Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions.

We study a free boundary problem of the form: ut = uxx + f(t, u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) – α(t) and g′(t) = −ux(t, g(t)) + β(t), where β(t) and α(t) are positive T-periodic functions, f(t, u) is a Fisher–KPP type of nonlinearity and T-periodic in t. This problem can be used to describe the spreading of a biological or chemical species in time-periodic environment, where free boundaries represent the spreading fronts of the species. We study the asymptotic behaviour of bounded solutions. There are two T-periodic functions α0(t) and α*(t; β) 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, that is, h(t) – g(t) → +∞ and u(t, ⋅ + ct) → 1 with $c\in (-\overline{l},\overline{r})$, where $ \overline{l}:=\frac{1}{T}\int_{0}^{T}l(s)ds$, $\overline{r}:=\frac{1}{T}\int_{0}^{T}r(s)ds$, the T-periodic functions −l(t) and r(t) are the asymptotic spreading speeds of g(t) and h(t) respectively (furthermore, r(t) > 0 > −l(t) when 0 < β < α < α0; r(t) = 0 > −l(t) when 0 < β < α = α0; $0 \gt \overline{r} \gt -\overline{l}$ when 0 < β < α0 < α < α*); (i-2) vanishing, that is, $\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$ and $\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$, where $\mathcal {T}$ is some positive constant; (i-3) transition, that is, g(t) → −∞, h(t) → −∞, $0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$ and u(t, ⋅) → V(t, ⋅), where V is a T-periodic solution with compact support. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

We consider the two-dimensional problem of crystal growth in a forced flow. A dendrite is placed in a Hele-Shaw cell with insulating walls and grows due to undercooling. We neglect the surface energy in the Gibbs–Thomson relation. The problem is formulated in terms of analytic functions similarly to closely related work on the viscous fingering problem of Saffman and Taylor. We derive a solvability condition for the existence of a steady-state needle-like solidification front in the limit of small Peclet number, Pe = V∞l/a, where V∞ is the characteristic velocity of the melt, 2l is the channel width, and a is the thermal diffusivity of the liquid. The velocity of the crystallization front is directly proportional to the hydrodynamic velocity V∞ and undercooling, while the dendrite width ld does not depend upon the physical parameters, and indeed, ld = l.

This paper deals with a new method to determine the dependence of the electrical conductivity of metals or semiconductors on temperature. It is based on the fact that the current–voltage relationship is easily measurable. This inverse problem is solved by the classical Abel integral equation.

The purpose of this work is to investigate the properties of spreading speeds for the monotone semiflows. According to the fundamental work of Liang and Zhao [(2007) Comm. Pure Appl. Math.60, 1–40], the spreading speeds of the monotone semiflows can be derived via the principal eigenvalue of linear operators relating to the semiflows. In this paper, we establish a general method to analyse the sign and the continuity of the spreading speeds. Then we consider a limiting case that admits no spreading phenomenon. The results can be applied to the model of cellular neural networks (CNNs). In this model, we find the rule which determines the propagating phenomenon by parameters.

We consider short-time existence, uniqueness, and regularity for a moving boundary problem describing Stokes flow of a free liquid drop driven by surface tension. The surface tension coefficient is assumed to be a nonincreasing function of the surfactant concentration, and the surfactant is insoluble and moves by convection along the boundary. The problem is reformulated as a fully nonlinear, nonlocal Cauchy problem for a vector-valued function on a fixed reference manifold. This problem is, in general, degenerate parabolic. Existence and uniqueness results are obtained via energy estimates in Sobolev spaces of sufficiently high order. In the two-dimensional case, the problem is strictly parabolic, and we prove instantaneous smoothing of the free boundary, using maximal regularity results in little Hölder spaces.

We consider the thermal insulation property of homogeneous anisotropically heat-conducting bodies, i.e. those whose thermal tensor (matrix) A is constant throughout the body but not generally a constant times the identity. This anisotropy is a common feature of nano-composite materials. We propose using the principal Dirichlet eigenvalue λ of the associated elliptic differential operator −∇ ⋅ A∇ as a simple measurement for the insulating ability of the material, because the time scale of thermal flow is of the order of 1/λ. 1/λ is a generalization of the ‘R-value’ used in engineering practice as a measure of the insulating ability of isotropic conductors. If the thermal tensor A depends on parameters, e.g. inherited from nanostructure, then so does λ. It is important to know how λ depends on the parameters. For the material to be a good insulator, λ should be suppressed. But calculation or estimation of this principal elliptic eigenvalue, particularly over ranges of parameter values, is not a simple task. The focus of this paper is estimation – by fitted formulas and new exact bounds – of the principal elliptic Dirichlet eigenvalue of ellipses (2D) and ellipsoids (3D) using only simple expressions in the eigenvalues of the matrix A. Our simplest approximations and bounds avoid even the calculation of matrix eigenvalues and use in 2D merely the trace and determinant of A, and in 3D the trace and determinant of A and the trace of A2. The new bounds are shown to imply an extremal property in homogenization theory and a new condition for enhancement in Taylor dispersion. Recently, Zheng, Forest, Lipton, Zhou and Wang published formulas for the thermal tensor A in the case of strong extensional fiber-type flow of a polymer with dilute rod-like nano-inclusions, including explicitly the influence of the probability distribution of inclusion orientation. Our results are combined with these formulas to quantify the effect of variations of the probability distribution on the insulating ability of the composite.

Rapid solidification fronts are studied using a phase field model. Unlike slow moving solutions which approximate the Mullins–Sekerka free boundary problem, different limiting behaviour is obtained for rapidly moving fronts. A time-dependent analysis is carried out for various cases and the leading order behaviour of solidification front solutions is derived to be one of several travelling wave problems. An analysis of these problems is conducted, leading to expressions for front speeds in certain limits. The dynamics leading to these travelling wave solutions is derived, and conclusions about stability are drawn. Finally, a discussion is made of the relationship to other solidification models.

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

We regret that the paper by J. M. Greenberg, ‘Models of elastic–perfectly plastic materials’, vol. 1, part 2, pp. 131–150, was repeated in error as pp. 225–244 in part 3.

The publisher apologizes to the author and readers for this unfortunate error.

In this paper, we give a precise description of the rescaling behaviours of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behaviour is given in terms of many conserved quantities as well.

In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$, it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

Precise knowledge of magnetic fields is crucial in many medical imaging applications such as magnetic resonance imaging (MRI) or magnetic particle imaging (MPI), as they form the foundation of these imaging systems. Mathematical methods are essential for efficiently analysing the magnetic fields in the entire field-of-view. In this work, we propose a compact and unique representation of the magnetic fields using real solid spherical harmonic expansions, which can be obtained by spherical t-designs. To ensure a unique representation, the expansion point is shifted at the level of the expansion coefficients. As an application scenario, these methods are used to acquire and analyse the magnetic fields of an MPI system. Here, the field-free-point of the spatial encoding field serves as the unique expansion point.

In this paper, we consider the monotone travelling wave solutions of a reaction–diffusion epidemic system with nonlocal delays. We obtain the existence of monotone travelling wave solutions by applying abstract existence results. By transforming the nonlocal delayed system to a non-delayed system and choosing suitable small positive constants to define a pair of new upper and lower solutions, we use the contraction technique to prove the asymptotic stability (up to translation) of monotone travelling waves. Furthermore, the uniqueness and Lyapunov stability of monotone travelling wave solutions will be established with the help of the upper and lower solution method and the exponential asymptotic stability.

A simple computational procedure is described for the evaluation of the modified Bennett functions, which arise in the multiple Fourier analysis for bang-bang controls with memory.

This paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.

We study the geodesic interpolating spline with a biharmonic regulariser for solving the landmark image registration problem. We show existence of solutions, discuss uniqueness and show how the problem can be efficiently solved numerically. The main advantage of the geodesic interpolating spline is that it provides a diffeomorphism and we show this is preserved under our numerical approximation.