# Existence, uniqueness and stability of an inverse problem for two-
dimensional convective Brinkman-Forchheimer equations with the integral
overdetermination 111 1,2Department of Mathematics, Indian Institute of
Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667,
INDIA.e-mail: Pardeep Kumar<EMAIL_ADDRESS>e-mail: Manil T. Mohan:
<EMAIL_ADDRESS><EMAIL_ADDRESS>*Corresponding author. Key words:
Convective Brinkman-Forchheimer equations, inverse source problem, integral
overdetermination condition, contraction mapping theorem, well-posedness.
Mathematics Subject Classification (2020): Primary 35R30; Secondary 35Q35,
35Q30.

Pardeep Kumar1 and Manil T. Mohan2*

###### Abstract.

In this article, we study an inverse problem for the following convective
Brinkman-Forchheimer (CBF) equations:

$\displaystyle\boldsymbol{u}_{t}-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla
p=\boldsymbol{F}:=f\boldsymbol{g},\ \ \ \nabla\cdot\boldsymbol{u}=0,$

in a bounded domain $\Omega\subset\mathbb{R}^{2}$ with smooth boundary
$\partial\Omega$, where $\alpha,\beta,\mu>0$ and $r\in[1,3]$. The investigated
inverse problem consists of reconstructing the vector-valued velocity function
$\boldsymbol{u}$, the pressure field $p$ and the scalar function $f$. For the
divergence free initial data $\boldsymbol{u}_{0}\in\mathbb{L}^{2}(\Omega)$, we
prove the existence of a solution to the inverse problem for two-dimensional
CBF equations with the integral overdetermination condition, by showing the
existence of a unique fixed point for an equivalent operator equation (using
an extension of the contraction mapping theorem). Moreover, we establish the
uniqueness and Lipschitz stability results of the solution to the inverse
problem for 2D CBF equations with $r\in[1,3]$.

###### Contents

  1. 1 Introduction
    1. 1.1 The mathematical model and the direct problem
    2. 1.2 The inverse problem
    3. 1.3 Technical difficulties
    4. 1.4 Main results
  2. 2 Mathematical formulation
    1. 2.1 Function spaces
    2. 2.2 Equivalent formulation
  3. 3 Proof of Theorem 1.2
    1. 3.1 Existence
    2. 3.2 Uniqueness
    3. 3.3 Stability
  4. A Energy estimates

##  1\. Introduction

The convective Brinkman-Forchheimer (CBF) equations describe the motion of
incompressible fluid flows in a saturated porous medium (cf. [5]). This
article’s primary goal is to discuss the well-posedness of an inverse problem
to CBF equations in two dimensions for a divergence free initial data in
$\mathbb{L}^{2}(\Omega)$.

###  1.1. The mathematical model and the direct problem

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with a smooth boundary
$\partial\Omega$ (at least $\mathrm{C}^{2}$-boundary). The CBF equations are
given by

$\displaystyle\boldsymbol{u}_{t}-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla
p=\boldsymbol{F}$ $\displaystyle:=f\boldsymbol{g},\ \text{ in }\
\Omega\times(0,T),$ (1.1) $\displaystyle\nabla\cdot\boldsymbol{u}$
$\displaystyle=0,\ \ \ \ \text{ in }\ \Omega\times(0,T),$ (1.2)

with initial condition

$\displaystyle\boldsymbol{u}=\boldsymbol{u}_{0},\ \text{ in }\
\Omega\times\\{0\\},$ (1.3)

and boundary condition

$\displaystyle\boldsymbol{u}=\boldsymbol{0},\ \ \text{ on }\
\partial\Omega\times[0,T).$ (1.4)

Here $\boldsymbol{u}(x,t)\in\mathbb{R}^{2}$ represents the velocity field at
position $x$ and time $t$, $p(x,t)\in\mathbb{R}$ denotes the pressure field
and $\boldsymbol{F}(x,t)\in\mathbb{R}^{2}$ stands for the external forcing.
The constant $\mu$ denotes the positive Brinkman coefficient (effective
viscosity), the positive constants $\alpha$ and $\beta$ stand for the Darcy
coefficient (permeability of porous medium) and the Forchheimer coefficient
(proportional to the porosity of the material), respectively (cf. [23]). The
absorption exponent $r\in[1,\infty)$ and the cases, $r=3$ and $r>3$, are known
as the critical exponent and the fast growing nonlinearity, respectively. For
$\alpha=\beta=0$, we obtain the classical $2\mathrm{D}$ Navier-Stokes
equations (NSE). Thus, one can consider the equations (1.1)-(1.4) as a
modification (by introducing an absorption term
$\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$) of the
classical NSE. Thus, one may consider the model (1.1)-(1.4) as NSE with
damping. In order to obtain the uniqueness of the pressure $p$, one can impose
the condition $\int_{\Omega}p(x,t)\/\mathrm{d}\/x=0,$ for $t\in[0,T]$. The
model given in (1.1)-(1.4) is recognized to be more accurate when the flow
velocity is too large for the Darcy’s law to be valid alone, and apart from
that, the porosity is not too small, so that we call these types of models as
_non-Darcy models_ (cf. [30]). It has been proved in Proposition 1.1, [18]
that the critical homogeneous CBF equations have the same scaling as NSE only
when $\alpha=0$ and no scale invariance property for other values of $\alpha$
and $r$.

Let us now discuss some results available in the literature on the global
solvability of the system (1.1)-(1.4) (direct problem). The existence and
uniqueness of weak as well as strong solutions in two and three dimensional
bounded domains are available in [3, 16, 23, 31], etc., for the global
solvability results in periodic domains and whole space, one can refer to [7,
18, 43], etc. The Navier-Stokes problem modified an absorption term
$|\boldsymbol{u}|^{r-1}\boldsymbol{u}$, for $r>1$, in bounded domains with
compact boundary is considered in [3]. The existence of Leray-Hopf weak
solutions, for any dimension $d\geq 2$, and its uniqueness for $d=2$ is
established in [3]. For $d=2,r\in[1,\infty)$ and $d=3,r\in[3,\infty)$
($2\beta\mu\geq 1$ for $r=3$), the existence and uniqueness of global Leray-
Hopf weak as well as strong solution is established in [31].

###  1.2. The inverse problem

Even though the direct problem is important, but it requires the knowledge of
physical parameters such as the Brinkman coefficient $\mu$, Darcy coefficient
$\alpha$, Forchheimer coefficient $\beta$ and the forcing term
$\boldsymbol{F}:=f\boldsymbol{g}$. When in addition to the solution of the
equation, recovery of some physical properties of the investigated object or
the effects of the external sources are needed, we require inverse problems to
determine a coefficient or to handle the right hand side of the differential
equation arising in mathematical modeling of physical phenomena. In such
modeled problems, it is more efficient to consider the inverse source
problems. However, posing an inverse problem requires some additional
information of the solution besides the given initial and boundary conditions.
An additional information on a solution of the inverse problem can be integral
overdetermination condition, which is the case considered in this paper. The
integral overdetermination condition considered in this work is given by

$\displaystyle\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=\varphi(t),\
\ t\in[0,T],$ (1.5)

where $\varphi(t)$ is the measurement data, which stands for the average
velocity on the domain $\Omega$ and $\boldsymbol{\omega}$ is given quantity,
which corresponds to the type of device used to measure the velocity.

Let the vector-valued external forcing $\boldsymbol{F}$ appearing in (1.1) be
represented by

$\boldsymbol{F}(x,t):=f(t)\boldsymbol{g}(x,t),$

where the scalar function $f$ is unknown and $\boldsymbol{g}$ is a given
vector-valued function. The investigation of the nonlinear inverse problem in
this article consists of reconstructing the vector-valued velocity function
$\boldsymbol{u}$, the pressure field $p$ and the scalar function $f$ from the
system (1.1)-(1.4), with the integral overdetermination condition (1.5), and
the given functions $\boldsymbol{u}_{0},\boldsymbol{\omega},\varphi$ and
$\boldsymbol{g}$.

The inverse problems with final overdetermination (cf. [6, 17, 20, 21, 39],
etc.) and integral overdetermination (cf. [2, 15, 22, 24, 26, 33, 34, 35]
etc.) conditions have been well studied in the literature. The existence and
uniqueness of the generalized solution of the inverse problem for the
nonlinear nonstationary system of Navier-Stokes equations with integral
overdetermination is investigated in [41]. The existence results of an inverse
problem to NSE with both the integral as well as final overdetermination data
are proved in [34] by using Schauder’s fixed point theorem in two and three
dimensions. The global well-posedness of the inverse source problem for
parabolic systems is examined in [36]. Under the assumption that the initial
data $\boldsymbol{u}_{0}\in\mathbb{H}$ and the viscosity constant is
sufficiently large, the authors in [14] proved that an inverse problem for 2D
NSE with the final overdetermination data is well-posed. To prove the same,
they have used Tikhonov’s fixed point theorem. In [13], the authors considered
an inverse problem of determining a viscosity coefficient in NSE by observing
data in a neighborhood of the boundary. For an extensive study on various
inverse problems corresponding to NSE and related models, where one requires
to reconstruct the density of external forces or some coefficients of the
equations on the basis of final or functional overdetermination, we refer the
interested readers to [4, 9, 10, 11, 12, 19, 21, 25, 29, 34, 37, 42], etc.,
and references therein.

The existence and uniqueness of an inverse problem for three-dimensional
nonlinear equations of Kelvin-Voigt fluids is proved in [24]. Using the
contraction mapping principle, a local in time existence and uniqueness result
of an inverse problem for Kelvin-Voigt fluid flow equations with memory term
and integral overdetermination condition is obtained in [26]. The authors in
[2] established the local unique solvability result of an inverse problem for
generalized Kelvin-Voigt equation with $p$-Laplacian and damping term with
integral overdetermination condition. Recently, the well-posedness of an
inverse problem for 2D and 3D convective Brinkman-Forchheimer equations with
final overdetermination data using Schauder’s fixed point theorem is examined
in [27]. The results in [27] are obtained by assuming sufficient smoothness on
the given data ($\boldsymbol{u}_{0}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V}$).

###  1.3. Technical difficulties

We emphasize that, due to a technical difficulty in working with bounded
domains, the method employed in [14, 34, 41], etc. (for the initial data
$\boldsymbol{u}_{0}\in\mathbb{H}$) may only be applicable for $\ r\in[1,3]$ in
this article. Note that in the case of bounded domains,
$\mathrm{P}_{\mathbb{H}}(|\boldsymbol{u}|^{r-1}\boldsymbol{u})$
($\mathrm{P}_{\mathbb{H}}$ is the Helmholtz-Hodge orthogonal projection, see
subsection 2.1, [27]) need not be zero on the boundary, and
$\mathrm{P}_{\mathbb{H}}$ and $-\Delta$ are not necessarily commuting (for a
counter example, see Example 2.19, [38]). Furthermore, while taking the inner
product with $-\Delta\boldsymbol{u}$ in (1.1),
$-\Delta\boldsymbol{u}\cdot\boldsymbol{n}|_{\partial\Omega}\neq 0$ in general
and the term with pressure will not vanish (see [23]). As a result, the
equality ([18])

$\displaystyle\int_{\Omega}(-\Delta\boldsymbol{u}(x))\cdot|\boldsymbol{u}(x)|^{r-1}\boldsymbol{u}(x)\/\mathrm{d}\/x$
$\displaystyle=\int_{\Omega}|\nabla\boldsymbol{u}(x)|^{r-1}\boldsymbol{u}(x)\/\mathrm{d}\/x+4\bigg{(}\frac{r-1}{(r+1)^{2}}\bigg{)}\int_{\Omega}|\nabla|\boldsymbol{u}(x)|^{\frac{r+1}{2}}|^{r}\/\mathrm{d}\/x$
$\displaystyle=\int_{\Omega}|\nabla\boldsymbol{u}(x)|^{2}|\boldsymbol{u}(x)|^{r-1}\/\mathrm{d}\/x+\frac{r-1}{4}\int_{\Omega}|\boldsymbol{u}(x)|^{r-3}|\nabla|\boldsymbol{u}(x)|^{2}|^{2}\/\mathrm{d}\/x,$

may not be helpful in bounded domains.

###  1.4. Main results

By a solution of the inverse problem (1.1)-(1.5), we mean by a triplet
$(\boldsymbol{u},p,f)$ such that

$\displaystyle\boldsymbol{u}\in\mathrm{L}^{\infty}(0,T;\mathbb{H})\cap\mathrm{L}^{2}(0,T;\mathbb{V})\cap\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1}),\
p\in\mathrm{L}^{\frac{r+1}{r}}(0,T;\mathrm{L}_{0}^{2}(\Omega)),\
f\in\mathrm{L}^{2}(0,T),$

and in addition, all the relations (1.1)-(1.5) hold. In order to prove the
existence and uniqueness of solutions for the above investigated inverse
problem, we use the method developed in [34, 41]. Note that the work [34, 41]
does not take into account of the stability of the solution. The major goals
of this article are

  1. (i)

the existence and uniqueness of the solution,

  2. (ii)

the stability of the solution in the norm of corresponding function spaces,

to the inverse problem (1.1)-(1.5) under the assumptions
$\boldsymbol{u}_{0}\in\mathbb{H},\ \boldsymbol{g}\in\mathrm{C}([0,T];\
\mathbb{L}^{2}(\Omega)),\ \varphi\in\mathrm{H}^{1}(0,T)$, using an extension
of the contraction mapping theorem. In contrast to the results obtained for
CBF equations in [27], the well-posedness of the generalized solution of the
inverse problem holds for the initial data $\boldsymbol{u}_{0}\in\mathbb{H}$
in two dimensions for $r\in[1,3]$ and the additional information given in the
form of integral overdetermination condition.

######  Definition 1.1 ([41]).

Let $r\in[1,3]$. A triplet $(\boldsymbol{u},p,f)$ is said to be a _weak
solution_ of a nonlinear inverse problem (1.1)-(1.5) if

$\boldsymbol{u}\in\mathrm{L}^{\infty}(0,T;\mathbb{H})\cap\mathrm{L}^{2}(0,T;\mathbb{V})\cap\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1}),\
p\in\mathrm{L}^{\frac{r+1}{r}}(0,T;\mathrm{L}_{0}^{2}(\Omega)),$
$\int_{\Omega}\boldsymbol{u}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x\in\mathrm{H}^{1}(0,T),\
\text{ for all }\ \mathbf{v}\in\mathbb{V};\ \ f\in\mathrm{L}^{2}(0,T),$

and they satisfy the integral identity

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\int_{\Omega}\boldsymbol{u}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x+\mu\int_{\Omega}\nabla\boldsymbol{u}(x,t)\cdot\nabla\mathbf{v}(x)\/\mathrm{d}\/x$
$\displaystyle+\int_{\Omega}(\boldsymbol{u}(x,t)\cdot\nabla)\boldsymbol{u}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x+\alpha\int_{\Omega}\boldsymbol{u}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x$
$\displaystyle+\beta\int_{\Omega}|\boldsymbol{u}(x,t)|^{r-1}\boldsymbol{u}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x=\int_{\Omega}f(t)\boldsymbol{g}(x,t)\cdot\mathbf{v}(x)\/\mathrm{d}\/x,$
(1.6)

along with the initial condition (1.3) and the integral overdetermination
condition (1.5), for any $\mathbf{v}\in\mathbb{V}$.

In this definition, we have incorporated the incompressibility of the fluid
velocity and the boundary condition (1.4) as in Definition 1.1 in the sense
that the function $\boldsymbol{u}(\cdot,t)\in\mathbb{V}$ for a.e. $t\in[0,T]$.
Let us now state the main result in this work on the well-posedness of
solutions of the inverse problem (1.1)-(1.5).

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{2},\ \boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\varphi\in\mathrm{H}^{1}(0,T)$

$\text{ and }\
\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|\geq
g_{0}>0,\ \ (g_{0}\equiv\text{constant}),\ \ \ 0\leq t\leq T,$

and in addition, $\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty}$
and the compatibility condition

$\displaystyle\int_{\Omega}\boldsymbol{u}_{0}(x)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=\varphi(0),$
(1.7)

be satisfied. Then under the assumptions of Theorem 3.2 (see below), the
following assertions are satisfied for the inverse problem (1.1)-(1.5):

  1. (i)

There exists a unique weak solution $(\boldsymbol{u},p,f)$ to the inverse
problem (1.1)-(1.5).

  2. (ii)

Let $(\boldsymbol{u}_{i},p_{i},f_{i})$ $(i=1,2)$ be two solutions of the
inverse problem (1.1)-(1.5) corresponding to the input data
$(\boldsymbol{u}_{0i},\varphi_{i},\boldsymbol{g}_{i})\ (i=1,2)$. Then there
exists a constant $C$ such that

$\displaystyle\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{\infty}(0,T;\mathbb{H})}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{2}(0,T;\mathbb{V})}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}$
$\displaystyle\quad+\|p_{1}-p_{2}\|_{\mathrm{L}^{\frac{r+1}{r}}(0,T;\mathrm{L}^{2}(\Omega))}+\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\leq
C\bigg{(}\|\boldsymbol{u}_{01}-\boldsymbol{u}_{02}\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|(\boldsymbol{g}_{1}-\boldsymbol{g}_{2})(t)\|_{\mathbb{L}^{2}}+\|\varphi_{1}-\varphi_{2}\|_{\mathrm{H}^{1}(0,T)}\bigg{)},$
(1.8)

where $C$ depends on the input data, $\mu,\alpha,\beta,r$,$T$ and $\Omega$.

The rest of the paper is organized as follows: In section 2, we first state
and prove the relation between the solvability of the inverse problem
(1.1)-(1.5) and an equivalent operator equation of the second kind (Theorem
2.2). In section 3, we prove our main result, that is, Theorem 1.2, by first
showing the existence of a solution to the equivalent operator equation by
using the extension of the contraction mapping theorem and then demonstrating
the uniqueness and stability of the solution to the inverse problem. In
Appendix A, we deduce a number of a-priori estimates that are needed to handle
the inverse problem (1.1)-(1.5).

##  2\. Mathematical formulation

In this section, we state and prove Theorem 2.2, which converts the inverse
problem (1.1)-(1.5) into an equivalent nonlinear operator equation of the
second kind (2.3) and prove their equivalence. We commence this section by
introducing function spaces and some standard notations (cf. Section 2.1, [31,
32]), which will be used throughout the paper.

###  2.1. Function spaces

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain with smooth boundary
$\partial\Omega$. Let $\mathrm{C}_{0}^{\infty}(\Omega;\mathbb{R}^{2})$ be the
space of all infinitely differentiable functions ($\mathbb{R}^{2}$-valued)
with compact support in $\Omega\subset\mathbb{R}^{2}$. Let us define

$\displaystyle\mathcal{V}$
$\displaystyle:=\\{\boldsymbol{u}\in\mathrm{C}_{0}^{\infty}(\Omega,\mathbb{R}^{2}):\nabla\cdot\boldsymbol{u}=0\\},$
$\displaystyle\mathbb{H}$ $\displaystyle:=\text{the closure of }\ \mathcal{V}\
\text{ in the Lebesgue space
}\mathbb{L}^{2}(\Omega)=\mathrm{L}^{2}(\Omega;\mathbb{R}^{2}),$
$\displaystyle\mathbb{V}$ $\displaystyle:=\text{the closure of }\ \mathcal{V}\
\text{ in the Sobolev space
}\mathbb{H}^{1}(\Omega)=\mathrm{H}^{1}(\Omega;\mathbb{R}^{2}),$
$\displaystyle\widetilde{\mathbb{L}}^{p}$ $\displaystyle:=\text{the closure of
}\ \mathcal{V}\ \text{ in the Lebesgue space
}\mathbb{L}^{p}(\Omega)=\mathrm{L}^{p}(\Omega;\mathbb{R}^{2}),$

for $p\in(2,\infty]$. Then, under some smoothness assumptions on the boundary,
we characterize the spaces $\mathbb{H}$, $\mathbb{V}$,
$\widetilde{\mathbb{L}}^{p}$ and $\widetilde{\mathbb{L}}^{\infty}$ as
$\mathbb{H}=\\{\boldsymbol{u}\in\mathbb{L}^{2}(\Omega):\nabla\cdot\boldsymbol{u}=0,\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\Omega}=0\\},$
where $\boldsymbol{n}$ is the unit outward drawn normal to $\partial\Omega$,
and $\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\Omega}$ should be
understood in the sense of trace in $\mathbb{H}^{-1/2}(\partial\Omega)$ (cf.
Theorem 1.2, Chapter 1, [40]), with the norm
$\|\boldsymbol{u}\|_{\mathbb{H}}^{2}:=\int_{\Omega}|\boldsymbol{u}(x)|^{2}\/\mathrm{d}\/x,$
$\mathbb{V}=\\{\boldsymbol{u}\in\mathbb{H}_{0}^{1}(\Omega):\nabla\cdot\boldsymbol{u}=0\\},$
with the norm
$\|\boldsymbol{u}\|_{\mathbb{V}}^{2}:=\int_{\Omega}|\nabla\boldsymbol{u}(x)|^{2}\/\mathrm{d}\/x$
(since $\Omega$ is a bounded smooth domain),
$\widetilde{\mathbb{L}}^{p}=\\{\boldsymbol{u}\in\mathbb{L}^{p}(\Omega):\nabla\cdot\boldsymbol{u}=0,\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\Omega}=0\\},$
with the norm
$\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{p}}^{p}:=\int_{\Omega}|\boldsymbol{u}(x)|^{p}\/\mathrm{d}\/x$,
and
$\widetilde{\mathbb{L}}^{\infty}=\\{\boldsymbol{u}\in\mathbb{L}^{\infty}(\Omega):\nabla\cdot\boldsymbol{u}=0,\boldsymbol{u}\cdot\boldsymbol{n}\big{|}_{\partial\Omega}=0\\},$
with the norm
$\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{\infty}}:=\operatorname*{ess\,sup}\limits_{x\in\Omega}|\boldsymbol{u}(x)|$,
respectively. Let $(\cdot,\cdot)$ denote the inner product in the Hilbert
space $\mathbb{H}$ and $\langle\cdot,\cdot\rangle$ represents the duality
pairing between the spaces $\mathbb{V}$ and its dual $\mathbb{V}^{\prime}$,
and $\widetilde{\mathbb{L}}^{p}$ and its dual
$\widetilde{\mathbb{L}}^{\frac{p}{p-1}}$. The following well-known
Ladyzhenkaya inequality (cf. Lemma 1, [28])

$\|\boldsymbol{u}\|_{\mathbb{L}^{4}}\leq
2^{1/4}\|\boldsymbol{u}\|_{\mathbb{L}^{2}}^{1/2}\|\nabla\boldsymbol{u}\|_{\mathbb{L}^{2}}^{1/2},\
\text{ for all }\ \boldsymbol{u}\in\mathbb{H}_{0}^{1}(\Omega)$

is used repeatedly in the paper. Wherever needed, we assume that
$p_{0}\in\mathrm{L}^{2}_{0}(\Omega),$ where
$\mathrm{L}^{2}_{0}(\Omega):=\mathopen{}\mathclose{{}\left\\{p\in\mathrm{L}^{2}(\Omega):\int_{\Omega}p(x)\/\mathrm{d}\/x=0}\right\\}$.
In the sequel, $C$ denotes a generic constant which may take different values
at different places.

Later discussions of the equivalent formulation of an inverse problem
(1.1)-(1.5) are based on the works [34, 41], in which the authors established
a relationship between the solvability of the inverse problem for NSE in two
dimensions and an operator equation of the second kind.

###  2.2. Equivalent formulation

We require the following input data function requirements:

$\boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty},\
\varphi\in\mathrm{H}^{1}(0,T),$ $\text{and}\ \
\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|\geq
g_{0}>0,\ \ (g_{0}\equiv\text{constant}),\ \ \ 0\leq t\leq T.$

Under the above assumptions, we derive an operator equation of the second kind
for the scalar function $f$. We define the nonlinear operator
$\mathcal{A}:\mathrm{L}^{2}(0,T)\to\mathrm{L}^{2}(0,T)$ by

$\displaystyle(\mathcal{A}f)(t):=\frac{1}{g_{1}(t)}\bigg{\\{}\int_{\Omega}\big{(}\mu\nabla\boldsymbol{u}\cdot\nabla\boldsymbol{\omega}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}\cdot\boldsymbol{\omega}+\alpha\boldsymbol{u}\cdot\boldsymbol{\omega}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}\cdot\boldsymbol{\omega}\big{)}\/\mathrm{d}\/x+\varphi^{\prime}(t)\bigg{\\}},$
(2.1)

where $\boldsymbol{u}$ is obtained by using the weak solution of the direct
problem (1.1)-(1.4) and

$\displaystyle
g_{1}(t)=\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x.$
(2.2)

We analyze the following nonlinear operator equation of the second kind over
the space $\mathrm{L}^{2}(0,T)$:

$\displaystyle f=\mathcal{A}f.$ (2.3)

The solvability of the inverse problem (1.1)-(1.5) is inevitably connected to
the fixed points of the operator $\mathcal{A}$. For this purpose, we first
consider a closed ball $\mathcal{D}$ in space $\mathrm{L}^{2}(0,T)$ with
center $\varphi^{\prime}/g_{1}$ such that

$\displaystyle\mathcal{D}=\big{\\{}f:f\in\mathrm{L}^{2}(0,T)\text{ and
}\|f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}\leq a\big{\\}},$ (2.4)

where

$g_{1}(t)=\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x.$

######  Remark 2.1.

Since the equation (1.1) is taken in the divergence free space $\mathbb{V}$,
the pressure $p(\cdot,\cdot)$ does not appear in the equations (1.1) and
(2.1). The pressure $p(\cdot,\cdot)$ can be recovered from the equation (1.1)
after reconstructing the pair $(\boldsymbol{u},f)$.

The relation between the solvability of the inverse problem (1.1)-(1.5) and
the nonlinear operator equation of the second kind (2.3) is verified by the
following theorem.

######  Theorem 2.2.

Let $\Omega\subset\mathbb{R}^{2},\ \boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\varphi\in\mathrm{H}^{1}(0,T)$

$\text{ and }\
\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|\geq
g_{0}>0,\ \ (g_{0}\equiv\text{constant}),\ \ \ 0\leq t\leq T.$

Then the following assertions hold true:

  1. (i)

If the inverse problem (1.1)-(1.5) has a solution, then the operator equation
(2.3) also has a solution.

  2. (ii)

If the operator equation (2.3) has a solution and the compatibility condition
(1.7) is satisfied, then there exists a solution of the inverse problem
(1.1)-(1.5).

###### Proof.

Let us first prove (i). Let the inverse problem (1.1)-(1.5) has a solution,
say $(\boldsymbol{u},p,f)$. Multiplying both sides of (1.1) by
$\boldsymbol{\omega}(x)$, and integrating by parts, we obtain

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\mu\int_{\Omega}\nabla\boldsymbol{u}(x,t)\cdot\nabla\boldsymbol{\omega}(x)\/\mathrm{d}\/x$
$\displaystyle\quad+\int_{\Omega}(\boldsymbol{u}(x,t)\cdot\nabla)\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\alpha\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x$
$\displaystyle\quad+\beta\int_{\Omega}|\boldsymbol{u}(x,t)|^{r-1}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=f(t)\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x.$
(2.5)

From the definition (2.1) of the operator $\mathcal{A}$, the integral
overdetermination condition (1.5), and assumption (2.2), one can easily deduce
from (2.2) that

$\mathcal{A}f=f.$

This implies that the function $f$ solves the operator equation (2.3); thereby
proving (i).

We proceed to prove (ii). Let the operator equation (2.3) has a solution, say
$f\in\mathrm{L^{2}}(0,T)$. Upon substituting $f$ into (1.1), we use the system
(1.1)-(1.4) to find a unique weak solution $(\boldsymbol{u}(\cdot),p(\cdot))$
of the direct problem (1.1)-(1.4). We claim that the function
$\boldsymbol{u}(\cdot)$ satisfies the integral overdetermination condition
(1.5). By inserting $\mathbf{v}(x)=\boldsymbol{\omega}(x)$ in (1.1), and using
assumption (2.2), we arrive at

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\mu\int_{\Omega}\nabla\boldsymbol{u}(x,t)\cdot\nabla\boldsymbol{\omega}(x)\/\mathrm{d}\/x$
$\displaystyle\quad+\int_{\Omega}(\boldsymbol{u}(x,t)\cdot\nabla)\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\alpha\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x$
$\displaystyle\quad+\beta\int_{\Omega}|\boldsymbol{u}(x,t)|^{r-1}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=f(t)g_{1}(t).$
(2.6)

On the other hand, since the function $f$ is a solution of the operator
equation (2.3), we have

$\displaystyle\varphi^{\prime}(t)+\mu\int_{\Omega}\nabla\boldsymbol{u}(x,t)\cdot\nabla\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\int_{\Omega}(\boldsymbol{u}(x,t)\cdot\nabla)\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x$
$\displaystyle\quad+\alpha\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x+\beta\int_{\Omega}|\boldsymbol{u}(x,t)|^{r-1}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=f(t)g_{1}(t).$
(2.7)

Subtracting the equation (2.2) from the equation (2.2), we find

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x-\varphi^{\prime}(t)=0.$
(2.8)

Integrating the equation (2.8) with respect to time $t$ from $0$ to $t$, and
then using the compatibility condition (1.7), we obtain

$\displaystyle\int_{\Omega}\boldsymbol{u}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x=\varphi(t),$

which is the integral overdetermination condition (1.5). Having obtained this,
we can say that the triplet $(\boldsymbol{u},p,f)$ is a weak solution of the
inverse problem (1.1)-(1.5) and this proves the second assertion of the
theorem. ∎

##  3\. Proof of Theorem 1.2

The energy estimates obtained in Appendix A allow us to demonstrate the
existence and uniqueness of solution to the inverse problem (1.1)-(1.5) as
well as the stability of the solution obtained. For proving the existence of a
solution to the inverse problem (1.1)-(1.5), by using Theorem 2.2, it suffices
to show that the nonlinear operator $\mathcal{A}$ has a fixed point in
$\mathcal{D}$ and this follows from an application of the extension of the
contraction mapping theorem. Subsequently, arguments for the existence and
uniqueness of the solution of the inverse problem (1.1)-(1.5) are based on the
works [34, 41], where the existence and uniqueness of a solution of the
inverse problem for 2D NSE has been investigated, by exploiting an extension
of the contraction mapping theorem.

######  Theorem 3.1 (Theorem 2.4, [1]).

Let $(\mathbb{X},d)$ be a complete metric space and let
$\mathcal{A}:\mathbb{X}\to\mathbb{X}$ be a mapping such that for some positive
integer $k$, $\mathcal{A}^{k}$ is a contraction on $\mathbb{X}$. Then,
$\mathcal{A}$ has a unique fixed point.

###  3.1. Existence

The solvability of the inverse problem (1.1)-(1.5) is inevitably connected to
the fixed points of the operator $\mathcal{A}$. Now, we demonstrate that the
operator $\mathcal{A}$ has a unique fixed point. The following theorem
provides sufficient conditions under which the operator $\mathcal{A}$ maps the
closed ball $\mathcal{D}$ into itself.

######  Theorem 3.2.

Let $\Omega\subset\mathbb{R}^{2},\ \boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty},\
\varphi\in\mathrm{H}^{1}(0,T)$ and

$\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|\geq
g_{0}>0,\ \ (g_{0}\equiv\text{constant}),\ \ \ 0\leq t\leq T.$

  1. (i)

For $r\in(2,3]$, if

$\displaystyle m_{1}<a,$ (3.1)

where

$\displaystyle m_{1}$
$\displaystyle=\frac{C}{g_{0}}\bigg{[}T\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\qquad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\qquad+T^{\frac{3-r}{r-1}}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}^{\frac{4}{(r-1)}}$
$\displaystyle\qquad\times\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}^{\frac{2(r-2)}{(r-1)}}\bigg{]}^{\frac{1}{2}},$
(3.2) $\widetilde{a}=a+\|\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)},$

and $a$ is the radius of the closed ball $\mathcal{D}$.

  2. (ii)

For $r\in[1,2]$, if

$\displaystyle m_{2}<a,$ (3.3)

where

$\displaystyle m_{2}$
$\displaystyle=\frac{CT^{1/2}}{g_{0}}\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\quad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}$
$\displaystyle\quad+\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}^{r}\bigg{\\}}.$
(3.4)

Then the operator $\mathcal{A}$ maps the closed ball $\mathcal{D}$ into
itself.

###### Proof.

Let $f$ be an arbitrary fixed function in $\mathcal{D}$, then by the
definition of the ball $\mathcal{D}$, we deduce that

$\displaystyle\|f\|_{\mathrm{L}^{2}(0,T)}\leq\widetilde{a}.$ (3.5)

Case I: _ $r\in(2,3]$._ The norm of the function
$\mathcal{A}f-\varphi^{\prime}/g_{1}$ can be estimated as follows:

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle=\bigg{(}\int_{0}^{T}|\mathcal{A}f-\varphi^{\prime}/g_{1}|^{2}\/\mathrm{d}\/t\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{g_{0}}\bigg{[}\int_{0}^{T}\bigg{(}\mu\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}\|\boldsymbol{u}(t)\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}$
$\displaystyle\quad+\alpha\|\boldsymbol{\omega}\|_{\mathbb{H}}\|\boldsymbol{u}(t)\|_{\mathbb{H}}+\beta\|\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r}}^{r}\bigg{)}^{2}\/\mathrm{d}\/t\bigg{]}^{\frac{1}{2}}.$
(3.6)

We estimate
$\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r}}^{2r}\/\mathrm{d}\/t$
by using interpolation and Hölder’s inequalities as

$\displaystyle\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r}}^{2r}\/\mathrm{d}\/t$
$\displaystyle\leq\int_{0}^{T}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{\frac{2}{r(r-1)}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{(r-2)(r+1)}{r(r-1)}}}\right)^{2r}\/\mathrm{d}\/t$
$\displaystyle\leq\int_{0}^{T}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{\frac{4}{(r-1)}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{2(r-2)(r+1)}{(r-1)}}}\right)\/\mathrm{d}\/t$
$\displaystyle\leq
T^{\frac{3-r}{r-1}}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{\frac{4}{(r-1)}}\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t}\right)^{\frac{2(r-2)}{(r-1)}}.$
(3.7)

Substituting the estimate (3.1) in (3.1), we obtain

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\quad\leq\frac{C}{g_{0}}\bigg{[}T\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\}}^{2}$
$\displaystyle\qquad+T^{\frac{3-r}{r-1}}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{\frac{4}{(r-1)}}\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t}\right)^{\frac{2(r-2)}{(r-1)}}\bigg{]}^{\frac{1}{2}}.$
(3.8)

With reference to (A.1) and (A.1), it can be seen from (3.1) that

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\quad\leq\frac{C}{g_{0}}\bigg{[}T\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|f\|_{\mathrm{L^{2}}(0,T)}\bigg{)}$
$\displaystyle\qquad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L^{2}}(0,T)}^{2}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\qquad+T^{\frac{3-r}{r-1}}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|f\|_{\mathrm{L^{2}}(0,T)}\bigg{)}^{\frac{4}{(r-1)}}$
$\displaystyle\qquad\times\bigg{\\{}\frac{1}{2\beta}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L^{2}}(0,T)}^{2}\bigg{)}\bigg{\\}}^{\frac{2(r-2)}{(r-1)}}\bigg{]}^{\frac{1}{2}}.$
(3.9)

Using (3.5) in (3.1), we deduce that

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}\leq
m_{1},$ (3.10)

where $m_{1}$ is defined in ((i)). One can take $T$ sufficiently small such
that $m_{1}<a$ (if needed $a$ can be chosen sufficiently large so that
$a>\|\boldsymbol{u}_{0}\|_{\mathbb{H}}$). For the case $r=3$, apart from
smallness of $T$, one may need to restrict the input data also. Using this
relation, the estimate (3.10) immediately implies that the nonlinear operator
$\mathcal{A}$ brings the ball $\mathcal{D}$ into itself.

Case II: _ $r\in[1,2]$._ From (3.1), the norm of the function
$\mathcal{A}f-\varphi^{\prime}/g_{1}$ can be estimated as follows

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\quad\leq\frac{C}{g_{0}}\bigg{[}T\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\}}^{2}$
$\displaystyle\qquad+|\Omega|^{2-r}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2r}\/\mathrm{d}\/t\bigg{]}^{\frac{1}{2}}$
$\displaystyle\quad\leq\frac{CT^{1/2}}{g_{0}}\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}$
$\displaystyle\qquad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{r}\bigg{\\}},$
(3.11)

where $|\Omega|$ is the Lebesgue measure of $\Omega$. Substituting the
estimates (A.1) and (A.1) in the inequality (3.1), we get

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\quad\leq\frac{CT^{1/2}}{g_{0}}\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|f\|_{\mathrm{L^{2}}(0,T)}\bigg{)}$
$\displaystyle\qquad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L^{2}}(0,T)}^{2}\bigg{)}$
$\displaystyle\qquad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|f\|_{\mathrm{L^{2}}(0,T)}\bigg{)}^{r}\bigg{\\}}.$
(3.12)

Using the relation (3.5) in (3.1) results to

$\displaystyle\|\mathcal{A}f-\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}\leq
m_{2},$ (3.13)

where $m_{2}$ is defined in ((ii)). One can choose $T$ sufficiently small so
that $m_{2}<a$. Considering this relationship, the estimate (3.13) implies
that the nonlinear operator $\mathcal{A}$ brings the ball $\mathcal{D}$ into
itself.

As a result, for $r\in[1,3]$, the nonlinear operator $\mathcal{A}$ maps the
ball $\mathcal{D}$ into itself, completing the proof of the theorem. ∎

The next theorem establishes the existence of a solution to the inverse
problem (1.1)-(1.5).

######  Theorem 3.3.

Let $\Omega\subset\mathbb{R}^{2},\ \boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty},\
\varphi\in\mathrm{H}^{1}(0,T)$ and

$\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x,t)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|\geq
g_{0}>0,\ \ (g_{0}\equiv constant),\ \ \ 0\leq t\leq T.$

Also assume that the nonlinear operator $\mathcal{A}$ maps the ball
$\mathcal{D}$ into itself. Then there exists a positive integer $k$ such that
the operator $\mathcal{A}^{k}$ is a contraction mapping in the ball
$\mathcal{D}$.

###### Proof.

Let both of the functions $f_{1}$ and $f_{2}$ belong to the ball
$\mathcal{D}$. Let $\boldsymbol{u}_{1}$ and $\boldsymbol{u}_{2}$ be the
functions corresponding to the coefficients $f_{1}$ and $f_{2}$, respectively.
By the definition of the operator $\mathcal{A}$, we have

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle=\int_{0}^{T}\mathopen{}\mathclose{{}\left|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right|^{2}\/\mathrm{d}\/t$
$\displaystyle\leq\frac{1}{g_{0}^{2}}\int_{0}^{T}\bigg{(}\mu\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}$
$\displaystyle\quad\times\big{(}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\big{)}+\alpha\|\boldsymbol{\omega}\|_{\mathbb{H}}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}$
$\displaystyle\quad+\beta\|\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\||\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)\|_{\mathbb{L}^{1}}\bigg{)}^{2}\/\mathrm{d}\/t$
$\displaystyle\leq\frac{C}{g_{0}^{2}}\bigg{[}T\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}$
$\displaystyle\quad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\int_{0}^{T}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)}\right\|_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t\bigg{]}.$
(3.14)

Let us define $h(\boldsymbol{u}):=|\boldsymbol{u}|^{r-1}\boldsymbol{u}$. Then,
by applying Taylor’s formula (Theorem 7.9.1, [8]), we obtain (cf. [32])

$\displaystyle\int_{0}^{T}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right\|_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t$
$\displaystyle=\int_{0}^{T}\bigg{\|}\int_{0}^{1}h^{\prime}(\theta\boldsymbol{u}_{1}+(1-\theta)\boldsymbol{u}_{2})\/\mathrm{d}\/\theta(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\bigg{\|}_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t$
$\displaystyle\leq\int_{0}^{T}\sup_{0<\theta<1}\big{\|}\big{(}\boldsymbol{u}_{1}(t)-\boldsymbol{u}_{2}(t)\big{)}\big{|}\theta\boldsymbol{u}_{1}(t)+(1-\theta)\boldsymbol{u}_{2}(t)\big{|}^{r-1}$
$\displaystyle\quad+(r-1)\big{(}\theta\boldsymbol{u}_{1}(t)+(1-\theta)\boldsymbol{u}_{2}(t)\big{)}\big{|}\theta\boldsymbol{u}_{1}(t)+(1-\theta)\boldsymbol{u}_{2}(t)\big{|}^{r-3}$
$\displaystyle\quad\times\big{(}(\theta\boldsymbol{u}_{1}(t)+(1-\theta)\boldsymbol{u}_{2}(t))\cdot(\boldsymbol{u}_{1}(t)-\boldsymbol{u}_{2}(t))\big{)}\big{\|}_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t$
$\displaystyle\leq
C\int_{0}^{T}\sup_{0<\theta<1}\big{\|}|\theta\boldsymbol{u}_{1}(t)+(1-\theta)\boldsymbol{u}_{2}(t)|^{r-1}\big{\|}_{\mathbb{H}}^{2}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/t$
$\displaystyle\leq
C\int_{0}^{T}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{2(r-1)}}^{2(r-1)}+\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{2(r-1)}}^{2(r-1)}}\right)\/\mathrm{d}\/t$
$\displaystyle\leq
C\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\int_{0}^{T}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{2(r-1)}}^{2(r-1)}+\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{2(r-1)}}^{2(r-1)}}\right)\/\mathrm{d}\/t.$
(3.15)

Case I: _ $r\in(2,3]$._ An application of Gagliardo-Nirenberg’s and Hölder’s
inequalities in (3.1) yields

$\displaystyle\int_{0}^{T}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)}\right\|_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t$
$\displaystyle\quad\leq
C\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\int_{0}^{T}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2(r-2)}+\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2(r-2)}}\right)\/\mathrm{d}\/t$
$\displaystyle\quad\leq
C\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2(r-2)}\/\mathrm{d}\/t$
$\displaystyle\qquad+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2(r-2)}\/\mathrm{d}\/t\bigg{)}$
$\displaystyle\quad\leq
CT^{3-r}\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}$
$\displaystyle\qquad+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}\bigg{\\}}.$
(3.16)

Substituting the relations (A.2) and (3.1) in (3.1), we arrive at

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle\quad\leq\frac{C}{g_{0}^{2}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right)$
$\displaystyle\qquad\times\bigg{[}T\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\qquad+T^{3-r}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{\\{}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}$
$\displaystyle\qquad+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}\bigg{\\}}\bigg{]}.$
(3.17)

The relation (3.1) simplifies to

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}^{2}\leq
m_{3}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2},$ (3.18)

where

$\displaystyle m_{3}$
$\displaystyle=\frac{CT^{3-r}}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right)$
$\displaystyle\quad\times\bigg{[}T^{r-2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{\\{}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}$
$\displaystyle\quad+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}\bigg{\\}}\bigg{]}.$

Using the energy estimates (A.1) and (A.1), the quantity $m_{3}$ can be
bounded by

$\displaystyle m_{3}$
$\displaystyle\leq\frac{CT^{3-r}}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{\\}}$
$\displaystyle\quad\times\bigg{[}T^{r-2}\bigg{\\{}\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}2\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}$
$\displaystyle\quad\times\mathopen{}\mathclose{{}\left(\|f_{1}\|_{\mathrm{L}^{2}(0,T)}+\|f_{2}\|_{\mathrm{L}^{2}(0,T)}}\right)\bigg{)}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}\bigg{\\}}^{2}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{\\{}2\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\mathopen{}\mathclose{{}\left(\|f_{1}\|_{\mathrm{L}^{2}(0,T)}^{2}+\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}}\right)\bigg{\\}}^{r-1}\bigg{]}.$
(3.19)

Since the functions $f_{1}$ and $f_{2}$ lie within the ball $\mathcal{D}$, a
combination of the inequalities (3.18) and (3.1) gives the estimate

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}\leq\mathopen{}\mathclose{{}\left(m_{4}T^{3-r}}\right)^{1/2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)},$
(3.20)

where

$\displaystyle m_{4}$
$\displaystyle=\frac{C}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{\\}}$
$\displaystyle\quad\times\bigg{[}T^{r-2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}^{r-1}\bigg{]}\
\text{ and }$ $\displaystyle\widetilde{a}$
$\displaystyle=a+\|\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)},$

and $a$ is the radius of the ball $\mathcal{D}$. Note that $m_{4}$ is
expressed only in terms of the input data.

In accordance with the assumption made, the operator $\mathcal{A}$ maps the
ball $\mathcal{D}$ into itself which makes it possible to define for any
positive integer $k$, the $k^{\mathrm{th}}$ degree of the operator
$\mathcal{A}$. Let this operator be denoted by the symbol $\mathcal{A}^{k}$.
Using the standard technique of mathematical induction on $k$ the inequality
(3.20) validates the estimate

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}^{k}f_{1}-\mathcal{A}^{k}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}\leq\bigg{(}\frac{m_{4}^{k}T^{(3-r)k}}{k!}\bigg{)}^{1/2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}.$
(3.21)

It is visible that

$\displaystyle\big{(}m_{4}^{k}T^{(3-r)k}/k!\big{)}^{1/2}\rightarrow 0\ \
\text{ as }\ \ k\rightarrow\infty.$

This implies that there exists a positive integer $k_{0}$ such that

$\displaystyle\big{(}m_{4}^{k_{0}}T^{(3-r)k_{0}}/k_{0}!\big{)}^{1/2}<1.$

Due to estimate (3.21), one can conclude that the operator
$\mathcal{A}^{k_{0}}$ is a contracting mapping in the ball $\mathcal{D}$. Case
II: _ $r\in[1,2]$._ Applying Hölder’s inequality in (3.1) gives

$\displaystyle\int_{0}^{T}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right\|_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t$
$\displaystyle\quad\leq
C\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\int_{0}^{T}\mathopen{}\mathclose{{}\left(|\Omega|^{2-r}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2(r-1)}+|\Omega|^{2-r}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2(r-1)}}\right)\/\mathrm{d}\/t$
$\displaystyle\quad\leq
CT\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2(r-1)}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2(r-1)}\bigg{)},$
(3.22)

where $|\Omega|$ is the Lebesgue measure of $\Omega$. Substitution of the
relations (A.2) and (3.1) in (3.1) results to

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle\leq\frac{CT}{g_{0}^{2}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right)$
$\displaystyle\quad\times\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}^{r-1}\bigg{\\}}^{2}.$

From the above estimate, it follows that

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}^{2}\leq
m_{5}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2},$ (3.23)

where

$\displaystyle m_{5}$
$\displaystyle=\frac{CT}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right)$
$\displaystyle\quad\times\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}\bigg{)}^{r-1}\bigg{\\}}^{2}.$

Using the estimates (A.1) and (A.1), the quantity $m_{5}$ can be bounded as

$\displaystyle m_{5}$
$\displaystyle\leq\frac{CT}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{\\}}\bigg{[}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}$
$\displaystyle\quad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}2\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\big{(}\|f_{1}\|_{\mathrm{L}^{2}(0,T)}+\|f_{2}\|_{\mathrm{L}^{2}(0,T)}\big{)}\bigg{)}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}2\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\big{(}\|f_{1}\|_{\mathrm{L}^{2}(0,T)}+\|f_{2}\|_{\mathrm{L}^{2}(0,T)}\big{)}\bigg{)}^{r-1}\bigg{]}^{2}.$
(3.24)

A combination of the inequalities (3.23) and (3.1) gives the following
estimate:

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}f_{1}-\mathcal{A}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}\leq(m_{6}T)^{1/2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)},$
(3.25)

where

$\displaystyle m_{6}$
$\displaystyle=\frac{C}{g_{0}^{2}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\exp\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\widetilde{a}^{2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{\\}}$
$\displaystyle\quad\times\bigg{[}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}+\|\nabla\boldsymbol{\omega}\|_{\mathbb{L}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\quad+\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\widetilde{a}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}^{r-1}\bigg{]}^{2}\
\text{ and }$ $\displaystyle\widetilde{a}$
$\displaystyle=a+\|\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)},$

and $a$ is the radius of the ball $\mathcal{D}$. The quantity $m_{6}$ is also
expressed in terms of the input data only.

Using the same arguments those made in the case for $r\in(2,3]$, we can define
the operator $\mathcal{A}^{k}$. Using the standard technique of mathematical
induction on $k,$ the inequality (3.25) validates the estimate

$\displaystyle\mathopen{}\mathclose{{}\left\|\mathcal{A}^{k}f_{1}-\mathcal{A}^{k}f_{2}}\right\|_{\mathrm{L}^{2}(0,T)}\leq\bigg{(}\frac{m_{6}^{k}T^{k}}{k!}\bigg{)}^{1/2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}.$
(3.26)

It is clear that

$\displaystyle\big{(}m_{6}^{k}T^{k}/k!\big{)}^{1/2}\rightarrow 0\ \ \text{ as
}\ \ k\rightarrow\infty.$

Therefore, there exists a positive integer $\tilde{k}_{0}$ such that

$\displaystyle\big{(}m_{6}^{\tilde{k}_{0}}T^{\tilde{k}_{0}}/\tilde{k}_{0}!\big{)}^{1/2}<1.$

From the estimate (3.26), it is immediate that the operator
$\mathcal{A}^{\tilde{k}_{0}}$ is a contracting mapping in the ball
$\mathcal{D}$. ∎

Let us now prove the existence of a solution $(\boldsymbol{u},p,f)$ to the
inverse problem (1.1)-(1.5).

###### Proof of Theorem 1.2 (i)-Existence.

Since the operator $\mathcal{A}^{k}$ is a contraction mapping in the ball
$\mathcal{D}$, an application of the extension of the contraction mapping
theorem (Theorem 3.1) yields the operator $\mathcal{A}$ has a unique fixed
point. Hence, the inverse problem (1.1)-(1.5) has a solution, which completes
the proof. ∎

###  3.2. Uniqueness

Let us now prove the uniqueness of the solution obtained in Theorem 1.2 (i).

###### Proof of Theorem 1.2 (i)-Uniqueness.

Assume on the contrary that there are two distinct solutions,
$(\boldsymbol{u}_{1},p_{1},f_{1})$ and $(\boldsymbol{u}_{2},p_{2},f_{2})$ of
the inverse problem (1.1)-(1.5) such that both $f_{1}$ and $f_{2}$ lie within
the ball $\mathcal{D}$.

We wish to prove that $f_{1}$ does not coincide with $f_{2}$ a.e. on $[0,T]$.
Indeed, if $f_{1}$ is equal to $f_{2}$ a.e. on $[0,T]$, then
$(\boldsymbol{u}_{1},p_{1})$ will coincide with $(\boldsymbol{u}_{2},p_{2})$
a.e. in $\Omega\times[0,T],$ which is in accordance with the uniqueness
theorem for the direct problem.

We begin by analyzing the first triplet $(\boldsymbol{u}_{1},p_{1},f_{1})$.
Using similar arguments as in the proof of Theorem 2.2, we can conclude that
the function $f_{1}$ represents a solution to the equation (2.3). Similar
arguments also prove that the function $f_{2}$ also solves the same equation
(2.3). But we have just proved the uniqueness of the solution to the equation
(2.3) in $\mathcal{D}$. Thus, we have arrived at the conclusion that the
assumption about the existence of two distinct solutions
$(\boldsymbol{u}_{i},p_{i},f_{i})\ (i=1,2)$ does not hold. Hence, the inverse
problem (1.1)-(1.5) has a unique solution, which completes the proof. ∎

We will now give an example (motivated from [41]) to show that the class of
functions satisfying the conditions of part (i) of the Theorem 1.2 is not
empty.

######  Example 3.4.

Let $\Omega\subset\mathbb{R}^{2},\ \boldsymbol{u}_{0}\in\mathbb{H},\
\boldsymbol{g}=\boldsymbol{g}(x),\ \boldsymbol{g}\in\mathbb{L}^{2}(\Omega),\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty},\
\varphi\in\mathrm{H}^{1}(0,T)$ and

$\mathopen{}\mathclose{{}\left|\int_{\Omega}\boldsymbol{g}(x)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x}\right|=g_{0}>0.$

If we consider

$\displaystyle\int_{\Omega}\boldsymbol{g}(x)\cdot\boldsymbol{\omega}(x)\/\mathrm{d}\/x\equiv\varphi(t),$

that is, $\varphi\equiv\text{constant}$, and if we set $a=1$, it is clear to
observe that $\widetilde{a}=1$, since
$\widetilde{a}=a+\|\varphi^{\prime}/g_{1}\|_{\mathrm{L}^{2}(0,T)}$. Case I: _
$r\in(2,3]$._ The inequality (3.1) has the following form:

$\displaystyle\frac{C}{g_{0}}\bigg{[}T\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\qquad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}\bigg{\\}}^{2}$
$\displaystyle\qquad+T^{\frac{3-r}{r-1}}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}^{\frac{4}{(r-1)}}$
$\displaystyle\qquad\times\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}^{\frac{2(r-2)}{(r-1)}}\bigg{]}^{\frac{1}{2}}<1.$
(3.27)

Clearly, as $T\to 0^{+}$, the left-hand side of (3.4) approaches zero for
$r\in(2,3)$. As a result, there exists a time $T^{\prime}>0$ such that for any
$T\in(0,T^{\prime}],$ the estimate (3.4) holds true. Based on this reasoning,
it appears reasonable to investigate the inverse problem (1.1)-(1.5), with
$T\in(0,T^{\prime}]$ and $a=1$, the radius of the ball $\mathcal{D}$.

For $r=3$, the condition becomes
$\frac{C}{g_{0}}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\|\boldsymbol{g}(0)\|_{\mathbb{L}^{2}}^{2}}\right)^{\frac{1}{2}}<1$.
Under this smallness assumption on the data, one can obtain the existence and
uniqueness of the inverse problem (1.1)-(1.5) for any $T\in(0,T^{\prime}]$.
Case II: _ $r\in[1,2]$._ The inequality (3.3) has the following form:

$\displaystyle\frac{CT^{1/2}}{g_{0}}\bigg{\\{}\big{(}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}+\|\boldsymbol{\omega}\|_{\mathbb{H}}\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\quad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)}$
$\displaystyle\quad+\boldsymbol{\omega}\|_{\mathbb{H}^{2}}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)}^{r}\bigg{\\}}<1.$
(3.28)

Clearly, as $T\to 0^{+}$, the left-hand side of (3.4) approaches zero. Thus,
there exists a time $T^{\prime}>0$ such that for any $T\in(0,T^{\prime}],$ the
estimate (3.4) holds true. Based on this reasoning, it seems acceptable to
investigate the inverse problem (1.1)-(1.5), with $T\in(0,T^{\prime}]$ and
$a=1$, the radius of the ball $\mathcal{D}$.

Hence, for $r\in[1,3]$, it is apparent that the inverse problem with these
input data satisfies the conditions of part (i) of Theorem 1.2.

###  3.3. Stability

In order to get a result on the stability, we first provide some supporting
Lemmas. Let $(\boldsymbol{u}_{i},p_{i},f_{i})$ $(i=1,2)$ be the solutions of
inverse problem (1.1)-(1.5) corresponding to the given data
$(\boldsymbol{u}_{0i},\varphi_{i},\boldsymbol{g}_{i})$ $(i=1,2)$ and set

$\displaystyle\boldsymbol{u}$
$\displaystyle:=\boldsymbol{u}_{1}-\boldsymbol{u}_{2},\ \ \ \ \ \
p:=p_{1}-p_{2},\ \ \ \ \ f:=f_{1}-f_{2},$ $\displaystyle\boldsymbol{u}_{0}$
$\displaystyle:=\boldsymbol{u}_{01}-\boldsymbol{u}_{02},\ \ \
\varphi:=\varphi_{1}-\varphi_{2},\ \ \ \ \
\boldsymbol{g}:=\boldsymbol{g}_{1}-\boldsymbol{g}_{2}.$

The following lemma establishes the stability of the velocity field
$\boldsymbol{u}(\cdot)$ of the solution of the inverse problem.

######  Lemma 3.5.

Let $\boldsymbol{u}_{0i}\in\mathbb{H},\
\boldsymbol{g}_{i}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega))$ and
$f_{i}\in\mathrm{L}^{2}(0,T)$, for $i=1,2$. Then, the following estimate
holds:

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\mu\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t+\frac{\beta}{2^{r-2}}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t$
$\displaystyle\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\|f\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)},$
(3.29)

where $C$ depends on the input data, $\mu,\alpha,\beta,r,T$ and $\Omega$.

###### Proof.

Subtracting the equations (1.1) for $\\{\boldsymbol{u}_{i},p_{i},f_{i}\\}\
(i=1,2)$, we deduce that

$\displaystyle\boldsymbol{u}_{t}(t)-\mu\Delta\boldsymbol{u}(t)+(\boldsymbol{u}_{1}(t)\cdot\nabla)\boldsymbol{u}(t)+(\boldsymbol{u}(t)\cdot\nabla)\boldsymbol{u}_{2}(t)+\alpha\boldsymbol{u}(t)$
$\displaystyle\quad+\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)}\right)+\nabla
p(t)=f(t)\boldsymbol{g}_{1}(t)+f_{2}(t)\boldsymbol{g}(t),$ (3.30)

for a.e. $t\in[0,T]$ in $\mathbb{V}^{\prime}$. Taking the inner product with
$\boldsymbol{u}(\cdot)$ in the equation (3.3), we obtain

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\mu\|\boldsymbol{u}(t)\|_{\mathbb{V}}^{2}+\alpha\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}$
$\displaystyle=(f(t)\boldsymbol{g}_{1}(t)+f_{2}(t)\boldsymbol{g}(t),\boldsymbol{u}(t))-\langle(\boldsymbol{u}(t)\cdot\nabla)\boldsymbol{u}_{2}(t),\boldsymbol{u}(t)\rangle$
$\displaystyle\quad-\beta\langle|\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t),\boldsymbol{u}(t)\rangle.$
(3.31)

Using Hölder’s and Young’s inequalities, we estimate
$|(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g},\boldsymbol{u})|$ as

$\displaystyle|(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g},\boldsymbol{u})|$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(|f|\|\boldsymbol{g}_{1}\|_{\mathbb{L}^{2}}+|f_{2}|\|\boldsymbol{g}\|_{\mathbb{L}^{2}}}\right)\|\boldsymbol{u}\|_{\mathbb{H}}$
$\displaystyle\leq\frac{1}{\alpha}\mathopen{}\mathclose{{}\left(|f|^{2}\|\boldsymbol{g}_{1}\|_{\mathbb{L}^{2}}^{2}+|f_{2}|^{2}\|\boldsymbol{g}\|_{\mathbb{L}^{2}}^{2}}\right)+\frac{\alpha}{2}\|\boldsymbol{u}\|_{\mathbb{H}}^{2}.$
(3.32)

Making use of Hölder’s, Ladyzhenskaya’s, and Young’s inequalities, we have

$\displaystyle|((\boldsymbol{u}\cdot\nabla)\boldsymbol{u}_{2},\boldsymbol{u})|$
$\displaystyle\leq\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{4}}^{2}\|\nabla\boldsymbol{u}_{2}\|_{\mathbb{H}}$
$\displaystyle\leq\sqrt{2}\|\boldsymbol{u}\|_{\mathbb{H}}\|\boldsymbol{u}\|_{\mathbb{V}}\|\boldsymbol{u}_{2}\|_{\mathbb{V}}\leq\frac{\mu}{2}\|\boldsymbol{u}\|_{\mathbb{V}}^{2}+\frac{1}{\mu}\|\boldsymbol{u}\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{2}\|_{\mathbb{V}}^{2}.$
(3.33)

An estimate similar to (A.13) gives

$\displaystyle\beta(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2},\boldsymbol{u})$
$\displaystyle\geq\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{1}|^{\frac{r-1}{2}}\boldsymbol{u}\big{\|}_{\mathbb{H}}^{2}+\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{2}|^{\frac{r-1}{2}}\boldsymbol{u}\big{\|}_{\mathbb{H}}^{2}$
$\displaystyle\geq\frac{\beta}{2^{r-1}}\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}.$
(3.34)

Substituting (3.3)-(3.3) in (3.3), and then integrating from $0$ to $t$, we
obtain

$\displaystyle\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\mu\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/s+\alpha\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/s+\frac{\beta}{2^{r-2}}\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/s$
$\displaystyle\leq\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{2}{\mu}\int_{0}^{t}\|\boldsymbol{u}_{2}(s)\|_{\mathbb{V}}^{2}\|\boldsymbol{u}(s)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/s$
$\displaystyle\quad+\frac{2}{\alpha}\int_{0}^{t}\mathopen{}\mathclose{{}\left(|f(s)|^{2}\|\boldsymbol{g}_{1}(s)\|_{\mathbb{L}^{2}}^{2}+|f_{2}(s)|^{2}\|\boldsymbol{g}(s)\|_{\mathbb{L}^{2}}^{2}}\right)\/\mathrm{d}\/s,$
(3.35)

for all $t\in[0,T]$. Applying Gronwall’s inequality in (3.3), it can be seen
that

$\displaystyle\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}$
$\displaystyle\leq\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{2}{\alpha}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{g}_{1}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\bigg{\\}}$
$\displaystyle\quad\times\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right),$

for all $t\in[0,T]$. Thus, from (3.3), it is immediate that

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\mu\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t+\frac{\beta}{2^{r-2}}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t$
$\displaystyle\leq\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{2}{\alpha}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{g}_{1}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\bigg{\\}}$
$\displaystyle\quad\times\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right)$
$\displaystyle\leq\bigg{\\{}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{2}{\alpha}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{g}_{1}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\bigg{\\}}$
$\displaystyle\quad\times\exp\bigg{\\{}\frac{1}{\mu^{2}}\bigg{(}\|\boldsymbol{u}_{02}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}_{2}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\bigg{\\}}$
$\displaystyle\leq
C\big{(}\mu,\alpha,\beta,r,\|\boldsymbol{u}_{02}\|_{\mathbb{H}},\|f_{2}\|_{\mathbb{L}^{2}},\|\boldsymbol{g}_{2}\|_{0},T\big{)}\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\|f\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)},$

and (3.5) follows. ∎

The next lemma establishes the stability of the scalar function $f$ of the
solution of the inverse problem.

######  Lemma 3.6.

Let $\boldsymbol{u}_{0i}\in\mathbb{H},\
\boldsymbol{\omega}\in\mathbb{H}^{2}(\Omega)\cap\mathbb{V},\
\nabla\boldsymbol{\omega}\in\widetilde{\mathbb{L}}^{\infty}$,
$\boldsymbol{g}_{i}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega)),\
\varphi_{i}\in\mathrm{H}^{1}(0,T)$, and $f_{i}\in\mathrm{L}^{2}(0,T)$, for
$i=1,2$. Then, the following estimate holds:

$\displaystyle\|f\|_{\mathrm{L}^{2}(0,T)}\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+\|\varphi\|_{\mathrm{H}^{1}(0,T)}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)},$
(3.36)

where $C$ depends on the input data, $\mu,\alpha,\beta,r,T$ and $\Omega$.

###### Proof.

Taking the inner product with $\boldsymbol{\omega}(x)$ in the equation (3.3),
we obtain

$\displaystyle\int_{\Omega}\boldsymbol{u}_{t}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x-\int_{\Omega}\mu\Delta\boldsymbol{u}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x+\int_{\Omega}(\boldsymbol{u}_{1}\cdot\nabla)\boldsymbol{u}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x+\int_{\Omega}(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}_{2}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x$
$\displaystyle\quad+\alpha\int_{\Omega}\boldsymbol{u}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x+\beta\int_{\Omega}\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right)\cdot\boldsymbol{\omega}\/\mathrm{d}\/x$
$\displaystyle\quad=\int_{\Omega}f\boldsymbol{g}_{1}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x+\int_{\Omega}f_{2}\boldsymbol{g}\cdot\boldsymbol{\omega}\/\mathrm{d}\/x.$
(3.37)

From the above equation, we have the following estimate:

$\displaystyle|f(t)|^{2}g_{0}^{2}$ $\displaystyle\leq
2|f_{2}(t)|^{2}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|\boldsymbol{\omega}\|_{\mathbb{H}}^{2}+C\bigg{[}|\varphi^{\prime}(t)|^{2}+\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}^{2}$
$\displaystyle\quad+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}^{2}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\big{(}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}+\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\big{)}+|\varphi(t)|^{2}\bigg{]}$
$\displaystyle\quad+C\|\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}^{2}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)}\right\|_{\mathbb{L}^{1}}^{2}.$
(3.38)

Integrating the estimate (3.3) from $0$ to $T$, we deduce that

$\displaystyle g_{0}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}$ $\displaystyle\leq
2\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|\boldsymbol{\omega}\|_{\mathbb{H}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}+C\bigg{[}\|\varphi\|_{\mathrm{L}^{2}(0,T)}^{2}+\|\varphi^{\prime}\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle\quad+T\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}^{2}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle\quad+C\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\int_{0}^{T}\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t)}\right\|_{\mathbb{L}^{1}}^{2}\/\mathrm{d}\/t.$
(3.39)

Case I: _ $r\in(2,3]$._ Using the estimate (3.1) in (3.3), one can deduce that

$\displaystyle g_{0}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}$ $\displaystyle\leq
2\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|\boldsymbol{\omega}\|_{\mathbb{H}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}+C\bigg{[}\|\varphi\|_{\mathrm{L}^{2}(0,T)}^{2}+\|\varphi^{\prime}\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle\quad+T\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}^{2}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle\quad+CT^{3-r}\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}$
$\displaystyle\quad+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t\bigg{)}^{r-2}\bigg{\\}}\bigg{]}.$
(3.40)

Substituting the estimates (A.1), (A.1), and (3.5) in (3.3), we arrive at

$\displaystyle\|f\|_{\mathrm{L}^{2}(0,T)}^{2}\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\|\varphi\|_{\mathrm{L}^{2}(0,T)}^{2}+\|\varphi^{\prime}\|_{\mathrm{L}^{2}(0,T)}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\bigg{)},$

and finally we get

$\displaystyle\|f\|_{\mathrm{L}^{2}(0,T)}\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+\|\varphi\|_{\mathrm{H}^{1}(0,T)}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)},$

which is the estimate (3.36). Case II: _ $r\in[1,2]$._ Substituting the
estimate (3.1) in (3.3), we obtain

$\displaystyle g_{0}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}$ $\displaystyle\leq
2\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|\boldsymbol{\omega}\|_{\mathbb{H}}^{2}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}+C\bigg{[}\|\varphi\|_{\mathrm{L}^{2}(0,T)}^{2}+\|\varphi^{\prime}\|_{\mathrm{L}^{2}(0,T)}^{2}$
$\displaystyle\quad+T\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{\\{}\|\Delta\boldsymbol{\omega}\|_{\mathbb{H}}^{2}+\|\nabla\boldsymbol{\omega}\|_{\widetilde{\mathbb{L}}^{\infty}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2}\bigg{)}\bigg{\\}}\bigg{]}$
$\displaystyle\quad+CT\|\boldsymbol{\omega}\|_{\mathbb{H}^{2}}^{2}\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\bigg{(}\sup_{t\in[0,T]}\|\boldsymbol{u}_{1}(t)\|_{\mathbb{H}}^{2(r-1)}+\sup_{t\in[0,T]}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{H}}^{2(r-1)}\bigg{)}.$
(3.41)

Using the estimates (A.1), (A.1), and (3.5) in (3.3), one can easily deduce
the estimate (3.36), which completes the proof. ∎

###### Proof of part (ii) of Theorem 1.2.

_Stability of the pressure field $p$:_ The equation (3.3) can be used directly
to demonstrate the stability of the pressure field $p$. Taking divergence on
both sides of the equation (3.3), we deduce that

$\displaystyle-\Delta p$
$\displaystyle=-\nabla\cdot(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g})+\big{\\{}\nabla\cdot\big{[}(\boldsymbol{u}_{1}\cdot\nabla)\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}_{2}+\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right)\big{]}\big{\\}},$

in the weak sense. The above equation yields

$\displaystyle p$
$\displaystyle=(-\Delta)^{-1}\big{\\{}\nabla\cdot\big{[}-(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g})+\nabla\cdot(\boldsymbol{u}_{1}\otimes\boldsymbol{u})$
$\displaystyle\quad+\nabla\cdot(\boldsymbol{u}\otimes\boldsymbol{u}_{2})+\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right)\big{]}\big{\\}}.$
(3.42)

Taking $\mathrm{L}^{2}$-norm on both sides in the equation (3.3), and then
using elliptic regularity (Cattabriga’s regularity theorem), Hölder’s
inequality and Taylor’s formula, we have

$\displaystyle\|p\|_{\mathrm{L}^{2}}$
$\displaystyle=\big{\|}(-\Delta)^{-1}\big{\\{}\nabla\cdot\big{[}-(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g})+\nabla\cdot(\boldsymbol{u}_{1}\otimes\boldsymbol{u})$
$\displaystyle\qquad+\nabla\cdot(\boldsymbol{u}\otimes\boldsymbol{u}_{2})+\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right)\big{]}\big{\\}}\big{\|}_{\mathrm{L}^{2}}$
$\displaystyle\leq
C\big{\|}\big{\\{}\nabla\cdot\big{[}-(f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g})+\nabla\cdot(\boldsymbol{u}_{1}\otimes\boldsymbol{u})$
$\displaystyle\qquad+\nabla\cdot(\boldsymbol{u}\otimes\boldsymbol{u}_{2})+\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right)\big{]}\big{\\}}\big{\|}_{\mathbb{H}^{-2}}$
$\displaystyle\leq
C\|f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g}\|_{\mathbb{V}^{\prime}}+C\|\boldsymbol{u}_{1}\otimes\boldsymbol{u}\|_{\mathbb{H}}+C\|\boldsymbol{u}\otimes\boldsymbol{u}_{2}\|_{\mathbb{H}}$
$\displaystyle\qquad+C\beta\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right\|_{\mathbb{V}^{\prime}}$
$\displaystyle\leq
C\|f\boldsymbol{g}_{1}+f_{2}\boldsymbol{g}\|_{\mathbb{L}^{2}}+C\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}\|_{\widetilde{\mathbb{L}}^{4}}+\|\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{4}}}\right)\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{4}}$
$\displaystyle\qquad+C\mathopen{}\mathclose{{}\left\||\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}}\right\|_{\widetilde{\mathbb{L}}^{\frac{r+1}{r}}}$
$\displaystyle\leq
C|f|\|\boldsymbol{g}_{1}\|_{\mathbb{L}^{2}}+C|f_{2}|\|\boldsymbol{g}\|_{\mathbb{L}^{2}}+C\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}\|_{\widetilde{\mathbb{L}}^{4}}+\|\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{4}}}\right)\|\boldsymbol{u}\|_{\widetilde{\mathbb{L}}^{4}}$
$\displaystyle\quad+C\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{r+1}}\mathopen{}\mathclose{{}\left(\|\boldsymbol{u}_{1}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r-1}+\|\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r-1}}\right),$
(3.43)

where we have used the embedding
$\mathbb{V}\subset\widetilde{\mathbb{L}}^{r+1}\subset\mathbb{H}\equiv\mathbb{H}^{\prime}\subset\widetilde{\mathbb{L}}^{\frac{r+1}{r}}\subset\mathbb{V}^{\prime}$.
Taking the $(\frac{r+1}{r})^{\text{th}}$ power on both sides in (3.3) and then
integrating the resulting inequality from $0$ to $T$, followed by applying
Hölder’s inequality, we deduce that

$\displaystyle\int_{0}^{T}\|p(t)\|_{\mathrm{L}^{2}}^{\frac{r+1}{r}}\/\mathrm{d}\/t$
$\displaystyle\leq
C\sup_{t\in[0,T]}\|\boldsymbol{g}_{1}(t)\|_{\mathbb{L}^{2}}^{\frac{r+1}{r}}\int_{0}^{T}|f(t)|^{\frac{r+1}{r}}\/\mathrm{d}\/t+C\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{\frac{r+1}{r}}\int_{0}^{T}|f_{2}(t)|^{\frac{r+1}{r}}\/\mathrm{d}\/t$
$\displaystyle\quad+C\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{\frac{r+1}{r}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{\frac{r+1}{r}}\/\mathrm{d}\/t+C\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{\frac{r+1}{r}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{\frac{r+1}{r}}\/\mathrm{d}\/t$
$\displaystyle\quad+C\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{(r-1)(r+1)}{r}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{r+1}{r}}\/\mathrm{d}\/t$
$\displaystyle\quad+C\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{(r-1)(r+1)}{r}}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{\frac{r+1}{r}}\/\mathrm{d}\/t$
$\displaystyle\leq
CT^{\frac{r-1}{2r}}\sup_{t\in[0,T]}\|\boldsymbol{g}_{1}(t)\|_{\mathbb{L}^{2}}^{\frac{r+1}{r}}\|f\|_{\mathrm{L}^{2}(0,T)}^{\frac{r+1}{r}}+CT^{\frac{r-1}{2r}}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{\frac{r+1}{r}}\|f_{2}\|_{\mathrm{L}^{2}(0,T)}^{\frac{r+1}{r}}$
$\displaystyle\quad+CT^{\frac{r-1}{2r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{4}\/\mathrm{d}\/t\bigg{)}^{\frac{r+1}{4r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{4}\/\mathrm{d}\/t\bigg{)}^{\frac{r+1}{4r}}$
$\displaystyle\quad+CT^{\frac{r-1}{2r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{4}\/\mathrm{d}\/t\bigg{)}^{\frac{r+1}{4r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{4}}^{4}\/\mathrm{d}\/t\bigg{)}^{\frac{r+1}{4r}}$
$\displaystyle\quad+C\bigg{(}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t\bigg{)}^{\frac{1}{r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{1}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t\bigg{)}^{\frac{r-1}{r}}$
$\displaystyle\quad+C\bigg{(}\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t\bigg{)}^{\frac{1}{r}}\bigg{(}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t\bigg{)}^{\frac{r-1}{r}}.$

Substituting the energy estimates (A.1), (3.5) and (3.36) in the above
estimate, we arrive at

$\displaystyle\|p\|_{\mathrm{L}^{\frac{r+1}{r}}(0,T;\mathrm{L}^{2}(\Omega))}$
$\displaystyle\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}+\|f\|_{\mathbb{L}^{2}}\bigg{)}$
$\displaystyle\leq
C\bigg{(}\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+\|\varphi\|_{\mathrm{H}^{1}(0,T)}+\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\bigg{)},$

where $C$ depends on the input data, $\mu,\alpha,\beta,r$,$T$ and $\Omega$. We
can see that the solution depends continuously on the data from the stability
estimate of the pressure field $p$ and Lemmas 3.5-3.6. We infer the Lipschitz
stability of the solution $(\boldsymbol{u},p,f)$ as

$\displaystyle\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{\infty}(0,T;\mathbb{H})}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{2}(0,T;\mathbb{V})}+\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathrm{L}^{r+1}(0,T;\widetilde{\mathbb{L}}^{r+1})}$
$\displaystyle\quad+\|p_{1}-p_{2}\|_{\mathrm{L}^{\frac{r+1}{r}}(0,T;\mathrm{L}^{2}(\Omega))}+\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}$
$\displaystyle\leq
C\bigg{(}\|\boldsymbol{u}_{01}-\boldsymbol{u}_{02}\|_{\mathbb{H}}+\sup_{t\in[0,T]}\|(\boldsymbol{g}_{1}-\boldsymbol{g}_{2})(t)\|_{\mathbb{L}^{2}}+\|\varphi_{1}-\varphi_{2}\|_{\mathrm{H}^{1}(0,T)}\bigg{)},$

which completes the proof of part (ii) of Theorem 1.2. ∎

##  Appendix A Energy estimates

Since the existence and uniqueness of weak solutions for the system
(1.1)-(1.4) are known, we derive a number of a-priori estimates for the
solutions. To obtain the energy estimates of the solutions of CBF equations
(1.1)-(1.4), we assume that $\boldsymbol{u}_{0}\in\mathbb{H}$,
$\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega))$ and
$f\in\mathrm{L^{2}}(0,T)$. The next lemma provides the usual energy estimates
for the CBF equations (1.1)-(1.4).

######  Lemma A.1.

Let $(\boldsymbol{u}(\cdot),p(\cdot))$ be the unique weak solution of the CBF
equations (1.1)-(1.4) and $\boldsymbol{u}_{0}\in\mathbb{H}$. Then, the
following estimate holds:

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}\leq\|\boldsymbol{u}_{0}\|_{\mathbb{H}}+T^{1/2}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|f\|_{\mathrm{L}^{2}(0,T)},$
(A.1)

and

$\displaystyle\sup_{t\in[0,T]}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+2\mu\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t+\alpha\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/t+2\beta\int_{0}^{T}\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t$
$\displaystyle\leq\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f\|_{\mathrm{L}^{2}(0,T)}^{2}.$
(A.2)

###### Proof.

Taking the inner product with $\boldsymbol{u}(\cdot)$ in the equation (1.1)
and use the fact that
$\langle(\boldsymbol{u}\cdot\nabla)\boldsymbol{u},\boldsymbol{u}\rangle=0$ and
$\langle\nabla p,\boldsymbol{u}\rangle=0$ to obtain

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\mu\|\boldsymbol{u}(t)\|_{\mathbb{V}}^{2}+\alpha\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+\beta\|\boldsymbol{u}(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}=(f\boldsymbol{g}(t),\boldsymbol{u}(t)),$
(A.3)

for a.e. $t\in[0,T]$. Simplifying (A.3), we get

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}\leq|f(t)|\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}\|\boldsymbol{u}(t)\|_{\mathbb{H}},$

which further reduces to

$\displaystyle\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|\boldsymbol{u}(t)\|_{\mathbb{H}}\leq|f(t)|\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}.$
(A.4)

Integrating the inequality (A.4) from $0$ to $t$, and taking supremum over $0$
to $T$ on both sides, one can easily deduce (A.1).

Using Hölder’s and Young’s inequalities, we estimate
$|(f\boldsymbol{g},\boldsymbol{u})|$ as

$\displaystyle|(f\boldsymbol{g},\boldsymbol{u})|\leq|f|\|\boldsymbol{g}\|_{\mathbb{L}^{2}}\|\boldsymbol{u}\|_{\mathbb{H}}\leq\frac{1}{2\alpha}|f|^{2}\|\boldsymbol{g}\|_{\mathbb{L}^{2}}^{2}+\frac{\alpha}{2}\|\boldsymbol{u}\|_{\mathbb{H}}^{2}.$
(A.5)

Substituting (A.5) in (A.3), and then integrating it from $0$ to $t$, we find

$\displaystyle\|\boldsymbol{u}(t)\|_{\mathbb{H}}^{2}+2\mu\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/s+\alpha\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/s+2\beta\int_{0}^{t}\|\boldsymbol{u}(s)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/s$
$\displaystyle\quad\leq\|\boldsymbol{u}_{0}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\int_{0}^{t}|f(s)|^{2}\|\boldsymbol{g}(s)\|_{\mathbb{L}^{2}}^{2}\/\mathrm{d}\/s,$

for all $t\in[0,T]$ and (A.1) follows. ∎

The next lemma establishes stability estimates for the CBF equations
(1.1)-(1.4).

######  Lemma A.2.

Let $(\boldsymbol{u}_{1}(\cdot),p_{1}(\cdot))$ be the solution of the direct
problem (1.1)-(1.4) corresponding to the external forcing
$f_{1}\boldsymbol{g}$ and the initial velocity $\boldsymbol{u}_{01}$; and
$(\boldsymbol{u}_{2}(\cdot),p_{2}(\cdot))$ be the solution of the same problem
corresponding to the external forcing $f_{2}\boldsymbol{g}$ and the initial
velocity $\boldsymbol{u}_{02}$. Then, for
$\boldsymbol{u}_{01},\boldsymbol{u}_{02}\in\mathbb{H},$
$f_{1},f_{2}\in\mathrm{L}^{2}(0,T)$ and
$\boldsymbol{g}\in\mathrm{C}([0,T];\mathbb{L}^{2}(\Omega))$, the following
estimate holds:

$\displaystyle\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}+\mu\int_{0}^{T}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t$
$\displaystyle\quad+\alpha\int_{0}^{T}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/t+\frac{\beta}{2^{r-2}}\int_{0}^{T}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/t$
$\displaystyle\leq\bigg{(}\|\boldsymbol{u}_{01}-\boldsymbol{u}_{02}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right).$
(A.6)

Moreover, for $\boldsymbol{u}_{01}=\boldsymbol{u}_{02}$, we have

$\displaystyle\sup_{t\in[0,T]}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}$
$\displaystyle\leq\bigg{(}\frac{1}{\alpha}\sup_{t\in[0,T]}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}\|f_{1}-f_{2}\|_{\mathrm{L}^{2}(0,T)}^{2}\bigg{)}\exp\mathopen{}\mathclose{{}\left(\frac{2}{\mu}\int_{0}^{T}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/t}\right).$
(A.7)

###### Proof.

It is clear that the pair
$\big{(}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(\cdot),(p_{1}-p_{2})(\cdot)\big{)}$
satisfies the following system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})_{t}-\mu\Delta(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})+\alpha(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})+\nabla(p_{1}-p_{2})&=\boldsymbol{F}_{k},\
\ \ \text{in}\ \Omega\times(0,T),\\\
\nabla\cdot(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})&=0,\ \ \ \ \ \text{in}\
\Omega\times(0,T),\\\ \boldsymbol{u}_{1}-\boldsymbol{u}_{2}&=\boldsymbol{0},\
\ \ \ \ \text{on}\ \partial\Omega\times[0,T),\\\
\boldsymbol{u}_{1}-\boldsymbol{u}_{2}&=\boldsymbol{u}_{01}-\boldsymbol{u}_{02},\
\ \text{in}\ \Omega\times\\{0\\},\end{aligned}}\right.$ (A.8)

in $\mathbb{V}^{\prime}$, where

$\displaystyle\boldsymbol{F}_{k}=(f_{1}-f_{2})\boldsymbol{g}-(\boldsymbol{u}_{1}\cdot\nabla)\boldsymbol{u}_{1}+(\boldsymbol{u}_{2}\cdot\nabla)\boldsymbol{u}_{2}-\beta\big{(}|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2}\big{)}.$
(A.9)

Taking the inner product with $(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(\cdot)$
to the first equation in (A.8), we obtain

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}+\mu\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{V}}^{2}+\alpha\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}$
$\displaystyle=((f_{1}-f_{2})(t)\boldsymbol{g}(t),(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t))-\langle((\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\cdot\nabla)\boldsymbol{u}_{2}(t),(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\rangle$
$\displaystyle\quad-\beta\langle|\boldsymbol{u}_{1}(t)|^{r-1}\boldsymbol{u}_{1}(t)-|\boldsymbol{u}_{2}(t)|^{r-1}\boldsymbol{u}_{2}(t),(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\rangle,$
(A.10)

for a.e. $t\in[0,T]$. It is worth emphasizing that, for $r\geq 1$ (see Sec.
2.4, [31, 32])

$\displaystyle\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2},\boldsymbol{u}_{1}-\boldsymbol{u}_{2}}\right)$
$\displaystyle\geq\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{1}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}+\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{2}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}.$
(A.11)

It can be easily seen that

$\displaystyle\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}$
$\displaystyle=\int_{\Omega}|\boldsymbol{u}_{1}(x)-\boldsymbol{u}_{2}(x)|^{r-1}|\boldsymbol{u}_{1}(x)-\boldsymbol{u}_{2}(x)|^{2}\/\mathrm{d}\/x$
$\displaystyle\leq
2^{r-2}\int_{\Omega}\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}(x)|^{r-1}+|\boldsymbol{u}_{2}(x)|^{r-1}}\right)|\boldsymbol{u}_{1}(x)-\boldsymbol{u}_{2}(x)|^{2}\/\mathrm{d}\/x$
$\displaystyle\leq
2^{r-2}\mathopen{}\mathclose{{}\left(\big{\|}|\boldsymbol{u}_{1}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}+\big{\|}|\boldsymbol{u}_{2}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}}\right).$

From the above inequality, we have

$\displaystyle\frac{2^{2-r}\beta}{2}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\leq\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{1}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}+\frac{\beta}{2}\big{\|}|\boldsymbol{u}_{2}|^{\frac{r-1}{2}}(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\big{\|}_{\mathbb{H}}^{2}.$
(A.12)

Thus, from (A) and (A.12), one can easily deduce that

$\displaystyle\beta\mathopen{}\mathclose{{}\left(|\boldsymbol{u}_{1}|^{r-1}\boldsymbol{u}_{1}-|\boldsymbol{u}_{2}|^{r-1}\boldsymbol{u}_{2},\boldsymbol{u}_{1}-\boldsymbol{u}_{2}}\right)\geq\frac{\beta}{2^{r-1}}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}.$
(A.13)

Using the Cauchy-Schwarz and Young’s inequalities, we get

$\displaystyle\mathopen{}\mathclose{{}\left((f_{1}-f_{2})\boldsymbol{g},\boldsymbol{u}_{1}-\boldsymbol{u}_{2}}\right)\leq\frac{1}{2\alpha}|f_{1}-f_{2}|^{2}\|\boldsymbol{g}\|_{\mathbb{L}^{2}}^{2}+\frac{\alpha}{2}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathbb{H}}^{2}.$
(A.14)

Making use of Hölder’s, Ladyzhenskaya’s, and Young’s inequalities, we have

$\displaystyle\langle((\boldsymbol{u}_{1}-\boldsymbol{u}_{2})\cdot\nabla)\boldsymbol{u}_{2},\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\rangle$
$\displaystyle\leq\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\widetilde{\mathbb{L}}^{4}}^{2}\|\boldsymbol{u}_{2}\|_{\mathbb{V}}$
$\displaystyle\leq\sqrt{2}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathbb{H}}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathbb{V}}\|\boldsymbol{u}_{2}\|_{\mathbb{V}}$
$\displaystyle\leq\frac{\mu}{2}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathbb{V}}^{2}+\frac{1}{\mu}\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{2}\|_{\mathbb{V}}^{2}.$
(A.15)

Substituting (A.13)-(A) in (A), we obtain

$\displaystyle\frac{1}{2}\frac{\/\mathrm{d}\/}{\/\mathrm{d}\/t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}+\frac{\mu}{2}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{V}}^{2}$
$\displaystyle\quad+\frac{\alpha}{2}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}+\frac{\beta}{2^{r-1}}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}$
$\displaystyle\leq\frac{1}{2\alpha}|(f_{1}-f_{2})(t)|^{2}\|\boldsymbol{g}(t)\|_{\mathbb{L}^{2}}^{2}+\frac{1}{\mu}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{2}(t)\|_{\mathbb{V}}^{2}.$

Integrating the above inequality from $0$ to $t$, we deduce

$\displaystyle\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(t)\|_{\mathbb{H}}^{2}+\mu\int_{0}^{t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(s)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/s$
$\displaystyle\quad+\alpha\int_{0}^{t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(s)\|_{\mathbb{H}}^{2}\/\mathrm{d}\/s+\frac{\beta}{2^{r-2}}\int_{0}^{t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(s)\|_{\widetilde{\mathbb{L}}^{r+1}}^{r+1}\/\mathrm{d}\/s$
$\displaystyle\leq\|\boldsymbol{u}_{01}-\boldsymbol{u}_{02}\|_{\mathbb{H}}^{2}+\frac{1}{\alpha}\int_{0}^{t}|(f_{1}-f_{2})(t)|^{2}\|\boldsymbol{g}(s)\|_{\mathbb{L}^{2}}^{2}\/\mathrm{d}\/s$
$\displaystyle\quad+\frac{2}{\mu}\int_{0}^{t}\|(\boldsymbol{u}_{1}-\boldsymbol{u}_{2})(s)\|_{\mathbb{H}}^{2}\|\boldsymbol{u}_{2}(s)\|_{\mathbb{V}}^{2}\/\mathrm{d}\/s,$
(A.16)

for all $t\in[0,T]$. Applying Gronwall’s inequality in (A) and then taking
supremum over $0$ to $T$ on both sides, we finally have (A.2). One can easily
deduce (A.2) by taking $\boldsymbol{u}_{01}=\boldsymbol{u}_{02}$ in (A.2) and
it completes the proof. ∎

Acknowledgments: P. Kumar and M. T. Mohan would like to thank the Department
of Science and Technology (DST), India for Innovation in Science Pursuit for
Inspired Research (INSPIRE) Faculty Award (IFA17-MA110).

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