arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 253 records · Page 14Linked to original sources

arXiv2004

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or {\em exponential} with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.\\ We apply our results to the 2D Navier - Stokes equations forced by a white noise.

math.DS↗

arXiv2004

Invariant manifolds for stochastic partial differential equations

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional {\em random} dynamical systems generated by {\em stochastic} partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

math.DS↗

arXiv2004

Smooth stable and unstable manifolds for stochastic partial differential equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds.

math.DS↗

arXiv2004

Exponential Stability Of The Quasigeostrophic Equation Under Random Perturbations

he quasigeostrophic model describes large scale and relatively slow fluid motion in geophysical flows. We investigate the quasigeostrophic model under random forcing and random boundary conditions. We first transform the model into a partial differential equation with random coefficients. Then we show that, under suitable conditions on the random forcing, random boundary conditions, viscosity, Ekman constant and Coriolis parameter, all quasigeostrophic motion approach a unique stationary state exponentially fast. This stationary state corresponds to a unique invariant Dirac measure

math.AP↗

arXiv2004

Generalization of the Second Bogolyubov's Theorem for Non-Almost Periodic Systems

The article is devoted to the generalization of the second Bogolyubov's theorem to non-almost periodic dynamical systems. We prove the analog of the second Bogolyubov's theorem for recurrent or pseudo recurrent dynamical systems in Banach spaces. Namely, we obtain the relation between a recurrent dynamical system and its averaged dynamical system. We also study existence of recurrent and pseudo recurrent motions (including special cases of periodic, quasi-periodic and almost periodic motions) in related nonautonomous systems.

math.DS↗

arXiv2004

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier-Stokes Equations

The article is devoted to the study of non-autonomous Navier-Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier-Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic,almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier-Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier-Stokes equations.

math.DS↗

arXiv2004

Dynamics of a Coupled Atmosphere-Ocean Model

We consider a coupled atmosphere-ocean model, which involves hydrodynamics, thermodynamics and nonautonomous interaction at the air-sea interface. First, we show that the coupled atmosphere-ocean system is stable under the external fluctuation in the atmospheric energy balance relation. Then, we estimate the atmospheric temperature feedback in terms of the freshwater flux, heat flux and the external fluctuation at the air-sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Finally, we prove that the coupled atmosphere-ocean system has time-periodic, quasiperiodic and almost periodic motions, whenever the external fluctuation in the atmospheric energy balance relation is time-periodic, quasiperiodic and almost periodic, respectively.

math.DS↗

arXiv2004

Ergodic Dynamics of the Stochastic Swift-Hohenberg System

The Swift-Hohenberg fluid convection system with both local and nonlocal nonlinearities under the influence of white noise is studied. The objective is to understand the difference in the dynamical behavior in both local and nonlocal cases. It is proved that when sufficiently many of its Fourier modes are forced, the system has a unique invariant measure, or equivalently, the dynamics is ergodic. Moreover, it is found that the number of modes to be stochastically excited for ensuring the ergodicity in the local Swift-Hohenberg system depends {\em only} on the Rayleigh number (i.e., it does not even depend on the random term itself), while this number for the nonlocal Swift-Hohenberg system relies additionally on the bound of the kernel in the nonlocal interaction (integral) term, and on the random term itself

math.DS↗

arXiv2004

A Markov jump process approximation of the stochastic Burgers equation

We consider the stochastic Burgers equation $ \dnachd{t} ψ(t,r) = Δψ(t,r) + \nabla ψ^2(t,r)+\sqrt{γψ(t,r)} η(t,r) $ with periodic boundary conditions, where $t \ge 0,$ $r \in [0,1],$ and $η$ is some space-time white noise. A certain Markov jump process is constructed to approximate a solution of this equation.}

math.DS↗

arXiv2004

Ergodic Dynamics of the Coupled Quasigeostrophic Flow-Energy Balance System

The authors consider a mathematical model for the coupled atmosphere-ocean system, namely, the coupled quasigeostrophic flow-energy balance model. This model consists of the large scale quasigeostrophic oceanic flow model and the transport equation for oceanic temperature, coupled with an atmospheric energy balance model. After reformulating this coupled model as a random dynamical system (cocycle property), it is shown that the coupled quasigeostrophic-energy balance fluid system has a random attractor, and under further conditions on the physical data and the covariance of the noise, the system is ergodic, namely, for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.

math.DS↗

arXiv2004

Enstrophy and Ergodicity of Gravity Currents

We study a coupled deterministic system of vorticity evolution and salinity transport equations, with spatially correlated white noise on the boundary. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain an asymptotic estimate of enstrophy evolution, and then show that the system is ergodic under suitable conditions on mean salinity input flux on the boundary, Prandtl number and covariance of the noise.

math.AP↗

arXiv2001

Dynamics of the Thermohaline Circulation under Wind forcing

The ocean thermohaline circulation, also called meridional overturning circulation, is caused by water density contrasts. This circulation has large capacity of carrying heat around the globe and it thus affects the energy budget and further affects the climate. We consider a thermohaline circulation model in the meridional plane under external wind forcing. We show that, when there is no wind forcing, the stream function and the density fluctuation (under appropriate metrics) tend to zero exponentially fast as time goes to infinity. With rapidly oscillating wind forcing, we obtain an averaging principle for the thermohaline circulation model. This averaging principle provides convergence results and comparison estimates between the original thermohaline circulation and the averaged thermohaline circulation, where the wind forcing is replaced by its time average. This establishes the validity for using the averaged thermohaline circulation model for numerical simulations at long time scales.

math.AP↗

arXiv2001

Enstrophy Dynamics of Stochastically Forced Large-Scale Geophysical Flows

Enstrophy is an averaged measure of fluid vorticity. This quantity is particularly important in {\em rotating} geophysical flows. We investigate the dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under random wind forcing. We obtain upper bounds on the enstrophy, as well as results establishing its Hölder continuity and describing the small-time asymptotics.

math.AP↗

arXiv2001

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results. We apply our results to the 2D Navier - Stokes equations forced by a white noise.

nlin.CD↗

arXiv2001

Ergodicity of Stochastically Forced Large Scale Geophysical Flows

We investigate the ergodicity of 2D large scale quasigeostrophic flows under random wind forcing. We show that the quasigeostrophic flows are ergodic under suitable conditions on the random forcing and on the fluid domain, and under no restrictions on viscosity, Ekman constant or Coriolis parameter. When these conditions are satisfied, then for any observable of the quasigeostrophic flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.

math.AP↗

arXiv2001

Probabilistic Dynamics of Two-Layer Geophysical Flows

The two-layer quasigeostrophic flow model is an intermidiate system between the single-layer 2D barotropic flow model and the continuously stratified, 3D baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. The wind forcing acts only on the top layer. We consider the two-layer quasigeostrophic model under stochastic wind forcing. We first transformed this system into a coupled system of random partial differential equations and then show that the asymptotic probabilistic dynamics of this system depends only on the top fluid layer. Namely, in the probability sense and asymptotically, the dynamics of the two-layer quasigeostrophic fluid system is determinied by the top fluid layer, or, the bottom fluid layer is slaved by the top fluid layer. This conclusion is true provided that the Wiener process and the fluid parameters satisfy a certain condition. In particular, this latter condition is satisfied when the trace of the covariance operator of the Wiener process is controled by a certain upper bound, and the Ekman constant $r$ is sufficiently large. Note that the generalized time derivative of the Wiener process models the fluctuating part of the wind stress forcing on the top fluid layer, and the Ekman constant $r$ measures the rate for vorticity decay due to the friction in the bottom Ekman layer.

math.AP↗

arXiv2000

The 3D Quasigeostrophic Fluid Dynamics under Random Forcing on Boundary

The three-dimensional baroclinic quasigeostrophic flow model has been widely used to study basic mechanisms in oceanic flows and climate dynamics. In this paper, we consider this flow model under random wind forcing and time-periodic fluctuations on fluid boundary (the interface between the oceans and the atmosphere). The time-periodic fluctuations are due to periodic rotation of the earth and thus periodic exposure of the earth to the solar radiation. After establishing the well-posedness of the baroclinic quasigeostrophic flow model in the state space, we demonstrate the existence of the random attractors, again in the state space. We also discuss the relevance of our result to climate modeling.

math.AP↗

arXiv2000

Symbolic Representations of Iterated Maps

This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general subshift over usually an uncountable alphabet. It is shown that at most the second order representation is enough for a continuous map. In particular, it is shown that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study of subshift structure of a general symbolic dynamics system. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps, piecewise continuous maps are discussed, and as examples, a representation of the Gauss map via a full shift over a countable alphabet and representations of interval exchange transformations as subshifts of infinite type are given.

nlin.CD↗