Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions

Abstract

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a 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.2000 Mathematics Subject Classification. 35B40, 35B42.therefore, the problem of finding the IM for the 3D Cahn-Hilliard equation with periodic boundary conditions becomes non-trivial and to the best of our knowledge, has been not considered before.The next theorem gives the main result of the paper.Theorem 1.1. For infinitely many values of N ∈ N there exists an N -dimensional IM M N for the Cahn-Hilliard problem (1.4) with periodic boundary conditions which is the graph of a Lipschitz continuous function over the N -dimensional space spanned by the first N eigenvectors of the Laplacian. Moreover, this function is C 1+ε -smooth for some small ε = ε(N ) > 0 and the manifold possesses the so-called exponential tracking property, see Section 3 for the details.

On the hyperbolicity properties of inertial manifolds of reaction–diffusion equations

“…As was already mentioned, the technique in [6] permits one to derive the existence of a normally hyperbolic inertial manifold in an appropriate state space for Eqs. There is a suspicion that, for an appropriate choice of the phase space and the family of admissible nonlinearities, the validity of the principle of spatial averaging and the sparseness of the spectrum of the Laplace operator in the scalar reaction-diffusion equations are necessary and sufficient for the existence of a normally hyperbolic inertial manifold and an absolutely normally hyperbolic inertial manifold, respectively.…”

“…The methods in [6] permit one to establish that the validity of the abstract version of the principle of spatial averaging [20] implies the existence of a normally hyperbolic inertial manifold in the state space of the parabolic problem (1.1). For the reaction- Moreover, from the viewpoint of applications to chemical kinetics, the degrees of the polynomials should not exceed 3.…”

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions

“…So, this trick actually allows to check the cone property for the case where P N w H 2 per ≤ 2R. On the other hand, together with the control (4.14), this gives us the control of H 2−κ -norm in the estimates related with the cone property, see also [5,20] and the proof of Theorem 4.7 below.…”

“…Note that the proof of this theorem differs essentially from the one given in the first part of our study for the case of Dirichlet boundary conditions. In particular, we have to use a special cut-off procedure similar to the one developed in [10] (see also [5,9,20]) for the so-called spatial averaging method as well as the graph transform and invariant cones instead of the Perron method. The extra term u is added only in order to have dissipativity and the global attractor in the periodic case as well and is not essential for IMs.…”

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

“…We restrict ourselves to the discussion of the absolute normal hyperbolicity only by two reasons. First, the IMs constructed by the Perron method are usually absolutely normally hyperbolic (although, non-absolute normally hyperbolic IMs naturally arise when the alternative method based on the invariant cones is used, e.g., for methods involving the so-called spatial averaging, see [23,20,38]). Second, the absolute normal hyperbolicity can be relatively easily extended to the non-compact case and the proper extension (suitable for IMs) of nonabsolute hyperbolicity to the non-compact case requires the replacing of exponents in (4.4) by more complicated functions, see [10].…”