Continuous Data Assimilation with Blurred-in-Time Measurements of the Surface Quasi-Geostrophic Equation*

Michael S.JOLLY Vincent R.MARTINEZ Eric J.OLSON Edriss S.TITI

(Dedicated to Professor Andrew J.Majda on the occasion of his 70th birthday)

Abstract An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time.The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average.Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows,provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine,that the approximating solution converges exponentially fast to the observed solution over time.In particular,the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account.Two key ingredients in our analysis are additional boundedness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

Keywords Data assimilation,Nudging,Time-Averaged observables,Surface quasigeostrophic equation

1 Introduction

The surface quasi-geostrophic(SQG for short)equation models the dynamics of the potential temperature on the two-dimensional horizontal boundaries of the three-dimensional quasigeostrophic equations,which,in turn,are approximations to the shallow water equations in the limit of small Rossby number where the inertial forces are an order of magnitude smaller than the Coriolis and pressure forces.This is the regime of strong rotation,where the time scales associated with atmospheric flow over long distances are much larger than the time scales associated with the Earth’s rotation(cf.[38]).The model of focus in our study of data assimilation is the subcritically dissipative SQG equation subject to periodic boundary conditions over the fundamental domain T2=[-π,π]2.In non-dimensionalized variables,it is given by

Since their introduction into the mathematical community by Constantin,Majda and Tabak[15],the subcritical,critical and supercritical SQG equations have been thoroughly studied.Well-posedness and global regularity in various function spaces has been resolved in all but the supercritical case(cf.[9,16-18,20,21,34-35,39]),and also for certain inviscid regularizations(cf.[33]).The long-time behavior in the subcritical and critical has been studied as well and in particular,a global attractor theory has been established for them (cf.[10,12,13,16,19,29]).These equations have been used to simulate the production of fronts in geophysical flows and in spite of being a scalar model in two dimensions,possess solutions that behave in ways that are strikingly similar to fully three-dimensional flows.Therefore,(1.1)provide a physically-relevant dynamical context in which to analyze the performance of our model for data assimilation,that also supplies additional analytical difficulties that requires us to further develop the theoretical foundations of our approach.

Given a geophysical equation that describes some aspect of reality,the ability to predict the future using this equation requires an initial condition that accurately represents the current physical state.Although weather data has been collected nearly continuously in time since the 1960s,this data represents,at best,an incomplete picture of the current state of the atmosphere.Thus,rather than an exact initial condition,in practice one has a time series of low-resolution observations.Moreover,due to the nature of the measuring devices,the data itself may contain noise as well as systematic errors.Of particular interest to our present study is the fact that nearly all physical instrumentation produces measurements which are manifestly blurred in time.For example,the heat capacity of a thermometer naturally averages temperatures as they change over time while the rotational inertial of an anemometer similarly averages velocities.Time averages in satellite images result from finite shutter speeds and further averages result when satellite data is obtained by comparing two subsequent images.Blocher [6] shows both analytically and computationally that noisy,blurred-in-time observations of the X variable can be used to synchronize two copies of the three-dimensional Lorenz system of ordinary differential equations(ODEs for short)up to a factor of the variance of the noise,see also[7].As the analysis of the SQG equation is more complicated,we do not consider noise or systematic errors in this work,as this was studied in[5]and[26],but instead focus solely on how to assimilate data that has been subject to a moving time average.

The idea of finding the current physical state by combining a time-series of partial observations with knowledge about the dynamics dates back to a 1969 paper of Charney,Halem and Jastrow [11].Doing this optimally is the subject of data assimilation.Data assimilation has received considerable attention in both its theoretical development and practical use for the prediction of the weather (cf.Kalnay [30] and references therein).The approach of interest in this article computes an approximation using a “auxiliary system” obtained by taking the original model,which is assumed to coincide with the observations in the absence of measurement error,and applying feedback control based on the observations.This feedback control serves to nudge the solution towards the unknown but observed solution no matter what original initial condition was chosen for it.In theory,one could then integrate the approximate solution forward in time to obtain a good approximation of the current physical state.This approximation would then serve as an initial condition for subsequent forecasts.

The auxiliary system described above was first proposed as an approach to data assimilation for the model problem of the two-dimensional incompressible Navier-Stokes equations by Azouani,Olson and Titi in [2].In that work,exponential convergence of the approximating solution to the observed solution was shown under general conditions in which the observations were assumed to be taken continuously and instantaneously in time.By now this approach has been studied for several other physical systems such as the one-dimensional Chaffee-Infante equation,the two-dimensional Boussinesq,the three-dimensional Brinkman-Forchheimer extended Darcy equations,the three-dimensional Bénard convection in porous media,and the three-dimensional Navier-Stokes α-model (cf.[1-3,22-24,36]).Notably,Farhat,Lunasin and Titi in [25],recently verified,in the case of the three-dimensional planetary geostrophic model,an earlier conjecture of Charney that posited that in simple atmospheric models,the temperature history determines all other state variables.The effects of noisy data were studied by Bloemker,Law,Stuart and Zygalakis [8] and Bessaih,Olson and Titi [5].A case related to the study undertaken by this paper,where observations are taken at discrete moments in time,rather than continuously,and with systematic deterministic errors,was studied in [26],while fully discretized versions were considered in [27].Postprocessing methods were also applied to further ameliorate errors in this downscaling algorithm and in particular,obtain error bounds which are uniform-in-time (cf.[37]).See also [4] for a study into the continuous-time extended Kalman-Bucy filter in the setting of stochastic nonlinear ODEs.Observational measurements that have been blurred in time are studied here.

In continuation of the work in [28],we combine a feedback control based on time-averaged modal observables with the dynamics of the 2π-periodic subcritical SQG equation to obtain

Here μ is a relaxation parameter,Jδh(θ)represents an idealized interpolant based on modal measurements with observation resolution h along with a moving time average over intervals of width δ that represents the blur intrinsic to the measuring device used to obtain the data.It is natural to suppose that the observed solution,θ,represents the long-time evolution of the SQG equations,which is to say that θ belongs to the global attractor and therefore exists backward in time for all t ＜0.For our analysis,however,it is sufficient to go back only as far as t=-2δ.We therefore make the milder assumption that θ(·,-2δ)belongs to an absorbing ball for (1.1)with a sufficient regularity.Note also that in order to construct the data assimilation algorithm given by(1.2),we have assumed that the SQG equation is known in addition to the exact value of κ.What is not known,of course,is the initial condition for η represented by the function g(x,t).Theoretically speaking one might as well take g(x,t)= 0; however,any 2π-periodic function with mean zero that lies in the aforementioned absorbing set would be fine.Therefore,there may be better choices for g in practice.In particular,if we take g(x,t)= θ(x,t)for t ∈(-2δ,0],then Jδh(η)= Jδh(θ)in (1.2),so that η(x,t)= θ(x,t),for all t ＞0; we refer the reader to Subsection 4.1 to help clarify this fact.Although there would be no need for data assimilation if θ(x,t)were already known,this cancellation is necessary to obtain the important mathematical property that,in the absence of noise or model error,η exactly synchronizes with θ over time.

