Inertial Manifolds for von Karman Plate Equations

Inertial manifolds associated with nonlinear plate models governed by dynamical von Karman equations are considered. Three different dissipative mechanisms are discussed: viscous, structural and thermal damping. Though the systems considered are subject to some dissipation, the overall dynamics may not be dissipative. This means that the energy may not be decreasing. The main result of the paper establishes the existence of an inertial manifold subject to the spectral gap condition for linearized problems. The validity of the spectral gap condition depends on the geometry of the domain and the type of damping. It is shown that the spectral gap condition holds for plates of rectangular shape. In the case of viscous damping, which is associated with hyperbolic-like dynamics, it is also required that the damping parameter be sufficiently large. This last requirement is not needed for other types of dissipation considered in the paper.     

Approximate Inertial Manifolds to the Newton-Boussinesq Equations

Approximate inertial manifolds are related to the study of long time behaviour of dissipative partial differential equa tions. In this paper, we construct two approximate inertial manifolds for the two dimensional Newton-Boussinesq Equations. The orders of approximations of these manifolds to the global attractor are derived.     

弱阻尼KdV方程的近似惯性流形

本文给出了弱阻尼KdV方程近似惯性流形族的一个构造方法，得到的近似惯性流形是一个弱Lipschitz连续的流形，它对吸引子的指数渐近速度比平坦惯性流形相应的速率要高。     

Inertial Manifolds for Weakly and Strongly Dissipative Hyperbolic Equations

One considers mixed boundary value problems for a quasilinear hyperbolic equation with a weak, as well as strong, dissipation. The nonlinear function in the equation is assumed Lipschitz continuous. For each of these problems one obtains the conditions on the Lipschitz constant that ensure the existence of inertial manifolds.     

广义Kuramoto—Sivashinsky型方程的惯性流形

本文及续文对一类广义Kuramoto-Sivashinsky型方程证明了惯性流形的存在性，并对其维数作了估计同时研究该方程的不变锥性质（ICP）和轨线的强挤压性（SSP）。     

J－J方程组的渐近惯性流形*

本文构造了Liapunov泛函,得出了Josephson结中的sine-Gordon方程组的高模态的衰减性质,从而给出了渐近惯性流形。     

强阻尼非线性波方程的近似惯性流形

本文研究一类强阻尼非线性波方程．通过交换的方法，我们构造了方程的近似惯性流形序列，并得到了对于方程的整体吸引子的逼近估计．     

非线性Sobolev Galpern方程的近似惯性流形

近似惯性流形概念与耗散偏微分方程的长时间行为研究有关, 该文对非线性Sobolev Galpern方程构造了两个近似惯性流形. 证明了非平滑近似惯性流形Σ和平滑近似惯性流形Σ_0=P_mH对整体吸引子有相同的逼近阶数.     

Approximate Inertial Manifolds of Strongly Damped Nonlinear Wave Equations

In this paper we consider a class of strongly damped nonlinear wave equations. By the transformation of unknown functions and decomposition of operators, we construct a family of approximate inertial manifolds, and obtain the estimate of orders of approximation of such manifolds to solution orbits.     

具有拟周期外力的非自治时滞发展方程的近似惯性流形

该文研究了一类具有拟周期外力的非自治时滞发展方程, 通过延伸相平面将非自治系统转化为自治系统, 再证明相应的自治系统的时滞惯性流形的存在性, 并在时滞惯性流形的基础上构造了非自治发展方程的近似惯性流形.     

拟周期时滞耗散半线性波方程的惯性流形

本文研究了具有拟周期外力作用的时滞波方程解的长时间性态,利用斜积半流方法,在扩展相空间中将非自治系统提升成自治系统,利用算子半群理论及Lyapunov-Perron方法在一定的谱间隙条件和充分小的时滞假设下,证明了拟周期半线性时滞波方程惯性流形的存在性.     

Chemotaxis-Growth方程的近似惯性流形

The long-time behaviour of solution to chemotaxis-growth system with Neumann condition is considered in this paper.The approximate inertial manifolds of such equations are constructed based on the contraction principle,and the orders of approximations of the manifolds to the global attractor are derived.     

Approximate Inertial Manifolds to the Generalized Symmetric Regularized Long Wave Equations with Damping Term

In the present paper, we construct two approximate inertial manifolds for the generalized symmetric regularized long wave equations with damping term. The orders of approximations of these manifolds to the global attractor are derived.     

Attractors for a von Karman equation with memory

In this paper a von Karman equation with memory,utt + α?2u- γ?utt- integral from n=-∞ to t μ(t- s)?2u(s)ds = [u, F(u)] + h is considered. This equation was considered by several authors. Existing results are mainly devoted to global existence and energy decay. However, the existence of attractors has not yet been considered. Thus, we prove the existence and uniqueness of solutions by using Galerkin method, and then show the existence of a finitedimensional global attractor.     

Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?ⁿ. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center‐stable subspaces. By rewriting the Euler equation as an ODE on an infinite‐dimensional manifold of volume‐preserving maps in W^{k, q} the unstable (and stable) manifolds of v₀ are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v₀ in the sense that arbitrarily small W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.     

Attractors for non-dissipative irrotational von Karman plates with boundary damping

Long-time behavior of solutions to a von Karman plate equation is considered. The system has an unrestricted first-order perturbation and a nonlinear damping acting through free boundary conditions only.This model differs from those previously considered (e.g. in the extensive treatise (Chueshov and Lasiecka, 2010 [11])) because the semi-flow may be of a non-gradient type: the unique continuation property is not known to hold, and there is no strict Lyapunov function on the natural finite-energy space. Consequently, global bounds on the energy, let alone the existence of an absorbing ball, cannot be a priori inferred. Moreover, the free boundary conditions are not recognized by weak solutions and some helpful estimates available for clamped, hinged or simply-supported plates cannot be invoked.It is shown that this non-monotone flow can converge to a global compact attractor with the help of viscous boundary damping and appropriately structured restoring forces acting only on the boundary or its collar.