Suppression of Chemotactic Singularity via Viscous Flow with Large Buoyancy
Zhongtian Hu Department of Mathematics, Duke University, Durham, NC 90320,
USA; email<EMAIL_ADDRESS>
###### Abstract

In this work, we study the Keller-Segel equation coupling to the
(Navier-)Stokes equation with low Reynolds number and Lions boundary condition
$\omega|_{\partial\Omega}=0$. The coupling of both equations is realized via a
buoyancy force. We show that for initial cell density with arbitrarily large
mass, the solution remains regular for all times in the regime of sufficiently
large buoyancy and viscosity. The major blowup suppression mechanism is a
norm-stabilizing property possessed by a “static problem,” where the full
problem can be seen as a perturbation of this static model.

##  1 Introduction

We consider the parabolic-elliptic Keller-Segel equation in periodic channel
$\Omega=\mathbb{T}\times[0,\pi]$ subject to the influence of a buoyancy-driven
(Navier-)Stokes equation:

$\begin{cases}\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,&\text{
in }\Omega,\\\ \partial_{t}u+\delta u\cdot\nabla
u-\frac{1}{\operatorname{Re}}\Delta u+\nabla p=\operatorname{Ra}\rho
e_{2},&\text{ in }\Omega,\\\ \operatorname{div}(u)=0&\text{ in }\Omega,\\\
\partial_{2}\rho\big{|}_{\partial\Omega}=0,\;u_{2}|_{\partial\Omega}=0,\;\omega|_{\partial\Omega}:=\nabla^{\perp}\cdot
u|_{\partial\Omega}=0,\\\ \rho(0,x)=\rho_{0}(x)\geq
0,\;u(0,x)=u_{0}(x).\end{cases}$ (1.1)

In this system, $\rho$ denotes the cell density, $u$ denotes the fluid
velocity, and $p$ denotes the fluid pressure. Moreover,
$\rho_{M}=|\Omega|^{-1}\int_{\Omega}\rho dx$, $(-\Delta_{N})^{-1}$ denotes the
inverse homogeneous Neumann Laplacian,
$\nabla^{\perp}=(-\partial_{2},\partial_{1})$, $\operatorname{Re}$ is the
Reynolds number, and $\operatorname{Ra}$ is the Rayleigh number characterizing
the relative strength of buoyancy force. Finally, $\delta\in\\{0,1\\}$
dictates whether the fluid flow conforms to a Stokes or Navier-Stokes
equation.

The first equation above i.e. Keller-Segel equation characterizes chemotaxis,
namely motions of micro-organisms that are dictated by a random walk and a
drift induced by certain chemoattractant. In the meantime, the chemotaxis
phenomenon takes place in the ambience of a viscous fluid, which is
characterized by a (Navier-)Stokes equation. In sundry realistic situations,
the micro-organisms and the fluid interact with each other via various means
(e.g. [24]). In this work, we focus on a scenario where the cells and fluid
interact through buoyancy. Such interaction, in particular, emerges in the
fluid equation through the buoyancy forcing term $\operatorname{Ra}\rho
e_{2}$. Moreover, since we work in a periodic channel with boundaries, we
would like to emphasize the boundary conditions: on one hand, we assert the
classical homogeneous Neumann boundary condition for the cell density. On the
other hand, we impose the Lions boundary condition onto the fluid equation:
$u\cdot n|_{\partial\Omega}=0,\;\omega|_{\partial\Omega}=0$. This particular
boundary condition, as a special case of the more general Navier boundary
condition, is prevalent in simulations of flows in the presence of rough
boundaries, such as in hemodynamics. It was first introduced and rigorously
studied by J.-L. Lions [39] and P.-L. Lions [40]. We refer the readers to
works such as [3, 36, 22, 12, 52] and references therein for a more thorough
treatment regarding Navier-Stokes equations equipped with Navier boundary
condition.

