We derive a class of Navier–Stokes–Cahn–Hilliard systems that models two-phase flows with mass transfer coupled to the process of chemotaxis. These thermodynamically consistent models can be seen as the natural Navier–Stokes analogues of earlier Cahn–Hilliard–Darcy models proposed for modelling tumour growth, and are derived based on a volume-averaged velocity, which yields simpler expressions compared to models derived based on a mass-averaged velocity. Then, we perform mathematical analysis on a simplified model variant with zero excess of total mass and equal densities. We establish the existence of global weak solutions in two and three dimensions for prescribed mass transfer terms. Under additional assumptions, we prove the global strong well-posedness in two dimensions with variable fluid viscosity and mobilities, which also includes a continuous dependence on initial data and mass transfer terms for the chemical potential and the order parameter in strong norms.

This paper concerns the drainage of a thin liquid lamella into a Plateau border. Many models for draining soap films assume that their gas-liquid interfaces are effectively immobile. Such models predict a phenomenon known as “marginal pinching”, in which the film tends to pinch in the margin between the lamella and the Plateau border. We analyse the opposite extreme, in which the gas-liquid interfaces are assumed to be stress-free. We apply a nonlocal coordinate transformation that transforms the nonlinear governing equations into the linear heat equation. We thus obtain a travelling-wave solution and show that it is globally attractive. We therefore find that no marginal pinching occurs in this case: the film always approaches a monotonic profile at large times. This behaviour reflects a marked qualitative difference between a film with immobile interfaces and one with perfectly mobile interfaces. In the former case, suction of fluid into the Plateau border is localised, while in the latter it is instantaneously transmitted throughout the film.

An algorithm, based on a discrete nonlinear model, is presented for evaluation of the critical shear stress required to move a dislocation through a lattice. The stability of solutions of the corresponding evolution problem is analysed. Numerical results provide upper and lower bounds for the critical shear stress.

Let , N ≥2 be a bounded smooth domain and α > 1. We are interested in the singular elliptic equationwith Neumann boundary conditions. In this paper, a complete description of all continuous radially symmetric solutions is given. In particular, we construct nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N = 2 and α = 3, the equation is used to model steady states of van der Waals force driven thin films of viscous fluids. We also consider the physical problem when total volume of the fluid is prescribed.

Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during the last five years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffusion-type differential equations model that describes how the number of criminals evolves in a specific area. We propose a more detailed form of this model that allows for two distinct criminal types associated with major and minor crime. Additionally, we examine a stochastic variant of the model that represents more realistically the ‘generation’ of new criminals. Numerical solutions from both models are presented and compared with actual crime data for the Greater Manchester area. Agreement between simulations and actual data is satisfactory. A preliminary statistical analysis of the data also supports the model's potential to describe crime.

We formulate a number of open problems for time-harmonic inverse electromagnetic scattering theory focusing on uniqueness theorems, the determination of the support of a scattering object and the determination of material parameters

The effects of the thin air layer entering play when a water droplet impacts on otherwise still water or on a fixed solid are studied theoretically with special attention on surface tension and on post-impact behaviour. The investigation is based on the small density and viscosity ratios of the two fluids. In certain circumstances, and in particular for droplet Reynolds numbers below a critical value which is about ten million, the air-water interaction depends to leading order on lubricating forces in the air coupled with potential flow dynamics in the water. The nonlinear integro-differential system for the evolution of the interface and induced pressure is studied for pre-impact surface tension effects, which significantly delay impact, and for post-impact interaction phenomena which include significant decrease of the droplet spread rate. Above-critical Reynolds numbers are also considered.

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problem

formula here

Here u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time-dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.

In this paper, we explore some of the various issues that may occur in attempting to fit a dynamical systems (either agent- or continuum-based) model of urban crime to data on just the attack times and locations. We show how one may carry out a regression analysis for the model described by Short et al. (2008, Math. Mod. Meth. Appl. Sci.) by using simulated attack data from the agent-based model. It is discussed how one can incorporate the attack data into the partial differential equations for the expected attractiveness to burgle and the criminal density to predict crime rates between attacks. Using this predicted crime rate, we derive a likelihood function that one can maximise in order to fit parameters and/or initial conditions for the model. We focus on carrying out data assimilation for two different parameter regions, namely in the case where stationary and non-stationary crime hotspots form. It is found that the likelihood function is ‘flat’ for large ranges of parameters, and that this has major implications for crime forecasting. Hence, we look at how one might carry out a goodness-of-fit and forecasting analysis for crime rates given the range of parameter fits. We show how one can use the Kolmogorov–Smirnov statistic to assess the goodness-of-fit. The dynamical systems analysis of the partial differential equations proves invaluable to understanding how the crime rate forecasts depend on the parameters and their sensitivity. Finally, we outline several interesting directions for future research in this area where we believe that the combination of dynamical systems modelling, analysis, and data assimilation can prove effective in developing policing strategies for urban crime.

