Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Recently, considerable theoretical and computational evidence has accumulated supporting the remarkable similarities between the long-time evolution of solutions of dissipative partial differential equations (PDEs) and solutions of finite-dimensional dynamical systems, or ordinary differential equations (ODEs). For the latter, numerous studies have discovered and analyzed complex dynamical bifurcations of finite vector fields [CoE, De, Sch, GH, MeP, ChH, BPV]. Computer simulations for the dynamics of many dissipative PDEs evidence an equally rich complexity [HN1, HN2, HNZ, BLMcLO]. The connection between the long-time behavior of finite differential systems and that of PDEs was first established by the discovery that dissipative PDEs possess a finite number of asymptotic degrees of freedom: they have a compact, universal attractor X with finite Hausdorff and fractal dimension (modulo some regularity conditions) [BV, BV1, CF1, CFT, DO, He, HI, HMO, MeP, MP, NST, NST1, T]. Estimates on the number of such degrees of freedom have been obtained for two- and three-dimensional turbulent continuum flows [CF1, CFT, CFMT]. Still, such results do not imply that, for a given dissipative PDE, the asymptotic behavior and in particular the universal attractor X coincide with those of an appropriate differential equation. Recently, it has been shown that for certain dissipative PDEs this is indeed the case.

The long-time behavior of dissipative partial differential systems is characterized by the presence of a universal attractor X toward which all trajectories converge. This is the largest bounded set in the phase space of the system on which the backward-in-time initial value problem has bounded solutions. The structure of X may be very complicated even in the case of simple ordinary differential equations: X may be a fractal or parafractal set (i.e., a compact set for which the Hausdorff and fractal dimensions are different). In the case of dissipative partial differential equations, although the phase space (in the function space) is an infinite-dimensional Hilbert space, X has finite fractal dimension (see [CF, CFT]). However, the already complex nature of X is in this case further complicated by the infinite degrees of freedom of the ambient space.

Let u(t) = S(t)u_o be a solution of where we choose R(u) of the form R(u) = B(u, u)+Cu + f, with a constant f ∈ H, a linear operator C, and a bilinear operator B (both of lower differential order than A).

Our aim is to investigate the time evolution of the position of Ker(I − P(t)) relative to the fixed system of coordinates given by w₁ w₂,… w _n ,… (the eigenvectors of A).

Let u _i ,(t) = S(t)u _i ^o , i = 1, 2, be two solutions of (2.1). Then their difference w = u₁(t) − u₂(t) satisfies the equation where .

One of the features of a dissipative equation of type (2.1), (2.2) is the existence of compact absorbing sets. More precisely, there exists Y ⊂ H satisfying

Let X be the global attractor of the dissipative system under consideration. Recall that X is the largest set in H with the properties

Suppose Σ is an n-dimensional integral surface in Y, that is, an n-dimensional manifold without boundary that is positively invariant. Let, for each u ∈Σ P(u) denote the projector on the tangent space T _u (Σ) to Σ at u. Let us assume that the surface is blocked in the sense that and that λ _n = Λ _m which satisfies condition (3.13). Let us consider u _o ∈ H and assume that the distance between u_o and Σ is attained at some u₁ ∈ Σ Then, clearly P(u₁)(u₁ − u₁)= 0. Let us consider the trajectories S(t)u_o, S(t)u₁. Their difference w(t) = S(t)u_o − S(t)u₁ satisfies (4.1). Denoting Λ(t) = (Aw(t), w(t))|w(t)|², we have as in Chapter 4:

Let Σ₀ be an m-dimensional smooth manifold in θY for some fixed θ ∈ [1, ∞), let u₀ ∈Σ₀, and let u = ϕ(α) be a local parametrization of Σ₀ in a neighborhood of u₀, where a α =(α₁,…,α _m ) runs over a neighborhood of 0 in ℝ ^m and u₀ = ϕ(0). The infinitesimal volume element of S(t)Σ₀ at S(t)u₀ is |υ ₁ (t) ^ … ^ υ _m (t)| where υ _i (ts) evolve according to (2.3) and υ _i (0) = ∂ϕ(α)/∂α _i |_α=o. Using (2.7) and (2.9) we deduce the equation (see [CF1]) where P(t) is the projector on the tangent space to S(t)Σ₀ at S(t)u₀. Thus the volume element will decay exponentially if

