dissipative length scale estimates

# Dissipative length scale estimates for turbulent flows - a Wiener algebra approach

A. Biswas Department of Mathematics and Statistics
University of Maryland, Baltimore County
Baltimore, MD 21250.
M. S. Jolly Department of Mathematics
Indiana University
Bloomington, IN 47405
V. Martinez Department of Mathematics
Indiana University
Bloomington, IN 47405
and  E. S. Titi Department of Mathematics and Department of Mechanical and Aerospace Engineering
University of California
Irvine, California 92697
Also:
Department of Computer Science and Applied Mathematics
Weizmann Institute of Science
Rehovot, 76100, Israel
corresponding author
July 26, 2019
###### Abstract.

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier-Stokes system on an -torus. This estimate improves or matches previously known estimates provided that certain bounds on the initial data are satisfied. It is argued that for 2D or 3D turbulent flows, the initial data is guaranteed to satisfy these hypothesized bounds on a significant portion of the 2D global attractor or the 3D weak attractor. In these scenarios, the estimate obtained for 3D generalizes and improves upon that of [9], while in 2D, the estimate matches the best known one found in [26]. A key feature in the approach taken here, is the choice of the Wiener algebra as the phase space, i.e. the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms.

###### Key words and phrases:
Navier-Stokes equations, turbulence, radius of analyticity
###### 2010 Mathematics Subject Classification:
35Q30, 76D05, 76F02, 76N10

A. Biswas]abiswas@umbc.edu M. S. Jolly]msjolly@indiana.edu V. Martinez]vinmarti@indiana.edu E. S. Titi] etiti@math.uci.edu

## 1. Introduction

The conventional theory of turbulence posits the existence of certain universal length scales of paramount importance. For instance, according to Kolmogorov, there exists a dissipation length scale, , beyond which the viscous effects dominate the nonlinear coupling. This length scale can be characterized by the exponential decay of the energy density. Consequently, one expects the dissipation wave-number, , to majorize the inertial range where energy consumption is largely governed by the nonlinear effects and dissipation can be ignored.

In [13, 9] it is shown that as characterized by Gevrey norms, the (uniform) radius of spatial analyticity, here denoted , provides a lower bound for the dissipation length scale, i.e., . The space analyticity radius has been well-studied over the years, especially after the pioneering work of Foias and Temam in [15], where they presented a novel Gevrey norm approach to establish analyticity of solutions to NSE in both space and time. An advantage of this approach is that it avoids having to make cumbersome recursive estimates on derivatives. Consequently, it has become a standard tool in estimating the analyticity radius for various equations (cf. [12, 33, 32, 29, 3, 2, 27, 28]).

Kolmogorov’s theory for 3D turbulence asserts that

 (1.1) λd∼λε:=(ν3/ε)1/4,

where is viscosity and is the mean energy dissipation rate per unit mass.

For 3D decaying turbulence, it is shown in [9] that

 (1.2) λa∼κ−10(κ0~λε)4,

where is as in (1.1), except that the energy dissipation rate is a supremum in time rather than an averaged quantity (see (3.11), (3.14)). The more significant discrepancy is a power of 4 versus a power of 1 in (1.1). Our improvement is done under the 2/3-power law assumption (3.16) on the energy spectrum for a forced, turbulent flow, by means of an ensemble average with respect to an invariant measure. It is valid on a portion of the attractor (weak in the 3D case); the significance of which is quantified in terms of this measure. Ultimately, we conclude that

 (1.3) λa≳pκ−10(κ0λε)59/24

holds with probability , where the suppressed constant in the inequality tends to as . Similarly, a heuristic scaling argument by Kraichnan for 2D turbulence leads to

 (1.4) λd∼λη:=(ν3/η)1/6,

where is the mean enstrophy dissipation rate per unit mass. We show that if the 2D power law (3.26) for the energy spectrum holds, then

 (1.5) λa≳pκ−10(κ0λη)2

up to a logarithm in .

These improved estimates actually follow from more general bounds on the radius of analyticity which require the solution to satisfy a certain “smallness” condition. Those conditions are met under the power law assumptions, when averaged with respect to an invariant measure. Kukavica [26] achieved the same bound in 2D up to a logarithmic correction on all of the attractor using complex analytic techniques, interpolating between norms of the initial data and the complexified solution, and invoking the theory of singular integrals.

The approach in [26] was actually a modification of the approach in [21], where it was shown that . It is interesting to ask if these estimates can be obtained by working exclusively in frequency-space using Fourier techniques, rather than in physical space with the norm. Indeed, this is an impetus of our work.

The technique applied here combines the use of Gevrey norms with the semigroup approach of Weissler [37]. Motivated by recent developments, we work over a subspace of the Wiener algebra, whose norm is a Sobolev-Gevrey-type norm in (see (2.11)). This norm and approach was applied in [4] to study spatial analyticity and Gevrey regularity of solutions to the NSE. However, the resulting estimate on the spatial radius of analyticity was not optimal for large data. This approach is refined here to obtain a sharper estimate for such data. The advantage of working in the Wiener algebra, , i.e. the Banach algebra of functions whose Fourier series converge absolutely, was explored in [33], where a sharp estimate on the radius of analyticity was obtained, for instance, for real steady states of the nonlinear Schrödinger equations. More recently, these -based Gevrey norms were also applied to the Szegö equation in [20] and the quasi-linear wave equation in [22]. In [20], an essentially sharp estimate on the radius is obtained there as well. While these works used energy-like approaches, the effectiveness and robustness of as a working space to study analyticity has become increasingly clear.

There are several advantages to our approach. First, our method is quite elementary. Since is embedded in , we essentially recover the results of [21] and [26] without resorting to complex-analytic techniques and the theory of singular integrals, while furthermore allowing for rougher initial data. Secondly, this approach also applies to the case , thereby unifying the results of [9], [15], [21], and [26] . Thirdly, no logarithmic corrections appear in our estimates initially; they only appear when specializing to the context of 3D or 2D turbulence (see (3.1.2)). Finally, the method is rather robust and applies to a wide class of active and passive scalar equations with dissipation, including the quasigeostrophic (QG) equations. Note that in the case of QG with supercritical dissipation, the method will only accommodate subanalytic Gevrey regularity (see [31]).

## 2. Preliminaries

The Navier-Stokes system in for is given by

 (2.1) ⎧⎨⎩ut−νΔu+u⋅∇u+∇p=F ∇⋅u=0u(x,0)=u0(x)

where and are given, and and are unknown. We assume that are all -periodic and mean-zero, and that is divergence-free.

We will use the so-called wave-vector form of (2.1), which is simply (2.1) written in terms of its Fourier coefficients

 (2.2) ⎧⎪⎨⎪⎩ddt^u(k,t)=−νκ20|k|2^u(k,t)+B[→u,→u](k,t)+ˆf(k,t)k⋅^u(k,t)=0^u(k,0)=^u0(k),

where , such that , and , where is the Helmholtz-Leray orthogonal projection, i.e. projection onto divergence-free vector fields,

 (2.3) P(^u(k)eiκ0k⋅x)=(^u(k)−(k|k|⋅^u(k))k|k|)eiκ0k⋅x,(k∈Zn).

Recall also that the mean zero condition forces for all . The bilinear term has Fourier coefficients given by

 (2.4) B[→u,→v](k,t)eiκ0k⋅x:=iκ0P⎛⎜⎝∑ℓ∈Zn∖{→0}(k⋅^u(ℓ,t))^v(k−ℓ,t)eiκ0k⋅x⎞⎟⎠,

Note that will denote the sequence .

Observe that

 (2.5) |ˆPu(k)|≲|^u(k)|,

and also that the following basic convolution estimate holds

 (2.6) |B[→u,→v](k)|≲κ0|k|(|→u|∗|→v|)(k) for all k∈Zn.

Since we will be working with (2.2), we choose an appropriate sequence space as our ambient space. Define

 (2.7) K

where . For define

 (2.8) Vσ

where

 (2.9) ∥→u∥σ :=κσ0∑k∈Zn|k|σ|^u(k)|.

and denotes an element of . Observe that when , the norm on agrees with that on the Wiener algebra, i.e.

 (2.10) (νκ0)−1∥→u∥0=∥u∥W,

where is the continuous function whose Fourier coefficients are given by . In fact, we have , for all .

For , we define the (analytic) Gevrey norm of by

 (2.11) ∥→u∥λ,σ :=κσ0∑k∈Zneλκ0|k||k|σ|^u(k)|

for . Observe that has the physical dimension of length.

For a time-dependent sequence such that , for all , we define the (analytic) Gevrey norm of by

 (2.12) ∥→u(t)∥λ(t),σ :=κσ0∑k∈Zneλ(t)κ0|k||k|σ|^u(k,t)|,

where is increasing and sublinear, i.e. for all . Observe that

 (2.13) ∥u(t)∥W≲σλ(t)φ(σ)κ−σ0νκ0∥→u(t)∥λ(t),σ,

for all and , where if and otherwise.

It is well-known that the Gevrey norm characterizes analyticity, a fact stated more precisely in the following proposition (cf. [29], [23]):

###### Proposition 1.

Let .

1. If , then admits an analytic extension on ;

2. If has an analytic extension on , then for all .

In particular, if a function has finite Gevrey norm, then the Fourier modes decay exponentially. Indeed, if , then

 (2.14) |^u(k)|≤e−λ|k||k|−σ∥→u∥λ,σ.
###### Definition 1.

If is analytic, then we define

 (2.15) λmax=sup{λ′>0:∥→u∥λ′,σ<∞}

to be the the maximal (uniform) radius of spatial analyticity of . Moreover, due to (2.14) we have .

###### Remark 2.

For convenience, we adopt the following conventions for the rest of the paper.

1. We will usually write simply as , which is the function whose Fourier series have modes , for .

2. By or , or when the context is clear, simply , we shall mean the time-dependent sequence , unless otherwise specified.

3. We will use to suppress extraneous absolute constants or physical parameters. In some instances, the dependence of these constants will be indicated as subscripts on .

4. We will also use the notation to denote that the two-sided relation and holds.

For and , we define

 (2.16) M0 :=κ−σ0νκ0∥u0∥σ, (2.17) Mf :=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩κ−σ0ν2κ30(νκ20∫Tf0∥f(s)∥qλ(s),σ ds)1/q,1≤q<∞κ−σ0ν2κ30sup0≤t≤Tf∥f(t)∥λ(t),σq=∞

and

 (2.18) M :=M0+Mf.

For any dimension , the Grashof number is defined as

 (2.19) G:=κn/20ν2κ30sup0≤t≤Tf∥f(t)∥L2.

Observe that and are dimensionless. One can show that when is time-independent and has only finitely many modes, i.e. , where

 (2.20) P¯κf:=∑|k|≤¯κ/κ0^f(k)eiκ0k⋅x,

then is comparable to up to a constant depending on only , a fixed parameter , and , where satisfies

 (2.21) sup|y|≤λf∥f( ⋅+iy)∥L2<∞;

see Proposition 23 in Appendix.

Now suppose that data and are given such that . Let be the Stokes operator, , where is defined as in (2.3). Then the heat kernel, , is the Fourier multiplier defined by

 (2.22) ˆeνtAu(k):=e−νtκ20|k|2^u(k),

or equivalently, . We will use two notions of solutions to (2.2).

###### Definition 2.

For , a mild solution to (2.2) is any function such that

 (2.23) ∫t0e−ν(t−s)κ20|k|2|B[→u,→u](k,s)| ds<∞,

for all , and

 (2.24) →u(t)=e−νtA→u0+∫t0e−ν(t−s)A→Pf(s) ds−∫t0e−ν(t−s)A→B[→u,→u](s) ds,

for all .

###### Definition 3.

For , a weak solution to (2.2) is any function such that

 (2.25) B[→u,→u](k,t)<∞

for all and a.e. and

 (2.26) ddt^u(k,t)=−νκ20|k|2^u(k,t)−B[^u,^u](k,t)+^f(k,t)

holds for all and a.e. .

The fact that Definition 3 is equivalent to the usual definition of weak solution for a periodic flow can be found in [36].

Finally, we define the regularity that we ultimately seek to establish.

###### Definition 4.

A mild or weak solution of (2.2) is Gevrey regular if there exists and sublinear such that

 (2.27) sup0≤t≤T∥→u(t)∥λ(t),σ<∞.

## 3. Main Theorems

We first state a result for a general force.

###### Theorem 3.

Let and and be as defined in (2.18). Suppose that and are given such that . Then for some , there exists a mild solution to (2.2), which is also a Gevrey regular weak solution, with radius of analyticity at time satisfying

 (3.1) λa≳κ−10⋅{M−1/(1−2|σ|/q′),1

where . Moreover, there exists a constant such that if , then one may take . In this case, the solution exists for all and the radius of analyticity at time satisfies

 (3.2) λa≳√νt.

In the case where the forcing is time-independent and has finitely many modes, we can express the estimate on the radius of analyticity in terms of the Grashof number, provided a “smallness” condition on the solution holds.

###### Theorem 4.

Suppose that is time-independent and satisfies . If

 (3.3) ∥u0∥W≲G1/2,

then for some , there exists a unique weak solution to (2.1) such that is Gevrey regular and the radius of analyticity at time satisfies

 (3.4) λa≳¯κ,κ0κ−10G−1/2.

The following estimate is not as sharp, but holds under a weaker “smallness” condition.

###### Theorem 5.

Suppose that is time-independent and satisfies . If

 (3.5) ∥A−3/8u0∥W≲κ−3/40G11/16,

where with periodic boundary conditions, is the Stokes operator, then for some , there exists a weak solution to (2.1) such that is Gevrey regular and the radius of analyticity at time satisfies

 (3.6) λa≳¯κ,κ0κ−10G−59/64.
###### Remark 6.

One can also have in Theorem 3 (see its proof in Section 7). In fact, a more general version of Theorem 4 and 5 is proved in Section 7 (see Theorem 20).

The estimate on in Theorem 3 can be compared to the one in [4] when . However, in that work their choice of (as in the Definition 4) yielded instead the estimate

 (3.7) λa≳κ−10M−2/(1−|σ|),

which is less sharp than the corresponding estimate in (3.1) when is large.

One should also note that if is too small, then the global attractor in 2D becomes trivial (cf [6, 30]). Physically, this corresponds to the case of decaying turbulence. Nevertheless, if is sufficiently small, then is allowed, in which case the solution exists globally in time with radius that grows without bound in time as .

Uniqueness of weak solutions to (2.1) is guaranteed in two-dimensions, but in 3D is still an open question. There are, however, cases where the uniqueness is guaranteed in any dimension (see [36] pp. 298-99). In particular, as long as , the solution of Theorem 3 is unique in the class of weak solutions.

In the case where the force is identically zero, one can employ energy techniques as in [9], [15] and obtain

 (3.8) λa≥Cκ−10∥u0∥W

where represents the radius of analyticity at some time strictly less than the maximal time of existence. The constant here can be explicitly identified as , where is the nontrivial solution to

 (2γ)−1log(1+γ)−(1+γ)−1=0.

Note that (3.8) is precisley the estimate in (3.1) (up to an absolute constant). The energy approach, however, encounters technical difficulties when one includes forcing on infinitely many scales. The reader is referred to [31] for additional details.

In [31], the estimates are also done in for . In particular, when , the result of [9] is generalized to include forcing on all scales, and the estimate on the radius is the same as the one derived there (up to an absolute constant). One can make an argument similar to the one presented in Section 3.1 that would justify the corresponding assumption on the initial data, but working on the 3D weak attractor. For background on the weak attractor, see [7] or [17].

Finally, the techniques used to prove Theorem 4 apply equally well to the vorticity formulation of Navier-Stokes, the case of fractional dissipation, and a wide class of active and passive scalar equations, including 2D dissipative QG equations, (see [31]). These techniques also apply to the case (see [2]). For more results on the subcritical QG, see for instance [5], where analyticity is established for arbitrary initial data in , or [11], where a local smoothing effect is exploited to establish analyticity, or [2], where analytic Gevrey regularity is established for several other equations as well. For results on the analyticity of solutions for critical QG equations, see [10] and [24]. For results on the regularity of passive scalar equations see [34] or [35]. The classical Hilbert space techniques of [15] have also been successfully applied to the Euler equations (see [27] and [29]).

### 3.1. Application to Turbulent Flows

In this subsection, we show how our results in Theorems 4, 5 improves the known estimates for for turbulent flows. While their “smallness” assumptions may not hold on all of the 2D global (3D weak) attractor, in the context of turbulence, one can expect these conditions to hold on average, in a precise sense.

The statistical theory of turbulence concerns relations between quantities that are averaged, either with respect to time or over an ensemble of flows, e.g. results from repeated experiments. It is remarkable that these two seemingly different approaches are in fact related.

The mathematical equivalent of a large time average is rigorously expressed in terms of Banach limits. Following [17], define the space by

 (3.9) H:={(^u(k))k∈Zn∈(Cn)Zn:∥→u∥ℓ2<∞}∩K.

Let be a real-valued weakly continuous function on . Then for any weak solution of (2.2) on , there exists a probability measure for which

 (3.10) ⟨Φ⟩:=∫HΦ(u) dμ(u)=LimT→∞1T∫T0Φ(u(t)) dt,

where is a Hahn-Banach extension of the classical limit. The measure is called a time-average measure of . Note that neither nor are unique. The use of surmounts the technical difficulty that the limit in the usual sense may not exist. If is weak solution to the 2D NSE, then by regularity of such solutions, one can work in the strong topology on . Moreover, by uniqueness, one can show that is in fact invariant with respect to the corresponding semigroup, i.e. for all , for all measurable sets . Thus, a time average measure is also a so-called stationary statistical solution of the NSE. For a more detailed background see [17].

We now specialize to the cases of 3D and 2D turbulence, and interpret the main theorems in those settings.

#### 3.1.1. 3D Turbulence

The mean energy dissipation rate per unit mass is defined as

 (3.11) ε:=νκ30⟨∥∇u∥2L2⟩.

In 3D, Kolomogorov argued that because one can ignore nonlinear effects in the dissipation range, the length scale indicating where dissipation is the dominant effect should depend solely on and . By a simple dimensional argument, one then arrives at

 (3.12) λε=(ν3ε)1/4.

In other words, according to Kolmogorov, for turbulent flows in 3D, with given in (3.12). We will now describe the best known rigorous result in this direction.

In [9], the radius of analyticity was estimated in terms of as

 (3.13) λa≳(νκ0)3ε% sup.

where

 (3.14) εsup:=νκ30sup0≤t≤T∗/2∥∇u(t)∥2L2.

represents the largest instantaneous energy dissipation rate (per unit mass) up to time , and is the maximal time of existence of a regular solution. A heuristic argument is given to support as in [9], then (3.13) becomes

 (3.15) λd≳κ−10(κ0~λε)4,where~λε=(ν3εsup)1/4

It is not presently known if remains finite beyond . Hence, it is not possible to obtain an estimate of the smallest length scale for an arbitrary weak solution. In fact, it is not possible to extend these estimates on the weak attractor either since it is not known whether or not a trajectory, i.e. a weak solution defined for all , is regular. However, it is well-accepted that statements regarding length scales in turbulence actually concern “averages” and not specific trajectories (cf. [14, 16, 18, 1], or [17, 19] for introductory approaches). Indeed, this is the thrust of our current discussion.

In addition to the dissipation range and wave number, another basic tenet in the Kolmogorov theory of turbulence is the so-called power law for the energy spectrum. More specifically, let denote the wave number in which energy is injected into the flow, i.e., . Denote the Kolmogorov wave-number . Then the range of wave-numbers is known as the inertial range in which the effect of viscosity is negligible. The nonlinear (inertial) term simply transfers the energy injected into the flow through the inertial range at a rate of . Moreover, defining the quantity

 eκ1,κ2:=κ30⟨∥(Pκ2−Pκ1)u∥2L2⟩,

the well-celebrated Kolmogorov’s power law asserts that a turbulent flow must satisfy the relation

 (3.16) eκ,2κ ∼ε2/3/κ2/3, for κ∈[¯κ,κε].

Additionally, it is also known that if the Grashof number is sufficiently small, then the flow is not turbulent and the attractor in this case consists of only one point. In view of this discussion, we define a flow to be turbulent if the Kolmogorov power law holds and the Grashof number is sufficiently large, i.e.

 (3.17) G≳(¯κκ0)3/2.

It is shown in [7] that for such a flow one necessarily has the bounds

 (3.18) ν2κ0(κ0¯κ)5/2G≲⟨ ∥u∥2L2⟩≲ν2κ0(κ0¯κ)G, (3.19) ν2κ0(κ0¯κ)11/4G3/2≲⟨∥ A1/2u∥2L2⟩≲ν2κ0(κ0¯κ)1/2G3/2.

The following is the main result of this section which improves upon the estimate in [9] for 3D turbulent flows.

###### Theorem 7.

Let be a time-average measure for a 3D turbulent flow and let . There exists a set with such that

 λd(u)≳pκ−10(κ0λε)59/24 for all u∈S.
###### Proof.

Recall that Theorem 5 ensures that

 (3.20) λa≳κ−10G−59/64,

provided that the initial data satisfies

 (3.21) ∥A−3/8u0∥W≲κ−3/40G11/16.

We argue that (3.21) is guaranteed to hold on a significant portion of the 3D weak attractor, . We now quantify the likelihood that (3.21) occurs within with respect to any time-average measure .

First, observe that by Proposition 25 with , one has the inequality

 (3.22) ∥A−3/8u∥W≲∥u∥1/4L2∥A1/2u∥3/4L2.

Let and define the following sets

 Ap :={u∈Aw:∥u∥L2≳√2pνκ−1/20(κ0¯κ)1/2G1/2} Bp :={u∈Aw:∥A1/2u∥L2≳√2pνκ1/20(κ0¯κ)1/4G3/4}.

Then by (3.18), (3.19), and Chebyshev’s inequality

 μ(Ap)≤p2 and μ(Bp)≤p2.

We note that the support of is contained in (see [17]), so that (3.18) and (3.19) ensure that these inequalities are not trivial. It follows that

 μ((Aw∖Ap)∩(Aω∖Bp))≥1−p.

This combined with (3.22) implies that

 (3.23) μ{u∈Aw:∥A−3/8u0∥W≲√2pνκ1/40G11/16}≥1−p.

Then Theorem 5 gives

 (3.24) μ{u∈Aw:λa(u)≳pκ−10G−59/64}≥1−p,

where suppresses a constant which tends to as . Finally, observe that (3.19) implies that , so that . Therefore

 (3.25) μ{u∈Aw:λa(u)≳pκ−10(κ0λε)59/24)}≥1−p,

where denotes the radius of analyticity of at time .

In particular, we have just shown that for any , the radius of analyticity for the corresponding solution at time is bounded below by

 λa≳pκ−10(κ0λε)59/24,

provided that we are in the turbulent scenario described above. ∎

#### 3.1.2. 2D Turbulence

In the Kraichnan theory of 2D turbulence enstrophy is also dissipated, and it does so at a mean rate per unit mass given by

 η=νκ20⟨|Au|2⟩.

Two key wave numbers are

 κη:=(ην2)1/6∼(⟨∥Au∥2L2⟩L2ν2)1/6,  κσ :=(⟨∥Au∥2L2⟩⟨∥A1/2u∥2L2⟩)1/2,

where is the Stokes operator.

