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Accessible Unlicensed Requires Authentication 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

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.


Communicated by Paul Rabinowitz


Funding source: National Science Foundation

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. (0.1)

    suph>0JhϕLpϕLp, for any p(1,), ϕLperp(𝕋2),

  2. (0.2)

    JhϕLph2/p-2/qϕLq, for any 1qp<, ϕLperq(𝕋2),

  3. (0.3)

    JhϕH˙βh-βϕL2, for any β0, ϕLper2(𝕋2).

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

  1. (1.1)

    ϕ-JhϕL2hβϕH˙β, for any ϕH˙perβ(𝕋2) (Vβ if Jh is (A.8)),

  2. (1.2)

    ϕ-JhϕH˙-βhβϕL2, for any ϕLper2(𝕋2) (Lper2(𝕋2)𝒵 if Jh is (A.8)).

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

  1. (2.1)

    ϕ-JhϕH˙αhβ-αϕH˙β, for any ϕH˙perβ(𝕋2),β>α,

  2. (2.2)

    ΛβJhϕ=JhΛβϕ, for any ϕH˙perβ(𝕋2),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

Qα=[ih,(i+1)h)×[jh,(j+1)h),where α=(i,j)𝒥.

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

𝒬:={Qα}α𝒥.

Consider the functions

χα(x):=𝟙Qα(x)andψα(x):=k2χα(x+2πk).

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)=(ρϵ*ψα)(x),x𝕋2,

with the function ρϵ(x):=ϵ-2ρ(xϵ), where ρ is given by

ρ(x):={K0exp(-11-|x|2),|x|<1,0,|x|1,

and K0>0 is the absolute constant given by

K0-1=|x|<1exp(-11-|x|2)𝑑x.

Now suppose that N9 and ϵ=h10. For each α=(i,j)𝒥, let us also define the augmented squares, Q^α and Qα(ϵ), by

Q^α:=[(i-1)h,(i+2)h]×[(j-1)h,(j+2)h]andQα(ϵ):=Qα+B(0,ϵ)

so that QαQα(ϵ)Q^α for each α𝒥, and the “core”, Cα(ϵ), by

Cα(ϵ):=Qα(ϵ)ααQα(ϵ),α𝒥.

Finally, for ϕL1(𝕋2), define

ϕ:=14π2𝕋2ϕ(x)𝑑x.

Then we have the following proposition, which follows from the definition of ψ~α. We note that properties (i)–(iii) below can be found in [5], while property (iv) can be proved by using a characterization of the Sobolev space norm (see Remark A.1 below and rescaling Proposition A.5 (below) and using the fact that {ψα} satisfies (i)–(iii) (see Remark A.1 below and the proof of Corollary A.6 for relevant details).

Proposition A.2

Let N9, h:=L/N, and ϵ:=h/10. The collection {ψα}αJ forms a smooth partition of unity satisfying

  1. (i)

    0ψ~α1 and sptψ~α(Qα(ϵ)+(2π)2),

  2. (ii)

    ψ~α=1 for all x(Cα(ϵ)+(2π)2) and α𝒥ψ~α(x)=1 for all x2,

  3. (iii)

    ψ~α=(h/(2π))2 and 4h/5ψ~αL2(Ω)6h/5,

  4. (iv)

    supα𝒥ψ~αH˙βh1-β for all β0.

Remark A.1

When σ0, let [σ] denote the greatest integer such that [σ]σ. Then define

(A.1)ϕH˙~σ2:=0<|𝐤|[σ]𝐤ϕL22+|𝐤|=[σ]𝐤ϕH˙σ-[σ]2,𝐤2{},𝐤=xk1yk2,

where for 0<β<1, we define

(A.2)ϕH˙~β2:=𝕋2[-π,π)2|ϕ(x+y)-ϕ(y)|2|x|2+2β𝑑x𝑑y.

Then H˙~σ is equivalent to (2.2) when σ0 (cf. [1, 9] and Proposition A.5). Indeed, there exists an absolute constant, C>0, such that for all ϕHperσ(𝕋2) with σ0, we have

C-1ϕH˙~σ2ϕH˙σ2CϕH˙~σ2.

Therefore, to see (iv), let ψ~α𝐤:=(𝐤ρ)h/10*ψα, for 𝐤2{}. Observe that

(A.3)𝐤ψ~α=(𝐤ρh/10)*ψ~α=10|𝐤|h|𝐤|ψ~α𝐤,

and by Young’s convolution inequality we have

(A.4)ψ~α𝐤L2C𝐤h,

where C𝐤 depends on 𝐤ρ, but not h. On the other hand, by (A.2) one can show that

(A.5)ψ~α𝐤Hβ-[β]Ch1-β+[β],

where C depends only on ρ,β, but not h. Thus, from (A.1), (A.3), (A.4), and (A.5) that for β>0, we have

ψ~αH˙β2C0<|𝐤|[β]𝐤ψ~αL22+C|𝐤|=[β]𝐤ψ~αHβ-[β]2
C0<|𝐤|[β]|h|-2|𝐤|ψ~α𝐤L22+C|𝐤|=[β]|h|-2[β]ψ~α𝐤Hβ-[β]2
Ch2-2β,

as desired.

For ϕLloc1(Ω), define

ϕQ=1a(Q)Qϕ(x)𝑑xandϕ~Qα=1a~(Qα)𝕋2ϕ(x)ψ~α(x)𝑑x,

where a(Q) denotes the area of Q and

a~(Qα):=𝕋2ψ~α(x)𝑑x.

At this point, let us emphasize that ψ~α are non-negative for each α𝒥. Observe that for each α𝒥, there exists an absolute constant c>0, independent of h,α,ϵ, such that

(A.6)c-1a~(Qα)a(Q),a(Q)a(Q^α),a(Q)a(Qα(ϵ))c,Q{Qα,Qα(ϵ),Q^α}.

Finally, we define the smooth volume element interpolant by

(A.7)h(ϕ):=α𝒥ϕ~Qαψ~α

and the “shifted” smooth volume element interpolant by

(A.8)Ih(ϕ):=α𝒥ϕ~Qαψ¯α,ψ¯α:=ψ~α-ψ~α.

Observe that we have the following relation between h and Ih.

Lemma A.3

Let Ih,Ih be given by (A.7), (A.8). Let ϕLloc1(T2). Then:

  1. (i)

    Ihϕ=hϕ-hϕ,

  2. (ii)

    Ihϕ=0,

  3. (iii)

    ΛβIhϕ=Λβhϕ for β>0.

Now let us prove Proposition A.1 when Jh is given by either h or Ih, as defined by (A.7), (A.8), respectively.

Proof of Proposition A.1: Part I.

We will establish properties (0.1)(0.3) for Jh given by either (A.7) or (A.8). It will suffice to consider Jh=h given by (A.7). Indeed, by Lemma A.3, the fact that

(A.9)ϕLpϕLp,ϕLp(𝕋2),1p,

and the triangle inequality, we have that the properties (0.1)(0.3) applied to Jh=h easily imply the corresponding properties for Jh=Ih given by (A.8).

Suppose then that Jh=h. Observe that for each x𝕋2 and α𝒥, we have that

nα(x):=#{Qα(ϵ):xQα(ϵ) for some α𝒥 and ψ~α(x)0}

is independent of h. In particular,

supα𝒥supxΩnα(x)=n0

for some fixed positive integer n0, independent of N,h. It follows that

(A.10)(α𝒥ϕ~Qαψ~α)pCpα𝒥ϕ~Qαpψ~αp

for some absolute constant C>0 depending only on n0.

We prove property (0.1). For 1p, it follows from (A.6), (A.10), and Hölder’s inequality that

hϕLppCpα𝒥|ϕ~Qα|pψ~αLppCpα𝒥(a(Qα(ϵ))a~(Qα))pϕLp(Qα(ϵ))pCpϕLpp

for some absolute constant C>0 that depends on n0.

Next, we prove property (0.2). Suppose pq. It follows from (A.6), (A.10), and the Cauchy–Schwarz inequality that

hϕLppCpα𝒥|ϕ~Qα|pψ~αLpp
Cpα𝒥1a~(Qα)p/qa(Qα(ϵ))ϕLq(Qα(ϵ))p
Cph2-2p/qϕLqp-qα𝒥ϕLq(Qα(ϵ))q
Cph2-2p/qϕLqp

for some absolute constant C>0, depending on n0.

To prove property (0.3), it follows from Proposition A.2 (iv), (A.6), and the Cauchy–Schwarz inequality that

hϕH˙β2Cα𝒥ϕ~Qα2ψ~αH˙β2Ch-2βα𝒥a(Qα)ϕ~Qα2Ch-2βϕL22,

as desired. Note that the absolute constant above depends on ψ~α, but is independent of h. ∎

To prove part II of Proposition A.1, we will require the following two results, the first of which is a fractional Poincaré-type inequality. The second provides an alternate characterization of Sobolev norms.

Lemma A.4

Lemma A.4 ([45])

Let QR2 be a closed square. Let 1qp< and δ,ρ(0,1). Then for ϕLp(Q), we have

1a(Q)Q|ϕ(x)-ϕQ|q𝑑xa(Q)q(δ/2-1/p)(QQB(x,ρ|Q|1/2)|ϕ(x)-ϕ(y)|p|x-y|2+δp𝑑y𝑑x)q/p,

where the suppressed absolute constant is independent of ϕ.

Proposition A.5

Proposition A.5 ([9])

Let 0<β<1. Then for ϕH˙perβ(T2), we have

ϕH˙β(𝕋2)2𝕋2[-π,π)2|ϕ(x+y)-ϕ(y)|2|x|2+2β𝑑x𝑑y.

We then have the following corollary.

Corollary A.6

Let 0<δ<1 and QR2 a closed square. Then for ϕHperδ(Q), we have

Q|ϕ(x)-ϕQ|2𝑑x|Q|2δϕH˙δ(Q)2.

Proof.

Let Q2 be a closed square and let x0 denote its center. Let Q0 denote the same square, but centered at the origin. Let ϕHperδ(Q) and ρ14 and let ϕx0(x)=ϕ(x+x0). Then observe by translating and rescaling, we have

QQB(x,ρ|Q|1/2)|ϕ(x)-ϕ(y)|p|x-y|2+δp𝑑y𝑑xQ0B(0,ρ|Q|1/2)|ϕx0(x)-ϕx0(x+y)|p|y|2+δp𝑑y𝑑x
=Q0B(0,ρ|Q|1/2)|ϕx0(x+y)-ϕx0(y)|p|x|2+δp𝑑x𝑑y
=|Q|1-δp/2(2π)2-δp𝕋2B(0,ρ)|ϕ~x0(x+y)-ϕ~x0(y)|p|x|2+δp𝑑x𝑑y
|Q|1-δp/2(2π)2-δp𝕋2[-π,π)2|ϕ~x0(x+y)-ϕ~x0(y)|p|x|2+δp𝑑x𝑑y,

where ϕ~(x)=ϕ((|Q|1/2/(2π))x). Thus, by setting p=q=2, then applying Lemma A.4 and Proposition A.5, we obtain

Q|ϕ(x)-ϕQ|2𝑑x(2π)2δ-2ϕ~x0H˙δ(𝕋2)2|Q|2δϕH˙δ(Q)2,

as desired. ∎

The next result adapts Corollary A.6 to modified local spatial averages. In particular, given a square Q2 and ϵ>0, define Q(ϵ)=Q+B(0,ϵ). Suppose that ψ~C(2) is an arbitrary non-negative function satisfying 0ψ~1, sptψQ(ϵ), and ψ~|Q>0. Then, given ϕLloc1(2), we define

a~(Q):=ψ~(x)𝑑xandϕ~Q:=1a~(Q)ϕ(x)ψ~(x)𝑑x.

Corollary A.7

Let 0<δ<1, and QR2 a closed square. Then for ϕHperδ(Q), we have

(A.11)Q|ϕ(x)-ϕ~Qα|2𝑑x(a(Qα)+ϵ2)2δϕHδ(Q)2.

Proof.

First observe that

ϕQ-ϕ~Q=1a~(Q)(ϕQ-ϕ(x))ψ~(x)𝑑x.

Then by Hölder’s inequality, we have

|ϕQ-ϕ~Q|21a~(Q)|ϕ(x)-ϕQ|2ψ(x)𝑑x1a~(Q)Q(ϵ)|ϕ(x)-ϕQ|2𝑑x.

It follows then from Minkowski’s inequality and convexity that

Q|ϕ(x)-ϕ~Q|2C2(Q|ϕ(x)-ϕQ|2𝑑x+a(Q)a~(Q)Q(ϵ)|ϕ(x)-ϕ~Q|2𝑑x).

Therefore, by (A.6) and Corollary A.6, we obtain (A.11). ∎

Finally, we are ready to complete the proof of Proposition A.1 when Jh is given by (A.7) or (A.8).

Proof of Proposition A.1: Part II.

First suppose that Jh=h is given by (A.7). The case β=1 follows from the classical Poincarè inequality, so let β(0,1) and ϕH˙perβ(𝕋2). Thus, by Proposition A.2, (A.10), and Corollary A.7, it follows that

(A.12)ϕ-hϕL22α𝒥ϕ-ϕ~QαL2(Qα(ϵ))2h2βα𝒥ϕH˙β(Qα(ϵ))2h2βϕH˙β2,

which proves property (1.1). To establish property (1.2), first observe that for g,hLloc1(𝕋2) we have

(h-h~Qα)ψ~α,g~Qα=g~Qαh(x)ψα(x)𝑑x-h~Qαa~(Qα)g~Qα=0,

and by symmetry

h~Qα,(g-g~Qα)ψ~α=0.

It then follows from this and Proposition A.2 (ii) that for gLloc2(𝕋2), we have

ϕ-hϕ,g=α𝒥ϕ-ϕ~Qα,(g-g~Qα)ψ~α
=α𝒥ϕ,(g-g~Qα)ψ~α
(A.13)=ϕ,g-hg.

Thus, given gH˙perβ(𝕋2), it follows from (A.13), Proposition A.2, and the Cauchy–Schwarz inequality that

|ϕ-hϕ,g|=α𝒥ϕL2(Qα(ϵ))(g-g~Qα)ψ~αL2(Qα(ϵ))ϕL2(α𝒥(g-g~Qα)ψ~αL2(Qα(ϵ))2)1/2.

Then Corollary A.7 implies

g-g~QαL2(Qα)2h2βgHβ(Qα)2.

Therefore, by duality we have

(A.14)ϕ-hϕ,gH˙-β=supgH˙β=1|ϕ-hϕ,g|hβϕL2,

which is precisely property (1.2).

Now let Jh=Ih be given by (A.8). To show that property (1.1) holds, first observe that h1=1 and Ih1=0. Given ϕLloc1(𝕋2) such that ϕ=0, it follows from the fact that h,Ih are linear and Lemma A.3 (i) that

ϕ-Ihϕ=ϕ-ϕ-Ih(ϕ-ϕ)
=(ϕ-hϕ)+hϕ-hϕ-ϕ-hϕ
(A.15)=(ϕ-hϕ)-ϕ-hϕ.

Thus, property (1.1) for Jh=Ih and ϕVβ follows from Minkowski’s inequality, (A.9), and (A.12). To see property (1.2) for Jh=Ih, simply observe that if g=0, then (A.15) implies that

(A.16)ϕ-Ihϕ,g=ϕ-hϕ,g-ϕ-hϕ,g=ϕ-hϕ,g.

Recall that Lemma A.3 (ii) shows that Ihϕ𝒵 for any ϕLloc1(𝕋2). In particular, ϕ-Ihϕ𝒵. Thus, given ϕLper2(𝕋2)𝒵, it follows from duality and (A.16) that

ϕ-IhϕH˙-β=supgVβgH˙β=1|ϕ-Ihϕ,g|=supgVβgH˙β=1|ϕ-hϕ,g|.

Arguing as we did for (A.14), we have that Jh=Ih satisfies property (1.2), as desired. ∎

A.2 Modal Projection (Type II)

Here we let Jh be given by projection onto Fourier modes. The projection can be given by either rough or smooth cut-offs in the frequency side. The “rough projection” will be given by convolution with the square Dirichlet kernel, while the “smooth projection” will be given by Littlewood–Paley projection. As in the previous subsection, we work with rescaled variables, so that the 2π-periodic box is given by Ω=𝕋2=[-π,π]2.

Rough Modal Projection.

Let N>0. For k2, k=(k1,k2), denote by ϕ^(k) the k-th Fourier wave-number and define the “rough modal projection” by PN by

(A.17)(PNϕ)(x1,x2):=(DN*ϕ)(x1,x2),

where

DN(x1,x2):=|k1|N|k2|Neikx

is the two-dimensional Dirichlet kernel. In particular, we have

DN(x1,x2)=DN(x1)DN(x2),DN(x)={sin((N+1/2)x)sin(x/2),x𝕋2{},2N+1,x=.

Let us now prove Proposition A.1 with

(A.18)Jh:=PN,h=2πN,N16.

Proof of Proposition A.1.

It is classical that Jh defined by (A.18) this way satisfies property (0.1) with constant independent of h (cf. [42]). One also has the following estimate on the Dirichlet kernel for q(1,) (cf. [42]):

(A.19)DNLq(𝕋2)(2N+1)2/q,

where q and q are Hölder conjugates and the suppressed constant depends on q.

To show that PN satisfies property (0.2), we apply Young’s convolution inequality and (A.19) to obtain

PNϕLppDNL2p/(p+2)pϕL2p(2N+1)p-2ϕL2ph2-pϕL2p.

To prove property (0.3), we apply Plancherel and estimate as follows:

PNϕH˙α2CN2α|kj|N|ϕ^(k)|2=CN2αϕL2Ch-2αϕL2.

Clearly, for β0, ΛβIh=IhΛβ by the Plancherel theorem, which proves property (2.2).

To prove property (2.1), let β>α. Then from (v), the Cauchy–Schwarz inequality, and the Plancherel theorem, it follows that

ϕ-IhϕH˙α2|k|>N|k|2α|ϕ^(k)|2|k|>N|k|2(α-β)|k|2β|ϕ^(k)|2Nα-βΛβϕL22hβ-αϕH˙β2,

as desired. ∎

Smooth Modal Projection.

We define this projection by the Littlewood–Paley decomposition. We presently give a brief review of this decomposition. More thorough treatments can be found in [7, 29, 59, 61]. We state the decomposition for 2 for convenience, but point out that it is also valid in the case 𝕋2 for periodic distributions. In particular, the Bernstein inequality (Proposition A.8) stated below also hold in 𝕋2 provided that one redefine the Littlewood–Paley blocks, k, in a suitable manner (cf. [29]).

Let ψ0 be a smooth, radial bump function such that ψ0(ξ)=1 when [|ξ|14]2, and

0ψ01andsptψ0=[|ξ|12].

Define ϕ0(ξ):=ψ0(ξ/2)-ψ0(ξ). Observe that

0ϕ01andsptϕ0=[14|ξ|1].

Now for each integer j0, define

ϕj(ξ):=ϕ0(ξ2-j).

Then, in view of the above definitions, we clearly have

sptϕj=[2j-2|ξ|2j].

If we let ϕ-1:=ψ0 and ϕj0 for j<-1, observe that

(A.20)jϕj(ξ)=1for ξ2.

One can then define

kg:=ϕk*g,~kg:=|k-|2g,Skg:=kg,Tk:=I-Sk,

where ϕk:=ϕˇk is the inverse Fourier transform of ϕk. We call the operators k Littlewood–Paley projections.

One can show that (A.20) implies that

g=j-1jgfor all g𝒮(2),

where 𝒮(2) is the space of tempered distributions over 2.

For N>0, define Jh by

(A.21)Jh:=SN,h=2-N.

That properties (0.1)(0.3) and (2.1)(2.2) are satisfied by Ih defined in this way follows from the Bernstein inequalities (cf. [7]).

Proposition A.8

Proposition A.8 (Bernstein inequalities)

Let 1pq and gS(R2). Then for βR and j-1 we have

ΛβjgLq2jβjgLp,jgLq22j(1/p-1/q)jgLp.

For σ0 and j-1, we have

ΛβSjgLq2βj+2j(1/p-1/q)SjgLp,

where the suppressed absolute constants depend only on β,ϕˇ0,ψˇ0.

Indeed, let us prove Proposition A.1 for Jh=SN given by (A.21).

Proof of Proposition A.1.

Property (0.1) and property (0.2) follow immediately from the Bernstein inequalities. For property (0.3), simply observe that for β0, we have

SNϕHβ222βNϕL22.

To prove property (2.1), observe that for α,β, with βα, we have

ϕ-SNϕHα2jN+12-2j(β-α)22βjjϕL222-2N(β-α)ϕH˙β2.

Also, property (2.2) holds simply by applying the Fourier convolution theorem. Therefore, Jh given by (A.21) is of Type II. ∎

Acknowledgements

The authors would like to thank the Institute of Pure and Applied Mathematics at UCLA, where part of this work was performed. The authors would also like to thank Cecilia Mondaini for insightful discussions in the course of this work.

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Received: 2016-12-10
Accepted: 2016-1-4
Published Online: 2017-1-13
Published in Print: 2017-2-1

© 2017 by De Gruyter