Second-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at $\infty$. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at $\infty$.

We consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

We investigate the role of projections in norming a $C^*$-algebra by a type II1 subfactor. Applications are given for factors with property $\varGamma$ and for free-product factors.

We present an abstract approach to the construction of holomorphic functional calculi for unbounded operators and apply it to the special case of sectorial operators. In effect, we obtain a calculus for a much larger class of functions than was known before, including certain meromorphic functions. We discuss the role of topology. Then we prove in detail a composition rule $(f\circ g)(A)=f(g(A))$ which is the main result of the paper. This is done in such a way that the proof can easily be transferred to functional calculi for other classes of operators.

A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

We investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

given a compact $n$-dimensional immersed riemannian manifold $m^n$ in some euclidean space we prove that if the hausdorff dimension of the singular set of the gauss map is small, then $m^n$ is homeomorphic to the sphere $s^n$.

also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with a small set of points of zero gauss–kronecker curvature are topologically the sphere minus a finite number of points. a characterization of the $2n$-catenoid is obtained.

An optimization problem for the fundamental eigenvalue $\lam_0$ of the Laplacian in a planar simply-connected domain that contains $N$ small identically-shaped holes, each of radius $\eps\ll 1$, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue $\lam_0$ is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains $N$ small traps. For small hole radii $\eps$, a two-term asymptotic expansion for $\lam_0$ is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations $x_{i}$, for $i=1,\ldots,N$, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize $\lam_0$. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes $\lam_0$. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize $\lam_0$. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer–Meinhardt reaction-diffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg–Landau theory of superconductivity.

A free boundary problem arising in a model for inviscid, incompressible shallow water entry at small deadrise angles is derived and analysed. The relationship between this novel free boundary problem and the well-known viscous squeeze film problem is described. An inverse method is used to construct explicit solutions for certain body profiles and to find criteria under which the splash sheet can ‘split’. A variational inequality formulation, conservation of certain generalized moments and the Schwarz function formulation are introduced.

Mathematical models for the prediction of the hydrodynamic pressure distribution and the force on a body entering liquid are investigated. Particular attention is paid to analytical models which are based on the velocity potential given by the classical Wagner theory. Formal use of the Wagner theory provides the loads on an entering body, which are higher than the measured ones. To improve the predictions, the higher order terms in the Bernoulli equation are taken into account within the generalized Wagner model and the Logvinovich model. It is shown that the Logvinovich model corresponds better to the experimental data than the generalized Wagner model. A rational derivation of the Logvinovich model is given in the paper for the two-dimensional case. The analytical models are tested against both numerical and experimental results.

We consider short-time existence, uniqueness, and regularity for a moving boundary problem describing Stokes flow of a free liquid drop driven by surface tension. The surface tension coefficient is assumed to be a nonincreasing function of the surfactant concentration, and the surfactant is insoluble and moves by convection along the boundary. The problem is reformulated as a fully nonlinear, nonlocal Cauchy problem for a vector-valued function on a fixed reference manifold. This problem is, in general, degenerate parabolic. Existence and uniqueness results are obtained via energy estimates in Sobolev spaces of sufficiently high order. In the two-dimensional case, the problem is strictly parabolic, and we prove instantaneous smoothing of the free boundary, using maximal regularity results in little Hölder spaces.

We solve the free boundary problem for the dynamics of a cylindrical, axisymmetric viscoelastic filament stretching in a gravity-driven extensional flow for the Upper Convected Maxwell and Oldroyd-B constitutive models. Assuming the axial stress in the filament has a spatial dependence provides the simplest coupling of viscoelastic effects to the motion of the filament, and yields a closed system of ODEs with an exact solution for the stretch rate and filament thickness satisfied by both constitutive models. This viscoelastic solution, which is a generalization of the exact solution for Newtonian filaments, converges to the Newtonian power-law scaling as $t \rightarrow \infty$. Based on the exact solution, we identify two regimes of dynamical behavior called the weakly- and strongly-viscoelastic limits. We compare the viscoelastic solution to measurements of the thinning filament that forms behind a falling drop for several semi-dilute (strongly-viscoelastic) polymer solutions. We find the exact solution correctly predicts the time-dependence of the filament diameter in all of the experiments. As $t \rightarrow \infty$, observations of the filament thickness follow the Newtonian scaling $1/\sqrt{t}$. The transition from viscoelastic to Newtonian scaling in the filament thickness is coupled to a stretch-to-coil transition of the polymer molecules.

