On an area-preserving evolution equation for plane curves

Gage, Michael E.

doi:10.1090/conm/051/848933

Cited by 170 publications

(204 citation statements)

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“…The example studied here is similar to, but not identical with, the one proposed in [Gage 1986], where it was suggested as an example which should exhibit a loss of embeddedness under the averaged mean curvature flow. A curve modeled on the one proposed by Gage was also numerically investigated in [Mayer and Simonett 2000], but the curve in the latter paper remains regular due to a slight variation of parameters as compared to the curve studied here.…”

Section: Introductionsupporting

confidence: 68%

“…In fact, the author had no idea at the time of writing of the former paper that with a slight change of parameters a singularity could be forced to occur. As of this writing the mathematical literature contains essentially two types of results about existence of smooth solutions: short-term existence results for arbitrary curves, and long-term existence results for convex curves [Gage 1986;Huisken 1987;Escher and Simonett 1998]. Escher and Simonett also show that if the embedded initial curve is close to a circle but not necessarily convex, existence is still guaranteed for all time, and the curve will approach some possibly other circle.…”

Section: Other Commentsmentioning

confidence: 99%

“…Of course, it has been conjectured for quite some time that the averaged mean curvature flow can drive curves to a loss of embeddedness; such a conjecture enters the literature with [Gage 1986]. This has recently analytically been proved to be correct [Mayer and Simonett 2000].…”

Section: Introductionmentioning

confidence: 99%

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A Singular Example for the Averaged Mean Curvature Flow

Mayer¹

2001

Experimental Mathematics

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Section: Introductionsupporting

confidence: 68%

Section: Other Commentsmentioning

confidence: 99%

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A Singular Example for the Averaged Mean Curvature Flow

Mayer¹

2001

Experimental Mathematics

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“…However, for the averaged mean curvature flow the behaviour is different. The following example was first mentioned in the paper of M. Gage [42] for the problem…”

Section: The Example Of Gage In Two Dimensionsmentioning

confidence: 99%

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Mayer

2000

Eur. J. Appl. Math

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Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow to treat such free boundary problems in both IR 2 and IR 3 . The advantage of such an approach is the re-usability of much of the setup for all of the different problems.As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity. IntroductionIn this paper we will study geometric evolution problems for surfaces driven by curvature. These moving boundary problems arise from models in physics and the material sciences, and they describe phase changes, or more generally, phenomena in fluid flow.The Hele-Shaw model (named after H.S. ) describes the pressure of two immiscible viscous fluids trapped between two parallel glass plates and has attracted considerable attention in the literature, both on the analytical side [23,27,28,29,33,38,51,54,63] and on the numerical side [2,6,7,8,12,25,26,49,50,65]. This list of references also encompasses work related to the one-sided Hele-Shaw model, which arises as a limit when the viscosity of one of the fluids approaches zero. Furthermore, the Hele-Shaw model has classically been considered on a bounded domain, but many of the references above also address the problem on an unbounded domain. We shall only consider the problem on all of IR n ; for the precise formulation see (4.4).The Mullins-Sekerka model was first proposed by (and much later named after) Mullins and Sekerka to study solidification of materials of negligible specific heat [58]. For a while this model was also called the Hele-Shaw model, leading to a somewhat confused literature. Pego [61], and then Alikakos, Bates, and Chen [3], and also Stoth [68], established this model as a singular limit of the Cahn-Hilliard equation [17], a fourth order partial differential equation modelling nucleation and coarsening phenomena in a melted binary alloy. The Mullins-Sekerka model is considered to be a good model to describe the stage of Ostwald ripening in phase transitions, which is the stage after the initial nucleation has 2 U. F. Mayer essentially been completed and where some particles grow at the cost of others in an effort to decrease interfacial energy (surface area). The literature focuses mainly on the modelling and analytical aspects of the problem [11,16,18,19,21,32,34,46,56,66,71,72] though there are numerical simulations for the two-dimensional case [10,73]. We shall consider the two-phase problem on all of IR n , see (4.3).The mean curvature flow is probably the most studied geometric moving boundary problem, it is in some way also the simplest: the normal velocity is equal to the mean curvature of the interface. This sha...

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“…The vp-mcf was introduced by Gage in [9] (for n = 1) and Huisken in [11] for n ≥ 2. The ap-mcf was first studied by Pihan [20] (for n = 1) and McCoy ( [15,16] for n ≥ 2.…”

Section: 2mentioning

confidence: 99%

Non-preserved curvature conditions under constrained mean curvature flows

Cabezas-Rivas

2016

Differential Geometry and its Applications

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Abstract. We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume-or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.Mathematics Subject Classification (2010) 53C44

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On an area-preserving evolution equation for plane curves

Cited by 170 publications

References 0 publications

A Singular Example for the Averaged Mean Curvature Flow

A Singular Example for the Averaged Mean Curvature Flow

A numerical scheme for moving boundary problems that are gradient flows for the area functional

Non-preserved curvature conditions under constrained mean curvature flows

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