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BY 4.0 license Open Access Published by De Gruyter January 13, 2017

A Data Assimilation Algorithm for the Subcritical Surface Quasi-Geostrophic Equation

  • Michael S. Jolly , Vincent R. Martinez and Edriss S. Titi EMAIL logo

Abstract

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood–Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.

1 Introduction and Main Results

Continuous data assimilation dates back to 1960s with the idea of Charney, Halem and Jastrow in [15], where they proposed that the equations of the atmosphere be used to process the observations, which were collected essentially continuously in time, in order to refine estimates for the current atmospheric state. Assuming that these equations represent the observed reality exactly, one can then use the improved estimate for the current state as initial data with which to integrate the model in time and thus, provide more accurate forecasts.

The data assimilation approach studied in this article is inspired by the one proposed in [6, 5], where the large scale observations are inserted in the physical model through the addition of a feedback control term that serves to relax the solution of the modified algorithm towards the reference solution of the original system associated with the data. Indeed, suppose that perfect, coarse spatial scale measurements, Jh(v), are given, where h>0 represents the spatial mesh size of the measurements and v represents a reference solution that evolved from some unknown initial value v(0)=v0 according to the evolution equation

d v d t = F ( v ) .

In this context, one can more generally view the observation operator, Jh, as an interpolation operator, i.e., a finite-rank linear operator satisfying certain approximation estimates (see Section 2.2). Then to “recover” v forward in time, one finds an approximating solution, w(t), which solves the following initial value problem:

(1.1) d w d t = F ( w ) - μ ( J h ( w ) - J h ( v ) ) , w ( 0 ) = w 0 ,

where w0 is arbitrary and μ=μ(h)>0 is the “nudging parameter”. The approximating solution, w, converges to the reference solution at an exponential rate depending on the size of μ. We remark that some of the important features of this algorithm are that it can be initialized arbitrarily, e.g. w0identically zero, and that Jh can be defined in a sufficiently general manner to accommodate observables in the form of Fourier modes, local spatial averages, nodal values, and suitable modifications of these. Indeed, in the original algorithm of Charney–Halem–Jastrow, the observables were inserted into the nonlinear term of the equation, which made certain choices of Jh difficult to implement. We also remark that in the case of noisy observations, i.e., Jh(v) replaced by Jh(v+W˙) in (1.1), where W is a white noise in time, then the resulting stochastic partial differential equation can in fact be viewed as the continuous-time limit of the 3DVAR filter (cf. [11]).

The synchronizing model (1.1) has been analyzed for several important physical models including the one-dimensional (1D) Chaffee–Infante equation, two-dimensional (2D) Navier–Stokes (NS) equation, 2D Boussinesq, three-dimensional (3D) Brinkman–Forchheimer extended Darcy equations, 3D Bénard convection in porous media, and 3D NS α-model, (cf. [2, 6, 5, 32, 33, 34, 35, 56]), while the case of contaminated measurements was studied in [11] for measurements given by Fourier modes, and [10] for the case of more general classes of measurements, including those given as local spatial averages or nodal values. More recently, problem (1.1) has been adapted to accommodate insertion of spatial data which is discrete-in-time [36] or blurred-in-time by a time-averaging process [46]. It is important to note that each of these cases present their own different analytical difficulties or have structural features that allow one to reduce measurements to a certain component. Other studies at the level of the partial differential equation (PDE) for variational or bayesian approaches to data assimilation can be found, for instance, in [12, 44], while avenues to improve them were explored in, for instance, [4, 55]. Lastly, we emphasize that of the studies taken thus far for the nudging model (1.1), this article is the first to address the mathematical difficulties brought on by the appearance of a nonlocal dissipation operator whose dissipativity is weaker than that provided by the full Laplacian. This operator appears naturally, albeit for mathematical analytical reasons, in the surface quasi-geostrophic (SQG) equation defined below in (1.3) (cf. [30]). To overcome these difficulties, we establish various boundedness and approximation properties for Jh in the scale of Lebesgue spaces and fractional Sobolev spaces. We believe that the elementary nature of the estimates gives a flexibility to these properties that should prove useful in further applications or other contexts.

This particular feedback control approach to data assimilation was inspired by the fact that many dissipative systems, e.g., 2D NS equations, possess a finite number of determining parameters, which is to say that if a projection of the difference of two solutions converge asymptotically in time to 0, then the difference of the solutions themselves must also converge to 0. This property was established for the 2D NS equations in the seminal paper of Foias and Prodi [37] in the modal case and then later extended to the nodal and volume elements case in [38, 48, 47, 49]. The case of general determining interpolants was introduced and investigated in [18, 19].

The physical model of interest in this article is a special case of the 3D quasi-geostrophic (QG) equation on a half-space. The 3D QG equation asserts the conservation of “potential vorticity” subject to a dynamical boundary condition. It is valid in 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, i.e., low Rossby number. It is the simplest model of such fluids with nontrivial dynamics that describes the departure from geostrophically balanced flows, i.e., where the Coriolis force and the horizontal pressure gradient are in balance (cf. [57]). An interesting consequence of this balance is that it imposes a planar dimensionality on the corresponding solution, and is thus the source for the strong 2D features of the otherwise 3D flow (cf. [20]). In the simplified scenario where the potential vorticity, q, is advected along material lines, one has that q0 solves its evolution equation exactly, but retains the evolution of its streamfunction through its boundary condition. Stated in non-dimensionalized variables and in terms of the potential vorticity streamfunction, Ψ=Ψ(𝐱,t), 𝐱=(x,y,z), we have

(1.2) Δ 3 D Ψ = 0 in Ω × { z > 0 } × { t > 0 } , z Ψ ( , 0 , t ) = θ ( , t ) , lim z Ψ ( , z , t ) = 0 ,

where

Δ 3 D = x 2 + y 2 + z 2

and the Neumann boundary condition, θ, satisfies the forced, dissipative SQG equation

(1.3) t θ + κ Λ γ θ + u θ = f , u = θ ,

and both Ψ and θ are subjected to periodic boundary conditions in the horizontal variables x,y over the fundamental periodic domain Ω=[-π,π]2. Here, γ(0,2) and θ represents the scalar surface temperature or buoyancy of a fluid, which is advected along the velocity vector field, u, and f is a given source term. The external forcing can be time-dependent provided that it is bounded in time with values in the relevant spatial space; for simplicity, we will deal with the case of f being time-independent. The velocity is related to θ by a Riesz transform, :=(-R2,R1), where the symbol of Rj is given by iξj/|ξ|. For 0<γ2, we denote the dissipative operator by Λγ:=(-Δ)γ/2, which is the operator whose symbol is |ξ|γ; it appears with γ=1 when accounting for viscous drag at the boundary, i.e., Ekman pumping, (cf. [20, 30]). In this paper, we will concern ourselves with the subcritically diffusive case, γ(1,2), which still preserves the analytical difficulties that we will encounter arising from the nature of its nonlocality, but simplifies the issue of well-posedness for the synchronizing model (1.4). We will assume that θ0,f are π-periodic in Ω with mean zero. In particular, θ is also π-periodic with mean zero since the mean-zero property is preserved by the evolution.

