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arxiv: 1907.00833 · v1 · pith:AWK4BLZAnew · submitted 2019-07-01 · 🧮 math.AP

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

Harald Garcke , Maximilian Rauchecker This is my paper

Pith reviewed 2026-05-25 11:58 UTC · model grok-4.3

classification 🧮 math.AP
keywords Mullins-Sekerka flowstability analysiscontact anglestationary solutionslinearized stabilitynonlinear stabilitytwo-dimensional flow
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The pith

Linear stability analysis shows that stationary solutions of the Mullins-Sekerka flow with 90 degree contact angle are exponentially stable or unstable depending on curvature, length, and boundary curvature.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper conducts a complete linearized stability analysis for stationary solutions of the Mullins-Sekerka flow with 90 degree contact angle in two space dimensions. The solutions considered are flat interfaces and arcs of circles. The stability changes based on the curvature and length of the solution as well as the curvature of the domain boundary at the contact points, leading to either exponential stability or instability. Additionally, an initial result on nonlinear stability is given for curved boundaries. Understanding this helps determine when interfaces in such flows remain stable or evolve away from equilibrium.

Core claim

We 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.

What carries the argument

The spectrum of the linearized operator around the stationary solutions, which determines exponential stability or instability based on parameters like curvature and boundary properties.

If this is right

  • Exponential stability holds for certain values of curvature, length, and boundary curvature.
  • Instability occurs for other combinations of these parameters.
  • Nonlinear stability can be established for curved boundaries under the appropriate conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These results could guide numerical simulations to test stability thresholds in specific geometries.
  • Similar analysis might apply to other interface evolution equations with contact angles.
  • The dependence on boundary curvature suggests design implications for domains in physical models.

Load-bearing premise

All stationary solutions are flat interfaces or circular arcs that satisfy the 90 degree contact angle condition exactly, and that the linearization around them is sufficient to determine the stability.

What would settle it

Observation of a stationary solution that is neither flat nor a circular arc with 90 degree contact angle, or experimental evidence of instability where the linear analysis predicts stability.

Figures

Figures reproduced from arXiv: 1907.00833 by Harald Garcke, Maximilian Rauchecker.

Figure 1
Figure 1. Figure 1: Different signs of ω1, ω2. Left: ω1 = ω2 < 0. Middle: ω1 = ω2 = 0. Right: ω1 = ω2 > 0. 4.1. Stability and instability results. Let us start with the left hand side case, where we can show exponential stability of h∗ = 0 for (4.2). Theorem 4.1. Let ω1, ω2 ≤ 0. Then h∗ = 0 is normally stable, that is, (1) The set of equilibria of (4.2) is the kernel of A, which is one-dimensional. (2) The eigenvalue zero is … view at source ↗
Figure 2
Figure 2. Figure 2: Construction of ¯g [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: κ∗ = 0. Exponential stability for all ω1, ω2 ≤ 0 regard￾less of L > 0 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: κ∗ = 0 and fixed ω1 = ω2 > 0. Exponential stability for small L > 0, instability for large L > 0. Σ∗ ∂Ω Σ∗ ∂Ω [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Fixed l > 0 and small κ∗ 6= 0. Exponential stability for ω1, ω2 ≤ 0. Σ∗ ∂Ω Σ∗ ∂Ω [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fixed l > 0 and small κ∗ 6= 0. Exponential stability for ω1, ω2 > 0 small. Instability for ω1, ω2 > 0 large. 7. Nonlinear stability In this section we show a first nonlinear stability result for the case of a flat stationary solution. We are concerned with the full transformed nonlinear problem (2.1) for the height function, which reads as ∂th = −a(h)JnΓh · ∇hηK, on Σ, η|Σ = K(h), on Σ, ∆hη = 0, in Ω\Σ, n … view at source ↗
Figure 8
Figure 8. Figure 8: Fixed l > 0 and ω1, ω2 = 0. Exponential stability for κ 2 ∗ small. Instability for κ 2 ∗ large. Σ∗ ∂Ω Σ∗ ∂Ω [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fixed κ∗ 6= 0 and ω1, ω2 = 0. Exponential stability for l > 0 small. Instability for l > 0 large. deduce nonlinear stability and convergence to an equilibrium solution at exponential rate. First we need to analyze the boundary condition (7.1)5. Since we work in two space dimensions, the condition (7.1)5 can be rewritten as n∂Ω(Θ( ˜ h(t)) · Rτh = 0, where Θ( ˜ h(t)) is defined by means of Θ( ˜ h(t))(x) := Θ… view at source ↗
Figure 10
Figure 10. Figure 10: Parametrization by [m 7→ m + u(m)] in Theorem 7.2. Remark 7.3. For situations in which I∗ is positive on X0 γ , we refer to the arguments of Section 4.1. We now apply the generalized principle of linearized stability [24] to the nonlinear evolution problem (7.6). Theorem 7.4. Let p ∈ (6, ∞), q ∈ (19/10, 2) ∩ (2p/(p + 1), 2), v∗ = 0, and I∗ positive on X0 γ . Let Ev be the set of equilibrium solutions to (… view at source ↗
read the original abstract

