Nonlinear stability results for the modified Mullins–Sekerka and the surface diffusion flow

Abstract

It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins-Sekerka or Hele-Shaw flow. The same result holds for three-dimensional periodic configurations that are strictly stable with respect to the sharp-interface Ohta-Kawaski energy. In this case, they are exponentially stable for the so-called modified Mullins-Sekerka flow.

The Surface Diffusion Flow with Elasticity in the Plane

“…In order to control the two last terms in (5.1) we need the following interpolation result on the evolving boundaries. The proof of the next lemma is precisely the same as [1,Lemma 4.7] and therefore we omit it.…”

“…A crucial role in this analysis is played by the energy identity proven in Proposition 5.3 and by the estimates on the flow provided by Theorem 4.4. Let us remark that such estimates allow us also to considerably simplify the arguments of [1] and to obtain stronger asymptotic convergence results. This paper is organized as follows.…”

“…They are very well studied in the physical and numerical literature, see for instance [26,29,40,41,42]. Concerning rigorous mathematical analysis, we refer to [6,8,10,17,21,25,28] for some existence, regularity and stability results 1 related to a variational model describing the equilibrium configurations of two-dimensional epitaxially strained elastic films, and to [9,16] for results in three-dimensions. A hierarchy of variational principles to describe equilibrium shapes in the aforementioned contexts has been introduced in [30].…”

The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

“…The focus of the paper is the long-time behaviour of flat flows in two dimensions. Previous results on the long-time convergence of volume preserving flows are mostly confined to the case of smooth solutions starting from specific classes of initial regular sets, see for instance [18,24,37] for the volume preserving mean curvature flow and [1,9,19,23] for the Mullins-Sekerka flow. For less general initial data, the long time behaviour of the volume preserving mean curvature flow starting from convex and star-shaped sets (see [6,26]) has been characterized only up to (possibly diverging in the case of [6]) translations.…”

The switching control method for the nonlinear response of seismic surface based on the local T-S model

“…The larger the number of fuzzy rules is, the higher the precision is. However, when the number of rules of fuzzy reasoning is large, the calculation is complex and in some cases, no common positive definite matrix can be found to meet the stability conditions (Acerbi et al, 2016;Guo et al, 2017).…”