On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations

Introduction

Consider the linear evolution equation

posed on a Banach space (B, ∥ • ∥ B ), where L(t) is a time-varying, closed linear operator (potentially unbounded). Let us assume (1) is globally well-posed and it gives rise to an evolution semigroup S(t), namely, a bounded family of solution operators S(t) : B → B, t 0, such that v(t) = S(t)v 0 is the unique solution to (1) for all fixed initial v 0 ∈ B. The Multiplicative Ergodic Theorem (MET) is a powerful tool for characterizing the asymptotic behavior of systems such as (1) in the case when L(t) depends on the value of some auxiliary stationary process, e.g., when L(t) is random with probabilistic law independent of t. In this setting and under some mild conditions on the operators S(t), the MET asserts that for 'typical' realizations of t → L(t), there exists a value λ 1 ∈ [-∞, ∞) and a finite-codimensional subspace F ⊂ B such that, for all v 0 ∈ B\F,

As B \ F is open and dense in B, it follows that the growth rate λ 1 is experienced by 'typical' initial v 0 . For further details and a review of the MET in this setting, see Sect. 2.1 below. Linear evolution equations such as (1) cover a broad variety of time-dependent dissipative linear PDE. In this paper, we consider the following example settings:

(i) the advection diffusion equation for a passive scalar advected by a velocity field evolving according to a "statistically stationary" evolution equation, e.g., the 2d Navier-Stokes equations with either time-periodic or stochastic forcing; and (ii) the first variation (linearization) of the 2d Navier-Stokes equations with either time-periodic or stochastic forcing.

We will discuss both of these examples in more detail in Sect. 1.1 below. Other relevant examples that can be treated (though not discussed) in this setting are: the kinematic dynamo equation governing the advection and diffusion of a magnetic field in a flow, as well as the first variation equation of a wide class of forced dissipative semilinear parabolic problems, including reaction diffusion equations, magnetohydrodynamics (MHD), and various dissipative wave equations (see [75] and [15] for examples and descriptions of these models and more).

Lyapunov exponents along scales of norms

For globally well-posed linear evolution equations as above, it is common to have global well-posedness on a scale of Banach spaces (B α , ∥ • ∥ B α ) for α ∈ [a, b] ⊂ R such that B β is embedded in B α for all α < β. Such scales of spaces might capture varying degrees of integrability, e.g., L p spaces, or spatial regularity, e.g., Sobolev spaces W r, p or Besov spaces B r p,q . Each B α comes equipped with its own norm ∥ • ∥ B α with respect to which one can compute Lyapunov exponents, and so apriori, the same evolution equation (1) might possess an entire range of Lyapunov exponents

depending on the scale parameter α.

It is evident from (2) that for a finite-dimensional Banach space, the choice of norm has no effect on the value of λ 1 : in this case, local compactness implies that all norms on B are equivalent up to a multiplicative constant which vanishes under the limit of 1 t log. Thus it is often said that in the finite dimensional setting, Lyapunov exponents are intrinsic to the underlying system in that they do not depend on the choice of norm. However, local compactness is false in infinite dimensions, and so it is possible that λ 1 (B α ) could depend nontrivially on the scale parameter α. This casts doubt on the 'intrinsic' nature of Lyapunov exponents in the infinite dimensional setting, especially when there is no natural or otherwise physically relevant choice for the space B α .

The following is an informal statement of the main result of this paper: Then, for all β ∈ [a, b] and v 0 ∈ B β \{0}, the limit

exists and does not depend on α.

This result affirms the idea that for such systems, the Lyapunov exponent is an intrinsic feature of the system, independent of the choice of norm ∥ • ∥ B α . For full statements, see Sect. 2.2. See Sect. 2.3 for a literature review of prior work on this topic.

Remark 1.1. (Time-transient behavior) Suppose that we are in the setting of the theorem above, and let ε > 0, β ∈ [a, b], and a vector v 0 ∈ B β be fixed; we define T ε,β (v 0 ) = min T > 0 :

1 t log ∥S(t)v 0 ∥ B βλ(v 0 ) < ε for all t T .

The value T ε,β (v 0 ) is, roughly speaking, the time it takes for the exponent λ(v 0 ) to be 'realized' in the norm of B β to within accuracy ε. While our main results give conditions under which the asymptotic value λ(v 0 ) is independent of the norm ∥ • ∥ B β , the value T ε,β (v 0 ) can and will, in general, depend on the norm. Indeed, a prime example of this dependence is passive scalar advection; see the discussion in Sect. 1.1.1 below. Nevertheless, under stronger but realistic conditions it is possible to compare these timescales between norms: see Sect. 2.1.4 and Corollary 2.8 in Sect. 2.2.1 for precise statements.

Applications

This paper contains several applications of the main result to systems of interest in fluid dynamics. We will discuss these applications and their physical relevance below, deferring detailed statements to Sect. 4. While several simplifying assumptions are made, e.g., working with periodic domains without boundary, we note that many of the main ideas discussed below remain valid in broader generality. Moreover, while we work with Hilbert regularity spaces H s , much of what we show can also be done in spaces like W s, p and B s p,q , which often carry more precise regularity information.

1.1.1. Lyapunov Exponents for Passive Scalar Advection Let d > 1 and let u : [0, ∞) × T d → R d be a time dependent, incompressible velocity field on the torus T d , which for the purposes of this discussion will be assumed to be C ∞ in x locally uniformly in t. Let f (t, x) be a solution to the passive scalar advection equation

for a given mean-zero initial scalar f 0 : T d → R. This equation models the advection of the scalar density f 0 (e.g., a dilute chemical concentration or small temperature variation) by a fluid with velocity field u(t, x) taking into account molecular diffusivity κ > 0. Equation ( 3) is globally well-posed on the Sobolev space H r for any r ∈ R, and so gives rise to a linear (nonautonomous) semiflow S(t) : H r → H r of bounded linear operators. Here, H r is viewed as a Hilbert space of mean-zero functions (or distributions, if r < 0) g : T d → R with the homogeneous Sobolev norm ∥ f ∥ H r = ∥(-) r/2 f ∥ L 2 and corresponding inner product (•, •) H r . When t → u(t, •) evolves according to some ergodic, stationary process, e.g., the Navier-Stokes equations with spatially regular, time-periodic or stochastic forcing, the Lyapunov exponent

exists in [-∞, ∞) with probability 1 for all sufficiently regular initial velocity profiles u 0 and for an open and dense set of initial scalars f 0 .

In the mathematical literature on advection diffusion, special interest is often taken in interpretations of the growth or decay of ∥ f (t)∥ H r for various values of r . When κ = 0, the H 1 norm is naturally connected to the strength of shear-straining in the fluid (see (5) below), while the H -1 norm and other negative Sobolev norms measure the degree to which the scalar f t has been uniformly "mixed" into the fluid-see, e.g., [22,53,58,74]-and are related to the decay of correlations of the Lagrangian flow associated to the velocity u. The addition of diffusion (κ > 0) somewhat complicates these interpretations: when advection generates small scales, diffusion can effect decay of the L 2 norm on time scales faster than the diffusive one. This is known in the mathematics literature as enhanced dissipation; this effect has been studied in both the physics [8,26,51,55,67] and (somewhat more recent) mathematics literature [2,3,6,7,18,29,81,89,90].

Despite such varied interpretations of the measurement of various norms, in this manuscript we prove the following (see Theorem 2.6 below for a precise statement):

Informal Theorem. Assume κ > 0 and that u(t, •) is an ergodic, stationary process and 1 0 ∥u(t, •)∥ H γ dt satisfies a moment condition for some γ > d 2 + 1. Then, λ(H s ) exists for all s ∈ [-γ , γ ] and does not depend on s. In particular,

For a full statement, see Theorem 4.2 in Sect. 4.2 below. To our knowledge, ours is the first proof of this fact for passive scalar advection, although we note that it has been predicted before, e.g., by numerical evidence in the recent paper [62], as well as in [60].

