We investigate a degenerate parabolic variational inequality arising from optimal continuous exercise perpetual executive stock options. It is also shown in Qin et al. (Continuous-Exercise Model for American Call Options with Hedging Constraints, working paper, available at SSRN: http://dx.doi.org/10.2139/ssrn.2757541) that to make this problem non-trivial the stock's growth rate must be no smaller than the discount rate. Well-posedness of the problem is established in Lai et al. (2015, Mathematical analysis of a variational inequality modeling perpetual executive stock options, Euro. J. Appl. Math., 26 (2015), 193–213), Qin et al. (2015, Regularity free boundary arising from optimal continuous exercise perpetual executive stock options, Interfaces and Free Boundaries, 17 (2015), 69–92), Song & Yu (2011, A parabolic variational inequality related to the perpetual American executive stock options, Nonlinear Analysis, 74 (2011), 6583-6600) for the case when the underlying stock's expected return rate is smaller than the discount rate. In this paper, we consider the remaining case: the discount rate is bigger than the growth rate but no bigger than the return rate. The existence of a unique classical solution as well as a continuous and strictly decreasing free boundary is proved.

In numerical linear stability investigations, the rates of change of the kinetic and thermal energy of the perturbation flow are often used to identify the dominant mechanisms by which kinetic or thermal energy is exchanged between the basic and the perturbation flow. Extending the conventional energy analysis for a single-phase Boussinesq fluid, the energy budgets of arbitrary infinitesimal perturbations to the basic two-phase liquid–gas flow are derived for an axisymmetric thermocapillary bridge when the material parameters in both phases depend on the temperature. This allows identifying individual transport terms and assessing their contributions to the instability if the basic flow and the critical mode are evaluated at criticality. The full closed-form energy budgets of linear modes have been derived for thermocapillary two-phase flow taking into account the temperature dependence of all thermophysical parameters. The influence of different approximations to the temperature dependence on the linear stability boundary of the axisymmetric flow in thermocapillary liquid bridges is tested regarding their accuracy. The general mechanism of symmetry breaking turns out to be very robust.

Biofilms are communities of bacteria that exhibit a multitude of multiscale biomechanical behaviours. Recent experimental advances have led to characterisations of these behaviours in terms of measurements of the viscoelastic moduli of biofilms grown in bioreactors and the fracture and fragmentation properties of biofilms. These properties are macroscale features of biofilms; however, a previous work by our group has shown that heterogeneous microscale features are critical in predicting biofilm rheology. In this paper, we use tools from statistical physics to develop a generative statistical model of the positions of bacteria in biofilms. Specifically, the model is a type of pairwise interaction model (PIM). We show through simulation that the macroscopic mechanical properties of biofilms depend on the choice of microscale spatial model. A key finding is that uniform and non-uniform sets of points lead to differing mechanical properties. This distinction appears not to have been previously considered in mathematical biofilm literature. We also found that realisations of a biologically informed PIM have realistic in silico mechanical properties, and have statistical properties that closely match experimental data. We also note that a Poisson spatial point process of suitable number density also yields realistic mechanical properties, but that the spatial distribution of points does not reflect those occurring in our experimentally observed biofilm.

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

The use of both higher-order amplitude corrections and complex rays to increase the accuracy of the approximation of the eigenvalues of Helmholtz equation on an annulus is demonstrated.

We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

In this paper, a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier–Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids; thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model verifies a dissipative entropy inequality up to a second-order term. Moreover, we propose a correction of the model – by taking into account the second-order extension for the pressure – that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first test, we compare the numerical results with the viscous bilayer shallow water model proposed in Narbona-Reina et al. (Comput. Model. Eng. Sci., 2009, Vol. 43, pp. 27–71). In the second test, the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test, the influence of the friction coefficient on the coastal area affected by the pollutant is studied.

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of type Mu = −N∇p + F with suitable matrices M and N with F depending on the right-hand side in the bulk term and in the boundary condition.

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

We provide a detailed mathematical analysis of a model for phase separation on biological membranes which was recently proposed by Garcke, Rätz, Röger and the second author. The model is an extended Cahn–Hilliard equation which contains additional terms to account for the active transport processes. We prove results on the existence and regularity of solutions, their long-time behaviour, and on the existence of stationary solutions. Moreover, we investigate two different asymptotic regimes. We study the case of large cytosolic diffusion and investigate the effect of an infinitely large affinity between membrane components. The first case leads to the reduction of coupled bulk-surface equations in the model to a system of surface equations with non-local contributions. Subsequently, we recover a variant of the well-known Ohta–Kawasaki equation as the limit for infinitely large affinity between membrane components.

