Numerical analysis of Cahn-Hilliard type equations on evolving surfaces

自1958年约翰·W·卡恩和约翰·E·希利亚德首次提出卡恩-希利亚德方程以来，人们对它进行了大量的研究。通常，这是在固定的欧几里得区域上研究的。这项研究考虑了两方面的改进，通过观察可能随时间演变的超曲面(即非欧几里德区域)。这给现有的理论增加了不小的复杂性。除了数学上的好奇心，还有真正的理由考虑这种设置，因为有物理上相关的模型将Cahn-Hilliard方程耦合到表面演化方程(例如，通过强迫平均曲率流)。在这项研究中，我的目标是：将具有对数势能的Cahn-Hilliard方程的数值分析结果应用于(定常)表面有限元的情况，并在演化区域上研究类似的方程。这包括对具有足够光滑位势的演化表面上的Cahn-Hilliard方程的全离散数值格式-因为目前的研究只考虑了这个问题的半离散格式。类似地，人们可以将Cahn-Hilliard方程耦合到Navier-Stokes方程，以获得一个可以用来模拟更丰富现象的系统。演化表面的Navier-Stokes方程一直是人们感兴趣的问题，但目前关于演化表面的Navier-Stokes-Cahn-Hilliard系统的研究要少得多。我的目标是研究一个类似于著名的Hohenberg和Halperin的“H模型”的Navier-Stokes-Cahn-Hilliard系统，在一个不断变化的表面上，从解析和数值两方面进行研究。这些问题被研究在固定的、平坦的区域上，但关于演化区域或超曲面的文献却相当少。目前，人们对演化超曲面上的偏微分方程组非常感兴趣，这增加了底层物理模型的复杂性和准确性。因此，对这些模型进行数值分析对于将要实施的实际方案是必要的。同样，这很好地用作演化曲面上的非线性偏微分方程的一个例子，我们可以获得关于它的定量信息。最近关于“从纯相中分离”现象的研究就是一个例子。资金是通过工程和物理科学研究理事会(EPSRC)提供的，该理事会涵盖数学科学的研究。此外，除了数学之外，Cahn-Hilliard方程还应用于许多现实世界的问题，如冶金、趋化性建模、RNA/蛋白质动力学和图像处理分析。同样，Navier-Stokes-Cahn-Hilliard系统也被用来模拟趋化作用(有平流)和肿瘤生长，以及脂质生物膜的动力学。

There has been much research into the Cahn-Hilliard equation since it was initially proposed by John W. Cahn and John E. Hilliard in 1958. Typically, this is studied on a stationary Euclidean domain. This research considers a twofold improvement, by looking at hypersurfaces (i.e. a non-Euclidean domain) which may be evolving in time. This adds non-trivial complications to the existing theory. Beyond mathematical curiosity, there is genuine reason to consider this set up, as there are physically relevant models coupling the Cahn-Hilliard equation to surface evolution equations (for example by a forced mean curvature flow). In this research I aim to: adapt results of the numerical analysis of the Cahn-Hilliard equation with a logarithmic potential energy to the case of (stationary) surface finite elements, and study similar equations on an evolving domain. This includes a fully discrete numerical scheme for the Cahn-Hilliard equation on an evolving surface with a sufficiently smooth potential - as current research only has considered semi-discrete schemes for this problem. Similarly, one can couple the Cahn-Hilliard equations to the Navier-Stokes equations to obtain a system which can be used to model richer phenomena. The evolving surface Navier-Stokes equations have been a recent topic of interest, but there is currently much less research regarding Navier-Stokes-Cahn-Hilliard systems on evolving surfaces. I aim to study a Navier-Stokes-Cahn-Hilliard system, similar to the well known "Model H" of Hohenberg and Halperin, on an evolving surface both analytically and numerically. These problems been studied for a stationary, flat domain but there is considerably less literature for evolving domains or hypersurfaces. There is currently much interest in partial differential equations on evolving hypersurfaces, this adds extra complexity and accuracy to the underlying physical model. As such, the numerical analysis of such models is necessary for practical schemes which will be implemented. Similarly, this serves nicely as an example of a nonlinear partial differential equation on an evolving surface for which we can obtain quantitative information. An example being recent work on the phenomenon of "separation from the pure phases". Funding is through the Engineering and Physical Sciences Research Council (EPSRC), which covers research in mathematical sciences. Moreover, the Cahn-Hilliard equation finds applications, outside of mathematics, to many real world problems such as in metallurgy, modelling chemotaxis, RNA/protein dynamics, and in image processing analysis. Likewise the Navier-Stokes-Cahn-Hilliard system has been used to model chemotaxis (with advection) and tumour growth, and the kinetics of lipid biomembranes.