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This work was initiated in the summer of 1985 while all of the authors were at the Center of Nonlinear Studies of the Los Alamos National Laboratory; it was then continued and polished while the authors were at Indiana Univer­ sity, at the University of Paris-Sud (Orsay), and again at Los Alamos in 1986 and 1987. Our aim was to present a direct geometric approach in the theory of inertial manifolds (global analogs of the unstable-center manifolds) for dissipative partial differential equations. This approach, based on Cauchy integral mani­ folds for which the solutions of the partial differential equations are the generating characteristic curves, has the advantage that it provides a sound basis for numerical Galerkin schemes obtained by approximating the inertial manifold. The work is self-contained and the prerequisites are at the level of a graduate student. The theoretical part of the work is developed in Chapters 2-14, while in Chapters 15-19 we apply the theory to several remarkable partial differ­ ential 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.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

Chapter 1. Presentation of the Approach and of the Main Results

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.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 2. The Transport of Finite-Dimensional Contact Elements

Let u(t) = S(t)uo be a solution of
$$\frac{{du}}{{dt}} + N\left( u \right) = \frac{{du}}{{dt}} + Au + R\left( u \right) = 0,$$
$$u\left( 0 \right) = {u_0},$$
where we choose R(u) of the form R(u) = B(u, u)+Cu + f, with a constant fH, a linear operator C, and a bilinear operator B (both of lower differential order than A).
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 3. Spectral Blocking Property

Our aim is to investigate the time evolution of the position of Ker(IP(t)) relative to the fixed system of coordinates given by w1 w2,… w n ,… (the eigenvectors of A).
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 4. Strong Squeezing Property

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
$$ \frac{d}{w} + \rlap{--}{\lambda}\left( t \right)w = 0 ,$$
$$w\left( 0 \right) = {w_0} = u_1^0 - u_2^0,$$
$$ \begin{gathered} \rlap{--}{\lambda}\left( t \right)g = Ag + Cg + B\left( {u\left( t \right),g} \right) + B\left( {g,u\left( t \right)} \right), \hfill \\ u\left( t \right) = \frac{1}{2}\left( {{{u}_{1}}\left( t \right) + {{u}_{2}}\left( t \right)} \right) \hfill \\ \end{gathered} $$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 5. Cone Invariance Properties

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 YH satisfying
$$Y{\text{ is convex, closed in }}H,{\text{ a bounded neighborhood or 0 in }}\mathcal{D}{\text{ (}}{A^{1/4}}{\text{) (in particular, }}Y{\text{ is compact in }}H).{\text{ }}$$
$${\text{For every }}\theta {\text{ }} \geqslant {\text{ 1 and any }}{u_o} \in \theta Y{\text{ the inequalities (3}}{\text{.7), (3}}{\text{.8), and (4}}{\text{.5a) are valid}}{\text{. The constants }}{k_{\text{1}}}{\text{, }}{k_{\text{2}}}{\text{, }}{k_{\text{3}}}{\text{, }}{k_{\text{5}}}{\text{, and }}{k_{\text{6}}}{\text{ depend on }}\theta {\text{ only}}{\text{. }}$$
$${\text{The set }}Y{\text{ is absorbing; i}}{\text{.e}}{\text{., if }}Z{\text{ is any bounded set in }}H{\text{, there exists a }}{{\text{t}}_{\text{o}}}{\text{ }} \geqslant {\text{ 0 (depending on }}Z{\text{) such that }}S{\text{(}}t{\text{)}}Z{\text{ }} \subset {\text{ }}Y{\text{ for }}t{\text{ }} \geqslant {\text{ }}{t_{\text{o}}}{\text{.}}$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 6. Consequences Regarding the Global Attractor

Let X be the global attractor of the dissipative system under consideration. Recall that X is the largest set in H with the properties
S(t)X = X for t ≥ 0,
X is bounded in H,
dist(S(t)uo,X) → 0 as t → ∞ for all uoH.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 7. Local Exponential Decay Toward Blocked Integral Surfaces

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
$$\lambda \left( {P\left( u \right)} \right) > \frac{{{\lambda _n} + {\lambda _{n + 1}}}}{2}{\text{ for all }}u$$
and that λ n = Λ m which satisfies condition (3.13). Let us consider u o H and assume that the distance between uo and Σ is attained at some u1 ∈ Σ Then, clearly P(u1)(u1u1)= 0. Let us consider the trajectories S(t)uo, S(t)u1. Their difference w(t) = S(t)uoS(t)u1 satisfies (4.1). Denoting Λ(t) = (Aw(t), w(t))|w(t)|2, we have as in Chapter 4:
$$\frac{d}{{dt}}|w\left( t \right){|^2}\left( {{k_4}\Lambda \left( t \right) - {k_7}} \right)|w{|^2} \leqslant 0.$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 8. Exponential Decay of Volume Elements and the Dimension of the Global Attractor

