Spontaneous potential well-logging is an important technique in petroleum exploitation. To make the corresponding log interpretation chart, it is supposed that the geometrical structure of the formation, the resistivity in each subdomain, and the spontaneous potential difference on each interface are all known; then in the direct problem, the spontaneous potential u = u(r, z) satisfies an elliptic boundary value problem with jump conditions on interfaces. At the joint points A and B of the interfaces (figure 2), the jumps of the spontaneous potential do not, in general, satisfy the compatibility condition. It turns out that it is impossible to find a piecewise H1 solution to the problem, and the standard finite element method cannot be applied to get an approximate solution. In this paper, by means of a method of removing the singularities at A and B, it is proved that the problem admits a unique weak solution that is piecewise W1,p for any fixed p with 1 ≤ p <2. Moreover, based on this method a numerical scheme is suggested, and some numerical examples and some conclusions of practical interest are given. The techniques used in this paper will find a wider applicability in other problems.

This paper considers the stability of melt-solid interfaces to eigenfunction perturbations for a system of equations which describe the melting and freezing of helium. The analysis is carried out in both planar and spherical geometries. The principal results are that when the melt is freezing, under certain far-field conditions, the interface is stable in the sense of Mullins and Sekerka. On the other hand, when the solid is melting (at least when the melting is sufficiently fast), the interface is unstable. In some circumstances these instabilities are oscillatory, with amplitude and growth rate increasing with surface tension and frequency. The last section considers the original problem of Mullins and Sekerka in the present notation.

We study the singular limit of the dimensionless phase-field equations

We consider two cases: either the space dimension is 1 and then ɛ tends to zero; or the solutions are radially symmetric and then both ɛ and α tend to zero. It turns out that, in the first case, the limiting functions solve the Stefan problem with kinetic undercooling, provided the initial temperature is small compared to the surface tension and the latent heat. In the second case, the limiting functions satisfy the Stefan problem coupled with the Gibbs–Thomson law for the melting temperature. We show, in addition, that the multiplicity of the interface is always one, in a sense to be explained at the end of § 1. As main tool we use energy type estimates, and prove that the formal first-order asymptotic expansion with respect to ɛ in fact gives an approximation of the exact solution. Our results hold without smoothness assumptions on the limiting Stefan problem.

We introduce the concept of B-determining equations of a system of partial differential equations that generalize the defining equations of the symmetry groups. We show how this concept may be applied to obtain exact solutions of partial differential equations. The exposition is reasonable self-contained, and supplemented by examples of direct physical importance, chosen from fluid mechanics.

We consider the non-local problem

which models the temperature when an electric current flows through a material with temperature dependent electrical resistivity f(u) > 0, subject to a fixed potential difference. It is found that for some special cases where f is decreasing and

so the problem can be scaled to make

then:(a) for λ < 8 there is a unique steady state which is globally asymptotically stable: (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, – 1 < x < 1.

Flows with free boundaries in a Hele-Shaw cell provide a unique opportunity to study non-linear boundary dynamics using rigorous analytic approaches. While of limited direct ‘practical value’, these studies give rise to a plethora of new phenomena and insights which may serve as beacons in the turbulent ocean of moving free boundaries and pattern forming. This paper gives a brief summary of the authors' studies of Hele-Shaw flows with free boundaries and some related problems based upon Richardson's approach. Some promising directions of further research are also discussed.

We consider an equilibrium problem for a thin inclusion in a shell. The faces of the inclusion are assumed to satisfy a non-penetration condition, which is an inequality imposed on the tangential shell displacements. The properties of the solution are studied, in particular, the smoothness of the stress field in the vicinity of the inclusion. The tangential displacements are proved to belong to the space H2 near the internal points of the inclusion. The character of the contact between the inclusion faces is described in terms of a suitable non-negative measure. The stability of the solution is investigated for small perturbations to the inclusion geometry.

A sharp estimate of the growth of solutions of the initial value problem for systems of the form

where Cj(t) are matrices with elements of power growth, is found. As a corollary of this result, it follows, for instance, that each solution of the initial value problem satisfies the estimate ‖u(t)‖ ≤ Cexp{γln2(1+|t|)} for some C > 0 and γ > 0.

The bifurcation from a normally conducting to a superconducting state as an external magnetic field is lowered is examined using the Ginzburg–Landau theory. Linear and weakly nonlinear stability analyses are performed near the bifurcation point, and the implications of the results for each of three examples is considered.

The main result of this paper is a complete description of local invertible transformations

relating one equation of the form uxy = F(x, y, u, ux, uy) to another equation of the form υxy = G(x, y, υ, υx, υy).

We describe the slow evolution of the wave speed and reaction temperature in a model of filtration combustion. In the counterflow configuration of the process, a porous solid matrix is converted to a porous solid product by injecting an oxidizing gas at high pressure into one end of a fresh sample of the solid while igniting it at the other end. The solid and gas react exothermically at high activation energy and, under favourable conditions, a self-sustaining combustion wave travels along the sample, converting reactants to product. Since the reaction rate depends on the gas pressure p in the pores, small gradients in p cause variations in the conditions of combustion, which, in turn, cause inhomogeneities in the physical properties of the product. We determine the slow evolution of the wave speed, the reaction temperature, and the mass flux of the gas downstream of the reaction zone. In the absence of a pressure gradient, there is a branch of steadily propagating solutions which has a fold. For planar disturbances on the slow time scale, we show that the middle part of the branch is unstable, with the change of stability occurring at the turning points of the branch. When the pressure gradient is nonzero, there are no steadily propagating solutions and the wave continually evolves. Conditions on the state of the gas at the inlet are described such that the variation in the wave speed and reaction temperature throughout the process can be minimized.

The existence, uniqueness and regularity of the solution to a one-dimensional linear thermoelastic problem with unilateral contact of the Signorini type are established. A finite element approximation is described, and an error bound is derived. It is shown that if the time step is O(h2), then the error in L2 in the temperature and in L∞ in the displacement is O(h). Some numerical experiments are presented.

It is shown that the influence of microstructure in the damage accumulation process leads to a nonlinear diffusion effect, with a strongly stress-dependent diffusion coefficient. A nonlinear parabolic equation with a source term is obtained for the damage parameter. This equation is relevant to blow-up and quenching problems well known to mathematicians with rupture corresponding to blow-up or quenching. However, the damage accumulation equation possesses an additional nonlinearity due to the non-healing of damage. Depending on the value of a dimensionless constant parameter (the ratio of a properly defined microstructural length-size to a characteristic length-size of the initial damage distribution), two essentially different types of damage accumulation process appear to be possible for a given initial damage distribution over the bar length. In processes of the first type, the damage accumulation remains non-homogeneous over the length of the bar, so that the lifetime for the whole specimen is determined by the maximal initial damage within the bar. For processes of the second type the damage distribution over the specimen at first becomes homogeneous (at least in a considerable part of specimen), and then the damage accumulation proceeds uniformly over all or part of the specimen. The lifetime for processes of the second type is essentially longer than the first. Results of a numerical experiment based on the proposed model are presented. In particular, the origin and development of damage propagation waves is demonstrated. Also, it is demonstrated that when there is substantial damage transfer, the ultimate value of the damage parameter in the life-time calculation is of no significance.

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation ℱ between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that ℱ satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique.