An arXiv preprint proposes a visco-elastic Mullins–Sekerka system with a prescribed constant contact angle. The paper treats the interface between phases as a sharp mathematical boundary and seeks weak solutions. Its main objects are a measurable phase indicator, oriented varifolds and an elastic displacement field.
This is a methods study, not an empirical investigation. The analysis concerns a mathematical partial differential equation and reports no outcomes from people, animals, cells or real materials. It also contains no experimental or numerical validation, so its findings describe what can be established about the proposed equations rather than measured physical effects.
A model built around energy
The system is derived as a gradient flow, a formulation in which the evolution is organized around an energy. That energy contains perimeter, capillary and elastic contributions, along with a second-gradient term used in the regularized model. The derivation is described as formal under sufficiently regular interface assumptions.
The proposed formulation is intended to describe visco-elastic phase dynamics while retaining a dissipative structure. The analysis identifies phase-mass conservation as a structural property and expresses dissipation through an energy-dissipation inequality.
Because the treatment is designed to include weak interfaces, the paper introduces a measure-valued solution concept. In that framework, oriented varifolds provide the geometric representation used alongside the phase indicator and displacement field. The formulation also includes a weak generalized Gibbs–Thomson law with an associated potential.
What the theorems establish
The existence result is built through an implicit time-discretization scheme. Rather than assuming a classical solution from the outset, the analysis constructs solutions within the measure-valued framework. The result establishes existence under the theorem’s assumptions; uniqueness of the constructed weak solutions is not reported.
The principal theorem is posed on a bounded smooth domain in spatial dimensions 2 or 3. Under its stated conditions, it produces a measurable phase indicator, oriented varifolds and a displacement field.
The varifold formulation includes a weak generalized Gibbs–Thomson relation and a sharp De Giorgi-type energy-dissipation inequality. The inequality is stated over almost all allowed pairs of times, making it a central part of the weak-solution concept rather than an experimental measurement.
The system’s mass-conservation property is also preserved in the mathematical construction. Here, conservation refers to phase mass within the proposed equations, not to an observation made in a physical experiment.
A narrower result for BV solutions
A separate theorem addresses BV solutions, using a bounded-variation framework for phase indicators and interfaces that need not be described by smooth functions. The paper proves existence only conditionally: it assumes energy conservation and a contact-angle condition whose cosine satisfies cos α = 0.
Those conditions are part of the theorem’s scope. The result does not cover general contact angles, and it does not remove the energy-conservation hypothesis. The paper therefore presents the BV conclusion as conditional rather than as a general existence statement.
The limits of the weak theory
The general formulation does not establish integer varifolds, generalized mean curvature or rectifiability. The authors identify the bounds needed for stronger varifold regularity as prohibitively difficult to verify in the distributional Gibbs–Thomson setting.
Weak–strong uniqueness is likewise not established as a general theorem. The authors frame it as an expected result in a more homogeneous setting, where a weak solution could potentially be shown to agree with a sufficiently regular one.
Taken together, the results provide a mathematical existence framework, not a complete regularity theory. The proved statements are limited to the specified domain, admissible initial data and additional assumptions attached to each theorem.
Research record
The paper reports no conflict of interests and says that no associated data accompany the manuscript.
The last author reports support from Graduiertenkolleg 2339 IntComSin of the Deutsche Forschungsgemeinschaft, with Project-ID 321821685.
Paper data and sources
Original title: On a visco-elastic Mullins-Sekerka System
Authors: Helmut Abels, Harald Garcke, Jonas Haselböck
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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