Numerical analysis of Cahn-Hilliard type equations on evolving surfaces

自从John W. Cahn和John E. 1958年的希利亚德。通常情况下，这是研究一个固定的欧几里德域。这项研究考虑了双重改进，通过查看可能随时间演变的超曲面（即非欧几里德域）。这给现有的理论增加了复杂性。除了数学上的好奇心之外，还有真正的理由考虑这种设置，因为有物理相关的模型将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：将对数势能的Cahn-Hilliard方程的数值分析结果应用于（静态）表面有限元的情况，并在演化域上研究类似的方程。这包括一个完全离散的数值方案的Cahn-Hilliard方程在一个不断发展的表面上具有足够光滑的潜力-目前的研究只考虑半离散计划，这个问题。类似地，可以将Cahn-Hilliard方程耦合到Navier-Stokes方程以获得可用于模拟更丰富现象的系统。演化曲面Navier-Stokes方程是近年来的研究热点，但目前关于演化曲面上Navier-Stokes-Cahn-Hilliard方程组的研究较少。我的目标是研究一个Navier-Stokes-Cahn-Hilliard系统，类似于著名的“H型”的Hohenberg和Halperin，在一个不断变化的表面上的分析和数值。这些问题已经研究了一个固定的，平坦的域，但有相当少的文献发展域或超曲面。目前，人们对演化超曲面上的偏微分方程有很大的兴趣，这给底层的物理模型增加了额外的复杂性和准确性。因此，这种模型的数值分析是必要的实际计划，将实施。类似地，这很好地作为演化曲面上的非线性偏微分方程的一个例子，我们可以获得定量信息。一个例子是最近对“从纯相分离”现象的研究。 资金是通过工程和物理科学研究理事会提供的，该理事会负责数学科学的研究。此外，卡恩-希利亚德方程在数学之外还可以应用于许多真实的世界问题，如冶金、建模趋化性、RNA/蛋白质动力学和图像处理分析。同样，Navier-Stokes-Cahn-Hilliard系统已被用于模拟趋化性（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.