On one hand, when the ambient fluid is absent (That is, we formally set
$\operatorname{Ra}=0$ and $u\equiv 0$ in (1.1)), we recover the classical
parabolic-elliptic Keller-Segel equation:

$\partial_{t}\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,\quad\text{
in }\Omega$ (1.2)

equipped with homogeneous Neumann boundary condition on $\rho$. The equation
(1.2) was first introduced by Patlak [44], and Keller and Segel [35]. Thus
far, there have been extensive works concerning well-posedness, singularity
formation, and asymptotic behavior of (1.2). We refer the interested readers
to the following list of works: [1, 4, 5, 6, 7, 8, 9, 19, 29, 27, 33, 42, 43].
A remarkable feature enjoyed by (1.2) is that the solution can form
singularity in finite time when dimension is greater than $1$. Particularly in
dimension $2$, (1.2) is $L^{1}$-critical. We illustrate this point when (1.2)
is set in $\mathbb{R}^{2}$. For any initial datum $\rho_{0}$ with finite
second moment, the solution is globally regular if the initial mass
$\|\rho_{0}\|_{L^{1}}<8\pi$, see e.g. [4, 7, 9, 33, 49]. If the initial mass
is strictly greater than $8\pi$, then there a singularity forms in finite
time, as seen in [7, 8, 33, 42]. A more detailed characterization of such
blowup solutions are also carried out [13, 14, 46, 47, 48].

On the other hand, it is curious to understand the behavior of Keller-Segel
equation under the influence of fluid advection, given the fact that most
chemotactic processes take place in ambient fluid. Besides possibly
complicated effects induced by the fluid, how regularizing effect of an
incompressible fluid advection, namely the term $u\cdot\nabla\rho$, can
suppress chemotactic singularity is under active investigation recently. It is
worth noticing that such regularizing phenomenon has been observed in various
fluid models, in the sense that a lack of/a weakened incompressible advection
contributes to the singularity formation, as seen in [15, 38, 28, 20, 11, 34].
In the regime of passive advection (i.e. the fluid velocity $u$ is given and
is not coupled with the Keller-Segel equation), Kiselev and Xu in [37] first
demonstrate that given any initial datum $\rho_{0}$, there exists a
relaxation-enhancing flow (see [17] for a precise definition) with
sufficiently large amplitude that can suppress the singularity formation. This
result is later generalized by [32] to a larger class of passive flows and
aggregation-diffusion equations with more general nonlinearities. Moreover, in
[2, 23], the authors exploit the enhanced dissipation phenomenon induced by
strong monotone shear flows, which creates a dimension-reduction effect that
suppresses blowup. The fast-splitting scenario induced by hyperbolic flows are
also utilized to prevent chemotactic blowup in [25, 26].

Comparing with how passive flows suppress chemotactic singularities, the study
of blowup suppressing property of active flows is far more scarce. Indeed,
there have been numerous studies regarding the regularity properties of
Keller-Segel equation coupling to active fluid models, and many of which
address the global regularity of solutions to such coupled systems. We
highlight that, among those results, either there is smallness assumption on
initial data (e.g. [10, 18, 41]), or the global regularity of both chemotaxis
equation and fluid equation might still hold if they are uncoupled (e.g. [50,
51]). It is worth noticing that the authors in [24, 53] study the blowup
suppression mechanism of Keller-Segel-Navier-Stokes equation near a strong
Couette flow. Despite focusing on different ways of coupling in between the
chemotaxis and the fluid equation, these works essentially recast the global
regularity problem of the Keller-Segel-Navier-Stokes equation to a nonlinear
stability problem of Navier-Stokes equation near a sufficiently strong Couette
flow.

