This paper is concerned with the persistence and extinction of a randomized non- autonomous logistic system. Su1cient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The critical value between weak persistence and extinction is obtained. The behaviors of the system in every coe1cient case are studied.

We consider a new model for vertical vibrations of an elastic string fixed at the ends. When the tension on the string is not constant, the Kirchhoff model is extensively investigated. In the present paper we obtain a model which is a perturbation of Kirchhoff's model. We prove that for every T > 0 a mixed problem for this new model is well posed in the interval [0, T), with restriction on the initial data, which depend on T. We apply the Galerkin method, multiplier techniques and compactness results to obtain the existence and uniqueness. For the numerical solution, we employ the finite element method and also introduce an implicit time discretization. Some numerical examples are presented to validate the numerical method and numerical experiments are presented to compare with the Kirchhoff model and the effect of coefficients in the string vibration.

We consider the evolution of the separation distance between two particles advected by a random velocity field with slowly decaying temporal and spatial correlations in the weak coupling regime. It has been shown in [A. Fannjiang and T. Komorowski, Ann. Appl. Probab. 10, 1100-1120, 2000] that the motion of a single particle converges to a fractional Brownian motion on a time scale $\delta^{-\gamma}$ with some $\gamma<2$, which is shorter than the classical diusive time scale $\delta^{-2}$ (see [H. Kesten, G. C. Papanicolaou, Commun. Math. Phys. 65, 97-128, 1979.]). In the present paper we prove that unlike the single particle position, the two-particle separation behaves diffusively, and evolves on the classical time scale $\delta^{-2}$, even when the random flow is slowly decorrelating in time and space. The results of this paper illustrate that the flows under consideration display both diffusive and superdiffusive transport on different time scales for various physical quantities.

We study large-time asymptotic behavior of classical solutions to an initial boundary value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on a bounded domain. Sufficient conditions are established under which classical solutions converge exponentially to constant states as time goes to infinity due to diffusion and boundary effects.

We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Using the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can "asymptotically" be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved with the Poincar4e-Bendixon theorem or by Hopf bifurcation.

This paper introduces a fast spectral algorithm for the quantum Boltzmann collision operator. In the usual spectral framework, one of the terms in the operator cannot be eval- uated efficiently. The new approach is based on the fundamental property of the exponential function which allows one to construct a new decomposition of the collision kernel to speed up the computation. Numerical results in 2-D and 3-D for both the Bose gas and Fermi gas are presented to illustrate the accuracy and efficiency of the method.

In this paper, we study the asymptotic behaviors of solution for two-dimensional stochastic non-Newtonian fluids with multiplicative noise. In particular, we prove the existence of random attractors in H under the condition 2< p<3.

A new model is introduced for describing the heat-conducting viscous uids over porous media. The innovative features of the presented model are the strongly nonlinear character given by temperature dependence of the physical parameters such as the viscosities, the permeability and complementary the thermal conductivity and thermal expansion. The ow velocities are small (for steady processes) and mainly driven by the pressure gradient in porous medium such that the Stokes-Darcy system is completed by the energy equation with the heat ux given by the Fourier law. The existence of solutions is established for the Stokes-Darcy-Fourier system either with the Beavers-Joseph-Saman or Beaver-Joseph interface boundary conditions. Both problems are solved by means a xed point procedure and Lagrange multiplier approach.

We consider time-continuous, reversible Markov processes on large or continuous state space. For a practical analysis of such processes it is often necessary to construct low dimensional approximations, like Markov State Models (MSM). MSM have been used for this purpose in several applications, particularly in molecular dynamics, see [16] for an example. One of the main goals of MSMs is the correct approximation of slow processes in the system. Recently, it was possible to understand under which conditions a MSM inherits the most dominant timescales of the original Markov process. However, all rigorous statements known have yet been concerned with the approximation of the absolutely slowest processes in the system, i.e., its dominant timescales. In this article, we will show that it is also possible to design MSMs to reproduce selected non-dominant timescales and which approximation quality can be achieved.

Solutions of constant-coeffcient nonlinear hyperbolic PDEs generically develop shocks, even if the initial data is smooth. Solutions of hyperbolic PDEs with variable coefficients can behave very differently. We investigate formation and stability of shock waves in a one-dimensional periodic layered medium by computational study of time-reversibility and entropy evolution. We find that periodic layered media tend to inhibit shock formation. For small initial conditions and large impedance variation, no shock formation is detected even after times much greater than the time of shock formation in a homogeneous medium. Furthermore, weak shocks are observed to be dynamically unstable in the sense that they do not lead to signicant long-term entropy decay. We propose a characteristic condition for admissibility of shocks in heterogeneous media that generalizes the classical Lax entropy condition and accurately predicts the formation or absence of shocks in these media.

The aim of this article is to derive a simplified sedimentation model thanks to an asymptotic analysis. We consider a two time scale erosion process due to tidal effects and we show that the approximation at the first order can model bed-load transport well. To this end, the simplified model is validated on numerical tests (evolution of a dune submitted to tidal effects in the ocean, run up near the coast) and compared to direct simulations that are very expensive in terms of computation time.

We propose transparent boundary conditions (TBCs) for the time–dependent Schroddinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson finite difference scheme. The presented discrete initial boundary-value problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schroddinger equation with a spatially dependent potetial include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method.

