Inertial manifolds for dissipative semiflows in banach spaces

Reduction of dimension of approximate intertial manifolds by symmetry

“…We first recall from [14,6,9] that an AIM, M, is a smooth finite dimensional manifold such that every solution enters and remains in a thin neighbourhood in a finite time, more precisely, there exist an integer m and a positive number e such that M is an m-dimensional manifold and dist(S(f)u 0 ,M) < e, for all t ^ t,…”

“…On the other hand, approximate inertial manifolds (AIM) are used to approximate the dynamics on the global attractors and inertial manifolds, see Jolly, Kevrekidis and Titi [6]; Foias, Jolly, Kevrekidis, Sell and Titi [14]; Sell [9], and the references therein. Since AIMs approximately describe the asymptotic dynamics of a given system, they are used both for dynamical studies and computational purposes.…”

“…In Section 4, we apply our abstract results to concrete evolution equations such as the KS equation and CH equation. There we shall be concerned with AIM's constructed by various well known methods: the nonlinear Galerkin method, Gamma method, and Euler-Galerkin method, see [9] for more details.…”

Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon

“…Now, we will prove that there exists an inertial manifold M (see a definition in Ref. [7]) for the semigroup S * (t) in the phase space Y = R 2 ×Ḣ 1 per (0, 1)×L 2 per (0, 1), i.e., a submanifold of Y such that (i)S * (t)M ⊂ M for every t ≥ 0, (ii) there exists δ > 0 satisfying that for every bounded set B ⊂ Y, there exists C(B) ≥ 0 such that dist(S(t), M) ≤ C(B)e −δt , t ≥ 0 see, for example, [7] and [23].…”

Finite-dimensional asymptotic behavior in a thermosyphon including the Soret effect

“…In this section we will consider the phase space Y"1;H Q K ;H Q K\ , m*2 and we will prove that there exists an inertial manifold M for the semigroup S*(t) in Y, i.e., a submanifold of Y such that see [1,9,14]. Assume then that ¹ 3H Q K with…”