We will assume that (1.2)governs the evolution of the approximating solution,η,used in our analysis of data assimilation for the SQG equation with observations that have been blurred in time and with 2π-periodic boundary conditions over T2.We will treat the subcritical case,when γ ∈(1,2).Our main results consist of the following two theorems:

(1)The data assimilation equations given by (1.2)are well posed (Theorem 3.1).

(2)For h sufficiently small,there exists a choice ofμand δ,for which the differences between η and θ vanish over time (Theorem 3.2).

Note that treating the critical case γ =1 would,of course,also be very interesting for any type of observational data.However,this is beyond the scope of our present analysis.

We defer formal statements of our theorems to Subsection 3,after we have defined the mathe-matical setting of our problem in Section 2.Let us point out,however,that the presence of the moving time average introduces certain analytical difficulties.Firstly,it is difficult to control temporal oscillations in the approximating solution and that arise due to deviations of the blurred-in-time observations from the exact values of the reference solution.For this,we must especially make use of more delicate boundedness properties of the interpolant operator,which we identify and prove in Subsection 2.2 and Appendix B,respectively.Second,a suitable non-local Gronwall inequality is required to control the difference between the approximating solution and the observed solution.Theorem 3.2 shows that these obstacles can indeed be surmounted provided that δ is small enough.In this regime,(1.2)achieves exact asymptotic synchronization at an exponential rate and therefore performs similarly to the case studied in[28],where the observations are not blurred in time.Lastly,we emphasize that our approach to the analysis of this problem renders transparent which errors arise from the delay and which arise from the blurring,as well as the manner in which these errors transfer from one timewindow to the next.Because of this,we are able to capture mathematically the role of the size of the averaging window.

2 Preliminaries

2.1 Function spaces:,Vσ,,,

Let 1 ≤p ≤∞,σ ∈R and T2=R2/(2πZ)=[-π,π]2.Let M denote the set of real-valued Lebesgue measurable functions over T2.Since we will be working with periodic functions,define

Let C∞(R2)be the class of functions which are infinitely differentiable on R2.Defineby

For 1 ≤p ≤∞,define the periodic Lebesgue spaces by

where

Let us also define

For any real number σ ≥0,define the homogeneous Sobolev space(T2),by

where

Similarly,for σ ≥0,we define the inhomogeneous Sobolev space(T2),by

where

Let V0?Z denote the set of trigonometric polynomials with mean zero over T2and set

where the closure is taken with respect to the norm given by(2.5).Observe that the mean-zero condition can be equivalently stated asThus,and ‖·‖Hσare equivalent as norms over Vσ.Moreover,by Plancherel’s theorem we have

Finally,for σ ≥0,we identify V-σas the dual space (Vσ)′of Vσ,which can be characterized as the space of all bounded linear functionals ψ on Vσrepresented by the Fourier coefficientswith duality paring

Given our use of non-dimensional variables and the 2π spatial periodicity of our functions,the Poincaré inequality may be written with a non-dimensional constant equal to one as

Moreover,we have the following continuous embeddings

Remark 2.1Since we will be working over Vσanddetermine equivalent norms over Vσ,we will often denotesimply by ‖·‖Hσfor convenience.Similarly,we will often abuse notation and denote(T2)simply by Lp.

2.2 General interpolant observables

We will consider general interpolant observables Jh,which are defined as those which satisfy certain boundedness and approximation-of-identity properties.The canonical examples of such observables include projection onto local spatial averages or projection onto finitely many Fourier modes.It was shown in [28] that such projections do in fact satisfy the properties we impose on Jh.

where C ＞0 represents a constant independent of φ,h.Note thatwhen q ＜p in which case the bound in (2.9)gets worse as h becomes smaller.In addition to (2.8)-(2.9),we will also suppose that Jhsatisfies the following approximation-of-identity properties

We will also require Jhto satisfy some boundedness properties.We verify in Appendix B that these properties hold for local spatial averages.They also hold for spectral projection,that is,projection onto finitely many lowest Fourier modes (see Remark B.1).To state these boundedness properties,we will adopt the following notation.For β1and β2non-negative integers we letwhere β1+β2=β,while if βj≥0 are real thenHererepresents the greatest integer less or equal β.Finally,if β ∈(-2,0),then Dβ:=Λβ,i.e.,the Riesz potential.

Now,given α ≥1,let ∈(α)be as in Proposition B.1(v)when α ∈[1,2)and identically 0 otherwise.Let Cα＞0 be a sufficiently large constant,depending possibly on α,and define

We assume that

We again emphasize that the above properties are consistent with those satisfied by the projection onto local spatial averages (see (B.11)-(B.12)in Appendix B).Furthermore,we again point out that they are also consistent with those satisfied by the spectral projection,up to possibly different constants(see Remarks 2.3 and B.1).For clarity of exposition,our analysis will be performed with the constants detailed above,though the conclusions are also true for Jhgiven by spectral projection.

Remark 2.2We are able to prove other boundedness properties in Appendix B in addition to the ones shown above.While our analysis requires us only to invoke properties (2.8)-(2.16),the additional boundedness properties asserted in Proposition B.3 may find use in other applications.

Remark 2.3In the case where Jhis given by the Littlewood-Paley spectral projection,i.e.,projection onto Fourier modesthen we replace CI(α,h)everywhere above by CS(α,h)according to the rule

Note that α=α(p)implicitly.One may thus refer to operators Jhwith constants CIas“Type I operators” and those with prefactors CSas “Spectral Type I operators”.Observe that in general we have CSCI,so all Spectral Type I operators are automatically Type I operators.We further observe that the Type II operators defined in [2],see also [5],using nodal-point measurements of the velocity field in physical space do not satisfy the above bounds.

Remark 2.4Note that in the estimates we perform below,the constant C ＞0 appearing in (2.11)may change line-to-line when invoking the above properties.Nevertheless,it can be fixed to be sufficiently large in the statement of the theorems where such constants appear.

2.3 Time-averaged interpolant observables

Suppose φ=φ(x,t).We define the time-averaged general interpolant operator Jδh,by

Due to the time-averaging,one must also control errors that arise from temporal deviations of the time-average from the instantaneous value.Indeed,observe that by the mean value theorem and by commuting ?τwith Jhwe have

We will make crucial use of (2.18)when we perform the a priori estimates.

Remark 2.5It may seem more natural to represent blurred-in-time measurements at time t by an average of the form

However,in this case the corresponding a feedback term obtained by using Iδh(η)in place of Jδh(η)in (1.2)would violate causality by introducing an integral over times in the future.We emphasize that the same interpolant operator must be used in the feedback as used for the measurements in order to maintain the property that g =θ for t ∈(-δ,0] implies η =θ for all times t ＞0 in the future.Therefore,the best we could do is insert the measurement Iδh(φ)into the model delayed in time by.This approach was taken in [6-7]for the Lorenz equations.In the present work,an additional delay has been inserted into the definition of Jδhφ to make the analysis more convenient.This allows the feedback control to be treated as a time-dependent force,thereby transforming what would have been partial integro-differential equations into merely partial differential equations.While any additional delay would achieve the same effect,for simplicity we choose its order to bewhich is the same as the delay already dictated by causality.

2.4 Calculus inequalities

We will make use of the following bound for the fractional Laplacian,which can be found for instance in [14,16,29].

Proposition 2.1Let p ≥2,0 ≤γ ≤2 and φ ∈C∞(T2).Then

We will also make use of the following calculus inequality for fractional derivatives (cf.[31-32] and references therein).

Proposition 2.2Let φ,ψ ∈C∞(T2),β ＞0 and p ∈(1,∞).Then we have that

Finally,we will frequently apply the following interpolation inequality,which is a special case of the Gagliardo-Nirenberg interpolation inequality and can be proven with Plancherel’s theorem and the Cauchy-Schwarz inequality.

Proposition 2.3Let φ ∈(T2)and 0 ≤α ≤β.Then

where C depends on α,β.

2.5 Well-posedness and Global attractor of the SQG equation

Let us recall the following well-posedness results of the SQG equation.In [18] it was shown that global strong solutions exist and that weak solutions are unique in the class of strong solutions.

Proposition 2.4(Global existence)Let 1 ＜γ ≤2 and σ ＞2-γ.Given T ＞0,suppose that θ0∈Vσand f satisfies

Proposition 2.5(Uniqueness)Let T ＞0 and 1 ＜γ ≤2.Suppose that θ0∈(T2)∩Z and f ∈L2(0,T;).Then for p ≥1,q ＞0 satisfying t

here is at most one solution to (1.1)such that θ ∈Lq(0,T;(T2)).

Let us recall the following estimates for the reference solution θ (cf.[16,29,39]).

Proposition 2.6Let γ ∈(0,2],σ ＞2-γ and θ0∈Vσ,f ∈Vσ-γ2∩(T2).Then there exists a constant C ＞0 such that for any p ≥2 satisfying 1-σ ＜＜γ-1,we have

It was shown in [29] for the subcritical range 1 ＜γ ≤2,that (1.1)has an absorbing ball in Vσand corresponding global attractor A ?Vσwhen σ ＞2-γ.In other words,there is a bounded set B ?Vσcharacterized by the property that for any θ0∈Vσ,there exists t0＞0 depending on‖θ0‖Hσsuch that S(t)θ0∈B for all t ≥t0.Here{S(t)}t≥0denotes the semigroup of the corresponding dissipative equation.

Proposition 2.7(Global attractor)Suppose that 1 ＜γ ≤2 and σ ＞2-γ.Let f ∈whereThen (1.1)has an absorbing ball BHσgiven by

for some ΘHσ＜∞.Moreover,the solution operator S =Sfof (1.1)given by S(t)θ0=θ(t)for t ≥0 defines a semigroup in the space Vσand possesses a global attractor A ?Vσ,i.e.,A is a compact,connected subset of Vσsatisfying the following properties:

(1)A is the maximal bounded invariant set;

(2)A attracts all bounded subsets in Vσin the topology of

3 Standing Hypotheses and Statements of Main Theorems

We will work under the following assumptions for the remainder of the paper.

Standing HypothesesAssume the following:

(H1)1 ＜γ ＜2;

(H2)σ ∈(2-γ,γ];

(H3)p ∈[1,∞] such that 1-σ ＜＜γ-1;

(H5)θ-2δ∈BHσ;

Observe that (H1)expresses the subcritical range of dissipation,while (H2)-(H5)ensure that we are in a regime of global strong solutions for (1.1)and that the global attractor exists.

Also observe that since γ ＜2,the range for σ in (H2)covers the natural spatial regularity class for strong solutions,e.g.Hγ.

On the other hand,from (H1)-(H5),Propositions 2.6-2.7 imply that

In particular,it immediately follows from (2.8)that

and from (2.12)that

for some constant CJ＞0.Also,for 1 ≤q ≤∞and α ∈R,let us define

Then for p given by (H3),the Sobolev embedding theorem and (H6)imply

Finally,we give exact mathematical statements of our main results.

Theorem 3.1Let θ be the unique global strong solution of (1.1)corresponding to initial data θ-2δhaving zero mean over T2.Then under the Standing Hypotheses,for all T ＞0,there exist a unique strong solutionsatisfying (1.2)with η(·,0)=g(·,0).

Theorem 3.2Under the hypotheses of Theorem 3.1,there exist constants c0,c′0＞0 such that if h,μ satisfy

and δ ＞0 is chosen sufficiently small,depending on h,then the solution η given by(1.2)satisfies

for some constant λ0∈(0,1).

Remark 3.1Note that the condition that δ ＞0 be sufficiently small can be described precisely by simultaneously satisfying (4.7)and (5.7)below.

Remark 3.2As we pointed out in Remark 2.3,since Spectral Type I operators satisfy all the properties of Type I operators,both Theorem 3.1 and 3.2 are also valid for Spectral Type I operators.In particular,they are valid when Jhis given by projection onto finitely many Fourier modes.

Remark 3.3The relationship between the full three-dimensional quasi-geostrophic equations and the SQG equation implies that being able to approximate θ by η,as in the conclusion of Theorem 3.2,is the same as synchronizing the corresponding three-dimensional solutions in which the potential vorticity is identically zero and the vertical motion eliminated.Therefore,in a way analogous to the discussion in [28],our theorem provides an example where timeaveraged data collected on a two-dimensional surface is sufficient to obtain synchronization in a three-dimensional domain.

Before we move on to the a priori analysis,we will set forth the following convention for constants.

