Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow
In this survey we present the state of the art about the asymptotic behavior and stability of the <i>modified Mullins</i>–<i>Sekerka flow</i> and the <i>surface diffusion flow</i> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we...
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-12-01
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| Series: | Mathematics in Engineering |
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| Online Access: | http://www.aimspress.com/article/doi/10.3934/mine.2022054?viewType=HTML |
| _version_ | 1797825693335158784 |
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| author | Serena Della Corte Antonia Diana Carlo Mantegazza |
| author_facet | Serena Della Corte Antonia Diana Carlo Mantegazza |
| author_sort | Serena Della Corte |
| collection | DOAJ |
| description | In this survey we present the state of the art about the asymptotic behavior and stability of the <i>modified Mullins</i>–<i>Sekerka flow</i> and the <i>surface diffusion flow</i> 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 <i>strict stability</i> 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 <i>strictly stable critical</i> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $. |
| first_indexed | 2024-03-13T10:56:28Z |
| format | Article |
| id | doaj.art-94d54c76234d45879ea9af857a4bf65e |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-03-13T10:56:28Z |
| publishDate | 2022-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-94d54c76234d45879ea9af857a4bf65e2023-05-17T01:06:56ZengAIMS PressMathematics in Engineering2640-35012022-12-0146110410.3934/mine.2022054Global existence and stability for the modified Mullins–Sekerka and surface diffusion flowSerena Della Corte 0Antonia Diana1Carlo Mantegazza21. Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands2. Scuola Superiore Meridionale, Università degli Studi di Napoli Federico Ⅱ, Italy3. Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" & Scuola Superiore Meridionale, Università degli Studi di Napoli Federico Ⅱ, ItalyIn this survey we present the state of the art about the asymptotic behavior and stability of the <i>modified Mullins</i>–<i>Sekerka flow</i> and the <i>surface diffusion flow</i> 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 <i>strict stability</i> 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 <i>strictly stable critical</i> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.http://www.aimspress.com/article/doi/10.3934/mine.2022054?viewType=HTMLnonlocal area functionalmullins–sekerka flowsurface diffusion flowglobal existenceasymptotic stability |
| spellingShingle | Serena Della Corte Antonia Diana Carlo Mantegazza Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow Mathematics in Engineering nonlocal area functional mullins–sekerka flow surface diffusion flow global existence asymptotic stability |
| title | Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow |
| title_full | Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow |
| title_fullStr | Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow |
| title_full_unstemmed | Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow |
| title_short | Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow |
| title_sort | global existence and stability for the modified mullins sekerka and surface diffusion flow |
| topic | nonlocal area functional mullins–sekerka flow surface diffusion flow global existence asymptotic stability |
| url | http://www.aimspress.com/article/doi/10.3934/mine.2022054?viewType=HTML |
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