Inertial manifolds for nonlinear evolutionary equations

Singularly perturbed 1D Cahn–Hilliard equation revisited

“…This allows to construct an inertial manifold which is globally realized as a graph for the semigroup generated by the modified equation and also by the original equation (see [45, p. 875]). Then, since the resulting modified equation is identical to the original one within the absorbing set, the intersection of this graph with the absorbing set also defines an inertial manifold (note that it is also an exponential attractor) for the original equation (cf., e.g., [15,31]). …”

Approximate inertial manifolds for the pattern formation Cahn-Hilliard equation

“…\q'-qi\+K\A{q 3 -q)\ (4 26) (thanks to (4 21)) Then, by substracting (4 1) from (4 23), we obtain The goal of this section is to dérive the existence of two more manifolds Ji 5 and M § which give better order approximations to the orbits than M^ and Jt± These manifolds will be constructed by considering in particular approximations of the second order time derivative of q (which up to now was neglected). Then, in (5.1), q m is neglected and the approximate value of q", namely q'{, is given by the resolution of…”

“…In particular, in space dimension n = = = 3, (0.1) possesses a global attractor [11]. Also, in the case where the spatial domain is a cube of St n y n = 1, 2, the existence of inertial manifolds has been derived [3,11] ; we recall that an inertial manifold [5] is a finite dimensional Lipschitz invariant manifold which attracts exponentially all the orbits as time goes to infinity. We will 464 M MARION consider hère Problem (0.1) posed on arbitrary bounded subsets of M n , n s* 3, and our aim is to show the existence of approximate inertial manifolds.…”

Center manifolds for smooth invariant manifolds

“…This method allows more detailed analysis on and near an invariant manifold especially when a natural coordinate system is available for a particular problem (e.g. [9], [20], [25], [30], [39], [56]). The Lyapunov-Perron method does not seem to directly apply to our problem.…”