We are now going to describe the set of initial data for our integral manifold. In order to make our treatment more transparent we shall assume that

Let Γ be the (n − l)-dimensional sphere {u|u = P _n u,|u| = R} in P _n H considered in Proposition 9.2. Let Σ be the integral manifold obtained with Γ as initial data:

Let be \(\bar \sum \) the inertial manifold constructed in Chapter 10. We recall that £ is smooth and that \(\bar \sum \) is parametrized by B _n = {u|P _n u = u, |u| ≤ R} through the Lipschitz function \(\Phi :{\bar B_n} \to \bar \sum \) of the Lipschitz constant 4/3. Thus, as long as p = P _n p, |p| < R,Φ (p)∈£, it follows that∂Φ(p)/∂p_ί, ί = 1,…, n, satisfy |∂Φ(p)/∂Φp _i | ≤ 4/3. Let B denote the ball in H, B = {u| |u| < R}.

Our aim here and the next two chapters is to prove two important properties of the inertial manifolds that are not (usually) satisfied by the attractors. In this chapter and Chapter 13 the property that we prove is the asymptotic completeness of the inertial manifold \(\bar \sum \) that we have constructed. We recall that the asymptotic completeness means that given any orbit of the dynamical system, we can find another orbit lying on \(\bar \sum \) that produces the same limit behavior at t → ∞.

In the previous chapter, Steps 1 to 3 do not depend on the validity of the asymptotic completeness property. In this chapter we assume that u(t) never belongs to \(\bar \sum \) since otherwise the result is obvious, and we argue by contradiction, assuming that Theorem 12.1 is not valid for u(t). All further steps in this chapter will hinge on the negation of Theorem 12.1. Without loss of generality we can consider a trajectory u(t) = S(t)u_o in θY ∩{u ∈H∣∣u∣≤R}. We assume that for every υ_o ∈ \(\bar \sum \), u(t) − υ(t) (where υ(t) = S(t)υ_o) does not converge to 0 as t → ∞. Thus for υ_o ∈ \(\bar \sum \) fixed, there exists ε > 0 and a sequence t _j → ∞ such that

We prove in the chapter the stability of the inertial manifolds constructed before with respect to perturbations. Three types of perturbations will be explicitly considered here: perturbations of the operators corresponding to a Galerkin approximity of the problem, perturbation of the viscosity parameter v, and perturbation of the right-hand side f (see(4.1)). Although we restrict ourselves to these three perturbations for the sake of simplicity, we believe that our perturbation results apply to more general situations.

We recall that in the case of the Kuramoto—Sivashinsky [HN, HN1,HNZ, NSh] equation on the space H of odd L-periodic functions, (du/dt)+Au+R(u)=0, we have with with the explicit time-independent ϕ defined in [FNST, FNST1], (Here as in the sequel c₀, c₁ … denote absolute constants; for instance, c₀ = (2π)⁴.) Also we shall consider L ≥ 1, the case L < 1 being of no interest.

The equation we will investigate here is of the form on H={u∈²(0,L): ∫ ₀ ^L u(x)dx=0,0≤ x ≤L}, where

In this chapter we consider the equation where on the space H = {u∈ L²(0,L): ∫ ₀ ^L udx = 0} (the same as in Chapter 16) with the periodic boundary conditions. This means in particular that A = (d⁴/dx⁴)u and N(u) = Au − A^1/2p(u), for u ∈ H⁴(0,L) (= the L²-Sobolev space of order 4) such that

In this section we illustrate the method developed in Chapters 3 to 13 on a simple semilinear reaction-diffusion equation in two space variables.

As an example of a parabolic reaction—diffusion equation with less stringent conditions than in Chapter 18, we briefly outline the construction of an inertial manifold for the Chaffee—Infante equation [H] in two dimensions: (we do not restrict ourselves to odd periodic functions). For λ > 1, this equation admits multiple nonconstant steady states besides u = 0 and u = ± 1.