Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.     

On Gopalsamy and Liu's conjecture for global stability in a population model

For a logistic equation with a piecewise constant argument which models the dynamics of a population of a single species undergoing a density-dependent harvesting, Gopalsamy and Liu (J. Math. Anal. Appl. 224 (1998) 59–80), have offered a condition of the growth rate of species and conjectured that this is a necessary and sufficient condition of not only the asymptotic stability but also the global asymptotic stability for the positive equilibrium of the equation. But until now, there were no mathematical answers except computer simulations.In this paper, we establish a mathematically rigorous proof of a partial affirmative answer of this conjecture.     

Spikes for the Gierer-Meinhardt System in Two Dimensions: The Strong Coupling Case

Numerical computations often show that the Gierer-Meinhardt system has stable solutions which display patterns of multiple interior peaks (often also called spots). These patterns are also frequently observed in natural biological systems. It is assumed that the diffusion rate of the activator is very small and the diffusion rate of the inhibitor is finite (this is the so-called strong-coupling case). In this paper, we rigorously establish the existence and stability of such solutions of the full Gierer-Meinhardt system in two dimensions far from homogeneity. Green's function together with its derivatives plays a major role.     

Physical entropy, information entropy and their evolution equations

Inspired by the evolution equation of nonequilibrium statistical physics entropy and the concise statistical formula of the entropy production rate, we develop a theory of the dynamic information entropy and build a nonlinear evolution equation of the information entropy density changing in time and state variable space. Its mathematical form and physical meaning are similar to the evolution equation of the physical entropy: The time rate of change of information entropy density originates together from drift, diffusion and production. The concise statistical formula of information entropy production rate is similar to that of physical entropy also. Furthermore, we study the similarity and difference between physical entropy and information entropy and the possible unification of the two statistical entropies, and discuss the relationship among the principle of entropy increase, the principle of equilibrium maximum entropy and the principle of maximum information entropy as well as the connection between them and the entropy evolution equation.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).     

Well posedness,large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.     

Positive pseudo almost periodic solutions for a delayed differential neoclassical growth model

This paper is concerned with a non-autonomous delayed differential neoclassical growth model with an oscillating death rate. Under proper conditions, we employ a novel argument to establish a criterion on the existence and global exponential stability of positive pseudo almost periodic solution, which improves and extends some known relevant results. Finally, we present an example along with its numerical simulations to demonstrate the validity of the proposed result.     

Variational principles and mixed multifractal spectra

We establish a ``conditional' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.

Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed' multifractal spectra.

     

Forced two layer beta-plane quasigeostrophic flow. Part I: Long-time existence and uniqueness of weak solutions

We consider a model of quasigeostrophic turbulence that has proven useful in theoretical studies of large scale heat transport and coherent structure formation in planetary atmospheres and oceans. The model consists of a coupled pair of hyperbolic PDEs with a forcing which represents domain-scale thermal energy source. Although the use to which the model is typically put involves gathering information from very long numerical integrations, little of a rigorous nature is known about long-time properties of solutions to the equations. In this first paper we define a notion of weak solution, and show using Galerkin methods the long-time existence and uniqueness of such solutions.     

A pre-order principle and set-valued Ekeland variational principle

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results.     

Stable and unstable flow regimes for active fluids in the periodic setting

Depending on the involved physiobiological parameters, stable or unstable behavior in active fluids is observed. In this paper a rigorous analytical justification of (in-)stability within the corresponding regimes is given. In particular, occurring instability for the manifold of ordered polar states caused by self-propulsion is proved. This represents the prerequisite for active turbulence patterns as observed in a number of applications. The approach is carried out in the periodic setting and is based on the generalized principle of linearized (in)-stability related to normally stable and normally hyperbolic equilibria.     

Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow

We present a rigorous numerical proof based on interval arithmetic computations categorizing the linearized and nonlinear stability of periodic viscous roll waves of the KdV–KS equation modeling weakly unstable flow of a thin fluid film on an incline in the small-amplitude KdV limit. The argument proceeds by verification of a stability condition derived by Bar–Nepomnyashchy and Johnson–Noble–Rodrigues–Zumbrun involving inner products of various elliptic functions arising through the KdV equation. One key point in the analysis is a bootstrap argument balancing the extremely poor sup norm bounds for these functions against the extremely good convergence properties for analytic interpolation in order to obtain a feasible computation time. Another is the way of handling analytic interpolation in several variables by a two-step process carving up the parameter space into manageable pieces for rigorous evaluation. These and other general aspects of the analysis should serve as blueprints for more general analyses of spectral stability.     

