Chapter 3 Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains

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The study of the asymptotic behavior of dynamical systems arising from mechanics and physics is a capital issue because it is essential for practical applications to be able to understand and even predict the long time behavior of the solutions of such systems. A dynamical system is a (deterministic) system that evolves with respect to the time. Such a time evolution can be continuous or discrete (i.e., the state of the system is measured only at given times, for example, every hour or every day). The chapter essentially considers continuous dynamical systems. While the theory of attractors for dissipative dynamical systems in bounded domains is rather well understood, the situation is different for systems in unbounded domains and such a theory has only recently been addressed (and is still progressing), starting from the pioneering works of Abergel and Babin and Vishik. The main difficulty in this theory is the fact that, in contrast to the case of bounded domains discussed above, the dynamics generated by dissipative PDEs in unbounded domains is (as a rule) purely infinite dimensional and does not possess any finite dimensional reduction principle. In addition, the additional spatial “unbounded” directions lead to the so-called spatial chaos and the interactions between spatial and temporal chaotic modes generate a space–time chaos, which also has no analogue in finite dimensions.

Introduction

The study of the asymptotic behavior of dynamical systems arising from mechanics and physics is a capital issue, as it is essential, for practical applications, to be able to understand, and even predict, the long time behavior of the solutions of such systems.

A dynamical system is a (deterministic) system which evolves with respect to the time. Such a time evolution can be continuous or discrete (i.e., one only measures the state of the system at given times, e.g., every hour or every day). We will essentially consider continuous dynamical systems in this survey.

In many situations, the evolution of the system can be described by a system of ordinary differential equations (ODEs) of the form

$$y^{\prime }=f(t,y),y=(y_1,\dots ,y_N),$$

together with the initial condition

$$y(\tau )=y_{\tau },\tau \in \mathbb{R}.$$

Assuming that the above Cauchy problem is well-posed, we can define a family of solving operators $U(t,\tau )$, $t⩾\tau$, $\tau \in \mathbb{R}$, acting on some subset Φ of ${\mathbb{R}}^N$ (called the phase space), i.e.,

$$U(t,\tau ):\Phi \to \Phi ,$$

$$y_{\tau }\mapsto y(t),$$

where $y(t)$ is the solution of (1.1), (1.2) at time t. It is easy to see that this family of operators satisfies

$$U(\tau ,\tau )=\mathrm{Id},U(t,s)○U(s,\tau )=U(t,\tau ),t⩾s⩾\tau ,\tau \in \mathbb{R},$$

where Id denotes the identity operator. We say that this family of operators forms a process. When the function f does not depend explicitly on the time (in that case, we say that the system is autonomous), we can write

$$U(t,\tau )=S(t-\tau ),$$

where the family of operators $S(t)$, $t⩾0$, satisfies

$$S(0)=\mathrm{Id},S(t)○S(s)=S(t+s),t,s⩾0.$$

We say that this family of solving operators $S(t)$, $t⩾0$, which maps the initial datum at $t=0$ onto the solution at time t, forms a semigroup. Furthermore, we say that the pair $(S(t),\Phi )$ (or $(U(t,\tau ),\Phi )$ for a nonautonomous system) is the dynamical system associated with our problem.

The qualitative study of such finite dimensional dynamical systems goes back to the pioneering works of Poincaré on the N-body problem in the beginning of the 20th century (see, e.g., [25]; see also [64] and the references therein for the study of discrete dynamical systems in finite dimensions). In particular, it was discovered, at the very beginning of the theory, that even relatively simple systems of ODEs can generate very complicated (chaotic) behaviors. Furthermore, these systems are extremely sensitive to perturbations, in the sense that trajectories with close, but different, initial data may diverge exponentially. As a consequence, in spite of the deterministic nature of the system, its temporal evolution is unpredictable on time scales larger than some critical value which depends on the error of approximation and on the rate of divergence of close trajectories, and can show typical stochastic behaviors.

