Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients
Xiao Wang Zhaohui Yang Xiongwei Liu

It is extremely difficult to establish the existence of almost periodic solutions for delay differential equations via methods that need the compactness conditions such as Schauder's fixed point theorem. To overcome this difficulty, in this paper, we employ a novel technique to construct a contraction mapping, which enables us to establish the existence of almost periodic solution for a delay differential equation system with time-varying coefficients. When the system's coefficients are periodic, coincide degree theory is used to establish the existence of periodic solutions. Global stability results are also obtained by the method of Liapunov functionals.

keywords: Almost periodic oscillation periodic oscillation global stability Liapunov functional delay differential equation
Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza
Olivia Prosper Omar Saucedo Doria Thompson Griselle Torres-Garcia Xiaohong Wang Carlos Castillo-Chavez
The lessons learned from the 2009-2010 H1N1 influenza pandemic, as it moves out of the limelight, should not be under-estimated, particularly since the probability of novel influenza epidemics in the near future is not negligible and the potential consequences might be huge. Hence, as the world, particularly the industrialized world, responded to the potentially devastating effects of this novel A-H1N1 strain with substantial resources, reminders of the recurrent loss of life from a well established foe, seasonal influenza, could not be ignored. The uncertainties associated with the reported and expected levels of morbidity and mortality with this novel A-H1N1 live in a backdrop of $36,000$ deaths, over 200,000 hospitalizations, and millions of infections (20% of the population) attributed to seasonal influenza in the USA alone, each year. So, as the Northern Hemisphere braced for the possibility of a potentially "lethal" second wave of the novel A-H1N1 without a vaccine ready to mitigate its impact, questions of who should be vaccinated first if a vaccine became available, came to the forefront of the discussion. Uncertainty grew as we learned that the vaccine, once available, would be unevenly distributed around the world. Nations capable of acquiring large vaccine supplies soon became aware that those who could pay would have to compete for a limited vaccine stockpile. The challenges faced by nations dealing jointly with seasonal and novel A-H1N1 co-circulating strains under limited resources, that is, those with no access to novel A-H1N1 vaccine supplies, limited access to the seasonal influenza vaccine, and limited access to antivirals (like Tamiflu) are explored in this study. One- and two-strain models are introduced to mimic the influenza dynamics of a single and co-circulating strains, in the context of a single epidemic outbreak. Optimal control theory is used to identify and evaluate the "best" control policies. The controls account for the cost associated with social distancing and antiviral treatment policies. The optimal policies identified might have, if implemented, a substantial impact on the novel H1N1 and seasonal influenza co-circulating dynamics. Specifically, the implementation of antiviral treatment might reduce the number of influenza cases by up to 60% under a reasonable seasonal vaccination strategy, but only by up to 37% when the seasonal vaccine is not available. Optimal social distancing policies alone can be as effective as the combination of multiple policies, reducing the total number of influenza cases by more than 99% within a single outbreak, an unrealistic but theoretically possible outcome for isolated populations with limited resources.
keywords: Control Theory Seasonal Influenza Basic Reproductive Number.
A numerical study of three-dimensional droplets spreading on chemically patterned surfaces
Hua Zhong Xiao-Ping Wang Shuyu Sun
We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and contact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.
keywords: two-phase flow. patterned surfaces asymptotic behavior contact line Contact angle hysteresis
Ning Ju Xiaoming Wang Jiahong Wu
Fluid phenomena are ubiquitous ranging from small scale blood flows in our body to large scale geophysical flows such as the Gulf Stream. The understanding of these phenomena is crucial to many applied areas such as meteorology, oceanography and aerospace industry. Partial Differential Equations (PDEs) are the most fundamental tools in studying fluid phenomena. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis and numerics of partial differential equations (PDEs) modeling fluids and complex fluids. It consists of twenty-one papers from invited speakers of a special sessions in the 7th AIMS Conference on Dynamical Systems and Differential Equations held at the University of Texas at Arlington, Texas, May 18 - 21, 2008.
   Topics covered by this collection of papers include the analysis and computations of solutions of PDEs modeling surface water waves, Navier-Stokes Equations and related ones including those modeling complex fluids and large scale geophysical flows. Some other related issues in fluid dynamics are also touched. The papers in this special issue are not grouped according to their topics but instead ordered alphabetically by the names of the first authors.

