Local solutions near the intersection of a free surface with a vertical wall are constructed numerically. Both gravity and surface tension are included in the dynamic boundary condition. It was shown that the solutions are characterized by the angle γ between the free surface and the wall at the separation point. There is a solution for each value of π/2<γ<π. As γ→π/2 and γ→π, the solutions reduce to the pure gravity solutions of Vanden-Broeck & Tuck [1].

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problem

ut =J*u−u−f (u),

the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.

We present a continuum theory for smectic liquid crystals which allows variable molecular tilt and layer spacing by relating the two. This compressible theory is employed to predict the effect of ‘homeotropic’ type ordering at the plates on a planar Sm C sample. Such a boundary condition induces dilation of individual Sm C layers near the bounding plates.

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.

This paper studies a vectorial problem in the calculus of variations arising in the theory of martensitic microstructure. The functional has an integral representation where the integrand is a non-convex function of the gradient with exactly four minima. We prove that the Young measure corresponding to a minimizing sequence is homogeneous and unique for certain linear boundary conditions. We also consider the singular perturbation of the problem by higher-order gradients. We study an example of microstructure involving infinite sequential lamination and calculate its energy and length scales in the zero limit of the perturbation.

In this paper we deal with the one-dimensional Stefan problem

ut−uxx =s˙(t)δ(x−s(t)) in ℝ ;× ℝ+, u(x, 0) =u0(x)

with kinetic condition s˙(t)=f(u) on the free boundary F={(x, t), x=s(t)}, where δ(x) is the Dirac function. We proved in [1] that if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some M>0 and γ∈(0, 1/4), then there exists a global solution to the above problem; and the solution may blow up in finite time if f(u)[ges ] Ceγ1[mid ]u[mid ] for some γ1 large. In this paper we obtain the optimal exponent, which turns out to be √2πe. That is, the above problem has a global solution if [mid ]f(u)[mid ][les ]Meγ[mid ]u[mid ] for some γ∈(0, √2πe), and the solution may blow up in finite time if f(u)[ges ] Ce√2πe[mid ]u[mid ].

The static electro-optic behaviour of achiral smectic C samples in the chevron geometry have been modelled using the continuum theory of Leslie et al. [1]. The model assumes that the layer tilt and smectic C director tilt angle are constant, and treats the layers at the chevron interface as an infinitely bound surface. Comparison of the predicted electro-optic behaviour with experimental results gives values for the bend (B1) and splay (B2) c-director elastic constants. However, more detailed optical studies show that surface and chevron interface terms become important at high values of applied electric field.

A fully nonlinear, degenerate parabolic equation arising in the theory of damage mechanics is shown to be well-posed. Its solutions blow up in finite time and, under suitable conditions on the initial configuration, the blow-up set, corresponding to the portion of the material which breaks at the blow-up time, is an interval of nonzero measure. In a special but physically relevant case the problem reduces to the study of the blow-up set of solutions of the quasilinear equation

The Pólya–Szegö dipole tensors are employed for the analysis of plane elasticity problems in non-homogeneous media. Classes of equivalence for defects (cavities and rigid inclusions) are specified for the Laplacian operator and elasticity equations: composite materials with defects of the same class have the same effective elastic moduli. Explicit asymptotic formulae for effective compliance matrices of dilute composites are presented.

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.

The motivation for this work arises from the study of the processes involved in the manufacturing of a class of composite materials, in particular, those that are obtained by injecting a resin through a porous preform. A one-dimensional model that describes the non-isothermal filtration of an incompressible fluid is presented, and it also includes the possibility of curing, i.e. the polymerization of the penetrating resin. It comes out a fully coupled system consisting of the heat diffusion equation, Darcy's law and an equation related to the kinetics of the chemical reaction. The system is regarded as a free boundary problem for the heat equation with non-constant discontinuous coefficients. Its weak formulation is studied and the local existence of solutions is proved.

We review the main outcomes of a continuum theory for the equilibrium of the interface between a nematic liquid crystal and an isotropic environment, in which the surface free energy density bears terms linear in the principal curvatures of the interface. Such geometric contributions to the energy occur together with more conventional elastic contribution, leading to an effective azimuthal anchoring of the optic axis, which breaks the isotropic symmetry of the interface. The theory assumes the interface to be fixed, as for a rigid cavity filled with liquid crystal, and so it does not apply to drops. It should be appropriate when the curvatures of the interface are small compared to that of the molecular interaction sphere. Also, interfaces bearing a sharp edge are encompassed within the theory; a line integral expresses the energy condensed along the edge: we see how it affects the equilibrium equations.

An ideal fibre-reinforced fluid is incompressible and inextensible along a family of material curves that are convected with the fluid. It is a model for continuous fibre-resin systems in the fluid state in which forming processes take place. Like liquid crystals, these fluids have strong directional properties. The kinematic and constitutive theory of ideal fibre-reinforced fluids is described, with particular reference to plane flows. The class of flows in which the fibres are aligned along the streamlines is considered, and an explanation is given for the observed prevalence of this class of flows.

A model for Poiseuille flow of nematic liquid crystals is examined where a layered structure of defects parallel to the flow is present. Constant gradient flows are discussed, together with symmetric and chevron configurations for such flow problems.

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.

In this paper a generalized model for the growth of avascular tumours is presented. The formulation leads naturally to the incorporation of free boundaries which define the outer tumour surface explicitly and various inner surfaces implicitly. A combination of numerical simulations, asymptotic analysis and perturbation techniques is used to study the model and yields results, which agree well with experimentally-observed phenomena.

We consider a model of the motion of a viscous dielectric liquid subjected to a DC electric field when the bulk conduction results from the presence of a dissociation-recombination process. It is shown that any weak solution approaches a neighbourhood of a spatially homogeneous steady state with radius r≈(d+ +d−)&14frac;, where d+, d− are the diffusion coefficients.

We consider a Boltzmann-like model of outgassing and contamination in a three-dimensional region V= V1∪V2∪V3. V1 is the region where the contaminant particles are produced, and V2 is the region where such particles migrate and interact with some inert gas. V3 is where contamination takes place because of the particles emanating from V2. In each of the three regions, the behaviour of the contaminant particles is represented by means of a Boltzmann-like equation. We show that such a problem has a unique positive strict solution, belonging to a suitable L1 Banach space X. Finally, a system of ordinary differential equations is derived which gives the evolution of the total number of contaminant particles in each of the three regions.