The sharp-interface limit of the Allen-Cahn equation

Abstract: The Allen-Cahn equation is one of the most basic reaction-diffusion equations, modeling a large variety of phase transition problems. In a suitable scaling limit, the transition becomes sharp and the system follows a geometric evolution equation - the mean curvature flow. The structure of this "sharp-interface limit" has received continuous attention from both the geometric and applied analysis communities over the last decades. In this talk, I will give an overview of some of the breakthroughs in this field and highlight a recent short convergence analysis in a joint work with Julian Fischer and Theresa Simon. Our proof is based on a new relative entropy for diffuse interface problems that allows us to prove the optimal convergence rate toward mean curvature flow.

analysis of PDEs

Audience: researchers in the discipline

( paper )

Comments: Atanas Stefanov, University of Alabama, Birminghan (stefanov@uab.edu)

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Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

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The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

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