We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

We consider a competition–diffusion system and study its singular limit as the interspecific competition rate tend to infinity. We prove the convergence to a Stefan problem with zero latent heat.

Although the European Journal of Applied Mathematics is a journal that has no official affiliation with any particular academic organisation, its aims overlap those of many institutes, centres, societies and, especially, conferences. Indeed, in our first ten years we have published three special issues related to conferences on topics that have been timely and important for the development of new mathematical ideas relevant to practical problems. Thus, when ICIAM, the world's largest industrial-and-applied mathematics conference, took place in Edinburgh, Scotland, in July 1999, we were only too happy to be associated with a gathering of so many people who all worked in the style of EJAM, and the organisers kindly allowed us to co-sponsor the conference reception. A highlight of this evening was Professor Graham Wilks' dramatic portrayal of Sir Isaac Newton, complete with apple. Accompanied by his wife, Graham sang the following verse.

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.

When a thin layer of normal (non-superconducting) material is placed between layers of superconducting material, a superconducting-normal-superconducting junction is formed. This paper considers a model for the junction based on the Ginzburg–Landau equations as the thickness of the normal layer tends to zero. The model is first derived formally by averaging the unknown variables in the normal layer. Rigorous convergence is then established, as well as an estimate for the order of convergence. Numerical results are shown for one-dimensional junctions.

This special issue of the European Journal of Applied Mathematics is devoted to papers on free boundary problems, largely for Hele-Shaw or Stokes flows. The majority of the papers were presented at a meetingSupport from the London Mathematical Society is gratefully acknowledged. on these topics held in Oxford in August 1998, 100 years after H. S. Hele-Shaw first described his cell [2], and 40 years on from the classic experiment of Saffman & Taylor [4]. The conference delegates sent a message of good wishes to P. Ya Polubarinova-Kochina (below), whose paper in 1945 [3], together with that of Galin in the same year [1], may be said to have initiated the modern study of the Hele-Shaw free boundary problem. We learned with great sadness of Polubarinova-Kochina's death this year, at the age of 100; her active scientific career spanned some 75 years.

As part of the preparation for the conference, with the assistance of K. A. Gillow we have assembled a 600-paper bibliography on Hele-Shaw and Stokes flow; it can be found at www.maths.ox.ac.uk/˜howison/Hele-Shaw/. We hope it will be useful to the many researchers in these areas, which even after a century of investigation will clearly retain their mathematical and practical interest for many years to come.

In this paper we consider the problem of hedging options in the presence of cost in trading the underlying asset. This work is an asymptotic analysis of a stochastic control problem, as in Hodges & Neuberger and Davis, Panas & Zariphopoulou;. We derive a simple expression for the ‘hedging bandwidth’ around the Black–Scholes delta; this is the region in which it is optimal not to rehedge. The effect of the costs on the value of the option, and on the width of this hedging band is of a significantly greater order of magnitude than the costs themselves. When costs are proportional to volume traded, rehedging should be done to the edge of this band; when there are fixed costs present, trading should be done to an optimal point in the interior of the no-transaction region.

We study the interior spike solutions to a steady state problem of the shadow system of the Gierer–Meinhardt system arising from biological pattern formation. We first show that at a non-degenerate peak point the interior spike solution is locally unique, and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow system is unstable. Furthermore, the metastability of such solutions is analysed.

We consider two problems related to the b–l and b–ε models of propagation of turbulent bursts. We show that these equations admit some particular exact solutions which reduce to a finite-dimensional dynamical system. This makes it possible to describe a singular effect of finite-time extinction, and in particular, nonsymmetric solutions which do not exhibit symmetrization in the asymptotic behaviour. We show that in the multi-dimensional equation related to the b–l model, the nonsymmetric extinction behaviour is governed by the first-order equation. For the b–ε model with α=β=1 and γ<1, using such particular solutions, we establish that the ω-limit set of all the rescaled extinction orbits is essentially infinite-dimensional.

