Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup {S(t)}t ≥ 0. Suppose that {S(t)}t ≥ 0 possesses a global attractor $${\mathcal{A}}$$. We show that, for any generalized Banach limit LIMT → ∞ and any probability distribution of initial conditions $${\mathfrak{m}_0}$$, that there exists an invariant probability measure $${\mathfrak{m}}$$, whose support is contained in $${\mathcal{A}}$$, such that$$\int_{X} \varphi(x) {\rm d}\mathfrak{m}(x) = \underset{t \rightarrow \infty}{\rm LIM}\frac{1}{T} \int_0^T \int_X \varphi(S(t) x) {\rm d}\mathfrak{m}_0(x) {\rm d}t,$$for all observables φ living in a suitable function space of continuous mappings on X.This work is based on the framework of Foias et al. (Encyclopedia of mathematics and its applications, vol 83. Cambridge University Press, Cambridge, 2001); it generalizes and simplifies the proofs of more recent works (Wang in Disc Cont Dyn Syst 23(1–2):521–540, 2009; Lukaszewicz et al. in J Dyn Diff Eq 23(2):225–250, 2011). In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when {S(t)}t ≥ 0 does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and thus restricts the phase space X to the case of a reflexive Banach space.Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail. We first consider the Navier-Stokes equations with memory in the diffusion terms. This is the so called Jeffery’s model which describes certain classes of viscoelastic fluids. We then consider a family of neutral delay differential equations, that is equations with delays in the time derivative terms. These systems may arise in the study of wave propagation problems coming from certain first order hyperbolic partial differential equations; for example for the study of line transmission problems. For the second example the phase space is $${X= C([-\tau,0],\mathbb{R}^n)}$$, for some delay τ > 0, so that X is not reflexive in this case.