This paper is concerned with some mathematical aspects of magnetic resonance imaging (MRI) of the beating human heart. In particular, we investigate the so-called retrospective gating technique which is a non-triggered technique for data acquisition and reconstruction of (approximately) periodically changing organs like the heart. We formulate the reconstruction problem as a moment problem in a Hilbert space and give the solution method. The stability of the solution is investigated and various error estimates are given. The reconstruction method consists of temporal interpolation followed by spatial Fourier inversion. Different choices for the Hilbert space ℋ of interpolating functions are possible. In particular, we study the case where ℋ is (i) the space of bandlimited functions, or (ii) the space of spline functions of odd degree. The theory is applied to reconstructions from synthetic data as well as real MRI data.

We study of a new type of multi-bump blowup solutions of the Ginzburg–Landau equation. Multi-bump blowup solutions have previously been found in numeric simulations, asymptotic analysis and were proved to exist via geometric construction. In the geometric construction of the solutions, the existence of two types of multi-bump solutions was shown. One type is exponentially small at ξ=0, and the other type of solutions is algebraically small at ξ=0. So far, the first type of solutions were studied asymptotically. Here, we analyse the solutions which are algebraically small at ξ=0 by using asymptotic methods. This construction is essentially different from the existing one, and ideas are obtained from the geometric construction. Hence, this is a good example of where both asymptotic analysis and geometric methods are needed for the overall picture.

The high Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a non-linear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The comparison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient.

We consider a spherical particle levitating above a liquid bath owing to the Leidenfrost effect, where the vapour of either the bath or sphere forms an insulating film whose pressure supports the sphere’s weight. Starting from a reduced formulation based on a lubrication-type approximation, we use matched asymptotics to describe the morphology of the vapour film assuming that the sphere is small relative to the capillary length (small Bond number) and that the densities of the bath and sphere are comparable. We find that this regime is comprised of two formally infinite sequences of distinguished limits which meet at an accumulation point, the limits being defined by the smallness of an intrinsic evaporation number relative to the Bond number. These sequences of limits reveal a surprisingly intricate evolution of the film morphology with increasing sphere size. Initially, the vapour film transitions from a featureless morphology, where the thickness profile is parabolic, to a neck–bubble morphology, which consists of a uniform pressure bubble bounded by a narrow and much thinner annular neck. Gravity effects then become important in the bubble leading to sequential formation of increasingly smaller neck–bubble pairs near the symmetry axis. This process terminates when the pairs closest to the symmetry axis become indistinguishable and merge. Subsequently, the inner section of that merger transitions into a uniform-thickness film that expands radially, gradually squishing increasingly larger neck–bubble pairs into a region of localised oscillations sandwiched between the uniform film and what remains of the bubble whose radial extent is presently comparable to the uniform film; the neck–bubble pairs farther from the axis remain essentially intact. Ultimately, the uniform film gobbles up the largest outermost bubble, whereby the morphology simplifies to a uniform film bounded by localised oscillations. Overall, the asymptotic analysis describes the continuous evolution of the vapour film from a neck–bubble morphology typical of a Leidenfrost drop levitating above a flat solid substrate to a uniform-film morphology which resembles that in the case of a large liquid drop levitating above a liquid bath.

We analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation ℱ between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that ℱ satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique.

Singular Systems Analysis (SSA), or time domain Principal Component Analysis (PCA), is most appropriately analysed in terms of local, moving-window spectral analysis. The behaviour of Empirical Orthogonal Functions (EOF) of this theory are examined, for continuously sampled data, in the limits of large and small window length, and for centre or end projection. Filters obtained by projecting on to these EOFs are shown to approximate local, linear band pass filters, where the EOFs depend upon the correlation structure (or the power spectral density) of the signal and the window length. Power in the spectra is not generally conserved, and projection to the endpoints of a window may not converge to the underlying signal in the absence of noise. The filters are independent of the phase of the Fourier transform, and are therefore unable to distinguish dynamically between a signal and a surrogate (phase-randomized) transform of it. Iteration of such local filters using a prediction error-based stopping criterion can and does lead to improved results, but the choice of window length must be made a priori. Hence, we introduce an iterative local filter with the window length being determined as part of the filtering procedure. This involves the determination of the predictability of the projected time series, and hence allows SSA to be used in a genuinely nonlinear way.

The discrete non-linear Schrödinger equation is one of the most important inherently discrete models, having a crucial role in the modelling of a great variety of phenomena, ranging from solid-state and condensed-matter physics to biology. In this paper, a class of discrete non-linear Schrödinger equations are considered. Using critical point theory in combination with periodic approximations, we establish some new sufficient conditions on the existence results for solitons of the equation. The classical Ambrosetti–Rabinowitz superlinear condition is improved.

We consider the following system of equations: where the spatial average ⟨ B ⟩ = 0 and μ > σ > 0. This system plays an important role as a Ginzburg-Landau equation with a mean field in several areas of the applied sciences and the steady-states of this system extend to periodic steady-states with period L on the real line which are observed in experiments. Our approach is by combining methods of nonlinear functional analysis such as nonlocal eigenvalue problems and the variational characterization of eigenvalues with Jacobi elliptic integrals. This enables us to give a complete classification of all stable steady-states for any positive L.