Fast Reaction, Slow Diffusion, and Curve Shortening

Rubinstein, Jacob; Sternberg, Peter; Keller, Joseph B.

doi:10.1137/0149007

Cited by 288 publications

(207 citation statements)

References 10 publications

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“…If the transition region between two stable states is sufficiently narrow, the dynamics can often be approximated by an interface motion [25,51]. Examples include the solid/liquid interface in solidification [22,41,42], the combustion front separating burnt and unburnt combustants [25,45,51], and the motion of heteroclinic fronts connecting steady states in reaction-diffusion models [10,25,52].…”

Section: Introductionmentioning

confidence: 99%

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Bernoff

Bertozzi

1995

Physica D: Nonlinear Phenomena

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Section: Introductionmentioning

confidence: 99%

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Bernoff

Bertozzi

1995

Physica D: Nonlinear Phenomena

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“…We believe that efforts should be concentrated on qualitative behavior, asymptotic and numerical solutions. Similar equations have been studied by several authors (You and Kavet 2000, Lu et al 2007, Rubinstein et al 1989, Bernoff and Bertozzi 1995, Myers 1998.…”

Section: Discussionmentioning

confidence: 52%

A new analytical formulation of retention effects on particle diffusion processes

Bevilácqua

Galeão

Costa

2011

An. Acad. Bras. Ciênc.

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The ultimate purpose of this paper is to present a new analytical formulation to simulate diffusion with retention in a reactive medium under stable thermodynamic conditions. The analysis of diffusion with retention in a continuum medium is developed after the solution of an equivalent problem using a discrete approach. The new law may be interpreted as the reduction of all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect namely a circulatory motion. It is remarkable that the governing equation requires a fourth order differential term as suggested by the discrete approach. The relative fraction of diffusion particles β is introduced as a control parameter in the diffusion-retention law as suggested by the discrete approach. This control parameter is essential to avoid retention isolated from the diffusion process. Two matrices referring to material properties are introduced and related to the real phenomenon through the circulation hypothesis. The governing equation may be highly non-linear even if the material properties are constant, but the retention effect is a function of the concentration level, that is, β is a function of the concentration.

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“…It was shown in [25] that limit of the rescaled solutions u (x, t ), → 0 + of the AllenCahn equation yields motion by mean curvature. The modification of the MBO scheme proposed in [9] by Esedoglu and Tsai particularly interesting.…”

Section: Background On Variational Methods In Image Segmentation and mentioning

confidence: 99%

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

Kostic¹,

Bertozzi²

2012

J Sci Comput

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Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation version of the well-known Maximum Penalized Likelihood Estimation (MPLE) model, as well as a minimization algorithm based on thresholding dynamics originally proposed by Merriman et al. (The Computational Crystal Growers, pp. 73-83, 1992). We also present some computational examples, including one with actual residential burglary data from the San Fernando Valley.

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Fast Reaction, Slow Diffusion, and Curve Shortening

Cited by 288 publications

References 10 publications

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

A new analytical formulation of retention effects on particle diffusion processes

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

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