A mathematical analysis reports global quasi-strong solutions for a Voigt-regularized model of incompressible two-phase viscous flow, along with an energy equality for those solutions. The theorem is stated under the model's assumptions, admissible initial data and compatibility conditions, so the result is a conditional existence statement within the equations.
The document is an arXiv preprint, version 1, dated 26 Aug 2026. It is not an experiment or a data study. It examines a deterministic system of partial differential equations in a smooth bounded domain Ω in two or three dimensions, with two fluids allowed to have different densities. No empirical sample is involved, and the paper states that no datasets were generated or analysed.
A coupled picture of two-phase flow
The model is a diffuse-interface system with phase fields in the bulk and on the surface. Its fluid dynamics couple Navier-Stokes-Voigt velocity dynamics to a convective Cahn-Hilliard phase equation, with a singular potential, generalized Navier-slip conditions and dynamic boundary conditions.
The phase variables are central to the analysis. In the stated solution class, the fields satisfy |φ|<1 and |ψ|<1 almost everywhere. A related strict-separation statement places them inside (-1+σ, 1-σ), keeping them away from the singular endpoints used by the potential.
The regularizing ingredient is the Voigt term with α>0. The analysis records an H1(Ω) estimate for the time derivative of velocity, ∂t v, and that time regularity appears in the quasi-strong class used for the existence theorem.
What the theorem establishes
In practical terms, quasi-strong is a label for a solution with specified spatial and temporal control. The paper states v∈L∞(0,T;H1div(Ω)), ∂tv∈L2(0,T;H1div(Ω)), and φ∈L∞(0,T;H2). These are analytical regularity statements, not measurements taken from a physical flow.
Existence is not the only conclusion. The paper reports an energy equality for the constructed solutions and states that sufficiently regular solutions satisfy a mass-conservation property. Both statements describe identities within the model, not empirical measurements.
The existence argument is presented through approximation schemes and a semi-Galerkin method, alongside the global quasi-strong result reported for the regularized system.
Appendix A adds a narrower result for the viscously regularized bulk-surface convective Cahn-Hilliard problem with prescribed velocity fields. It states unique weak solvability for that subsystem. The claim concerns the auxiliary phase subsystem, not an empirical test of the complete flow model.
Where the result stops
The scope becomes more important at the parameter limits. The case (K,L)∈{0}×[0,+∞] is explicitly not covered in the present work.
For the endpoint analysis, the route as L→0 requires matched densities, ρ1=ρ2>0. By contrast, the L→+∞ route requires no matched-density assumption, according to the analysis. The analysis does not establish the unmatched-density passage as L→0.
These qualifications keep the headline result in perspective. It applies to the specified PDE model on smooth bounded domains in dimensions two and three, under the paper's assumptions, initial-data requirements and parameter regimes. It is a theorem about that defined setting, not a general empirical claim about two-phase flows.
For readers outside mathematical analysis, the practical message is limited but clear: the preprint reports a global existence result with an energy equality for a regularized bulk-surface two-phase model. It does not report data, experiments or measured performance, so its contribution is to the analysis of the equations themselves.
Paper data and sources
Original title: Global Quasi-strong Solutions to the Voigt Regularization of a Diffuse Interface Model for Incompressible Two-phase Flows with Bulk-surface Interaction
Authors: Patrik Knopf, Maoyin Lv, Hao Wu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text