On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow

LeCrone, Jeremy; Simonett, Gieri

doi:10.1137/120883505

Cited by 13 publications

(21 citation statements)

References 48 publications

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“…In [15] the short time existence for hinged curves is obtained with different methods. Maximal regularity methods for short time existence of geometric problems has been studied for instance in [9], [4], [12] or [1]. The main theorem of this paper yields a unique solution of the elastic flow to initial curves in a certain trace space which embeds merely in C 2,ε for some ε > 0, if the initial curve can be written as a normal graph over some fixed reference curve, see Assumption 1.3 and Theorem 1.2 for details.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Short time existence for the elastic flow of clamped curves

Spener

2017

Mathematische Nachrichten

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“…Proof. Existence of maximal solutions is proved in the same way as Proposition 2.2 of [27], whereas the claim for global solutions differs slightly from Proposition 2.3 of [27] due to the fact that we do not have compact embedding of little-Hölder spaces over the non-compact manifold C r . In particular, well-posedness follows from [12, Theorems 3.1 and 4.1(c)], [36, Proposition 2.2(c)], and Proposition 3.1, while the semiflow properties follow from [12] when µ < 1 and from [5] in case µ = 1.…”

Section: 3mentioning

confidence: 91%

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

LeCrone

Simonett

2016

Journal of Differential Equations

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We study the surface diffusion flow acting on a class of general (non-axisymmetric) perturbations of cylinders Cr in IR 3 . Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially-unbounded) surfaces defined over Cr via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that Cr is normally stable with respect to 2π-axially-periodic perturbations if the radius r > 1,and unstable if 0 < r < 1. Stability is also shown to hold in settings with axial Neumann boundary conditions.

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“…(T3) Σ has a tubular neighborhood. (b) All of the manifolds considered in [20,21] are (URT)-hypersurfaces. In particular, the infinite cylinder with radius r > 0,…”

Section: Urt-hypersurfacesmentioning

confidence: 99%

“…These flows have been studied by several authors for compact (closed) hypersurfaces. In this setting, existence, regularity, and qualitative behavior of solutions have been analyzed in [13,14,20,27,33,36,37] for the surface diffusion flow, and in [9,17,18,19,24,25,26,32,35] for the Willmore flow, to mention just a few publications.…”

Section: Introductionmentioning

confidence: 99%

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

LeCrone¹,

Shao²,

Simonett³

2020

DCDS-S

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We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are C 1+α -regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existence for initial surfaces which are C 1+α -close to a sphere, and we prove that these solutions become spherical as time goes to infinity.2010 Mathematics Subject Classification. 35K55, 53C44, 54C35, 35B65, 35B35 .

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On Well-Posedness, Stability, and Bifurcation for the Axisymmetric Surface Diffusion Flow

Cited by 13 publications

References 48 publications

Short time existence for the elastic flow of clamped curves

Short time existence for the elastic flow of clamped curves

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

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