While it is still not fully understood how to describe the motion of fluids in the regime of low Rossby number, the SQG equation has many interesting features which are relevant both physically and mathematically [43]. For instance, experiments [8] have shown that the energy spectra of rotating fluids exhibit a power law scaling different from the one derived through the 2D NS equations, but consistent with the one derived through the SQG equation [20]. On the other hand, mathematically, the SQG equation has striking resemblance to the 3D NS equations. Indeed, by applying to (1.3) one uncovers a mechanism for vortex stretching analogous to the one found in the vorticity formulation of the 3D NS equations. Thus, the SQG equation provides a two-dimensional scalar model which exhibits three-dimensional phenomena and challenges. Since its introduction into the mathematical community by Constantin, Majda and Tabak [23], the inviscid equation, along with its critical (γ=1) and supercritical (γ<1) counterparts, have been studied extensively, and by now, well-posedness in various function spaces and global regularity has been resolved in all but the supercritical case (cf. [13, 25, 26, 24, 28, 31, 53, 54, 58]). The long-time behavior and existence of a global attractor has also been studied in both the subcritical and critical cases (cf. [14, 17, 21, 27, 24, 50]). In particular, in [50], Ju established the existence of a global attractor, 𝒜, for (1.3) in the subcritical regime γ(1,2), which we recall in Section 2.4. In the context of the nudging scheme induced by (1.4), we will assume that the noiseless observables are sampled from an absorbing ball that contains 𝒜 (see Section 3).

In this paper, data assimilation achieved through

(1.4) t η + κ Λ γ η + v η = f - μ J h ( η - θ ) , v = η ,

subject to periodic boundary conditions over Ω=[-π,π]2, as in (1.3). We recall that μ is the nudging parameter and Jh with h>0 is an interpolating operator based on coarse spatial measurements that satisfies certain approximation properties. We will first establish in Theorem 1 the global existence and uniqueness of strong solutions to (1.4) in a Sobolev class of functions for certain time-independent source terms, provided that μ and h are jointly chosen appropriately. The estimates required to prove Theorem 1 will be performed in Section 3. We then show that if μ is chosen large enough and h is correspondingly taken sufficiently small, then the unique solution of (1.4) synchronizes at an exponential rate with the reference solution, θ, for any γ(1,2) (see Section 2.2). In particular, we show that

(1.5) η ( t ) - θ ( t ) L 2 ( Ω ) 0 at an exponential rate as t .

Convergence of the corresponding three-dimensional streamfunctions then follows immediately from this (Corollary 3). To see this, let Ψ be given by (1.2) and Ψη be given by

(1.6) Δ 3 D Ψ η = 0 in Ω × { z > 0 } × { t > 0 } , z Ψ η ( , 0 , t ) = η ( , t ) , lim z Ψ η ( , z , t ) = 0 .

Then (1.5) implies

( Ψ η ( t ) - Ψ ( t ) ) L 2 ( Ω × + ) 0 exponentially as t .

Indeed, since the Dirichlet-to-Neumann map determines the relations Ψ|z=0=-Λ-1θ and Ψ|z=0=-Λ-1η, upon taking the L2-scalar product of (1.2) with Ψ, integrating by parts, and applying the boundary conditions it follows that

( Ψ ( t ) - Ψ η ( t ) ) L 2 ( Ω × + ) 2 = | Ω ( η ( t ) - θ ( t ) ) Λ - 1 ( η ( t ) - θ ( t ) ) 𝑑 x 𝑑 y | C η ( t ) - θ ( t ) L 2 ( Ω ) 2

for some absolute constant C>0. We refer to [60], for example, for details regarding the Neumann problem. One may thus conclude, at least in the simplified scenario described above, that measurements on only the boundaryΩ×{z=0}+3 are required for the synchronization of the streamfunction over the entire 3D domain (1.4). Studies on data assimilation on simple forecast models, in which some state variable observations are not available as input were carried out in [41, 40], for instance. It was observed that although the nonlinearity in the models can mediate couplings across all length scales, the full state of the system can nevertheless be recovered by employing coarse mesh measurements of only some the state variables so long as one uses a good dynamical model and data assimilation algorithm (cf. [15, 35]). Our result, therefore, rigorously confirms this understanding in a precise way, e.g., employing only surface measurements to recover the three-dimensional stream function, through the SQG model (1.3) with the corresponding nudging equation (1.4).

2 Preliminaries

2.1 Function Spaces Lperp, Vσ, Hperσ, H˙perσ, Cper

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

per := { ϕ : ϕ ( x , y ) = ϕ ( x + 2 π , y ) = ϕ ( x , y + 2 π ) = ϕ ( x + 2 π , y + 2 π ) a.e. } .

Let C(𝕋2) denote the class of functions which are infinitely differentiable over 𝕋2. We define Cper(𝕋2) by

C per ( 𝕋 2 ) := C ( 𝕋 2 ) per .

For 1p, we define the periodic Lebesgue spaces by

L per p ( 𝕋 2 ) := { ϕ per : ϕ L p < } ,

where

ϕ L p := ( 𝕋 2 | ϕ ( x ) | p 𝑑 x ) 1 / p , 1 p < , and    ϕ L := ess sup x 𝕋 2 | ϕ ( x ) | .

Let us also define

(2.1) 𝒵 := { ϕ L per 1 : 𝕋 2 ϕ ( x ) 𝑑 x = 0 } .

Let ϕ^(𝐤) denote the Fourier coefficient of ϕ at wave-number 𝐤2. For any real number σ, we define the homogeneous Sobolev space, H˙perσ(𝕋2), by

(2.2) H ˙ per σ ( 𝕋 2 ) := { ϕ L per 2 ( 𝕋 2 ) : ϕ H ˙ σ < } ,

where

𝐤 2 { } | 𝐤 | 2 σ | ϕ ^ ( 𝐤 ) | 2 .

We define the inhomogeneous Sobolev space, Hperσ(𝕋2), by

H per σ ( 𝕋 2 ) := { ϕ L per 2 ( 𝕋 2 ) : ϕ H σ < } ,

where

(2.3) ϕ H σ 2 := 𝐤 2 ( 1 + | 𝐤 | 2 ) σ | ϕ ^ ( 𝐤 ) | 2 .

Let 𝒱0𝒵 denote the set of trigonometric polynomials with mean zero over 𝕋2 and set

(2.4) V σ := 𝒱 0 ¯ H σ ,

where the closure is taken with respect to the norm given by (2.3). Observe that the mean-zero condition can be equivalently stated as ϕ^()=0. Thus, H˙σ and Hσ are equivalent as norms over Vσ. Moreover, by Plancherel’s theorem we have

ϕ H ˙ σ = Λ σ ϕ L 2 .

Finally, for σ0, we identify V-σ as the dual space, (Vσ), of Vσ, which can be characterized as the space of all bounded linear functionals, ψ, on Vσ such that

ψ H ˙ - σ < .