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Simulations of 3-D domain wall structures in thin films
Xiao-Ping Wang Ke Wang Weinan E
Using the Gauss-Seidel projection method developed in [4] and [17], we simulate the three dimensional domain wall structures for thin films at various thickness. We observe transition from Néel wall to cross-tie wall and to Bloch wall as the thickness is increased. Periodic structures for cross-tie wall are also studied. The results are in good agreement with the experimental observations. Hysteresis loops are calculated for samples of various sizes. In particular, we study the effect of cross-tie wall in the switching process. These simulations have demonstrated high efficiency of the Gauss-Seidel projection method.
keywords: Magnetic domain wall Gauss-Seidel projection method.
A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems
Xiaocai Wang Junxiang Xu Dongfeng Zhang
In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, andprove that if the frequency mapping
$ω(y) ∈ \mathbb{R}^n$
and normal frequency mapping
$λ(y) ∈ \mathbb{R}$
satisfy that
$\text{deg} (ω/λ ,\mathcal{O},ω_0/λ_0)≠ 0,$
$ω_0 =ω(y_0)$
$λ_0 = λ(y_0)$
satisfy Melnikov's non-resonance conditions for some
, then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrablereversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].
keywords: KAM iteration lower-dimensional tori topological degree
Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus
Xiaojie Wang
This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in the context of truncated SPDEs. The obtained formula that enjoys the absence of the irregular term involved with the unbounded operator is then applied to a parabolic SPDE. Under certain mild assumptions on the nonlinearity, we treat a full discretization based on the spectral Galerkin spatial approximation and provide an easy weak error analysis, which does not rely on Malliavin calculus.
keywords: exponential Euler scheme weak convergence. additive noise Semi-linear stochastic partial differential equations
New asymptotic analysis method for phase field models in moving boundary problem with surface tension
Jie Wang Xiaoqiang Wang
In this paper, we give an asymptotic analysis of the phase field Allen-Cahn and Cahn-Hilliard models of free surfaces with surface tension. Unlike the traditional approach that approximates the solution by the so-called matched asymptotic expansion involving outer expansion, inner expansion and matching, our new approach utilizes a uniform double asymptotic expansion to expand the whole phase field function directly. Although the main result is not new, we would like to emphasize that we derive the result under a uniform double asymptotic expansion. Thus, in this paper the detailed structure of the phase field functions in the equilibrium state is obtained, and the consistency of the phase field models with the corresponding sharp interface models is discussed, including the free surface Allen-Cahn model, Cahn-Hilliard model, and the Allen-Cahn model with volume constraint. The explicit asymptotic expansion of the phase field function reveals rich details of its structures. Moreover, it nicely explains some unusual phenomena we observed in numerical experiments. The theory introduced in this paper can be applied to guide the future modeling and simulation of other moving boundary problems by phase field models.
keywords: Allen-Cahn equation Hele-Shaw problem double asymptotic expansion asymptotic analysis Cahn-Hilliard equation numerical simulation Phase field mean curvature flow approximation.
Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control
Bangyu Shen Xiaojing Wang Chongyang Liu
In fed-batch culture, feeding substrates is to provide sufficient nutrition and reduce inhibitions simultaneously for cells growth. Hence, when and how much to feed substrates are important during the process. In this paper, a nonlinear impulsive controlls system, in which the volume of feeding is taken as the control function, is proposed to formulate the fed-batch fermentation process.In the system, both impulsive moments and jumps size of state are state-dependent. Some important properties of the system are investigated. To maximize the concentration of target product at the terminal time, an optimal control model involving the nonlinear state-dependent impulsive controlled system is presented.The optimal control problem is subject to the continuous state inequality constraint and the control constraint. The existence of optimal control is also obtained. In order to derive the optimality conditions, the optimal control model is transcribed into an equivalent one by treating the constraints. Finally, the optimality conditions of the optimal control model are obtained via calculus of variations.
keywords: Nonlinear state-dependent impulsive controlled system; optimal control; optimality conditions; Fed-batch culture.
Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations
Alain Miranville Xiaoming Wang
Our aim in this article is to derive an upper bound on the dimension of the attractor for Navier-Stokes equations with nonhomogeneous boundary conditions. In space dimension two, for flows in general domains with prescribed tangential velocity at the boundary, we obtain a bound on the dimension of the attractor of the form $c\mathcal{R} e^{3/2}$, where $\mathcal{R} e$ is the Reynolds number. This improves significantly on previous bounds which were exponential in $\mathcal{R} e$.
keywords: energy dissipation rate Taylor-Couette flows Hausdorff and fractal dimensions global attractor Reynolds number. channel flows background flows functional invariant set Navier-Stokes equations

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