Additional discussion and context

The case κ = 0. Key to the validity of ( 4) is compactness of the solution linear operators S t : H r → H r for (3) when κ > 0. A clear example is provided in the case κ = 0; in this case, (3) is still globally well-posed on H r for all r , and by the method of characteristics one has

for all f 0 ∈ L 2 , where ϕ t : T d is the Lagrangian flow associated to the velocity field u. Note that in this case, S(t) : L 2 → L 2 cannot be compact, as it is unitary:

by incompressibility. When the Lagrangian flow ϕ t associated to u has a positive Lyapunov exponent on a positive volume, i.e.,

then ∥ f (t)∥ H 1 can grow exponentially fast, hence λ 1 (H 1 ) > 0. It was recently shown by the authors and J. Bedrossian in [5] that when u solves the stochastic Navier-Stokes equations with nondegenerate, white-in-time forcing, the LHS of (6) has full Lebesgue measure with probability 1. 1 L 2 and H 1 decay rates. When κ > 0, standard heat equation energy estimates for the L 2 norm immediately imply

and therefore our result implies the same holds true for λ 1 (H 1 ). At first glance, this might be surprising in light of the tendency of (32) to form large gradients. It is however consistent with the energy estimate

We emphasize, though, that λ 1 (H 1 ) < 0 refers only to time asymptotic behavior, and does not 1 Despite a wealth of numerical evidence, in the absence of noise it is a notoriously challenging open problem to prove that positivity of Lyapunov exponents for incompressible systems of practical interest. This is already the case for low-dimensional discrete-time toy models [19] of Lagrangian flow such as the Chirikov standard map [16], for which the analogue of ( 6) is a wide-open problem-see, e.g., the discussion in [10,21].

rule out transient growth in H 1 on some (κ-dependent, potentially quite long) time scale after which the diffusion dominates (c.f. Remark 1.1 and Sects. 2.1.4, 2.2.1). H -1 decay rates. In [3][4][5], the authors proved the following exponential decay estimate for (32) when u solves the 2d stochastic Navier-Stokes equation (or any of a large class of noisy evolution equations):

Here the deterministic constant γ > 0 is independent of κ and the random variable D κ 1 has κ-independent expectation. Using that S(t) instantly regularizes H -1 to H 1 for t > 0, this readily implies that

In light of our main result and the results of [3], we conclude that, for the stochastic Navier-Stokes equations and related models, all Sobolev norms (including L 2 ) eventually decay no slower than the uniform-in-κ exponential decay rate γ > 0 (perhaps after an initial κ-dependent period of transient growth). This κ → 0 singular limit bears a striking similarity to the stochastic stability of the so-called Ruelle-Pollicott resonances associated to stationary hyperbolic flows [27].

Remark 1.2. We emphasize that we are not the first to apply the MET and related ideas to passive scalar advection. Froyland et al. have developed data-driven algorithms for identifying coherent structures in incompressible fluids [33], with applications in the forecasting of oceanic features such as persistent gyres in the Atlantic ocean [78]. Justifying the use of these algorithms required extending the MET for compositions of possibly noninjective linear operators, addressed in [32] in finite dimensions and, e.g., [38] in infinite dimensions. Additional applications of the MET in this vein include the exploration of almost-sure statistical properties for random compositions of mappings [23,24].

Lyapunov Exponents for the Navier-Stokes Equations

Let u(t, x) be a mean-zero divergence free velocity field solving the Navier-Stokes equations on the periodic box T 2 ,

where F is some spatially smooth, white-in-time or time-periodic forcing term, ν > 0 is fixed, and p denotes the pressure that enforces the divergence-free condition. Under appropriate conditions on the forcing, for all r 0 this nonlinear evolution equation gives rise to a stochastic semiflow of C 1 Frechet-differentiable mappings t ω : H r → H r , where H r denotes the Sobolev space of H r (weakly) divergencefree fields (r = 0 corresponding to L 2 ). Here, ω denotes the history of the driving path. Given an initial u 0 = u(0, •) and an initial divergence-free v 0 ∈ H r , the v 0 derivative v t = (D u 0 t )v 0 solves the linearized Navier-Stokes equations

with initial data v 0 . When F is deterministic and time-periodic or when F is stochastic and white-in-time, the MET applies: under mild additional conditions, the H r Lyapunov exponent

exists with probability 1 and for 'typical' initial velocity fields u 0 , where v 0 is drawn from an open and dense subset of H r . 2 Since we are working with a first variation equation, the value λ 1 (H r ) represents the asymptotic exponential rate at which nearby trajectories converge (λ 1 < 0) or diverge (λ 1 > 0) in the H r norm as time progresses.

In the study of the 2d Navier-Stokes equations it is often useful to formulate the equation in terms of vorticity w = curl u,

or in terms of the stream function ψ = -1 curl u, which is the Hamiltonian for the velocity field u = ∇ ⊥ ψ = (-∂ y ψ, ∂ x ψ). Depending on the variable considered, it is natural to study the associated growth of the perturbation in L 2 of the associated variable. Hence, measuring the linearization in H 1 , L 2 or H -1 corresponds to measuring the linearization in L 2 for the vorticity, velocity, or stream function formulations of the equation.

In the inviscid case (ν = 0), it is known that the stability of the equation is strongly dependent on the whether one is considering L 2 of vorticity, velocity or the stream function with some perturbation being stable in L 2 of the stream function, but not in L 2 of velocity or vorticity due to the generation of high-frequencies due to mixing effects. For the viscid problem ν > 0, we prove in this paper the following: Informal Theorem. Assume that u(t, •) solve the Navier-Stokes equations with forcing F (either time-periodic or white-in-time) and that the resulting process on velocity fields is (statistically) stationary and ergodic. Assume 1 0 ∥u(t, •)∥ H γ +2 dt has finite moments for some γ > 2. Then, λ(H s ) exists for all s ∈ [-γ + 1, γ + 1] and does not depend on s. In particular,

For full details, see Theorem 4.12 in Sect. 4.3. There is a long and extensive literature on the linear stability or instability of stationary (time independent) solutions to the Euler and Navier-Stokes equations; see, e.g., the textbooks [14,25,42,73,86]. Lyapunov exponents, which can be viewed as analogous to spectra for nonstationary flows, have been employed extensively in the study of semilinear parabolic problems such as Navier-Stokes, for instance in providing upper bounds on the dimension of the global attractors-see, e.g., [17,31,76]. Ruelle and Takens proposed dynamical chaos, of which a positive Lyapunov exponent is a natural hallmark, as a mechanism involved in the transition to turbulence [56,71]. To this end, Ruelle established an extension of smooth ergodic theory to dissipative parabolic problems such as Navier-Stokes [70] (see also, e.g., [11,52,54]). For numerical studies of Lyapunov exponents in turbulent regimes, see, e.g., [20,87].

Plan for the paper Sect. 2 covers necessary background from ergodic theory and a full statement of our main abstract result, Theorem 2.6, the full proof of which is given in Sect. 3. Full statements and proofs of the assertions in Sect. 1.1 above are given Sect. 4.

2.1.2.

The MET Let B be a separable Banach space with norm ∥ • ∥ B . The MET concerns cocycles of operators, which for our purposes are compositions of the form

where A : X → L(B), L(B) the space of bounded operators on B, is the generator of the cocycle. We view the composition A n x as being "driven" by the dynamics T : X → X . The MET describes the asymptotic exponential growth rates

where they exist, as x ranges over m-typical initial conditions in X and v ∈ B \ {0}.