Fibre drawing is an important industrial process for synthetic polymers and optical communications. In the manufacture of optical fibres, precise diameter control is critical to waveguide performance, with tolerances in the submicron range that are met through feedback controls on processing conditions. Fluctuations arise from material non-uniformity plague synthetic polymers but not optical silicate fibres which are drawn from a pristine source. The steady drawing process for glass fibres is well-understood (e.g. [11, 12, 20]). The linearized stability of steady solutions, which characterize limits on draw speed versus processing and material properties, is well-understood (e.g. [9, 10, 11]). Feedback is inherently transient, whereby one adjusts processing conditions in real time based on observations of diameter variations. Our goal in this paper is to delineate the degree of sensitivity to transient fluctuations in processing boundary conditions, for thermal glass fibre steady states that are linearly stable. This is the relevant information for identifying potential sources of observed diameter fluctuation, and for designing the boundary controls necessary to alter existing diameter variations. To evaluate the time-dependent final diameter response to boundary fluctuations, we numerically solve the model nonlinear partial differential equations of thermal glass fibre processing. Our model simulations indicate a relative insensitivity to mechanical effects (such as take-up rates, feed-in rates), but strong sensitivity to thermal fluctuations, which typically form a basis for feedback control.

This paper addresses short-time existence and uniqueness of a solution to the N-dimensional Hele–Shaw flow problem with surface tension as driving mechanism. Global existence in time and exponential decay of the solution near equilibrium are also proved. The results are obtained in Sobolev spaces Hs with sufficiently large s. The main tools are perturbations of a fixed reference domain, linearization with respect to these perturbations, a quasilinearization argument based on a geometric invariance property, and a priori energy estimates.

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

The problem of extrapolating asymptotic perturbation-theory expansions in powers of a small variable to large values of the variable tending to infinity is investigated. The analysis is based on self-similar approximation theory. Several types of self-similar approximants are considered and their use in different problems of applied mathematics is illustrated. Self-similar approximants are shown to constitute a powerful tool for extrapolating asymptotic expansions of different natures.

We undertake the numerical analysis of a reaction-diffusion system of ‘$\lambda$–$\omega$’ type [26]. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case [11]. We establish a priori estimates and error bounds for a semi-discrete and a fully discrete finite element approximation. The theoretical results are illustrated and verified via the numerical solution of periodic plane waves in one space dimension. Experiments in two space dimensions led to‘target patterns’ and spiral wave break-up.

Game-theoretic p-Laplacian or normalized p-Laplacian operator is a version of classical variational p-Laplacian which was introduced recently in connection with stochastic games called Tug-of-War with noise (Peres et al. 2008, Tug-of-war with noise: A game-theoretic view of the p-laplacian. Duke Mathematical Journal145(1), 91–120). In this paper, we propose an adaptation and generalization of this operator on weighted graphs for 1 ≤ p ≤ ∞. This adaptation leads to a partial difference operator which is a combination between 1-Laplace, infinity-Laplace and 2-Laplace operators on graphs. Then we consider the Dirichlet problem associated to this operator and we prove the uniqueness and existence of the solution. We show that the solution leads to an iterative non-local average operator on graphs. Finally, we propose to use this operator as a unified framework for interpolation problems in signal processing on graphs, such as image processing and machine learning.

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins–Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow us to treat such free boundary problems in both IR2 and IR3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method, we compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

We consider the Gierer–Meinhardt system with precursor inhomogeneity and two small diffusivities in an interval

$$\begin{equation*}\left\{\begin{array}{ll}A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm]\tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm]A' (-1)= A' (1)= H' (-1) = H' (1) =0,\end{array}\right.\end{equation*}$$

$$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad\end{equation*}$$

$$\begin{equation*}\tau\geq 0\mbox{ and$\tau$ is independent of $\epsilon$.}\end{equation*}$$

$O(\sqrt{D})$

We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.