2. Notation and functional setting

Let $\Omega =[0,2\pi ]^2$ and $L^p_\sigma =L^p_\sigma (\Omega )$, for $p\in [1,\infty ]$, denote the space of $p$-integrable (in the sense of Lebesgue), mean-free, solenoidal vector fields over $\Omega$, which are periodic in each direction; its norm is given by

(2.1)\begin{equation} \lVert u\rVert_{L^p}^p:=\int_{\Omega} |u(x)|^p\,{\rm d}x,\quad \text{for}\ p\in[1,\infty),\quad \lVert u\rVert_{L^\infty}:=\operatorname{ess\ sup}_{x\in\Omega}|u(x)|, \end{equation}

where $\operatorname {ess\ sup}$ denotes the essential supremum. We let $H^k_\sigma =H^k_\sigma (\Omega )$ denote the space of periodic, mean-free, solenoidal vector fields over $\Omega$ whose weak derivatives (in the sense of Sobolev) up to order $k$ belong to $L^2_\sigma$; its norm is given by

(2.2)\begin{equation} \lVert u\rVert_{H^k}^2:=\sum_{|\alpha|\leq k}\int_{\Omega}|\partial^\alpha u(x)|^2\,{\rm d}x, \end{equation}

where $\alpha \in (\mathbb {N}\cup \{0\})^2$ is a multi-index. We will abuse notation and use $L^2_\sigma, H^k_\sigma$ to denote the corresponding spaces for scalar functions as well; in this case, $|u|$ is interpreted as absolute value, rather than Euclidean norm.

We will make use of the functional form of the NSE, which is given by

(2.3)\begin{equation} \frac{{\rm d}}{{\rm d}t}u+\nu Au+B(u,u)=Pf, \end{equation}

where

\[ Au={-}P\Delta u\quad\text{and}\quad B(u,v)=P(u\cdotp\nabla) v, \]

and $P$ denotes the Leray–Helmholtz projection, that is, the orthogonal projection onto divergence-free vector fields; we refer to $A$ as the Stokes operator. Note that by orthogonality, $\lVert P \rVert _{L^2\rightarrow L^2}\leq 1$. Moreover,

(2.4)\begin{equation} \widehat{Pu}(n):=\hat{u}(n)-\frac{n\cdotp\hat{u}(n)}{|n|^2}n,\quad n\in\mathbb{Z}^2\setminus\{(0,0)\}, \end{equation}

where $\hat {u}(n)$ denotes the Fourier coefficient of $u$ corresponding to wavenumber $n\in \mathbb {Z}^2$. Also, powers of $A$ can be defined spectrally via:

(2.5)\begin{equation} \widehat{A^{k/2}u}(n):=|n|^k\hat{u}(n). \end{equation}

Hence, $P$ commutes with $\Delta$ in the setting of periodic boundary conditions. Recall that for $u\in H^1_\sigma$ the Poincaré inequality states:

(2.6)\begin{equation} \lVert u\rVert_{L^2}\leq \lVert A^{1/2} u\rVert_{L^2}=\lVert A^{1/2}u\rVert_{L^2}. \end{equation}

It follows that for each integer $k\geq 1$, there exists a constant $c_k>0$ such that

(2.7)\begin{equation} c_k^{{-}2}\lVert u\rVert_{H^k}^2\leq \sum_{|\alpha|=k}\int_\Omega|\partial^\alpha u(x)|^2\,{\rm d}x\leq c_k^2\lVert u\rVert_{H^k}^2, \end{equation}

whenever $u\in H^k_\sigma$. In particular, by the Parseval identity, it follows that there also exist constants $c_k>0$ such that

(2.8)\begin{equation} c_k^{{-}1}\lVert u\rVert_{H^k}\leq \lVert A^{k/2}u\rVert_{L^2}\leq c_k\lVert u\rVert_{H^k}, \end{equation}

for all $u\in H^k_\sigma$.

Given $t_0\geq 0$ and $f\in L^\infty (t_0,\infty ;L^2_\sigma )$, we define the Grashof-type number by

(2.9)\begin{equation} \tilde{G}:=\frac{\sup_{t\geq t_0}\lVert f(t)\rVert_{L^2}}{\kappa_0^2\nu^2}, \end{equation}

where $\kappa _0=(2\pi )/L$, where $L$ is the linear length of the domain; since $\Omega =[0,2\pi ]^2$, we see that $\kappa _0=1$. We note that the Grashof number is traditionally denoted by $G$, that is, undecorated, when the force is independent of time. Since we allow for time-dependence in the force, we will distinguish between the two notations by making use of tilde. See, for instance [Reference Balci, Foias and Jolly5], where this distinction is also maintained.

Observe that $Pf=f$, for any $f\in L^2_\sigma$. One has the following classical results regarding the existence theory for (2.3) [Reference Constantin and Foias26, Reference Foias, Manley, Rosa and Temam41, Reference Temam65, Reference Temam66].

Theorem 2.1 Let $t_0\geq 0$ and $f\in L^\infty (t_0,\infty ;L^2_\sigma )$. For all $u_0\in H^1_\sigma$, there exists a unique $u\in C([t_0,T];H_\sigma ^1)\cap L^2(t_0,T;H^{2})$ satisfying (2.3), for all $T>0$, such that $({{\rm d}}/{{\rm d}t})u\in L^2(t_0,T;L^2_\sigma )$ and

(2.10)\begin{equation} \lVert A^{1/2}u(t)\rVert_{L^2}^2\leq \lVert A^{1/2} u(t_0)\rVert_{L^2}^2\,{\rm e}^{-\nu (t-t_0)}+\nu^2\tilde{G}^2(1-{\rm e}^{-\nu(t-t_0)}), \end{equation}

for all $t\geq t_0\geq 0$.

