Phase transitions and generalized motion by mean curvature

Evans, Lawrence C.; Soner, H. Meté; Souganidis, Panagiotis E.

doi:10.1002/cpa.3160450903

Cited by 467 publications

(354 citation statements)

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“…This sharp interface model arises, for example, as a singular limit of the Allen-Cahn equation [4], see the paper by Evans, Soner, and Souganidis [36]. Physically, the Allen-Cahn equation describes the motion of phase-antiphase boundaries between two grains in a solid material.…”

Section: Introductionmentioning

confidence: 99%

“…There is also a volume preserving version of the mean curvature flow, for which one subtracts the average of the mean curvature from the normal velocity. A plethora of analytical results and theories of weak solutions exists, for example, see [20,22,30,35,36,37,43,44,52,53,57,67], and for numerical solutions [1,9,13,14,15,60,69], to name but a few.The algorithm proposed below is a front-tracking boundary-integral method. It has the advantage that one does not have to differentiate across the front, as compared to a level-set approach.…”

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confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

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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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“…We will consider only smooth motions, which are well-defined if T is sufficiently small [4]. Since singularities may develop in finite time, one may need to consider the evolution in the sense of viscosity solutions [5,25]. The evolution of Ω(t) is closely related to the minimization of the following energy:…”

Section: Introductionmentioning

confidence: 99%

“…The convergence of ∂Ω (t) to ∂Ω(t) has been proved for smooth motions [10,17] and in the general case without fattening [5,25]. The convergence rate has been proved to behave as O( 2 |log | 2 ).…”

Section: Introductionmentioning

confidence: 99%

Phase field method for mean curvature flow with boundary constraints

Bretin

Perrier

2012

ESAIM: M2AN

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“…After interfaces are generated, they move and their motion is asymptotic to motion by mean curvature on a time scale O(ε −2 ); see [8] or [3] for the smooth case. Loss of regularity occurs quite often for motion by mean curvature, and a number of mathematical devices have been invented to extend solutions beyond the onset of singularities; a Hamilton-Jacobi approach has been described in [10]; a geometric measure theory solution has been given in [14].…”

Section: Introductionmentioning

confidence: 99%