We consider a Keller–Segel model that describes the cellular chemotactic movement away from repulsive chemical subject to logarithmic sensitivity function over a confined region in ${{\mathbb{R}}^n},\,n \le 2$ . This sensitivity function describes the empirically tested Weber–Fecher’s law of living organism’s perception of a physical stimulus. We prove that, regardless of chemotaxis strength and initial data, this repulsive system is globally well-posed and the constant solution is the global and exponential in time attractor. Our results confirm the ‘folklore’ that chemorepulsion inhibits the formation of non-trivial steady states within the logarithmic chemotaxis model, hence preventing cellular aggregation therein.

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

A simple method of reducing a parabolic partial differential equation to canonical form if it has only one term involving second derivatives is the following: find the general solution of the first-order equation obtained by ignoring that term and then seek a solution of the original equation which is a function of one more independent variable. Special cases of the method have been given before, but are not well known. Applications occur in fluid mechanics and the theory of finance, where the Black-Scholes equation yields to the method, and where the variable corresponding to time appears to run backwards, but there is an information-theoretic reason why it should.

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.

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for non-linear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully non-linear case and open research questions.

The consistency of a non-local anisotropic Ginzburg–Landau type functional for data classification and clustering is studied. The Ginzburg–Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.

This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

The goal of this note is to explain the process of computing the lowest energy level achieved by an elastic crystal subject to homogeneous boundary deformation. This analysis follows mainly the lines of Chipot and Kinderlehrer or Fonesca but we expect that it will lead to some new results for some other energy functional having different invariance properties. The mathematical interest of the result also lies in the fact that the lowest energy level computed this way coincides with the relaxed energy functional (see Fonesca). This relaxed energy is determined by a known thermodynamic quantity, Ericksen' subenergy. Within this review we will also analyse some of the mathematical and physical aspects of these highly oscillatory problems.

We consider a phenomenological model proposed by Barenblatt [1] for non-equilibrium two-phase flow in porous media. In the case of zero total flow it reduces to a pair of equations:

where Φ(σ) is a non-decreasing (not necessarily increasing) smooth function defined in the interval 0 ≤ σ ≤ 1. We consider initial-boundary problems for this system in which the initial data are given only for s, and the boundary data only for ς (which corresponds to the physical sense of the model). The degenerate character of the system allows us to apply simple topological methods. We show that the boundary problem is well-posed, in the sense that there exits a unique (weak) solution which satisfies the maximum principle and depends continuously on the initial data. The solution is no less smooth than the initial and boundary data.

Motivated by the technology of magnetically targeted drug and gene delivery, in which a magnetic field is used to direct magnetic carrier particles from the circulation to a target site, we develop a continuum model for the motion of particles (magnetic carriers) subject to an external body force (magnetic field) in a flow of a concentrated suspension of a species of neutrally buoyant particles (blood). An advection–diffusion equation describes the evolution of the carrier particles as they advect in the flow under the action of an external body force, and diffuse as a result of random interactions with the suspension of neutrally buoyant particles (shear-induced diffusion). The model is analysed for the case in which there is steady Poiseuille flow in a cylindrical vessel, the diffusive effects are weak and there is weak carrier uptake along the walls of the vessel. The method of matched asymptotic expansions is used to show that carriers are concentrated in a boundary layer along the vessel wall and, further, that there is a carrier flux along this layer which results in a sub-layer, along one side of the vessel, in which carriers are even more highly concentrated. Three distinguished limits are identified: they correspond to cases for which (i) the force is sufficiently weak that most particles move through the vessel without entering the boundary layers along the walls of the vessel and (ii) and (iii) to a force which is sufficiently strong that a significant fraction of the particles enter the boundary layers and, depending upon the carrier absorption from the vessel walls, there is insignificant/significant axial carrier flux in these layers.