We consider two-dimensional bubbles in a corner flow in a Hele–Shaw cell of a viscous incompressible fluid that occupies the complement to a bubble. We discuss the governing equations, some basic properties of the free interface of the bubbles, their geometry, and construct explicit solutions that present asymmetric long bubbles analogous to the famous Saffman–Taylor fingers in a wedge of arbitrary angle $\alpha\in (0,2\pi)$.

We examine a number of initial boundary value problems for a paradigm sixth-order degenerate parabolic equation of the form $h_t=(h^n h_{xxxxx})_x$ which arises when considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. Analytical and numerical methods are exploited to characterise the solutions, which turn out to be rather sensitive to the value of $n$.

We use lubrication theory on the flow equations for nematic liquid crystals to derive a simple model describing the evolution of the film height under gravity, in the case of finite surface “anchoring energy” at the free surface and at the rigid substrate. This means that the molecules of the nematic have a preferred alignment at interfaces, modelled by a single-well potential surface energy (first introduced by Rapini & Papoular [9]). This paper generalises the earlier work of Ben Amar & Cummings, in which the orientation of the nematic liquid crystal molecules is effectively specified at both surfaces (strong anchoring; isotropic surface tension). Additional terms, analogous in some sense to Marangoni terms, are introduced into the PDE governing the film height evolution. The stability of the derived model is considered, and stability criteria are presented and discussed. The existence of static, drop-like solutions to the model is also briefly considered.

Capillary waves on fluid sheets are computed in the presence of a uniform electric field acting horizontally with respect to the undisturbed configuration. The fluid is taken to be inviscid, incompressible and nonconducting. In previous work (Papageorgiou & Vanden-Broeck [14]) symmetric travelling waves were investigated. In this paper we show that there are in addition antisymmetric waves. These waves are calculated numerically for arbitrary amplitudes and wavelengths and the effect of the electric field is studied. The numerical procedure is based on a reformulation of the problem as a system of nonlinear integro–differential equations.

The effects of the thin air layer entering play when a water droplet impacts on otherwise still water or on a fixed solid are studied theoretically with special attention on surface tension and on post-impact behaviour. The investigation is based on the small density and viscosity ratios of the two fluids. In certain circumstances, and in particular for droplet Reynolds numbers below a critical value which is about ten million, the air-water interaction depends to leading order on lubricating forces in the air coupled with potential flow dynamics in the water. The nonlinear integro-differential system for the evolution of the interface and induced pressure is studied for pre-impact surface tension effects, which significantly delay impact, and for post-impact interaction phenomena which include significant decrease of the droplet spread rate. Above-critical Reynolds numbers are also considered.

A class of exact mathematical solutions describing distributed regions of uniform vorticity attached to two solid walls meeting at an angle $2\alpha$ is derived. Exterior to the uniform vorticity region the flow is quiescent and irrotational. The mathematical method used is a generalization of ideas original presented in Crowdy [4] combined with elements of conformal mapping theory associated with a differential equation method due to Polubarinova-Kochina traditionally applied in finding the solution to various free boundary problems.

The modeling of the motion of a contact line, the triple point at which solid, liquid and air meet, is a major outstanding problem in the fluid mechanics of thin films [2, 9]. In this paper, we compare two well-known models in the specific context of Marangoni driven films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. The two models considered here have subtle differences in their description, requiring a careful treatment when comparing traveling waves and effective contact angles. Numerical results exhibit broad agreement between the two models, but the closest comparison can be done only for a rather restricted range of parameters. The driven film context gives contact angle results quite different from the case of a film moving under the action of gravity alone. The numerical technique for exploring phase portraits for the third order ODE is also used to tabulate the kinetic relation and nucleation condition, information that can be used with the underlying hyperbolic conservation law to explain the rich combination of wave structures observed in simulations of the PDE and in experiments [3, 15].

We consider the dynamics of multi-component heat-conducting viscous incompressible flow in a plane domain when the viscosity and thermal conductivity of the medium depend on temperature. The dynamics of the flow is governed by an initial-boundary value problem for the Navier–Stokes system with heat conduction and heat transfer taken into account. The existence of a generalized global solution with velocity field and temperature of Hopf's class has been established in conjunction with the estimate of fractional smoothness of the order 1/2 in the time variable.