Such behaviors have first been observed and established for the pendulum equation perturbed by time periodic external forces, namely,

$$y^{″}(t)+\sin (y(t))(1+\mathrm{ϵ}\sin (\omega t))=0,$$

$\mathrm{ϵ},\omega >0$. Another, important, example is the Lorenz system, obtained by truncation of the Navier–Stokes equations (more precisely, one considers here a three-mode Galerkin approximation (one in velocity and two in temperature) of the Boussinesq equations),

$$x^{\prime }=\sigma (y-x),$$

$$y^{\prime }=-xy+rx-y,$$

$$z^{\prime }=xy-bz,$$

where the positive constants σ, r, and b correspond to the Prandtl number, the Rayleigh number, and the aspect ratio, respectively; in the original work of Lorenz (see [146]), these numbers take the values 10, 28, and $\frac{8}{3}$, respectively. This system gives an approximate description of a two-dimensional layer of fluid heated from below: the warmer fluid formed at the bottom tends to rise, creating convection currents, which is similar to what is observed in the atmosphere. For a sufficiently intense heating, the time evolution has a sensitive dependence on the initial conditions, thus representing a very irregular (chaotic) convection. This fact was used by Lorenz to justify the so-called “butterfly effect”, a metaphor for the imprecision of weather forecast. Other well-known relatively simple systems which exhibit chaotic behaviors are the Minea system [170] and the Rössler system [202].

Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space having a very complicated geometric structure, e.g., locally homeomorphic to the Cartesian product of ${\mathbb{R}}^m$ and some Cantor set, which thus accumulates the nontrivial dynamics of the system, the so-called strange attractor (see, e.g., [27]). One noteworthy feature of a strange attractor is its dimension. First, in order for the sensitivity to initial conditions to be possible on the strange attractor, this dimension has to be strictly greater than 2, so that the dimension of the phase space has to be greater than 3; let us assume, for simplicity, that this dimension is equal to 3, as in the Lorenz system. Then the volume of the strange attractor must be equal to 0; indeed, in systems having a strange attractor, one observes a contraction of volumes in the phase space. Thus, the dimension of a strange attractor is noninteger, strictly between 2 and 3, and we need to use other dimensions than the Euclidean dimension to measure it. Several dimensions, which are not equivalent and yield different values of the dimension in concrete applications, can be used (roughly speaking, some notions of dimensions are related to the connectedness of the sets that one measures, others are related to the way that these sets are embedded into the ambient space, for instance). We will mainly consider in this article the box-counting (or entropy) dimension (see below; see also [84]), which we will call the fractal dimension. Other possible notions of dimensions are the Hausdorff dimension or the Lyapunov dimension (see [84]). Thus, the main features of a strange attractor are

• the trajectories (at least those starting from a neighborhood) are attracted to it;

• close, but different, trajectories may diverge;

• it has a noninteger (fractal) dimension (for instance, for the Lorenz system, numerical investigations show that this dimension is close to, but greater than, 2, namely, 2.05… , which means that there is a “strong” contraction of volumes).

Now, for a distributed system whose initial state is described by functions depending on the spatial variable, the time evolution is usually governed by a system of partial differential equations (PDEs). In that case, the phase space Φ is (a subset of) an infinite dimensional function space; typically, $\Phi =L^2(\Omega )$ or $L^{\infty }(\Omega )$, where Ω is some domain of ${\mathbb{R}}^N$. We will thus speak of infinite dimensional dynamical systems.

A first, important, difference, when compared with ODEs, is that the analytical structure of a PDE is much more complicated. In particular, we do not have a unique solvability result in general, or such a result can be very difficult to obtain. We can, for instance, mention the three-dimensional Navier–Stokes equations, for which a proper global well-posedness result is not known yet (see, e.g., [218]). Nevertheless, the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from mechanics and physics, and it is therefore natural to investigate whether the features mentioned above for dynamical systems generated by systems of ODEs, and, in particular, the strange attractor, generalize to systems of PDEs.

Such behaviors can be observed in a large class of PDEs which exhibit some energy dissipation and are called dissipative PDEs. Roughly speaking, the highly complicated behaviors observed in such systems usually arise from the interaction of the following mechanisms:

• energy dissipation in the higher part of the Fourier spectrum;

• external energy income in its lower part (in order to have nontrivial dynamics, the system has to also account for the energy income);

• energy flux from the lower to the higher Fourier modes, due to the nonlinear terms of the equations.

As already mentioned, this class of PDEs contains a large number of equations from mechanics and physics; we can mention for instance reaction–diffusion equations, the incompressible Navier–Stokes equations, pattern formation equations (e.g., the Cahn–Hilliard equation in materials science and the Kuramoto–Sivashinsky equation in combustion), and damped wave equations.

It is worth emphasizing once more that the phase space is an infinite dimensional function space. However, experiments showed that, as in the case of finite dimensional dynamical systems, the trajectories are localized, up to some transient process, in a “thin” invariant subset of the phase space having a very complicated geometric structure, which thus accumulates all the essential dynamics of the system.

From a mathematical point of view, this led to the notion of a global attractor (see [22], [49], [51], [119], [136], [137], [197], [211], and [217]; see also [15] and [195] for some historical comments). Assuming that the problem is well-posed and that the system is autonomous (i.e., that the time does not appear explicitly in the equations), we have, as in the finite dimensional case, the semigroup $S(t)$, $t⩾0$, acting on the phase space Φ, which maps the initial condition onto the solution at time t. Then we say that $\mathcal{A}\subset \Phi$ is the global attractor for $S(t)$ if

• it is compact in Φ;

• it is invariant, i.e., $S(t)\mathcal{A}=\mathcal{A}$, $\forall t⩾0$;

• $\forall B\subset \Phi$ bounded,

  $$\underset{t\to +\infty }{\lim }\mathrm{dist}(S(t)B,\mathcal{A})=0,$$

  where dist denotes the Hausdorff semi-distance between sets (we assume that Φ is a metric space with distance d) defined by

  $$\mathrm{dist}(A,B):=\underset{a\in A}{\sup }\underset{b\in B}{\inf }d(a,b).$$

This is equivalent to the following: $\forall B\subset \Phi$ bounded, $\forall \mathrm{ϵ}>0$, $\exists t_0=t_0(B,\mathrm{ϵ})$ such that $t⩾t_0$ implies $S(t)B\subset {\mathcal{U}}_{\mathrm{ϵ}}$, where ${\mathcal{U}}_{\mathrm{ϵ}}$ is the ϵ-neighborhood of $\mathcal{A}$.

We note that it follows from (ii) and (iii) that the global attractor, if it exists, is unique. Furthermore, it follows from (i) that it is essentially thinner than the initial phase space Φ; indeed, in infinite dimensions, a compact set cannot contain a ball and is nowhere dense. It is also not difficult to prove that the global attractor is the smallest (for the inclusion) closed set enjoying the attraction property (iii); it thus appears as a suitable object in view of the study of the long time behavior of the system. It is also the maximal bounded invariant set. We finally note that $\mathcal{A}$ attracts all the trajectories (uniformly with respect to bounded sets of initial data), and not just those starting from a neighborhood. The global attractor is sometimes called the maximal or the universal attractor (which is reasonable in view of the above considerations), although these denominations are less used nowadays.

It has also been early conjectured that the invariant attracting sets mentioned above, and, in particular, the global attractor, should be, in a proper sense, finite dimensional and that the dynamics, restricted to these sets, should be effectively described by a finite number of parameters. The notions of dimensions mentioned above, and, in particular, the fractal dimension, should again be appropriate to measure the dimension of these sets. So, when this conjecture is true, the effective dynamics, restricted to the global attractor, is finite dimensional, even though the initial phase space is infinite dimensional. This also suggests that such systems cannot produce any new dynamics which are not observed in finite dimensions, the infinite dimensionality only bringing (possibly essential) technical difficulties.

