Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.Comment: 33 pages, minor typos corrected, accepted versio

On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behavior of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem can be shown. Using the approach of Colli and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems.Comment: 34 page

Mobilising management control systems to manage stakeholder relationship and sustainability challenges: an empirical case study of a major construction product company

An emerging stream of management accounting research has investigated how management accounting techniques can be used to manage sustainability. Because of the transformative nature of sustainability, the subject area embraces a significant breadth of issues across different research disciplines. While literature suggests there is a lack of definitive texts for thinking about sustainability in management accounting studies, scholars see potential to help organisations identify social and environmental risks and opportunities by using extant accounting mechanisms and engaging with business. The research identifies literature that borrows insights from Professor Freeman s Stakeholder Theory (1984) and recognises its potential to enrich the current understanding of accounting for sustainability. Therefore, the aim of this research is to augment Simons Levers of Control (LOC) framework to incorporate greater depth on the role of interactions between managers and stakeholders in the implementation of (sustainability) management control systems. The research undertakes a qualitative approach that uses empirical case study data from a major construction product company. Using interview data from 29 semi- structured interviews (where 7 interviews were conducted with external stakeholders) and documentary evidence (such as sustainability reports and policy) to provide a useful track of what has happened within the case organisation. LOC is employed as the theoretical lens to investigate how managers use management control systems (MCS) for sustainability-related, strategic decision-making. The research provides empirical findings that indicate Stakeholder Theory can augment Simons LOC framework, as an analytic framework that used by the managers. The research responds to calls to investigate the potential of using extant accounting mechanisms to cope with sustainability challenges through exploring the relationships between: (a) sustainability, (b) MCS, and (c) strategic decision-making. The empirical material is an example used to illustrate the theoretical framework in one case study organisation. Firstly, the research identifies organisational use of MCS to address stakeholders sustainability requirement for stakeholder management. This study explores a range of MCS used by the case organisation to consider and weigh stakeholders concerns and expectations in the process of stakeholder management. Aligning with Simons LOC literature, strong evidence is found to suggest MCS can be used in a dynamic and multi-layered way in managing stakeholders sustainability requirements. Secondly, the influences of the external use of MCS on the case organisation s sustainability strategy have been revealed. Empirical evidence provides comprehensive understanding of three key stakeholders levels of influence (i) top- down, (ii) mediated and (iii) collaborative relationship, to case organisation s sustainability strategy. Each is provided with specific strategic motivation(s) and is achieved through the mobilisation of MCS between the case organisation and stakeholders. Lastly, the research comprehends the use of MCS by the managers to manage sustainability-related strategic decision-making. Findings show that MCS are used in a variety of ways to manage strategy formation and strategy implementation. Both positive and negative controls are important for an intended sustainability strategy to be successful. The academic literature suggests that interactive control systems can be deployed to manage strategy formation, as originated by Simons (1995). This research suggests that such systems are also used by external stakeholders, namely for strategic decision-making; and this allows managers to cope with both the fluidity of a sustainability strategy, as well as responding to external changes

Phase field modelling of surfactants in multi-phase flow

A diffuse interface model for surfactants in multi-phase flow with three or more fluids is derived. A system of Cahn-Hilliard equations is coupled with a Navier-Stokes system and an advection-diffusion equation for the surfactant ensuring thermodynamic consistency. By an asymptotic analysis the model can be related to a moving boundary problem in the sharp interface limit, which is derived from first principles. Results from numerical simulations support the theoretical findings. The main novelties are centred around the conditions in the triple junctions where three fluids meet. Specifically the case of local chemical equilibrium with respect to the surfactant is considered, which allows for interfacial surfactant flow through the triple junctions

On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $\varphi_p$, $\varphi_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p$, $\varphi_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a quasistatic diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector $(\varphi_p,\varphi_d)$ is constrained to remain in the range of physically admissible values. On the other hand, in view of the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of $\varphi_p$ and $\varphi_d$. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.Comment: 41 page

Measuring the Interdependence of Banks in Hong Kong

This paper assesses systemic linkages among banks in Hong Kong using the risk measure "CoVaR" derived from quantile regression. The CoVaR measure captures the co-movements of banks¡¯ default risk by taking into account their nonlinear relationship when the banks are in distress. Based on equity price information, our estimation results show that the default risks of the banks were interdependent during the recent crisis. Although local banks are generally smaller, their systemic importance is found to be similar to their international and Mainland counterparts, which may be due to a higher degree of commonality in the risk profile of local banks. Regarding the impact of external shocks on the banks, international banks are more likely to be affected by the equity price fall in the US market, while local banks are relatively more responsive to funding liquidity risk.Value-at-Risk, Systemic Risk, Risk Spillovers, Quantile Regression