Approximate Inertial Manifolds of Exponential Order
نویسندگان
چکیده
A fairly general class of nonlinear evolution equations with a self-adjoint or non self-adjoint linear operator is considered, and a family of approximate inertial manifolds (AIMs) is constructed. Two cases are considered: when the spectral gap condition (SGC) is not satisfied and an exact inertial manifold is not known to exist the construction is such that the AIMs have exponential order, while when the SGC is satisfied (and hence there exists an exact inertial manifold) the construction is such that the AIMs converge exponentially to the exact inertial manifold.
منابع مشابه
Reduced Order Control based on Approximate Inertial Manifolds
A reduced-order method based on approximate inertial manifolds is applied to optimal control problems in infinite-dimensional state spaces. A detailed analysis of the method is given for the linear quadratic regulator problem. The method can also be applied to higher-order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay.
متن کاملApproximate inertial manifolds for the pattern formation Cahn-Hilliard equation
An approximate inertial manifold for an évolution équation is a finite dimensional smooth manifold such that the orbits enter, after a transient time, a very thin neighbourhood of the manifold In this paper, we consider the Cahn-Hilliard équation and we present a method which allows to construct several approximate inertial manifolds providing better and better order approximations to the orbit...
متن کاملInertial manifolds under multistep discretization
Finite-dimensional inertial manifolds attract solutions to a nonlinear parabolic diierential equation at an exponential rate. In this paper inertial manifolds for multistep discretizations of such equations are studied. We provide an existence result for inertial manifolds under multistep discretization and show that these inertial manifolds converge to the inertial manifold of the original equ...
متن کاملApproximate Inertial Manifolds for Nonlinear Parabolic Equations via Steady - State Determining Mapping Yuncheng
ABSTRACT. For nonlinear parabolic evolution equations, it is proved that, under the assumptions oflocal Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. I...
متن کاملA Perturbation Approach for Approximate Inertial Manifolds
We present an explicit form for the construction of approximate inertial manifolds (AIMs) for a class of nonlinear dissipative partial differential equations by using a perturbation technique. We investigate two numerical examples of the reaction diffusion equation with polynomial nonlinearity and non-polynomial nonlinearity to show comparison of accuracy for our perturbation method with other ...
متن کامل