99%

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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A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

Cited by 5 publications

References 13 publications

A survey of the elastic flow of curves and networks

A survey of the elastic flow of curves and networks

A Survey of the Elastic Flow of Curves and Networks

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

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