Small mass asymptotic for the motion with vanishing friction

We consider the small mass asymptotic (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.     

Langevin equation of a fluid particle in wall-induced turbulence

We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-induced turbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer). The analysis is based on the asymptotic behavior at a large Reynolds number. We use the Lagrangian Kolmogorov theory, recently derived asymptotic expressions for the spatial distribution of turbulent energy dissipation, and also newly derived reciprocity relations analogous to the Onsager relations supplemented with recent measurement results. The long-time limit of the derived Langevin equation yields the diffusion equation for admixture dispersion in wall-induced turbulence.     

Smoluchowski–Kramers approximation in the case of variable friction

We consider the small mass asymptotics (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski–Kramers approximation. Some applications of the Smoluchowski–Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered. Bibliography: 15 titles.     

A simple 1D model of inviscid fluid-solid interaction

We analyze a one-dimensional fluid-particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

Analysis of the backward-euler/langevin method for molecular dynamics

This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward-Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ω_c, which may be set equal to kT/h to simulate quantum-mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward-Euler/Langevin method is considered, an integral equation for the equilibrium phase-space density is derived, and an asymptotic analysis of that integral equation in the limit Δt → 0 is performed. The result of this asymptotic analysis is a second-order partial differential equation for the equilibrium phase-space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.     

Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.where $B,C,D\in C^{d\times d},q\in (0,1)$,and $B$ is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L-stable Runge-Kutta method can preserve the above-mentioned stability properties.     

Asymptotic stability and decay rate of solutions for a nonlinear wave equation with variable coefficients

This paper is concerned with investigating the global asymptotic behavior of the solution to a nonlinear wave equation with variable coefficients. Moreover an estimate of the rate of decay of the solution is obtained.     

On a reaction–advection–diffusion equation with Robin and free boundary conditions

ABSTRACT

A reaction–advection–diffusion equation with variable intrinsic growth rate, Robin and free boundary conditions is investigated in this paper. Firstly, we present a spreading–vanishing dichotomy for the asymptotic behavior of the solutions of the equation. Then, we obtain criteria for spreading and vanishing, and get an estimate for the asymptotic spreading speed of the spreading front. Moreover, numerical simulation is also given to illustrate the impact of the expansion capacity on the free boundary.     

三阶非线性中立型变时滞动力方程的渐近性与振动性(英文)

考虑了时标上三阶非线性中立型变时滞动力方程的渐近性和振动性.利用广义Ric-cati技巧与完全平方技巧,获得了方程所有解渐近和振动的准则.所得结果推广了已有三阶动力方程的结果,给出了一些例子加以说明本文的主要结论.     

On a nonlocal diffusion model with Neumann boundary conditions

We study a nonlocal diffusion model analogous to heat equation with Neumann boundary conditions. We prove the existence and uniqueness of solutions and a comparison principle. Furthermore, we analyze the asymptotic behavior of the solutions as the temporal variable goes to infinity and the boundary datum depends only on a spacial variable.     

Remarks on the beam evolution equations in noncylindrical domains

In this article, we present results concerning the existence, uniqueness and the asymptotic behavior of solutions for a beam evolution equation with variable coefficients in noncylindrical domains.     

On the one‐dimensional relativistic induction equation

The induction equation of relativistic magnetohydrodynamics is considered as a singular perturbation problem for small magnetic diffusivity. When the quantities depend on a single space variable, the resulting hyperbolic equation may be studied with techniques of asymptotic analysis. Different approximations are found for initial, intermediate, and large times. The last case is the most difficult; the approximate magnetic flux function satisfies a certain parabolic equation. This equation is studied from the viewpoint of energy dissipation, providing clues on the behavior of the electric and magnetic fields. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions

We investigate a nonlinear autonomous parabolic partial differential equation in one space variable subject to Neumann boundary conditions on a compact interval. The object of our study is to determine the asymptotic behavior of solutions. Our methods are borrowed from the Liapunov theory of stability for dynamical systems. We give conditions under which a solution has a nonempty ω-limit set. We show that any such ω-limit set consists solely of equilibrium solutions. We render criteria for asymptotic stability and for instability of an equilibrium solution. We examine the possibility of escape behavior.     

Connection between the Fokker-Planck-Kolmogorov and nonlinear Langevin equations

We recall the general proof of the statement that the behavior of every holonomic nonrelativistic system can be described in terms of the Langevin equation in Euclidean (imaginary) time such that for certain initial conditions, the different stochastic correlators (after averaging over the stochastic force) coincide with the quantum mechanical correlators. The Fokker-Planck-Kolmogorov (FPK) equation that follows from this Langevin equation is equivalent to the Schrödinger equation in Euclidean time if the Hamiltonian is Hermitian, the dynamics are described by potential forces, the vacuum state is normalizable, and there is an energy gap between the vacuum state and the first excited state. These conditions are necessary for proving the limit and ergodic theorems. For three solvable models with nonlinear Langevin equations, we prove that the corresponding Schrödinger equations satisfy all the above conditions and lead to local linear FPK equations with the derivative order not exceeding two. We also briefly discuss several subtle mathematical questions of stochastic calculus.     

Asymptotic behavior on the Milne problem with a force term

This paper studies the existence, uniqueness and asymptotic behavior of the solution for a half-space linearized stationary Boltzmann equation with an external force term, in the case of a specified incoming distribution at the boundary and a given mass flux. Without the external force, the solution of the stationary Boltzmann equation has been proved to tend toward a constant state, which is independent of the space variable. Due to the presence of the external force, we show that the solution tends to some function which depends on the space variable.     

Maximum likelihood estimation for small noise multiscale diffusions

We study the problem of parameter estimation for stochastic differential equations with small noise and fast oscillating parameters. Depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter, we consider three different regimes. For each regime, we construct the maximum likelihood estimator and we study its consistency and asymptotic normality properties. A simulation study for the first order Langevin equation with a two scale potential is also provided.     

Asymptotic preserving HLL schemes

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1396–1422, 2011     

一类具有边界摄动的非线性泛函椭圆型方程奇摄动问题

研究了具有边界摄动的非线性泛函椭圆型方程奇摄动边值问题．在适当的条件下．利用伸长变量、微分不等式理论，讨论了问题解的渐近性态和原问题解的存在唯一性。