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

Serena Della Corte, Antonia Diana, Carlo Mantegazza*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Downloads (Pure)

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 W2,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.

Original languageEnglish
Pages (from-to)1-104
Number of pages104
JournalMathematics In Engineering
Volume4
Issue number6
DOIs
Publication statusPublished - 2022

Keywords

  • Asymptotic stability
  • Global existence
  • Mullins–Sekerka flow
  • Nonlocal Area functional
  • Surface diffusion flow

Fingerprint

Dive into the research topics of 'Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow'. Together they form a unique fingerprint.

Cite this