For simplicity, we assume below that A x is compact for all x ∈ X ; otherwise, we make no additional assumptions, e.g., on the injectivity of A x (we follow the convention that log 0 = -∞). Many proofs of the MET in this setting exist; the following is taken from [52]; see also [72].

Theorem 2.2. (MET for compact cocycles) Let T : (X, F , m) be an mpt. Assume that A : X → L(B) is strongly measurable 6 and that A(x) is a compact linear operator on B for all x ∈ X . Lastly, assume the log-integrability condition

Then, for every λ c > -∞, there exists a (i) function r λ c :

4 That such weak * limits exist follows by compactness of A. That such limiting measures are T -invariant is straightforward to check: see, e.g., Lemma 2.2.4 of [80]. The above procedure is often referred to as the Krylov-Bogolyubov argument for the existence of T -invariant measures [48].

5 E.g., Proposition 4.3.2 of [80] and the Krein-Milman Theorem, paragraph I.A.22 in [85]. 6 When B is separable, we say that x → A x is strongly measurable if it is Borel measurable w.r.t. the strong operator topology on L(B), or equivalently, when x → A x v is a Borel measurable mapping for each fixed v ∈ B. For a summary of alternative measurability requirements for the MET, see, e.g., [79]. and (iii) at m-a.e. x ∈ X a filtration

by closed, finite-codimensional, measurably varying 7 subspaces F i (x), Fλ c (x) such that

for m-a.e. x ∈ X. The functions r λ c (x), λ i (x) are constant along m-a.e. trajectory, as are the codimensions M i (x) := codim F i+1 (x) < ∞. Moreover, when T : (X, F , m) is ergodic, r λ c and the values λ 1 , • • • , λ r λc are constant over m-a.e. x ∈ X.

It is immediate that the F i (x) are invariant in the sense that

. 8 Note that we allow the inclusion to be strict. Observe also that

is the codimension of F i+1 (x) in F i (x); we refer to d i (x) as the multiplicity of λ i (x).

Lyapunov Exponents

The values {λ i } are called Lyapunov exponents, while the collection of them is referred to as the Lyapunov spectrum, in analogy with the spectrum of a single closed operator. The value λ c is a cutoff, past which we do not resolve the spectrum further, while adjusting the value λ c lower can potentially 'uncover' additional Lyapunov spectrum (i.e., r λ c increases as λ c decreases). Define

To simplify the discussion below, assume T : (X, F , m) is ergodic, so that r and the {λ i } are constants. We distinguish three scenarios: 7 Throughout, we consider the space of closed subspaces of B with the Hausdorff metric d Haus of unit spheres; see (17) for details. Here, we are asserting that x → F i (x) is Borel measurable w.r.t. the topology induced by d H .

8 Some authors refer to the property (9) as equivariance.

(a) No Lyapunov exponents are uncovered (r λ c = 0 for all values of cutoff λ c ). In this case,

for a.e. x ∈ X and all v ∈ B. When this occurs, we follow the convention

uncovered. Each exponent corresponds to a member of the filtration

x ∈ X (see, e.g., discussion after Corollary 2.2 in [70]) implies that lim i λ i = -∞, and each exponent corresponds to a member of the filtration

In this case we follow the convention r = ∞.

We note that in all three scenarios, the codimension codim

Remark 2.3. When T : (X, F , m) is nonergodic, the limiting value r (x) in (11) depends on x ∈ X . In particular, X can be subdivided into the T -invariant (possibly empty) sets {r (x) = 0}, {1 r (x) < ∞} and {r (x) = ∞} along which each of scenarios (a)-(c) holds, respectively. Remark 2.4. These scenarios are analogous to the situation for the spectrum σ (K ) of a compact linear operator K on B: (a) when σ (K ) = {0} (e.g., K is a compact shift operator); (b) when σ (K ) is a finite set containing {0} (e.g., K is finite rank); and (c) when σ (K ) is countable and accumulates only at {0} (e.g., K = -1 is the inverse Laplacian on L 2 ([0, 1]) with Dirichlet boundary conditions). Indeed, when A x ≡ K is a fixed compact operator not depending on x, the λ i are precisely the logarithms of the absolute values of the elements of σ (K ), while the F i are direct sums of the corresponding generalized eigenspaces.

Example 2.5. Let T : B → B be a continuous mapping as in Example 2.1 admitting a compact invariant set A ⊂ B and an invariant Borel probability m. Assume in addition that T is C 1 Frechet differentiable, and that the derivative D x T is a compact linear operator (as is the case for a broad class of dissipative parabolic evolution equations [76]). Theorem 2.2 applies to the cocycle generated by A(x) = D x T ∈ L(B) (note that by our assumptions, x → log + |D x T | is a continuous function and A is compact, so (8) holds automatically). It follows that for m-a.e. x ∈ X and for all v ∈ B, the limit

exists, and if finite, equals one of the values λ i (x).

Rate at Which Lyapunov Exponents are "Realized"

The MET guarantees convergence of the exponential rates λ(x, v) for x ∈ X, v ∈ B as in equation ( 7), but this convergence can be badly nonuniform in x ∈ X . While little can be said at this level of generality, we can at least quantify this nonuniformity as we show below.

To fix ideas, assume (T, m) is ergodic and r > 0 (scenarios (b) or (c) in Sect. 2.1.3). Fix an i ∈ {1, . . . , r }, so by the statement of the MET we have that λ(x, v) = λ i for all v ∈ F i (x)\F i+1 (x). With additional work, it is possible to show (see [11]) that for any ε > 0, one has

where

is finite for m-a.e. x ∈ X . For corresponding lower bound, note that the convergence of λ(x, v) to λ i should be slower as v approaches F i+1 (x). To account for this, given v ∈ B \ {0} and a closed subspace

Then,

where

is again m-almost surely finite. Define

The sets ℓ ⊂ X are sometimes referred to as uniformity sets or Pesin sets: when ε is chosen sufficiently small, for any fixed ℓ > 1 we have

) bounded away from 0, up to the multiplicative constant ℓ and ignoring the slowly-growing factors e nε . This can be very useful, e.g., in smooth ergodic theory where the exponential expansion/contraction along various directions of B is used to construct stable/unstable manifolds of smooth systems (see references below). Unfortunately, despite their importance, little else can be said about D ε without additional assumptions.

Additional Background and Context for the MET

The MET for stationary compositions of d × d matrices was first proved by Oseledets [63] in the late 60's, although investigations on the properties of IID products of d × d matrices date from the early 60's [35,36]. There are now many proofs available-see, e.g., [66,69,82] and [32]. Since then the MET has been extended in several directions, e.g., to the asymptotic behavior of random walks on semisimple Lie groups [43] and on spaces of nonpositive curvature [44].

One of the most significant impacts of the MET has been in smooth ergodic theory, the study of the ergodic properties of differentiable mappings. For such systems, the MET implies the existence of stable and unstable subspaces in the moving frames along "typical" trajectories of the dynamics. Pesin discovered [65,69] soon after that these could be used in the construction of stable and unstable manifolds, generalizing directly from the classical theory of stable/unstable manifolds for equilibria and periodic orbits. This development is at the core of our contemporary understanding of chaotic dynamical systems and the fractal geometry of strange attractors [28]. For more discussion, see, e.g., the the textbook [1] or the surveys [64,84,88].

