Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.     

Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case

We prove the existence of solutions for a Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space C_γ(H). Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

On the HSL-flow

We introduce a natural geometric fourth order flow associated to Hamiltonian stationary submanifolds in Kähler–Einstein manifolds. Afterwards we study some of its properties and show short-time existence. In case of Hamiltonian stationary submanifolds with bounded second fundamental form evolving in flat space we obtain an existence time estimate (Theorem 3.7).     

On the convergence of sequences of stationary jump Markov processes

This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups T_N(t) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (T_N(t), π_N) to the stationary limit one.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Existence of optimal stationary policies in deterministic optimal control

This paper considers deterministic discrete-time optimal control problems over an infinite horizon involving a stationary system and a nonpositive cost per stage. Various results are provided relating to existence of an ?-optimal stationary policy, and existence of an optimal stationary policy assuming an optimal policy exists.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u ₀ satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u ₀. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.     

Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case

We perform a qualitative investigation of critical Hamilton–Jacobi equations, with stationary ergodic Hamiltonian, in dimension 1. We show the existence of approximate correctors, give characterizing conditions for the existence of correctors, provide Lax-type representation formulae and establish comparison principles. The results are applied to look into the corresponding effective Hamiltonian and to study a homogenization problem. In the analysis a crucial role is played by tools from stochastic geometry such as, for instance, closed random stationary sets.     

STATIONARY OSCILLATIONS OF SECOND ORDER PERIODIC SYSTEMS UNDER STOCHASTIC NOISE INTERFERENCE

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An asymptotic‐induced one‐dimensional model to describe fires in tunnels II: the stationary problem

We study stationary solutions of a one‐dimensional low‐Mach‐number model derived in Gasser and Struckmeier (Math. Meth. Appl. Sci. 2002; 25 (14): 1231) to describe fire events in long tunnels. The existence of solutions of the corresponding stationary model is shown to be equivalent to the existence of solutions of an algebraic problem. Multiple solutions are shown to be possible. The relation between different formulations of the problem is analysed. Weak and special distributional solutions are considered. Finally, numerical examples of realistic tunnel data with single and multiple solutions of the stationary problem are given. Copyright © 2003 John Wiley & Sons, Ltd.     

A Penalty Approach to Optimal Control of Allen-Cahn Variational Inequalities: MPEC-View

A scalar Allen-Cahn-MPEC problem is considered and a penalization technique is applied to show the existence of an optimal control. We show that the stationary points of the penalized problems converge to some stationary points of the limit problem, which however are weaker than C-stationarity conditions.     

Foliations by stationary disks of almost complex domains

We study the problem of existence of stationary disks for domains in almost complex manifolds. As a consequence of our results, we prove that any almost complex domain which is a small deformation of a strictly linearly convex domain D⊂Cn with standard complex structure admits a singular foliation by stationary disks passing through any given internal point. Similar results are given for foliations by stationary disks through a given boundary point.     

被捕食者带有第三边值的捕食模型的正稳态解的存在性

该文研究了一类被捕食者带有第三边值的捕食模型. 首先获得了它存在正稳态解的充要条件是a>mb+d₁λ₁; 然后研究了它的正稳态解的局部稳定性和唯一性;最后讨论了充分大的扩散参数对它的正稳态解的存在性的影响.     

Equivalence of Lyapunov stability criteria in a class of Markov decision processes

We are concerned with Markov decision processes with countable state space and discrete-time parameter. The main structural restriction on the model is the following: under the action of any stationary policy the state space is acommunicating class. In this context, we prove the equivalence of ten stability/ergodicity conditions on the transition law of the model, which imply the existence of average optimal stationary policies for an arbitrary continuous and bounded reward function; these conditions include the Lyapunov function condition (LFC) introduced by A. Hordijk. As a consequence of our results, the LFC is proved to be equivalent to the following: under the action of any stationary policy the corresponding Markov chain has a unique invariant distribution which depends continuously on the stationary policy being used. A weak form of the latter condition was used by one of the authors to establish the existence of optimal stationary policies using an approach based on renewal theory.This research was supported in part by the Third World Academy of Sciences (TWAS) under Grant TWAS RG MP 898-152.     

The Burgers equation with affine linear noise: Dynamics and stability

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.     

Stationary points and equilibria

Almost all existence results in mathematical economics and game theory rely on some type of fixed point theorem. However, in many cases it is much easier to apply an equivalent stationary point theorem. In this survey paper we show for various general equilibrium and game-theoretical models the usefulness of the stationary point approach to prove existence of equilibrium. We also consider path following methods to find a stationary point and illustrate such methods by an example. Finally, we discuss some refinements of stationary points and their application to solve the problem of equilibrium selection.     

Periodic Solution and Stationary Distribution of Stochastic Predator–Prey Models with Higher-Order Perturbation

In this paper, two stochastic predator–prey models with general functional response and higher-order perturbation are proposed and investigated. For the nonautonomous periodic case of the system, by using Khasminskii’s theory of periodic solution, we show that the system admits a nontrivial positive T-periodic solution. For the system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent.