Nonlinear Stability of Stationary Solutions for Surface Diffusion with Boundary Conditions

Garcke, Harald; Itô, Kazuo; Kohsaka, Yoshihito

doi:10.1137/070694752

Cited by 11 publications

(20 citation statements)

References 23 publications

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“…lines, which are the stationary states in this case. In 15 the same authors showed nonlinear stability results for the above situation.…”

Section: Introductionmentioning

confidence: 81%

Linearized stability analysis of surface diffusion for hypersurfaces with boundary contact

Depner

2012

Mathematische Nachrichten

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Abstract. The linearized stability of stationary solutions for surface diffusion is studied. We consider hypersurfaces that lie inside a fixed domain, touch its boundary with a right angle and fulfill a no-flux condition. We formulate the geometric evolution law as a partial differential equation with the help of a parametrization from Vogel [Vog00], which takes care of a possible curved boundary. For the linearized stability analysis we identify as in the work of Garcke, Ito and Kohsaka [GIK05] the problem as an H −1 -gradient flow, which will be crucial to show self-adjointness of the linearized operator. Finally we study the linearized stability of some examples.

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“…lines, which are the stationary states in this case. In 15 the same authors showed nonlinear stability results for the above situation.…”

Section: Introductionmentioning

confidence: 81%

Linearized stability analysis of surface diffusion for hypersurfaces with boundary contact

Depner

2012

Mathematische Nachrichten

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“…We now use (5.16) (and of course (5.12)) to geṫ 18) where in the last equality we have used the fact that w t = H t + 4γ v t on ∂E t . Therefore from (2.5), (5.17) and (5.18) we get d dt…”

Section: Proofs Of Technical Lemmasmentioning

confidence: 99%

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

Acerbi

Fusco

Julin

et al. 2019

J. Differential Geom.

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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.

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“…is a smooth diffeomorphism onto its image and X(., .) is a curvilinear coordinate system, see also [9], [15]. In particular, X(p, 0) = p, p ∈ Σ, hence D p X = I on Σ, as well as…”

Section: Curvilinear Coordinates and Hanzawa Transformmentioning

confidence: 99%

Stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact

Garcke¹,

Rauchecker²

2019

Preprint

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We first give a complete linearized stability analysis around stationary solutions of the Mullins-Sekerka flow with 90 • contact angle in two space dimensions. The stationary solutions include flat interfaces, as well as arcs of circles. We investigate the different stability behaviour in dependence of properties of the stationary solution, such as its curvature and length, as well as the curvature of the boundary of the domain at the two contact points. We show that the behaviour changes in terms of these parameters, ranging from exponential stability to instability.We also give a first result on nonlinear stability for curved boundaries.

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Nonlinear Stability of Stationary Solutions for Surface Diffusion with Boundary Conditions

Cited by 11 publications

References 23 publications

Linearized stability analysis of surface diffusion for hypersurfaces with boundary contact

Linearized stability analysis of surface diffusion for hypersurfaces with boundary contact

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

Stability analysis for stationary solutions of the Mullins-Sekerka flow with boundary contact

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