A part of Ruelle's work in [70] was a version of the MET for stationary compositions of Hilbert space operators. By now, there are many works extending the MET to various infinite-dimensional settings. Highlights include extensions to stationary products of compact linear operators on a Banach space [57], dropping the compactness assumption [77], versions of the MET suited to the first variation equations of SPDE [52,72], and a version of the MET for compositions of operators drawn from a vonNeumann algebra [12]; see also, e.g., [9,37,79].

Statement of Main Results

To start, we will assume

and (V, ∥ • ∥ V ) are separable by a standard argument. Additionally, (1) T : (X, F , m) is an mpt;

(2) A : X → L(B) is strongly measurable and A(x) ∈ L(B) is compact for all x ∈ X ; (3) The restriction A(x)| V has range contained in V and is a compact linear operator V → V , and moreover, x → A(x)| V is strongly measurable X → L(V ); and finally, (4) The operator A(x) satisfies the log-integrability condition regarded on both

Under ( 1)-( 4), the MET as in Theorem 2.2 applies to A n x regarded as a cocycle on either B or V . Below, for either of W = B or V we write λ W i , r W and F W i for the objects in Theorem 2.2 applied to A n

x regarded as a cocycle on W . Theorem 2.6. Under ( 1)-( 4), we have that r V (x) = r B (x) for m-a.e. x ∈ X. Writing r (x) for this common value, the following holds for all i 1 and m-a.e. x ∈ {r i}:

In particular, for m-a.e. x ∈ X and all v ∈ V , we have that

Note also that scenarios (a), (b) and (c) above are carried over from B to V . For instance, in the ergodic case, λ B 1 = -∞, r B = 0 (our convention for scenario (a)) holds if and only if λ

(ii) When T is the time-1 mapping for a dissipative semilinear parabolic problem, e.g., the 2d Navier-Stokes equations, one typically works with a scale of Banach spaces

, it is often the case that D x T : B → B is compact, and that D x T restricted to V maps into V and is similarly compact; this is a consequence of parabolic regularity for the first variation (linearization) equation. In particular, Theorem 2.6 applies. See Sect. 4 for more details in the case of 2d Navier-Stokes.

Comparison of Uniformity Sets

It is natural to attempt to compare the rate at which Lyapunov exponents are realized between the norms of B and V . Fix i 1 and assume m{r i} > 0. Fix ε > 0 and let D W ε , D W ε , W = B, V be as in ( 12), (14), respectively (note that we have not assumed (T, m) is ergodic, so

We prove such a comparison under the following additional assumption:

(5) For all x ∈ X , the range of A x : B → B is contained in V and is bounded as a linear operator

Moreover, assume that for some p > 3 we have that

Corollary 2.8. Assume the setting of Theorem 2.6 and additionally that (16) above holds. For any δ > 0, there exists a function K δ :

hold for any ε > 0 and m-a.e. x ∈ {r i}. The function K δ satisfies the moment estimate {r i}

where ϕ(x)

That is, while usually one has little control over D W ε , D W ε , these terms are comparable between W = B and V , in a way that can be made explicit in terms of the L p -norm of log + ∥A • ∥ B→V . Viewing V as a "higher regularity" subspace of B (c.f. the discussion in Sect. 1.1), condition (5) has the connotation that A x regularizes initial data from B into V . This condition is natural for linear cocycles derived from dissipative parabolic PDE, and holds for all the applications covered in Sect. 4 below.

Proof of Corollary 2.8 assuming Theorem 2.6. We restrict attention to the lower bound for D B ε ; the upper bound on D V ε is easier and omitted for brevity. Moreover, we assume below that (T, m) is ergodic, so the value r i and the Lyapunov exponents λ i are almost-surely constant (the non-ergodic case is treated similarly and is omitted for brevity).

To start, observe that for v ∈ F V i (x), we have

Since V is dense in B and

The above estimate holds uniformly over v ∈ F V i (x), and so by density of

we conclude the same holds for v ∈ F B i (x). It remains to bound ∥A T n x ∥ B→V from above. Below, we write " p , p,q " for bounds up to a universal multiplicative constant depending only on p, and/or q and independent of all other parameters, e.g., δ. With G n := {x ∈ X :

is almost-surely finite by the Borel-Cantelli Lemma, with the tail estimate

For m-a.e. x, we now have

Plugging K δ into our previous estimate, we conclude

ε+δ , as desired. It remains to estimate the q-th moment of log + K δ , where from here on q < p is fixed. We have

Using Hölder's inequality on each summand and that m • T -1 = m, we obtain

The sum in the RHS is finite iff q < p( p- 3) p-1 (note the right-hand quantity is > 0 iff p > 3).

Preliminaries

Let (B, ∥ • ∥ B ) be a Banach space. Let Gr(B) denote the Grassmanian of B, i.e., the set of closed subspaces of B. For k ∈ N, write Gr k (B) for the set of kdimensional subspaces of B and Gr k (B) for the set of closed, k-codimensional subspaces.

Throughout, Gr(B) is endowed with the metric topology coming from the Hausdorff distance

Equation ( 17) is the usual Hausdorff distance between two closed subsets of a metric space. In this case, we are taking the usual Hausdorff distance of the unit spheres S E := {e ∈ E : ∥e∥ B = 1}. Note that for v ∈ B and S ⊂ B we write dist B (v, S) = inf s∈S ∥v -s∥ B for the minimal distance between v and S in the ∥ • ∥ B norm. We note that in d B Haus , the sets Gr k (B), Gr k (B) are clopen in Gr(B) for all k 1. For additional background, see Section IV.2 of [45].

, noting 1 α(E) < ∞ automatically by compactness of S E when dim E < ∞ (in particular, the sup is a max).

For our purposes, we will require some degree of control over α(E) as E varies: [45]), hence α(E) < ∞ automatically. Lemma 3.2 goes further, asserting that the B and V norms are uniformly equivalent for all E close enough to E 0 in d Haus .

Proof of Lemma 3.2. Let E ∈ Gr(V ) and assume d V Haus (E, E 0 ) < δ for some δ > 0 to be specified. Let v ∈ E, ∥v∥ V = 1 be so that

Using that ∥v 0 ∥ B α(E 0 ) -1 , we see that

On taking δ > 0 sufficiently small (depending only on α(E 0 ) and ε > 0), we conclude α(E) α(E 0 ) + ε as desired.

It is also useful to compare the Hausdorff distances d B Haus , d V Haus :

The desired conclusion now follows from the following standard inequality (c.f. Section IV.2 of [45]):

Determinants in Banach Spaces

The following is an assignment to each E ∈ Gr k (B) a "volume element" along E. We note that the measure Vol B E on E is characterized uniquely by (i) translation invariance and (ii) the normalization (18). 9 It follows from this unique characterization that det B (A|E) is well-defined irrespective of the choice S ⊂ E. Below we recall some basic properties of det B (•|•); for proofs and additional background, see [9].

In what follows, we define the minimum-norm of A| E by

Note that if A| E is injective, then finite-dimensionality of E implies that A| E : E → A(E) is invertible, and therefore we have

(a) For bounded linear operators A 1 , A 2 : B → B, we have that

(b) If A : B → B is a bounded linear operator and A| E injects, then

The following compares det B and det V :

Let A : B → B be a bounded linear operator for which A(E) ⊂ V and A| E injects. Then,

Proof. Let D B E denote the unit ball of E in the ∥ • ∥ B norm. First, we show that, for all Borel K ⊂ E, we have

To see this, observe that by uniqueness of Haar measure, there exists c > 0 such that that Vol

To obtain the first inequality, observe that by definition of α(E),

For the second inequality, since Vol

Measurable Selection of Projectors

Next, we will need the following Lemmas on the geometry of Banach spaces. Below, given a splitting B = E ⊕F into closed subspaces E, F ⊂ V , we define π E//F to be the unique oblique projection onto E with kernel F. Lemma 3.9. (Corollary III.B.11 in [85]) Let F ∈ Gr k (B) for k 1, then there exists E ∈ Gr k (B)with

The following is a version of Lemma 3.9 for a measurable family F(x), x ∈ X, of finite-codimensional spaces. Lemma 3.10. Let (X, F ) be a measurable space and let F : X → Gr k (B) be a measurable family.

Then, there exists a measurable family E : X → Gr k (B) such that for all x ∈ X, (i) V = E(x) ⊕ F(x), (ii) the mapping x → π E(x)//F(x) is strongly measurable; and (iii) we have that ∥π

Proof. Separability of B * implies separability of Gr k (B) (Lemma B.12 in [38]; see also Chapter IV, §2.3 of [45]). Let {F n } ⊂ Gr k (B) be a countable dense sequence and for each n let E n be a k-dimensional subspace of B such that ∥π

and recursively define

By the density of {F n } and measurability of x → F(x), it holds that {S n } is a countable partition of X by F -measurable sets. Now, define

follows from Lemma B.18 of [38]. To estimate ∥π E(x)//F(x) ∥ B , Proposition 2.7 of [9] implies that for x ∈ S n ,

Below we record various additional measurability properties, used freely and without further mention in Sect. 3.3, below. Lemma 3.11. Let (B, ∥ • ∥ B ) be a separable Banach space and let M : X → L(B) be a strongly measurable mapping. [38]) If F : X → Gr(B) is measurable, then x → ∥M(x)| F(x) ∥ B is measurable. Consequently, the function x → ∥M(x)∥ B is measurable. 2. (Lemma A.5 of [38]) If M ′ : (X, F ) → L(B) is another strongly measurable mapping, then M ′ • M : (X, F ) → L(B), x → M ′ (x) • M(x) is also strongly measurable.

Quotient Cocycle Construction

Assume r V i for some fixed i 1, and define F(x) := F V i+1 (x). By Lemma 3.10, there exists a measurably-varying family x → E(x) of closed, finite-dimensional complements in V to F(x), equipped with a measurably-varying family of projectors x → π ⊥

We define Âx : E(x) → E(T x), Ân

x : E(x) → E(T n x) as follows:

We note that invariance of F(x) under A x as in (9) implies the identities

Below we write det V ( Ân x ) below for the determinant of Ân x : E(x) → E(T n x). In view of the expression (22), we have that det

Lemma 3.12. For m-a.e. x ∈ X we have that

The proof uses the following corollary to the MET, which we recall here. Recall that ∠ V (v, F) ∈ [0, π/2] is the angle between v ∈ V \ {0} and a subspace F ⊂ V . For a nontrivial subspace E ⊂ V , we write

Note that if E and F share a nontrivial subspace E ∩ F, then ∠ V min (E, F) = 0.

Corollary 3.13. For m-a.e. x ∈ X and for any complement E to F(x) in V , we have that

The proof of Corollary 3.13 is contained in, e.g., paragraph "Proof of (h)" in the proof of Theorem 16 in [39].

Proof of Lemma 3.12. To start, by (23) and multiplicativity of the determinant (Lemma 3.5(a)), we have that

It suffices to check that

By Lemma 3.5(b), we have

where

is the minimum norm defined by (19). The RHS is uniformly bounded in x, n from above by a constant in k, so it remains to bound the LHS from below. For this, we use the following estimate which related the norm of π E//F v to the distance between v and F. Claim 3.14. Let E, F be complementary closed subspaces of the Banach space V . Then

Consequently, the above claim implies that

x (E(x)), F(T n x)). Applying Corollary 3.13 completes the proof.

Proof of Proposition 3.8(a)

To start, we check unconditionally that

Using Proposition 3.7(b) and the density of

). Applying now Proposition 3.6, we have

For this, observe that with the measurable selection x → E(x) as above, we have that α(E(x)) < ∞ m-almost everywhere.

Choose C 0 > 0 large enough so that m(U ) 99 100 , where U := {α(E(x)) C 0 } ⊂ X.

Observe that m(T -1 U ∩ U ) 98 100 > 0 by T -invariance of m. By the Poincaré Recurrence Theorem, for m-a.e. x ∈ U ∩ T -1 U , there is a sequence of times n k → ∞ such that T n k x ∈ U ∩ T -1 U for all k. Fixing such an x and sequence (n k ), we estimate

where in the first line we used Lemma 3.12 and in the second we used Proposition 3.6. The first lim inf goes to 0, as α(E(T n+1 x)) C 0 along the sequence of times n = n k . For the second term, we estimate

using Lemma 3.5(a) and ( 22) in the first line, Proposition 3.6 in the second line, and Lemma 3.5(b) in the third line. By construction, along n = n k we have α(E(T n x)) C 0 . To control the ∥A T n x ∥ V term, define g(x) := log + ∥A x ∥ V . By our assumptions, we have log + g ∈ L 1 (m). By a standard corollary of the Birkhoff Ergodic Theorem (see, e.g., Theorem 1.14 in [83]), it holds that

In all, we conclude V

, which in conjunction with (24) implies

Proof of Proposition 3.8(b) Since ( 24) holds for all k, it suffices to check that

The proof below is parallel to that of Proposition 3.8(a), the most notable change being that we use a quotient cocycle parallel to the space F(x) := F B i+1 (x). The following is an analogue of Lemma 3.10 above.

and (c) the projector π⊥

The proof requires the following.

Lemma 3.16. For all m, k 1 there exists C m,k > 0 such that the following holds. Lemma 3.16. We induct on the codimension k. The base case k = 1 is Corollary 2.5 in [61]. Assume the induction hypothesis for k and fix

for all i m. Let us now bound sin ∠ B min (E ′ , V i ) from below, where E ′ := E + ⟨v⟩ and ⟨v⟩ is the line spanned by v. We compute:

Proof of Claim. The proof is parallel to that of Lemma 3.10. Assume for now that either r V = ∞ or r V < ∞ and M V r V M B i ; we address the alternative case at the end. Fix j r V so that M

) , with C 2,k as in Lemma 3.16, and

Refine to a partition S m,n , m, n 1 of X such that S m,n ⊂ Sm,n for all m, n.

Finally, using density of

. This step can be justified using, e.g., continuity of E → π E//F as E ranges over the set of complements to F ∈ Gr k (B) (Lemma B.18 in [38]).

One now sets

the estimate on the B-norm of π ⊥ x = π Ẽ(x)// F(x) is completely parallel to that in Lemma 3.10 and is omitted. That Ẽ(x) ∩ F(x) = {0} follows from a similar argument.

The proof is now complete when either

i the proof is simpler: every E 0 ∈ Gr k (V ) satisfies the right-hand equation in (25) by Proposition 3.7(b). So, in this case it suffices to apply Lemma 3.10 directly to obtain a complement Ẽ(x) to F(x).

Completing the Proof of Theorem 2.6

To prove Theorem 2.6 we first establish the following claims.

Claim 3.17. If r V = 0, then r B = 0.

Proof of Claim. Pursuing a contradiction, observe that if λ B 1 > -∞, then by the density of V ⊂ B, for m-a.e. x there exists v ∈ V \{0}, ∥v∥ V = 1 so that

That r V = 0 implies the LHS limit exists for m-a.e. x and equals -∞, a contradiction.

Claim 3.18. (a) For all i 1, we have that r V i if and only if r B i. (b) If r V i, then λ V j = λ B j and d V j = d B j for all 1 j i.

Proof of Claim 3.18. We will prove below, by induction on i, that r V i implies r B i and λ V j = λ B j , d V j = d B j for all 1 j i. The proof that r B i implies r V i is identical on exchanging the roles of V and B below; further details are omitted.