From (2.10), we see that for $t_0=0$ and all $t>0$ sufficiently large, depending on $\lVert A ^{1/2}u_0\rVert _{L^2}$, one has:

(2.11)\begin{equation} \lVert A^{1/2}u(t)\rVert_{L^2}\leq \sqrt{2}\nu \tilde{G}=:{\tilde{c}}_1{\tilde{R}}_1,\quad {\tilde{c}}_1=\sqrt{2}. \end{equation}

Let ${\tilde {B}}_1$ denote the ball of radius $\tilde {R}_1$, centred at the origin in $L^2_\sigma$. Observe that $A^{1/2}u_0\in \tilde {B}_1$ implies via (2.10) that $u(t;u_0,f)\in \tilde {B}_1$, for all $t\geq 0$, where $u(t;u_0,f)$ denotes the unique solution of (2.3) with initial data $u_0$ and external forcing $f$. Moreover, there exists a constant ${\tilde {c}}_2>0$ such that if $f\in L^\infty (0,\infty ;H^1_\sigma )$ and $Au_0\in \alpha {\tilde {B}}_2$, where $\alpha >0$ is arbitrary, ${\tilde {B}}_2$ denotes the ball of radius $\tilde {R}_2={\tilde {c}}_2\nu (\tilde {G}+\tilde {\sigma }_1)\tilde {G}$, centred at the origin in $L^2_\sigma$, and $\alpha {\tilde {B}}_2$ the same ball of radius $\alpha {\tilde {R}}_2$, then

(2.12)\begin{equation} \lVert Au(t)\rVert_{L^2}^2\leq (1+\alpha^2)\tilde{c}_2^2\nu^2(\tilde{\sigma}_1+\tilde{G})^2\tilde{G}^2, \end{equation}

for all $t\geq 0$, where $\tilde {\sigma }_1$ denotes a ‘shape factor’ defined by

(2.13)\begin{equation} \tilde{\sigma}_1:=\frac{\sup_{t\geq t_0}\lVert A^{1/2} f(t)\rVert_{L^2}}{\sup_{t\geq t_0}\lVert f(t)\rVert_{L^2}}. \end{equation}

In other words, $Au_0\in \alpha {\tilde {B}}_2$ implies $Au(t)\in (1+\alpha ^2)^{1/2}{\tilde {B}}_2$, for all $t\geq 0$. Observe that $\tilde {\sigma }_1\geq 1$ by Poincaré's inequality. In particular, if $Au_0\in {\tilde {B}}_2$, then

(2.14)\begin{equation} \lVert Au(t)\rVert_{L^2}\leq \sqrt{2}{\tilde{c}}_2\nu(\tilde{G}+\tilde{\sigma}_1)\tilde{G}=\sqrt{2}{\tilde{R}}_2. \end{equation}

Bounds sharper than (2.14) were established in [Reference Dascaliuc, Foias and Jolly27] in the setting where $f$ was time-independent. In this particular case, one has that:

(2.15)\begin{equation} \lVert Au(t)\rVert_{L^2}\leq c_2\nu(\sigma_{1}^{1/2}+G)G=:R_2, \end{equation}

holds for all $t\geq 0$, for some $c_2>0$, provided that $u_0\in B_2$, the ball of radius $R_2$ in $H^2_\sigma$. Here, $G$ denotes the Grashof number, which is simply given by (2.9) when $f$ is time-independent. Similarly, $\sigma _1$ is given by (2.13) when $f$ is time-independent. For the sake of completeness, we supply a short proof of (2.15) in Appendix A.

3. Description of the algorithm

We consider the following feedback control system:

(3.1)\begin{equation} \frac{{\rm d}}{{\rm d}t}v+\nu Av+B(v,v)=h-\mu P_N(v-u), \end{equation}

where $h$, possibly time-dependent, is given. The well-posedness theory and synchronization properties of this model were originally developed in [Reference Azouani, Olson and Titi4] for a more general class of observables, which includes projection onto finitely many Fourier modes as a special case, in [Reference Bessaih, Olson and Titi7, Reference Blömker, Law, Stuart and Zygalakis12] in the setting of noisy observations, while the issue of synchronization in higher-order topologies was studied in [Reference Biswas, Brown and Martinez8, Reference Biswas and Martinez10]. In the idealized setting considered in this article, the existence, uniqueness results in [Reference Azouani, Olson and Titi4] will suffice for our purposes. This is stated in the following theorem.

Theorem 3.1 Let $t_0\geq 0$ and $h\in L^\infty (t_0,\infty ;L^2_\sigma )$. Let $u$ denote the unique solution of (2.3) corresponding to initial data $u_0\in H^1_\sigma$ guaranteed by theorem 2.1. There exists a constant ${\tilde {c}}>0$ such if $\mu,N$ satisfy

(3.2)\begin{equation} \mu \leq {\tilde{c}}\nu N^2 \end{equation}

then given $v_0\in H^1_\sigma$, there exists a unique solution, $v$, to the initial value problem corresponding to (3.1) such that

(3.3)\begin{equation} v\in C([t_0,T];H^1_\sigma)\cap L^2(t_0,T; H^2_\sigma)\quad\text{and}\quad \frac{{\rm d}}{{\rm d}t}v\in L^2(t_0,T;L^2_\sigma), \end{equation}

for any $T>0$.

The feedback control system (3.1) was originally conceived as a way to assimilate observations on the flow field into the equations of motion in order to reconstruct the unobserved scales of motion. There exists a considerable body of research analysing the extent to which this is possible in various situations in hydrodynamics such as Rayleigh–Bénard convection [Reference Altaf, Titi, Gebrael, Knio, Zhao and McCabe3, Reference Cao, Jolly, Titi and Whitehead14, Reference Farhat, Glatt-Holtz, Martinez, McQuarrie and Whitehead32–Reference Farhat, Lunasin and Titi35, Reference Farhat, Lunasin and Titi37], turbulence [Reference Albanez, Nussenzveig Lopes and Titi2, Reference Chen, Li and Lunasin19, Reference Clark Di Leoni, Mazzino and Biferale23, Reference Farhat, Lunasin and Titi38, Reference Larios and Pei58, Reference Yu, Giorgini, Jolly and Pakzad67, Reference Zauner, Mons, Marquet and Leclaire68], geophysical fluids [Reference Albanez and Benvenutti1, Reference Desamsetti, Dasari, Langodan, Titi, Knio and Hoteit30, Reference Farhat, Lunasin and Titi36, Reference Jolly, Martinez, Olson and Titi52, Reference Jolly, Martinez and Titi53, Reference Pei64], dispersive equations [Reference Jolly, Sadigov and Titi54, Reference Jolly, Sadigov and Titi55], as well as various numerical analytical and computational studies [Reference Blocher, Martinez and Olson13, Reference Celik, Olson and Titi18, Reference Diegel and Rebholz31, Reference Foias, Mondaini and Titi42, Reference García-Archilla and Novo48–Reference Ibdah, Mondaini and Titi51, Reference Larios, Rebholz and Zerfas59].