Starting from the pioneering works of Ladyzhenskaya (see, e.g., [135], [136], and the references therein), this finite dimensional reduction, based on the global attractor, has been given solid mathematical grounds in the past decades for dissipative systems in bounded domains. In particular, the existence of the finite dimensional global attractor has been proven for many classes of dissipative PDEs, including the examples mentioned above. We refer the reader to [22], [49], [119], [136], [137], [197], [211], and [217] for extensive reviews on this subject.

Now, the global attractor may present several defaults. Indeed, it may attract the trajectories at a slow rate. Furthermore, in general, it is very difficult, if not impossible, to express the convergence rate in terms of the physical parameters of the problem. This can be seen on the following real Ginzburg–Landau equation in one space dimension:

$${\partial }_{t}u-\nu {\partial }_x^{2}u+u^3-u=0,x\in [0,1],\nu >0,$$

$$u(0,t)=u(1,t)=-1,t⩾0,$$

see Remark 2.25. A second drawback, which can also be seen as a consequence of the first one, is that the global attractor may be sensitive to perturbations; a given system is only an approximation of reality and it is thus essential that the objects that we study are robust under small perturbations. Actually, in general, the global attractor is upper semicontinuous with respect to perturbations, i.e.,

$$\mathrm{dist}({\mathcal{A}}_{\mathrm{ϵ}},{\mathcal{A}}_0)\to 0\text{as}\mathrm{ϵ}\to 0^{+},$$

where ${\mathcal{A}}_0$ is the global attractor associated with the nonperturbed system and ${\mathcal{A}}_{\mathrm{ϵ}}$ that associated with the perturbed one, $\mathrm{ϵ}>0$ being the perturbation parameter. Very roughly speaking, this property means that the global attractor cannot explode under small perturbations. Now, the lower semicontinuity, i.e.,

$$\mathrm{dist}({\mathcal{A}}_0,{\mathcal{A}}_{\mathrm{ϵ}})\to 0\text{as}\mathrm{ϵ}\to 0^{+},$$

which, roughly speaking, means that the global attractor cannot implode also, is much more difficult to prove (see, e.g., [195]). Furthermore, this property may not hold. This can already be seen in finite dimensions by considering the following ODE (see [195]):

$$x^{\prime }=(1-x^2)(1-{\lambda }^2),\lambda \in [-1,1].$$

Then, when $\lambda =0$, ${\mathcal{A}}_{\lambda }=[0,1]$, whereas ${\mathcal{A}}_{\lambda }=\{1\}$ for $\lambda <0$ and ${\mathcal{A}}_{\lambda }=[-\sqrt{\lambda },1]$ for $\lambda >0$. Thus, there is a bifurcation phenomenon at $\lambda =0$ and the global attractor is not lower semicontinuous at $\lambda =0$. It thus follows that the global attractor may change drastically under small perturbations. Furthermore, in many situations, the global attractor may not be observable in experiments or in numerical simulations. This can be due to the fact that it has a very complicated geometric structure, but not necessarily. Indeed, we can again consider the above Ginzburg–Landau equation. Then, due to the boundary conditions, $\mathcal{A}=\{-1\}$. Now, this problem possesses many metastable “almost stationary” equilibria which live up to a time $t_{\star }\sim {\mathrm{e}}^{{\nu }^{-1/2}}$. Thus, for ν small, one will not see the global attractor in numerical simulations. Finally, in some situations, the global attractor may fail to capture important transient behaviors. This can be observed, e.g., on some models of one-dimensional Burgers equations with a weak dissipation term (see [28]). In that case, the global attractor is trivial, it is reduced to one exponentially attracting point, but the system presents very rich and important transient behaviors which resemble some modified version of the Kolmogorov law. We can also mention models of pattern formation equations in chemotaxis for which one observes important transient behaviors, i.e., patterns, which are not contained in the global attractor (see [215]).