It is shown in [6], that if the well-recognized power law

 (3.26) eκ,2κ=⟨∥P2κQκu∥2L2⟩∼η2/3κ2,

holds on over the inertial range and if

 (3.27) κ––i≤4κη,⟨∥A1/2Pκ––iu∥2L2⟩≲⟨∥A1/2Qκ––iu∥L2⟩,G≳(¯κ/κ0)2,

then

 (3.28) ν2κ20(¯κκ0)−1G≲⟨∥A1/2u∥2L2⟩≲ν2κ20(¯κκ0)G(lnG)3/2 (3.29) ν2κ40(¯κκ0)−3/2G3/2(lnG)3/2≲⟨∥Au∥2L2⟩≲ν2κ40(¯κκ0)3/2G3/2(lnG)3/4.

This is to say that on average is of order on the global attractor. As in the 3D case, we can make this precise in terms of probabilities.

First, observe that by the “time-averaged” Brézis-Gallouët inequality (see Proposition 24)

 (νκ0)2⟨∥u0∥2W⟩≲⟨∥A1/2u0∥2L2⟩(1+ln(κσ/κ0)).

Hence, (3.28) and (3.29) imply that

 ⟨∥u0∥2W⟩≲LG,

where

 L:=(¯κ/κ0)(lnG)3/2[1+ln((¯κ/κ0)5/2G1/2(lnG)3/4)],

As before, Chebyshev’s inequality then implies

 (3.30) μ{u∈A:∥u∥W≲√LpG1/2}≥1−p,

for any , provided that either (3.26) and (3.27) hold. Therefore, we can conclude by Theorem 4 that

 (3.31) μ{u∈A:λa≳κ−10G−1/2}≥1−p,

where the constant inside depends only on , , and logarithms of . Since by (3.29)

 λη=(ν3η)1/6≤1κ0(κ0¯κ)1/4G−1/4,

we have the following

###### Theorem 8.

Let be a time-invariant measure for a 2D turbulent flow and let . There exists a set with such that

 λd(u)≳pκ−10(κ0λη)2 for all u∈S.
###### Remark 9.

There are also 3D versions of a time-averaged Brézis-Gallouët inequality, i.e. Proposition 24, which accomodate the endpoint cases of the Agmon-type inequality in Proposition 25, namely, and . However, neither of these cases fit within our discussion. Indeed, in the case , one must have some control over the quantity , which is not presently known. On the other hand, although we do have control over the quantity in 3D, in this case the Brézis-Gallouët inequality will only provide an estimate for the quantity , which lies outside of the range allowed by Theorem 20. Let us lastly note that if one could control , at least on average, then one could argue as before and apply Theorem 20 to obtain the estimate .

## 4. Outline of Proofs of Main Theorems

Following [4], our approach is to use a contraction mapping argument. Fix , , and . Define the spaces

 (4.1) X :={u( ⋅)∈C([0,T];Vσ):∥u∥X<∞}, (4.2) Y :={u( ⋅)∈C((0,T];Vσ+β):∥u∥Y<∞}, (4.3) Z :=X∩Y,

where are equipped with the norms

 (4.4) ∥u∥X :=κ−σ0νκ0⋅sup0≤t≤T∥u(t)∥√νt,σ, (4.5) ∥u∥Y :=νβ/2κ−σ0νκ0⋅sup0

and . Then are Banach spaces with continuously. Observe moreover that these norms are dimensionless.

By the Duhamel principle, the solution that we seek will be a fixed point of the operator defined by

 (4.7) (Su( ⋅))(t):=e−νtAu0+∫t0e−ν(t−s)APf(s) dsΦ(t)−∫t0e−ν(t−s)AB[u(s),u(s)] dsw(t).

In particular, we establish the existence of such a function in the closed subset given by

 (4.8) E:={u∈Z:∥u−Φ∥Z≤C},

for some , which satisifes . To do so, we will invoke the following existence theorem whose proof can be found in [4].

###### Theorem 10.

Suppose that and that for some . If and whenever and , for given by either

 (4.9) w(t)=∫t0e−ν(t−s)AB[u(s),v(s)] ds  or  w(t)=∫t0e−ν(t−s)AB[v(s),u(s)] ds,

then there exists a unique such that

 (4.10) u=Φ−∫