Remark 3.4In the estimates that follow below,c and C will generically denote positive constants,which depend only on other non-dimensional scalar quantities,and may change lineto-line in the estimates.We emphasize that in the estimates we perform below,the constants c and C may change in magnitude from line-to-line,but as the equations were fully nondimensionalized from the beginning they will never carry any physical dimensions.

4 A Priori Estimates

4.1 Initial value problem and proof of Theorem 3.1

We recouch(1.2)as a sequence of initial value problems over consecutive time intervals.Once we have defined the setting properly,we may immediately prove Theorem 3.1 by appealing to Propositions 2.4-2.5.

Observe that owing to the delay in the interpolant operator Jδh,we must initialize the averaging process.By (H1)-(H5)and Proposition 2.7,we may assume that θ is the strong solution of (1.1)with initial data starting at t=-2δ such that θ-2δ∈BHσ.

For any k ≥-2 set

Let η(-1)(·,t)=g(·,t)for t ∈I-1.Then we may express a solution η of

as the sum

where for each k ≥0,η(k)satisfies

Hence,over each interval Ikwe may view the termin(4.3)as a smooth,time-dependent forcing term and (4.3)as an initial value problem over Ikwith initial data η0(x)= η(x,δk).The proof of Theorem 3.1 follows readily.

Proof of Theorem 3.1We proceed by induction on k.For k = 0,from (H6)we have that η(·,0)= g(·,0)∈Vσ.Since we assume the Standing Hypotheses,we have thatholds for all T ＞0 (by (2.8)and (2.12)),so that we may apply Propositions 2.4-2.5 to deduce existence and uniqueness of a strong solution η(0)over I0to(4.3).Suppose unique strong solutions to(4.3)exist for all ?=0,··· ,k.Consider (4.3)over Ik+1.Observe that by hypothesis η(k+1)(·,δk+1)= η(k)(·,δk+1)∈Vσandhold once again by(2.8)and (2.12).Therefore,we apply Propositions 2.4-2.5 to guarantee existence and uniqueness of a strong solution η(k+1)to (4.3)over Ik+1,completing the proof.

In the remainder of Section 4 we establish uniform-in-time estimates for η in L2,Lpand Hσ.As we will see,the synchronization property will rely crucially on these uniform estimates.To obtain uniform Hσestimates,we perform a bootstrap from L2to Lp,then from Lpto Hσ.Once we have collected the requisite uniform bounds,we proceed to Section 5 and the proof of Theorem 3.2.

4.2 Uniform L2 estimates

In this section,we will ultimately obtain L2estimates for the solution η of (4.2)that are uniform in time.In this work,any bound of this type shall be referred to as a “good” bound.The main result in this section is the “good” bound stated as Proposition 4.1 below.We emphasize that the structure of the analysis in Subsections 4.2.2-4.2.4 will be mimicked in Section 5 when we establish the synchronization property.

We begin by introducing some notation that will be convenient when expressing the necessary bounds in our proofs.Let

where Γ2,kis the function of δ ＞0 given by

Note that Γ2,kand consequentlyare increasing functions of δ.Therefore,any upper bounds given by the constants defined in(4.4)and(4.5)for a particular δ =δ0continue to hold when δ ＜δ0.We shall immediately make use of this property to show that the hypotheses on δ in Proposition 4.1 stated below are not vacuous.

Proposition 4.1There exist constants c0,c1＞0,with c1depending on c0,such that if h,μ satisfy

and δ is chosen such that

as well as

and

Observe that both sides of the inequalities given by (4.7)-(4.8)depend on δ.This is,as already mentioned,becausedepends on δ.However,sinceappears in the denominator of the right-hand side and is an increasing function of δ,it is easy to see that there must be a δ ＞0 which satisfies both these inequalities.

To prove Proposition 4.1,we employ three preliminary lemmas.First,in Subsection 4.2.1 we establish bounds in L2which are uniform in each time interval Ik,but ultimately depend on k.Throughout this work we will refer to any bounds that depend on k as “rough” bounds.Such bounds are insufficient on their own but needed in order to close estimates later.Then in Subsection 4.2.2,we establish time-derivative estimates to control the temporal oscillations that emanate from the feedback term (see Subsection 4.2.2).The third lemma is a non-local Gronwall inequality that ensures uniform bounds provided that the window of time-averaging is sufficiently small; its proof is deferred to Appendix A.This Gronwall inequality will be used again to establish the synchronization property in Section 5.We finally prove Proposition 4.1 in Subsection 4.2.4.

Remark 4.1We will often exchange the quantity μ for the quantity κh-γvia the relation(4.6),in order to emphasize that δ and μ ultimately depend only on h (and ΘLp)alone.

4.2.1 Rough L2 estimates

We will first establish the following “rough” a priori bound.We omit most of the details,though they can easily be gleaned from the proof of Proposition 4.1.An alternative form of Lemma 4.1 is given by Corollary 4.1 stated below,which will be convenient to use in the proof of Proposition 4.1 later.

Lemma 4.1Letbe given by (2.21),(3.1),(4.4),respectively.There exists a constant C0＞0,independent of k,such that

where

ProofSuppose t ∈Ikfor some k ≥0.We perform standard energy estimates to obtain

Observe that by the Cauchy-Schwarz inequality and (2.8)we have

Returning to (4.13)and applying these facts along with (3.2),we obtain

Finally,by integrating (4.14)over [δk,t] for t ∈Ikwe arrive at

which can be simplified to (4.11)using (4.4),as desired.

Corollary 4.1Let k ＞0.Suppose that for each 0 ≤? ≤k,there exists M?＞0 such that

Then there exists a constant C0＞0,independent of k,such that

While δ can be chosen in these bounds so that the size ofis small,this alone does not suffice to obtain uniform-in-time bounds for ‖η(t)‖L2upon iteration in k,which will be crucial in establishing the synchronization property.Nevertheless,these “rough”bounds will be useful in order to close our estimates and achieve uniform bounds later.

4.2.2 Control of temporal oscillations at fixed spatial scale

We recall from (2.18)that we will require estimates for the time-derivative ?tη,but only over length scalesh where h measures the spatial resolution of the observables.

Lemma 4.2Let k ＞0.Suppose there exists M?＞0 such that

Let c0＞0 be any constant such that

Then there exists a constant C0＞0,depending on c0,but independent of k,such that

holds for all t ∈(-2δ,δk+1],and

holds for all t ∈Ik+1.

ProofBy (H1)we have＜1.Therefore,by (2.11),see also (B.16),we have

Now,applying Jhto (1.2),using the fact that v is divergence free,and then taking thenorm we have

By (H1),(H7),(2.15),(2.21),(3.1)and (4.16)we may estimate

For the quadratic term apply (2.16),the Cauchy-Schwarz inequality and the fact that R⊥is a bounded operator in L2to estimate

and

Upon collecting these estimates,returning to (4.20),we apply (3.1)and (4.4)to obtain

for t ∈(-2δ,δk+1],as well as

for t ∈Ik+1.Note that in collecting the terms we have used the fact that all constants and variables have been non-dimensionalized so that,for example,terms such as 1+and 1+Mk+RL2make sense.Thus,upon squaring both sides of these inequalities,then applying Young’s inequality and (4.17),we arrive at (4.18)and (4.19).

4.2.3 Growth during initial transient period

Due to the delay,we must quantify bounds over the initial transient period during which the feedback effects from large scales can amplify the solution.Consider the definition of Γ2,kfor k =-1,0,1,··· given by (4.5).Observe that

By (3.5),Lemma 4.1 and Corollary 4.1 we have

It then follows from (4.21)that

for any ρ ≥0.

As we will see,the choice of ρ will be dictated by the estimates (4.37)and (4.41)below.In anticipation of this,consider the third definition of (4.4)given by

Then (4.22)implies

Therefore,the conclusion of Proposition 4.1 is that there is a choice of ρ such that the bound given by (4.24)propagates beyond the initial transient period,provided that δ is chosen small enough.In particular,Proposition 4.1 provides a more precise version of(4.24),which not only allows this bound to propagate through all times t ＞2δ,but in such a way that it eventually“forgets” the initializing function g as well.

We are now ready to prove Proposition 4.1.

4.2.4 Proof of Proposition 4.1

We proceed by induction on k.As we shall see shortly,by Lemma A.1(ii),it suffices to show for k ≥2 and t ∈Ikthat

We proceed in three steps.Step I proves the base case when k = 2 while Step II provides the induction step thereby completing the induction.Finally,Step III uses (4.25)along with Lemma A.1(ii)to obtain (4.9)-(4.10)which finishes the proof.

I Base caseLet k =2 and suppose t ∈I2.By Corollary 4.1 and (4.24)we have

It then follows from (4.24)and the second condition of (4.7)that

Multiply (4.3)by η,integrate over T2,and apply (2.18)to obtain

where

and

Observe that by (2.10),Cauchy-Schwarz inequality,Young’s inequality and (3.2)we have

and

Further estimating I1,I2and I4using (4.6),(3.2)and (4.4)gives

To estimate I3,apply Fubini’s theorem,Parseval’s theorem,the Cauchy-Schwarz inequality,(2.8)and Young’s inequalities in the following sequence of estimates,

Let

Observe that S(t)=S0+S1+S2(t),where for ? ≥0,we have defined

Returning to (4.27)and applying (4.28)and (4.6),we have

To obtain bounds on S0and S1define

so that,upon simplifying (4.18)with (4.6),we obtain from Lemma 4.2 and (4.26)that

To bound S2(t)for t ∈I2,observe that by Lemma 4.2 and (4.26)we have

where,upon simplifying (4.19)with (4.6),we have defined

and

Combining (4.33)and (4.34)then gives

Observe that since O2(δ2)≤O1(δ2),it follows from the third condition on δ in (4.7)that

Thus,upon returning to (4.28),we have

By applying the resulting bounds on S(t)in (4.31),we have for t ∈I2that

Now observe that (4.8)ensures that (A.2)holds in Lemma A.1 with

Applying Lemma A.1(i)then gives

which finishes the proof of the base case.

II Induction stepSuppose k ≥2 and for each ?=2,··· ,k and t ∈I?that

We show the bound corresponding to ?=k+1 holds for t ∈Ik+1.

As already demonstrated,our choice of δ has been chosen so that the hypotheses of Lemma A.1 hold for the differential inequality (4.36).These hypotheses are also satisfied for the modified inequality obtained by replacing δ2by δ?for ?=2,··· ,k which we write as

for t ∈I?.Now,dropping the integral in (4.38)and rewriting the last term yields

so that by iterating part (ii)of Lemma A.1 for ?=2,··· ,k we obtain

By Corollary 4.1 it follows that

Thus,by the second condition in (4.7)we have

Now proceed exactly as in the base case,this time making use of the bounds (4.40)-(4.41).Indeed,we may derive (4.31)as before.Then,since t ∈Ik+1,we may split the time integral over three regions

Over Ik-1and Ik,Lemma 4.2 and (4.41)imply (4.33)for Sk-1and Sk.Over Ik+1,we have(4.41),so that Lemma 4.2 implies(4.34)for Sk+1(t).We then deduce(4.35)for t ∈Ik+1,which leads to the differential inequality(4.39)with ?=k+1.Applying Lemma A.1(i)as before then yields

for t ∈Ik+1thus completing the induction.

III Finish the proofWe have already obtained (4.9)for all values of k by iterating Lemma A.1(ii)as part of the induction step.To obtain (4.10)drop the first term in (4.25)and keep the integral.Consequently,we may then deduce that

Since the first condition in(4.6)-(4.7)together imply e(μ2)δ≤2,it follows from(4.4)and (4.22)that

This completes the proof.

Remark 4.2We point out that the energy estimates in Lpand Hσwill not proceed along these lines,the reason being that even if one were to do so,the resulting bounds would still not be independent of h.So long as these bounds are uniform-in-time,however,we will be able to use them strengthen the topology of convergence in which the synchronization takes place via interpolation.We will thus be content with rather modest bounds in Lpand Hσ.

4.3 L2 to Lp uniform bounds

We will prove the following “good” bound.

Proposition 4.2Let FLp,ΘLp,ML2be given by(2.20),(3.1)and(4.4),respectively.Define

where

Let c0＞0 be any constant.Suppose that

Then there exists a constant C0＞0,depending on c0,such that

In particular,

where

ProofObserve that by (3.5),we have

For t ≥0,the evolution of ‖η(t)‖Lpis obtained by multiplying (4.2)by η|η|p-2,integrating over T2,applying Proposition 2.1,H?lder’s inequality,Young’s inequality and (2.20)to obtain

Applying H?lder’s inequality,the Fubini-Tonelli theorem,(2.9)with q =2 and Proposition 4.1 we have

observe that

Note that the constant(2π)γcarries the units of Lγ; however,as L=2π throughout this paper we avoid keeping track of the dimensions in this case,and simply denote the prefactor C(2π)γby C.By interpolation,Young’s inequality and H?lder’s inequality we have

Upon combining (4.49)-(4.51),(4.45)and returning to (4.48),we arrive at

An application of (4.46)and Gronwall’s inequality completes the proof.

