# Determining the viscosity of the Navier-Stokes equations from observations
of finitely many modes

Animikh Biswas and Joshua Hudson

###### Abstract.

In this work, we ask and answer the question: when is the viscosity of a fluid
uniquely determined from spatially sparse measurements of its velocity field?
We pose the question mathematically as an optimization problem using the
determining map (the mapping of data to an approximation made via a nudging
algorithm) to define a loss functional, the minimization of which solves the
inverse problem of identifying the true viscosity given the measurement data.
We give explicit _a priori_ conditions for the well-posedness of this inverse
problem. In addition, we show that smallness of the loss functional implies
proximity to the true viscosity. We then present an algorithm for solving the
inverse problem and prove its convergence.

##  1\. Introduction

In many instances, the general form of a particular dynamical system is
commonly derived from well-accepted fundamental physical, biological or
epidemiological principles. However, frequently these systems have one or
multiple parameters and their values play a crucial role in its dynamical
evolution. For instance, in geophysical models such as the Navier–Stokes or
the Boussinesq systems, physical parameters such as the Reynolds, Raleigh or
Prandtl numbers determine whether or not the system approaches an equilibrium
or transitions to chaos, while in the SEIRD model of transmission of an
epidemic, the associated parameters determine whether the disease is
eradicated or becomes endemic. Other examples include reservoir modelling [11]
and determining the transmissivity coefficient of groundwater, which captures
its ability to move across an aquifer. It is a central parameter in these
models that must be estimated in some way [7, 19] in order to make accurate
predictions.

Due to its ubiquity in applications, there has been a recent surge of interest
in parameter determination and estimation problems _from finitely many partial
observations_ of the system under consideration. For instance, in case of a
geophysical model with a state variable $\theta(\bm{x},t)$, depending on a
spatial variable $\bm{x}$ and time $t$, such observations may take the form of
nodal values $\\{\theta(\bm{x}_{i},t)\\}_{i=1}^{N}$ where
$\\{\bm{x}_{i}\\}_{i=1}^{N}$ is a finite set of points in the spatial domain.
These observations can either be collected continuously in time or at a
discrete sequence of time points $t_{1}<t_{2}<\cdots$. Other examples of
commonly used observational data include finitely many Fourier coefficients
$\\{\widehat{\theta}(n,t):|n|\leq N\\}$ or volume elements
$\\{\bar{\theta}_{i}\\}_{i=1}^{N}$ where
$\bar{\theta}_{i}=\int_{\Omega_{i}}\theta(\bm{x},t)\\}$ and $\Omega_{i}$ is a
subset of the whole spatial domain $\Omega$. For finite dimensional dynamical
systems, such partial observations constitute a time series of observations on
a subset of the variables. In the SEIRD model of epidemic transmission, data
may not be available for certain variables such as the exposed population due
to asymptomatic transmission, thus leading to partial observations in these
cases.

The above mentioned examples can be mathematically formulated as follows. Let
$\\{{\bm{u}}(t)\\}$ be a dynamical system parameterized by $\nu$ evolving on a
Hilbert space $\mathcal{H}$ according to the equation

${\displaystyle\frac{\text{d}{\bm{u}}}{\text{dt}}}=F_{\bm{\nu}}({\bm{u}}(t)),\>{\bm{u}}(\cdot)\in\mathcal{H},\>{\bm{\nu}}\in\mathbb{R}^{d}.$
(1.1)

The problem under consideration then is the estimation of the parameter
${\bm{\nu}}\in\mathbb{R}^{d}$ based on observations $\\{\cal
O({\bm{u}}(t))\\}$, where $\cal O:\cal H\longrightarrow\cal\mathbb{R}^{N}$ is
an adequate linear (or nonlinear) operator and $t\in[t_{0},T],T\leq\infty$.
More generally, one may have observations on a discrete set of points $\\{\cal
O({\bm{u}}(t_{i})\\}_{i=1}^{\infty}$ and the observations may be contaminated
with error. When ${\bm{\nu}}$ is known, recovering the state variable from the
observational data is commonly referred to as data assimilation. When
${\bm{\nu}}$ is unknown, we may view this as a parameter estimation problem
embedded in data assimilation. This has led to various computational
strategies, often ad hoc, using machine learning [14, 15], Bayesian estimation
methods related to various extensions of Kalman filtering employed in data
assimilation, or other on the fly numerical methods based on the nudging
scheme for data assimilation [4, 13, 1]. In this context, we note first that
the parameter identification problem is not in general well-posed even when
the state variable is fully observable, i.e. when the solution ${\bm{u}}$ in
(1.1) is known exactly. As described below, this can be seen in the multi-
parameter identification problem addressed in [13].

Let $\Omega=[0,2\pi]$ and

$\cal H=\\{f:\Omega\rightarrow\mathbb{R},\>f\ \text{is}\
2\pi-\text{periodic},\>\int_{\Omega}f=0\\}$

and $A=-\partial_{xx}$ on $\cal H\cap\mathbb{H}^{2}$. Consider now the problem
of determining the parameter vector $(\lambda_{1},\lambda_{2})$ from the
solution $u$ of the equation

$u_{t}-\lambda_{1}Au+\lambda_{2}A^{2}u=f,$ (1.2)

which is an illustrative example of the type of parameter identification
problems discussed in [13]. If one takes $u=u_{\delta}$ to be an (time-
independent) eigenfunction of $A$ corresponding to the eigenvalue $\delta$,
and $f=c\delta u_{\delta}$, then $u_{\delta}$ solves the equation (1.2)
provided $(\lambda_{1}-\lambda_{2}\delta)=c$. However, _one cannot determine
the parameters_ $\lambda_{1}$ and $\lambda_{2}$, even if one knows $u$
completely on the entire domain, $\Omega$; in a neighborhood of this solution,
the inverse problem of determining the parameter $(\lambda_{1},\lambda_{2})$
is ill-posed. Thus, the question whether or not the parameter ${\bm{\nu}}$ can
be determined in (1.1) from partial observations (i.e. whether or not the
parameter-to-data map is one-to-one) is even more delicate.

###  1.1. Overview of Results

Here, for the specific case of determining the viscosity parameter for the
two-dimensional Navier–Stokes equations (2D NSE), we consider the fundamental
questions inherent in rigorous justification of all of the parameter
estimation methods mentioned before, namely, when can one recover the
parameter from observational data and what is the regularity property of the
inverse map.

In addition, we define an algorithm for solving the inverse problem using the
determining map. Our algorithm can be seen as an improvement over the
algorithm originally presented in [4] due to the replacement of the _a
posteriori_ verifiable condition with an _a priori_ condition for the update
to have a nonzero denominator. Our rigorous convergence criteria is also an
improvement over the recent work in [12] (which was developed simultaneously
with this work) for the same reason; in addition, our criteria is independent
of the magnitude of the observable error, whereas the condition in [12]
requires more data as the observable error decreases.

In Section 2 we will establish some notation and review the basic theory
necessary for studying the Navier–Stokes equations. We will also briefly
review the nudging data assimilation algorithm originally presented in [2] for
the Navier–Stokes equations. Then, in Section 3, we define the determining
map, extending its domain from that of [3] to include viscosity as a variable,
and prove some of its regularity properties.

In Section 4, we frame the viscosity recovery inverse problem as a PDE
constrained optimization problem, and provide rigorous conditions for the
inverse problem to have a unique solution. Then, in Section 6, we prove that
the loss function controls the viscosity error, which we leverage in Section 7
to define an algorithm that converges to the true viscosity, solving the
inverse problem.

##  2\. Preliminaries

###  2.1. The Navier–Stokes equations and their functional form

The Navier–Stokes equations for a homogeneous, incompressible Newtonian fluid
in two dimensions are given by

$\frac{\partial{\bm{u}}}{\partial
t}-\nu\Delta{\bm{u}}+({\bm{u}}\cdot\nabla){\bm{u}}+\nabla
p=\bm{g},\quad\nabla\cdot{\bm{u}}=0,$ (2.1)

where ${\bm{u}}=(u_{1},u_{2})$ and $p$ are the unknowns and denote the
velocity vector field and the pressure, respectively, while $\nu>0$ and
$\bm{g}$ are given and denote the kinematic viscosity parameter and the body
forces applied to the fluid per unit mass, respectively. In this work, we
consider the Navier–Stokes equations defined on the spatial domain

$\Omega=[0,L]\times[0,L],$

with periodic boundary conditions.

Let $\hat{{\bm{u}}}(\bm{k}),\bm{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ denote the
$\bm{k}$-th Fourier coefficient of ${\bm{u}}$. Given finitely many Fourier
coefficients $\\{\hat{\bm{u}}(\bm{k}):|\bm{k}|\leq N\\}$, we address the
question as to when the viscosity $\nu$ be determined from this finite
observational data and whether the data-to-viscosity map is Lipschitz.

We now proceed to recall the basic functional setting of the NSE, a systematic
development of which can be found in [6, 17, 18]. Let $\mathcal{V}$ be the
space of test functions, given by

$\mathcal{V}=\left\\{\bm{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{2}\,:\,\bm{\varphi}\,\text{
is an $L$-periodic trigonometric polynomial, }\right.\\\
\left.\nabla\cdot\bm{\varphi}=0,\int_{\Omega}\bm{\varphi}({\bm{x}}){\text{\rm
d}}{\bm{x}}=0\right\\}.$ (2.2)

We denote by $H$ and $V$ the closures of $\mathcal{V}$ with respect to the
norms in $(L^{2}(\Omega))^{2}$ and $(H^{1}(\Omega))^{2}$, respectively.
Moreover, we denote by $H^{\prime}$ and $V^{\prime}$ the dual spaces of $H$
and $V$, respectively. As usual, we identify $H$ with $H^{\prime}$, so that
$V\subseteq H\subseteq V^{\prime}$ with the injections being continuous and
compact, with each space being densely embedded in the following one. The
duality action between $V^{\prime}$ and $V^{\prime}$ is denoted by
$\langle\cdot,\cdot\rangle_{V^{\prime},V}$.

The inner product in $H$ is given by

$({\bm{u}}_{1},{\bm{u}}_{2})=\int_{\Omega}{\bm{u}}_{1}\cdot{\bm{u}}_{2}\,{\text{\rm
d}}{\bm{x}}\quad\forall{\bm{u}}_{1},{\bm{u}}_{2}\in H,$

with the corresponding norm denoted by
$|{\bm{u}}|:=\|{\bm{u}}\|_{L^{2}}=({\bm{u}},{\bm{u}})^{1/2}$. In $V$, we
consider the following inner product:

$(\\!({\bm{u}}_{1},{\bm{u}}_{2})\\!)=\int_{\Omega}\nabla{\bm{u}}_{1}:\nabla{\bm{u}}_{2}\,{\text{\rm
d}}{\bm{x}}\quad\forall{\bm{u}}_{1},{\bm{u}}_{2}\in V,$

where it is understood that $\nabla{\bm{u}}_{1}:\nabla{\bm{u}}_{2}$ denotes
the component-wise product between the tensors $\nabla{\bm{u}}_{1}$ and
$\nabla{\bm{u}}_{2}$. The corresponding norm in $V$ is given by
$\|{\bm{u}}\|:=\|\nabla{\bm{u}}\|_{L^{2}}=(\\!({\bm{u}},{\bm{u}})\\!)^{1/2}$.
The fact that $\|\cdot\|$ defines a norm on $V$ follows from the Poincaré
inequality, given in (2.10), below.

For every subspace $\Lambda\subset(L^{1}(\Omega))^{2}$, we denote

$\dot{\Lambda}_{\textnormal{per}}=\left\\{\bm{\varphi}\in\Lambda\,:\,\bm{\varphi}\mbox{
is $L$-periodic and }\int_{\Omega}\bm{\varphi}({\bm{x}}){\text{\rm
d}}{\bm{x}}=0\right\\}.$

Observe that $H$ is a closed subspace of $(\dot{L}^{2}(\Omega))^{2}$. Let
$P_{\sigma}$ denote the Helmholtz-Leray projector, which is defined as the
orthogonal projection from $(\dot{L}^{2}_{\textnormal{per}}(\Omega))^{2}$ onto
$H$. Applying $P_{\sigma}$ to (2.1), we obtain the following equivalent
functional formulation of the Navier–Stokes equations.

######  System 2.1 (Navier–Stokes in functional form).

$\frac{{\text{\rm d}}{\bm{u}}}{{\text{\rm d}}t}+\nu
A{\bm{u}}+B({\bm{u}},{\bm{u}})=\bm{f}\mbox{ in }V^{\prime},$ (2.3)

where $\bm{f}=P_{\sigma}\bm{g}$.

The bilinear operator $B:V\times V\to V^{\prime}$ is defined as the continuous
extension of

$B({\bm{u}},{\bm{v}})=P_{\sigma}[({\bm{u}}\cdot\nabla){\bm{v}}]\quad\forall{\bm{u}},{\bm{v}}\in\mathcal{V},$

and $A:D(A)\subset V\to V^{\prime}$, the Stokes operator, is the continuous
extension of

$A{\bm{u}}=-P_{\sigma}\Delta{\bm{u}}\quad\forall{\bm{u}}\in\mathcal{V}.$

In fact, in the case of periodic boundary conditions, we have $A=-\Delta$.

We recall that $D(A)=V\cap(\dot{H}^{2}_{\textnormal{per}}(\Omega))^{2}$ and
that $A$ is a positive and self-adjoint operator with compact inverse.
Therefore, the space $H$ admits an orthonormal basis
$\\{\bm{{\bm{\phi}}}_{j}\\}_{j=1}^{\infty}$ of eigenfunctions of $A$
corresponding to a non-decreasing sequence of eigenvalues
$\\{\lambda_{j}\\}_{j=1}^{\infty}$, where
$\lambda_{j}\in\\{\kappa_{0}^{2}|{\bm{k}}|^{2},{\bm{k}}\in\mathbb{Z}^{2}\setminus\\{0\\}\\}$
and $\lambda_{1}:=\kappa_{0}^{2}=(2\pi/L)^{2}$. For a periodic function
${\bm{\phi}}$ on $\Omega$, we recall the Parseval’s identity:

$|{\bm{\phi}}|^{2}=L^{2}\sum_{\bm{k}}|\hat{{\bm{\phi}}}_{\bm{k}}|^{2},$

where $\hat{{\bm{\phi}}}_{\bm{k}}$ denotes the Fourier coefficient
corresponding to ${\bm{k}}=(k_{1},k_{2})\in\mathbb{Z}^{2}$. Furthermore,
${\bm{\phi}}\in L^{2}(\Omega)$ satisfies the mean-free condition
$\int_{\Omega}{\bm{\phi}}=0$ if and only if $\hat{{\bm{\phi}}}_{0}=0$.

Given a normed space $\mathcal{X}$ with norm $\|\cdot\|_{\mathcal{X}}$, let
$C_{b}(\mathcal{X})$ denote the continuous bounded mappings from $\mathbb{R}$
to $X$, and define the norm
$\|{\bm{\phi}}\|_{C_{b}(X)}=\sup_{t\in\mathbb{R}}\|{\bm{\phi}}(t)\|_{\mathcal{X}}.$
In particular, we have the norms $\|\cdot\|_{C_{b}(H)}$ and
$\|\cdot\|_{C_{b}(V)}.$

###  2.2. Global attractor for the Navier–Stokes equations

It is well-known that, given ${\bm{u}}_{0}\in H$, there exists a unique
solution ${\bm{u}}$ of (2.3) on $[0,\infty)$ such that
${\bm{u}}(0)={\bm{u}}_{0}$ and

${\bm{u}}\in C([0,\infty);H)\cap L_{\textnormal{loc}}^{2}([0,\infty);V)\
\mbox{and}\ {\displaystyle\frac{\text{d}{\bm{u}}}{\text{dt}}}\in
L^{2}_{loc}([0,\infty);V^{\prime}).$ (2.4)

Moreover, we also have ${\bm{u}}\in C((0,\infty);D(A))$ (see, e.g., [6,
Theorem 12.1]). Therefore, equation (2.3) has an associated semigroup
$\\{S_{\nu}(t)\\}_{t\geq 0}$, where, for each $t\geq 0$, $S_{\nu}(t):H\to H$
is the mapping given by

$S_{\nu}(t){\bm{u}}_{0}={\bm{u}}_{\nu}(t),$ (2.5)

with ${\bm{u}}_{\nu}$ being the unique solution of (2.3) on $[0,\infty)$
satisfying ${\bm{u}}_{\nu}(0)={\bm{u}}_{0}$ and (2.4). For simplicity, we will
drop the subscript $\nu$ as it will be understood from context and simply
write $S(t)$ and ${\bm{u}}(t)$ instead.

Recall that a bounded set $\mathcal{B}\subset H$ is called _absorbing_ with
respect to $\\{S(t)\\}_{t\geq 0}$ if, for any bounded subset $B\subset H$,
there exists a time $T=T(B)$ such that $S(t)B\subset\mathcal{B}$ for all
$t\geq T$. The existence of a bounded absorbing set for (2.3) is a well-known
result; therefore, a global attractor $\mathcal{A}_{\nu}$ of (2.3) exists, and
is uniquely defined by any of the equivalent conditions given below.

######  Definition 2.2 (Global Attractor).

Let $\mathcal{B}\subset H$ be a bounded absorbing set with respect to
$\\{S(t)\\}_{t\geq 0}$. Then the global attractor, $\mathcal{A}_{\nu}$, exists
as given by any of the following equivalent definitions:

  1. (1)

$\mathcal{A}_{\nu}=\bigcap_{t\geq 0}S(t)\mathcal{B}.$

  2. (2)

$\mathcal{A}_{\nu}$ is the largest compact subset of $H$ which is invariant
under the action of the semigroup $\\{S(t)\\}_{t\geq 0}$, i.e.,
$S(t)\mathcal{A}_{\nu}=\mathcal{A}_{\nu}$ for all $t\geq 0$.

  3. (3)

$\mathcal{A}_{\nu}$ is the minimal set that attracts all bounded sets.

  4. (4)

$\mathcal{A}_{\nu}$ is the set of all points in $H$ through which there exists
a globally bounded trajectory ${\bm{u}}(t)$, $t\in\mathbb{R}$, with
$\sup_{t\in\mathbb{R}}\|{\bm{u}}(t)\|_{L^{2}}<\infty$.

Also, recall the definition of the (dimensionless) Grashof number, given by

$G_{\nu}=\frac{\|\bm{f}\|_{L^{2}}}{(\nu\kappa_{0})^{2}}.$ (2.6)

In the periodic case, the following bounds hold on the global attractor,
$\mathcal{A}_{\nu}$:

$\|{\bm{u}}\|_{L^{2}}\leq\nu
G_{\nu},\quad\|\nabla{\bm{u}}\|_{L^{2}}\leq\nu\kappa_{0}G_{\nu}\quad\forall{\bm{u}}\in\mathcal{A}_{\nu},$
(2.7)

and

$uu\|A{\bm{u}}\|_{L^{2}}\leq
c_{2}\nu\kappa_{0}^{2}(G_{\nu}+c_{0}^{-2})^{3}\quad\forall{\bm{u}}\in\mathcal{A}_{\nu},$
(2.8)

where $c_{0}$ is the constant given below in (2.16), and
$c_{2}=2137\,c_{0}^{4}$. The proof of (2.7) can be found in any of the
references listed above ([6, 17, 18]), and the proof of (2.8) is given in [10,
Lemma 4.4].

In particular, we will make use of the following fact wich follows directly
from (2.7) and (2.8): if ${\bm{u}}\in\mathcal{A}_{\nu}$, then ${\bm{u}}\in
C_{b}(H)$ and ${\bm{u}}\in C_{b}(V)$.

###  2.3. Data Assimilation via Nudging

The data assimilation problem we consider is defined as follows: given
measurements of a reference solution of (2.3) on the attractor, and a guess
for the fluid viscosity, contruct an approximation of the solution. This kind
of problem is typically known as data assimilation, but can also be thought of
as an inverse problem (mapping measurements of the velocity field back to the
full velocity field) or a regression problem (interpolating the measurements
in a way that generalizes to the entire space and time domain).

There are many approaches to solving data assimilation problems. Extensions of
the Kalman filter for nonlinear problems, like the extended Kalman filter, or
ensemble Kalman filter are common, as well as 3DVAR and 4DVAR, or more
recently, machine learning approaches such as Physics Informed Neural Networks
[14, 15]. The approach we consider, often called nudging, benefits from a
rigorous mathematical framework for establishing convergence results after
being studied by Azouani, Olsen and Titi in 2014 [2].

The idea can be simply stated as follows: given data collected on a variable,
$y$, and a method of interpolating the data, putting them together, we have an
interpolant operator $I_{h}$, where $h$ is a measure of the resolution of the
data, and $I_{h}(y)$ is the interpolation of the data. If $y$ satisfies a
differential equation $\dot{y}=F(y)$, then the nudging algorithm is to solve

$\dot{\tilde{y}}=F(y)+\mu\left(I_{h}(y)-I_{h}(\tilde{y})\right)$

for an approximation, $\tilde{y}$. The algorithm is succesful when $\tilde{y}$
converges to $y$.

In [2], the authors prove results for interpolant operators satisfying one of
two approximation criteria. The first kind of interpolant operator (and most
strict) has the following property:

######  Definition 2.3 (Type 1 Interpolant Operator).

$|{\bm{\phi}}-I_{h}({\bm{\phi}})|\leq
c_{1}h|\nabla{\bm{\phi}}|\quad\forall{\bm{\phi}}\in(\dot{H}^{1}(\Omega))^{2}$

In this work, we focus on the case where the measurements are exact values for
finitely many spectral coefficients, which are then interpolated to give
approximations in the range of the modal projection, $P_{N}.$ Specifically,
for any $N\geq 1$, $P_{N}$ is defined by:

$\widehat{P_{N}({\bm{\phi}})}_{\bm{k}}=\begin{cases}\widehat{{\bm{\phi}}}_{\bm{k}},\quad|{\bm{k}}|<N\\\
0,\quad|{\bm{k}}|\geq N\end{cases}\quad\forall{\bm{k}}\in\mathbb{Z}^{2}.$

The modal projection is an example of a Type 1 interpolant operator, as can
easily be verified.

$|{\bm{\phi}}-P_{N}{\bm{\phi}}|^{2}\leq\frac{1}{\kappa_{0}^{2}}\frac{1}{N^{2}}\|{\bm{\phi}}\|^{2}$
(2.9)

In particular, for any ${\bm{\phi}}\in H$ (which by the mean-free condition
implies $P_{1}{{\bm{\phi}}}=0$) we have the Poincaré inequality

$|{\bm{\phi}}|=|{\bm{\phi}}-P_{1}{{\bm{\phi}}}|\leq\frac{1}{\kappa_{0}}\|{\bm{\phi}}\|.$
(2.10)

We explicitly state the nudging data assimilation algorithm for our current
setting in System 2.11.

######  System 2.4 (Navier–Stokes with data feedback).

Given ${\bm{\phi}}\in C_{b}(V)$ (the data) and an approximate viscosity,
$\gamma$, find ${\bm{v}}\in C_{b}(V)$ satisfying

$\frac{d}{dt}{\bm{v}}+\gamma
A{\bm{v}}+B\left({\bm{v}},{\bm{v}}\right)=\bm{f}+\mu({\bm{\phi}}-P_{N}{\bm{v}}).$
(2.11)

When solving the data assimilation problem, we typically have
${\bm{\phi}}=P_{N}{\bm{u}}$, where ${\bm{u}}\in\mathcal{A}_{\nu}$. However,
(2.11) remains valid in the general setting with ${\bm{\phi}}\in C_{b}(V)$,
and is the basis for the definition of the determining map in Section 3.

###  2.4. Standard Inequalities

We now review several standard inequalities which are used throughout the
subsequent sections. We often use Young’s inequality to establish estimates,

$a^{p}b^{1-p}\leq pa+(1-p)b,\quad\forall a,b\geq 0,\;p\in[0,1].$ (2.12)

In particular, we frequently use Young’s inequality in the following form:

$ab\leq\frac{1}{2\epsilon}a^{2}+\frac{\epsilon}{2}b^{2},\quad\forall a,b\geq
0,\;\epsilon>0.$ (2.13)

We also make use of Jensen’s inequality,

$\left(\frac{1}{L}\int_{0}^{L}{\bm{\phi}}(x)dx\right)^{2}\leq\frac{1}{L}\int_{0}^{L}{\bm{\phi}}^{2}(x)dx$
(2.14)

We use Ladyzhenskaya’s inequality (for periodic boundary conditions) in the
following form to obtain estimates of the nonlinear term in (2.11) (a proof is
provided in the Appendix):

$\|{\bm{\phi}}\|_{L^{4}(\Omega)}\leq|{\bm{\phi}}|^{\frac{1}{2}}\left(\tfrac{1}{L}|{\bm{\phi}}|+\|{\bm{\phi}}\|\right)^{\frac{1}{2}}$
(2.15)

######  Remark 2.5.

Note that, using (2.10), we can rewrite (2.15) in the more usual form

$\|{\bm{\phi}}\|_{L^{4}(\Omega)}\leq
c_{0}|{\bm{\phi}}|^{\frac{1}{2}}\|{\bm{\phi}}\|^{\frac{1}{2}}$ (2.16)

where $c_{0}=\sqrt{1+\tfrac{1}{2\pi}}$.

Using (2.15), we have the following estimate:

$|b\left({\bm{u}},{\bm{v}},{\bm{w}}\right)|\leq\|{\bm{u}}\|_{L^{4}(\Omega)}|\nabla{\bm{v}}|\|{\bm{w}}\|_{L^{4}(\Omega)}\\\
\leq|{\bm{u}}|^{\frac{1}{2}}\left(\tfrac{1}{L}|{\bm{u}}|+\|{\bm{u}}\|\right)^{\frac{1}{2}}\|{\bm{v}}\||{\bm{w}}|^{\frac{1}{2}}\left(\tfrac{1}{L}|{\bm{w}}|+\|{\bm{w}}\|\right)^{\frac{1}{2}}$
(2.17)

where
$b\left({\bm{u}},{\bm{v}},{\bm{w}}\right):=\left(B\left({\bm{u}},{\bm{v}}\right),{{\bm{w}}}\right)$.

##  3\. Determining Map

The determining map was first defined in [8, 9] as the mapping of data to the
corresponding solution of (2.11). It was exploited in [3] for statistical data
assimilation. We extend the definition of the determining map to include
viscosity as an input and establish its Lipschitz property in all its
arguments, which plays a pivotal role in establishing the results in Section 4
and Section 7.

######  Definition 3.1 (Determining Map).

Fix an arbitrary lower bound $\nu_{0}>0$ for the viscosity, and fix an
arbitrary radius $R>0$ for a ball $B_{R}(0)\subset C_{b}(V)$. The determining
map, $\mathbb{W}:(\nu_{0},\infty)\times B_{R}(0)\to C_{b}(V)$, is the mapping
of viscosity and data to the corresponding solution of (2.11) on the
attractor.

For example, $\mathbb{W}(\tilde{\nu},P_{N}{\bm{u}})$ is the solution of (2.11)
with $\gamma=\tilde{\nu}$ and ${\bm{\phi}}=P_{N}{\bm{u}}$. However, we don’t
in general require that the data ${\bm{\phi}}$ come from a solution of (2.3).

We begin by deriving sufficient conditions for the determining map to be a
well-defined Lipschitz continuous mapping. These conditions are placed on
$\mu$ and $N$. Note that in contrast to the usual analysis taken wherein $N$
depends on $\mu$, we derive our results in such a way that the lower bound for
$N$ comes directly from the data, forcing, and $\nu_{0}$, and $\mu$ is chosen
to be larger than a lower bound which grows with $N^{2}$. Specifically, we
parameterize $\mu$ in terms of a (non-dimensional) constant $\mu_{0}>0$ as
follows:

$\mu=\mu_{0}\nu_{0}\kappa_{0}^{2}N^{2}.$ (3.1)

######  Lemma 3.2.

Fix $\nu_{0}\in\mathbb{R}_{+}$ and suppose that

$\mu_{0}\geq 1\quad\text{i.e.}\quad\mu\geq\nu_{0}\kappa_{0}^{2}N^{2}.$ (3.2)

For any $\bm{f},{\bm{\phi}}\in C_{b}(V),$ and for any $1\leq N\in\mathbb{R},$
if

$\gamma\in[\nu_{0},\infty)$

and ${\bm{v}}$ is a corresponding solution of (2.11), then the following
bounds are satisfied:

$\|{\bm{v}}\|_{C_{b}(H)}^{2}\leq\frac{2}{(\nu_{0}\kappa_{0}^{2}N^{2})^{2}}\|\bm{f}\|_{C_{b}(H)}^{2}+2\mu_{0}^{2}\|{\bm{\phi}}\|_{C_{b}(H)}^{2}\leq(M_{H}({\bm{\phi}}))^{2},$
(3.3)

and

$\|{\bm{v}}\|_{C_{b}(V)}^{2}\leq\frac{2}{(\nu_{0}\kappa_{0}^{2}N^{2})^{2}}\|\bm{f}\|_{C_{b}(V)}^{2}+2\mu_{0}^{2}\|{\bm{\phi}}\|_{C_{b}(V)}^{2}\leq(M_{V}({\bm{\phi}}))^{2}.$
(3.4)

Here, given fixed $\bm{f}\in C_{b}(V),$ $\nu_{0},\kappa_{0}>0,$ and
$\mu_{0}\geq 1$, we define

$M_{H}({\bm{\phi}})=\sqrt{2}\sqrt{\frac{1}{(\nu_{0}\kappa_{0}^{2})^{2}}\|\bm{f}\|_{C_{b}(H)}^{2}+\mu_{0}^{2}\|{\bm{\phi}}\|_{C_{b}(H)}^{2}}$
(3.5)

and

$M_{V}({\bm{\phi}})=\sqrt{2}\sqrt{\frac{1}{(\nu_{0}\kappa_{0}^{2})^{2}}\|\bm{f}\|_{C_{b}(V)}^{2}+\mu_{0}^{2}\|{\bm{\phi}}\|_{C_{b}(V)}^{2}}.$
(3.6)

###### Proof.

Let ${\bm{v}}$ be a solution of (2.11). Then taking the inner product of
(2.11) with ${\bm{v}}$, we have

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\gamma\|{\bm{v}}\|^{2}=\left<\bm{f},{\bm{v}}\right>+\mu\left<{\bm{\phi}},{\bm{v}}\right>-\mu|P_{N}{\bm{v}}|^{2}.$

Therefore, for any choice of $c_{1},c_{2}>0$,

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\gamma\|{\bm{v}}\|^{2}+\mu|P_{N}{\bm{v}}|^{2}\leq|\bm{f}||{\bm{v}}|+\mu|{\bm{\phi}}||{\bm{v}}|\leq\frac{1}{2c_{1}}|\bm{f}|^{2}+\frac{\mu}{2c_{2}}|{\bm{\phi}}|^{2}+\frac{c_{1}+\mu
c_{2}}{2}|{\bm{v}}|^{2}\\\
\leq\frac{1}{2c_{1}}|\bm{f}|^{2}+\frac{\mu}{2c_{2}}|{\bm{\phi}}|^{2}+\frac{c_{1}+\mu
c_{2}}{2}|P_{N}{\bm{v}}|^{2}+\frac{1}{\kappa_{0}^{2}}\frac{c_{1}+\mu
c_{2}}{2N^{2}}\|{\bm{v}}\|^{2},$

where to obtain the last inequality, we used
$|{\bm{v}}|^{2}=|P_{N}{\bm{v}}|^{2}+|(I-P_{N}){\bm{v}}|^{2}$ and (2.9).
Collecting terms, we have

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\left(\gamma-\frac{1}{\kappa_{0}^{2}}\frac{c_{1}+\mu
c_{2}}{2N^{2}}\right)\|{\bm{v}}\|^{2}+\left(\mu-\frac{c_{1}+\mu
c_{2}}{2}\right)|P_{N}{\bm{v}}|^{2}\leq\frac{1}{2c_{1}}|\bm{f}|^{2}+\frac{\mu}{2c_{2}}|{\bm{\phi}}|^{2}.$

Let $\epsilon,\delta\in(0,1),$ and define

$r=\frac{\nu_{0}\kappa_{0}^{2}N^{2}}{\mu}.$

Choosing

$c_{1}=2\mu r\delta(1-\epsilon)\quad\text{ and }\quad c_{2}=2r(1-\delta),$

we have

$c_{1}+\mu c_{2}=2\mu r(1-\delta\epsilon)$

and the previous differential inequality becomes

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\left(\gamma-\nu_{0}(1-\delta\epsilon)\right)\|{\bm{v}}\|^{2}+\left(\mu-\nu_{0}\kappa_{0}^{2}N^{2}(1-\delta\epsilon)\right)|P_{N}{\bm{v}}|^{2}\leq\frac{1}{2c_{1}}|\bm{f}|^{2}+\frac{\mu}{2c_{2}}|{\bm{\phi}}|^{2}.$
(3.7)

Observe that

$\|{\bm{v}}\|^{2}=\|P_{N}{\bm{v}}\|^{2}+\|(I-P_{N}){\bm{v}}\|^{2}\geq\kappa_{0}^{2}N^{2}|(I-P_{N}){\bm{v}}|^{2},$

where to obtain the last inequality we used (2.9). Consequently,

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\nu_{0}\delta\epsilon\|{\bm{v}}\|^{2}+\left(\mu-\nu_{0}\kappa_{0}^{2}N^{2}(1-\delta\epsilon)\right)|P_{N}{\bm{v}}|^{2}\\\
\geq\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\nu_{0}\kappa_{0}^{2}N^{2}\delta\epsilon|{\bm{v}}|^{2}+\left(\mu-\nu_{0}\kappa_{0}^{2}N^{2}\right)|P_{N}{\bm{v}}|^{2}.$
(3.8)

Note that by imposing the constraint $\epsilon,\delta\in(0,1),$ we have
ensured that $\gamma-\nu_{0}(1-\delta\epsilon)\geq\nu_{0}\delta\epsilon>0.$
Using (3.7), (3.8) and (3.2), and by dropping the last term in (3.8), we
obtain

$\frac{d}{dt}|{\bm{v}}|^{2}+\beta|{\bm{v}}|^{2}\leq\frac{1}{c_{1}}|\bm{f}|^{2}+\frac{\mu}{c_{2}}|{\bm{\phi}}|^{2},$

with $\beta:=2\nu_{0}\kappa_{0}^{2}N^{2}\delta\epsilon.$

So, by Gröwall’s inequality,

$|{\bm{v}}(t)|^{2}\leq
e^{-\beta(t-s)}|{\bm{v}}(s)|^{2}+\int_{s}^{t}e^{-\beta(t-\tau)}\left(\frac{1}{c_{1}}|\bm{f}(\tau)|^{2}+\frac{\mu}{c_{2}}|{\bm{\phi}}(\tau)|^{2}\right)d\tau\\\
\leq
e^{-\beta(t-s)}|{\bm{v}}(s)|^{2}+\left(\frac{1}{c_{1}}\|\bm{f}\|_{C_{b}(H)}^{2}+\frac{\mu}{c_{2}}\|{\bm{\phi}}\|_{C_{b}(H)}^{2}\right)\frac{1}{\beta}\left(1-e^{-\beta(t-s)}\right)$

We get the stated result by choosing $\epsilon=\frac{2}{3}$ and
$\delta=\frac{3}{4}$ (so $c_{1}=\frac{1}{2}\nu_{0}\kappa_{0}^{2}N^{2}$ and
$c_{2}=\frac{1}{2}\frac{\nu_{0}\kappa_{0}^{2}N^{2}}{\mu}$), and taking the
limit as $s\to-\infty.$

Taking the inner product of (2.11) with $A{\bm{v}}$ and proceeding similarly,
but this time using the enstrophy cancelation property
$(B({\bm{v}},{\bm{v}}),A{\bm{v}})=0$, we obtain (3.4). ∎

We now derive the main Lipschitz continuity property of the determining map
which is essential to the proofs of the subsequent results.

######  Lemma 3.3.

Let ${\bm{v}}_{1}$ and ${\bm{v}}_{2}$ be solutions of (2.11) with viscosity
$\gamma_{1}\geq\nu_{0}$ and $\gamma_{2}\geq\nu_{0}$ and data ${\bm{\phi}}_{1}$
and ${\bm{\phi}}_{2}$ respectively. Let $p\in[0,1]$ and set

$\alpha=\frac{\gamma_{1}+\gamma_{2}}{2}\left(1-\frac{1}{2}\left(2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}+\gamma_{2}}\right)^{p}\right).$

Then, $\alpha\in\left[\frac{1}{2}\nu_{0},\bar{\gamma}\right]$ where
$\bar{\gamma}:=\frac{1}{2}(\gamma_{1}+\gamma_{2})$. Moreover, if

$\mu_{0}\geq\max\left\\{1,\frac{\alpha}{\nu_{0}}\right\\},$ (3.9)

and

$N\geq\frac{4}{\alpha\kappa_{0}}M_{V}({\bm{\phi}}_{i}),i=1,2.$ (3.10)

then

$\|{\bm{v}}_{1}-{\bm{v}}_{2}\|_{C_{b}(H)}^{2}\leq\frac{5}{2}\frac{|\gamma_{1}-\gamma_{2}|^{2-p}}{\left(\frac{\gamma_{1}+\gamma_{2}}{2}\right)^{1-p}}\frac{\|\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})\|_{C_{b}(V)}^{2}}{\alpha\kappa_{0}^{2}N^{2}}+\frac{5}{2}\mu_{0}^{2}\left(\frac{\nu_{0}}{\alpha}\right)^{2}\|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}\|_{C_{b}(H)}^{2}.$
(3.11)

###### Proof.

Observe that

$\alpha=\bar{\gamma}-\frac{1}{2}\bar{\gamma}^{1-p}|\gamma_{1}-\gamma_{2}|^{p}=\bar{\gamma}\left(1-\frac{1}{2}\left(2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}+\gamma_{2}}\right)^{p}\right),$

and note that for all $p\in[0,1],$

$\bar{\gamma}\geq\alpha\geq\bar{\gamma}\left(1-\frac{1}{2}\max\left\\{1,2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}+\gamma_{2}}\right\\}\right)=\min\left\\{\frac{1}{2}\bar{\gamma},\min\left\\{\gamma_{1},\gamma_{2}\right\\}\right\\}\geq\frac{1}{2}\nu_{0}.$
(3.12)

Thus, $\alpha\in\left[\frac{1}{2}\nu_{0},\bar{\gamma}\right]$

Now let ${\bm{v}}_{1}$ and ${\bm{v}}_{2}$ be solutions of (2.11) with
viscosity $\gamma_{1}$ and $\gamma_{2}$ and data ${\bm{\phi}}_{1}$ and
${\bm{\phi}}_{2}$ respectively. We now analyze the difference
${\bm{w}}:={\bm{v}}_{1}-{\bm{v}}_{2},$ and for convenience of presentation, we
define the average quantities
$\bar{{\bm{v}}}=\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})$ and
$\bar{\gamma}=\frac{1}{2}(\gamma_{1}+\gamma_{2}).$

Writing the evolution equation for ${\bm{w}}$ using (2.11),

$\partial_{t}{\bm{w}}+\bar{\gamma}A{\bm{w}}+(\gamma_{1}-\gamma_{2})A\bar{{\bm{v}}}+B\left(\bar{{\bm{v}}},{\bm{w}}\right)+B\left({\bm{w}},\bar{{\bm{v}}}\right)=\mu({\bm{\phi}}_{1}-{\bm{\phi}}_{2})-\mu
P_{N}{\bm{w}}$

and taking the inner product with ${\bm{w}}$ and using (2.17), we have

$\frac{1}{2}\frac{d}{dt}|{\bm{w}}|^{2}+\bar{\gamma}\|{\bm{w}}\|^{2}+\mu|P_{N}{\bm{w}}|^{2}=-(\gamma_{1}-\gamma_{2})\left<A\bar{{\bm{v}}},{\bm{w}}\right>+\mu\left<({\bm{\phi}}_{1}-{\bm{\phi}}_{2}),{\bm{w}}\right>-b\left({\bm{w}},\bar{{\bm{v}}},{\bm{w}}\right)\\\
\leq|\gamma_{1}-\gamma_{2}|\|\bar{{\bm{v}}}\|\|{\bm{w}}\|+\mu|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}||{\bm{w}}|+\|\bar{{\bm{v}}}\||{\bm{w}}|(\tfrac{1}{L}|{\bm{w}}|+\|{\bm{w}}\|)$

Using (2.13), for any $p\in[0,1]$, we can write

$|\gamma_{1}-\gamma_{2}|\|\bar{{\bm{v}}}\|\|{\bm{w}}\|\leq\frac{1}{2}\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\frac{1}{2}\bar{\gamma}^{1-p}|\gamma_{1}-\gamma_{2}|^{p}\|{\bm{w}}\|^{2},$

therefore,

$\frac{1}{2}\frac{d}{dt}|{\bm{w}}|^{2}+\left(\bar{\gamma}-\frac{1}{2}\bar{\gamma}^{1-p}|\gamma_{1}-\gamma_{2}|^{p}\right)\|{\bm{w}}\|^{2}+\mu|P_{N}{\bm{w}}|^{2}\\\
\leq\frac{1}{2}\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\mu|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}||{\bm{w}}|+\|\bar{{\bm{v}}}\||{\bm{w}}|(\tfrac{1}{L}|{\bm{w}}|+\|{\bm{w}}\|).$

For the remaining terms, using (2.13) and (2.9), for any $c_{1}>0$ we have

$\mu|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}||{\bm{w}}|\leq\frac{\mu}{2c_{1}}|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}|^{2}+\frac{\mu
c_{1}}{2\kappa_{0}^{2}N^{2}}\|{\bm{w}}\|^{2}+\frac{\mu
c_{1}}{2}|P_{N}{\bm{w}}|^{2},$

and

$\|\bar{{\bm{v}}}\||{\bm{w}}|(\tfrac{1}{L}|{\bm{w}}|+\|{\bm{w}}\|)\\\
\leq\left(\frac{1}{\kappa_{0}^{2}N^{2}L}+\frac{1}{\kappa_{0}N}\right)\|\bar{{\bm{v}}}\|\|{\bm{w}}\|^{2}+\left(\frac{1}{L}+\frac{\kappa_{0}N}{2}\right)\|\bar{{\bm{v}}}\||P_{N}{\bm{w}}|^{2}.$

Therefore,

$\frac{1}{2}\frac{d}{dt}|{\bm{w}}|^{2}+\left(\alpha-\frac{\mu
c_{1}}{2\kappa_{0}^{2}N^{2}}-\left(\frac{1}{\kappa_{0}^{2}N^{2}L}+\frac{1}{\kappa_{0}N}\right)\|\bar{{\bm{v}}}\|\right)\|{\bm{w}}\|^{2}\\\
+\left(\mu-\frac{\mu
c_{1}}{2}-\left(\frac{1}{L}+\frac{\kappa_{0}N}{2}\right)\|\bar{{\bm{v}}}\|\right)|P_{N}{\bm{w}}|^{2}\\\
\leq\frac{1}{2}\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\frac{\mu}{2c_{1}}|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}|^{2}$

and after choosing $c_{1}=\alpha\frac{\kappa_{0}^{2}N^{2}}{\mu}$ and
simplifying,

$\frac{1}{2}\frac{d}{dt}|{\bm{w}}|^{2}+\left(\frac{1}{2}\alpha-\left(1+\frac{1}{2\pi
N}\right)\frac{\|\bar{{\bm{v}}}\|}{\kappa_{0}N}\right)\|{\bm{w}}\|^{2}\\\
+\left(\mu-\frac{1}{2}\alpha\kappa_{0}^{2}N^{2}-\frac{1}{L}\left(1+\pi
N\right)\|\bar{{\bm{v}}}\|\right)|P_{N}{\bm{w}}|^{2}\\\
\leq\frac{1}{2}\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\frac{\mu^{2}}{2\alpha\kappa_{0}^{2}N^{2}}|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}|^{2}.$

Now, by (3.10) and (3.4), we have

$N\geq\frac{4}{\alpha\kappa_{0}}\|\bar{{\bm{v}}}\|,$ (3.13)

so,

$\frac{1}{2}\alpha-\left(1+\frac{1}{2\pi
N}\right)\frac{\|\bar{{\bm{v}}}\|}{\kappa_{0}N}\geq\frac{1}{2}\alpha-\left(1+\frac{1}{2\pi
N}\right)\frac{\alpha}{4}=\frac{1}{4}\alpha\left(1-\frac{1}{2\pi
N}\right)>\frac{1}{5}\alpha>0.$ (3.14)

Therefore, we can apply (2.9), obtaining

$\frac{1}{2}\frac{d}{dt}|{\bm{w}}|^{2}+\left(\frac{1}{2}\alpha-\left(1+\frac{1}{2\pi
N}\right)\frac{\|\bar{{\bm{v}}}\|}{\kappa_{0}N}\right)\kappa_{0}^{2}N^{2}|{\bm{w}}|^{2}\\\
+\left(\mu-\alpha\kappa_{0}^{2}N^{2}+\frac{1}{2}\kappa_{0}N\|\bar{{\bm{v}}}\|\right)|P_{N}{\bm{w}}|^{2}\\\
\leq\frac{1}{2}\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\frac{\mu^{2}}{2\alpha\kappa_{0}^{2}N^{2}}|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}|^{2}.$

By assumption (3.9) we can drop the third term. Then, after simplifying and
applying the estimate (3.14) to the second term on the left hand side, we have

$\frac{d}{dt}|{\bm{w}}|^{2}+\frac{2}{5}\alpha\kappa_{0}^{2}N^{2}|{\bm{w}}|^{2}\\\
\leq\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|^{2}+\frac{\mu^{2}}{\alpha\kappa_{0}^{2}N^{2}}|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}|^{2}.$

From here, for any $-\infty<s<t$ we can apply Grönwall’s inequality,

$|{\bm{w}}(t)|^{2}\leq
e^{-\frac{2}{5}\alpha\kappa_{0}^{2}N^{2}(t-s)}|{\bm{w}}(s)|^{2}\\\
+\bar{\gamma}^{p-1}|\gamma_{1}-\gamma_{2}|^{2-p}\|\bar{{\bm{v}}}\|_{C_{b}(V)}^{2}\frac{5}{2}\frac{1}{\alpha\kappa_{0}^{2}N^{2}}\left(1-e^{-\frac{2}{5}\alpha\kappa_{0}^{2}N^{2}(t-s)}\right)\\\
+\frac{\mu^{2}}{\alpha\kappa_{0}^{2}N^{2}}\|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}\|_{C_{b}(H)}^{2}\frac{5}{2}\frac{1}{\alpha\kappa_{0}^{2}N^{2}}\left(1-e^{-\frac{2}{5}\alpha\kappa_{0}^{2}N^{2}(t-s)}\right)$

and taking the limit as $s\to-\infty$ we get the stated result. ∎

In the previous lemma, the lower bound for $N$ depends on the value of
$\alpha$, which depends on the viscosities. If we impose an upper bound,
$\nu_{1}$, on the viscosities in addition to the lower bound, $\nu_{0}$, we
can easily replace this dependence to be on $\nu_{1}$ and $\nu_{0}$.

######  Corollary 3.4.

Let ${\bm{v}}_{1}$ and ${\bm{v}}_{2}$ be solutions of (2.11) with viscosity
$\gamma_{1}$ and $\gamma_{2}$ and data ${\bm{\phi}}_{1}$ and ${\bm{\phi}}_{2}$
respectively, and suppose that $\gamma_{1},\gamma_{2}\in[\nu_{0},\nu_{1}]$,
where $\nu_{1}>\nu_{0}$. If

$\mu_{0}\geq\frac{\nu_{1}}{\nu_{0}},$ (3.15)

and

$N\geq\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{\phi}}_{i})$ (3.16)

for $i=1,2,$ then

$\|{\bm{v}}_{1}-{\bm{v}}_{2}\|_{C_{b}(H)}^{2}\leq
5\frac{|\gamma_{1}-\gamma_{2}|^{2-p}}{\left(\frac{\gamma_{1}+\gamma_{2}}{2}\right)^{1-p}}\frac{\|\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})\|_{C_{b}(V)}^{2}}{\nu_{0}\kappa_{0}^{2}N^{2}}+10\mu_{0}^{2}\|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}\|_{C_{b}(H)}^{2}.$

###### Proof.

Substituting for $\alpha$ using the bounds
$\frac{\nu_{0}}{2}\leq\alpha\leq\frac{\gamma_{1}+\gamma_{2}}{2}\leq\nu_{1}$
proves the corollary. ∎

We can also allow for $\gamma_{1}$ and $\gamma_{2}$ to be arbitrarily large,
as long as their relative difference is smaller than $1$.

######  Corollary 3.5.

Let ${\bm{v}}_{1}$ and ${\bm{v}}_{2}$ be solutions of (2.11) with viscosity
$\gamma_{1}\geq\nu_{0}$ and $\gamma_{2}\geq\nu_{0}$ and data ${\bm{\phi}}_{1}$
and ${\bm{\phi}}_{2}$ respectively. If

$\mu_{0}\geq 1,$ (3.17)

and

$N\geq\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{\phi}}_{i})$ (3.18)

for $i=1,2,$ and if

$\frac{|\gamma_{1}-\gamma_{2}|}{\frac{1}{2}(\gamma_{1}+\gamma_{2})}\leq 1,$

then

$\|{\bm{v}}_{1}-{\bm{v}}_{2}\|_{C_{b}(H)}^{2}\leq
5\left(\frac{|\gamma_{1}-\gamma_{2}|}{\frac{\gamma_{1}+\gamma_{2}}{2}}\right)^{2-p}\frac{\|\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})\|_{C_{b}(V)}^{2}}{\kappa_{0}^{2}N^{2}}+10\mu_{0}^{2}\|{\bm{\phi}}_{1}-{\bm{\phi}}_{2}\|_{C_{b}(H)}^{2}.$

###### Proof.

Using the same notation as before, we have by assumption
$\frac{|\gamma_{1}-\gamma_{2}|}{\bar{\gamma}}\leq 1,$ so
$\frac{1}{2}\nu_{0}\leq\alpha\leq\nu_{0},$ hence (3.9) and (3.10) are
satisfied by (3.17) and (3.18). Also,

$\alpha\geq\bar{\gamma}\left(1-\frac{1}{2}\max\left\\{1,2\frac{|\gamma_{1}-\gamma_{2}|}{\gamma_{1}+\gamma_{2}}\right\\}\right)=\frac{1}{2}\bar{\gamma}.$

Substituting this lower bound for $\alpha$ in (3.11) along with the more
general lower bound, $\alpha>\frac{1}{2}\nu_{0}$, gives the result. ∎

We can now show the well-posedness of the determining map.

######  Theorem 3.6.

Let $R>0$ and $\nu_{0}>0$, and let $\bm{f}\in C_{b}(V).$ If

$N\geq\frac{8}{\nu_{0}\kappa_{0}}\sqrt{2}\left(\frac{1}{\nu_{0}\kappa_{0}^{2}N^{2}}\|\bm{f}\|_{C_{b}(V)}+\mu_{0}R\right),$
(3.19)

for any $\mu_{0}\geq 1$ with $\mu$ given by (3.1), then $\mathbb{W}$ as
defined by Definition 3.1 is well-defined.

###### Proof.

The existence of solutions on the attractor for any given pair
$(\gamma,{\bm{\phi}})$ on the domain of $\mathbb{W}$ can be shown by first
taking a Galerkin truncation and proceeding to the limit as is done in the
classical case for the 2D Navier–Stokes, and so we omit the details here.

The uniqueness of solutions follows from Corollary 3.5: for any
$(\gamma,{\bm{\phi}})\in(\nu_{0},\infty)\times B_{R}(0)$, note that the
conditions of the corollary are satisfied. Therefore, given two solutions
${\bm{v}}_{1}$ and ${\bm{v}}_{2}$ of (2.11) corresponding to $\gamma$ and
${\bm{\phi}}$, by Corollary 3.5 we have

$\|{\bm{v}}_{1}-{\bm{v}}_{2}\|_{C_{b}(H)}=0,$

therefore, by the embedding $V\subset H$,

$\|{\bm{v}}_{1}-{\bm{v}}_{2}\|_{C_{b}(V)}=0.$

∎

######  Remark 3.7.

Theorem 3.6 also provides an alternate proof of the classical result that
there are finitely many determining modes for the Navier–Stokes equations. To
see that this is true, suppose ${\bm{u}}$ and ${\bm{v}}$ are two solutions of
(2.3) in $C_{b}(V)$ with viscosity $\nu$, which agree at low modes, i.e.
$P_{N}{\bm{u}}=P_{N}{\bm{v}}$ for all $t\in\mathbb{R}$. If $N$ satisfies the
conditions of Theorem 3.6 with $\nu_{0}=\nu,$
$R=\max\\{\|{\bm{u}}\|_{C_{b}(V)},\|{\bm{v}}\|_{C_{b}(V)}\\},$ and
$\mu_{0}=1$, then

${\bm{u}}=\mathbb{W}(\nu,P_{N}{\bm{u}})=\mathbb{W}(\nu,P_{N}{\bm{v}})={\bm{v}}.$

##  4\. The Parameter Recovery Inverse Problem

Our goal in this section is to define and study the following optimization
problem.

######  Problem 4.1.

Let $P_{N}{\bm{u}}$ be given on $\mathbb{R}$. Define the cost functional,
$\mathcal{L}$, by

$\mathcal{L}(\gamma)=\|P_{N}\mathbb{W}(\gamma,P_{N}{\bm{u}})-P_{N}{\bm{u}}\|_{C_{b}(H)}.$
(4.1)

Find $\gamma^{*}>0$ such that

$\mathcal{L}(\gamma^{*})=\min_{\gamma>0}\mathcal{L}(\gamma).$

As motivation, consider the following:

######  Fact 4.2.

If ${\bm{u}}\in C_{b}(V)$ is a solution of (2.3) with viscosity $\nu$, then

$\min_{\gamma>0}\|\mathbb{W}(\gamma,P_{N}{\bm{u}})-{\bm{u}}\|_{C_{b}(H)}$

has a unique solution, namely $\nu$, provided that $\mu$ and $N$ satisfy the
conditions of Theorem 3.6 with $R=\|{\bm{u}}\|_{C_{b}(V)},$ and that
$\bm{f}\not\equiv 0$.

###### Proof.

Choosing $\gamma=\nu$ makes the cost exactly $0$, as ${\bm{u}}$ solves (2.11)
when $\gamma=\nu$, and therefore, by Theorem 3.6,
$\mathbb{W}(\nu,P_{N}{\bm{u}})={\bm{u}}$. To see that this solution is unique,
suppose that ${\bm{v}}:=\mathbb{W}(\gamma,P_{N}{\bm{u}})$ satisfies
$\|{\bm{v}}-{\bm{u}}\|_{C_{b}(H)}=0.$ Then

$\frac{d}{dt}{\bm{v}}+\gamma
A{\bm{v}}+B\left({\bm{v}},{\bm{v}}\right)=\bm{f}+P_{N}{\bm{u}}-P_{N}{\bm{v}}.$

Taking an inner-product with ${\bm{v}}$, we obtain

$\frac{1}{2}\frac{d}{dt}|{\bm{v}}|^{2}+\gamma\|{\bm{v}}\|^{2}=\left<\bm{f},{\bm{v}}\right>+\left<P_{N}{\bm{u}},{\bm{v}}\right>-|P_{N}{\bm{v}}|^{2},$

whereas, by the definition of ${\bm{u}}$ we have

$\frac{d}{dt}{\bm{u}}+\nu A{\bm{u}}+B\left({\bm{u}},{\bm{u}}\right)=\bm{f},$

and taking an inner-product with ${\bm{u}}$, we get

$\frac{1}{2}\frac{d}{dt}|{\bm{u}}|^{2}+\nu\|{\bm{u}}\|^{2}=\left<\bm{f},{\bm{u}}\right>.$

Now, by assumption, $|{\bm{v}}-{\bm{u}}|=0$ for each $t$, so
$\left<\bm{f},{\bm{v}}\right>=\left<\bm{f},{\bm{u}}\right>$,
$\left<P_{N}{\bm{u}},{\bm{v}}\right>=|P_{N}{\bm{v}}|^{2}$, and
$\|{\bm{u}}\|=\|{\bm{v}}\|.$ Therefore, taking the difference of the two
energy equations, we see that for all $t\in(-\infty,\infty)$,

$(\gamma-\nu)\|{\bm{u}}\|=0.$

This can only hold if ${\bm{u}}\equiv 0$ or if $\gamma=\nu$. However,
${\bm{u}}\equiv 0$ only if $\bm{f}\equiv 0$. Therefore $\gamma=\nu$. ∎

Note that the minimization problem presented in Fact 4.2 is not useful,
because to calculate the cost we would need to know ${\bm{u}}$ exactly. In
which case, rather than solving the minimization problem using
$\mathbb{W}(\cdot,\cdot)$, we could compute $\nu$ directly by averaging the
energy equation over $[-T,T]$ for any $T>0$:

$\nu=\frac{\int_{-T}^{T}\left<\bm{f},{\bm{u}}\right>-\left(|{\bm{u}}(T)|^{2}-|{\bm{u}}(-T)|^{2}\right)}{\int_{-T}^{T}\|{\bm{u}}\|^{2}}.$

The point of posing Problem 4.1 is to obtain a similar result while requiring
a minimal amount of data.

##  5\. Uniqueness of the Inverse Problem

Given the same data, the Lipschitz properties we proved in Section 3 show that
closeness in the viscosity implies smallness of the cost function,
$\mathcal{L}$. Now that we are considering the inverse problem, we are
interested in the converse: when does smallness in the cost function imply
closeness to the true viscosity? We give conditions for the converse to be
true in Section 6, but first, we directly address the simpler question of
uniqueness of the solution, $\gamma=\nu$, to Problem 4.1.

In the following lemma, we provide an inequality bounding the difference
between two viscosities which, given the same data ${\bm{\phi}}$, map (via
$\mathbb{W}$) to trajectories with identical projections on $range(P_{N}).$ We
leverage this result to show the uniqueness of solutions of Problem 4.1 in the
subsequent theorem.

######  Lemma 5.1.

Let $\bm{f},{\bm{\phi}}\in C_{b}(V)$ and
$\gamma_{1},\gamma_{2}\in[\nu_{0},\nu_{1}]$, and fix
$\mu_{0}\geq\frac{\nu_{1}}{\nu_{0}}$ and
$N\geq\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{\phi}}).$ If

$\|P_{N}\mathbb{W}(\gamma_{1},{\bm{\phi}})-P_{N}\mathbb{W}(\gamma_{2},{\bm{\phi}})\|_{C_{b}(H)}=0,$
(5.1)

then either $|\gamma_{1}-\gamma_{2}|=0,$ or for any $p\in[0,1]$,

$\left(\frac{|\gamma_{1}-\gamma_{2}|}{\frac{\gamma_{1}+\gamma_{2}}{2}}\right)^{p/2}\|{\bm{\psi}}\|_{C_{b}(H)}\,N\leq\frac{2\sqrt{5c}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\sqrt{\ln(N+1)}M_{H}({\bm{\phi}})M_{V}({\bm{\phi}}),$
(5.2)

where
${\bm{\psi}}:=P_{N}\mathbb{W}(\gamma_{1},{\bm{\phi}})=P_{N}\mathbb{W}(\gamma_{2},{\bm{\phi}}).$

###### Proof.

Let ${\bm{v}}_{1}=\mathbb{W}(\gamma_{1},{\bm{\phi}})$ and
${\bm{v}}_{2}=\mathbb{W}(\gamma_{2},{\bm{\phi}})$, and assume that
$P_{N}{\bm{v}}_{1}=P_{N}{\bm{v}}_{2}=:{\bm{\psi}}$. Let
${\bm{w}}={\bm{v}}_{1}-{\bm{v}}_{2}$ (note that $P_{N}{\bm{w}}$ is therefore
$0$). Writing (2.11) in Fourier space, for any ${\bm{k}}\in\mathbb{Z}^{2}$,
$|{\bm{k}}|\leq N$, we have

$\frac{d}{dt}\widehat{{\bm{v}}_{1}}({\bm{k}})+\gamma_{1}\kappa_{0}^{2}|{\bm{k}}|^{2}\widehat{{\bm{v}}_{1}}({\bm{k}})+i\kappa_{0}P_{\sigma}\sum_{{\bm{h}}}({\bm{k}}\cdot\widehat{{\bm{v}}_{1}}({\bm{h}}))\widehat{{\bm{v}}_{1}}({\bm{k}}-{\bm{h}})=\widehat{\bm{f}}({\bm{k}})+\mu\widehat{{\bm{\phi}}}({\bm{k}})-\mu\widehat{{\bm{v}}_{1}}({\bm{k}})$

and

$\frac{d}{dt}\widehat{{\bm{v}}_{2}}({\bm{k}})+\gamma_{2}\kappa_{0}^{2}|{\bm{k}}|^{2}\widehat{{\bm{v}}_{2}}({\bm{k}})+i\kappa_{0}P_{\sigma}\sum_{{\bm{h}}}({\bm{k}}\cdot\widehat{{\bm{v}}_{2}}({\bm{h}}))\widehat{{\bm{v}}_{2}}({\bm{k}}-{\bm{h}})=\widehat{\bm{f}}({\bm{k}})+\mu\widehat{{\bm{\phi}}}({\bm{k}})-\mu\widehat{{\bm{v}}_{2}}({\bm{k}}).$

Subtracting these equations and using the fact that
$\widehat{{\bm{v}}_{1}}({\bm{k}})=\widehat{{\bm{v}}_{2}}({\bm{k}})=\widehat{{\bm{\psi}}}({\bm{k}})$
and $\widehat{{\bm{w}}}({\bm{k}})=0$ for $|{\bm{k}}|\leq N$, we find

$(\gamma_{1}-\gamma_{2})\kappa_{0}^{2}|{\bm{k}}|^{2}\widehat{{\bm{\psi}}}({\bm{k}})+i\kappa_{0}P_{\sigma}\sum_{{\bm{h}}}({\bm{k}}\cdot\widehat{{\bm{v}}_{1}}({\bm{h}}))\widehat{{\bm{w}}}({\bm{k}}-{\bm{h}})+i\kappa_{0}P_{\sigma}\sum_{{\bm{h}}}({\bm{k}}\cdot\widehat{{\bm{w}}}({\bm{h}}))\widehat{{\bm{v}}_{2}}({\bm{k}}-{\bm{h}})=0.$

Next we estimate the convolutions. First, we may write

$|\gamma_{1}-\gamma_{2}|\kappa_{0}|{\bm{k}}|^{2}|\widehat{{\bm{\psi}}}({\bm{k}})|\leq\sum_{{\bm{h}}}|{\bm{k}}||\widehat{{\bm{v}}_{1}}({\bm{h}})||\widehat{{\bm{w}}}({\bm{k}}-{\bm{h}})|+\sum_{{\bm{h}}}|{\bm{k}}||\widehat{{\bm{v}}_{2}}({\bm{k}}-{\bm{h}})||\widehat{{\bm{w}}}({\bm{h}})|.$

Then, applying the Cauchy-Schwarz inequality and Parseval’s theorem, we obtain

$|\gamma_{1}-\gamma_{2}|\kappa_{0}|{\bm{k}}|^{2}|\widehat{{\bm{\psi}}}({\bm{k}})|\leq|{\bm{k}}|\frac{1}{|\Omega|}|{\bm{v}}_{1}||{\bm{w}}|+|{\bm{k}}|\frac{1}{|\Omega|}|{\bm{v}}_{2}||{\bm{w}}|.$

Now, for any ${\bm{k}}\neq 0$, we can multiply both sides by
$\frac{|\widehat{{\bm{\psi}}}({\bm{k}})|}{|{\bm{k}}|^{2}}|\Omega|$,

$|\gamma_{1}-\gamma_{2}|\kappa_{0}|\widehat{{\bm{\psi}}}({\bm{k}})|^{2}|\Omega|\leq\frac{1}{|{\bm{k}}|}|\widehat{{\bm{\psi}}}({\bm{k}})|(|{\bm{v}}_{1}|+|{\bm{v}}_{2}|)|{\bm{w}}|,$

and taking the sum over ${\bm{k}}\in\mathbb{Z}^{2},0<|{\bm{k}}|\leq N,$ and
using the Cauchy-Schwarz inequality,

$|\gamma_{1}-\gamma_{2}|\kappa_{0}|P_{N}{\bm{\psi}}|^{2}\leq\left(\sum_{0<|{\bm{k}}|\leq
N}\frac{1}{|{\bm{k}}|^{2}}\right)^{\frac{1}{2}}\frac{1}{\sqrt{|\Omega|}}|P_{N}{\bm{\psi}}|(|{\bm{v}}_{1}|+|{\bm{v}}_{2}|)|{\bm{w}}|.$

Using the estimate

$\sum_{0<|{\bm{k}}|\leq N}\frac{1}{|{\bm{k}}|^{2}}\leq 8+2\pi(1+\ln(N))\leq
c\ln(N+1),$

and noting that ${\bm{\psi}}=P_{N}{\bm{\psi}}$, we have

$|\gamma_{1}-\gamma_{2}|\kappa_{0}\sqrt{|\Omega|}|{\bm{\psi}}|^{2}\leq\sqrt{c\ln(N+1)}|{\bm{\psi}}|(|{\bm{v}}_{1}|+|{\bm{v}}_{2}|)|{\bm{w}}|.$

Finally, by Corollary 3.4,

$|{\bm{w}}(t)|\leq\sqrt{5}\frac{|\gamma_{1}-\gamma_{2}|^{1-p/2}}{\left(\frac{\gamma_{1}+\gamma_{2}}{2}\right)^{1/2-p/2}}\frac{\|\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})\|_{C_{b}(V)}}{\sqrt{\nu_{0}}\kappa_{0}N},$

and using Lemma 3.2, we can bound the norms involving ${\bm{v}}_{1}$ and
${\bm{v}}_{2}$. The resulting inequality is

$\left(\frac{|\gamma_{1}-\gamma_{2}|}{\frac{\gamma_{1}+\gamma_{2}}{2}}\right)^{p/2}|\gamma_{1}-\gamma_{2}||{\bm{\psi}}|^{2}\\\
\leq|\gamma_{1}-\gamma_{2}||{\bm{\psi}}|\frac{2\sqrt{5}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\frac{\sqrt{c\ln(N+1)}}{N}M_{H}({\bm{\phi}})M_{V}({\bm{\phi}}),$

so either $|\gamma_{1}-\gamma_{2}|=0$ (taking care in the case where $p=0$),
or

$\left(\frac{|\gamma_{1}-\gamma_{2}|}{\frac{\gamma_{1}+\gamma_{2}}{2}}\right)^{p/2}|{\bm{\psi}}|N\leq\frac{2\sqrt{5c}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\sqrt{\ln(N+1)}M_{H}({\bm{\phi}})M_{V}({\bm{\phi}}).$

This holds for all time, so we may take the supremum over all time and get the
stated result. ∎

In the previous lemma no conclusions can be drawn regarding the viscosity in
the event that the data are zero in $C_{b}(H)$ (compare to the nonzero
requirement of Fact 4.2). However, given a nonzero function ${\bm{\phi}}\in
C_{b}(H)$, we always have $\|P_{n}({\bm{\phi}})\|_{C_{b}(H)}>0$ for some
$n>1$. Therefore, given $0\neq{\bm{\phi}}\in C_{b}(H)$, let

$n_{0}({\bm{\phi}})=\sup\\{n\in\mathbb{R}\>|\>\|P_{n}({\bm{\phi}})\|_{C_{b}(H)}=0\\}.$
(5.3)

Then $1\leq n_{0}({\bm{\phi}})<\infty$, and
$\|P_{N}({\bm{\phi}})\|_{C_{b}(H)}>0$ for any $N>n_{0}({\bm{\phi}}).$

We now give conditions for Problem 4.1 to have a unique solution.

######  Theorem 5.2.

Let ${\bm{u}}\in C_{b}(V)$ be a solution of (2.3) with viscosity
$\nu\in[\nu_{0},\nu_{1}]$, where $0<\nu_{0}<\nu_{1}<\infty$ are given, and
suppose that ${\bm{u}}\neq 0$. Choose $\mu_{0}\geq\frac{\nu_{1}}{\nu_{0}}.$ If

$N>\max\left\\{\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{u}}),\>n_{0}({\bm{u}})\right\\}$

and

$\frac{N}{\sqrt{\ln(N+1)}}>\frac{2\sqrt{5c}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\frac{M_{H}({\bm{u}})M_{V}({\bm{u}})}{\|P_{N}{\bm{u}}\|_{C_{b}(H)}},$
(5.4)

then $\nu$ is the _unique_ solution of Problem 4.1 on $[\nu_{0},\nu_{1}]$.

###### Proof.

By the assumption that ${\bm{u}}\in C_{b}(V),$ we may take
$R=\|P_{N}{\bm{u}}\|_{C_{b}(V)}\leq\|{\bm{u}}\|_{C_{b}(V)}<\infty,$ in Theorem
3.6 and conclude that ${\bm{u}}=\mathbb{W}(\nu,P_{N}{\bm{u}}).$

Let $\gamma\in[\nu_{0},\nu_{1}]$ and suppose that $\mathcal{L}(\gamma)=0.$
Then by Lemma 5.1 with $p=0$, either $\nu=\gamma$ or

$\|P_{N}{\bm{u}}\|_{C_{b}(H)}\,N\leq\frac{2\sqrt{5c}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\sqrt{\ln(N+1)}M_{H}(P_{N}{\bm{u}})M_{V}(P_{N}{\bm{u}})\\\
\leq\frac{2\sqrt{5c}}{\nu_{0}\kappa_{0}^{2}\sqrt{|\Omega|}}\sqrt{\ln(N+1)}M_{H}({\bm{u}})M_{V}({\bm{u}}).$

This inequality is invalidated by (5.4), so we conclude that $\nu=\gamma$. ∎

In Theorem 5.2, condition (5.4) depends on the solution ${\bm{u}}$ on which we
only have knowledge of its finite-dimensional projection $P_{N}u$. The
previous Theorem can be reformulated in terms of a dynamical property namely
the distance between zero and the attractor, $\mathcal{A}$.

######  Theorem 5.3.

Assume $\nu\in[\nu_{0},\nu_{1}]$ with $\nu_{0}>0$ and assume
$\bm{0}\notin\mathcal{A}_{\nu}$. Let

$G=\max\left\\{\frac{\|\bm{f}\|_{L^{2}}}{\nu_{0}^{2}\kappa_{0}^{2}},\frac{\|\bm{f}\|_{\mathbb{H}^{1}}}{\nu_{0}^{2}\kappa_{0}^{3}}\right\\}\
\mbox{and}\
\delta=d_{H}(\bm{0},\mathcal{A}_{\nu})=\inf_{{\bm{u}}\in\mathcal{A}_{\nu}}|{\bm{u}}|.$

Then $n_{0}({\bm{u}})\lesssim\frac{\nu_{1}G}{\delta}$ where
$G:=\frac{\|\bm{f}\|_{L^{2}}}{\nu_{0}^{2}\kappa_{0}^{2}}$. Moreover, $\nu$ is
the unique solution of Problem 4.1 provided

$N\gtrsim\max\\{\frac{\nu_{1}G}{\delta},\frac{\nu_{1}}{\nu_{0}}G\sqrt{1+\frac{\nu_{1}}{\nu_{0}}}\\}\
\mbox{and}\
\frac{N}{\sqrt{\ln(N+1)}}\gtrsim\frac{\nu_{1}^{2}(1+\frac{\nu_{1}}{\nu_{0}})G}{\nu_{0}\delta}.$
(5.5)

###### Proof.

Note that in view of (2.9). we have

$|P_{n}{\bm{u}}-{\bm{u}}|^{2}\leq\frac{1}{\kappa_{0}^{2}N^{2}}\|{\bm{u}}\|^{2}\lesssim\frac{\nu_{1}^{2}G^{2}}{N^{2}},$
(5.6)

where, with $G_{\nu}$ as in (2.6), we used the classical bounds (2.7) yielding

$\sup_{{\bm{u}}\in\mathcal{A}_{\nu}}|{\bm{u}}|\leq\nu G_{\nu}\leq\nu_{1}G\
\mbox{and}\
\sup_{{\bm{u}}\in\mathcal{A}_{\nu}}\|{\bm{u}}\|\leq\nu\kappa_{0}G_{\nu}\leq\nu_{1}\kappa_{0}G.$
(5.7)

Then, provided $N\gtrsim\frac{\nu_{1}G}{\delta}$, from (5.6), we have
$|P_{N}{\bm{u}}-{\bm{u}}|^{2}\leq\frac{1}{2}\delta^{2}$. Using the equality
$|{\bm{u}}|^{2}=|P_{N}{\bm{u}}|^{2}+|(I-P_{N}){\bm{u}}|^{2}$ and the
inequality $|{\bm{u}}|^{2}\geq\delta^{2}$ for ${\bm{u}}\in\mathcal{A}_{\nu}$,
it follows that $|P_{N}{\bm{u}}|^{2}\geq\frac{1}{2}\delta^{2}$. It follows
from the definition (5.3) that $n_{0}({\bm{u}})\leq\frac{\nu_{1}G}{\delta}$.

Let $\mu_{0}=\frac{\nu_{1}}{\nu_{0}}$. Since $\nu\in[\nu_{0},\nu_{1}]$. it
follows readily from (5.7) and the definition of $M_{H}({\bm{u}})$ and
$M_{V}({\bm{u}})$ in (3.5) and (3.6) that

$M_{H}({\bm{u}})\lesssim\nu_{1}\sqrt{1+\frac{\nu_{0}}{\nu_{1}}}\,\widetilde{G}\
\mbox{and}\
M_{H}({\bm{u}})\lesssim\nu_{1}\sqrt{1+\frac{\nu_{0}}{\nu_{1}}}\kappa_{0}\,\widetilde{G}.$

The conclusion now follows immediately from Theorem 5.2 by inserting the bound
$|P_{N}{\bm{u}}|^{2}\geq\frac{1}{2}\delta^{2}$ in (5.4). ∎

We can easily see that the condition ${\bm{u}}\neq 0$ is necessary for the
unique recovery of $\nu$ by inspecting (2.3): if ${\bm{u}}\equiv 0$ satisfies
(2.3) for a given force $\bm{f}$, then it does so with any value for $\nu$
because $Au\equiv 0$, hence there is no unique $\nu$ to recover. As evidenced
by the following example, the condition $N>n_{0}({\bm{u}})$ in Theorem 5.2 is
also necessary.

######  Example 5.4.

Fix ${\bm{k}}\in\mathbb{Z}^{2}/\\{0\\}$ and $c\in\mathbb{C}/\\{0\\}$ and
define $\bm{f}:\Omega\to V$ by

$\bm{f}({\bm{x}})=c{\bm{k}}^{\perp}e^{i\kappa_{0}{\bm{k}}\cdot{\bm{x}}}+\bar{c}{\bm{k}}^{\perp}e^{-i\kappa_{0}{\bm{k}}\cdot{\bm{x}}},\quad\forall{\bm{x}}\in\Omega,$

(where ${\bm{k}}^{\perp}$ is a unit vector orthogonal to ${\bm{k}}$, for
example, $[-k_{2},k_{1}]^{T}/|{\bm{k}}|$), i.e. $\bm{f}$ is a time-autonomous
“single mode” force in $V$. Then

${\bm{u}}\equiv\frac{1}{\nu\kappa_{0}^{2}|{\bm{k}}|^{2}}\bm{f}$

is a stationary solution of (2.3) for any $\nu>0$. Note that

$\|\bm{f}\|=2\sqrt{2}\pi|{\bm{k}}||c|,$

so, by choosing $c=\frac{1}{2\sqrt{2}\pi|{\bm{k}}|}$, we have $\|\bm{f}\|=1$
for any choice of ${\bm{k}}$, and

$\|{\bm{u}}\|=\frac{1}{\nu\kappa_{0}^{2}|{\bm{k}}|^{2}},$

so

$M_{V}(P_{N}{\bm{u}})=\sqrt{2}\sqrt{\frac{1}{(\nu\kappa_{0}^{2}N^{2})^{2}}+\mu_{0}^{2}\|P_{N}{\bm{u}}\|^{2}}\leq\frac{\sqrt{2}}{\nu\kappa_{0}^{2}}\sqrt{\frac{1}{N^{4}}+\mu_{0}^{2}\frac{1}{|{\bm{k}}|^{4}}}.$

Now, choosing $\mu_{0}>1$ and
$N\geq\frac{8\sqrt{2}}{\nu^{2}\kappa_{0}^{3}}\sqrt{1+\mu_{0}^{2}}$, for any
$\gamma\in[\frac{\nu}{\mu_{0}},\mu_{0}\nu]$,

$\mathbb{W}(\gamma,P_{N}{\bm{u}})=\frac{1}{\gamma\kappa_{0}^{2}|{\bm{k}}|^{2}+\mu}(\bm{f}+\mu
P_{N}{\bm{u}}).$

Therefore,

$\|\mathbb{W}(\gamma,P_{N}{\bm{u}})-{\bm{u}}\|_{C_{b}(H)}=\frac{|\bm{f}|}{\gamma\kappa_{0}^{2}|{\bm{k}}|^{2}+\mu}\frac{1}{\nu}\begin{cases}|\gamma-\nu|,&\text{
if }|{\bm{k}}|<N\\\
\left|\gamma-\nu+\frac{\mu}{\kappa_{0}^{2}|{\bm{k}}|^{2}}\right|,&\text{ if
}|{\bm{k}}|\geq N\end{cases}$

and

$\|P_{N}\mathbb{W}(\gamma,P_{N}{\bm{u}})-P_{N}{\bm{u}}\|_{C_{b}(H)}=\begin{cases}\frac{|\bm{f}|}{\gamma\kappa_{0}^{2}|{\bm{k}}|^{2}+\mu}\frac{|\gamma-\nu|}{\nu},&\text{
if }|{\bm{k}}|<N\\\ 0,&\text{ if }|{\bm{k}}|\geq N\end{cases}.$

So, if $|{\bm{k}}|\geq N$, then $P_{N}{\bm{u}}=0,$ and
$\|P_{N}\mathbb{W}(\gamma,P_{N}{\bm{u}})-P_{N}{\bm{u}}\|_{C_{b}(H)}=0$ while
$|\nu-\gamma|$ can be arbitrarily large. Hence, we have found cases with $N$
arbitrarily large and ${\bm{u}}\neq 0$ where there are infinitely many
solutions of Problem 4.1 because $P_{N}{\bm{u}}=0$.

##  6\. Lipschitz Continuity of the Inverse Problem

In the previous section we derived conditions for Problem 4.1 to have a unique
solution. In so doing, we found conditions on $N$ for Problem 4.1 to be a
well-defined mapping of data, $P_{N}{\bm{u}}$, to viscosity. We now provide
conditions for this mapping, which can be thought of as the inverse of the
determining map, $\mathbb{W}(\cdot,P_{N}{\bm{u}})^{-1}$, to be Lipschitz
continuous; i.e., we derive bounds on the viscosity discrepancy in terms of
$\mathcal{L}$. In addition, we propose an algorithm for solving Problem 4.1,
which is revealed by the proof of the Lipschitz continuity.

When $\mathcal{L}\geq 0$, the rate of change of the difference between two
trajectories corresponding to different viscosities is dynamic, and needs to
be integrated. As a consequence, the following results explicitly involve the
time-derivative of the data, $P_{N}{\bm{u}}$.

Before focusing on the case where $\phi=P_{N}{\bm{u}}$ for some solution
${\bm{u}}$ of (2.3), we first obtain a bound valid for the determining map in
the general setting.

######  Lemma 6.1.

Let ${\bm{\phi}}\in C_{b}(V)$ and $\gamma_{1},\gamma_{2}\in[\nu_{0},\nu_{1}]$,
where $0<\nu_{0}<\nu_{1}<\infty$ are given. Let
$\mu_{0}\geq\frac{\nu_{1}}{\nu_{0}},$ and

$N>\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{\phi}}).$

Let ${\bm{v}}_{1}=\mathbb{W}(\gamma_{1},{\bm{\phi}})$ and
${\bm{v}}_{2}=\mathbb{W}(\gamma_{2},{\bm{\phi}})$. Then for any interval of
time $[s,t]\subset\mathbb{R}$,

$|\gamma_{1}-\gamma_{2}|\left(\inf_{[s,t]}|P_{N}{\bm{v}}_{1}|^{2}-\frac{1}{N}M_{1}\right)\leq
M_{2}\|P_{N}({\bm{v}}_{1}-{\bm{v}}_{2})\|_{C_{b}(H)},$

where

$M_{1}=\frac{2\sqrt{5}c_{0}^{2}}{\nu_{0}\kappa_{0}^{2}}\;M_{H}({\bm{\phi}})\,(M_{V}({\bm{\phi}}))^{2}$

and

$M_{2}=\mu\frac{1+\delta^{N^{2}}}{1-\delta^{N^{2}}}\|A^{-1}P_{N}{\bm{v}}_{1}\|_{C_{b}(H)}+\|A^{-1}\partial_{t}P_{N}{\bm{v}}_{1}\|_{C_{b}(H)}+\gamma_{2}\|P_{N}{\bm{v}}_{1}\|_{C_{b}(H)},$

and $\delta=e^{-\mu_{0}\nu_{0}\kappa_{0}^{2}(t-s)}$.

###### Proof.

Define ${\bm{w}}={\bm{v}}_{1}-{\bm{v}}_{2}$ and
${\bm{\psi}}=P_{N}{\bm{v}}_{1}.$ Writing the evolution equation for ${\bm{w}}$
using (2.11),

$\partial_{t}{\bm{w}}+\gamma_{2}A{\bm{w}}+(\gamma_{1}-\gamma_{2})A{\bm{v}}_{1}+B\left({\bm{v}}_{1},{\bm{w}}\right)+B\left({\bm{w}},{\bm{v}}_{2}\right)+\mu
P_{N}{\bm{w}}=0,$

and taking an inner-product with respect to $A^{-1}{\bm{\psi}}$, we have

$\left<\partial_{t}{\bm{w}},A^{-1}{\bm{\psi}}\right>+\gamma_{2}\left<{\bm{w}},{\bm{\psi}}\right>+(\gamma_{1}-\gamma_{2})|{\bm{\psi}}|^{2}\\\
+b\left({\bm{v}}_{1},{\bm{w}},A^{-1}{\bm{\psi}}\right)+b\left({\bm{w}},{\bm{v}}_{2},A^{-1}{\bm{\psi}}\right)+\mu\left<P_{N}{\bm{w}},A^{-1}{\bm{\psi}}\right>=0.$

Using the product rule, we have
$\frac{d}{dt}\left<{\bm{w}},A^{-1}{\bm{\psi}}\right>=\left<\partial_{t}{\bm{w}},A^{-1}{\bm{\psi}}\right>+\left<{\bm{w}},\partial_{t}A^{-1}{\bm{\psi}}\right>$
and using the facts that $P_{N}{\bm{\psi}}={\bm{\psi}}$, $P_{N}$ is self-
adjoint, and $P_{N}$ commutes with $A^{-1}$, we can rewrite the last
differential inequality as

$\frac{d}{dt}\left<A^{-1}P_{N}{\bm{w}},{\bm{\psi}}\right>-\left<A^{-1}P_{N}{\bm{w}},\partial_{t}{\bm{\psi}}\right>+\gamma_{2}\left<P_{N}{\bm{w}},{\bm{\psi}}\right>+(\gamma_{1}-\gamma_{2})|{\bm{\psi}}|^{2}\\\
+b\left({\bm{v}}_{1},{\bm{w}},A^{-1}{\bm{\psi}}\right)+b\left({\bm{w}},{\bm{v}}_{2},A^{-1}{\bm{\psi}}\right)+\mu\left<A^{-1}P_{N}{\bm{w}},{\bm{\psi}}\right>=0.$

Let ${\bm{\beta}}:=\left<A^{-1}P_{N}{\bm{w}},{\bm{\psi}}\right>$. Then

$\dot{{\bm{\beta}}}+\mu{\bm{\beta}}-\left<A^{-1}P_{N}{\bm{w}},\partial_{t}{\bm{\psi}}\right>+\gamma_{2}\left<P_{N}{\bm{w}},{\bm{\psi}}\right>+(\gamma_{1}-\gamma_{2})|{\bm{\psi}}|^{2}\\\
+b\left({\bm{v}}_{1},{\bm{w}},A^{-1}{\bm{\psi}}\right)+b\left({\bm{w}},{\bm{v}}_{2},A^{-1}{\bm{\psi}}\right)=0,$

so, multiplying by the integrating factor $e^{\mu(\tau-s)}$ and taking the
time-integral over the interval $[s,t]$, we have

${\bm{\beta}}(t)-e^{-\mu(t-s)}{\bm{\beta}}(s)+(\gamma_{1}-\gamma_{2})\int_{s}^{t}e^{-\mu(t-\tau)}|{\bm{\psi}}|^{2}\>d\tau\\\
=\int_{s}^{t}e^{-\mu(t-\tau)}\left(\left<A^{-1}P_{N}{\bm{w}},\partial_{t}{\bm{\psi}}\right>-\gamma_{2}\left<P_{N}{\bm{w}},{\bm{\psi}}\right>\right)\;d\tau\\\
-\int_{s}^{t}e^{-\mu(t-\tau)}\left(b\left({\bm{v}}_{1},{\bm{w}},A^{-1}{\bm{\psi}}\right)+b\left({\bm{w}},{\bm{v}}_{2},A^{-1}{\bm{\psi}}\right)\right)\;d\tau$
(6.1)

Therefore,

$|\gamma_{1}-\gamma_{2}|\frac{1}{\mu}\left(1-e^{-\mu(t-s)}\right)\inf_{[s,t]}|{\bm{\psi}}|^{2}\leq|{\bm{\beta}}(t)|+e^{-\mu(t-s)}|{\bm{\beta}}(s)|\\\
+\frac{1}{\mu}\left(1-e^{-\mu(t-s)}\right)\sup_{[s,t]}\left(|A^{-1}\partial_{t}{\bm{\psi}}|+\gamma_{2}|{\bm{\psi}}|\right)\|P_{N}{\bm{w}}\|_{C_{b}(H)}\\\
+\frac{1}{\mu}\left(1-e^{-\mu(t-s)}\right)\sup_{[s,t]}\left((\|{\bm{v}}_{1}\|_{L^{4}}+\|{\bm{v}}_{2}\|_{L^{4}})\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{L^{4}}\right)\|{\bm{w}}\|_{C_{b}(H)}.$

Note that for all $t$,

$|{\bm{\beta}}(t)|\leq|P_{N}{\bm{w}}||A^{-1}{\bm{\psi}}|\leq\|P_{N}{\bm{w}}\|_{C_{b}(H)}\|A^{-1}{\bm{\psi}}\|_{C_{b}(H)}<\infty,$

and letting
$\delta^{N^{2}}=e^{-\mu(t-s)}=e^{-\mu_{0}\nu_{0}\kappa_{0}^{2}N^{2}(t-s)}$ (so
$t-s=\frac{\ln(\frac{1}{\delta})}{\mu_{0}\nu_{0}\kappa_{0}^{2}}$), we can
replace the previous inequality with

$|\gamma_{1}-\gamma_{2}|\inf_{[s,t]}|{\bm{\psi}}|^{2}\\\
\leq\left(\mu\frac{1+\delta^{N^{2}}}{1-\delta^{N^{2}}}\|A^{-1}{\bm{\psi}}\|_{C_{b}(H)}+\|A^{-1}\partial_{t}{\bm{\psi}}\|_{C_{b}(H)}+\gamma_{2}\|{\bm{\psi}}\|_{C_{b}(H)}\right)\|P_{N}{\bm{w}}\|_{C_{b}(H)}\\\
+\sup_{[s,t]}\left((\|{\bm{v}}_{1}\|_{L^{4}}+\|{\bm{v}}_{2}\|_{L^{4}})\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{L^{4}}\right)\|{\bm{w}}\|_{C_{b}(H)}.$

Now, by Corollary 3.4 with $p=0$,

$\|{\bm{w}}\|_{C_{b}(H)}\leq\sqrt{5}\frac{|\gamma_{1}-\gamma_{2}|}{\left((\gamma_{1}+\gamma_{2})/2\right)^{1/2}}\frac{\|\frac{1}{2}({\bm{v}}_{1}+{\bm{v}}_{2})\|_{C_{b}(V)}}{\sqrt{\nu_{0}}\kappa_{0}N}\\\
\leq\frac{\sqrt{5}}{2}\frac{|\gamma_{1}-\gamma_{2}|}{\nu_{0}\kappa_{0}N}(\|{\bm{v}}_{1}\|_{C_{b}(V)}+\|{\bm{v}}_{2}\|_{C_{b}(V)}),$

so

$|\gamma_{1}-\gamma_{2}|\left(\inf_{[s,t]}|{\bm{\psi}}|^{2}-\frac{1}{N}\widetilde{M}_{1}\right)\leq
M_{2}\|P_{N}{\bm{w}}\|_{C_{b}(H)}$

where

$\widetilde{M}_{1}=\frac{\sqrt{5}}{2\nu_{0}\kappa_{0}}(\|{\bm{v}}_{1}\|_{C_{b}(V)}+\|{\bm{v}}_{2}\|_{C_{b}(V)})\sup_{[s,t]}\left((\|{\bm{v}}_{1}\|_{L^{4}}+\|{\bm{v}}_{2}\|_{L^{4}})\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{L^{4}}\right)$

and

$M_{2}=\mu\frac{1+\delta^{N^{2}}}{1-\delta^{N^{2}}}\|A^{-1}{\bm{\psi}}\|_{C_{b}(H)}+\|A^{-1}\partial_{t}{\bm{\psi}}\|_{C_{b}(H)}+\gamma_{2}\|{\bm{\psi}}\|_{C_{b}(H)}.$

Finally, we use (2.15) to bound the $L^{4}$ norms,

$\sup_{[s,t]}\left(\|{\bm{v}}_{1}\|_{L^{4}}+\|{\bm{v}}_{2}\|_{L^{4}}\right)\leq
c_{0}\sup_{[s,t]}\left(|{\bm{v}}_{1}|^{\frac{1}{2}}\|{\bm{v}}_{1}\|^{\frac{1}{2}}+|{\bm{v}}_{2}|^{\frac{1}{2}}\|{\bm{v}}_{2}\|^{\frac{1}{2}}\right)\\\
\leq
c_{0}\left(\|{\bm{v}}_{1}\|_{C_{b}(H)}^{\frac{1}{2}}\|{\bm{v}}_{1}\|_{C_{b}(V)}^{\frac{1}{2}}+\|{\bm{v}}_{2}\|_{C_{b}(H)}^{\frac{1}{2}}\|{\bm{v}}_{2}\|_{C_{b}(V)}^{\frac{1}{2}}\right)$

and (additionally using (2.10)),

$\sup_{[s,t]}\left(\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{L^{4}}\right)\leq
c_{0}\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{C_{b}(H)}^{\frac{1}{2}}\|A^{-\frac{1}{2}}{\bm{\psi}}\|_{C_{b}(V)}^{\frac{1}{2}}\\\
\leq
c_{0}\frac{1}{\kappa_{0}^{\frac{1}{2}}}\|{\bm{\psi}}\|_{C_{b}(H)}^{\frac{1}{2}}\frac{1}{\kappa_{0}^{\frac{1}{2}}}\|{\bm{\psi}}\|_{C_{b}(V)}^{\frac{1}{2}}\leq\frac{c_{0}}{\kappa_{0}}\|{\bm{v}}_{1}\|_{C_{b}(H)}^{\frac{1}{2}}\|{\bm{v}}_{1}\|_{C_{b}(V)}^{\frac{1}{2}}$

so that we can replace $\widetilde{M}_{1}$ with $M_{1}$ using Lemma 3.2:

$\widetilde{M}_{1}\leq\frac{\sqrt{5}}{2\nu_{0}\kappa_{0}}\;2M_{V}({\bm{\phi}})\;2c_{0}\sqrt{M_{H}({\bm{\phi}})}\sqrt{M_{V}({\bm{\phi}})}\;\frac{c_{0}}{\kappa_{0}}\sqrt{M_{H}({\bm{\phi}})}\sqrt{M_{V}({\bm{\phi}})}=M_{1}.$

∎

In the following Theorem, we apply Lemma 6.1 to the case where ${\bm{v}}_{1}$
is a solution of (2.3) with viscosity $\nu$.

######  Theorem 6.2.

Let ${\bm{u}}\in C_{b}(V)$ be a solution of (2.3) with viscosity
$\nu\in[\nu_{0},\nu_{1}]$, where $0<\nu_{0}<\nu_{1}<\infty$ are given, and
suppose that for some $n\geq 1$ and a time interval, $[s,t]$,

$\inf_{[s,t]}|P_{n}{\bm{u}}|^{2}>0.$

Choose $\mu_{0}\geq\frac{\nu_{1}}{\nu_{0}}.$ If

$N>\max\left\\{\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{u}}),\>n,\>2\frac{M_{1}}{\inf_{[s,t]}|P_{N}{\bm{u}}|^{2}}\right\\},$

then for any $\gamma\in[\nu_{0},\nu_{1}]$,

$|\nu-\gamma|\leq
2\frac{M_{2}}{\inf_{[s,t]}|P_{N}{\bm{u}}|^{2}}\mathcal{L}(\gamma),$

where $M_{1}$ and $M_{2}$ are defined as in Lemma 6.1 with
${\bm{v}}_{1}={\bm{u}}$ and $\phi=P_{N}{\bm{u}}$.

###### Proof.

By the assumption that ${\bm{u}}\in C_{b}(V),$ we may take
$R=\|P_{N}{\bm{u}}\|_{C_{b}(V)}\leq\|{\bm{u}}\|_{C_{b}(V)}<\infty,$ in Theorem
3.6 and conclude that ${\bm{u}}=\mathbb{W}(\nu,P_{N}{\bm{u}}).$ The rest is an
immediate consequence of Lemma 6.1. ∎

######  Remark 6.3.

We chose to state Theorem 6.2 with the _a priori_ assumption that there exists
an interval of time where finitely many modes of ${\bm{u}}$ are bounded away
from zero. However, given the continuity of ${\bm{u}}$ in time, the weaker
assumption that ${\bm{u}}\neq 0$ is sufficient to guarantee the existence of
such a time interval.

To see this, if ${\bm{u}}\neq\bm{0}$. there exists $t_{0}$ such that
${\bm{u}}(t_{0})\neq\bm{0}$. This implies that there exists an $\epsilon>0$
such that $\inf_{t_{0}-\epsilon,t_{0}+\epsilon]}|{\bm{u}}(t)|\geq\delta$.
Proceeding as in the proof of Theorem 5.3, it follows that there exists
$n_{0}$ sufficiently large such that
$\inf_{[t_{0}-\epsilon,t_{0}+\epsilon]}|P_{n_{0}}{\bm{u}}(t)|\geq\frac{1}{2}\delta$.
One can in fact show that if $\bm{0}\notin\mathcal{A}_{\nu}$, then there
exists $\delta>0$ and $t_{0}>0$ such that
$\inf_{[t_{0},\infty)}|P_{n_{0}}{\bm{u}}(t)|\geq\delta$, where $n_{0}$ depends
only on the Grashoff number and the distance of $\bm{0}$ from the attractor.

##  7\. Algorithm for Solving the Inverse Problem

So far we have shown the validity of Problem 4.1 as a means of parameter
recovery. We now present a method of computing its solution. In [4], the
authors proposed the following update for the approximate viscosity, $\gamma$,
as part of an algorithm to recover the true viscosity, $\nu$, using the
available data only:

$\nu\approx\gamma+\frac{\frac{1}{2}\frac{1}{t-s}\left(|P_{N}{\bm{w}}(t)|^{2}-|P_{N}{\bm{w}}(s)|^{2}\right)+\mu\left<|P_{N}{\bm{w}}|^{2}\right>_{s}^{t}}{\left<\left<P_{N}A\mathbb{W}(\gamma,P_{N}{\bm{u}}),P_{N}{\bm{w}}\right>\right>_{s}^{t}},$
(7.1)

where $P_{N}{\bm{w}}=P_{N}{\bm{u}}-P_{N}\mathbb{W}(\gamma,P_{N}{\bm{u}})$.
Rigorous conditions for the convergence of this algorithm were obtained in
[12].

We provide a proof for the convergence of a slightly modified version of this
algorithm, which is presented in Algorithm 1. This new algorithm is based on
the approach taken in Lemma 6.1. In comparison to (7.1), the new update
formula (see the definition of $\Gamma$ in Algorithm 1) is computed using
inner-products with the data, $P_{N}{\bm{u}}$, and the time derivative of the
data, as opposed to the observable error, $P_{N}{\bm{u}}-P_{N}{\bm{w}}$.
Convolutions are taken with the decaying exponential factor $\tau\mapsto
e^{-\mu\tau}$ over a time interval, $[s,t]$; with $\mu$ fixed, we denote this
operation by $\left<\cdot\right>_{s}^{t}$,

$\left<\phi\right>_{s}^{t}:=\frac{1}{t-s}\int_{s}^{t}e^{-\mu(t-\tau)}\phi(\tau)\;d\tau,\quad\forall\phi\in
C([s,t];\mathbb{R}).$

Algorithm 1 Viscosity inference from data using the determining map

1:$0<\nu_{0}<\nu_{1}$ such that $\nu\in[\nu_{0},\nu_{1}]$ $\triangleright$
bounds for the unknown viscosity, $\nu$

2:$N,\mu_{0}$ $\triangleright$ both chosen sufficiently large (see Theorem
7.1)

3:$P_{N}{\bm{u}}(t)\quad\forall t\in\mathbb{R}$ $\triangleright$ observations
of the reference solution

4:$\>s<t\>$ such that $\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}>0$

5:$\epsilon_{1}\in(0,\nu_{0})$ $\triangleright$ the convergence factor (also
limited by $N$)

6:$\epsilon_{2}>0$ $\triangleright$ a stopping tolerance for the viscosity
error

7:$\gamma\leftarrow\gamma_{0}$ $\triangleright$ an initial guess for the true
viscosity

8:repeat

9: $\gamma_{0}\leftarrow\gamma$

10: $\gamma\leftarrow\textsc{$\Gamma$}(\gamma)$

11:until
$|\gamma-\gamma_{0}|/|\nu_{1}-\nu_{0}|\leq\epsilon_{2}/(\nu_{1}-\nu_{0}+\epsilon_{1})$
$\triangleright$ the stopping condition

12:return $\gamma$

13:procedure $\Gamma$($\gamma$)

14: ${\bm{v}}\leftarrow\mathbb{W}(\gamma,P_{N}{\bm{u}})$ $\triangleright$
compute the solution of the determining map

15: ${\bm{\psi}}\leftarrow P_{N}{\bm{u}}-P_{N}{\bm{v}}$ $\triangleright$
compute the observable velocity error

16:
$c_{1}\leftarrow\frac{1}{t-s}\left(\left<A^{-1}{\bm{\psi}}(t),P_{N}{\bm{u}}(t)\right>-e^{-\mu(t-s)}\left<A^{-1}{\bm{\psi}}(s),P_{N}{\bm{u}}(s)\right>\right)$

17:
$c_{2}\leftarrow-\left<\left<A^{-1}{\bm{\psi}},\partial_{t}P_{N}{\bm{u}}\right>\right>_{s}^{t}$

18:
$c_{3}\leftarrow\gamma\left<\left<{\bm{\psi}},P_{N}{\bm{u}}\right>\right>_{s}^{t}$

19: return
$\gamma-\frac{c_{1}+c_{2}+c_{3}}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}$
$\triangleright$ return a new approximation for the viscosity

20:end procedure

The benefit of this new update formula is that the condition for the
denominator to be nonzero is an _a priori_ condition depending on the
observations of the reference solution only, as opposed to an _a posteriori_
condition which depends on the observable error, and so cannot be verified
until the update is computed.

Conditions for the convergence of Algorithm 1 are given in the following
theorem.

######  Theorem 7.1.

Let ${\bm{u}}$ be a solution of (2.3) with viscosity
$\nu\in[\nu_{0},\nu_{1}]$, and suppose there exists an $n>1$ and a time
interval $[s,t]$ over which

$\left<|P_{n}{\bm{u}}|^{2}\right>_{s}^{t}>0.$

Let $\epsilon_{1}\in(0,\nu_{0})$ and fix
$\mu_{0}\geq\frac{\nu_{1}+\epsilon_{1}}{\nu_{0}-\epsilon_{1}}.$ If

$N>\max\left\\{\frac{8}{\nu_{0}\kappa_{0}}M_{V}({\bm{u}}),\>n,\>\frac{\nu_{1}-\nu_{0}+\epsilon_{1}}{\epsilon_{1}}\frac{M_{1}}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}\right\\},$

where
$M_{1}:=\frac{2\sqrt{5}c_{0}^{2}}{\nu_{0}\kappa_{0}^{2}}\;M_{H}({\bm{u}})\,(M_{V}({\bm{u}}))^{2},$
then, with $\Gamma$ as defined in Algorithm 1, for any choice of
$\gamma\in[\nu_{0}-\epsilon_{1},\nu_{1}+\epsilon_{1}]$,

$|\nu-\Gamma(\gamma)|<\frac{\epsilon_{1}}{\nu_{1}-\nu_{0}+\epsilon_{1}}|\nu-\gamma|<|\nu-\gamma|,$

and $\Gamma(\gamma)\in[\nu_{0}-\epsilon_{1},\nu_{1}+\epsilon_{1}].$ Therefore,
$\Gamma$ can be applied iteratively, and the result converges to $\nu$.

Furthermore, let $\epsilon_{2}>0$ be a tolerance for stopping. We can infer
closeness to the true viscosity by examining the residuals:

$\frac{|\Gamma^{k+1}(\gamma)-\Gamma^{k}(\gamma)|}{(\nu_{1}-\nu_{0})}\leq\frac{\epsilon_{2}}{\nu_{1}-\nu_{0}+\epsilon_{1}}\implies|\Gamma^{k}(\gamma)-\nu|\leq\epsilon_{2}.$

###### Proof.

Let ${\bm{v}}=\mathbb{W}(\gamma,P_{N}{\bm{u}})$, and let
${\bm{w}}={\bm{u}}-{\bm{v}}=\mathbb{W}(\nu,P_{N}{\bm{u}})-\mathbb{W}(\gamma,P_{N}{\bm{u}})$.
Proceeding as in Lemma 6.1 up to equation (6.1), we obtain

${\bm{\beta}}(t)-e^{-\mu(t-s)}{\bm{\beta}}(s)+(\nu-\gamma)\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}=\left<\left<A^{-1}P_{N}{\bm{w}},\partial_{t}P_{N}{\bm{u}}\right>-\gamma\left<P_{N}{\bm{w}},P_{N}{\bm{u}}\right>\right>_{s}^{t}\\\
-\left<b\left({\bm{u}},{\bm{w}},A^{-1}P_{N}{\bm{u}}\right)+b\left({\bm{w}},{\bm{v}},A^{-1}P_{N}{\bm{u}}\right)\right>_{s}^{t}$

where ${\bm{\beta}}:=\left<A^{-1}P_{N}w,P_{N}{\bm{u}}\right>.$ Moving the
first term on the right hand side over to the left, and dividing by the term
multipying $(\nu-\gamma)$, we have

$\nu-\Gamma(\gamma)=-\frac{1}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}\left(\left<b\left({\bm{u}},{\bm{w}},A^{-1}P_{N}{\bm{u}}\right)\right>_{s}^{t}+\left<b\left({\bm{w}},{\bm{v}},A^{-1}P_{N}{\bm{u}}\right)\right>_{s}^{t}\right)$

where $\Gamma(\gamma)$ is as defined in Algorithm 1.

We then take the absolute value of both sides and estimate the remaining terms
on the right hand side as was done in Lemma 6.1, and obtain

$|\nu-\Gamma(\gamma)|\leq\frac{1}{N}\frac{M_{1}}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}|\nu-\gamma|.$

Let $\delta>0$. Taking
$N>\frac{1}{\delta}\frac{M_{1}}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}$, we
have

$|\nu-\Gamma(\gamma)|\leq\delta|\nu-\gamma|.$

Choosing $\delta<1$ ensures $\Gamma(\gamma)$ is closer to $\nu$ than $\gamma$,
but we also need to ensure that $\Gamma(\gamma)$ satisfies the feasibility
condition: $\Gamma(\gamma)\in[\nu_{0}-\epsilon_{1},\nu_{1}+\epsilon_{1}]$.
From the previous inequality and the feasibility condition imposed on
$\gamma$, as well as the assumption $\nu\in[\nu_{0},\nu_{1}]$, we have

$\nu_{0}-\delta(\nu_{1}-\nu_{0}+\epsilon_{1})\leq\Gamma(\gamma)\leq\nu_{1}+\delta(\nu_{1}-\nu_{0}+\epsilon_{1}).$

Therefore, the feasibility condition for $\Gamma(\gamma)$ is satisfied with
$\delta=\frac{\epsilon_{1}}{\nu_{1}-\nu_{0}+\epsilon_{1}}.$ This expression
for $\delta$ is an increasing function of $\epsilon_{1}\in(0,\nu_{0})$, and so
attains its max when $\epsilon_{1}=\nu_{0}$. Therefore,
$\delta=\frac{\epsilon_{1}}{\nu_{1}-\nu_{0}+\epsilon_{1}}<\frac{\nu_{0}}{\nu_{1}}<1$,
and iterating $\Gamma$ results in convergence to $\nu$:

$|\Gamma^{k}(\gamma)-\nu|\leq\delta^{k}|\gamma-\nu|\to 0\>\text{ as
}\>k\to\infty.$

Now, for the stopping condition, note that

$|\Gamma^{k}(\gamma)-\nu|=|\Gamma^{k}(\gamma)-\Gamma^{k+1}(\gamma)+\Gamma^{k+1}(\gamma)-\nu|\leq|\Gamma^{k+1}(\gamma)-\Gamma^{k}(\gamma)|+|\Gamma^{k+1}(\gamma)-\nu|$

so,

$|\Gamma^{k+1}(\gamma)-\Gamma^{k}(\gamma)|\geq|\Gamma^{k}(\gamma)-\nu|-|\Gamma^{k+1}(\gamma)-\nu|\geq(1-\delta)|\Gamma^{k}(\gamma)-\nu|.$

Therefore, if after $k$ iterations we have
$|\Gamma^{k}(\gamma)-\nu|>\epsilon_{2}$, then

$|\Gamma^{k+1}(\gamma)-\Gamma^{k}(\gamma)|>(1-\delta)\epsilon_{2}=\frac{(\nu_{1}-\nu_{0})\epsilon_{2}}{\nu_{1}-\nu_{0}+\epsilon_{1}}.$

We get the stopping condition from the last statement by taking its
contrapositive. ∎

In Theorem 7.1, we first chose the convergence factor $\epsilon_{1}$ and then
chose $N$. However, we can instead let the data dictate the convergence
factor; if

$N>\frac{\nu_{1}}{\nu_{0}}\frac{M_{1}}{\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}}$

then

$\frac{(\nu_{1}-\nu_{0})M_{1}}{N\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}-M_{1}}<\nu_{0},$

so we can choose $\epsilon_{1}$ such that

$\nu_{0}>\epsilon_{1}\geq\frac{(\nu_{1}-\nu_{0})M_{1}}{N\left<|P_{N}{\bm{u}}|^{2}\right>_{s}^{t}-M_{1}}\implies|\nu-\Gamma(\gamma)|<\frac{\epsilon_{1}}{\nu_{1}-\nu_{0}+\epsilon_{1}}|\nu-\gamma|.$

##  8\. Conclusions

We extended the definition of the determining map to include viscosity as a
variable. We then studied the inverse problem of determining the viscosity
from the velocity data using the determining map, and found conditions for the
well-posedness of the inverse problem, as well as conditions for the
regularity of the inverse mapping, and defined an iterative algorithm for
obtaining its solution.

One consequence of our results is that, on the attractor, viscosity is
determined, just like all higher modes, from the low modes only. Therefore, we
have extended the concept of determining modes to include a parameter of the
equation as being determined from finitely many low modes. This also has
consequences for how different attractors (indexed by the viscosity) can
intersect, as no two solutions on the attractor can overlap for any interval
of time without the viscosities and solutions being identical

For simplicity of presentation, we chose to frame our results as well as
Algorithm 1 in terms of solutions on the attractor, which requires having data
for the reference solution for all time and computing the solution of the
determining map for all time. However, our results easily extend to the
initial value problem, where the data are only required on a bounded interval
of time, and the determining map is solved as an initial value problem with an
arbitrary initial condition. Algorithm 1 can also be modified as an “on-the-
fly” algorithm, as was done in [4].

## References

  * [1] D. A. F. Albanez and M. J. Benvenutti  Parameter analysis in continuous data assimilation for three-dimensional Brinkman-Forchheimer-extended Darcy,  _preprint_ (2022).  https://doi.org/10.48550/ARXIV.2210.11432
  * [2] A. Azouani, E. Olson and E. S. Titi,  Continuous data assimilation using general interpolant observables.  _J Nonlinear Sci_ 24 (2014), 277–304.  https://doi.org/10.1007/s00332-013-9189-y.
  * [3] A. Biswas, C. Foias, C. F. Mondaini and E. S. Titi,  Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations,  _Annales de l’institut henri poincaré c, analyse non linéaire_ , Elsevier. (2018)
  * [4] E. Carlson, J. Hudson and A. Larios,  Parameter recovery for the 2 dimensional Navier–Stokes equations via continuous data assimilation,  SIAM J. Sci. Comput. 42 (2020), no. 1, A250-A270.
  * [5] T. Colin, A. Iollo, J-B. Lagaert and O. Saut.  An inverse problem for the recovery of the vascularization of a tumor,  _J. Inverse Ill-Posed Probl._ 22 (2014), 759–786.
  * [6] P. Constantin and C. Foias,  _Navier–Stokes Equations_ ,  Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
  * [7] R. E. Ewing and T. Lin,  A class of parameter estimation techniques for fluid flow in porous media,  Adv. Water Resour. 1991 14, 89–97,  Parameter identification in ground water flow, transport, and related processes, part I.
  * [8] C. Foias, M. S. Jolly, R. Kravchenko and E. S. Titi,  A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case,  _Journal of Mathematical Physics_ 53 (2012), 115623.  https://doi.org/10.1063/1.4766459.
  * [9] C. Foias, M. S. Jolly, R. Kravchenko and E. S. Titi,  A unified approach to determining forms for the 2D Navier-Stokes equations – the general interpolants case,  _Russian Mathematical Surveys_ 69 (2014), 359–381.  https://doi.org/10.1070/RM2014v069n02ABEH004891.
  * [10] C. Foias, M.S. Jolly, R. Lan, R. Rupam, Y. Yang and B. Zhang,  _Time analyticity with higher norm estimates for the 2D Navier–Stokes equations_ ,  IMA J. Appl. Math., 80 (2015), pp. 766–810.
  * [11] M. A. Iglesius and C. Dawson,  An iterative representer-based scheme for data inversion in reservoir modeling,  _Inverse Problems_ 25 (2009) 035006 (34pp).
  * [12] Vincent R. Martinez,  Convergence Analysis of a Viscosity Parameter Recovery Algorithm for the 2D Navier–Stokes Equations,  _Nonlinearity_ 35 (2022), 2241.  https://doi.org/10.1088/1361-6544/ac5362.
  * [13] B. Panchev, J. P. Whitehead and S. A. McQuarrie,  Concurrent multiparameter learning demonstrated on the Kuramoto-Sivashinsky equation,  _SIAM J. Sci. Comput._ 44 (2022), no. 5, A2974-A2990.
  * [14] M. Raissi and G. E. Karniadakis,  Hidden physics models: machine learning of nonlinear partial differential equations.  _J. Comput. Phys._ 357 (2018), 125-141.
  * [15] M. Raissi, P. Perdikaris and G. E. Karniadakis,  Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.  _J. Comput. Phys._ 378 (2019), 686-707.
  * [16] R. Temam,  _Navier–Stokes Equations and Nonlinear Functional Analysis_ , 2nd ed.,  CBMS-NSF Regional Conference Series in Applied Mathematics, 66, SIAM, Philadelphia, PA, 1995.
  * [17] R. Temam,  _Infinite-Dimensional Dynamical Systems in Mechanics and Physics_ , 2nd ed.,  Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.
  * [18] R. Temam,  _Navier–Stokes Equations. Theory and Numerical Analysis_ , Studies in Mathematics and its Applications, 3rd edition,  North-Holland Publishing Co., Amsterdam-New York, 1984. Reedition in the AMS Chelsea Series, AMS, Providence, 2001.
  * [19] W W-G Yeh,  Review of parameter identification procedures in groundwater hydrology: the inverse problem,  Water Resour. Res. 1986, 22 95–108.

##  Appendix A Proof of Ladyzhenskaya’s inequality

For completeness, we provide a proof of Ladyzhenskaya’s inequality, valid on
the torus in 2D.

$\|{\bm{\phi}}\|_{L^{4}(\Omega)}\leq|{\bm{\phi}}|^{\frac{1}{2}}\left(\tfrac{1}{L}|{\bm{\phi}}|+\|{\bm{\phi}}\|\right)^{\frac{1}{2}}$
(A.1)

###### Proof.

Let ${\bm{\phi}}\in C([0,L];\mathbb{R})$ be periodic (i.e.
${\bm{\phi}}(L)={\bm{\phi}}(0)$ and ${\bm{\phi}}$ can be extended to
$\mathbb{R}$ via the relation ${\bm{\phi}}(x+nL):={\bm{\phi}}(x)\>\forall
x\in[0,L],n\in\mathbb{N}$). Denote the mean of ${\bm{\phi}}$ by
$\overline{{\bm{\phi}}}=\frac{1}{L}\int_{0}^{L}{\bm{\phi}}(x)dx.$ Then by the
Mean Value Theorem, there is a point $x_{0}\in[0,L]$ such that
${\bm{\phi}}(x_{0})=\overline{{\bm{\phi}}}.$

For any $x\in[0,L]$, we have

$\frac{1}{2}{\bm{\phi}}^{2}(x)=\frac{1}{2}\overline{{\bm{\phi}}}^{2}+\int_{x_{0}}^{x}{\bm{\phi}}(z){\bm{\phi}}^{\prime}(z)dz\leq\frac{1}{2}\overline{{\bm{\phi}}}^{2}+\int_{x_{0}}^{x}|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|dz,$

and by periodicity,

$\frac{1}{2}{\bm{\phi}}^{2}(x)=\frac{1}{2}\overline{{\bm{\phi}}}^{2}-\int_{x}^{x_{0}+L}{\bm{\phi}}(z){\bm{\phi}}^{\prime}(z)dz\leq\frac{1}{2}\overline{{\bm{\phi}}}^{2}+\int_{x}^{x_{0}+L}|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|dz,$

so

${\bm{\phi}}^{2}(x)\leq\overline{{\bm{\phi}}}^{2}+\int_{x_{0}}^{x_{0}+L}|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|dz=\overline{{\bm{\phi}}}^{2}+\int_{0}^{L}|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|dz.$

This easily extends to vector valued ${\bm{\phi}}\in C([0,L];R^{2}):$

$|{\bm{\phi}}(x)|^{2}={\bm{\phi}}_{1}^{2}(x)+{\bm{\phi}}_{2}^{2}(x)\leq\overline{{\bm{\phi}}_{1}}^{2}+\int_{0}^{L}|{\bm{\phi}}_{1}(z)||{\bm{\phi}}_{1}^{\prime}(z)|dz+\overline{{\bm{\phi}}_{2}}^{2}+\int_{0}^{L}|{\bm{\phi}}_{2}(z)||{\bm{\phi}}_{2}^{\prime}(z)|dz,$

which we can simplify by applying (2.14),

$\overline{{\bm{\phi}}_{1}}^{2}+\overline{{\bm{\phi}}_{2}}^{2}=\left(\frac{1}{L}\int_{0}^{L}{\bm{\phi}}_{1}(x)dx\right)^{2}+\left(\frac{1}{L}\int_{0}^{L}{\bm{\phi}}_{2}(x)dx\right)^{2}\\\
\leq\frac{1}{L}\int_{0}^{L}{\bm{\phi}}_{1}^{2}(x)dx+\frac{1}{L}\int_{0}^{L}{\bm{\phi}}_{2}^{2}(x)dx=\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(x)|^{2}dx$

and Cauchy-Schwarz,

$|{\bm{\phi}}_{1}(z)||{\bm{\phi}}_{1}^{\prime}(z)|+|{\bm{\phi}}_{2}(z)||{\bm{\phi}}_{2}^{\prime}(z)|=\begin{bmatrix}|{\bm{\phi}}_{1}(z)|\\\
|{\bm{\phi}}_{2}(z)|\end{bmatrix}\cdot\begin{bmatrix}|{\bm{\phi}}_{1}^{\prime}(z)|\\\
|{\bm{\phi}}_{2}^{\prime}(z)|\end{bmatrix}\leq|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|$

to obtain

$|{\bm{\phi}}(x)|^{2}\leq\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(z)|^{2}dz+\int_{0}^{L}|{\bm{\phi}}(z)||{\bm{\phi}}^{\prime}(z)|dz.$

Now, if ${\bm{\phi}}\in C(\Omega;\mathbb{R}^{2})$, using the estimates for a
one-dimensional domain, for any $(x,y)\in\Omega,$ we have

$|{\bm{\phi}}(x,y)|^{4}=|{\bm{\phi}}(x,y)|^{2}|{\bm{\phi}}(x,y)|^{2}\\\
\leq\left(\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(z,y)|^{2}dz+\int_{0}^{L}|{\bm{\phi}}(z,y)||\partial_{1}{\bm{\phi}}(z,y)|dz\right)\\\
\left(\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(x,z)|^{2}dz+\int_{0}^{L}|{\bm{\phi}}(x,z)||\partial_{2}{\bm{\phi}}(x,z)|dz\right),$

so

$\|{\bm{\phi}}\|_{L^{4}(\Omega)}^{4}=\int_{0}^{L}\int_{0}^{L}|{\bm{\phi}}(x,y)|^{4}\,dx\,dy\\\
\leq\int_{0}^{L}\left(\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(z,y)|^{2}dz+\int_{0}^{L}|{\bm{\phi}}(z,y)||\partial_{1}{\bm{\phi}}(z,y)|dz\right)dy\\\
\int_{0}^{L}\left(\frac{1}{L}\int_{0}^{L}|{\bm{\phi}}(x,z)|^{2}dz+\int_{0}^{L}|{\bm{\phi}}(x,z)||\partial_{2}{\bm{\phi}}(x,z)|dz\right)dx\\\
\leq\left(\frac{1}{L}|{\bm{\phi}}|^{2}+|{\bm{\phi}}||\partial_{1}{\bm{\phi}}|\right)\left(\frac{1}{L}|{\bm{\phi}}|^{2}+|{\bm{\phi}}||\partial_{2}{\bm{\phi}}|\right)\\\
\leq\left(\frac{1}{L}|{\bm{\phi}}|^{2}+|{\bm{\phi}}|\|{\bm{\phi}}\|\right)\left(\frac{1}{L}|{\bm{\phi}}|^{2}+|{\bm{\phi}}|\|{\bm{\phi}}\|\right),$

where the second to last line follows from the Cauchy-Schwarz inequality. ∎