Traveling near-bichromatic solutions supported at resonant frequencies are computed in a family of nonlinear model equations. Both the `Wilton's Ripple', where a wave resonates with its rst harmonic, as well higher harmonic resonances are computed. Wilton's expansion is calculated for these solutions to all orders, and used as the basis for a perturbative numerical method for computing near-bichromatic traveling waves. The perturbative numerical method relies on the analyticity of solutions with respect to wave amplitude. A non-perturbative continuation method is used both to verify convergence of the perturbative method and to compute large amplitude solutions which are outside the radius of convergence.

We establish existence of periodic travelling-wave solutions to a generalized Boussi- nesq system by using the topological degree theory for positive operators defined on a cone in an appropriate Banach space. Furthermore, we derive a high-accurate pseudospectral solver based on a Fourier decomposition to construct numerical approximations of these stationary solutions. The numerical simulations are in perfect agreement with the theoretical results and new travelling-wave solutions of the system are computed which do not belong to the family of solutions proved to exist.

We develop a multiscale tailored finite point method (MsTFPM) for second order elliptic equations with rough or highly oscillatory coeffcients. The finite point method has been tailored to some particular properties of the problem, so that it can capture the multiscale solutions using coarse meshes without resolving the fine scale structure of the solution. Several numerical examples in one- and two-dimensions are provided to show the accuracy and convergence of the proposed method. In addition, some analysis results based on the maximum principle for the one-dimensional problem are proved.

A hydrodynamic model of open vesicles in solution is presented to study the enlargement and shrinkage of a pore in biological lipid membrane. The vesicle is modeled by diffusive interfaces. Transport equations permitting consistent treatment of the pore and pore rim are intro- duced. Dynamic simulations implemented by the finite difference method show the evolution of a pore in stretched vesicles. Simulation results include direct visualization of the membrane shape, water motion, and dissipation of energy. Comparison is made with data obtained from microscopy experiments.

Recently, liquid-gas flows involved in droplets, bubbles, and thin films on solid surfaces with thermal and wettability gradients have attracted widespread attention because of the many physical processes involved and their promising potential for applications in biology, chemistry, and industry. Various new physical effects have been discovered at fluid-solid interfaces by experiments and molecular dynamics simulations, e.g., fluid velocity slip, temperature slip (Kapitza resistance), mechanical-thermal cross coupling, etc. There have been various models and theories proposed to explain the experimental and numerical observations. However, to the best of our knowledge, a continuum hydrodynamic model capable of predicting the temperature and velocity profiles of liquid-gas flows on non-isothermal, heterogeneous solid substrates is still absent. The purpose of this work is to construct a continuum model for simulating the liquid-gas flows on solid surfaces that are flat and rigid, and may involve wettability gradient and thermal gradient. This model is able to describe fluid velocity slip, temperature slip, and mechanical- thermal coupling that may occur at fluid-solid interfaces. For this purpose, we first employ the diffuse interface modeling to formulate the hydrodynamic equations for one-component liquid- gas flows in the bulk region. This reproduces the dynamic van der Waals theory of Onuki [Phys. Rev. Lett., 94: 054501, 2005]. We then extend s method [Z. Naturforsch. A, 22: 1269-1280, 1967] to formulate the boundary conditions at the fluid-solid interface that match the hydrodynamic equations in the bulk. The effects of the solid surface curvature are also briefly discussed in the appendix. The guiding principles of our model derivation are the conservation laws and the positive definiteness of entropy production together with the Onsager reciprocal relation. The derived model is self-consistent in the sense that the boundary conditions are mathematically demanded by the bulk equations. A finite difference scheme is presented for numerically solving the model system. We show that some widely used boundary conditions can actually be recovered by taking appropriate limits. We also point out that the framework presented here for modeling two-phase flows on solid surfaces, from bulk equations to boundary conditions, is in a form that can be readily generalized to model other fluid-solid interfacial phenomena.

The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H^{-1}. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated, in the one-dimensional case, for the piecewise constant finite element approximation of the porous medium equation with the \delta-distribution as initial value. As a byproduct, L^p-stability of the H^{-1}-orthogonal projection onto the space of piecewise constant functions is shown for the one-dimensional case.

In this paper, we compose explicit two-stage Runge-Kutta schemes of strong order one for solutions of stochastic dierential equations driven by jump-diusion processes. By using rooted trees, we obtain the convergence rate. Our numerical tests verify our theoretical results.

We derive a continuum model for the dynamics of a dislocation array that consists of dislocations in different slip planes. In the continuum model, the dislocation array is represented by a continuous surface, of which there are many dislocations in a unit area at the scale of the continuum model. The continuum model is derived rigorously from the discrete model of the dynamics of the constituent dislocations in the array using asymptotic analysis. The obtained continuum model contains an integral over the dislocation array surface representing the long-range interaction of dislocations, and a local term that comes from the line tension effect of dislocations. The size-dependent effect due to dislocation line tension is accurately incorporated in the continuum model. Well-posedness of the continuum model is examined. Generalization to dislocation arrays in an elastically anisotropic medium is discussed.

We consider the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic Cahn-Hilliard dynamics through nite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios.