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 defined on a closed linear subspace of $H^1(Ω)$. We also proved that the family ${\Cal A}_ε$ of the corresponding attractors is upper semicontinuous at $ε=0$. In this paper we prove that, if $Ω$ satisfies some natural assumptions, then the limiting equation can be characterized as a reaction-diffusion equation on a finite topological graph. Moreover, there is a family $\Cal M_ε$ of inertial $C^1$-manifolds for $(E_ε)$, of some fixed finite dimension $ν$, and, as $ε\to 0$, the flow on $\Cal M_ε$ converges in the $C^1$-sense to the limit flow on $\Cal M_0$.
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).
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.
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…
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.
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…
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.
We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-Δ)^{\fracα{2}}$, with $0<α<2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $α$. Then we approximate the inertial manifold when a small normal diffusion $\varepsilon Δ$ (with $\varepsilon \in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and…
The existence of an inertial manifold for the modified Leray-$α$ model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.
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…
The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in \cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for…
We show the existence of an inertial manifold (i.e. a globally invariant, exponentially attracting, finite-dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.
Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of `splitting' the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy. In particular, we use these tools in order to establish the hyperbolicity of certain global solutions for non-autonomous parabolic par…
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…
In this paper we obtain $C^{1,θ}$-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.
The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier-Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.
The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the …
This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.
We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the recently suggested method of spatio-temporal averaging.
The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by G. Sell and J. Mallet-Paret. We present a universal approach which covers the most part of known results obtained via this method as well as gives a number of new ones. Among our applications are reaction-diffusion equations, various types of generalized Cahn-Hilliard equations, including fractional and 6th order Cahn-Hilliard equations and several classes of modified Navier-Stokes equations including the Leray-$α$ regularization, h…
These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mané projection theorems. The recent counterexamples which show that the underlying dynamics may be in a sense infinite-dimensional if the spectral gap condition is violated as well as the discussion on the most important open problems are also included.
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 …
Technical University of Munich, School of Computation, Information and Technology（慕尼黑工业大学计算、信息与技术学院）；Leiden University, Mathematical Institute（莱顿大学数学研究所）
In this work we consider a dissipative reaction-diffusion equation in a $d$-dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in \cite{Arrieta-Santamaria-C0} and Shadowing theory.
