Zhenxin Liu
In this paper, stochastic inertial manifold for damped wave equations subjected to additive white noise is constructed by the Lyapunov-Perron method. It is proved that when the intensity of noise tends to zero the stochastic inertial manifold converges to its deterministic counterpart almost surely.
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Limiting behavior of inertial manifolds for stochastic differential equations driven by non-Gaussian Levy noise
本文研究了由非高斯α-稳定莱维噪声驱动的随机微分方程在α趋近于2时的极限行为，证明了解的收敛性及惯性面的概率收敛性。
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
For 3D reaction--diffusion equations, we study the problem of existence or nonexistence of an inertial manifold that is normally hyperbolic or absolutely normally hyperbolic. We present a system of two coupled equations with a cubic nonlinearity which does not admit a normally hyperbolic inertial manifold. An example separating the classes of such equations admitting an inertial manifold and a normally hyperbolic inertial manifold is constructed. Similar questions concerning absolutely normally hyperbolic inertial manifolds are discussed.
In this paper, we study the Gromov-Hausdorff stability and continuous dependence of the inertial manifolds under perturbations of the domain and equation. More precisely, we use the Gromov-Hausdorff distances between two inertial manifolds and two dynamical systems to consider the continuous dependence of the inertial manifolds and the stability of the dynamical systems on inertial manifolds induced by the reaction diffusion equations under perturbations of the domain and equation.
By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski--Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a…
Linear Algebra Problems Solved by Using Damped Dynamical Systems on the Stiefel Manifold
本文提出一种在斯蒂费尔流形上求解最优化问题的新方法，通过阻尼动力系统满足约束，并通过数值实验与最新共轭梯度法进行比较。
We propose a new notion of Partial Inertial Manifold to study the long-time asymptotic behavior of dissipative differential equations. As shown on an example, such manifolds may exist in the cases when the classical Inertial manifold does not exist (or not known to exist).
In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness of inertial manifolds with respect to small delays for a certain class of neutral differential equations. The method can be also applied to deal with more general small-delay systems. Finally, we apply our main results to the van der Pol oscillator model with small delay.
A numerical approach for the approximation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. This numerical approach is tested in two illustrative examples: one is for a system of stochastic ordinary differential equations and the other is for a stochastic partial differenti…
We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson theory. In the case $n=3$ we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.
A construction of a robust family of compact inertial manifolds is presented. The result aims to complete an analysis of certain types of attracting sets for a class of dissipative infinite dimensional dynamical systems. Application to a hyperbolically relaxed Chaffee-Infante reaction diffusion equation is also discussed.
We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.
In this note we prove the existence of an inertial manifold, i.e., a global invariant, exponentially attracting, finite-dimensional smooth manifold, for two different sub-grid scale $α$-models of turbulence: the simplified Bardina model and the modified Leray-$α$ model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of the large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differe…
A method is provided for approximating random slow manifolds of a class of slow-fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon…
In this article, we devote to the existence of an $N$-dimensional inertial manifold for the incompressible Navier-Stokes equations in $\mathbb{T}^{d}$ ($d=2,3$). Our results can be summarized as two aspects: Firstly, we construct an $N$-dimensional inertial manifold for the Navier-Stokes equations in $\mathbb{T}^{2}$; Secondly, we extend slightly the spatial averaging method to the abstract case: $\partial_{t}u+A^{1+α}u+A^αF(u)=f$ (here $0<α<1$, $A>0$ is a self-adjoint operator with compact inverse and $F$ is Lipschitz from a Hilbert space $\mathbb{H}$ to $\mathbb{H}$), and then verify the exi…
Smoothness of Inertial Manifold for the Burgers Equation
本文研究Burgers方程惯性流形的光滑扩展，通过新框架证明其长期行为可由显式光滑一阶ODE确定，提出构造C^{n,ε}光滑扩展的充分条件。
In this paper we obtain estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we estimate the distance in terms of the distance of the resolvent operators of the corresponding elliptic operators and the distance of the nonlinearities of the equations.
$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems
本文用基于单个解的 Lyapunov–Perron 方法刻画非自治系统的稳定与惯性流形，给出统一间隙条件并证明所得惯性流形为 $C^1$。
We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by a family of indefinite quadratic Lyapunov-like functionals) and asymptotic compactness. Under such conditions it is possible to reconstruct foliations as in the theory of normally hyperbolic manifolds. Our approach allows to unify many "practical" theories of local and nonlocal invariant manifolds, such as stable/unstable/center manifolds and inertial manifolds, and, moreover, it often leads to op…
Inertial manifolds for the nonlocal parabolic problem
本文针对一类非局部抛物问题，提出研究其惯性流形的抽象框架，通过调整非局部项得到谱间隙条件，成功证明了正方形域上二维非局部反应扩散方程的惯性流形存在性。
Inertial manifolds via spatial averaging: a control-theoretic perspective
本文从控制论视角建立泛函分析工具，在经典频率条件及空间平均原理下证明相关非自治 Hamiltonian 系统稳定 Lagrange 丛的存在性与一致非振荡性，以服务惯性流形研究。
A unified approach to compute foliations, inertial manifolds, and tracking initial conditions
提出基于压缩映射的统一算法，用于精确计算双曲不动点附近ODE的叶状结构、惯性流形及追踪初始条件，并在Kuramoto-Sivashinsky方程上验证。
Inertial manifolds for the two-dimensional hyperviscous Navier-Stokes equations
本研究针对二维周期域高粘性Navier-Stokes方程，确定惯性流形问题临界指数β=3/2，通过格点分布精细分析将理论拓展至β<3/2的超临界区，证明了β>17/12时惯性流形的存在性。
In this paper we study a family of semilinear reaction-diffusion equations on thin spatial domains, lying close to a lower dimensional submanifold $M$. As the thickness tends to zero, the domains collapse onto (a subset of) $M$. As it was proved in a previous paper (M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Stud. Math. 151), the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain abstract limit phase space. One of the objectives of this paper is to give more manageable characterizations of the…
Frequency theorem for parabolic equations and its relation to inertial manifolds theory
本文提出抛物型方程的频域定理，用于构造二次李雅普诺夫泛函，证明谱间隙条件是频域不等式特例，并推广惯性流形理论。
Slow Manifold Reduction for Inertial Particles with Quadratic Drag
研究高雷诺数下惯性粒子动力学中临界流形持续性问题，用爆破方法转换方程去奇异化，证明其在小扰动下持续并推导约化方程，通过例子展示有效性及在湍流扩散模型源反演中的应用。
The spectral concentration for damped waves on compact Anosov manifolds
本文研究紧致 Anosov 流形上阻尼波的特征值分布，利用 Anosov 测地流的中偏差原理证明大多数特征值虚部以对数速度集中于阻尼函数平均值附近，并将其应用于扭曲 Selberg zeta 函数零点的集中性。
An improved approach for estimating the dimension of inertial manifolds in chaotic distributed dynamical systems via analysis of angles between tangent subspaces
本文提出一种通过分析切空间角度改进的惯性流形维度估计方法，该方法减少计算资源需求并提高效率，适用于复杂动态系统的高维吸引子表征。
Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds
提出利用谱子流形（SSM）降低混沌系统维度的数据驱动方法，SSM作为低维惯性流形能准确预测短期混沌动力学并重现长期统计特征，在多个模型及屈曲梁实验中验证了其预测能力。
Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension estimates
本文在希尔伯特空间中研究时滞方程，构造非线性半群与求解算子，方法更初等，并探讨维数估计与惯性流形。
We rigorously derive multi-pulse interaction laws for the semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a "normal-hyperbolicity" condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the lineariz…
Frequency theorem and inertial manifolds for neutral delay equations
本文针对希尔伯特空间中无界代价的二次调节器问题，通过频率定理证明最优过程存在且代价由二次泛函给出，并应用于延迟方程，为构造惯性流形提供统一框架。
Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
该研究提出惯性流形神经算子（IMNO）及其平移等变变体IMNO-SE，利用耗散型时变PDE的低维结构提升长时预测性能，通过基准实验验证了其优势。
Let $Ω$ be an arbitrary smooth bounded domain in $\R^2$ and $ε>0$ be arbitrary. Squeeze $Ω$ by the factor $ε$ in the $y$-direction to obtain the squeezed domain $Ω_ε=\{(x,εy)\mid (x,y)\inΩ\}$. In this paper we study the family of reaction-diffusion equations $$ \alignedat 2 u_t&=Δu+f(u),&\quad &t>0, (x,y)\inΩ_ε\partial_{ν_ε} u&=0,& & t>0, (x,y)\in\partialΩ_ε,\endalignedat\tag $E_ε$ $$ where $f$ is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as $ε\to 0$, the equations $(E_ε)$ have a limiting equation which is an abstract semilinear parabolic equation def…
In structural dynamics, energy dissipative mechanisms with non-viscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non--oscillatory motion are called critical damping surfaces (or manifolds, since we can have a lot of parameters). In this paper a general metho…
For general nonlinear Klein-Gordon equations with dissipation we show that any finite energy radial solution either blows up in finite time or asymptotically approaches a stationary solution in $H^1\times L^2$. In particular, any global solution is bounded. The result applies to standard energy subcritical focusing nonlinearities $|u|^{p-1} u$, $1\textless{}p\textless{}(d+2)/(d-2)$ as well as any energy subcritical nonlinearity obeying a sign condition of the Ambrosetti-Rabinowitz type. The argument involves both techniques from nonlinear dispersive PDEs and dynamical systems (invariant manifo…
The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named {\it determining form}, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining …
Machine learning models have emerged as powerful tools in physics and engineering. Although flexible, a fundamental challenge remains on how to connect new machine learning models with known physics. In this work, we present an autoencoder with latent space penalization, which discovers finite dimensional manifolds underlying the partial differential equations of physics. We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations. We show that the resulting optimal latent space of the K-S equation is consistent with the dimension of the inertial ma…
We link nonlinear manifold learning techniques for data analysis/compression with model reduction techniques for evolution equations with time scale separation. In particular, we demonstrate a `"nonlinear extension" of the POD-Galerkin approach to obtaining reduced dynamic models of dissipative evolution equations. The approach is illustrated through a reaction-diffusion PDE, and the performance of different simulators on the full and the reduced models is compared. We also discuss the relation of this nonlinear extension with the so-called "nonlinear Galerkin" methods developed in the context…
We present examples of exponential stabilization for the damped wave equation on a compact manifold in situations where the geometric control condition is not satisfied. This follows from a dynamical argument involving a topological pressure on a suitable uncontrolled set.
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