Asymptotic behaviour of two-dimensional time-delayed Navier-Stokes equations

We consider the two-dimensional Navier-Stokes equations with a time-delayed convective term and a forcing term which contains some hereditary features. Some results on existence and uniqueness of solutions are established. We discuss the asymptotic behaviour of solutions and we also show the exponential stability of stationary solutions.

Asymptotic behaviour of the density in a parabolic SPDE

Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$.In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noiseparameter is obtained. The coefficients and the residue of the expansion are explicitly calculated.In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximatingprocess and the limit one is proved. Also a suitable local integration by parts formula is developped.

Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Asymptotic Behaviour of Colength of Varieties of Lie Algebras

This project was partially supported by RFBR, grants 99-01-00233, 98-01-01020 and 00-15-96128.

A Note on the Asymptotic Behaviour of a Periodic Multitype Galton-Watson Branching Process

2000 Mathematics Subject Classification: 60J80.

Asymptotic Behaviour of a Supercritical Galton-Watson Process with Controlled Binomial Migration

AMS subject classification: 60J80, 62F12, 62P10.

The Asymptotic Behaviour of the First Eigenvalue of Linear Second-Order Elliptic Equations in Divergente Form

2000 Mathematics Subject Classification: 35J70, 35P15.

Asymptotic gravity wave drag expressions for non-hydrostatic rotating flow over a ridge

Asymptotic expressions are derived for the mountain wave drag in flow with constant wind and static stability over a ridge when both rotation and non-hydrostatic effects are important. These expressions, which are much more manageable than the corresponding exact drag expressions (when these do exist) are found to provide accurate approximations to the drag, even when non-hydrostatic and rotation effects are strong, despite having been developed for cases where these effects are weak. The derived expressions are compared with approximations to the drag found previously, and their asymptotic behaviour in various limits is studied.

Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations

We consider second kind integral equations of the form x(s) - (abbreviated x - K x = y ), in which Ω is some unbounded subset of Rn. Let Xp denote the weighted space of functions x continuous on Ω and satisfying x (s) = O(|s|-p ),s → ∞We show that if the kernel k(s,t) decays like |s — t|-q as |s — t| → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∈ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∈ B(X0) then (I - K)-1 ∈ B(XP) for 0 < p < q, and (I- K)-1∈ B(Xq) if further conditions on k hold. Thus, if k(s, t) = O(|s — t|-q). |s — t| → ∞, and y(s)=O(|s|-p), s → ∞, the asymptotic behaviour of the solution x may be estimated as x (s) = O(|s|-r), |s| → ∞, r := min(p, q). The case when k(s,t) = к(s — t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I — K, for I — K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propaga-tion above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation mod-els, in particular, road traffic noise propagation along an infinite road surface sur-rounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Large time behaviour for functional differential equations with dominant eigenvalues of arbitrary order

The spectral theory for linear autonomous neutral functional differential equations (FDE) yields explicit formulas for the large time behaviour of solutions. Our results are based on resolvent computations and Dunford calculus, applied to establish explicit formulas for the large time behaviour of solutions of FDE. We investigate in detail a class of two-dimensional systems of FDE. (C) 2009 Elsevier Inc. All rights reserved.

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09

Phase sensitive pumping and coherence of superposition states

The continuous parametric pumping of a superconducting lossy QED cavity supporting a field prepared initially as a superposition of coherent states is discussed. In contrast to classical pumping, we verify that the phase sensitivity of the parametric pumping makes the asymptotic behaviour of the cavity field state strongly dependent on the phase theta of the coherent state \ alpha > = \ alpha \e(i theta)>. Here we consider theta = pi /4, -pi /4 and we analyse the evolution of the purity of the superposition states with the help of the linear entropy and fidelity functions. We also analyse the decoherence process quantitatively through the Wigner function, for both states, verifying that the decay is slightly modified when compared to the free decoherence case: for theta = -pi /4 the process is accelerated while for theta = pi /4 it is delayed.

Set-valued Markov chains and negative semitrajectories of discretized dynamical systems

Computer simulation of dynamical systems involves a phase space which is the finite set of machine arithmetic. Rounding state values of the continuous system to this grid yields a spatially discrete dynamical system, often with different dynamical behaviour. Discretization of an invertible smooth system gives a system with set-valued negative semitrajectories. As the grid is refined, asymptotic behaviour of the semitrajectories follows probabilistic laws which correspond to a set-valued Markov chain, whose transition probabilities can be explicitly calculated. The results are illustrated for two-dimensional dynamical systems obtained by discretization of fractional linear transformations of the unit disc in the complex plane.

Surface phenomena and hydrodynamic effects on the deposition of Pseudomonas fluorescens

Biofilm adhesion to metals (copper, aluminium and brass) was studied at two different velocities and pH values of 7 and 9. Both bacteria and metals showed negative surface charges at those values of pH, which tends to slow down adhesion. Film densities increased with the fluid velocity and were also affected by the pH and by the growth rate of the bacteria. Long duration tests based on heat transfer measurements were run at five different fluid velocities and at pH = 7, showing in general an asymptotic behaviour and a control of deposition by adhesion and growth phenomena.