Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants
Abstract
We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. Because of the non-standard, mixed-order strongly coupled structure of Model C, the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions are not applicable due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis, adapted to the temporal regularity of the source term, in a suitable Hilbert space, which yields an exact cancellation of the highest-order cross-couplings in the estimates. This crucial decoupling enables the rigorous application of the operator-valued Mikhlin multiplier theorem to obtain resolvent estimates on ℝ. Furthermore, we analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish local well-posedness for strong solutions.
Related
- Well-posedness of Strong Solutions for a Thermodynamically Consistent Diffuse-interface Model for Incompressible Two-phase Flows with a Soluble Surfactant
- Well-posedness, regularity and stability for linear Maxwell's equations with interface damping
- Simultaneous Sharp-Interface Limits for a Diffuse-Interface Landau--de Gennes Droplet Model
- Forward-Backward McKean-Vlasov System with Jumps and a Monotone Source With à Control-Shifted Interface : Well-Posedness, Stability, and Yosida Rates
- A Robust All-Mach Six-Equation Diffuse-Interface Method for Multiphase Flows with Surface Tension
BibTeX
@misc{abels2026local,
title = {{Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants}},
author = {Helmut Abels and Songzhuang Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03342},
eprint = {2610.03342},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03342},
}
RIS
TY - UNPB
TI - Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants
AU - Helmut Abels
AU - Songzhuang Chen
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03342
UR - https://arxiv.org/abs/2610.03342
AB - We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. Because of the non-standard, mixed-order strongly coupled structure of Model C, the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions are not applicable due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis, adapted to the temporal regularity of the source term, in a suitable Hilbert space, which yields an exact cancellation of the highest-order cross-couplings in the estimates. This crucial decoupling enables the rigorous application of the operator-valued Mikhlin multiplier theorem to obtain resolvent estimates on ℝ. Furthermore, we analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish local well-posedness for strong solutions.
ER -