On Dekster’s angle measure in Minkowski spaces

Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between and 8.     

Non-simultaneous blow-up in localized reaction–diffusion equations

This article deals with blow-up solutions in reaction–diffusion equations coupled via localized exponential sources, subject to null Dirichlet conditions. The optimal and complete classification is obtained for simultaneous and non-simultaneous blow-up solutions. Moreover, blow-up rates and blow-up sets are also discussed. It is interesting that, in some exponent regions, blow-up phenomena depend sensitively on the choosing of initial data, and the localized nonlinearities play important roles in the blow-up properties of solutions.     

On the stationary Navier–Stokes flows around a rotating body

Consider the stationary motion of an incompressible Navier–Stokes fluid around a rotating body $ \mathcal{K} = \mathbb{R}^3 \, \backslash \, {\Omega}$ which is also moving in the direction of the axis of rotation. We assume that the translational and angular velocities U, ω are constant and the external force is given by f = div F. Then the motion is described by a variant of the stationary Navier–Stokes equations on the exterior domain Ω for the unknown velocity u and pressure p, with U, ω, F being the data. We first prove the existence of at least one solution (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega)}$ and ${u \in L_3, \infty (\Omega)}$ under the smallness condition on ${|U| + |\omega| + ||F||_{L_{3/2, \infty} (\Omega)}}$ . Then the uniqueness is shown for solutions (u, p) satisfying ${\nabla u, p \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}$ and ${u \in L_{3, \infty} (\Omega) \cap L_{q*, r} (\Omega)}$ provided that 3/2 <? q <? 3 and ${{F \in L_{3/2, \infty} (\Omega) \cap L_{q, r} (\Omega)}}$ . Here L _q,r (Ω) denotes the well-known Lorentz space and q* =? 3q /(3 ? q) is the Sobolev exponent to q.     

On slow flows of the full nonlinear Doi–Edwards polymer model

The full Doi–Edwards model constitutive equation derived by Palierne (Phys Rev Lett 93:136001-1–136001-4, (2004) is discussed in detail. The corresponding configurational probability equation is next solved for slow flows, and the solution is used to calculate the material constants: zero-shear viscosity and the normal stress differences.     

Azimuthal equatorial capillary–gravity flows in spherical coordinates

We work in the setting of spherical coordinates to prove existence of free-surface capillary–gravity azimuthal equatorial flows which allow for variations in the vertical direction (so currents can be accommodated) but with no meridional flow. We perform the task by deriving an implicit equation, which determines the pressure at the surface if the free surface is known, and vice-versa (given the current profile). The solutions of this equation are found by using the implicit function theorem in a proper setting, which incorporates the observed symmetry about the Equator and takes advantage of the freedom in choosing the reference for the length of the Earth’s radius at the Equator.     

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.     

On the Navier–Stokes flows for heat-conducting fluids with mixed boundary conditions

We consider a coupled model for steady flows of viscous incompressible heat-conducting fluids with temperature dependent material coefficients in a fixed three-dimensional open cylindrical channel. We introduce the Banach spaces X and Y to be the space of possible solutions of this problem and the space of its data, respectively. We show that the corresponding operator of the problem acting between X and Y is Fréchet differentiable. Applying the local diffeomorphism theorem we get the local solvability results for a variational formulation.     

On the action of the Iwahori–Hecke algebra on modular invariants

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of the Takayasu modules.     

On the representation of localized functions in ℝ2 by the Maslov canonical operator

We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being the Maslov canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.     

Coupled lattice Boltzmann method for simulating electrokinetic flows: A localized scheme for the Nernst–Plank model

We present a coupled lattice Boltzmann method (LBM) to solve a set of model equations for electrokinetic flows in micro-/nano-channels. The model consists of the Poisson equation for the electrical potential, the Nernst–Planck equation for the ion concentration, and the Navier–Stokes equation for the flows of the electrolyte solution. In the proposed LBM, the electrochemical migration and the convection of the electrolyte solution contributing to the ion flux are incorporated into the collision operator, which maintains the locality of the algorithm inherent to the original LBM. Furthermore, the Neumann-type boundary condition at the solid/liquid interface is then correctly imposed. In order to validate the present LBM, we consider an electro-osmotic flow in a slit between two charged infinite parallel plates, and the results of LBM computation are compared to the analytical solutions. Good agreement is obtained in the parameter range considered herein, including the case in which the nonlinearity of the Poisson equation due to the large potential variation manifests itself. We also apply the method to a two-dimensional problem of a finite-length microchannel with an entry and an exit. The steady state, as well as the transient behavior, of the electro-osmotic flow induced in the microchannel is investigated. It is shown that, although no external pressure difference is imposed, the presence of the entry and exit results in the occurrence of the local pressure gradient that causes a flow resistance reducing the magnitude of the electro-osmotic flow.     

On system reliability in stress–strength setup

In this paper we study stress–strength reliability for a general coherent system. The exact expression as well as bounds and approximations for system reliability are presented. We also illustrate the estimation procedure for exponential stress–strength distributions.     

On Hardy–Rellich type inequalities in RN

In this note we introduce a new class of Hardy–Rellich type inequalities and explicitly obtain their corresponding sharp constants. Our approach suggests definitions of new Sobolev spaces and embedding results.     

On Absorbing Cycles in Min–Max Digraphs

We consider smooth finite dimensional optimization problems with a compact, connected feasible set M and objective function f. The basic problem, on which we focus, is: how to get from one local minimum to all the other ones. To this aim we introduce a bipartite digraph as follows. Its nodes are formed by the set of local minima and maxima of f|_M, respectively. Given a smooth Riemannian (i.e. variable) metric, there is an arc from a local minimum x to a local maximum y if the ascent (semi-)flow induced by the projected gradients of f connects points from a neighborhood of x with points from a neighborhood of y. The existence of an arc from y to x is defined with the aid of the descent (semi-)flow. Strong connectedness of ensures that, starting from one local minimum, we may reach any other one using ascent and descent trajectories in an alternating way. In case that no inequality constraints are present or active, it is well known that for a generic Riemannian metric the resulting min-max digraph is indeed strongly connected. However, if inequality constraints are active, then there might appear obstructions. In fact, we show that may contain absorbing two-cycles. If one enters such a cycle, one cannot leave it anymore via ascent and descent trajectories. Moreover, the cycles being constructed are stable with respect to small perturbations (in the C¹-topology) of the Riemannian metric.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.