Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

Serena Della Corte; Antonia Diana; Carlo Mantegazza; Serena Della Corte; Antonia Diana; Carlo Mantegazza

doi:10.3934/mine.2022054

Mathematics in Engineering

2022, Volume 4, Issue 6: 1-104. doi: 10.3934/mine.2022054

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Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

1.
Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands
2.
Scuola Superiore Meridionale, Università degli Studi di Napoli Federico Ⅱ, Italy
3.
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" & Scuola Superiore Meridionale, Università degli Studi di Napoli Federico Ⅱ, Italy

Received: 26 July 2021 Accepted: 10 November 2021 Published: 21 December 2021

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth strictly stable critical set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.
- nonlocal Area functional,
- Mullins–Sekerka flow,
- surface diffusion flow,
- global existence,
- asymptotic stability
Citation: Serena Della Corte, Antonia Diana, Carlo Mantegazza. Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow[J]. Mathematics in Engineering, 2022, 4(6): 1-104. doi: 10.3934/mine.2022054

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Abstract

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth strictly stable critical set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.

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