Recently in a series of works [31, 30] by the author joining with Kiselev and
Yao, we analyze how buoyancy effects in fluid equations suppress chemotactic
singularities nonlinearly and dynamically. In [31], the authors investigated
the Keller-Segel equation evolving in ambient porous media under the influence
of buoyancy. The authors demonstrated that a coupling with the porous media
equation via an arbitrarily weak buoyancy constant suffices to arrest any
potential chemotactic blowup. The key argument in [31] is a careful analysis
of a potential energy and the coercive term
$\|\partial_{1}\rho\|_{H^{-1}_{0}}$ in its time derivative. The authors
observed that this $H^{-1}_{0}$ norm has to be small, and induces an
anisotropic mixing effect along $x_{1}$-direction. This effect renders the
system quasi-one-dimensional and therefore suppresses finite-time blowup. On
the other hand, [30] studies how Stokes-Boussinesq flow with strong buoyancy
suppresses blowup of a Keller-Segel equation equipped with zero Dirichlet
boundary condition. A rather soft argument in [30] shows that the flow fosters
an exponential decay in cell density in the regime of large buoyancy thanks to
an increased interaction with a cold boundary.

In this work, we plan to investigate (1.1) in the parameter regime
$\operatorname{Ra}\gg\frac{1}{\operatorname{Re}}\gg 1$, which characterizes a
fluid with large viscosity and exerting significant buoyancy force onto the
micro-organisms. In fact, an understanding of cell movements in fluids with
low Reynolds number is of biological importance (see [45]), and we focus on
the regime where the buoyancy effect is salient. To make the problem more
amenable to analysis, it is convenient to write $B=\operatorname{Re}^{-1}$ and
$g=\operatorname{Ra}/B$. Another handy transformation is that in periodic
channel $\Omega$, one can reduce the Lions boundary condition in the following
way: the no-flux boundary is equivalent to $u_{2}|_{\partial_{\Omega}}=0$.
Since $\partial_{1}$ is the tangential derivative on $\partial\Omega$, we have
the extra boundary constraint $\partial_{1}^{k}u_{2}|_{\partial\Omega}=0$ for
all $k\in\mathbb{N}$. Thus, we may reduce:

$\omega|_{\partial\Omega}=\partial_{2}u_{1}|_{\partial\Omega}=0.$

To sum up, we will work with the following equivalent, yet more convenient
form of (1.1) for the rest of our analysis:

$\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,\quad\text{
in }\Omega,$ (1.3a) $\partial_{t}u+\delta u\cdot\nabla u-B\Delta u+\nabla
p=Bg\rho e_{2},\quad\operatorname{div}(u)=0,\quad\text{ in }\Omega,$ (1.3b)
$\partial_{2}\rho\big{|}_{\partial\Omega}=0,\;u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0,$
(1.3c) $\rho(0,x)=\rho_{0}(x)\geq 0,\;u(0,x)=u_{0}(x).$ (1.3d)

The main result of this work is that the regular solution of (1.3) with
arbitrary large mass is in fact globally regular given both parameters $B$ and
$g$ sufficiently large. The precise statement is given as follows:

######  Theorem 1.1.

Let $\rho_{0}\in C^{\infty}(\Omega)$ be nonnegative and $u_{0}\in
C^{\infty}(\Omega)\cap W$, where the precise definition of $W$ is given in
Section 2.2. For $\delta=0,1$, there exists
$g_{\delta}=g_{\delta}(\rho_{0})>0$ sufficiently large that the following
holds: for any $g\geq g_{\delta}$, there exists
$B_{\delta}=B_{\delta}(\rho_{0},u_{0},g)$ such that for any $B\geq
B_{\delta}$, (1.3) has a unique, regular, and global-in-time solution.

######  Remark 1.1.

With the appearance of time derivative and advection term in the fluid
equation (1.3b), the problem suffers from a loss of a Biot-Savart law which
directly connects density $\rho$ and fluid velocity $u$. In the parameter
regime specified in Theorem 1.1, however, one can view (1.3) as a suitable
perturbation of a static problem (2.7), which possesses a straightforward
Biot-Savart law that can be analyzed with tools developed in [31].

######  Remark 1.2.

We compare this result with [30]. Firstly, the two works study different
scenarios: [30] investigates a Keller-Segel equation with zero Dirichlet
boundary condition in an arbitrary regular domain, while we study the Keller-
Segel equation with classical zero Neumann boundary condition in a periodic
strip. Secondly, the blowup-suppressing mechanism in the two works are
different. In [30], the blowup is suppressed by an increased interaction
between cells and the cold boundary induced by a strong buoyancy-driven flow.
This mechanism forces the $L^{2}$ norm of cell density exponentially decay in
time. In our work, we observe that an anisotropic, nonlinear mixing effect
similar to that exhibited in [31] dominates the dynamics. It stabilizes the
$L^{2}$ norm of cell density around a possibly large, albeit finite, number.

Acknowledgement. The author acknowledges partial support from NSF-DMS grants
2306726, and 2304392. The author would like to cordially thank Professor
Alexander Kiselev for his kind guidance and support throughout the years. He
also thanks Professor Siming He for numerous insightful discussions,
suggestions, and for pointing out several mistakes in an early draft of this
work. Finally, the author thanks the hospitality of the Chinese University of
Hong Kong, where part of this work was completed.

##  2 Preliminaries and Main Strategy

###  2.1 Notation and Conventions

We will denote the usual $L^{p}$ norm by $\|\cdot\|_{L^{p}}$ or
$\|\cdot\|_{L^{p}(\cdot)}$. The second notation will be adopted if we would
like to emphasize the domain. Similarly, we will denote Sobolev $H^{s}$ norm
by $\|\cdot\|_{s}$ for $s\geq 1.$ We define the space $H^{1}_{0}(\Omega)$ to
be the completion of $C^{\infty}_{c}(\Omega)$ with respect to norm
$\|\cdot\|_{1}$, and consider its dual space $H^{-1}_{0}(\Omega)$. Moreover,
the following equivalence is standard:

$\|f\|_{H^{-1}_{0}}=C(\Omega)\left(\int_{\Omega}f(-\Delta_{D})^{-1}fdx\right)^{1/2},$

where $C(\Omega)$ is a constant that only depends on domain $\Omega$. For the
rest of this paper, we simply regard $C(\Omega)=1$ as the exact value of this
constant is inessential.

Recall the mean of $\rho$ on $\Omega:$
$\rho_{M}=\frac{1}{2\pi^{2}}\int_{\Omega}\rho(x,t)\,dx.$ In particular, we
note that $\rho_{M}$ is nonnegative and a conserved quantity along evolution
(see Proposition 4.1). We also consider the following decomposition to any
function $\rho\in L^{2}(\Omega)$ that would play a crucial role in our
analysis of the static problem: we write $\rho=\bar{\rho}+\tilde{\rho}$, where

$\bar{\rho}(x_{2})=\frac{1}{2\pi}\int_{\mathbb{T}}\rho(x_{1},x_{2})dx_{1},\;\tilde{\rho}=\rho-\bar{\rho}.$

Note that $\bar{\rho}$ is exactly the projection onto the zeroth mode
corresponding to direction $x_{1}$, and $\tilde{\rho}$ the orthogonal
complement. We remark that $\bar{\rho}-\rho_{M}$ and $\tilde{\rho}$ are
orthogonal in $L^{2}(\Omega)$:

$\int_{\Omega}(\bar{\rho}(x_{2})-\rho_{M})\tilde{\rho}(x_{1},x_{2})dx_{1}dx_{2}=0.$

One can connect the 1D and 2D $L^{2}$ norm for $\bar{\rho}-\rho_{M}$ by a
simple relation
$\|\bar{\rho}-\rho\|_{L^{2}_{\Omega}}^{2}=2\pi\|\bar{\rho}-\rho\|_{L^{2}_{[0,\pi]}}^{2}$.
In the rest of the paper, we will only use the 1D $L^{2}$ norm for
$\bar{\rho}$. We always refer to the 1D $L^{2}$ norm when we write
$\|\bar{\rho}-\rho_{M}\|_{L^{2}}$.

