Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guaranteed.

The time-varying flow in which fluid is withdrawn from or added to a reservoir of infinite or arbitrary finite depth through a point sink or source of variable strength beneath a free surface is considered. Backed up by some analytic work, a numerical method is used, and the results are compared with previous work on steady and unsteady flows. In the case of withdrawal for an impulsively started flow, it is found that the critical flow rate increases with reservoir depth, although it changes little as the depth increases beyond double the sink submergence depth. The largest flow rate at which steady solutions can evolve in source flows follows a similar pattern although at a considerably higher value. Simulations indicate that some of the previously calculated steady state solutions at higher flow rates may be unstable, if they exist at all.

In this paper, we study the following non-local problem:

\begin{equation*} \begin{cases} \displaystyle u_t=d{1\over\rho}\nabla\cdot(\rho V\nabla u)+b(\bar{u}-u)+ g(x) u^2(1-u) &\displaystyle \quad \textrm{in} \; \Omega\times (0,\infty),\\[3pt] \displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times (0,\infty),\\[3pt] \displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times (0,\infty).\vspace*{-2pt} \end{cases} \end{equation*}

The special lead article in this issue of EJAM, by Profs. John Ockendon and Brian Sleeman, is dedicated to Prof. Joseph B. Keller (referred to as Joe by his friends), Emeritus professor of Stanford University, who was one of the greatest applied mathematicians of our time. Joe died in September 2016. A workshop in honour of Joe was held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in March 2017, focussing on some of the astonishingly wide range of topics in the mathematical sciences and continuum mechanics that Joe made substantial contributions to over his long career. This article discusses some of these pioneering contributions, describes some modern developments as discussed by the workshop speakers, and provides a more personal tribute to Joe, his history, and to his large influence on the international applied mathematics community. May the spirit of Joe, and his flavour of applied mathematics, continue to be reflected in EJAM.

Complex biological products, such as those used to treat various forms of cancer, are typically produced by mammalian cells in bioreactors. The most important class of such biological medicines is proteins. These typically bind to sugars (glycans) in a process known as glycosylation, creating glycoproteins, which are more stable and effective medicines. The glycans are large polymers that are formed by a long sequence of enzyme catalysed reactions. This sequence is not always completed, thus leading to a heterogeneous glycoprotein distribution. A better comprehension of this distribution could lead to more efficient production of high quality drugs. To understand how the manufacturing process can affect the extent of glycosylation of protein, a non-linear ODE model of glycoprotein production is developed which describes the bioreactor configuration as well as the protein production and glycosylation reactions within the cell. The entire system evolves eventually to a stable steady state. The earlier evolution is critical however, as the amount of product produced and its quality varies over time. The model is considered as two coupled systems: the bioreactor submodel and the glycosylation submodel. To investigate the early time evolution within the bioreactor submodel, analytical and numerical properties are derived using matched asymptotic expansions and a finite difference scheme for a range of initial conditions. This leads to qualitatively different regimes for aglycosylated protein production, which affect the glycosylation submodel. The discrete glycoprotein distribution is approximated as continuous and written as a first-order PDE, with good agreement between the discrete and continuous models. The PDE is found to admit shocks, but the existence of these shocks is dependent on the early time evolution within the bioreactor submodel and leads to higher levels of glycosylation at early time. This suggests that changing the bioreactor configuration can lead to higher quality product at certain times.

This paper establishes the existence of smooth solutions for the Doi–Edwards rheological model of viscoelastic polymer fluids in shear flows. The problem turns out to be formally equivalent to a K-BKZ equation but with constitutive functions spanning beyond the usual mathematical framework. We prove, for small enough initial data, that the solution remains in the domain of hyperbolicity of the equation for all t≥0.

This work considers the problem of binary classification: given training data x1, . . ., xn from a certain population, together with associated labels y1,. . ., yn ∈ {0,1}, determine the best label for an element x not among the training data. More specifically, this work considers a variant of the regularized empirical risk functional which is defined intrinsically to the observed data and does not depend on the underlying population. Tools from modern analysis are used to obtain a concise proof of asymptotic consistency as regularization parameters are taken to zero at rates related to the size of the sample. These analytical tools give a new framework for understanding overfitting and underfitting, and rigorously connect the notion of overfitting with a loss of compactness.

We investigate a non-isothermal diffuse-interface model that describes the dynamics of two-phase incompressible flows with thermo-induced Marangoni effect. The governing PDE system consists of the Navier--Stokes equations coupled with convective phase-field and energy transport equations, in which the surface tension, fluid viscosity and thermal diffusivity are allowed to be temperature dependent functions. First, we establish the existence and uniqueness of local strong solutions when the spatial dimension is two and three. Then, in the two-dimensional case, assuming that the L∞-norm of the initial temperature is suitably bounded with respect to the coefficients of the system, we prove the existence of global weak solutions as well as the existence and uniqueness of global strong solutions.

Over the two days 2–3 March 2017, about 80 mathematicians and friends gathered in Cambridge to celebrate the life and work of Joseph Bishop Keller (1923–2016), one of the pre-eminent applied mathematicians of the 20th century. Joe, as he was known throughout the world, made pioneering contributions to a wide range of natural phenomena and developed fundamental mathematical techniques with which to understand them. Twenty-four talks were presented at the meeting, given by mathematicians who have either worked with Joe or have been influenced by his work. Rather than summarise each presentation, we have collated all the contributions under the headings of waves, fluids, solids, chemistry and biology, and finally some history.

