We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are such flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are N such disks in a general position, the evolution of the fluid region is described by a conformal map involving 2N−1 time-dependent parameters whose variation is governed by N invariants and N−1 first order differential equations. When N=2, or when the problem enjoys some special features of symmetry, the moving boundary of the fluid domain during the motion can be determined by solving purely algebraic equations, the solution of a single differential equation being needed only to link a particular boundary shape to a particular time. The analysis is aided by exploiting a connection with Hele-Shaw free boundary flows when the zero-surface-tension model is employed. If the initial configuration for the Stokes flow problem can be produced by injection (or suction) at N points into an initially empty Hele-Shaw cell, as can the N-disk configuration referred to above, then so can all later configurations; the points where the fluid must be injected move, but the amount to be injected at each of the N points remains invariant. The efficacy of our solution procedure is illustrated by a number of examples, and we exploit the method to show that the free boundary in such a Stokes flow driven by surface tension alone may pass through a cusped state.

The Ginzburg–Landau model for superconductivity is examined in the one-dimensional case. First, putting the Ginzburg–Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg–Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg–Landau energy is analysed and different convergence results are obtained, according to the exterior magnetic field. Numerical computations illustrate the various behaviours.

Following our preceding papers [1, 2] concerning semi-infinite superconducting films, we consider new a priori estimates on the exterior magnetic field h=A′(0), when (f, A) is a solution of the corresponding Ginzburg–Landau system. The main new results concern the limit as κ→∞, but we prove also the existence of a finite superheating field. We also discuss recent results [3] concerning the superheating field in the large κ limit, and show how to relate these formal solutions to suitable subsolutions and supersolutions giving the existence of a solution for h<1/√2 and κ large enough. We also analyse the same problem by variational techniques and get the existence of a locally stable solution for h<1/√2 and any κ>0.

The Lawrence–Doniach model for superconducting layered compounds in which the Ginzburg–Landau order parameters in adjacent layers are coupled by Josephson tunneling is considered. The main purpose of this paper is to demonstrate that the time-dependent Ginzburg–Landau model is the limit of the Lawrence–Doniach model as the layer spacing goes to zero.

A singular perturbation method is applied to a system of two weakly coupled strongly non-linear but non-identical oscillators. For certain parameter regimes, stable localized solutions exist for which the amplitude of one oscillator is an order of magnitude smaller than the other. The leading-order dynamics of the localized states is described by a new system of coupled equations for the phase difference and scaled amplitudes. The degree and stability of the localization has a non-trivial dependence on coupling strength, detuning, and the bifurcation parameter. Three distinct types of localized behaviour are obtained as solutions to these equations, corresponding to phase-locking, phase-drift, and phase-entrainment. Quantitative results for the phases and amplitudes of the oscillators and the stability of these phenomena are expressed in terms of the parameters of the model.

We consider both stationary and time-dependent heat equations for a non-convex body or a collection of disjoint conducting bodies with Stefan-Boltzmann radiation conditions on the surface. The main novelty of the resulting problem is the non-locality of the boundary condition due to self-illuminating radiation on the surface. Moreover, the problem is nonlinear and in the general case also non-coercive. We show that the non-local boundary value problem admits a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. This result is then applied to prove existence under some hypotheses that guarantee the existence of sub- and supersolutions. Some special cases where the problem is coercive are also discussed. Finally, the analysis is extended to cases with nonlinear material properties.

We consider the dynamic behaviour of a thermoviscoelastic body which may come into frictional contact with a rigid obstacle. The frictional contact is modelled by general contact and friction laws which include as special cases the power law normal compliance condition and the corresponding generalization of Coulomb's law of dry friction. The stress-strain constitutive relation is assumed to be of Kelvin-Voigt type and the frictional heat generation on the contact surface is taken into account. In this setting we establish the existence of a solution to a weak version of the energy-elasticity system which consists of a parabolic equation coupled with a variational inequality.