We study Hele–Shaw flows with a moving boundary and multipole singularities. We find that such flows can be defined only on a finite time interval. Using a complex variable approach, we construct a family of explicit solutions for a single multipole. These solutions turn out to have the maximal possible lifetime in a certain class of solutions.

We also discuss the generalized Hele-Shaw model in which surface tension at the moving boundary is considered, and develop a method of finding steady shapes. This method yields new one-parameter families of stationary solutions. In the Appendix we discuss a connection between these solutions and a variational problem of potential theory.

The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.

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.

We study the problem of designing a periodic interface between two homogeneous materials with different impedance properties, in such a way that time-harmonic waves incident on the interface over a given range of angles have minimal total reflected energy. It is shown that the problem can be ‘relaxed’ to include continuously varying profiles. A simple gradient descent minimization scheme is proposed and examples from several numerical calculations are given.

A mathematical model for the hydrodynamic flow and fibre deflections in a wet-spinning process is presented. The model is characterized by two parameters: α which is a characteristic ratio of the hydrodynamic force to the tension force acting on a fibre, and β which is a characteristic ratio of the fibre drag to the fluid inertia. The model is applied to three situations: (i) the spin bath of the process where α ≪ 1 and β ≫ 1, (ii) the preheat bath where α is order 1 and β ≫ 1, (iii) a fibre spreading process where α is order 1 and β is order 1 or smaller. In each case, analytical results are obtained for the fluid flow and fibre deflections and compared with experiment.

This paper is concerned with some mathematical aspects of magnetic resonance imaging (MRI) of the beating human heart. In particular, we investigate the so-called retrospective gating technique which is a non-triggered technique for data acquisition and reconstruction of (approximately) periodically changing organs like the heart. We formulate the reconstruction problem as a moment problem in a Hilbert space and give the solution method. The stability of the solution is investigated and various error estimates are given. The reconstruction method consists of temporal interpolation followed by spatial Fourier inversion. Different choices for the Hilbert space ℋ of interpolating functions are possible. In particular, we study the case where ℋ is (i) the space of bandlimited functions, or (ii) the space of spline functions of odd degree. The theory is applied to reconstructions from synthetic data as well as real MRI data.

We consider the behaviour of the interface (free boundary) between fresh and salt water in a porous medium (a reservoir). The salt water is below the interface (with respect to the direction of gravity) and is stagnant. The fresh water is above the interface and moves towards the wells which are present in the reservoir. We give a description of the corresponding flow problem leading to a weak variational formulation involving a parameter Q which is related to the strength of the wells. We show that Q is a critical parameter in the following sense: there exists Qcr > 0 such that for Q < Qcr a smooth interface exists which is monotone with respect to Q. For Q = Qcr, a free boundary with one or more singularities (cusps) will occur at a positive distance from the wells. The global analysis for the problem (existence, uniqueness, monotonicity) is given here for two and three dimensional flow situations. The local cusp analysis is two-dimensional, and will be discussed in Part II.

This paper deals with the mathematical characterization of microstructure in elastic solids. We formulate our ideas in terms of rank-one convexity and identify the set of probability measures for which Jensen's inequality for this type of functions holds. This is the set of laminates. We also introduce generalized convex hulls of sets of matrices and investigate their structure.

A stationary phase method is developed for the asymptotic evaluation, as R → ∞, of oscillatory sums of the form

It is extended to multidimensional sums. Numerical comparisons demonstrate the accuracy of the asymptotic approximations. The results are applied to the practical estimation of the number of lattice points in large domains in ℝ2.

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.

An analysis is given of brine transport through a porous medium, which incorporates the effect of volume changes due to variations in the salt concentration. Two specific situations are investigated which lead to self-similarity. We develop the existence and uniqueness theory for the corresponding ordinary differential equations, and give a number of qualitative properties of the solutions. In particular, we present an asymptotic expression for the solution in terms of the relative density difference (ρs−ρf)/ρf. Finally, we show some numerical results. It is found that the volume changes have a noticeable effect on the mass transport only when salt concentrations are large.

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.

The mathematical model of the flow of a viscous lubricant between elastic bearings leads to the study of a highly non-linear and non-local elliptic variational inequality. We discuss the existence of a solution by using an a prioriL∞-estimate. This method allows us to solve a large class of problems, including those arising from the linear Hertzian theory, and yields new existence results for the cases of a pressure-dependent viscosity or the inclusion of a load constraint. For small data the uniqueness of the solution holds, and we show that in the cylindrical journal bearing problem with small eccentricity ratio, the free boundary is given by two disjoint differentiable arcs close to the free boundary of the first-order approximate solution.