Given $h$, we may thus obtain from (3.1) an approximate reconstruction of the high modes of $u$ via $Q_Nv=(I-P_N)v$. We therefore propose the following algorithm: let $h=f_0$ denote the initial guess for the forcing field, defined for all $t\geq t_0$, for some fixed initial time $t_0\geq 0$, and let $v^0$ to be an arbitrary initial state; $f_0$ is considered to be the approximation at stage $n=0$ that is prescribed by the user and may, in fact, be chosen arbitrarily. Suppose that $P_Nf_0=f_0$. Then, at stage $n=1$, we consider:

(3.4)\begin{align} \frac{{\rm d}}{{\rm d}t}v_1+\nu Av_1+B(v_1,v_1)=f_0-\mu P_N(v_1-u),\quad v_1(t_0)=v_1^0,\quad t\in I_0:=[t_0,\infty). \end{align}

We define the first approximation to the flow field by

(3.5)\begin{equation} u_1=P_Nu+Q_Nv_1, \quad \text{for}\ t\in I_0. \end{equation}

By design, $u_1$ will relax towards $u$ after a transient period, $\rho _1:=t_1-t_0$, that is proportional to the relaxation timescale, $\mu ^{-1}$, where $t_1\gg t_0$, but only up to an error of size $O(g_0)$, where $g_0:=f_0-f$, which represents the ‘model error’. Only after this period has transpired we will extract the first approximation to $f$ via the formula:

(3.6)\begin{align} f_1(t)& :=\frac{{\rm d}}{{\rm d}t}P_Nu_1+\nu AP_Nu_1+P_NB(u_1,u_1),\nonumber\\ & \quad \text{for all}\ t\in I_1:=[t_0+\rho_1,\infty),\ \text{for some}\ \rho_1\gg 0. \end{align}

Observe that $P_Nf_1=f_1$. To obtain new approximations in subsequent stages, we proceed recursively: suppose that at stage $n-1$, a force, $f_{n-1}=P_Nf_{n-1}$ over $t\in I_{n-1}:=[t_{n-2}+\rho _{n-1},\infty )$, has been produced, where the relaxation period satisfies $\rho _{n-1}\gg 0$, and an arbitrary initial state, $v_n^0$, has been given. Then, consider:

(3.7)\begin{align} \frac{{\rm d}}{{\rm d}t}v_n+\nu Av_n+B(v_n,v_n)=f_{n-1}-\mu P_N(v_n-u),\quad v_n(t_{n-1})=v_n^0,\quad t\in I_{n-1}. \end{align}

Let $u_n=P_Nu+Q_Nv_n$, for $t\in I_{n-1}$, and define the approximation to the force at stage $n$ by

(3.8)\begin{align} f_n(t)& :=\frac{{\rm d}}{{\rm d}t}P_Nu_n+\nu AP_Nu_n+P_NB(u_n,u_n),\nonumber\\ & \quad \text{for all}\ t\in I_n:=[t_{n-1}+\rho_n,\infty),\ \text{for some}\ \rho_n\gg 0. \end{align}

This procedure produces a sequence of forces $f_1|_{I_1}, f_2|_{I_2}, f_3|_{I_3},\dots$ that approximates $f$ on time intervals $I_n$ whose left-hand endpoints are increasing with $n$. In particular, the sequence $\{f_n|_{I_n}\}_n$ asymptotically approximates $f$.

The key step to ensuring convergence of the generated sequence $\{f_n\}_{n\geq 1}$ to the true forcing, $f$, is to control model errors, $g_n:=f_n-f$, at each stage, in terms of the synchronization errors, $w_n$, which, in turn, are controlled by the model error from the previous stage; this will be guaranteed to be the case after transient periods of length $\rho _n:=t_n-t_{n-1}$, which allows relaxation in (3.4) to take place. However, it will be shown that this convergence can only be guaranteed to occur on the ‘observational subspace’ $P_NL^2_\sigma$, for $N$ sufficiently large. Indeed, a crucial observation at this point is that if $f=P_Nf$, then $f_n=P_Nf_n$, for all $N\geq 1$. We refer the reader to remarks 4.6 and 5.2 for additional remarks on the basic expectations for recovering force from low-mode data and the underlying limitations of this algorithm. We refer the reader to remark 7.1 for a more detailed discussion on the size of the transient periods, $\rho _n$.

Remark 3.2 We remark that the first guess, $f_0$, need not be arbitrary and can be chosen to be:

(3.9)\begin{equation} f_0:=\frac{{\rm d}}{{\rm d}t}P_Nu+\nu AP_Nu+P_NB (P_Nu,P_Nu), \end{equation}

as suggested in (1.2), where $t_{0}>0$ is chosen sufficiently large so that $u(t_0)$ is contained in an absorbing ball for (2.3). Similarly, the initial states, $v_n^0$, at each stage need not be arbitrary. Since $\{P_Nu(t)\}_{t\geq t_0}$ is assumed to be given, one can initialize the system governing $v_n$ at time $t=t_{n-1}$ with $v_n^0=P_Nu(t_{n-1})$ at each stage. These natural choices would presumably aid in the implementation of the proposed algorithm; we refer the reader to remark 7.1 for additional remarks related to this point.

Before outlining the proofs, we rigorously state the main results of the article.

4. Statements of main results

Let $A^{1/2}u_0\in \tilde {B}_1$ and $Au_0\in \tilde {B}_2$ and $f\in C([t_0,\infty );L^2_\sigma )\cap L^\infty (t_0,\infty ;L^2_\sigma )$, for some $t_0\geq 0$, where $\tilde {B}_1 ,\tilde {B}_2$ were defined as in § 2. Let $u$ denote the unique strong solution of (2.3) corresponding to forcing $f$ and initial data $u_0$ guaranteed by theorem 2.1 corresponding to initial velocity $u_0$ and external forcing $f$. Given $t_0\geq 0$, let $\gamma _0$ denote the initial relative error defined by

(4.1)\begin{equation} \gamma_0:=\frac{\sup_{t\geq t_0}\lVert f_0(t)-f(t)\rVert_{L^2}}{\sup_{t\geq t_0}\lVert f(t)\rVert_{L^2}}, \end{equation}

where $f_0$ is a user-prescribed initial guess for the force which satisfies $f_0\in C([t_0,\infty );L^2_\sigma )\cap L^\infty (t_0,\infty ;L^2_\sigma )$. Recall that the assumptions on the initial data imply [via (2.10), (2.15)] that $A^{1/2}u(t)\in \tilde {B}_1$ and $Au(t)\in 2\tilde {B}_2$, for all $t\geq 0$.

Theorem 4.1 Suppose that $f=P_Nf$, for some $N\geq 1$, and that $\{P_Nu(t)\}_{t\geq t_0}$ is given. There exists a positive constant $c$, depending on $\gamma _0$, such that for any $\beta \in (0,1)$, if $N\geq 1$ satisfies:

(4.2)\begin{equation} \beta^{2}N^2>c(\tilde{\sigma}_1+\tilde{G})\tilde{G}^2, \end{equation}

then there exists a choice for the tuning parameter $\mu$ and increasing sequence of times $t_{n-1}\leq t_n$, for $n\geq 2$, such that

(4.3)\begin{equation} \sup_{t\geq t_n}\lVert f_n(t)-f(t)\rVert_{L^2}\leq \beta\left(\sup_{t\geq t_{n-1}}\lVert f_{n-1}(t)-f(t)\rVert_{L^2}\right), \end{equation}

for all $n\geq 1$, where $f_n=P_Nf_n$, $f_n\in C([t_0,\infty );L^2_\sigma )\cap L^\infty (t_0,\infty ;L^2_\sigma )$, and each $f_n$ is given by (3.8).

Note that when $f$ is time-dependent, theorem 4.1 only asserts recovery of the external force asymptotically in time. However, when the force is time-periodic or time-independent, then theorem 4.1 immediately implies that the external force is eventually recovered; we provide a statement of the time-independent case in the following corollary since the corresponding approximating forces are obtained by evaluating the sequence of approximating forces asserted in theorem 4.1 at certain times.

Corollary 4.4 Suppose that $f=P_Nf$, for some $N\geq 1$, is time-independent, and that $\{P_Nu(t)\}_{t\geq t_0}$ is given. There exists a positive constant $c_0$, depending on $\gamma _0$, such that for any $\beta \in (0,1)$, if $N\geq 1$ satisfies:

(4.4)\begin{equation} \beta^2N^2>c_0(\tilde{\sigma}_1+G)G^2, \end{equation}

then there exists a choice for the tuning parameter $\mu$ and an increasing sequence of times $t_{n-1}\leq t_n$ such that

(4.5)\begin{equation} \lVert f_n-f\rVert_{L^2}\leq \beta\lVert f_{n-1}-f\rVert_{L^2}, \end{equation}

for all $n\geq 1$, where $f_n=P_Nf_n$ and each $f_n$ is given by (3.8) evaluated at $t=t_n$.

In the setting of time-independent forcing, one can in fact ‘recycle’ the data provided that a sufficiently long time series is available. By ‘recycle’ we mean that once an approximation to the force is proposed by the algorithm at a given stage, we may use this proposed forcing to re-run the algorithm over the same time window to produce the subsequent approximation of the force in the following stage, and so on. In this way, the same data set is used over and over in order to generate new approximations to the force.

Theorem 4.5 Let $T>0$. Suppose that $f=P_Nf$, for some $N\geq 1$, is time-independent, and that $\{P_Nu(t)\}_{t\in [0,T)}$ is given, for some $t_0\geq 0$. There exists a positive constant $c_0$, depending on $\gamma _0$, such that for any $\beta \in (0,1)$, if $N\geq 1$ satisfies:

(4.6)\begin{equation} \beta^2N^2>c_0(\tilde{\sigma}_1+G)G^2, \end{equation}

then for each $n\geq 1$, there exists a choice for the tuning parameter $\mu$ and a sequence of times, $t_n>0$, such that if $T>t_0+t_n$, for all $n\geq 1$, then

(4.7)\begin{equation} \lVert f_n-f\rVert_{L^2}\leq \beta\lVert f_{n-1}-f\rVert_{L^2}, \end{equation}

for all $n\geq 1$, where $f_n=P_Nf_n$, $f_n\in L^2_\sigma$, and each is determined by a procedure similar to that in § 3, except that $v_n, u_n, f_n$ are always derived on the interval $[0,T)$.

The proof of theorem 4.5 follows along the same lines as that of theorem 4.1, except that one requires a few technical modifications of the set-up described in § 3; we supply the relevant details of these modifications in § 7.

5. Outline of the convergence argument

To prove theorem 4.1, the object of interest will be the error in the forcing, which we also refer to as ‘model error’. This is denoted by

(5.1)\begin{equation} g_n:=f_n-f, \end{equation}

where $f_n$ is generated from the scheme described in § 3. We claim the following: there exists a sequence of times $t_n\geq t_{n-1}$, for all $n\geq 1$, such that

(5.2)\begin{equation} \sup_{t\geq t_n}\lVert A^{1/2} g_n(t)\rVert_{L^2}\leq \beta\sup_{t\geq t_{n-1}}\lVert A^{1/2} g_{n-1}(t)\rVert_{L^2}, \end{equation}

for some $\beta \in (0,1)$. The lengths of the relaxation periods, $\rho _n:=t_n-t_{n-1}$, between the moments, $t_{n-1}, t_n$, at which we choose to reconstitute new forces, $f_{n-1}, f_n$, respectively, are prescribed to be sufficiently large in order to allow time for system (3.7) to relax. As we will see, the length of these relaxation periods, $\rho _n$, will essentially be determined by the relaxation parameter $\mu$; the reader is referred to remark 7.1 for a precise relation.

Let us denote the synchronization error by

\[ w_n:=v_n-u. \]

Observe that the evolution of $w_n$ over the time interval $[t_{n-1},\infty )$ is governed by

(5.3)\begin{align} \frac{{\rm d}}{{\rm d}t}w_n+\nu Aw_n+B(w_n,w_n)+DB(u)w_n& =g_{n-1}-\mu P_Nw_n, \nonumber\\ w_n(t_{n-1})& =v_n^0-u(t_{n-1}), \end{align}

where $DB(u)v:=B(u,v)+B(v,u)$. Let $B_N=P_NB$ and $DB_N=P_NDB$. Then, using (2.3), the facts that $P_Nf=f$ and $u_n=P_Nu+Q_Nv_N$, and (3.8), we obtain

(5.4)\begin{align} g_n& =\left(\frac{{\rm d}}{{\rm d}t}P_Nu_n+\nu AP_Nu_n+B_N(u_n,u_n)\right)-\left(\frac{{\rm d}}{{\rm d}t}P_Nu+\nu AP_Nu+B_N(u,u)\right)\nonumber\\ & =B_N(P_Nu+Q_Nv_n,P_Nu+Q_Nv_n)-B_N(P_Nu+Q_Nu,P_Nu+Q_Nu)\nonumber\\ & =B_N(Q_Nw_n,Q_Nw_N)+DB_N(u)Q_Nw_n, \end{align}

which holds over $[t_{n-1},\infty )$. In analogy to (1.5), we define the ‘Reynolds stress’ at stage $n$ by

(5.5)\begin{equation} \mathcal{R}_{N}^{(n)}:=B_N(Q_Nw_n,Q_Nw_n)+DB_N(u)Q_Nw_n=g_n. \end{equation}

Ultimately, (5.5) enables us to envision a recursion in the model error at each stage through the dependence of $\mathcal {R}_N^{(n)}$ on $\mathcal {R}_N^{(n-1)}$ via the synchronization error $w_n$. Hence, in order to prove that $\mathcal {R}_{N}^{(n)}$ vanishes in the limit as $n\rightarrow \infty$, we require sensitivity-type estimates, that is, estimates on $w_n$. The estimates will take on the following form:

(5.6)\begin{equation} \lVert A^{1/2} w_n(t)\rVert_{L^2}\leq \nu\left(\frac{\nu}{\mu}\right)^{1/2}O\left(\frac{\sup_{t\geq t_n}\lVert g_{n-1}(t)\rVert_{L^2}}{\nu^2}\right), \end{equation}

for all $t\geq t_n$, for some sufficiently large $t_n\geq t_{n-1}$. Before we go on to develop estimates of form (5.6) in § 6, let us first determine the precise manner in which their application will arise.

To estimate (5.5), we will invoke the following inequalities for $B(u,v)$, which follows from a direct application of Hölder's inequality:

(5.7)\begin{equation} \lVert B(u,v)\rVert_{L^2}\leq \min\{\lVert u\rVert_{L^\infty}\lVert A^{1/2}v\rVert_{L^2},\lVert u\rVert_{L^4}\lVert A^{1/2}v\rVert_{L^4}\}. \end{equation}

We will also make use of the fact that $P_N, Q_N$ commute with $A^{m/2}$, for all integers $m$, and that the following inequalities hold for all $N>0$, $\ell >0$, and $m\in \mathbb {Z}$:

(5.8)\begin{equation} \lVert A^{m/2}P_Nv\rVert_{L^2}\leq N^m\lVert P_Nv\rVert_{L^2},\quad \lVert A^{m/2}Q_Nv\rVert_{L^2}\leq N^{-\ell}\lVert A^{(m+\ell)/2}Q_Nv\rVert_{L^2}. \end{equation}

Lastly, we make the following elementary, but important observation for treating the first term appearing in (5.5): for vector fields $u=(u_1,u_2)$ and $v=(v_1,v_2)$, we have:

\[ B(u,v)_i=\sum_{j=1}^2P(u\cdotp\nabla v_i)=P\nabla\cdotp(uv_i),\quad i=1,2, \]

where $B(u,v)_i$ denotes the $i$-th component of the vector field $B(u,v)$. Thus, upon applying (5.8) and Hölder's inequality, we obtain:

(5.9)\begin{equation} \lVert B_N(u,v)\rVert_{L^2}^2\leq2\sum_{i=1}^2\lVert P_NA^{1/2}(uv_i)\rVert_{L^2}^2\leq 2N^2\sum_{i=1}^2\lVert uv_i\rVert_{L^2}^2\leq 2N^2\lVert u\rVert_{L^4}^2\lVert v\rVert_{L^4}^2. \end{equation}

From (5.5), we now apply (5.9), (5.7), (5.8), and interpolation to obtain

(5.10)\begin{align} \lVert \mathcal{R}_{N}^{(n)}\rVert_{L^2}& \leq \lVert P_NA^{1/2}(Q_Nw_n\otimes Q_Nw_n)\rVert_{L^2}+\lVert DB_N(u)Q_Nw_n\rVert_{L^2}\nonumber\\ & \leq CN\lVert Q_Nw_n\rVert_{L^4}^2+\lVert u\rVert_{L^\infty}\lVert A^{1/2} Q_Nw_n\rVert_{L^2}+\lVert Q_Nw_n\rVert_{L^4}\lVert A^{1/2} u\rVert_{L^4}\nonumber\\ & \leq CN\lVert A^{1/2}Q_N w_n\rVert_{L^2}\lVert Q_Nw_n\rVert_{L^2}+C\lVert Au\rVert_{L^2}^{1/2}\lVert u\rVert_{L^2}^{1/2}\lVert A^{1/2} Q_Nw_n\rVert_{L^2}\nonumber\\ & \quad+\lVert A^{1/2} Q_Nw_n\rVert_{L^2}^{1/2}\lVert Q_Nw_n\rVert_{L^2}^{1/2}\lVert Au\rVert_{L^2}^{1/2}\lVert A^{1/2} u\rVert_{L^2}^{1/2}\nonumber\\ & \leq C\lVert A^{1/2}Q_N w_n\rVert_{L^2}^2+C\lVert Au\rVert_{L^2}^{1/2}\lVert u\rVert_{L^2}^{1/2}\lVert A^{1/2} Q_Nw_n\rVert_{L^2}\nonumber\\ & \quad+\frac{C}{N^{1/2}}\lVert A^{1/2} Q_Nw_n\rVert_{L^2}\lVert Au\rVert_{L^2}^{1/2}\lVert A^{1/2} u\rVert_{L^2}^{1/2}\nonumber\\ & \leq C_0\nu(\tilde{\sigma}_1+\tilde{G})^{1/2}\tilde{G}\left(1+\frac{\lVert A^{1/2} Q_Nw_n\rVert_{L^2}}{\nu}\right)\lVert A^{1/2} Q_Nw_n\rVert_{L^2}, \end{align}

for some universal constant $C_0>0$, independent of $n$, for all $t\geq t_{n-1}$. Note that we also invoked the assumption that $u$ belongs to the absorbing ball in $H^1_\sigma$ and $H^2_\sigma$ [see (2.11), (2.14), respectively] in obtaining the final inequality. It is at this point that one applies (5.6) in order to properly close the recursive estimate. In order to rigorously carry out this argument, let us therefore prove that (5.6) indeed holds.

Remark 5.2 In the case when the force contains modes beyond those that are observed, one can identify an obstruction that precludes a proof in the manner described above. Suppose that $f\neq P_Nf$. Then, modify the ansatz (3.8) for the force at each stage by removing the projection onto low modes. In particular, re-define $f_n$ so that:

\[ f_n=\frac{{\rm d}}{{\rm d}t}u_n+\nu Au_n+B(u_n,u_n). \]

Then, the model error becomes:

(5.11)\begin{align} g_n=\frac{{\rm d}}{{\rm d}t}Q_Nw_n+\nu AQ_Nw_n+B(Q_Nw_n,Q_Nw_n)+DB(u)Q_Nw_n. \end{align}

Upon applying the complementary projection, $Q_N$, to (5.3), and combining the result with (5.11), we see that:

\begin{align*} Q_Ng_n& ={-}B^N(P_Nw_n,P_Nw_n)-B^N(P_Nw_n,Q_Nw_n)-B^N(Q_Nw_n,P_Nw_n)\\ & \quad -DB^N(u)P_Nw_n+Q_Ng_{n-1}, \end{align*}

where $B^N=Q_NB$ and $DB^N=Q_NDB$. Due to the presence of $Q_Ng_{n-1}$ on the right-hand side, one cannot expect to obtain a convergent recursive relation of the form $\lVert Q _Ng_n\rVert _{L^2}\leq \beta \lVert Q _Ng_{n-1}\rVert _{L^2}$, for some $\beta \in (0,1)$. Although one can establish an estimate of $Q_Ng_n$ from this relation that is of the form $\lVert Q _Ng_n-Q_Ng_{n-1}\rVert _{L^2}\leq O_N(\lVert A ^{1/2}Q_Nw_n\rVert _{L^2})$, where the suppressed constant has a favourable dependence on $N$, a subsequent analysis will nevertheless be unable to establish a suitable recursion relation since we will only ever have access to an estimate of the form (5.6).

6. Sensitivity analysis

We establish a more precise form of the crucial estimates (5.6), which form the bridge to the desired recursion for the model error at each stage of the approximation. For this, we recall the notation introduced in § 5. In particular, we prove the following.

Proposition 6.1 There exist universal constants $\overline {c}_0, \underline {c}_0\geq 1$ such that if $\mu, N$ satisfy

(6.1)\begin{equation} \underline{c}_0\big(\tilde{\sigma}_1+\tilde{G}\big)\tilde{G}^2\leq \frac{\mu}{\nu}\leq \overline{c}_0 N^2, \end{equation}

then for each $n\geq 1$, there exist relaxation periods, $\rho _n=t_n-t_{n-1}$, for some $t_n>t_{n-1}$, such that

(6.2)\begin{equation} \sup_{t\in I_n}\left(\frac{\lVert A^{1/2} w_n(t)\rVert_{L^2}}{\nu}\right) \leq \left(\frac{2C_1\nu}{\mu}\right)^{1/2}\left(\frac{\sup_{t\in I_{n-1}}\lVert g_{n-1}(t)\rVert_{L^2}}{\nu^2}\right), \end{equation}

where $I_n:=[t_{n-1}+\rho _n,\infty )$, for some universal constant $C_1\geq 1$, independent of $n$. Moreover, $\rho _n$ satisfies (6.5).

Proof. Fix $n\geq 1$. The enstrophy-balance for $w_n$ is obtained by taking the $L^2$-inner product of (5.3) by $Aw_n$, which yields:

(6.3)\begin{align} \frac{1}2\frac{{\rm d}}{{\rm d}t}\lVert A^{1/2} w_n\rVert_{L^2}^2+\nu\lVert Aw_n\rVert_{L^2}^2& ={-}\langle B(u,w_n),Aw_n\rangle-\langle B(w_n,u),Aw_n\rangle\nonumber\\ & \quad +\langle g_{n-1},Aw_n\rangle-\mu \lVert A^{1/2} P_Nw_n\rVert_{L^2}^2\nonumber\\ & =I+II+III+IV. \end{align}

Observe that by interpolation, the Cauchy–Schwarz inequality, Poincaré's inequality, and (2.11), (2.14), we may estimate:

\begin{align*} |I|& \leq C\lVert u\rVert_{L^\infty}\lVert A^{1/2} w_n\rVert_{L^2}\lVert Aw_n\rVert_{L^2}\\ & \leq C\lVert Au\rVert_{L^2}^{1/2}\lVert u\rVert_{L^2}^{1/2}\lVert A^{1/2} w_n\rVert_{L^2}\lVert Aw_n\rVert_{L^2}\\ & \leq C\mu\left(\frac{\nu}{\mu}\right)\big(\tilde{\sigma}_1+\tilde{G}\big)\tilde{G}^2\lVert A^{1/2} w_n\rVert_{L^2}^2+\frac{\nu}{100}\lVert Aw_n\rVert_{L^2}^2. \end{align*}

Similarly, we obtain:

\begin{align*} |II| & \leq C\lVert A^{1/2} w_n\rVert_{L^2}^{1/2}\lVert w_n\rVert_{L^2}^{1/2}\lVert Au\rVert_{L^2}^{1/2}\lVert A^{1/2}u\rVert_{L^2}^{1/2}\lVert Aw_n\rVert_{L^2}\\ & \leq C\mu\left(\frac{\nu}{\mu}\right)\big(\tilde{\sigma}_1+\tilde{G}\big)\tilde{G}^2\lVert A^{1/2} w_n\rVert_{L^2}^2+\frac{\nu}{100}\lVert Aw_n\rVert_{L^2}^2. \end{align*}

For the third term, we apply the Cauchy–Schwarz inequality to obtain:

\[ |III|\leq C\frac{\lVert g_{n-1}\rVert_{L^2}^2}{\nu}+\frac{\nu}{100}\lVert Aw_n\rVert_{L^2}^2. \]

Lastly, we have:

\begin{align*} IV& ={-}\mu\lVert A^{1/2} w_n\rVert_{L^2}^2+\mu\lVert A^{1/2} Q_Nw_n\rVert_{L^2}^2\\ & \leq{-}\mu\lVert A^{1/2} w_n\rVert_{L^2}^2+\frac{\mu}{N^2}\lVert Aw_n\rVert_{L^2}^2. \end{align*}

Upon combining the estimates for $I$–$IV$ and invoking (6.1), where $\underline {c}_0, \overline {c}_0\geq 1$ are chosen appropriately relative to the constants $C$ appearing above, we arrive at:

\[ \frac{{\rm d}}{{\rm d}t}\lVert A^{1/2} w_n\rVert_{L^2}^2+\frac{3}2\nu\lVert Aw_n\rVert_{L^2}^2+\frac{3}2\mu\lVert A^{1/2} w_n\rVert_{L^2}^2\leq C\nu^3\left(\frac{\lVert g_{n-1}\rVert_{L^2}}{\nu^2}\right)^2. \]

An application of Gronwall's inequality yields:

(6.4)\begin{align} \left(\frac{\lVert A^{1/2} w_n(t)\rVert_{L^2}}{\nu}\right)^2 & \leq {\rm e}^{-\mu(t-t_n)}\,{\rm e}^{-\mu\rho_n}\left(\frac{\lVert A^{1/2} w_n(t_{n-1})\rVert_{L^2}}{\nu}\right)^2\nonumber\\ & \quad + C_1\left(\frac{\nu}{\mu}\right)\left(\frac{\sup_{t\geq t_{n-1}}\lVert g_{n-1}(t)\rVert_{L^2}}{\nu^2}\right)^2, \end{align}

for all $t\geq t_n$, for some $C_1\geq 1$ independent of $n$. Now, recall that $w_n(t_{n-1})=v^0_n-u(t_{n-1})$. We choose $\rho _n>0$ such that:

(6.5)\begin{equation} \rho_n\geq\frac{1}{\mu}\ln\left[\left(\frac{\mu}{C_1\nu}\right)\left(\frac{\nu\lVert A^{1/2} w_n(t_{n-1})\rVert_{L^2}}{\sup_{t\in I_{n-1}}\lVert g_{n-1}(t)\rVert_{L^2}}\right)^2\right]. \end{equation}

Returning to (6.4), it follows that

\[ \sup_{t\in I_n}\left(\frac{\lVert A^{1/2} w_n(t)\rVert_{L^2}}{\nu}\right) \leq \left(\frac{2C_1\nu}{\mu}\right)^{1/2}\left(\frac{\sup_{t\in I_{n-1}}\lVert g_{n-1}(t)\rVert_{L^2}}{\nu^2}\right), \]

as desired.

7. Proofs of convergence

Let us assume that $f=P_Nf$. At the initializing stage, $n=0$, we consider any $f_0\in C([0,\infty );L^2_\sigma )\cap L^\infty ([0,\infty ;);L^2_\sigma )$. Recall that in each subsequent stage, we will produce a new approximation, $f_n$, for the force via the ansatz (3.8). We proceed by induction.

Proof Proof of theorem 4.1

Fix $\beta \in (0,1)$. We choose $c_0$ to satisfy:

\[ c_0\geq\max\left\{\underline{c}_0, 2C_1\left(\frac{C_0}{\beta}\right)^2\left[1+\left(\frac{2C_1}{\underline{c}_0(\tilde{\sigma}_1+\tilde{G})}\right)^{1/2}\gamma_0\right]^2\right\}, \]

where $C_0, C_1, \underline {c}_0$ are the universal constants appearing in (5.10), (6.1), (6.2), respectively, and $\gamma _0$ denotes the initial relative error defined by (4.1). Fix any $c_1\leq {\tilde {c}}$, where ${\tilde {c}}$ is the constant appearing in (3.2). Then, assume that $\mu, N$ satisfies (4.2) with $c=c_0c_1^{-1}$. Then, choose $\mu$ such that:

(7.1)\begin{equation} c_0(\tilde{\sigma}_1+\tilde{G})\tilde{G}^2<\beta^2\frac{\mu}{\nu}\leq c_1 \beta^2 N^2. \end{equation}

Observe that $N$ satisfies (4.2) with $c=c_0c_1^{-1}$. Since $c_0\geq \underline {c}_0$ by choice, it immediately follows that (6.1) also holds.

Let $n=1$. Denote $\mathcal {R}_N^{(0)}:=g_0=f_0-f$. Observe that from (2.9), (4.1), we have:

(7.2)\begin{equation} \sup_{t\geq t_0}\lVert \mathcal{R}_N^{(0)}(t)\rVert_{L^2}=\gamma_0\nu^2\tilde{G}. \end{equation}

Combining proposition 6.1, (5.10) for $n=1$, and (7.2) ensures that there exists a relaxation period $\rho _1>0$, for some $t_1\geq t_0$, such that

(7.3)\begin{align} \sup_{t\geq I_1}\lVert \mathcal{R}_N^{(1)}(t)\rVert_{L^2}& \leq (2C_1)^{1/2}C_0(\tilde{\sigma}_1+\tilde{G})^{1/2}\tilde{G}\left(1+\left(\frac{2C_1\nu}{\mu}\right)^{1/2}\gamma_0\tilde{G}\right)\left(\frac{\nu}{\mu}\right)^{1/2}\nonumber\\ & \quad \sup_{t\geq I_0}\lVert \mathcal{R}_N^{(0)}(t)\rVert_{L^2}, \end{align}

where $I_1=[t_0+\rho _1,\infty )$.

Further assume that $\mu$ satisfies:

(7.4)\begin{equation} \beta^2\frac{\mu}{\nu}\geq 2C_1C_0^2\left((\tilde{\sigma}_1+\tilde{G})^{1/2}+\left(\frac{2C_1}{\underline{c}_0}\right)^{1/2}\gamma_0\right)^2\tilde{G}, \end{equation}

where $C_1$ is the same constant appearing in (7.3). Then, (7.3), (6.1), and (7.4) imply:

(7.5)\begin{equation} \sup_{t\geq I_1}\lVert \mathcal{R}_N^{(1)}(t)\rVert_{L^2}\leq \beta\left(\sup_{t\geq I_0}\lVert \mathcal{R}_N^{(0)}(t)\rVert_{L^2}\right). \end{equation}

This establishes the base case.

Now, suppose that for all $k=1,\ldots, n$, there exist relaxation periods $\rho _k$, such that:

(7.6)\begin{equation} \sup_{t\geq I_k}\lVert \mathcal{R}_N^{(k)}(t)\rVert_{L^2}\leq \beta\sup_{t\geq I_{k-1}}\lVert \mathcal{R}_N^{(k-1)}(t)\rVert_{L^2}. \end{equation}

With (7.2), it follows that:

(7.7)\begin{equation} \sup_{t\geq I_k}\lVert \mathcal{R}_N^{(k)}(t)\rVert_{L^2}\leq \beta^k\gamma_0\tilde{G}\leq \gamma_0\tilde{G}, \end{equation}

for all $k=1,\dots, n$. We may thus deduce from proposition 6.1, (5.10), and (7.7) that

(7.8)\begin{align} \sup_{t\geq I_{n+1}}\lVert \mathcal{R}_N^{(n+1)}(t)\rVert_{L^2}& \leq (2C_1)^{1/2}C_0(\tilde{\sigma}_1+\tilde{G})^{1/2}\tilde{G}\left(1+\left(\frac{2C_1\nu}{\mu}\right)^{1/2}\gamma_0\tilde{G}\right)\left(\frac{\nu}{\mu}\right)^{1/2}\nonumber\\ & \quad \sup_{t\geq I_n}\lVert \mathcal{R}_N^{(n)}(t)\rVert_{L^2}. \end{align}

An application of (7.4) and (6.1) then implies

\[ \sup_{t\geq I_{n+1}}\lVert \mathcal{R}_N^{(n+1)}(t)\rVert_{L^2}\leq \beta\left(\sup_{t\geq I_n}\lVert \mathcal{R}_N^{(n)}(t)\rVert_{L^2}\right), \]

as desired. This completes the induction.

Remark 7.1 From the proof of theorem 4.1, we can obtain more informative estimates on the relaxation periods, $\rho _n$. Indeed, by (6.5) and (7.2), we see that the first relaxation period satisfies:

(7.9)\begin{equation} \rho_1\geq \frac{1}{\mu}\ln\left[\left(\frac{\mu}{C_1\nu}\right)\frac{1}{\gamma_0^2\tilde{G}^2}\left(\frac{\lVert A^{1/2}(v^0_1-u(t_0))\rVert_{L^2}}{\nu}\right)^2\right]. \end{equation}

In subsequent stages, $n\geq 1$, by (6.5) and (6.2) applied at the preceding stage, we may deduce that the relaxation periods satisfy:

(7.10)\begin{equation} \rho_n\geq \frac{1}{\mu}\ln\left(\frac{\sup_{t\in I_{n-2}}\lVert \mathcal{R}_N^{(n-2)}(t)\rVert_{L^2}}{\sup_{t\in I_{n-1}}\lVert \mathcal{R}_N^{(n-1)}(t)\rVert_{L^2}}\right)\geq{-}\frac{1}{\mu}\ln\beta. \end{equation}

One may then observe that (7.9) and (7.10) follow a consistent pattern if we allow ourselves to make special, but nevertheless natural choices for initial force ansatz, $f_0$, and the initial data, $v_1^0$, of the first nudged system. Indeed, suppose that the solution, $u$, of (2.3) possesses additional regularity, for instance, $Au(t_0)\in L^2_\sigma$. Now, initialize the nudged system (3.4) with $v_1^0=P_Nu(t_0)$ and let the initial guess $f_0$ be given by (1.2). Then, the first relaxation period satisfies:

(7.11)\begin{equation} \rho_1\geq \frac{1}{\mu}\ln\left[\left(\frac{c_1}{C_1}\right)\left(\frac{\nu^2}{\sup_{t\in I_0}\lVert \mathcal{R}_N^{(0)}(t)\rVert_{L^2}}\right)^2\right], \end{equation}

where we applied the Poincaré inequality, the upper bound in (7.1), and (2.9).

Now, we prove a variation on the time-independent case, which allows one to recycle the existing data.

Proof Proof sketch of theorem 4.5

We modify the algorithm in §3 as follows: let $J_0=I_0=[t_0,\infty )$. For convenience, we assume that $t_0=0$. Suppose that $\{P_Nu(t)\}_{t\in J_0}$ is known. At stage $n=1$, we suppose initial data $v_1^0$ is given and solve (3.4) to produce the solution $v_1$ over $J_0$. We then define (3.5) over $J_0$ and immediately generate $f_1(t)$ via (3.6) over $J_0$. To define the stage $1$ approximation to $f$, we evaluate $f_1(t)$ after a transient period of length $\rho _1>0$ to obtain $f_1:=f_1(\rho _1)$.

In subsequent stages $n\geq 1$, we generate $v_n$ over $J_0$ via (3.7), where $f_{n-1} :=f_{n-1}(\rho _{n-1})$ was generated from the preceding stage, $n-1$. We then define the stage-$n$ approximation to the true state over $J_0$ by $u_n=P_Nu+Q_Nv_n$. We generate a new force, $f_n(t)$, via the ansatz (3.8). The stage-$n$ approximation of $f$ is then given by $f_n(\rho _n)$, for some $\rho _n>0$.

To assess the error, we once again form the synchronization error, $w_n=v_n-u$, and the model error, $g_n(t)=f_n(t)-f$. For each $n\geq 1$, the identity (5.5) still holds. Consequently, (5.10) holds as well. The remaining ingredient to establish convergence is a time-independent force analogue of proposition 6.1.

Supposing that (6.1) holds with $\tilde {G}\mapsto G$, the proof of proposition 6.1 proceeds the same way, except that the analysis is carried out entirely over the interval $J_0$, to arrive at the analogue of (6.4):

(7.12)\begin{align} \left(\frac{\lVert A^{1/2} w_n(\rho_n)\rVert_{L^2}}{\nu}\right)^2 \leq {\rm e}^{-\mu\rho_n}\left(\frac{\lVert A^{1/2} w_n(0)\rVert_{L^2}}{\nu}\right)^2+ C_1\left(\frac{\nu}{\mu}\right)\left(\frac{\lVert g_{n-1}\rVert_{L^2}}{\nu^2}\right)^2. \end{align}

One chooses $\rho _n>0$ according to:

\[ \rho_n\geq \frac{1}{\mu}\left[\frac{\mu}{C_1\nu}\left(\frac{\nu\lVert A^{1/2}w_n(0)\rVert_{L^2}}{\lVert g_{n-1}\rVert_{L^2}}\right)^2\right], \]

so that (7.12) yields:

\[ \left(\frac{\lVert A^{1/2} w_n(\rho_n)\rVert_{L^2}}{\nu}\right)^2 \leq 2C_1\left(\frac{\nu}{\mu}\right)\left(\frac{\lVert g_{n-1}\rVert_{L^2}}{\nu^2}\right)^2. \]

It is at this time $t=\rho _n$, that we evaluate (3.8).

After these adjustments, it is now clear that the proof of theorem 4.5 follows an analogous manner to the proof theorem 4.1, mutatis mutandis.