Mathematical Models and Methods in Applied Sciences

In this contribution, the nonlinear dynamics of the surplus (net assets or reserve) process for a dividend-distributing company is studied in conjunction with the dynamics of its dividend equalization fund. The latter type of fund is maintained by leading insurance companies throughout the world and pays a special dividend for income that the investors lost because the dividend payment process was adversely affected for some or other reason. In our paper, a stochastic model for the related notion of a dividend equalization solvency ratio is derived. The ambient value of this ratio is an indication of the capacity of the insurer to pay dividends to shareholders especially when profit is low. The aforementioned analysis is, in turn, based on the construction of continuous time stochastic models for the dynamics of the surplus and total liabilities processes of an insurer. The discussions are reliant on principles arising within the asset-liability modelling paradigm.

In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation.

This paper follows the same spirit as the companion work,² where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.

In this paper, we study dynamics of the ground state and central vortex states in Bose–Einstein condensation (BEC) analytically and numerically. We show how to define the energy of the Thomas–Fermi (TF) approximation, prove that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in linear case, derive a second-order ordinary differential equation (ODE) which shows that time-evolution of the condensate width is a periodic function with/without a perturbation by using the variance identity, prove that the angular momentum expectation is conserved in two dimensions (2D) with a radial symmetric trap and 3D with a cylindrical symmetric trap for any initial data, and study numerically stability of central vortex states as well as interaction between a few central vortices with winding numbers ±1 by a fourth-order time-splitting sine-pseudospectral (TSSP) method. The merit of the numerical method is that it is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density, performs spectral accuracy for spatial derivatives and fourth-order accuracy for time derivative, and possesses "optimal" spatial/temporal resolution in the semiclassical regime. Finally we find numerically the critical angular frequency for single vortex cycling from the ground state under a far-blue detuned Gaussian laser stirrer in strong repulsive interaction regime and compare our numerical results with those in the literatures.

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ω_ε, which is the union of a domain Ω₀ and a large number 2N of thin rods with variable thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The Robin conditions are given on the lateral boundaries of the thin rods. Using the method of matched asymptotic expansions, we construct the asymptotic approximation for the solution as ε → 0 and prove the corresponding estimates in the Sobolev space H¹(Ω_ε).

This paper deals with the homogenization of a lubrication problem, using two-scale convergence techniques. We study in particular the Reynolds–Hertz model, which takes into account elastohydrodynamic deformations of the upper surface, when highly oscillating roughness effects occur. The difficulty arises when considering cavitation free boundary phenomena, leading to highly nonlinear and nonlocal problems.