We first give a complete linearized stability analysis around stationary solutions of the Mullins-Sekerka flow with $90^\circ$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript gives 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. The stability behaviour is investigated 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. The behaviour changes in terms of these parameters, ranging from exponential stability to instability. A first result on nonlinear stability for curved boundaries is also given.

Significance. If the derivations hold, the work supplies an explicit, parameter-dependent classification of linear stability for a parabolic free-boundary problem with fixed contact angle. This is useful for applications in materials science and provides concrete geometric criteria (interface length, curvature, boundary curvature at contacts) that determine stability versus instability. The self-contained linearization and the preliminary nonlinear result constitute clear strengths.

major comments (2)
  1. [Section 4 (linearized eigenvalue problem)] The transition from stability to instability for circular arcs is asserted to depend on length and boundary curvature, but the explicit spectral computation that establishes the sign change of the principal eigenvalue is not displayed in sufficient detail to verify the claimed threshold; an additional lemma or corollary with the characteristic equation would be required.
  2. [Section 6 (nonlinear stability)] The nonlinear stability statement for curved boundaries relies on a closeness assumption in a specific function space, yet the passage from linear decay to nonlinear control is only sketched; the precise interpolation or bootstrap argument that closes the estimate is missing and is load-bearing for the claim.
minor comments (2)
  1. [Introduction] The introduction would benefit from an explicit statement of the main theorems (including the precise parameter regimes for stability and instability) rather than a purely descriptive paragraph.
  2. Notation for the curvature parameters (interface curvature versus boundary curvature at contact points) should be unified across sections to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the work, and the recommendation of minor revision. We address each major comment below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Section 4 (linearized eigenvalue problem)] The transition from stability to instability for circular arcs is asserted to depend on length and boundary curvature, but the explicit spectral computation that establishes the sign change of the principal eigenvalue is not displayed in sufficient detail to verify the claimed threshold; an additional lemma or corollary with the characteristic equation would be required.

    Authors: We agree that the spectral computation merits a more explicit presentation to facilitate verification. In the revised version we will add a new corollary (Corollary 4.8) that isolates the characteristic equation satisfied by the eigenvalues of the linearized operator for circular-arc equilibria and explicitly tracks the sign of the principal eigenvalue as a function of arc length and boundary curvature at the contact points. This will make the stability threshold directly verifiable from the displayed equation. revision: yes

  2. Referee: [Section 6 (nonlinear stability)] The nonlinear stability statement for curved boundaries relies on a closeness assumption in a specific function space, yet the passage from linear decay to nonlinear control is only sketched; the precise interpolation or bootstrap argument that closes the estimate is missing and is load-bearing for the claim.

    Authors: We acknowledge that the bootstrap step from linear decay to nonlinear control is only outlined. In the revision we will expand the proof of Theorem 6.1 to include the full interpolation argument: we will state the precise interpolation inequalities between the linear decay rate and the quadratic nonlinear terms, show how the smallness assumption in the chosen function space absorbs the higher-order contributions, and close the a-priori estimate. This will render the nonlinear stability proof self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: standard linearized spectral analysis of explicitly characterized equilibria

full rationale

The derivation consists of identifying the stationary solutions (flat segments and circular arcs compatible with the 90° contact-angle condition), linearizing the Mullins-Sekerka evolution operator about these equilibria, and analyzing the spectrum of the resulting linear operator in dependence on curvature and length parameters. This is a self-contained PDE calculation that does not invoke fitted parameters renamed as predictions, self-citation load-bearing uniqueness theorems, or any reduction of the claimed stability statements to the input data by construction. The single nonlinear-stability result is likewise obtained by standard energy estimates without circular closure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical assumptions for the Mullins-Sekerka equation and linearization techniques without additional free parameters or invented entities.

axioms (1)
  • domain assumption Standard well-posedness and regularity assumptions for the Mullins-Sekerka free-boundary problem in two dimensions with 90° contact angle
    Invoked implicitly to justify the existence of the stationary solutions (flat and circular arcs) around which linearization is performed.

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