Therefore, we have the following continuous embeddings:

V σ V σ V 0 V - σ V - σ , 0 σ σ .

Remark 2.1

Since we will be working over Vσ and H˙σ, Hσ determine equivalent norms over Vσ, we will often denote H˙σ simply by Hσ for convenience. We will distinguish between the inhomogeneous and homogeneous Sobolev norm when we are outside of this context (see Appendix).

2.2 Interpolant Observables

We will consider two types of interpolant observables, which we refer to as Type I and Type II. We show in the Appendix that the interpolant given by local averages of the function over cubes that partition the domain, 𝕋2, are of Type I, and that modal projection onto finitely many, fixed wave-numbers are of Type II.

Let h>0, 1p, and Jh:Lp(𝕋2)Lp(𝕋2) be a linear operator. Suppose that Jh satisfies

(2.5) sup h > 0 J h ϕ L p C ϕ L p , 1 < p < ,
(2.6) J h ϕ L p C h 2 / p - 2 / q ϕ L q , 1 q p < ,
(2.7) J h ϕ H ˙ β C h - β ϕ L 2 , β 0 ,

where C>0 is an absolute constant independent of ϕ. Let us remark here that in our main theorems, we will impose that hh¯ for some h¯>0, where h¯ is the size of the required spatial resolution of the collected measurements (see Section 3). Thus, at least within the context of the theorems, only the operators Jh for hh¯ are considered, though numerical studies of the algorithm suggest that synchronization can occur for values of h¯ far larger than what is suggested by the analytical bounds given by the theorems (cf. [3, 39]).

Type I

In addition to (2.5)–(2.7), interpolants of Type I will also satisfy

(2.8) ϕ - J h ϕ L 2 C h β ϕ H ˙ β and ϕ - J h ϕ H ˙ - β C h β ϕ L 2 , β ( 0 , 1 ] .

Type II

In addition to (2.5)–(2.7), interpolants of Type II will also satisfy

(2.9) ϕ - J h ϕ H ˙ α C h β - α ϕ H ˙ β , β > α and Λ β J h ϕ = J h Λ β ϕ .

Observe that Type II interpolants are also Type I. We refer to the Appendix, where we provide examples of both Type I and II interpolants.

2.3 Inequalities for Fractional Derivatives

We will make use of the following bound for the fractional Laplacian, which can be found for instance in [22, 24, 50].

Proposition 1

Let p2, 0γ2, and ϕCper(T2). Then

𝕋 2 | ϕ | p - 2 ( x ) ϕ ( x ) Λ γ ϕ ( x ) 𝑑 x 2 p Λ γ / 2 ( | ϕ | p / 2 ) L 2 2 .

We will also make use of the following calculus inequality for fractional derivatives (cf. [51, 52] and references therein):

Proposition 2

Let ϕ,ψCper(T2), β>0, and p(1,). Let 1p=1p1+1p2=1p3+1p4, and p2,p3(1,). There exists an absolute constant C>0, depending only on σ,p,pi, such that

Λ β ( ϕ ψ ) L p C ψ L p 1 Λ β ϕ L p 2 + C Λ β ψ L p 3 ϕ L p 4 .

Finally, we will frequently apply the following interpolation inequality, which is a special case of the Gagliardo–Nirenberg interpolation inequality and can be proven by using the Plancherel theorem combined with Hölder’s inequality:

Proposition 3

Let ϕHperβ(T2) and 0α<β. There exists an absolute constant C>0, depending only on α,β, such that

Λ α ϕ L 2 C Λ β ϕ L 2 α β ϕ L 2 1 - α β .

2.4 L p -Bounds and Global Attractor of SQG Equation

Let us recall the following estimates for the reference solution θ (cf. [24, 50, 58]).

Proposition 4

Let κ>0, γ[0,2] and θ0,fLperp(T2)Z. Suppose that θLperp(T2) is a smooth solution of (1.3) such that θ(,0)=θ0(). There exists an absolute constant C>0 such that for any p2, we have

(2.10) θ ( t ) L p ( θ 0 L p - 1 C F L p ) e - C κ t + 1 C F L p , F L p := 1 κ f L p .

Moreover, for p=2 and fV-γ/2, we have

(2.11) θ ( t ) L 2 2 ( θ 0 L 2 2 - F H - γ / 2 2 ) e - κ t + F H - γ / 2 2 , F H - γ / 2 := 1 κ f H - γ / 2 .

It was shown in [50] that in the subcritical range, 1<γ2, equation (1.3) has an absorbing ball in Hperσ(𝕋2), for σ>2-γ, and corresponding global attractor, 𝒜Vσ. We recall that an Hperr(𝕋2)-absorbing set for a dissipative equation is a bounded set Hperr(𝕋2) characterized by the property that for any θ0Hperr(𝕋2), there exists t0=t0(θ0Hperr)>0 such that S(t)θ0 for all tt0, where {S(t)}t0 denotes the semigroup of the corresponding dissipative equation.

Proposition 5

Proposition 5 (Global attractor)

Suppose that 1<γ2 and σ>2-γ. Let fVσ-γ/2Lperp(T2), where 1-σ<2p<γ-1. Then (1.3) has an absorbing ball BHσ given by

(2.12) H σ := { θ V σ : θ H σ Θ H σ }

for some ΘHσ<. Moreover, the solution operator S(t)θ0=θ(t), t>0, of (1.3) defines a semigroup in the space Vσ and possesses a global attractor AVσ, i.e., A is a compact, connected subset of Vσ satisfying the following properties:

  1. 𝒜 is the maximal bounded invariant set,

  2. 𝒜 attracts all bounded subsets in V σ in the topology of H ˙ per σ .

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

Remark 2.2

In the estimates that follow below, c,C, will denote generic positive absolute constants, which depend only on other non-dimensional scalar quantities, and may change line-to-line in the estimates. We also use the notation AB and AB to denote the relations AcB and cBAc′′B, respectively, for some absolute constants c,c,c′′>0.

3 Main Results and A Priori Estimates

We will work in the following setting throughout this section.

Standing Hypotheses

Assume the following:

  1. 1 < γ 2 ,

  2. σ > 2 - γ ,

  3. p [ 1 , ] such that 1-σ<2p<γ-1, fixed,

  4. η 0 V σ ,

  5. f V σ - γ / 2 L p , time-independent,

  6. θ 0 H σ L p , where Hσ is the Hσ-absorbing ball with radius ΘHσ from Proposition 5,

  7. J h satisfies (2.5), (2.6), (2.7) and is either of Type I or Type II.

Observe that by Proposition 4, (H3) and (H5) immediately imply that the solution θ of (1.3) corresponding to initial data θ0 satisfies

(3.1) Θ L 2 := sup t > 0 θ ( t ) L 2 < and Θ L p := sup t > 0 θ ( t ) L p < .

We will first show (in Section 3.1) that smooth solutions to (1.4), η(t), satisfy

(3.2) M L 2 := sup t > 0 η ( t ) L 2 < ,

and (in Section 3.2) that this implies

(3.3) M L p := sup t > 0 η ( t ) L p < .