4.4 Uniform Hσ-estimates

As in the previous section,we obtain“good”Hσ-bounds without appealing to time-derivative estimates.

Proposition 4.3Let ML2be given by (4.23)and let ΘHσ,~MLpbe given by (2.22),(4.47),respectively.Define

as well as

Let c0＞0 be the constant given in Proposition 4.1.Suppose that

Then there exists a constant C0＞0,depending on c0,such that

ProofSuppose t ≥0.We multiply (4.2)by Λ2ση and integrate over T2to obtain

We estimate J1with H?lder’s inequality,interpolation and Young’s inequality as in [18,33],and invoke (4.52)to obtain

Note that (H1)-(H3)are needed for the interpolation.We interpolate once more to obtain

Thus,by Young’s inequality,we have

For J2,we make the familiar estimate through Parseval’s theorem,the Cauchy-Schwarz inequality,and then (4.53)to obtain

For J3and J4,we consider two cases:σ ≤and σ ＞.

Caseσ ≤It follows from Fubini’s theorem,H?lder’s inequality,(2.8)and the Poincar`e inequality that

Thus,by Young’s inequality we have

Similarly,since θ-2δ∈BL2by (H5),by (2.22)we have

Therefore,upon returning to (4.55),then applying the estimates for J1through J4and the Poincaré inequality gives

Then the Gronwall inequality implies

as desired.

Caseσ ＞Observe that by Fubini’s theorem,Plancherel’s theorem,H?lder’s inequality,(2.12),Proposition 4.1 and Young’s inequality we have

Similarly,since θ-2δ∈BL2by (H5),by (2.22)we have

Therefore,upon returning to (4.55),then applying the estimates for J1through J4and the Poincaré inequality gives

Then the Gronwall inequality and (4.54)implies

as desired.

5 Proof of Theorem 3.2

We are left to establish the synchronization of η to the reference solution θ.We point out that the uniform L2bounds will be used in a crucial way to establish suitable control on the time derivative and guarantee synchronization in a rather weak topology,i.e.,the H-12topology.We then make use of the uniform Lpand Hσ-bounds in order to strengthen the regularity of the convergence of the synchronization by interpolation.

Consider the difference ζ:= η-θ,where θ ∈BHσand η is the unique strong solution of(4.2).Observe that (3.5)ensures that ζ is defined for t ∈I-1.The evolution of ζ is given by

It will be convenient to work at the regularity level of the stream function of ζ.Thus,we define

5.1 Synchronization

Our main claim is the following.

Proposition 5.1Let ΘHσ,ΘL2,ΘLpand ML2be given by (2.22),(3.1)and (4.4).Define

and

There exist constants c0,c′0,c1,c2≥1 such that if h,μ satisfy

and δ is chosen to satisfy

then

To prove this,we proceed as in Subsection 4.2.4 and make some preparatory estimates.

5.1.1 Control of temporal oscillations at a fixed spatial scale

Lemma 5.1Let ΘL2and ML2be given by (3.1)and (4.23),respectively.Let c0＞0 be a constant.Suppose that

then there exists a constant C0＞0,depending on c0,such that

ProofLet t ＞-2δ.Applying Jhto (5.1)and taking the H-γ2-norm yields

Observe that by (H1),we have＜1,so that by (B.16),we have

By (2.14),(3.1),(5.2),the Cauchy-Schwarz inequality and (5.9)we have

To estimate the nonlinear terms,we apply (2.16),the Cauchy-Schwarz inequality,(3.1),Proposition 4.1,(5.2),interpolation and Young’s inequality.For instance,we have

Similarly

Therefore,by summing each of these estimates,we arrive at (5.10)as desired.

5.1.2 Growth during transient period

We introduce the following notation:Let α ∈(0,1)and ? ∈Z,then define

Observe that by the Poincar`e inequality,(4.22)implies

Clearly,one has

Then

We are now ready to prove the synchronization property.

5.2 Proof of Proposition 5.1

Proof of Proposition 5.1We proceed by induction on k with the base case,k = 1,as established by (5.11).Suppose that the following holds:

for t ∈I?and ?=0,··· ,k,where Ψ is given by (5.3).We show that this corresponding bound holds over Ik+1as well.

Let t ∈Ik+1,k ≥1.Multiply (5.1)by ψ and integrate over T2to obtain

Note that we have used the orthogonality property,i.e.,R⊥f·Rf =0.

We refer to [18,39] to estimate K1.In particular,by H?lder’s inequality,the Calder`on-Zygmund theorem and Sobolev embedding,H1pLq,we have

Thus,by Young’s inequality we obtain

where ΞLpis given by (5.3).We estimate K2with the Parseval’s theorem,the Cauchy-Schwarz inequality,(2.10),(5.2),interpolation and Young’s inequality to get

For K3,similar to (4.28),we estimate

Returning to(5.13)and combining K1through K3,then applying(5.6)with c0and c′0sufficiently large,we get

where

Let ? ∈{k-3,k-2,k-1,k}.By the second condition in (5.7),with c2chosen large enough,we have δμ≤C-1(ln 4),so that Lemma A.2 guarantees that

as well as

Thus,by Lemma 5.1 and (5.12),(5.16)-(5.17)we have

for some constant C ＞0.

Returning to (5.15)and combining (5.18)gives

Hence,provided that c1,c2are chosen sufficiently large with c2depending on c1,it follows from(5.7)that Lemma A.1(i)applies over t ∈Ik+1with

In particular,Lemma A.1(i)implies

By (5.12),we have

Also,we have

Since

it follows that

Observe that (5.7)with c1chosen sufficiently large ensuresThis establishes(5.12)for k+1.Through Lemma A.1(ii),we may iterate this bound to deduce(5.8),as desired.

5.3 Proof of Theorem 3.2

Under the Standing Hypotheses,Theorem 3.1 guarantees a unique,global strong solution η of(4.2).Let c0denote the maximum among all the constants,c0,c′0,appearing in Propositions 4.1 and 5.1.Then let c1,c2denote the maximum among all the c1,c2appearing in those propositions as well (possibly choosing c2larger).Suppose that μ,h satisfy

Choose δ so that (4.7),(5.7)are satisfied,and is chosen smaller than

Then (5.21)implies that (4.8)holds as well.Thus,upon applying Propositions 4.1 and 5.1,η satisfies

Observe that Propositions 2.7-4.3 then imply that

where

for some sufficiently large constant C0＞0.Therefore,for each σ′＜σ,by interpolation,there exists a constant λ0=λ0(σ′)∈(0,1)such that

Choosing σ′=0,yields the desired convergence in L2.