Assume first that r V 1: we will show that r B 1,

To start, by Proposition 3.8(a) we have

> -∞ and r B 1. Applying Proposition 3.8(b), we see that

To proceed, assume

while combining with (27) gives

Assume now the induction hypothesis that for some i 1, we have that r V i implies r B i and λ

, there is nothing to prove. If r V i + 1, we proceed: Proposition 3.8(a) implies

since M B i = M V i , it follows that r B i + 1, hence by Proposition 3.8(b) we have (29), we obtain

, then as before we obtain a contradiction, and conclude

Completing the proof of Theorem 2.6. We have already shown that if r V = 0, then r B = 0 and the proof is complete. We now consider the case r V = ∞; the case 0 < r V < ∞ is handled similarly and is omitted.

In this case, Claim 3.18 implies r B = ∞ and

for all i 1. It remains to check that the identity

holds for a.e. x ∈ X . To start, assume v ∈ F V i (x)\{0}. Then,

, and so lim n→∞

M e a n w h i l e , lim n→∞

. This completes the proof.

Remark 3.19. (The nonergodic case) We provide here a list of changes needed in the case when (T, m) is not ergodic. In the following steps, the main difficulty is to deal with the possibility that the values r W , λ W i , d W i , etc., all depend on x ∈ X . 1. To start, we can reduce to the case where r V , r B are both constant in x by restricting the measure m to sets of the form

for arbitrary pairs k B , k V ∈ {0, 1, 2, . . . , ∞}. Sets of this form are T -invariant (T -1 S = S up to m-measure zero sets) and so the restrictions m S (K ) := m(K ∩ S)/m(S) are T -invariant. Since there are at-most countably many such sets S, it suffices to prove Theorem 2.6 for each m S separately. 2. Suppose one has already restricted to a set of the form S for some fixed k B , k V .

The volume rates W k and multiplicities M W i are now functions of X , and so the analogue of Proposition 3.8 is to show that

As in the ergodic case, one starts by showing that k V i allows to construct a cocycle Ân

x , x ∈ S, quotienting along F(x) = F V i+1 (x). One can build the quotient spaces E(x) using Lemma 3.10 on restricting to sets of the form {M V i (x) = Const.}. The proof of Lemma 3.12 now proceeds with no real changes. The only change to the remainder of the proof of Proposition 3.8(a) is that one shows

(x) on a set of the form U ∩ T -1 U , where m(U ) > 1δ. Taking δ → 0 completes the proof of part (a); part (b) is treated similarly. 3. One checks that the case k B ̸ = k V leads to a contradiction, implying m{r V = r B } = 1. The arguments in Sect. 3.3 now carry over without substantive changes.

Skew Product Formulation

In what follows, we will describe a class of time dependent velocity fields that are subordinate to an ergodic measure preserving flow. The formulation we present is in the general setting of a skew-product, defined below, which providing a natural dynamical framework for systems evolving on H driven by an external forcing, either random or deterministic.

Let ( , F , P) be a probability space and let θ t : be a flow of measurable ergodic, P-preserving transformations on . Assume that for each t 0, we have a mapping τ t : × H of the form

where t • : ×H → H is measurable and u → t ω (u) is continuous for all ω ∈ . Moreover, we will assume t ω satisfies 0 ω (u) = u and the cocycle property

for all ω ∈ . Mappings of the form τ t are referred to as skew product flows over θ t : → . Conceptually, for each ω ∈ we view the (potentially non-invertible) map t ω : H as describing the evolution of a time-varying incompressible velocity field

From this perspective, the set encodes the set of nonautonomous driving or forcing paths ω = (ω(t)) t 0 and θ t denotes the time shift flow. Equation (30) reflects that the forcing path evolving from time t to time t + r is given by θ t ω.

This framework includes a variety of evolution equations on H, including the 2d Navier-Stokes equations with either stochastic (e.g. white in time forcing) or deterministic (e.g. time-periodic) driving terms as (see [50] Section 2.4.4 for a construction of such an RDS in the case of white in time forcing). Higher dimensional (d

3) examples of fluid motion can be considered by adding hyperviscosity or by Galerkin truncations. Other models can be formulated in terms of a skew-product flow and don't necessarily need to solve a fluid equation, e.g., time-stationary fields, time-periodic fields, and time dependent linear combinations u t (x) = k u k (x)z k (t) of fixed, time-independent vector fields {u k }, where {z k (t)} are a collection of processes on R (e.g., Ornstein Uhlenbeck processes).

Lastly, throughout what follows we will assume m is a τ t -invariant measure on

The measure m captures the statistics of typical velocity fields (u t ) with respect to the model, while (31) reflects that P is the law of the underlying driving. For additional discussion on ergodicity and invariance, see Sect. 2.1.1.

Passive Scalar Advection Linear Cocycle

Let τ t : × H → × H be as above. Given a fixed initial (ω, u 0 ) ∈ × H and t > 0, define u t := t ω (u 0 ). For κ > 0, we are interested in solutions ( f t ) t 0 to the passive scalar advection diffusion equation

for fixed initial mean-zero scalars f 0 : T d → R. Being a parabolic equation with Lipschitz velocity field, well-posedness of (32) H s for any s 0 is classical. One can extend to H -s by the density of H s in H -s , linearity of the equation, and the L 2 duality of H s and H -s .foot_3

Proposition 4.1. For any s ∈ R and (ω, u 0 ) ∈ × H, there is a semiflow of compact linear operators S t ω,u 0 : H s , t 0, such that f t := S t ω,u 0 f 0 is a solution to (32) with initial data f 0 ∈ H s . Moreover, the operators S t ω,u 0 form a linear cocycle over τ t : for r, t 0 we have S t+r ω,u 0 = S r θ t ω,u t • S t ω,u 0 for all (ω, u 0 ) ∈ × H. Our first result says that for measure-typical velocity fields (u t ), the asymptotic exponential growth (or decay) rate of ( f t ) exists for any initial scalar f 0 , and that 'typical' initial scalars see only a single exponential growth rate λ 1 independent of the H s norm one uses. Theorem 4.2. Let γ ′ γ > 1+d/2 and assume that there is an invariant measure m for τ t : × H γ that satisfies the mild moment condition

Let κ > 0 be fixed. Then the following hold:

(a) For m-a.e. (ω, u 0 ) ∈ ×H γ and for all s ∈ [-γ ′ , γ ′ ] and mean-zero f 0 ∈ H s , the global solution f t = S t ω,u 0 f 0 to (32) has the property that the limit

exists and is independent of s ∈ [-γ ′ , γ ′ ]. (b) If m is also ergodic, then there exists λ 1 ∈ R ∪ {-∞} and N ∈ Z 0 , each depending only on κ, with the following property: for all s ∈ [-γ ′ , γ ′ ], for m-a.e. (ω, u 0 ) ∈ × H, and for all f 0 chosen off of an N -codimensional subspace of H s , we have

Remark 4.3. In contrast to classical parabolic regularity theory, which gives H s regularity of f t for all s 0 as long as u t ∈ H locally uniformly in t, Theorem 4.2 requires more quantitative regularity estimates in terms of u t . Consequently, the range of s to which equality of exponents applies is constrained by the regularity of u where certain moments are available.

Proof. As an immediate implication of Lemmas 4.17 and 4.18 below, we obtain that for any -γ ′ s γ ′ , we have that S t ω,u 0 is a semiflow of compact linear operators in H s . By Lemmas 4.17 and 4.18, equation (33) implies the logarithmic moment estimate

which implies that the MET (Theorem 2.2) applies to S t ω,u 0 as a linear cocycle over (τ t , m) along integer times t. For the limits (34) taken along integer times, parts (a) and (b) now follow immediately from Theorem 2.6. To pass from discrete to continuous-time limits in part (a), it suffices 11 that the cocycle S t ω,u 0 satisfies

This too follows from Lemmas 4.17 and 4.18.

As discussed in Sect. 1.1.1, at κ = 0 equality of exponents does not hold. This suggests that as κ → 0 the "rate" at which the Lyapunov exponent is realized in H s depends heavily on s. For example, although the exponents in H 1 and L 2 agree and are negative as t → ∞, there is a κ-dependent transient timescale along which the H 1 norm increases before decay starts [60], while the L 2 norm can only decrease. Now we provide a way of quantifying this κ-dependence. For simplicity, we state the result in the case when (τ t , m) is ergodic and the comparison between L 2 and H s , s > 0.

For ε > 0, s ∈ [0, γ ′ ], define the Lyapunov regularity functions 12

Corollary 4.4. Assume the setting of Theorem 4.2, and in addition, that (τ t , m) is ergodic. Fix p > 3 and 0 < q < p 2 -3 p p-1 , and assume the moment condition

Then, for any δ, κ > 0 and -γ ′ s ′ < s γ ′ , there exists a function K s ′ ,s δ,κ :

and the following moment condition holds:

The proof is a straightforward consequence of Lemmas 4.17, 4.18 and Corollary 2.8.

Periodic Forcing

Below, we formulate evolution by the Navier-Stokes equations in the skew product formulation of Sect. 4.1.1 in the case of additive, time-periodic, spatially regular forcing. In this case, = S 1 , where the circle S 1 is parametrized by [0, 1) with the endpoints identified. The time shift θ t : is given by θ t ω = ω + t mod 1, while the measure P is normalized Lebesgue measure.

The following well-posedness and regularity results on Sobolev spaces are classical (see for instance [50,68,75]).

). For each fixed initial u 0 ∈ H 0 and for all ε > 0 there exists a unique solution u ∈ C([ε, T ]; H m ) ∩ L 2 ([ε, T ]; H m+1 ) for each ε > 0 and T 0. Moreover, there exists a constant C m such that the following inequality holds for each 0 t T :

(ii) For each 0 r m, there exists a continuous mapping t • : × H r → H r , over θ t , ( , F, P) such that u t = t ω (u 0 ) is the unique solution to (38) with initial data u 0 and forcing F t+ω . Moreover, t ω : H r → H r in injective and C 1 Fréchét differentiable for all ω ∈ , t 0. Since F t is periodic, it is natural to consider the time-one map 1 0 : L 2 since θ 1 ω = ω and therefore n

This mapping admits a compact global attractor A to which solutions converge: (a) The mapping 1 0 admits a compact global attractor A ⊂ H. Precisely, (i)

A is a homeomorphism, and (iii) for all u 0 ∈ H and ω ∈ , any subsequential limit u * = lim u n k of the trajectory (u n ) n 0 belongs to A.

(b) There exist invariant probability measures µ for 1 0 . Moreover, all such invariant measures are supported on A.

Proof sketch. For part (a), equation (39) implies that the time-1 semiflow 1 0 is dissipative on H, and so admits a global compact attractor A; see, e.g., Chapter 10 of [68] for further details. Injectivity of Given a 1 0 -invariant measure µ, we define an associated measure m on × H via dm(ω, u) = dµ ω (u)dP(ω), where µ

and

is the pushforward of the measure µ under the map ω 0 . We see that the measure m is an ergodic invariant measure for the skew-product flow τ t (ω, u) = (θ t ω, t ω (u)), and therefore we are in the general setup of Section 4.1.1. Proposition 4.7. Let µ be a probability measure on H, and define m and µ ω as in (40)

Proof sketch. Part 1 follows from the definitions and [50, Proposition 1.3.27]. For part 2, it is straightforward from the definitions that µ ω is ergodic for 1 ω : H for all ω ∈ . From here, it follows that any τ t -invariant function ψ(ω, u) is µ ωalmost surely independent of u, while ergodicity of P on implies almost-sure constancy in ω, so that in the end ψ itself is m-almost surely constant.

White-In-Time Forcing

Next we consider the white-in-time stochastically forced case. Specifically, we will assume that the forcing F is the time derivative of a Brownian process in H:

where {β j } are independent canonical 1d Wiener processes, {e j } forms an orthonormal basis for H, and the coefficients σ = {σ j } satisfy ∥σ ∥ 2 ℓ 2 = ∞ j=1 σ 2 j < ∞. This last condition ensures that the process ξ t = ξ(t, •) is a continuous process in H. Since ξ t is not differentiable in time, the Navier-Stokes equations must be interpreted in a time-integrated sense:

where equality holds in H -1 with probability 1.

In this setting we take to be the space C 0 (R + ; H) of continuous one-sided paths ω : R + → H vanishing at 0, with the standard Borel sigma algebra F and equipped with a Gaussian measure P whose projection onto basis elements e j through the map ω → ⟨e j , ω⟩ H is the canonical Wiener measure. We define the semiflow θ t : to be the shift map

which is easily seen to leave the measure P invariant.

The following well-posedness, regularity and construction of an RDS is wellknown (see e.g. [50] §2.4 ). Proposition 4.8. Let d = 2, γ > 1 + d 2 = 2, and suppose that F is of the form (41) where j j 2(γ -1) σ 2 j < ∞ (hence ξ t ∈ H γ -1 with probability 1). Then, there exists a measurable mapping t • : × H → H such that u t = t ω (u 0 ) is a strong pathwise solution to (38) (in the integral sense) with initial data u 0 and noise path ω = (ξ t ). Moreover, for P-a.e. ω, the mapping t ω : H → H is injective and C 1 Fréchét differentiable, and satisfies the cocycle property t+r ω = r θ t ω • t ω for all r, t 0, and for all ω ∈ . Lastly, ω → t ω only depends on ω| [0,t] . That t ω depends only on ω| [0,t] implies that it is Markovian, in the sense that u t = t ω is a Markov process. We say that a probability measure µ on H is a stationary measure for this Markov process if E( t ω ) * µ = µ . Existence of such a measure for dissipative RDS is generally guaranteed by a simple Krylov Bogoliubov argument (see, e.g. [50]). However, in contrast to the deterministic forcing case, it is often the case that this measure is in fact unique under fairly mild conditions on the noise (see e.g. [30,41]). It is well known that stationary measures µ are in one-to-one correspondence with the invariant measure m = P × µ on × H for the skew product flow τ t (ω, u) = (θ t ω, t ω (u)). Theorem 4.9. (a) (Theorem 4.2.9 [50]) Let µ be a stationary probability measure for (u t ). Then, m = P × µ is an invariant measure for the semiflow τ t : × H. (b) (Theorem I.2.1 [47]) If µ is the unique stationary measure for (u t ), then P × µ is ergodic for τ t .

Moreover depending on the regularity of F t one can obtain moment estimates of higher Sobolev norms with respect the the stationary measure µ. The following estimate is a consequence of [49] (also c.f. Exercise 2.5.8 [50]). Proposition 4.10. Suppose j j 2r σ 2 j < ∞ for some r γ . Then, any stationary measure µ for (38) satisfies the estimate

for all p 0.