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(t0 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])
$$\frac{1}{2}\frac{d}{{dt}}|{\upsilon _1}\left( t \right){|^2} + \left( {Tr A\left( t \right)P\left( t \right)} \right)|{\upsilon _1}\left( t \right) \wedge \cdots \wedge {\upsilon _m}\left( t \right){|^2} = 0,$$
where P(t) is the projector on the tangent space to S(t0 at S(t)u0. Thus the volume element will decay exponentially if
$$\mathop {\lim }\limits_{t \to \infty } \operatorname{int} \frac{1}{t}\int_o^t {Tr} \left( {A\left( s \right)P\left( s \right)} \right)ds > 0.$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 9. Choice of the Initial Manifold

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
$$\left( {B\left( {u,u} \right),u} \right) = 0{\text{ for all }}u \in \mathcal{D}{\text{(}}A{\text{)}}{\text{.}}$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 10. Construction of the Inertial Manifold

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:
$$\sum { = \bigcup\limits_{t > o}^{} {s(t)\Gamma .} } $$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 11. Lower Bound for the Exponential Rate of Convergence to the Attractor

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)/∂pί, ί = 1,…, n, satisfy |∂Φ(p)/∂Φp i | ≤ 4/3. Let B denote the ball in H, B = {u| |u| < R}.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 12. Asymptotic Completeness: Preparation

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 → ∞.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 13. Asymptotic Completeness: Proof of Theorem 12.1

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)uo in θY ∩{uH∣∣u∣≤R}. We assume that for every υo\(\bar \sum \), u(t) − υ(t) (where υ(t) = S(t)υo) does not converge to 0 as t → ∞. Thus for υo\(\bar \sum \) fixed, there exists ε > 0 and a sequence t j → ∞ such that
$$|u\left( {{t_j}} \right) - \upsilon \left( {{t_j}} \right)| \geqslant \varepsilon > 0,{\text{ for all }}j{\text{.}}$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 14. Stability with Respect to Perturbations

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.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 15. Application: The Kuramoto—Sivashinsky Equation

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
$$R\left( u \right)B\left( {u,u} \right) + Cu + f,$$
$$B\left( {u,\upsilon } \right) = u\frac{{d\upsilon }}{{dx}},$$
$$Cu = - {A^{1/2}}u + B(u,\varphi ) + B\left( {\varphi ,u} \right),$$
$$f = A\varphi + \psi {\text{ with}}\psi = \frac{{{d^2}\varphi }}{{d{x^2}}} + \varphi \frac{{d\varphi }}{{dx}},$$
with the explicit time-independent ϕ defined in [FNST, FNST1],
$${\Lambda _n} = {\lambda _n} = {c_0}{\left( {\frac{n}{L}} \right)^4},{\text{ }}n = 1,2,....$$
(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.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 16. Application: A Nonlocal Burgers Equation

The equation we will investigate here is of the form
$$\frac{{du}}{{dt}} + N\left( n \right) = 0,{\text{ }}N\left( u \right) = Au + R\left( u \right),$$
on H={u2(0,L): ∫ 0 L u(x)dx=0,0≤ xL}, where
$$R\left( u \right) = B\left( {u,u} \right) + \int , {\text{ }}\varphi {\text{ = 0,}}\psi {\text{ = }}f \ne 0,B\left( {u,\upsilon } \right) = \left( {u,\omega } \right)\upsilon ',{\text{ where }}\upsilon ' = d\upsilon /dx,{\text{ with a fixed }}\omega \in \ne {\text{0,}}$$
$$A = - \frac{{{d^2}}}{{d{x^2}}}{\text{ with periodic boundary conditions}}{\text{.}}$$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 17. Application: The Cahn—Hilliard Equation

In this chapter we consider the equation
$$\frac{\partial }{{\partial t}}u + \frac{{{\partial ^4}}}{{\partial {x^4}}}u + \frac{{{\partial ^2}}}{{\partial {x^2}}}p\left( u \right) = 0$$
$$p\left( {u = - {b_2}} \right){L^{ - 2}}u - {b_3}{L^{ - 1}}{u^2} - {b_4}{u^3}$$
on the space H = {uL2(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) = AuA1/2p(u), for uH4(0,L) (= the L2-Sobolev space of order 4) such that
$$ \begin{gathered} u\left( 0 \right) = u\left( L \right),{\text{ }}u'\left( 0 \right) = u'\left( L \right),{\text{ }}u''\left( 0 \right) = u\left( L \right), \hfill \\ {\text{ }}u'''\left( 0 \right) = u'''\left( L \right) \hfill \\ \end{gathered} $$
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 18. Application: A Parabolic Equation in Two Space Variables

In this section we illustrate the method developed in Chapters 3 to 13 on a simple semilinear reaction-diffusion equation in two space variables.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman

CHAPTER 19. Application: The Chaffee—Infante Reaction—Diffusion Equation

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:
$$\frac{{\partial u}}{{\partial t}} - \Delta u + \lambda \left( {{u^3} - u} \right) = 0,\;\lambda > {\text{ }}0,\Omega = {\left[ { - \pi , + \pi } \right]^2} = {T^2},{\text{ periodic boundary conditions, }}u\left( 0 \right) = {u_0}$$
(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.
P. Constantin, C. Foias, B. Nicolaenko, R. Teman


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