We study the semi-empirical b—ε model which describes the time evolution of turbulent spots in the case of equal diffusivity of the turbulent energy density b and the energy dissipation rate ε. We prove that the system of two partial differential equations possesses a solution, and that after some time this solution exhibits self-similar behaviour, provided that the system has self-similar solutions. The existence of such self-similar solutions depends upon the value of a parameter of the model.

Asymptotic methods are applied to the outdiffusion of a diffusant whose behaviour is governed by a power-law diffusivity. Both semi-infinite and finite domain problems are considered. The ‘fast’ diffusion (negative power-law) case is of particular interest, and the results are significantly different from those for the corresponding indiffusion problem.

Recently, the free boundary problem of quasistationary Stokes flow of a mass of viscous liquid under the action of surface tension forces has been considered by R. W. Hopper, L. K. Antanovskii, and others. The solution of the Stokes equations is represented by analytic functions, and a time dependent conformal mapping onto the flow domain is applied for the transformation of the problem to the unit disk. Two coupled Hilbert problems have to be solved there, which leads to a Fredholm boundary integral equation. The solution of this equation determines the time evolution of the conformal mapping. The question of the existence of a solution to this evolution problem for arbitrary (smooth) initial data has not yet been answered completely. In this paper, local existence in time is proved using a theorem of Ovsiannikov on Cauchy problems in an appropriate scale of Banach spaces. The necessary estimates are obtained in a way that is oriented at the a priori estimates for the solution given by Antanovskii. In the case of small deviations from the stationary solution represented by a circle, these a priori estimates, together with the local results, are used to prove even global existence of the solution in time.

Quite precise asymptotic estimates, in terms of the relaxation parameter and the time step, are derived for travelling wave solutions to a Stefan problem with phase relaxation and a semidiscrete counterpart. These estimates quantify the regularizing effects of phase relaxation and time discretization that give rise to thin transition layers as opposed to sharp interfaces. Layer width estimates, pointwise error estimates, and asymptotic expressions for the profile of the relevant physical variables are proved. Applications to a related nonlinear Chernoff formula are also given.

We consider the behaviour of a premixed flame anchored on a flat burner. For Lewis numbers L < L* < 1, stationary spatially periodic solutions corresponding to stationary cellular flames bifurcate from the basic solution which corresponds to a steady planar flame. We study the existence and stability of two-dimensional patterns which correspond to certain imposed symmetries by considering the evolution of N pairs of wave vectors, each of which is separated from the next by angle π/N. In the neighbourhood of the critical Lewis number L*, we derive evolution equations for the amplitudes corresponding to N = 2, which corresponds to square patterns, and N = 3, which corresponds to triangular or hexagonal patterns. We determine existence and stability results in terms of m∈(0, 1), the flow rate of the fuel, and K > 2/e, the scaled heat loss to the burner. Square patterns exist for L < L* and are stable for values of m and K above a stability boundary in the m−K plane, which has a maximum at K = K* ∼ 4.77, so that for K > K* square patterns are stable for all m. The stability of the square patterns does not vary with L. Hexagonal patterns exist for L < LH, where 1 > LH > L*. The size of the stability region increases with decreasing L < L*. For a range of values of L there is bistability, that is, for given parameter values rolls and hexagons are simultaneously stable, each with its own domain of attraction.

In this paper we examine the movement of hard contact lenses on the eye. In so doing, we take into account hydrodynamic forces underneath the lens, as well as surface tension forces at the lens periphery. This involves solving for the free surface of the tear film away from the lens in order to determine the magnitudes of the pressure and surface tension forces on the lens. The analysis, which assumes quasi-steady motion, is carried out in both two and three dimensions.

The Stefan problem with surface tension in the three-dimensional case with spherical symmetry is considered. We first establish the existence and uniqueness of the classical solution with surface tension and kinetic undercooling effects for all time, and then pass to the limit as the kinetic undercooling tends to zero. The limiting solution is the global-in-time weak solution and the local-in-time classical solution for the Stefan problem with surface tension. This solution cannot be the global-in-time classical solution. If S(t) is the radius of a solid ball in a supercooled liquid, then (1) there exists at least one point t* of discontinuity of the function S(t):

or (2) the continuous function S(t) cannot be absolutely continuous, and it maps some zero-measure set of (t*, T*) onto some set of Ω with a strictly positive measure.

The relationship between mathematical models of active parameters for transmission lines is studied. Treating transmission lines as a series of differential lumped circuits, we show that pairs of line parameters for inductance and capacitance per unit length must satisfy one of two constraints. One of these is a symmetry condition, which is satisfied by passive (i.e. constant) parameters. If a parameter pair satisfies either of these constraints, the energy per unit length and power loss in the line may be written as integrals of known functions, no matter what the pair's dependence on line current and voltage may be. Potential applications of these results to other subject areas are discussed.

Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstrate that some systems have no non-reducible solutions. I also demonstrate that certain elliptic systems have no non-reducible solutions.

In the focussing problem for the porous medium equation, one considers an initial distribution of material outside some compact set K. As time progresses material flows into K, and at some finite time T first covers all of K. For radially symmetric flows, with K a ball centred at the origin, it is known that the intermediate asymptotics of this focussing process is described by a family of self-similar solutions to the porous medium equation. Here we study the postfocussing regime and show that its onset is also described by self-similar solutions, even for nonsymmetric flows.