It is thus also important to construct and study larger objects which contain the global attractor, are more robust under perturbations, attract the trajectories at a fast (typically, exponential) rate, and are still finite dimensional. Two such objects have been proposed, namely, an inertial manifold (see [95]) and an exponential attractor (see [65]). We will discuss these objects in more details in the next sections, with an emphasis on exponential attractors (which are as general as global attractors).

An interesting question is whether one has a similar reduction principle for nonautonomous dissipative PDEs (in bounded domains). A first difference, compared with autonomous systems, is that both the initial and final times play an important role; assuming that the problem is well-posed, it defines a process $U(t,\tau )$, $t⩾\tau$, $\tau \in \mathbb{R}$, which maps the initial condition at time τ onto the solution at time t. For such systems, the notion of a global attractor is no longer adequate (in particular, we will not be able to construct proper time independent invariant sets), and one needs to consider other notions of attractors.

A first approach, initiated by Haraux (see [121]) and further studied and developed by Chepyzhov and Vishik (see, e.g., [45] and [49]), is based on the notion of a uniform attractor. Actually, in order to construct the uniform attractor, one considers, together with the initial equations, a whole family of equations. Then one proves the existence of the global attractor for a proper semigroup on an extended phase space, and, finally, projecting this global attractor onto the first component, one obtains the uniform attractor. The major drawback of this approach is that the extended dynamical system is essentially more complicated than the initial one, which leads, for general (translation compact, see Section 3; see also [45] and [49]) time dependences, to an artificial infinite dimensionality of the uniform attractor. This can already be seen on the following simple linear equation:

$${\partial }_{t}u-{\mathrm{\Delta }}_{x}u=h(t),u{|}_{\partial \Omega }=0,$$

in a bounded smooth domain Ω, whose dynamics is simple, namely, one has one exponentially attracting trajectory. However, for more or less general external forces h, the associated uniform attractor has infinite dimension and infinite topological entropy (see [49]).

Nevertheless, for periodic and quasiperiodic time dependences, one has in general finite dimensional uniform attractors (i.e., if the same is true for the global attractor of the corresponding autonomous system). Furthermore, one can derive sharp upper and lower bounds on the dimension of the uniform attractor, so that this approach is appropriate and relevant in those cases.

A second approach, which resembles the so-called kernel sections proposed by Chepyzhov and Vishik (see [44] and [49]), but was studied and developed independently, is based on the notion of a pullback attractor (see, e.g., [62], [129], and [207]). In that case, one has a time dependent attractor $\{\mathcal{A}(t),t\in \mathbb{R}\}$, contrary to the uniform attractor which is time independent. More precisely, a family $\{\mathcal{A}(t),t\in \mathbb{R}\}$ is a pullback attractor for the process $U(t,\tau )$ if

• the set $\mathcal{A}(t)$ is compact in Φ, $\forall t\in \mathbb{R}$;

• it is invariant, i.e., $U(t,\tau )\mathcal{A}(\tau )=\mathcal{A}(t)$, $\forall t⩾\tau$, $\tau \in \mathbb{R}$;

• it satisfies the following pullback attraction property:

  $$\forall B\subset \Phi \text{bounded},\forall t\in \mathbb{R},\underset{s\to +\infty }{\lim }\mathrm{dist}(U(t,t-s)B,\mathcal{A}(t))=0.$$