Small surface tension is known to select a discrete family from the continuum of Saffman–Taylor finger solutions with zero surface tension. Here a method developed recently in Chapman et al. [2] is employed in which a naive perturbation expansion in powers of the surface tension parameter is optimally truncated to show exponentially small terms being switched on across Stokes lines. These terms are responsible for finger selection.

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.

We present a general procedure for recursively improving the invariance of a numerical integrator under a symmetry group. If h is a symmetry, we construct the adjoint method h−1h. In each time step we apply either the original method or the adjoint method, according to a prescription based on the Thue–Morse sequence. The outcome is a solution sequence which displays progressively smaller symmetry errors, to any desired order in the time-step. The method can also be used to force the solution to stay close to a desired submanifold of phase space, while retaining structural properties of the original method.

The motion of a rigid particle in an inviscid non-uniformly vibrating ambient liquid is considered. The vibrations are caused by a dipole changing its strength in time. This model of a vibrator presents an asymptotic case of vibrations caused by a rigid sphere periodically changing its position when the radius and amplitude are small and the velocity is large. It is found that periodic oscillations with zero mean can cause the directed motion of a submerged particle even if its density equals that of the liquid. The direction of the motion is studied. It is shown that particles of density not less that of the ambient liquid are attracted by a vibrator. The direction of motion of lighter particles depends on their initial position.

We consider a new concept of weak solutions to the phase-field equations with a small parameter ε characterizing the length of interaction. For the standard situation of a single free interface, this concept (in contrast with the common one) leads to the well-known Stefan–Gibbs–Thomson problem as ε→0. For the case of a large number M(ε) (M(ε)→∞ as ε→0) of free interfaces, which corresponds to the ‘wave-train’ interpretation of a ‘mushy region’, this concept allows us to obtain the limit problem as ε→0.

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.

Uniqueness and solvability theorems are proved for the two-dimensional Neumann–Kelvin problem in the case when a body is totally submerged in a subcritical stream of finite depth fluid. A version of source method is developed to find a solution. The Green's identity coupling the solution with a solution of the problem with opposite stream direction is used to prove that the solution is unique.

Boundary value problems for linear transport equations sometimes require the explicit construction of solutions when boundary conditions are prescribed only on parts of the boundary and thus necessitate the construction of half-range expansions. In contrast to the standard eigenfunction expansion, a half-range expansion of a given function must be reconstructed just over half the domain, using just half of the eigenfunctions. The difficulty of such expansions arises because the eigenfunctions are not orthogonal, though they are complete, over half the domain, and there is no obvious method of obtaining the expansion coefficients. Here we use complex variable techniques to find explicit formulas for the coefficients of half-range expansions for regular, negative definite Sturm–Liouville operators. We prove that the half-range expansion formula is unique, and find the corresponding half-range Green functions.

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

New modulation equations for hexagonal patterns in reaction–diffusion systems are derived for parameter régimes corresponding to the onset of patterns. These systems include additional nonlinearities which are not present in Rayleigh–Bénard convection or Swift–Hohenberg type models. The dynamics of hexagonal and roll patterns are studied using a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations. The investigation demonstrates instabilities and new phenomena not found in other systems, and is applied to patterns of flame fronts in a certain model of burner stabilized flames.

The bifurcation of symmetric superconducting solutions from the normal solution is considered for the one-dimensional Ginzburg–Landau equations by the methods of formal asymptotics. The behaviour of the bifurcating branch depends upon the parameters d, the size of the superconducting slab, and κ, the Ginzburg–Landau parameter. It was found numerically by Aftalion & Troy [1] that there are three distinct regions of the (κ, d) plane, labelled S1, S2 and S3, in which there are at most one, two and three symmetric solutions of the Ginzburg–Landau system, respectively. The curve in the (κ, d) plane across which the bifurcation switches from being subcritical to supercritical is identified, which is the boundary between S2 and S1∪S3, and the bifurcation diagram is analysed in its vicinity. The triple point, corresponding to the point at which S1, S2 and S3 meet, is determined, and the bifurcation diagram and the boundaries of S1, S2 and S3 are analysed in its vicinity. The results provide formal evidence for the resolution of some of the conjectures of Aftalion & Troy [1].