These two bounds will then be used to show (in Section 3.3) that

M H σ := sup t > 0 η ( t ) H σ < .

With these estimates in hand and under the Standing Hypotheses, we establish in Section 4 short-time existence and uniqueness in the space Hσ in Section 4:

Theorem 1

Assume that (H1)(H7) hold. Let θ be the unique global strong solution of (1.3) corresponding to θ0. There exists ρ=ρ(h,σ,γ) (given by (3.24) below) such that if

(3.4) μ κ ρ ( h , σ , γ ) 1 ,

then for each T>0, there exists a unique strong solution ηL(0,T;Vσ)L2(0,T;H˙perσ+γ/2) of (1.4) such that

η ( t ) H σ M H σ , t [ 0 , T ] ,

for some quantity MHσ (given by (3.28) below) that depends only on μ,κ, fHσ-γ/2, and ΘHσ. Moreover, ηC([0,T];Vσ-ϵ) for all ϵ(0,σ+12).

Ultimately, the estimates we collect will also be used to ensure asymptotic synchronization of η to the reference solution θ.

Theorem 2

Assume that (H1)(H7) holds. Let θ be the unique global strong solution of (1.3) corresponding to θ0. Suppose that μ satisfies

(3.5) μ κ ( Θ L p κ ) γ / ( γ - 1 - 2 / p ) .

There exists an absolute constant cσ<1, depending on σ, such that if μ,h satisfy

(3.6) μ h γ κ ,

then

(3.7) η ( t ) - θ ( t ) L 2 2 O ( e - c σ μ t ) , t > 0 ,

where η is the unique global strong solution to (1.4) corresponding to η0.

As we discussed at the end of Section 1, Theorem 2 immediately implies the synchronization of the streamfunctions corresponding to θ and η.

Corollary 3

Assume that (H1)(H7) holds. Let θ and η be the unique global strong solutions of (1.3) and (1.4) corresponding to θ0 and η0, respectively. Let Ψ and Ψη denote the corresponding streamfunctions of θ and η, i.e., satisfying (1.2) and (1.6), respectively. Suppose that μ satisfies (3.5). Then there exists an absolute constant C>0 such that if μ and h satisfying (3.6), then

( Ψ η ( t ) - Ψ ( t ) ) L 2 ( Ω × + ) O ( e - c σ μ t ) .

Remark 3.1

We point out that to guarantee existence and uniqueness of strong solutions to (1.4), it suffices for h and μ to satisfy

(3.8) μ h γ κ .

To guarantee the synchronization property, (3.7), it suffices for h and μ to satisfy (3.5) in addition to (3.8).

Remark 3.2

Note that in the case where Jh is given by projection onto wave numbers of size N (see Section 2.2 and Appendix), conditions (3.5) with (3.4) provide an estimate on the number of modal observables that are sufficient for the algorithm to synchronize to the reference solution. We point out that this bound matches the scaling for the number of determining modes for (1.3) obtained in [16]. Indeed, denote by the smallest number with the property that if the difference of the modes up to size converge to 0, as t, then so must the difference of the solutions themselves. It is shown in [16] that

( Θ κ ) 1 / ( γ - 1 ) ,

where Θ is defined as the smallest constant for which {θLΘ} is an absorbing ball in L for (1.3).

On the other hand, if Jh is given by projection onto finitely many Fourier modes, i.e., of Type II, we may take ρ(h,σ,γ)=hγ, where h=2πN, so that (formally) setting p= in Theorem 2, the resulting condition on N becomes (see (4.4) and (4.5))

N ( Θ κ ) 1 / ( γ - 1 ) .

We note that for us, this choice for p is valid when σ>1.

In the case where Jh is given by local spatial averages over cubes of side-length h, we may also take ρ(h,γ)=hγ, when σγ2, so that (at least for σ>1), the number of local averages required to guarantee synchronization is proportional to Θ1/(γ-1).

We will perform energy estimates on the solutions to the following initial value problem given by

(3.9) t η + κ Λ γ η + v η = f - μ J h ( η - θ ) , v = η , η ( x , 0 ) = η 0 ( x ) ,

where κ is defined as in (1.3), h,μ are positive, absolute constants whose magnitudes are to be specified later, and where θ is a solution to (1.3) corresponding to θ0σ.

We note that in what follows, the estimates we perform are formal, though they may be done rigorously at the level of the equation with artificial viscosity, i.e., (1.4) with the additional term -νΔη on the left-hand side (see Section 4 for details). We emphasize that the estimates we obtain will be independent of γ and we pass to the limit as ν0.

3.1 Uniform L2-Estimates

For μ>0, let FH-γ/2 be given by (2.11), and ΘL2 by (3.1). Define

(3.10) R L 2 2 := C ( κ μ F H - γ / 2 2 + Θ L 2 2 ) ,

where C>0 is an absolute constant.

Proposition 6

There exist absolute constants c0,C0>0 with c0 depending on C0 such that if (3.10) holds with C=C0 and if μ,h satisfy

(3.11) μ h γ κ c 0 ,

then the following inequalities hold:

(3.12) η ( t 2 ) L 2 2 + κ t 1 t 2 e - μ ( t 2 - s ) η ( s ) H γ / 2 2 𝑑 s ( η 0 L 2 2 - R L 2 2 ) e - μ ( t 2 - t 1 ) + R L 2 2 ,
(3.13) η ( t 2 ) L 2 2 + κ t 1 t 2 η ( s ) H γ / 2 2 𝑑 s η ( t 1 ) L 2 2 + μ R L 2 2 ( t 2 - t 1 )

for all 0t1<t2. In particular, we have

(3.14) M L 2 ( t 1 , t 2 ) η ( t 1 ) L 2 + R L 2 ,

where ML2(t1,t2):=supt[t1,t2]η(t)L2.

Proof.

We multiply (3.9) by η and integrate over 𝕋2 to write

1 2 d d t η L 2 2 + κ Λ γ / 2 η L 2 2 + μ η L 2 2 f η 𝑑 x + μ ( η - J h η ) η 𝑑 x + μ J h θ η 𝑑 x
= I + 𝐼𝐼 + 𝐼𝐼𝐼 .

Observe that by the Cauchy–Schwarz and Young inequalities, as well as (2.5), we have

I 1 κ Λ - γ / 2 f L 2 2 + κ 4 Λ γ / 2 η L 2 2 , 𝐼𝐼𝐼 μ J h θ L 2 η L 2 C μ θ L 2 η L 2 C μ θ L 2 2 + μ 4 η L 2 2 .

On the other hand, by (2.8) or (2.9) we have

𝐼𝐼 μ η - J h η L 2 η L 2 C μ h γ / 2 Λ γ / 2 η L 2 η L 2 κ 4 Λ γ / 2 η L 2 2 + C h γ μ 2 κ η L 2 2 .

Combining these estimates with (3.1) yields

(3.15) 1 2 d d t η L 2 2 + κ 2 Λ γ / 2 η L 2 2 + μ ( 3 4 - C h γ μ κ ) η L 2 2 1 κ Λ - γ / 2 f L 2 2 + C μ Θ L 2 2 .