Existence and uniqueness of generalized stationary waves for viscous gas flow through a nozzle with discontinuous cross section

In this paper we study the existence and uniqueness of the generalized stationary waves for one-dimensional viscous isentropic compressible flows through a nozzle with discontinuous cross section. Following the geometric singular perturbation technique, we establish the existence and uniqueness of inviscid and viscous stationary waves for the regularized systems with mollified cross section. Then, the generalized inviscid stationary waves are classified for discontinuous and expanding or contracting nozzles by the limiting argument. Moreover, we obtain the generalized viscous stationary waves by using Helly?s selection principle. However, due to the choices of mollified cross section functions, there may exist multiple transonic standing shocks in the generalized stationary waves. A new entropy condition is imposed to select a unique admissible standing shock in generalized stationary wave. We show that, such admissible solution selected by the entropy condition, admits minimal total variation and has minimal enthalpy loss across the standing shock in the limiting process.     

泥沙的群体沉降速度

本文讨论了在浓度很高时候,泥沙的群体沉降速度.作者认为所谓泥沙的群体沉降速度问题从物理本质来说并不是真正的沉降速度问题,而是水在泥沙和水所组成的胶团形成的多孔介质中的渗流问题,也就是水流经这种多孔介质所形成的问题.作者根据这样的认识,写出了泥沙群体沉降速度的数学表达式,并且和黄委会水科所测量得到的实验数据进行了比较.比较的结果是令人满意的.     

Variational Approach to the Inviscid Compressible Fluid

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.      

Global stability and periodic oscillations for an SIV infection model with immune response and intracellular delays

In this paper, we consider the combined effects of cytotoxic T lymphocyte (CTL) responses on the competition dynamics of two Simian immunodeficiency virus (SIV) strains model. One of strains concerns a relatively slowly replicating and mildly cytopathic virus in the early infection (SIVMneCL8), the other is faster replicating and more cytopathic virus at later stages of the infection (SIVMne170). It is shown that the global dynamics of the ordinary differential equations can be determined by several threshold parameters, and we prove the global stability of the equilibria by rigorous mathematical analysis. To account for a series of infection mechanism leading to viral production, we incorporate time delays in the infection term. Using the methods of constructing suitable Lyapunov functionals and LaSalle’s invariance principle, we obtain the sufficient conditions for the global attractiveness of infection-free equilibrium with both virus strains going extinct, single-infection equilibrium with one of two virus strains out-competing the other one and the two strains coexisting infection equilibrium. We establish that the intracellular delays can destabilize the single-infection equilibrium leading to Hopf bifurcation and periodic oscillations. We show that introduction of immune responses is responsible for the coexistence of two virus strains and the intracellular delays may alter the two-strain competition results. Numerical simulations are presented to illustrate the theoretical conclusions.     

Small perturbations of an indefinite elliptic equation

We investigate a perturbed semilinear indefinite elliptic equation and show that the results known for the unperturbed equation still hold if the perturbation is sufficiently small. To this end, we use a continuity argument that allows us to establish the existence of two positive solutions even in the case where the strong maximum principle does not apply.     

随机利率模型下基于Tsallis熵分布的可转债定价

本文主要研究基于Tsallis熵分布且存在瞬时违约风险的情况下,随机利率服从Vasicek利率模型的可转换债券的定价问题。标的股票价格过程服从Tsallis熵分布的前提下,构建投资组合,利用无套利原理得到可转债价格所满足的偏微分方程,进一步采用有限元法得到可转债价格的数值解。根据长江证券、利欧股份以及吉林敖东股票的市场真实数据,利用Tsallis熵分布模拟收益率序列,并得到基于Tsallis熵分布的股价模型优于几何布朗运动模型下的最优参数,在此基础上,绘制股价基于Tsallis熵分布下三种标的股票所对应可转债的理论价格的三维图及与市场实际价格的对比图。研究结果发现,对应标的股票价格基于Tsallis熵分布下的可转债理论价格与市场真实价格更为接近。     

A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.     

Non‐linear stability for convection with quadratic temperature dependent viscosity

In this paper, we study the non‐linear stability of convection for a Newtonian fluid heated from below, where the viscosity of the fluid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Ra_c, the rest state of the fluid and the steady temperature distribution remains non‐linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16 :347–352). The central contribution of this paper lies in a simpler proof of non‐linear stability, than the ones in the current literature, by use of a suitable maximum principle argument. Copyright © 2006 John Wiley & Sons, Ltd.