We will denote a universal constant in the upper bounds by uppercase letters,
and that in lower bounds by lowercase letters. Constants without subscripts,
i.e. $c,C,\mathcal{C}$, are subject to change from line to line. Constants
with subscripts, i.e. $c_{i},C_{i},\mathcal{C}_{i}$, are fixed once they are
chosen. All such constants mentioned above only depend on domain $\Omega$ by
default. We will also use the notation $C(X)$ to signify the dependencies of a
constant. For two quantities $A$, $B$, we write $A\lesssim B$ to mean that
$A\leq CB$ for some universal constant $C$ that only depends on domain
$\Omega$. $A\gtrsim B$ is similarly defined. Lastly, we will use summation
convention throughout this paper. That is, the repeated indices are summed
over.

###  2.2 Steady Stokes Equation with Lions Boundary Condition

In this subsection, we lay out suitable functional-analytic settings for our
study of Stokes equation equipped with Lions boundary condition. We shall
first study the following static Stokes equation and derive its well-
posedness:

$\begin{cases}-\Delta u+\nabla p=f,&\text{ in }\Omega\\\
\operatorname{div}(u)=0,&\text{ in }\Omega\\\
u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0.\end{cases}$
(2.1)

In particular, we will derive suitable elliptic estimates similar to those
satisfied by the classical Stokes operator that corresponds to no-slip
boundary condition for velocity (see [16]).

We introduce the following function spaces:

$H=\\{v\in L^{2}(\Omega)\times
L^{2}(\Omega)\;:\;\int_{\Omega}v_{1}(x)dx=0\\},$ $V=\\{v\in
H^{1}(\Omega)\times
H^{1}(\Omega):\operatorname{div}v=0,\;v_{2}|_{\partial\Omega}=0,\;\int_{\Omega}v_{1}(x)dx=0\\},$
$W=\\{v\in V\cap(H^{2}(\Omega))^{2}:\partial_{2}v_{1}|_{\partial\Omega}=0\\},$

where we equip $H$ with $L^{2}$ topology, $V$ with topology induced by the
inner product $(u,v)_{V}=\int_{\Omega}\partial_{i}u_{j}\partial_{i}v_{j}dx$,
and $W$ with $H^{2}$ topology. Note that for $u\in V$, the following Poincaré
inequality holds:

######  Proposition 2.1.

Let $u\in V$. Then the following estimate holds:

$\|u\|_{L^{2}}\leq C_{p}\|\nabla u\|_{L^{2}},$ (2.2)

where $C_{p}$ denotes the Poincaré constant that only depends on the domain
$\Omega$.

Moreover, we have the following well-posedness and regularity result
concerning (2.1):

######  Proposition 2.2.

Assuming $f\in H$, then (2.1) admits a unique solution $u\in W$ with estimate

$\|u\|_{2}\leq C(\Omega)\|f\|_{L^{2}}.$ (2.3)

In fact, $u$ assumes the following explicit formula:

$u=\nabla^{\perp}(-\Delta_{D})^{-2}(\partial_{2}f_{1}-\partial_{1}f_{2}),$
(2.4)

where $\nabla^{\perp}:=(-\partial_{2},\partial_{1})$. In addition, if $f\in
H^{s}\cap H$, $s\geq 1$, we have the following improved regularity estimate:

$\|u\|_{s+2}\leq C(\Omega)\|f\|_{s}.$ (2.5)

We delay the proofs of the Propositions above to the appendix.

###  2.3 Main Strategy

In this section, we briefly sketch the idea of proving Theorem 1.1. Dividing
on both sides of (1.3b) by $B$, we have

$\frac{1}{B}(\partial_{t}u+\delta u\cdot\nabla u)-\Delta u+\frac{1}{B}\nabla
p=g\rho e_{2},\quad\operatorname{div}(u)=0,\quad\text{ in }\Omega$

In the regime of large $B$, it is natural to treat the term
$\frac{1}{B}(\partial_{t}u+\delta u\cdot\nabla u)$ perturbatively. In fact, it
is reasonable to guess that the following velocity law dominates the velocity
evolution:

$-\Delta u+\nabla p=g\rho e_{2},\;\operatorname{div}(u)=0,\quad\text{ in
}\Omega$ (2.6)

This motivates us to first understand the following system, which we will
later refer to as the static problem:

$\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,$
(2.7a) $-\Delta u+\nabla p=g\rho e_{2},\quad\operatorname{div}(u)=0,$ (2.7b)
$\partial_{2}\rho\big{|}_{\partial\Omega}=0,\;u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0,\;\rho(0,x)=\rho_{0}(x).$
(2.7c)

We remark that we do not specify the initial datum for $u$. Indeed, the
problem (2.1) is closed by itself, due to a remarkable property of
(2.7b)-(2.7c) that one may reduce them to the following compact form:

$u=-g\nabla^{\perp}(-\Delta_{D})^{-2}\partial_{1}\rho,$ (2.8)

thanks to Proposition 2.2. This renders no degree of freedom over choosing the
initial datum for $u$ in problem (2.1). From now on, we will work with the
more convenient form (2.8) instead of the original formulation.

We also remark that the static problem (2.7) is reminiscent of the Keller-
Segel-Incompressible-Porous-Medium (KS-IPM) equation studied in [31], whose
velocity law can be represented by:

$u=g\nabla^{\perp}(-\Delta_{D})^{-1}\partial_{1}\rho.$

In [31], the authors exploited an anisotropic mixing effect that is sufficient
to prevent chemotactic blowup. However, the velocity law (2.8) in this work
yields a more regular fluid flow, which induces a weaker mixing effect
comparing with the IPM flow. In this work, we consider large $g$ so that the
flow has large amplitude to generate sufficient mixing effect, so that we can
achieve a crucial norm stabilizing property of (2.7) (See Subsect. 4.3). That
is, the $L^{2}$ norm of $\rho$ stabilizes around a level $M_{0}$, whose size
only depends on the initial datum $\rho_{0}$ in the regime of $g$ sufficiently
large. In a more precise manner, we will show the following:

######  Proposition 2.3 (Norm-Stabilizing Property).

Consider problem (2.7) with regular, nonnegative initial data $\rho_{0}.$
Assume that $t_{0}\geq 0$ is inside the lifespan of the unique regular local
solution $\rho(t,x)$. Then there exist
$M_{0}=M_{0}(\rho_{M},\|\rho_{0}-\rho_{M}\|_{L^{2}})$, a time
$T_{*}=T_{*}(M_{0},\rho_{M})$, and
$\mathfrak{g}_{0}:=\mathfrak{g}_{0}(M_{0},\rho_{M})>0$, such that the
following statement holds: suppose that
$\|\rho(t_{0})-\rho_{M}\|_{L^{2}}^{2}=M_{0}$. Then for all
$g\geq\mathfrak{g}_{0}$, the regular solution $\rho(t,x)$ can be continued in
interval $[t_{0},t_{0}+T_{*}]$ with estimate

$\sup_{t\in[t_{0},t_{0}+T_{*}]}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq 2M_{0}.$
(2.9)

Moreover, there exists $T\in\left[t_{0}+\frac{T_{*}}{2},t_{0}+T_{*}\right]$
such that

$\|\rho(T)-\rho_{M}\|_{L^{2}}^{2}\leq\frac{M_{0}}{8}.$ (2.10)

Next, we employ a bootstrap argument to prove Theorem 1.1, which is a standard
practice in nonlinear stability problems. In fact, we would show that “the
full problem (1.3) is nonlinearly stable around the solution to the static
problem (2.7) in an $L^{2}$ sense.” Fixing initial condition $\rho_{0}$ and
$u_{0}$, we fix $M_{0}$ as in Remark 4.2. Define $\mathcal{T}_{0}$ to be the
largest time such that the following bootstrap assumption holds:

$\sup_{0\leq t\leq\mathcal{T}_{0}}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
9M_{0}.$ (A)

We prove the following improved estimate:

######  Proposition 2.4.

For (1.3) with $\delta=0,1$, given initial datum $\rho_{0}\in H^{1}$ that is
nonnegative and $u_{0}\in W$, there exists $g_{\delta}=g_{\delta}(\rho_{0})$
that the following holds: for any $g\geq g_{\delta}$, there exists
$B_{\delta}=B_{\delta}(\rho_{0},g)$ such that

$\sup_{0\leq t\leq\mathcal{T}_{0}}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
5M_{0}.$ (2.11)

######  Remark 2.1.

The explicit choices of parameters $M_{0},\mathfrak{g}_{0}$ are listed in
Remark 4.2. The choices of $g_{\delta},B_{\delta}$ for $\delta=0,1$ are listed
in Remark 5.4 and 6.1 respectively.

Now, we finish the proof of Therorem 1.1:

###### Proof of Theorem 1.1.

Combining Proposition 2.4 and the criterion in Theorem 3.2, we know that
$[0,\mathcal{T}_{0}]$ is both open and closed. This implies that
$\mathcal{T}_{0}=\infty$ and we have obtained global existence of (1.3). ∎

##  3 Local Well-Posedness and Regularity Criteria

In this section, we briefly comment on the local well-posedness for regular
solutions to problems (1.3) and (2.7), and establish $L^{2}$ regularity
criteria for both problems. We remark that local well-posedness results
concerning Keller-Segel equation coupling with various types of fluid
equations have been extensively studied (see e.g. [50, 51]). For this reason,
we are content with giving sketches of proofs or point the readers to suitable
references.

######  Theorem 3.1 (Local Well-Posedness for (2.7)).

Let $\rho_{0}\in C(\Omega)$ be nonnegative. Then there exists a time
$T=T(\rho_{0})>0$ such that $\rho(t,x)\in C^{\infty}((0,T)\times\Omega)$ is
the unique regular solution to (2.7), where $u$ is determined by formula
(2.8). If $(0,T)$ is the maximal lifespan of $\rho(t,x)$, then

$\lim_{t\to T}\|\rho(t,\cdot)\|_{L^{2}}=\infty.$

The proof of Theorem 3.1 is almost identical to that of Theorem 3.1 in [31],
which utilizes a explicit Biot-Savart law similar to (2.8) and a standard
fixed point argument.

We can also establish the following well-posedness result for the full
(Navier-)Stokes case (5.1):

######  Theorem 3.2 (Local Well-Posedness for (1.3)).

Let $\rho_{0}\in H^{1}(\Omega)$ be nonnegative and $u_{0}\in W$. Then there
exists a time $T=T(\rho_{0},u_{0})>0$ such that $\rho(t,x)\in
C^{\infty}((0,T)\times\Omega)$, $u\in
C^{\infty}((0,T);(C^{\infty}(\Omega))^{2}\cap W)$ is the unique regular
solution to (1.3). If $(0,T)$ is the maximal lifespan of $\rho(t,x)$, then

$\lim_{t\to T}\|\rho(t,\cdot)\|_{L^{2}}=\infty.$

One can prove the statement above either using a fixed point based argument as
in [50, 51], or an energy based method in the scale of Sobolev spaces as in
[30] with the help of elliptic estimates (2.3) and (2.5). The proof uses
standard Galerkin approximation scheme using basis described in Lemma 2.2 of
[12], while the only subtle part is the membership of $u(t,\cdot)\in W$ for
all times in lifespan $[0,T)$ given $u_{0}\in W$. This statement is indeed
true due to the following general fact:

######  Proposition 3.1.

Suppose $u(t,x)\in W$ is a smooth solution on $[0,T)$ to the following forced
Navier-Stokes equation

$\begin{cases}\partial_{t}u+u\cdot\nabla u-\Delta u+\nabla p=(0,f)^{T},&\text{
in }\Omega,\\\ \operatorname{div}(u)=0,&\text{ in }\Omega,\\\
u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0,\\\ u_{0}\in
W,\end{cases}$ (3.1)

where $f\in L^{2}((0,T)\times\Omega)$. Then
$\mathcal{M}(t):=\int_{\Omega}u_{1}(t,x)dx=0$ for all $t\in[0,T]$.