5.4 Concluding remarks

Depending on the type of measurement,the size of the averaging window that effectively blurs the observations in time may be quite different.For example,radiometers and hot-wire anemometers may produce data with averages in the microsecond range.Velocities obtained from mechanical weather-vane anemometers may be averaged with respect to a time window measured in seconds,while velocity data obtained from the Lagrangian trajectories of buoys placed in the ocean is likely to include time averages measured in hours if not days.Observations of temperatures are similar.As we saw,it is important for our analysis that the size of the time-averaging window is not too large.Intuitively speaking,the length of the averaging window should be smaller than any dynamically relevant timescales in the flow.Numerical computations involving the Lorenz system (cf.[6])show that synchronization occurs when the averaging window is of size δ = 0.25 which,poetically speaking,is about ten times smaller than the time it takes to travel around one wing of the butterfly.In the case of the fluids,we conjecture that the averaging window should be at least ten times smaller than the turnover time of the smallest physically relevant eddy.Alternatively,the largest averaging window such that our data assimilation algorithm leads to full recovery of the observed solution could be interpreted as a definition of the smallest physically relevant time scale.

We reiterate that a main motivation to consider a more realistic representation of physical observations is the reason for considering time averages.The additional δ delay introduced into equations (1.2)helps close the estimates in the analysis while being of the same magnitude as theδ2delay dictated by causality considerations in the feedback controller (see Remark 2.5).In practice,such a delay may also be used to advance an initial condition already obtained by data assimilation for a short time into the future to increase the stability of further predictions.However,this idea must be left for a different study.

A Appendix

To obtain the uniform estimates,we invoked a non-local Gronwall’s inequality,which ensured such bounds provided that the non-local term was sufficiently small.

Lemma A.1Let Φ,Ψ,F be non-negative,locally integrable functions on(t0,t0+δ]for some t0∈R and δ ＞0 such that

for some a,b,A,B ＞0.Suppose that δ,a,c satisfy

where we use the convention that= ∞if A = 0,B = 0,respectively.Then the following hold:

(i)For all t ∈(t0,t0+δ]:

(ii)If Φ satisfies

for some δ0＜t0,then (A.4)persists over t ∈(t0,t0+δ].

ProofMultiplying by the factor e(a2)κt,then integrating over [t0,t],we obtain

Observe that

Similarly

It follows that

provided that the first condition in (A.2)holds.This also holds with b,B,ψ,replacing a,A,Φ,respectively,provided the second condition in (A.2)holds.This implies (A.3).

Now assume that (A.1)holds over (t0,t0+δ)and that (A.4)holds over [δ0,t0],for some δ0＞0.Then applying (A.4)at t0to (A.3)we have

which simplifies to (A.4),as desired.

We also made use of the following lemma in order to control feedback effects that enter the present instant through a past time interval and ultimately,ensure synchronization(see(5.15)).

Lemma A.2Let ? ≥-1 and N ＞0.Let δ ＞0 and define δ?:=?δ and I?:=(δ?,δ?+1].Let Φ,Ψ be non-negative,locally integrable functions.Suppose that for some ? ≥-1,there exist constants a,b,Φ0＞0,independent of ?,N,such that

If δ satisfies

for some constant c ＞0,then there exists a constant CN＞0 such that

and

ProofObserve that by the mean value theorem

for some 0 ＜θ ＜1,depending on δ.

By assumption on ?′,?,and the fact that t ≤δ?′+1,we have

On the other hand,observe that

Upon application of (A.5),we have

Thus,by (A.9)we have

and we are done.

B Appendix

B.1 Partition of unity

Let us briefly recall the partition of unity constructed in [2] and used in [28].To this end,we define for φ ∈L1(T2),

Let N ＞0 be a perfect square integer and partition Ω into 4N squares of side-length h=Letand for each α ∈J,define the semi-open square

Let Q denote the collection of all Qα,i.e.,

so that Qα?Qα(∈)?for each α ∈J,and the “core” Cα(∈),by

Then there exists a collection of functions {ψα} satisfying the properties in Proposition B.1.Note that we will use the convention that when β is a positive integer,thenwhere β1+β2=β and βj≥0 are integers,while if β ＞0 is not an integer thenwhere [β]=[β1]+[β2],and finally,if β ∈(-2,0),then Dβ=Λβ.

Proposition B.1Let N ≥9,h:=and ∈:=.The collection {ψα}α∈Jforms a smooth partition of unity satisfying

Property (iii)was exploited in [28],but only in the case p = 2.We observe here,however,that it also holds for any p ∈[1,∞)since IQα≤≤1 and spt?(Qα(∈)+(2πZ)2).On the other hand,property (iv)for β ≥0 was sufficient for the purposes in [28].We will show here that it also holds β ∈(-2,0),i.e.,property (v)as well as the L∞estimate (vi).For this,we will appeal to the following elementary fact:

where we define

The relation(B.3)can be seen easily by appealing to the Fourier transform.Due to the subtleties of working with periodic functions,we include the details in Lemma B.1 below.To this end,let us define

Let us also denote the Fourier transform on T2,i.e.,for functions which are periodic with period 2π in x,y,by

and by Fλthe Fourier transform on λ-1T2,for λ ＞0,i.e.,for functions which are periodic with period λ-12π in x,y,by

Lemma B.1Let β ∈(-2,2].Then

(ii)Λβφ(λ·)(x)=λβ(Λβφ)(λx),for λ ＞0,and any β ∈R,provided that φ ∈.

ProofThe first property follows by a change of variables.Now observe that if φ ∈(T2)∩,then φ(λ·)∈(λ-1T2)∩with period 2πλ-1in x,y,whereis as in(2.1).Let=λk,for k ∈Z2.Then

It follows that for x ∈λ-1T2,we have

Let us now return to the proof of Proposition B.1(v)-(vii).For this,let

Proof of Proposition B.1(iv)through (vi).

Proof of (iv)for β ∈(-1,0).For convenience,let β ＞0.By Lemma B.1(ii),we have

It follows from the Hardy-Littlewood-Sobolev inequality that

We see now that from (B.5)we have

with constant independent of α and h,as desired.

Proof of (v).Let β ∈[1,2).We estimate by duality.Indeed,let χ ∈(T2)such thatThen since χ ∈,by Parseval’s theorem we have

where for the last inequality,we made use of the fact thati n h-1T2andis supported in a ball of area ～1.Thus

as desired.

Proof of (vi).The result is trivial when β =0 and k ＞0 simply by rescaling and observing thatis still supported in Qα(∈).

Suppose that β ∈(0,1).Now observe that for x ∈T2,Lemma B.1(ii)implies that

If x ∈2h-1Qα∩h-1T2,then |x-y|≤2 and we have

Thus |ΛβΨα(x)| ≤C for all x ∈h-1T2,which implies |Λβψα(x)|≤Ch-βfor all x ∈T2,where C is independent of α ∈J.This establishes (v).

To ultimately prove(2.12)-(2.14),we will exploit an additional property of the bump functions.For this,we will make use of the following short-hand for φ localized to the squares Qα(∈):

Lemma B.2Let β ∈(-∞,0)and φ ∈(T2).Then there exists a constant C ＞0 such that

ProofSuppose that β ∈(-∞,0).Observe that

Then by Parseval’s theorem,the Cauchy-Schwarz inequality and Proposition B.1(iv),we have

as desired.

B.2 Boundedness properties of volume element interpolants

where a(Q)denotes the area of Q and

Observe that for each α ∈J,there exists a constant c ＞0,independent of h,α,∈such that

We define the smooth volume element interpolant by

and the “shifted” smooth volume element interpolant by

We will make use of the following elementary fact for a “square-type”function.Let A be a finite index set and {Aα}α∈A?T2be a countable collection of sets such that for each x ∈T2,Define

Lemma B.3Let φ ∈L1(T2).There exists a constant C ＞0 such that

and

ProofLetSince N ＜∞,we have that for each x ∈T2,there are at most N sets Aαsuch that x ∈Aα.It follows that for each x ∈T2,there exists an integer C(x)＞0 such that C(x)≤N.In particular,we have

On the other hand,by Fubini’s theorem and the Cauchy-Schwarz inequality we have that

This completes the proof.

We immediately obtain the following corollary.

Corollary B.1Let K ∈(R2)such that K ≥0.Let φ ∈L1(T2)such that K *φ ∈L2(T2),

In particular,for β ∈(-2,0),we have

where (A,{Aα})is given by (J,{Qα(∈)})as in (B.2).

ProofObserve that

Therefore,by the non-negativity of K,the Cauchy-Schwarz inequality and (B.13)of Lemma B.3,we have

It then follows as a special case that(B.15)holds.Indeed,the Riesz potential,Λβ,β ∈(-2,0),has kernel K(x)～|x|-2+β,which is locally integrable and non-negative.

Proposition B.2Let Jhbe given by either (B.11)or (B.12).Given α ≥1,let ∈(α)be as in Proposition B.1(v)when α ∈[1,2),and identically 0 otherwise.Let C ＞0 and define

There exists a constant C ＞0,depending on α,such that:

(1)If (ρ,β)∈[0,∞)×[0,2),then

(2)If (ρ,β)∈[0,∞)×(-2,0],then

(3)If (ρ,β)∈(-2,0)×(-∞,0],then

ProofWe will prove the lemma for the case Jhgiven by (B.11).The case when Jhis given by (B.12)is similar.

Let (ρ,β)∈[0,∞)×[0,2).Then by Proposition B.1(iv)-(v),we have

where ∈(α)＞0 is chosen according to Proposition B.1(v)and C ＞0 is some constant,depending on α.

Hence,by (B.16)we have

where CI(β,h)is defined by (B.16),as desired.

Next,let (ρ,β)∈[0,∞)×(-2,0].We estimate as before,except that we apply Lemma B.2 and Corollary B.1 to obtain

as desired.

Finally,let (ρ,β)∈(-2,0)× (-∞,0].To prove (B.19),we proceed by duality.LetSince Jhis self-adjoint and φ ∈,it follows from Parseval’s theorem and(B.17)that

Thus,we have

as desired.

Proposition B.3Let Jhbe given by (B.11)or (B.12).Let CI(α,h)be defined as in (B.16).Define

Let ρ,β ∈R.There exists a constant C ＞0,depending only on ρ,β,such that:

(1)If ρ ≥0 and β =? is an integer,then

and

(2)If ρ ∈(-2,0),β ∈(-2,∞)and β′∈(-∞,β],then

On the other hand,if β =? is an integer,then

(3)For ρ ≥0 and β ∈(0,2)such that β1,we have

ProofLet ρ ≥0.By integrating by parts,Proposition B.1(iv),and the Cauchy-Schwarz inequality we have

which proves (B.23).

Similarly,estimating as before and applying Proposition B.1(vi)(instead of(iv))and H?lder’s inequality (instead of Cauchy-Schwarz)we have

Arguing as before,we ultimately arrive at (B.24).

For ρ ∈(-2,0)and β′∈(-∞,β],we proceed by duality.Indeed,let χ ∈(T2)withSince Jhis self-adjoint,by Parseval’s theorem we have

Then by Parseval’s theorem,the fact that φ ∈Z,the Cauchy-Schwarz inequality,the Poincaré inequality and (B.17)of Proposition B.2,we have

which implies (B.25).

On the other hand,to prove (B.26),let β = k be an integer.Since Jhis self-adjoint,upon integrating by parts,then applying H?lder’s inequality we obtain

Observe that

Therefore

which implies (B.26),as desired.

Finally,we prove (B.27).Let β ∈(0,2)be a non-integer.Then by Proposition B.1(iv)-(v),integration by parts,the fact that Λ is self-adjoint,and the Cauchy-Schwarz inequality we have

This completes the proof.

Remark B.1We point out that all of the above boundedness properties for Jhhold also when Jhis given by projection onto finitely many Fourier modes,in specific,when Jhis given by the Littlewood-Paley projection.The only difference is in the constants.Indeed,one may notice above that this “defect” between the spectral projection and the “volume-elements”projection can be traced to the fact the operator,Λβ,β ∈(-2,2),is a non-local operator;although its input may be compactly supported,the output need not have compact support.Generally speaking,the projection onto Fourier modes up to wave-number1hsatisfies convenient“orthogonality”properties,as captured by the Bernstein inequalities,that are not enjoyed by projection onto local spatial averages.The above boundedness properties then follow immediately from this inequality and the fact that differential operators will commute Jhwhen it is given as this projection.For this reason,we omit the details,but refer to [28],where the relevant estimates are carried out.

AcknowledgementsThe authors would like to thank the Institute of Pure and Applied Mathematics(IPAM)at UCLA for the warm hospitality where this collaboration was conceived.The authors are also thankful to Thomas Bewley,Aseel Farhat and Hakima Bessaih for the insightful discussions.