Surface Stokes-Cahn-Hilliard System: Analysis and Structure-Preserving Discretization
Abstract
We study the surface Stokes-Cahn-Hilliard system governing tangential flow and phase separation in viscous fluid membranes. The equations are posed on a closed, stationary C2 hypersurface Γ⊂ℝd+1, d=2,3. For constant viscosity and mobility and a class of regular double-well potentials, we prove global existence and uniqueness of weak solutions for initial phase fields in H1(Γ), with the velocity chosen orthogonal to the Killing fields. These solutions satisfy an energy equality and conserve mass. We establish Lipschitz continuous dependence in energy norms on every finite time interval, including for initial data with different means, and derive higher regularity results for d=2 under additional smoothness assumptions on the surface, potential, and initial data. We also develop a fully discrete unfitted finite element method that conserves the total phase and satisfies an exact discrete energy identity. The method combines a penalty on approximate Killing fields with projections in the transport and capillary terms to control these modes while preserving the energy balance. Numerical experiments examine the identification of Killing fields and illustrate phase separation on a torus and a liposome with experimentally informed material parameters.
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BibTeX
@misc{olshanskii2026surface,
title = {{Surface Stokes-Cahn-Hilliard System: Analysis and Structure-Preserving Discretization}},
author = {Maxim Olshanskii and Mansur Shakipov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06552},
eprint = {2610.06552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06552},
}
RIS
TY - UNPB
TI - Surface Stokes-Cahn-Hilliard System: Analysis and Structure-Preserving Discretization
AU - Maxim Olshanskii
AU - Mansur Shakipov
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06552
UR - https://arxiv.org/abs/2610.06552
AB - We study the surface Stokes-Cahn-Hilliard system governing tangential flow and phase separation in viscous fluid membranes. The equations are posed on a closed, stationary C^2 hypersurface Γ⊂ℝ^d+1, d=2,3. For constant viscosity and mobility and a class of regular double-well potentials, we prove global existence and uniqueness of weak solutions for initial phase fields in H^1(Γ), with the velocity chosen orthogonal to the Killing fields. These solutions satisfy an energy equality and conserve mass. We establish Lipschitz continuous dependence in energy norms on every finite time interval, including for initial data with different means, and derive higher regularity results for d=2 under additional smoothness assumptions on the surface, potential, and initial data. We also develop a fully discrete unfitted finite element method that conserves the total phase and satisfies an exact discrete energy identity. The method combines a penalty on approximate Killing fields with projections in the transport and capillary terms to control these modes while preserving the energy balance. Numerical experiments examine the identification of Killing fields and illustrate phase separation on a torus and a liposome with experimentally informed material parameters.
ER -