Inertial manifolds for partial differential evolution equations under time-discretization: Existence, convergence, and applications

Abstract

This work is devoted to the question of existence and convergence of inertial manifolds for evolution equations under time discretization. We show that provided the time step is sufficiently small and under the condition of existence of exact inertial manifolds (the spectral gap condition), the discretized problem do have an inertial manifold with the same dimension. We show the convergence of the approximated manifolds towards the exact one in a strong sense and we give an error estimate. Our applications include nondissipative equations, they are not limited to purely parabolic equations. We consider complex amplitude equations of the type of Ginzburg-Landau equation and also dissipative perturbations of Korteweg-de Vries equations.