This paper deals with the spherically symmetric Stefan problem in three space dimensions. The melting temperature satisfies the Gibbs–Thomson law. The solution is obtained as a limit of solutions of similar problems containing a small additional kinetic term in the melting temperature. Under some structural assumptions we show that the phase-change boundary has at most one discontinuity point t = T0 (see the corresponding result for the planar Stefan problem in Götz & Zaltzman (1995)). In the one-phase problem the discontinuity point always exists. At the time T0 the whole solid phase melts instantaneously. We study also the asymptotical stability (t → ∞) of stationary solutions satisfying boundary conditions of thermostat type.

This paper deals with a wave process induced by an acoustic impulse in a vertically inhomogeneous half-space. The solvability of the corresponding direct initial boundary value problem is proved. We also consider a class of one-dimensional inverse problems including, in particular, the classical inverse problem of the scattering theory, the transmission and reflection problems of seismology, etc. It is shown that all these inverse problems are equivalent in a certain sense, and therefore can be solved by identical methods. Theoretical results are illustrated by numerical examples.

In this paper, we present numerical simulations that demonstrate the effect of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension in surfactant-driven thin liquid films. Previous choices of the model EoS have been an ad-hoc decreasing function. Here, we instead propose an empirically motivated EoS; this provides a route to resolve some discrepancies and raises new issues to be pursued in future experiments. In addition, we test the influence of the choice of initial conditions and values for the non-dimensional groups. We demonstrate that the choice of EoS improves the agreement in surfactant distribution morphology between simulations and experiments, and influences the dynamics of the simulations. Because an empirically motivated EoS has regions with distinct gradients, future mathematical models may be improved by considering more than one timescale. We observe that the non-dimensional number controlling the relative importance of gravitational versus capillary forces has a larger influence on the dynamics than the other non-dimensional groups, but is nonetheless not a likely cause of discrepancy between simulations and experiments. Finally, we observe that the experimental approach using a ring to contain the surfactant could affect the surfactant and fluid dynamics if it disrupts the intended initial surfactant distribution. However, the fluid meniscus itself does not significantly affect the dynamics.

A problem for a backward–forward parabolic partial differential equation with a small diffusion coefficient ε is analyzed. The problem arises in describing the stochastic behaviour of a data-handling system with a large number of sources which turn on or off at random. We solve the problem asymptotically for ε small by using the ray method and the boundary layer method.

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

A spike solution is constructed on the infinite line for a sub-diffusive version of the Gierer–Meinhardt reaction – diffusion model. A non-local eigenvalue problem governs the spike's stability and is explicitly solvable for a certain choice of the kinetic parameters. Its solution generalises former results for the Gierer–Meinhardt model with regular diffusion, and the normal and anomalous systems' properties are juxtaposed. It is shown that a Hopf bifurcation occurs in the sub-diffusive system for larger values of the time parameter τo as compared to the normal counterpart, rendering the anomalous system more stable. Asymptotic solutions are obtained near important values of the diffusion anomaly index γ and collectively shown to be valid over most of the applicable range 0 < γ < 1. A bifurcation delay scenario is described for the sub-diffusive system, and the WKB exponent is computed analytically.

We assume that the Stefan problem with underheating has a classical solution until the moment of contact of two distinct free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with underheating, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution that is the solution of the heat equation.

Following the historical development of the theory of exact solutions for singularity-driven Hele-Shaw flows, this note demonstrates that the problem of two-dimensional viscous sintering preserves quadrature identities. This provides a unified theoretical perspective in which to understand the two separate free boundary problems. The result is established directly from the equations of motion without appeal to conformal mapping theory, although the result underlies the existence of exact conformal mapping solutions. The formulation leads to a concise, closed-form representation of the evolution equations for the parameters in the conformal mapping function. Some examples are given.

Waves on a neutrally buoyant intrusion layer moving into otherwise stationary fluid are studied. There are two interfacial free surfaces, above and below the moving layer, and a train of waves is present. A small amplitude linearized theory shows that there are two different flow types, in which the two interfaces are either in phase or else move oppositely. The former flow type occurs at high phase speed and the latter is a low-speed solution. Nonlinear solutions are computed for large amplitude waves, using a spectral type numerical method. They extend the results of the linearized analysis, and reveal the presence of limiting flow types in some circumstances.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.