Global Existence and Stability for the Modified Mullins-Sekerka and Surface Diffusion Flow

2021/07/26 by Serena Della Corte, Della Corte, Serena, Antonia Diana +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2107.12234

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.

Related

• Stability for the Surface Diffusion Flow
• Nonlinear stability results for the modified Mullins-Sekerka and the\n surface diffusion flow
• Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus
• A uniqueness and stability principle for surface diffusion
• The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions