TY - JOUR
T1 - Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow†
AU - Della Corte, Serena
AU - Diana, Antonia
AU - Mantegazza, Carlo
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - Asymptotic stability
KW - Global existence
KW - Mullins–Sekerka flow
KW - Nonlocal Area functional
KW - Surface diffusion flow
UR - http://www.scopus.com/inward/record.url?scp=85122990630&partnerID=8YFLogxK
U2 - 10.3934/mine.2022054
DO - 10.3934/mine.2022054
M3 - Article
AN - SCOPUS:85122990630
SN - 2640-3501
VL - 4
SP - 1
EP - 104
JO - Mathematics In Engineering
JF - Mathematics In Engineering
IS - 6
ER -