Upon applying (3.11) to (3.15), we arrive at

(3.16) d d t η L 2 2 + μ η L 2 2 + κ η H γ / 2 2 2 ( κ F H - γ / 2 2 + C μ Θ L 2 2 ) .

Hence, Gronwall’s inequality and (3.10) imply that

η ( t 2 ) L 2 2 + κ t 1 t 2 e - μ ( t 2 - s ) η ( s ) H γ / 2 2 𝑑 s η ( t 1 ) L 2 2 e - μ ( t 2 - t 1 ) + 2 R L 2 2 ( 1 - e - μ ( t 2 - t 1 ) ) .

On the other hand, integrating (3.16) over [t1,t2] gives (3.13), as desired. This completes the proof. ∎

3.2 L 2 to Lp Uniform Bounds

Let p>2 and μ>0. Let FLp be given by (2.10), ΘLp by (3.1), and MLp by (3.2). Define

(3.17) R L p p := C p ( κ p μ p F L p p + Θ L p p ) and R ~ L p p := C p ( κ p μ p ( F L p p + ( M L 1 p ) p ) + Θ L p p + h 2 - p M L 2 p ) ,

where C>0 is an absolute constant. We will show that Proposition 6 implies the following Lp-bounds.

Proposition 7

Suppose p>2 and μ>0. There exist absolute constants c0,C0>0, independent of p, such that if (3.17) holds with C=C0, and (3.11) holds, then

η ( t 2 ) L p p ( η ( t 1 ) L p p - p p μ p κ p R ~ L p p ) e - c 0 κ ( t 2 - t 1 ) + p p μ p κ p R ~ L p p

holds for all 0t1<t2. In particular, we have

(3.18) M L p ( t 1 , t 2 ) η ( t 1 ) L p + C 0 p μ κ R ~ L p ,

where MLp(t1,t2):=supt[t1,t2]η(t)Lp.

To prove Proposition 7, we will make use of the following identity.

Lemma 8

Let QT2 open and ϕL2(Q). Let ϕQ:=1a(Q)Qϕ𝑑x, where a(Q) denotes the area of Q. Then

ϕ - ϕ Q L 2 ( Q ) 2 = ϕ L 2 ( Q ) 2 - a ( Q ) ϕ Q 2 .

Proof.

Simply observe that (ϕ-ϕQ)2=ϕ2-2ϕϕQ+ϕQ2. Thus

ϕ - ϕ Q L 2 ( Q ) 2 = Q ϕ 2 𝑑 x - 2 ϕ Q Q ϕ 𝑑 x + a ( Q ) ϕ Q 2 = ϕ L 2 ( Q ) 2 - 2 a ( Q ) ϕ Q 2 + a ( Q ) ϕ Q 2 ,

as desired. ∎

Proof of Proposition 7.

Let p2. Upon multiplying (3.9) by η|η|p-2, integrating over 𝕋2, using the fact that vηη|η|p-2dx=0, then applying Hölder’s inequality, Proposition 1, and Young’s inequality, one arrives at

1 p d d t η L p p + 2 κ p Λ γ / 2 | η | p / 2 L 2 2 ( f L p + μ J h η L p + μ J h θ L p ) η L p p - 1
(3.19) κ C p - 1 ( p - 1 ) p - 1 p μ p κ p ( κ p μ p F L p p + J h η L p p + J h θ L p p ) + κ 2 p η L p p .

Observe that by (2.5) we have JhθLpCθLpCΘLp, which is finite by (3.1). By (2.6), Proposition 6, and (3.2) we have

(3.20) J h η L p C h 2 / p - 1 η L 2 C h 2 / p - 1 M L 2 .

From Corollary A.6 and Lemma 8, it follows that

(3.21) η L p p - ( 4 π 2 ) - 1 η L p / 2 p = | η | p / 2 - ( | η | p / 2 ) 𝕋 2 L 2 2 C Λ γ / 2 | η | p / 2 L 2 2 .

By interpolation of Lq-spaces and Young’s inequality, using the notation in (3.3), we have

(3.22) η L p / 2 p η L p p ( p - 2 ) p - 1 M L 1 p p - 1 C p ( p - 2 p - 1 ) p - 2 M L 1 p + π 2 η L p p

for some absolute constant C>0.

Thus, upon collecting (3.20), (3.21) and (3.22), we return to (3.19), and arrive at

1 p d d t η L p p + C κ p η L p p C p κ ( p - 1 ) p - 1 p μ p κ p ( R L p p + h 2 - p M L 2 p ) + κ C p p M L 1 p .

It then follows from Gronwall’s inequality and (3.17) that

η ( t 2 ) L p p η ( t 1 ) L p p e - C κ ( t 2 - t 1 ) + p p μ p κ p R ~ L p p ( 1 - e - C κ ( t 2 - t 1 ) ) ,

as desired. ∎

3.3 L p to Hσ Uniform Bounds

Let μ>0, ΘHσ be given by (2.12), and MLr by (3.18). Define

(3.23) F H σ - γ / 2 := 1 κ f H σ - γ / 2 , R H σ 2 := C ( κ μ F H σ - γ / 2 2 + Θ H σ 2 ) , Ξ r , α ( t 1 , t 2 ) := C ( M L r ( t 1 , t 2 ) κ ) 2 α γ - 1 - 2 / r ,

where C>0 is an absolute constant. For h>0, define

(3.24) ρ ( h , σ , γ ) := { h 2 σ , σ γ 2 , h γ , σ > γ 2 .

Proposition 9

There exist absolute constants C0,c0>0, with c0 depending on C0, such that if (3.23) holds, with C=C0 and h>0 satisfies

(3.25) μ κ ρ ( h , σ , γ ) c 0 ,

then when σγ2, we have that

(3.26) η ( t 2 ) H σ 2 [ η ( t 1 ) H σ 2 + Ξ p , σ ( t 1 , t 2 ) ( η ( t 1 ) L 2 2 - R L 2 2 ) - R H σ 2 ] e - μ ( t 2 - t 1 ) + Ξ p , σ ( t 1 , t 2 ) R L 2 2 + R H σ 2

holds for all 0t1<t2 and

(3.27) κ 0 t η ( s ) H σ + γ / 2 2 𝑑 s η 0 H σ 2 + Ξ p , σ ( 0 , t ) η 0 L 2 2 + μ t ( R H σ 2 + Ξ p , σ ( 0 , t ) R L 2 2 ) , t > 0 .

In particular, we have

(3.28) η ( t ) H σ 2 η 0 H σ 2 + R H σ 2 + Ξ p , σ M L 2 2 := M H σ 2 , t > 0 ,

where

(3.29) Ξ r , α := Ξ r , α ( 0 , ) .

On the other hand, when σ>γ2, then there exist absolute constants C0,c0>0 such that if (3.23) holds with C=C0 and h>0 satisfies (3.25), then

(3.30) η ( t ) H σ 2 η 0 H σ 2 + C 0 ( F H σ - γ / 2 2 + μ h 2 σ κ ( M L 2 2 + Θ L 2 2 ) ) + Ξ p , σ M H γ / 2 2 , t > 0 .

Proof.

Let t[t1,t2]. Multiply (3.9) by Λ2ση and integrate over 𝕋2 to obtain

1 2 d d t Λ σ η L 2 2 + κ Λ σ + γ / 2 η L 2 2 + μ Λ σ η L 2 2 = - v η Λ 2 σ η d x + f Λ 2 σ η 𝑑 x + μ ( η - J h η ) Λ 2 σ η 𝑑 x
+ μ J h θ Λ 2 σ η 𝑑 x
(3.31) = I + 𝐼𝐼 + 𝐼𝐼𝐼 + 𝐼𝑉 .

We estimate I and 𝐼𝐼 first. We will estimate 𝐼𝐼𝐼 and 𝐼𝑉 depending on whether Jh is of Type I or II.

To estimate I, we first use the facts that (vη)=vη and Λ-1, i.e., operator with symbol -iξ/|ξ|, to rewrite I with Parseval’s identity as

I = Λ σ - γ / 2 ( v η ) Λ σ + γ / 2 η 𝑑 x = Λ 1 + σ - γ / 2 ( v η ) Λ σ + γ / 2 η 𝑑 x .

Now, since 1<p< satisfies 1-σ<2p, Sobolev embedding ensures HσLp. Let q>1 be such that 1p+1q=12. By Proposition 2, the fact that and commute with Λ and are Calderón–Zygmund operators, i.e., vLrCηLr for all r(1,), and since H2/pLq, we may estimate I as

| I | Λ 1 + σ - γ / 2 ( v η ) L 2 Λ σ + γ / 2 η L 2
C Λ 1 + σ - γ / 2 η L q η L p Λ σ + γ / 2 η L 2
(3.32) C Λ 1 + σ - γ / 2 + 2 / p η L 2 η L p Λ σ + γ / 2 η L 2 .

By interpolation (Proposition 3), we have

(3.33) Λ 1 + σ - γ / 2 + 2 / p η L 2 Λ σ + γ / 2 η L 2 σ - ( γ - 1 - 2 / p ) σ Λ γ / 2 η L 2 γ - 1 - 2 / p σ .

Therefore, returning to (3.32), from (3.33) and Young’s inequality, we have

| I | C Λ σ + γ / 2 η L 2 2 σ - ( γ - 1 - 2 / p ) σ Λ γ / 2 η L 2 γ - 1 - 2 / p σ η L p
κ 8 Λ σ + γ / 2 η L 2 2 + 1 2 Ξ p , σ ( t 1 , t 2 ) κ Λ γ / 2 η L 2 2 , t [ t 1 , t 2 ] ,

where Ξp,σ(t1,t2) is given in (3.23). Note that this quantity is finite due from Proposition 7.

For 𝐼𝐼, we apply the Plancherel relation, the Cauchy–Schwarz inequality, and Young’s inequality, to obtain

| 𝐼𝐼 | 1 κ Λ σ - γ / 2 f L 2 2 + κ 4 Λ σ + γ / 2 η L 2 2 .

Now we estimate 𝐼𝐼𝐼 and 𝐼𝑉. We split the treatment of these terms into two cases: σγ/2 and σ>γ/2. Note that the estimates hold for both Type I and II operators, although we will only make use of Type I properties.

Case: σγ2. We estimate 𝐼𝐼𝐼 by applying the Cauchy–Schwarz inequality, the Poincaré inequality, (2.8), (3.24), (3.25) (with c0 sufficiently small), then applying Young’s inequality we estimate

| 𝐼𝐼𝐼 | μ η - J h η L 2 Λ 2 σ η L 2 C μ h σ Λ σ η L 2 Λ σ + γ / 2 η L 2
(3.34) κ 16 Λ σ + γ / 2 η L 2 2 + μ 4 Λ σ η L 2 2 .

Note that we used the fact that (H1) implies σ<1. On the other hand, using (2.8), the Cauchy–Schwarz and Poincaré inequalities, (3.24), and (3.25), we similarly estimate 𝐼𝑉 as

| 𝐼𝑉 | μ θ - J h θ L 2 Λ 2 σ η L 2 + μ Λ σ θ L 2 Λ σ η L 2
C μ μ h 2 σ κ θ H σ 2 + κ 16 Λ σ + γ / 2 η L 2 2 + C μ Λ σ θ L 2 2 + μ 4 Λ σ η L 2 2
C μ Θ σ 2 + κ 16 Λ σ + γ / 2 η L 2 2 + μ 4 Λ σ η L 2 2 .

Upon combining I𝐼𝑉, we arrive at

(3.35) d d t η H σ 2 + μ η H σ 2 + κ η H σ + γ / 2 2 2 κ F H σ - γ / 2 2 + 2 μ C Θ H σ 2 + Ξ p , σ ( t 1 , t 2 ) κ η H γ / 2 2 .

Thus, from Gronwall’s inequality applied over [t1,t2] we obtain

η ( t 2 ) H σ 2 η ( t 1 ) H σ 2 e - μ ( t 2 - t 1 ) + 2 R H σ 2 ( 1 - e - μ ( t 2 - t 1 ) ) + Ξ p , σ ( t 1 , t 2 ) ( κ t 1 t 2 e - μ ( t 2 - s ) η ( s ) H γ / 2 2 𝑑 s ) .

We may then apply Proposition 6 to bound the last term above and obtain (3.26).

On the other hand, letting t1=0 and t2=t, then integrating (3.35) over [0,t] gives

κ 0 t η ( s ) H σ + γ / 2 2 𝑑 s η 0 H σ 2 + 2 t μ R H σ 2 + Ξ p , σ ( 0 , t ) ( κ 0 t η ( s ) H γ / 2 2 𝑑 s ) ,

and we again use Proposition 6 to bound the last term and obtain (3.27). This establishes the case σγ2.

Case: σ>γ2. Since η0Hγ/2η0Hσ, it follows by estimating exactly as above that η is uniformly bounded in Hγ/2, provided that (3.25) holds with ρ=hγ. To obtain uniform estimates in Hσ, we do not insert a damping term, so that we replace (3.31) by

1 2 d d t Λ σ η L 2 2 + κ Λ σ + γ / 2 η L 2 2 = I + 𝐼𝐼 + 𝐼𝐼𝐼 + 𝐼𝑉 ,

where I, 𝐼𝐼, 𝐼𝑉 are as before and 𝐼𝐼𝐼 is given by

𝐼𝐼𝐼 = - μ J h η Λ 2 σ η 𝑑 x .

We treat I and 𝐼𝐼 the same. To estimate 𝐼𝐼𝐼 and 𝐼𝑉, we apply Plancherel’s theorem, the Cauchy–Schwarz inequality, (2.7), the Poincaré inequality, and Young’s inequality to obtain

| 𝐼𝐼𝐼 | μ J h η H σ - γ / 2 η H σ + γ / 2 C μ h - ( σ - γ / 2 ) η L 2 η H σ + γ / 2 C μ 2 h 2 σ - γ κ η L 2 2 + κ 8 η H σ + γ / 2 2 ,
| 𝐼𝑉 | C μ h 2 σ θ L 2 2 + μ 2 η H σ 2 .