Linearized Navier-Stokes

In either the time-periodically forced or stochastically forced setting, we will assume below that u t = t ω (u 0 ) on H. Our main goal is to study the linearized Navier-Stokes cocycle given by the Fréchét derivative D u 0 t ω which acts as a linear operator on divergence free velocity fields. Specifically, given an initial divergence free velocity v 0 ∈ H (viewed as an infinitesimal perturbation), the trajectory v t = (D u 0 t ω )v 0 satisfies the linearized or first variation equation

where q is the pressure enforcing the divergence-free constraint on v. Hence D u 0 t ω is the solution operator to the above linear equation and defines a compact linear co-cycle on H. By uniqueness of solutions to (42), the D u 0 t ω satisfy the cocycle property

To apply Theorem 2.6 in this setting, we want to treat D u 0 t ω as a cocycle over H s for a range of s, given a fixed base t ω on H = H γ , where γ > 2 is fixed. The following summarizes what is needed. Proposition 4.11. Fix γ ′ γ , ω ∈ , u 0 ∈ H and suppose u t = t ω (u 0 ) is a solution to the 2d Navier-Stokes equations (38) in the setting of either Proposition 4.5 or 4.8. Assume that for m-a.e. (ω, u 0

). Then: (a) The mapping D u 0 t ω : H extends 13 to a compact bounded linear operator on H s such that v t := D u 0 t ω v 0 is a solution to (42) with initial data v 0 ∈ H s . Equation (43) is satisfied as a cocycle on H s . (b) The mapping (ω, u 0 ) → D u 0 t ω is strongly measurable in H s . Proof sketch. Since (42) is parabolic (up to a compact perturbation), well-posedness in H s , s 0 is standard (see [42]), while well-posedness in H -s can be proved via a duality argument and using linearity of the equation. A priori estimates sufficient to deduce these statements for a range of s related to the regularity of u are presented in Sect. 4.4.3.

We are now in position to state our results on the Lyapunov exponents of 2d Navier-Stokes. Theorem 4.12. Let ν > 0 be fixed. Let τ t : H × be the skew product semiflow associated to the Navier-Stokes equations and let m be a τ t -invariant probability measure that satisfies the following moment condition for some γ ′ γ > 1+d/2 = 2:

Then: 13 When there is no confusion, we will abuse notation somewhat and write D u 0 t ω : H s → H s for the extended operator.

(a) For m-a.e. (ω, u 0 ) ∈ × H and for all s ∈ [-γ ′ + 1, γ ′ + 1] and mean-zero divergence free velocity fields v 0 ∈ H s , the global solution v t = D u 0 t ω v 0 to (32) has the property that the limit λ(ω, u 0 ; v 0 ) := lim

lim t→∞ 1 t log ∥v t ∥ H s . Proof. Lemmas 4.19 and 4.20 and the estimate (44) imply immediately that ∥D u 0 1 ω ∥ H s satisfies the analogue of the logarithmic moment estimate (35) for any s ∈ [-γ ′ + 1, γ ′ + 1]. The MET (Theorem 2.2) and Theorem 2.6 apply, implying convergence of the limits ( 45) and independence from s when taken along integer times. Passing from discrete to continuous time follows similarly, using the analogue of (36). Remark 4.13. Note that in light of the regularizing properties of Navier-Stokes the moment condition (44) for solutions to Navier stokes is ultimately a condition on the regularity of the force. In the periodically forced case, it is sufficient for F t ∈ H γ +1 for all t, due the to fact that there is an absorbing ball in H γ +2 . However, in the stochastically forced case, we require that F t belongs to H γ +2 , so that the moment bound (44) follows from Proposition 4.10 under the condition that j j 2(γ ′ +2) σ 2 j < ∞. We now apply Corollary 2.8 concerning Lyapunov regularity functions to the Navier-Stokes cocycle. For ε, ν > 0 and s ∈ [-γ ′ + 1, γ ′ + 1], let D H s ε,ν , D H s ε,ν : ×H → [1, ∞) denote the Lyapunov regularity functions for the cocycle D u 0 t ω , defined analogously to (37). Corollary 4.14. Assume the setting of Theorem 4.12, and in addition, that (τ t , m) is ergodic. Fix p > 3 and 0 < q < p 2 -3 p p-1 , and assume the moment condition

Then, for any δ, ν > 0 and -γ ′ + 1 s ′ < s γ ′ + 1, there exists a function K s ′ ,s δ,ν :

K s ′ ,s δ,κ D H s ε+δ,κ , and the following moment condition holds:

(log + K s ′ ,s δ,κ ) q dm p,q δ -( p-q) 1 + (ss ′ )| log ν| + I p .

Using this and that f t is smooth and u t is divergence-free gives the following energy estimate d dt

1 2 κ st ∥ f t ∥ 2 H st = -κ st ⟨ st f t , st (u t • ∇ f t )⟩ L 2 + κ st ⟨ st f t , (r log( √ κ 1 ) + κ ) st f t ⟩ L 2 -κ st ⟨ st f t , [ st , u t • ∇] f t ⟩ L 2 + κ st ∥ f t ∥ 2 H st κ st ∥ f t ∥ H st ∥[ st , u t • ∇] f t ∥ L 2 + ∥ f t ∥ 2 H st . Applying the commutator Lemma 4.15 with s = rt, assuming t ∈ [0, 1] and using that γ > d 2 + 1, we conclude that d dt κ st ∥ f t ∥ 2 H st (1 + ∥u t ∥ H γ ) κ st ∥ f t ∥ 2 H st . In particular, d dt log κ st ∥ f t ∥ 2 H st 1 + ∥u t ∥ H γ , and integrating t from 0 to 1 completes the proof.

Proof. As before, we focus below on the proof of (48) and omit that of (49). By an approximation argument, we can assume f 0 , u t are C ∞ . Our proof will use the L 2 duality of H -s with H s . To see this, let g 0 be a smooth, mean-zero function and let (g t ) solve the time reversed equation

We compute d dt ⟨g t , f 1-t ⟩ = ⟨∂ t g t , f 1-t ⟩ = -⟨g t , ∂ t f 1-t ⟩ = ⟨u 1-t • ∇g t , f 1-t ⟩ = -⟨g t , -u 1-t • ∇ f 1-t ⟩ = 0 using (i) that is self adjoint in the L 2 inner product and (ii) u • ∇ is skew-adjoint when u is divergence free. We conclude that ⟨ f 1 , g 0 ⟩ = ⟨ f 0 , g 1 ⟩ . Now,

∥g 1 ∥ H s , treating the g 0 under the sup as an initial condition for (50). By Lemma 4.17, it holds that ∥g 1 ∥ H s κ -s/2 exp c where w = curl u, and the velocity u is recovered by the Biot-Savart law u = -2 (∇ ⊥ w) =: K w, where here ∇ ⊥ = (-∂ y , ∂ x ) denotes the skew gradient. In this form, the first variation equation becomes

where v t = K η t .

Proof. Note that since w = ψ and ∇ ⊥ ψ = u, we can also rewrite the second term in 51 in the following more useful form:

Repeating previous computations, we have d dt 1 2 ν st ∥η t ∥ 2 H st = ν st ⟨ st η t , s log( √ ν ) + ν st η t ⟩ ν st ⟨ st η t , [ st , u t • ∇]η t ⟩ ν st ⟨ st η t , st ( u t • ∇) -2 η t ⟩ The first and second terms are bounded ν st (1 + ∥u t ∥ H γ )∥η t ∥ 2 H st as before, while by Lemma 4.16 we have that the third term is ν st ∥η∥ 2 H st ∥u t ∥ H γ +2 . Applying these inequalities and combining all terms, we have shown that d dt ν st ∥η t ∥ 2 H st ν st ∥η t ∥ 2 H st ∥u t ∥ H γ +2 .

The desired conclusion follows as before. The estimate when η 0 ∈ H s is similar and omitted.

Using a time reversed adjoint argument, we also have the analogue of Lemma 4.18, whose proof we omit. (53)