Plane, quasi-steady, free-boundary flows of an incompressible viscous fluid with surface tension in a container are considered. The mathematical problem is decomposed into an auxiliary elliptic problem for the Stokes system in a fixed flow domain, whose solution leads to the Cauchy problem for the free boundary with the so-called ‘normal velocity’ operator. By introducing the complex stress-stream function and applying time-dependent conformal mapping, the auxiliary problem is reduced to a boundary integral equation via consideration of two Hilbert problems for analytic functions in a unit disc. As an application, plane capillary flow with moving contact points is investigated asymptotically for small capillary numbers. We prove that in the case when a dynamic contact angle is equal to π, this problem is well-posed for a filling regime, and ill-posed for a drying one.

We consider a hybrid system consisting of a cable linked at its end to a rigid body. It is proved that such a hybrid system can be asymptotically stabilized by means of dissipative boundary feedbacks. Uniform decay estimates of energy are also established.

We examine the Black–Scholes partial differential equation for the pricing of a traded option (an American call option on an asset paying a continuous dividend) and make comparisons with other well known free boundary diffusion problems, such as the oxygen consumption problem. The pricing of American options can be viewed as a free boundary problem and is, therefore, inherently nonlinear. We consider the short and long time behaviour of the free boundary, present analytic results for the option value in such limits, and consider the formulation of the problem as a variational inequality, and its numerical solution.

Linear Fredholm operators (J, k) of the form u(x, t) = (J, k) v (x, t) ≡ J(t) v (x, t)+ k(t, s) v (x, s)ds can be found which map solutions v of the linear matrix system of n partial differential equations ∂u/∂t = DLv + Bv, into solutions u of the like system ∂u/∂t = DLu + Au, when the diagonal matrix D with positive elements and the matrices A and B commute with the linear, scalar operator L. For solution sets in appropriate function spaces, this mapping (J, k) is unique, independent of L, and 1–1 onto if it preserves initial values so that u(x, 0) = v(x, 0). When the set of solutions is restricted to those with zero initial values, this uniqueness aspect of (j, k) breaks down, and there are many linear maps of this Fredholm form which preserve the zero initial conditions and map all such solutions of the first equation in the appropriate function space into solutions of the second. When L is an unbounded operator like the Laplacian ∇2, the initial value problems have many solutions depending on the values of u and u on the boundaries of the region of the solutions, as well as their values in the region at time t = 0. Danckwerts introduced the concept of a ‘constant preserving’ map of Volterra form mapping solutions of the scalar diffusion equation ∂u/∂t = D∇2u, which are initially zero, onto solutions of a scalar diffusion equation ∂u/∂t = D∇2u + Au, with a linear, homogeneous, constant coefficient source term. This concept of a ‘constant preserving’ map extends to the nth order matrix Fredholm maps described. A map (J, k∗) is said to be constant preserving if for all constant n x n matrices C, (J, k∗)C = C, and hence J(t) + k∗(t, s)ds = I, where I is the unit matrix. In the restricted solution spaces where u(x, 0) and v(x, 0) are zero, there is a unique 1–1 onto Danckwerts map of this type transforming solutions of the v equation into solutions of the u equation. In the cases where the coupling matrix A is constant and B is zero, the kernel k∗ of the Danckwerts map can be expressed in terms of the kernel k for the more general Fredholm map associated with the same equations, but mapping the larger solution sets containing elements which need not vanish at time zero. The Danckwerts mapping is used to establish a generalized Ussing flux-ratio theorem for some coupled diffusion problems involving several chemically interacting components.

A minimization problem for partially wetting droplet profiles is considered, in which Van der Waal's forces have been taken into account via a singular ‘disjoining pressure’. When the singular disjoining pressure is neglected, energy minimization leads to Laplace's equation and Young's equation; once the singular disjoining pressure is included, this is no longer the case. Indeed, the free energy is no longer bounded from below. Introducing the notion of overtaking to compare the energies of configurations whose energies are arbitrarily large and negative, we demonstrate that if a configuration is not convex then it cannot be an absolute minimizer. If profiles are allowed to ‘double-over’ then there does not exist an absolute minimizer. Within the class of profiles which do not double-over, absolute minimizers are shown to exist; these minimizing profiles are not single-valued. The singular minimization problem is shown to be discontinuously dependent on the definition of the wetting profile in the neighbourhood of the contact points; the implications of this discontinuity are discussed.