Maximal regularity, the local inverse function theorem, and local well-posedness for the curve shortening flow

Boussandel, Sahbi; Chill, Ralph; Fašangová, Eva

doi:10.1007/s10587-012-0033-6

Cited by 4 publications

(2 citation statements)

References 27 publications

(20 reference statements)

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“…Note that in this proof of Theorem 3.12 we again prove the existence of a solution, but this (shorter) proof fails to provide time-uniform estimates which are needed to prove the uniqueness of the solution on the whole space X T and not only locally in a ball. Applying this method uniqueness could possibly be shown subsequently with energy methods, c.f [21,. Theorem 2].…”

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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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confidence: 99%

Short time existence for the elastic flow of clamped curves

Spener

2017

Mathematische Nachrichten

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“…Modulo isometries of R 2 , the planar curve γ together with its orientation is uniquely determined by its inclination angle. If the curve has length L > 0 and is parametrized by arc-length, the energy can be expressed in terms of an inclination angle θ : [0, L] → R and the density ρ : [0, L] → R by 2 ds, (1.2) using that κ = ∂ s θ. More precisely, we have…”

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confidence: 99%

A dynamic approach to heterogeneous elastic wires

Dall’Acqua¹,

Langer²,

Rupp³

2022

Preprint

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We consider closed planar curves with fixed length whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated L 2 -gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness and global existence of solutions for initial data in a weak regularity class and with arbitrary winding number.

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