Recent numerical evidence [8, 28, 33] suggests that in the Hele–Shaw suction problem with vanishingly small surface tension $\gamma$, the free boundary generically approaches the sink in a wedge-like configuration, blow-up occurring when the wedge apex reaches the sink. Sometimes two or more such wedges approach the sink simultaneously [33]. We construct a family of solutions to the zero-surface tension (ZST) problem in which fluid is injected at the (coincident) apices of an arbitrary number $N$ of identical infinite wedges, of arbitrary angle. The time reversed suction problem then models what is observed numerically with non-zero surface tension. We conjecture that (for a given value of $N$) a particular member of this family of ZST solutions, with special complex plane singularity structure, is selected in the limit $\gamma\,{\to}\,0$.

A vortex sheet is a surface across which the velocity field of incompressible and inviscid flows has a jump discontinuity. Mathematical and numerical studies reveal that a two-dimensional vortex sheet, which is governed by the Birkhoff–Rott equation, acquires a singularity in finite time without forming rolling-up spiral. On the other hand, numerical computation of a regularized Birkhoff–Rott equation shows that the vortex sheet evolves into a rolling-up doubly branched spiral. Because of the finite-time singularity, it is impossible to regard the rolling-up spiral as a solution of the Birkhoff–Rott equation as long as time is real. However, it may be possible to analytically continue the equation to the spiral along a path to get around the singularity in complex-time plane. In the present article, we consider singularities in complex-time plane for the regularized Birkhoff–Rott equation by numerical means. Distribution of the complex singularities and their limiting behaviour indicate that it is absolutely impossible to perform analytic continuation in complex-time domain to the spiral solution. Furthermore, we propose a simple model of a doubly branched spiral and investigate it mathematically. The model is successful in approximating the rolling-up motion of the vortex sheet. Comparing the vortex-sheet motion with the model indicate that the doubly branched spiral with infinite windings at the centre could be a solution of the Birkhoff–Rott equation beyond the singularity time.

In this paper we study a nonlocal parabolic/elliptic system which models thermistor behaviour in cases where heat losses to the surrounding gas play a significant role. The existence of time periodic solutions for the system is established through Faedo-Galerkin approximations and the Leray–Schauder degree theory. We show that for the small gas pressure case, the temperature of the time periodic solutions is positive. Moreover we consider the long time behaviour of the system and prove the existence of a uniform attractor. Finally, the finite dimensionality of the attractor is discussed.

We study travelling wave solutions of a one-dimensional two-phase Free Boundary Problem, which models premixed flames propagating in a gaseous mixture with dust. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative effects due to the presence of dust account for the divergence of the radiative flux entering the equation for temperature. This flux is modelled by the Eddington equation. In an appropriate limit the divergence of the flux takes the form of a nonlinear heat loss term. The resulting reduced model is able to capture a hysteresis effect that appears if the amount of fuel in front of the flame, or equivalently, the adiabatic temperature is taken as a control parameter.

In this work we describe some aspects of the dynamics of the Cahn–Hilliard equation. In particular, we consider the dynamics of ‘bubble’ solutions that is spherical interfaces which move superslowly towards the boundary without changing their shape. We show for the Cahn–Hilliard that the bubble drifts towards the closest point on the boundary provided it is sufficiently small. This is contrasted with the related mass conserving Allen–Cahn equation where size is not an issue.

‘Parametric dependence of exponents and eigenvalues in focusing porous media flows’ by D. G. Aronson, J. B. Van den Berg and J. Hulshof, European Journal of Applied Mathematics, Vol. 14 No. 4 August 2003, pp. 485–512