Then combining I𝐼𝑉, we apply the Poincaré’s inequality to arrive at

d d t η H σ 2 + κ η H σ 2 2 κ F σ , γ 2 + Ξ p , σ ( t 1 , t 2 ) κ η H γ / 2 2 + C μ h 2 σ ( μ h γ κ η L 2 2 + Θ L 2 2 ) .

Then by (3.25), Gronwall’s inequality, Propositions 6 and 7, we obtain (3.30). ∎

Remark 3.3

We observe that in the case that Jh is Type II, we need not treat the cases σγ2 and σ>γ2 separately. Indeed, for any σ>2-γ, observe that by the Cauchy–Schwarz inequality, (2.9), Young’s inequality, (3.24), (3.25), and c0 sufficiently small we may estimate

| 𝐼𝐼𝐼 | μ Λ σ ( η - J h η ) L 2 Λ σ η L 2
κ 8 Λ σ + γ / 2 η L 2 2 + C h γ μ 2 κ Λ σ η L 2 2 κ 8 Λ σ + γ / 2 η L 2 2 + μ 4 Λ σ η L 2 2 .

Using (2.5), (2.9), and since θ0Hσ, we estimate 𝐼𝑉 as

| 𝐼𝑉 | μ J h Λ σ θ L 2 Λ σ η L 2 C μ Θ H σ 2 + μ 4 Λ σ η L 2 2 .

Thus, we may combine these estimates with those for I,𝐼𝐼 and apply Gronwall’s inequality to arrive at (3.27).

4 Proofs of Main Theorems

For the proofs of both Theorems 1 and 2, we assume the Standing Hypotheses, (H1)(H7). Let FL2,FH-γ/2, FLp, Fσ-γ/2, ΘL2,ΘLp,ΘHσ, and RL2,RL2, RLp,RLp, RHσ,Ξr,α be defined as in Propositions 4, 6, 7, 9, and Proposition 5. Let C0 be the maximum among the constants, C0, appearing in Propositions 6, 7, 9, and let c0 be the minimum among the constants, c0, appearing there. We assume that c01. Suppose (3.4) holds with ρ given by (3.24). This implies that μ,h satisfy

(4.1) μ κ max { ρ ( h , σ , γ ) , h γ } c 0 .

We first prove Theorem 1, i.e., the short-time existence of strong solutions and establish uniqueness within this class of solutions. We will then prove Theorem 2, which establishes the synchronization property of the algorithm.

4.1 Proof of Theorem 1

Existence of Strong Solutions.

If η0,θ0Lper2(𝕋2)𝒵 and fV-γ/2, then assumption (4.1), and the a priori bounds of Propositions 4, 6 guarantee the existence of weak solutions (cf. [58]). To show that strong solutions exist, i.e., the solutions to (3.9) belong to Vσ, σ>2-γ, provided that η0Vσ, θ0σ, and fVσ-γ/2Lperp(𝕋2), where 1-σ<2p<γ-1, we need only establish a priori estimates. Indeed, by adding an artificial viscosity -νΔ to (3.9) and mollifying f by f(ν), we have global smooth solutions η(ν) (independent of γ) such that the weak limit η(ν)η, as ν0+, is a weak solution of (3.9) (cf. [26, 24]). Since (4.1) holds, the family satisfies precisely the same estimates performed above in establishing Propositions 6, 7, and 9. Consequently, these bounds are inherited in the limit, thus ensuring that η is a strong solution.

Uniqueness of Strong Solutions.

Let η1 and η2 be strong solutions of (3.9) with initial data η0(1),η0(2)Vσ, σ>2-γ, respectively, and θ the strong solution of (1.3) with initial data θ0σ. Let ζ:=η1-η2. Then the evolution of ζ is given by

{ t ζ + κ Λ γ ζ + ( ζ ) ζ + ( ζ ) η 2 + v 2 ζ = - μ J h ζ , x 𝕋 2 , t > 0 , ζ ( x , 0 ) = η 0 ( 1 ) ( x ) - η 0 ( 2 ) ( x ) , v 1 = η 1 , v 2 = η 2 .

We multiply by ψ:=-Λ-1ζ and integrate to obtain

1 2 d d t ψ H 1 / 2 2 + κ ψ H ( γ + 1 ) / 2 2 + μ ψ H 1 / 2 2 = v 2 ζ ψ d x = I + μ ( ζ - J h ζ ) ψ 𝑑 x = 𝐼𝐼 .

Note that we have used the orthogonality property (f)gΛ-1fdx=0.

We estimate I as follows. First observe that by integrating by parts and using the relation ψ=-Λ-1ζ, we may write

I = v 2 ζ ψ d x = ( v 2 Λ ψ ) ψ d x .

Thus (as in [58, p. 32]), we may apply Hölder’s inequality, the Calderòn–Zygmund theorem, and the Sobolev embedding H1/pLq to obtain

| I | C p η 2 L p ζ L q ψ L q C p η 2 L p ψ H 1 + 1 / p 2 ,

where 1p+2q=1. Since p>2γ-1 from (H3), by interpolation we have

ψ H 1 + 1 / p C ψ H ( γ + 1 ) / 2 1 + 2 / p γ ψ H 1 / 2 γ - 1 - 2 / p γ .

It follows that

| I | κ 4 ψ H ( γ + 1 ) / 2 2 + κ 2 p γ γ - 1 - 2 / p Ξ p , γ / 2 ψ H 1 / 2 2 ,

where Ξp,γ/2 is defined by (3.23) and (3.29), except in terms of η2Lp and with C=C0 there. Note that we will suppress the dependence of the constant on p. Also note that Ξp,γ/2< is guaranteed by Proposition 7 since μ satisfies (4.1).

The estimate of 𝐼𝐼 is, in fact, independent of the type of Jh. Indeed, if Jh is Type II, then Jh also satisfies (2.8).

So observe that by (2.8), the Cauchy–Schwarz and Young inequalities, (4.1), and by interpolation (Proposition 3) we have

| 𝐼𝐼 | μ ζ - J h ζ H - γ / 2 ψ H γ / 2
C μ h γ / 2 ψ H 1 ψ H γ / 2
C μ h γ / 2 ψ H ( γ + 1 ) / 2 ψ H 1 / 2
C μ 2 h γ κ ψ H 1 / 2 2 + κ 4 ψ H ( γ + 1 ) / 2 2 .

Thus, upon combining I, 𝐼𝐼, and (3.4), we may deduce

(4.2) d d t ψ H 1 / 2 2 + κ ψ H ( γ + 1 ) / 2 2 + 3 2 μ ψ H 1 / 2 2 κ Ξ p , γ / 2 ψ H 1 / 2 2 .

By Gronwall’s inequality, we have

ψ ( t ) H 1 / 2 2 ψ 0 H 1 / 2 2 e ( κ Ξ p , γ / 2 - ( 3 / 2 ) μ ) t ,

which establishes continuous dependence on initial conditions for (3.9) in the topology H-1/2 (since ψ=-Λ-1ζ). In fact, by interpolation (see (4.6)), we may establish continuous dependence in Hσ-ϵ for all ϵ(0,σ+1/2). In particular, if η0(1)=η0(2), then ζ00 and (4.2) implies ζ0, which establishes uniqueness of solutions.

This completes the proof of Theorem 1.

4.2 Proof of Theorem 2

Let η be the unique global strong solution of (3.9) with initial data η0Vσ, σ>2-γ, and let θ be the unique global strong solution of (1.3) with initial data θ0σ. We assume (3.5), which is

(4.3) μ c 0 κ ( Θ L p κ ) γ / ( γ - 1 - 2 / p ) ,

where c0 is the constant from (4.1) (whose magnitude is determined below). Let ζ:=η-θ and ψ:=-Λ-1ζ. Then, proceeding as in the proof of uniqueness from the previous section, we arrive at

(4.4) d d t ψ H 1 / 2 2 + μ ( 3 2 - Ξ p , γ / 2 ( θ ) κ μ ) ψ H 1 / 2 2 0 ,

where Ξp,γ/2(θ) is defined by (3.23) and (3.29), except with ΘLp replacing MLp, i.e.,

(4.5) Ξ p , γ / 2 ( θ ) := C ( Θ L p κ ) γ γ - 1 - 2 / p

for some absolute constant C>0. Note that Ξp,γ/2(θ) is independent of h,μ. Therefore, by (4.3) with c0chosen sufficiently small, it follows that

ψ H 1 / 2 2 ψ 0 H 1 / 2 2 e - μ t .

To upgrade the convergence, we need only interpolate since Proposition 9 ensures that η satisfies uniform bounds in Hσ. Indeed, let

M H σ := sup t 0 η ( t ) H σ .

Observe that for any σ>2-γ and 0<ϵ<σ+12, we have

(4.6) ψ ( t ) H σ + 1 - ϵ C ψ ( t ) H σ + 1 σ + 1 / 2 - ϵ σ + 1 / 2 ψ ( t ) H 1 / 2 ϵ σ + 1 / 2 .

Thus

η ( t ) - θ ( t ) H σ - ϵ 2 O ϵ ( e - μ ϵ t ) ,

where μϵ=ϵμ/(σ+12), and

O ϵ ( e - μ ϵ t ) := C ( M H σ + Θ H σ ) 2 ( σ + 1 / 2 - ϵ ) σ + 1 / 2 ψ 0 H 1 / 2 2 ϵ σ + 1 / 2 e - μ ϵ t .

In particular, this holds for ϵ=σ, so that

η ( t ) - θ ( t ) L 2 2 O σ ( e - c σ μ t ) ,

where cσ=σ/(σ+12). This establishes (3.7) upon rescaling, concluding the proof of Theorem 2.

Remark 4.1

Note that even though MHσ depends on h and μ, it will only affect the exponential rate, cσμ, up to a fixed, multiplicative factor. Thus, the synchronization still occurs at an exponential rate.


Dedicated to the memory of Professor Abbas Bahri



Communicated by Paul Rabinowitz


Award Identifier / Grant number: DMS-1418911

Award Identifier / Grant number: DMS-1109640

Award Identifier / Grant number: DMS-1109645

Funding source: Leverhulme Trust

Award Identifier / Grant number: VP1-2015-036

Funding source: Office of Naval Research

Award Identifier / Grant number: N00014-15-1-2333

Funding statement: Michael S. Jolly was supported by NSF grant DMS-1418911 and the Leverhulme Trust grant VP1-2015-036. The work of Edriss S. Titi was supported in part by the ONR grant N00014-15-1-2333 and the NSF grants DMS-1109640 and DMS-1109645.

A Appendix

In this section, we verify that volume element (see (A.7) and (A.8) below) and modal projection interpolants (see (A.18) and (A.21)) are of Type I and II, respectively. For convenience, we let Ω=𝕋2 denote the 2π-periodic box throughout, where 𝕋2=(/(2π))2, so that 𝕋2=[-π,π]2. Let 𝒵 and Vβ be the spaces defined by (2.1) and (2.4), respectively. The main claim is the following:

Proposition A.1

Suppose that Jh is a linear operator that is defined by either (A.7), (A.8), (A.17), (A.21), below. Then:

  1. sup h > 0 J h ϕ L p ϕ L p , for any p ( 1 , ) , ϕLperp(𝕋2),

  2. J h ϕ L p h 2 / p - 2 / q ϕ L q , for any 1 q p < , ϕLperq(𝕋2),

  3. J h ϕ H ˙ β h - β ϕ L 2 , for any β 0 , ϕLper2(𝕋2).

If Jh is defined by (A.7), (A.8), then for any β(0,1] we have

  1. ϕ - J h ϕ L 2 h β ϕ H ˙ β , for any ϕ H ˙ per β ( 𝕋 2 ) ( V β if J h is ( A.8 )),

  2. ϕ - J h ϕ H ˙ - β h β ϕ L 2 , for any ϕ L per 2 ( 𝕋 2 ) ( L per 2 ( 𝕋 2 ) 𝒵 if J h is ( A.8 )).

If Jh is defined by (A.18) or (A.21), then for any ϕH˙perβ(T2) we have

  1. ϕ - J h ϕ H ˙ α h β - α ϕ H ˙ β , for any ϕ H ˙ per β ( 𝕋 2 ) , β > α ,

  2. Λ β J h ϕ = J h Λ β ϕ , for any ϕ H ˙ per β ( 𝕋 2 ) , β .

Moreover, when Jh is given by (A.7), (A.8), or (A.21), then property (0.1) and property (0.2) are valid for p=1, and p=, respectively.

We then define Type I operators as any linear operator, Jh, that satisfy properties (0.1)(0.3) and properties (1.1)(1.2), while Type II operators are those that satisfy properties (0.1)(0.3) and properties (1.1)(1.2).

A.1 Local Averages (Type I)

Let us recall the following construction of a partition of unity from [5]. Let N>0 be a perfect square integer and partition Ω into 4N squares of side-length h=π/N. Let 𝒥={0,±1,±2,,±(N-1),-N}2 and for each α𝒥, define the semi-open square

Q α = [ i h , ( i + 1 ) h ) × [ j h , ( j + 1 ) h ) , where α = ( i , j ) 𝒥 .

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

𝒬 := { Q α } α 𝒥 .

Consider the functions

χ α ( x ) := 𝟙 Q α ( x ) and ψ α ( x ) := k 2 χ α ( x + 2 π k ) .

In particular, ψα is the 2π-periodic extension of the characteristic function, 𝟙Qα, of Qα to 2.

Given ϵ>0 fixed, we mollify ψα as

ψ ~ α ( x ) = ( ρ ϵ * ψ α ) ( x ) , x 𝕋 2 ,

with the function