发表机构
University of Houston（休斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究粘性流体膜表面Stokes-Cahn-Hilliard系统，证明弱解全局存在唯一性、能量守恒及正则性，并发展保结构的非拟合有限元方法，数值验证相分离。
AI 中文摘要
我们研究控制粘性流体膜中切向流动和相分离的表面Stokes-Cahn-Hilliard系统。方程定义在闭的、静止的$C^2$超曲面$\Gamma\subset\mathbb{R}^{d+1}$上，其中$d=2,3$。对于恒定粘度和迁移率以及一类正则双阱势，我们证明了初始相场在$H^1(\Gamma)$中时弱解的全局存在性和唯一性，速度选择为与Killing场正交。这些解满足能量等式并保持质量守恒。我们建立了在能量范数下任意有限时间区间上的Lipschitz连续依赖性，包括对具有不同均值的初始数据，并在$d=2$时在表面、势和初始数据的额外光滑性假设下推导了更高正则性结果。我们还开发了一种完全离散的非拟合有限元方法，该方法保持总相并满足精确的离散能量恒等式。该方法将对近似Killing场的惩罚与输运项和毛细项中的投影相结合，以控制这些模式同时保持能量平衡。数值实验检验了Killing场的识别，并展示了在环面和脂质体上使用实验信息材料参数的相分离。
英文摘要
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 $Γ\subset\mathbb{R}^{d+1}$, $d=2,3$. 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.
Comments28 pages, 3 figures