A three-dimensional model of the varistor device is proposed. The thermal and electric conductivity of the material are taken to depend, in addition to the electric potential, on the temperature. Two theorems of existence and uniqueness of solutions for the boundary-value problem which determine the potential and the temperature inside the device are proposed. Levy–Caccioppoli global inversion theorem is used for the proof.

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

Entropy maximising spatial interaction models have been widely exploited in a range of disciplines and applications: from trade and migration flows to the spread of riots and the understanding of spatial patterns in archaeological sites of interest. When embedded into a dynamic system and framed in the context of a retail model, the dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be addressed analytically. In this paper, we present some analysis of the continuous retail model and the corresponding discrete version, which yields insights into the effect of space on the evolving system, and an understanding of why certain retail centres are more successful than others. The slowly developing growths and the fast explosive growths that are of particular concern are explained in detail.

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 study a free boundary problem for the Fisher-KPP equation: ut = uxx + f(u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) − α and g′(t) = −ux(t, g(t)) + β for 0 < β < α. This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We investigate the asymptotic behaviour of bounded solutions. There are two parameters α0 and α* with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β < α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, i.e., h(t) − g(t) → +∞ and u(t, ⋅ + ct) → 1 with c ∈ (cL, cR), where cL and cR are the asymptotic spreading speed of g(t) and h(t), respectively, (cR > 0 > cL when 0 < β < α < α0; cR = 0 >cL when 0 < β < α = α0; 0 > cR > cL when α0 < α < α* and 0 < β < α0); (i-2) vanishing, i.e., limt→Th(t) = limt→Tg(t) and limt→T u(t, x) = 0, where T is some positive constant; (i-3) transition, i.e., g(t) → −∞, h(t) → −∞, 0 < limt→∞[h(t) − g(t)] < +∞ and u(t, x) → V*(x − c*t) with c* < 0, where V*(x − c*t) is a travelling wave with compact support and which satisfies the free boundary conditions. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

When a towed sonar array is straight, it has the difficulty that it cannot distinguish a contact on the left from one at the same angle on the right. When the array is not straight and its shape known, we calculate the probability that the left–right ambiguity is resolved correctly, using the Neyman–Pearson hypothesis testing framework, assuming a delay-sum beamformer, a single-frequency contact, and Gaussian noise. We also initially consider the noise field to be uncorrelated and show that the evaluation of the probability of correct resolution reduces to evaluating a one-dimensional integral. We find, as expected, that the probability increases, as the signal-to-noise ratio and the lateral deviation of the array from straight increase. For demonstration purposes, we also evaluate the probability of correct resolution for two representative shapes the array might assume in practice. Finally, we consider a more realistic, correlated noise field and we show that the initial assumption of an uncorrelated noise field provides a good approximation when the lateral deviation of the array is sufficiently large.

We consider a viscoelastic body occupying a smooth bounded domain $\Omega\subset \mathbb{R}^3$ under the effect of a volumic traction force g. The macroscopic displacement vector from the equilibrium configuration is denoted by u. Inertial effects are considered; hence, the equation for u contains the second-order term utt. On a part ΓD of the boundary of Ω, the body is fixed and no displacement may occur; on a second part ΓN ⊂ ∂Ω, the body can move freely; on a third portion ΓC ⊂ ∂Ω, the body is in adhesive contact with a solid support. The boundary forces acting on ΓC due to the action of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a non-linear ordinary differential equation settled on ΓC and describing the evolution of the delamination order parameter z. Following the lines of a new approach outlined in Bonetti et al. (2015, arXiv:1503.01911) and based on duality methods in Sobolev–Bochner spaces, we define a suitable concept of weak solution to the resulting system of partial differential equations. Correspondingly, we prove an existence result on finite-time intervals of arbitrary length.

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.

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes system:

$$\begin{align*} \gamma \tfrac{\partial u}{\partial t} + \beta v \cdot \nabla u - \nabla \cdot \left( \nabla w \right) & = 0 \,, \quad w= -\gamma \Delta u + \gamma ^{-1} \Psi ' (u) - \tfrac12 \alpha c'(\cdot,u) | \nabla \phi |^2\,, \\ \nabla \cdot (c(\cdot,u) \nabla \phi) & = 0\,,\quad \begin{cases} -\Delta v + \nabla p = \varsigma w \nabla u, \\ \nabla \cdot v = 0, \end{cases} \end{align*}$$

$\mathbb{R}_{>0}$

$\mathbb{R}_{\geq0}$

We consider a non-local version of the Cahn–Hilliard equation characterized by the presence of a fractional diffusion operator, and which is subject to fractional dynamic boundary conditions. Our system generalizes the classical system in which the dynamic boundary condition was used to describe any relaxation dynamics of the order-parameter at the walls. The proposed fractional dynamic boundary condition appears to be more general in the sense that it incorporates non-local effects which were completely ignored in the classical approach. We aim to deduce well-posedness and regularity results as well as to establish the existence of finite-dimensional attractors for this system.