One can prove that, in general, $\mathcal{A}(t)$ has finite dimension, $\forall t\in \mathbb{R}$, see, e.g., [38] and [139]. Now, the attraction property essentially means that, at time t, the attractor $\mathcal{A}(t)$ attracts the bounded sets of initial data coming from the past (i.e., from −∞). However, in (iii), the rate of attraction is not uniform in t, so that the forward convergence does not hold in general (see nevertheless [35], [40], and [138] for cases where the forward convergence can be proven). We can illustrate this on the following nonautonomous ODE:

$$y^{\prime }=f(t,y),$$

where $f(t,y):=-y$ if $t⩽0$, $(-1+2t)y-t{y}^2$ if $t\in [0,1]$, and $y-y^2$ if $t⩾1$. Then one has the existence of a pullback attractor $\{\mathcal{A}(t),t\in \mathbb{R}\}$, namely, $\mathcal{A}(t)=\{0\}$, $\forall t\in \mathbb{R}$. However, for $t⩾1$, every trajectory, different from {0}, starting from a small neighborhood of 0, will leave this neighborhood, never to enter it again. This clearly contradicts our intuitive understanding of attractors.

So, these two theories of attractors for nonautonomous systems do not yield an entirely satisfactory finite dimensional reduction principle, contrary to the autonomous case, since we have either an artificial infinite dimensionality or no forward attraction in general. We will see below that the construction of exponential attractors allows to overcome the main drawback of pullback attractors, namely, the problem of the forward attraction, as proven in [73]; indeed, one has an exponential uniform control on the rate of attraction. This yields a satisfactory reduction principle for nonautonomous dynamical systems associated with dissipative PDEs in bounded domains.

Now, while the theory of attractors for dissipative dynamical systems in bounded domains is rather well understood, the situation is different for systems in unbounded domains and such a theory has only recently been addressed (and is still progressing), starting from the pioneering works of Abergel [1] and Babin and Vishik [21]. The main difficulty in this theory is the fact that, in contrast to the case of bounded domains discussed above, the dynamics generated by dissipative PDEs in unbounded domains is (as a rule) purely infinite dimensional and does not possess any finite dimensional reduction principle. Furthermore, the additional spatial “unbounded” directions lead to the so-called spatial chaos and the interactions between spatial and temporal chaotic modes generate a space–time chaos which also has no analogue in finite dimensions.

As a consequence, most of the ideas and methods of the classical (finite dimensional) theory of dynamical systems do not work here (as such systems have infinite Lyapunov dimension, infinite topological entropy, …). Thus, we are faced with dynamical phenomena with new levels of complexity which do not have analogues in the finite dimensional theory and we need to develop a new theory in order to describe such phenomena in an accurate way.

It is also interesting to note that, in the case of bounded domains, the dimension of the global attractor grows at least linearly with respect to the volume of the spatial domain and, thus, for sufficiently large domains, the reduced dynamical system may be too large for reasonable investigations. Furthermore, as shown in [16], the spatial complexity of the system (e.g., the number of topologically different equilibria) grows exponentially with respect to the volume of the spatial domain. Therefore, even in the case of relatively small dimensions, the reduced system can be out of reach of reasonable investigations, due to its extremely complicated structure. As a consequence, it seems more natural, at least from a physical point of view, to replace large bounded domains by their limit unbounded ones (e.g., the whole space or cylindrical domains), which, of course, requires a systematic study of dissipative dynamical systems associated with PDEs in unbounded domains.

We will discuss such (for most of them new) developments in Section 5 of this survey.

In a last section, we will briefly discuss extensions of the theory of attractors to ill-posed problems, with an emphasis on the so-called trajectory attractor, see, e.g., [46], [47], [49], and [210]. Indeed, for many interesting problems, including the three-dimensional Navier–Stokes equations, various types of damped hyperbolic equations (e.g., damped wave equations with supercritical nonlinearities), … , the well-posedness of the solution operator $S(t)$ has not been proven yet or/and the proper choice of the phase space is not known. Furthermore, e.g., for dissipative systems with non-Lipschitz nonlinearities or for systems arising from the dynamical approach of elliptic boundary value problems in unbounded domains, nonuniqueness results and the ill-posedness of the associated solution operator are known.