###### Proof.

Using the velocity equation, we may compute that

$\displaystyle\mathcal{M}^{\prime}$
$\displaystyle=\int_{\Omega}\partial_{t}u_{1}dx=-\int_{\Omega}u\cdot\nabla
u_{1}dx+\int_{\Omega}\Delta u_{1}-\int_{\Omega}\partial_{1}pdx$
$\displaystyle=\int_{\partial\Omega}u_{2}u_{1}dS+\int_{\partial\Omega}\partial_{2}u_{1}dS$
$\displaystyle=0,$

where we used $\operatorname{div}(u)=0$ and periodicity of $p$ in $x_{1}$
coordinate in the third equality, as well as boundary condition satisfied by
$u$ in the final equality. The proof is thus completed by observing that
$\mathcal{M}(0)=0$ since $u_{0}\in W$. ∎

######  Remark 3.1.

We note that for both problems (1.1) and (2.7), it suffices to control only
the evolution of $L^{2}$ norm of cell density $\rho$ to extend regular
solutions in time.

##  4 Analysis of the Static Problem

In this section, we perform a detailed analysis of (2.7), which will serve as
a starting point prior to the study of the full problem (1.3). In Section 4.1,
we establish a series of a priori estimates concerning both cell density
$\rho$ and velocity field $u$ of the static problem. In Section 4.2, we show a
Nash-type inequality that elucidates an anisotropic mixing effect in the
regime of large $g$, which has a similar flavor to that in [31]. Finally, we
prove Proposition 2.3 in Section 4.3.

###  4.1 Key A Priori Estimates

We first establish several fundamental energy estimates concerning the system
(2.7). While the estimates corresponding to the Keller-Segel part are
standard, estimates corresponding to the fluid part (2.7b) turn out to be
crucial in establishing the norm-stabilizing property and global well-
posedness of (2.7).

####  4.1.1 Cell Density Estimates

As a preparation, we consider the Keller-Segel equation advected by a general
incompressible flow, which may or may not depend on the solution $\rho$
itself:

$\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,\;\rho(0,x)=\rho_{0}(x)\geq
0,\;\partial_{2}\rho|_{\partial\Omega}=0,$ (4.1)

where $u$ is a smooth, divergence-free vector field that satisfies no-flux
boundary condition. The first proposition reveals that (4.1) always conserves
mass.

######  Proposition 4.1.

Assume $\rho(t,x)$ to be a regular, nonnegative solution to (4.1) on $[0,T]$.
Then $\|\rho(t,\cdot)\|_{L^{1}}=\|\rho_{0}\|_{L^{1}}$ for any $t\in[0,T]$.

The next result gives several variants of $L^{2}$ energy estimates of (4.1),
which play a major role in establishing the norm-stabilizing property of
(2.7).

######  Proposition 4.2.

Assume $\rho(t,x)$ to be a regular, nonnegative solution to (4.1) on
$[t_{0},t_{0}+T]$ for some $t_{0}\geq 0$ and $T>0$. Then for any
$t\in(t_{0},t_{0}+T)$, there exists a universal constant $C>0$ such that

$\frac{d}{dt}\|\rho-\rho_{M}\|_{L^{2}}^{2}+\|\nabla\rho\|_{L^{2}}^{2}\leq
C_{0}\|\rho-\rho_{M}\|_{L^{2}}^{4}+2\rho_{M}\|\rho-\rho_{M}\|_{L^{2}}^{2}.$
(4.2)

Furthermore, there exists $M_{0}=M_{0}(\rho_{M})\geq\rho_{M}$ sufficiently
large that if $\|\rho(t,\cdot)-\rho_{M}\|_{L^{2}}^{2}\geq\frac{M_{0}}{8}$ for
all $t\in[t_{0},t_{0}+T]$, then there exists a universal constant
$C_{1},C_{2},C_{3}$ such that

$\displaystyle\frac{d}{dt}\|\rho-\rho_{M}\|_{L^{2}}^{2}+\|\nabla\rho\|_{L^{2}}^{2}$
$\displaystyle\leq C_{1}\|\rho-\rho_{M}\|_{L^{2}}^{4},$ (4.3)

and

$\displaystyle\frac{d}{dt}\|\rho-\rho_{M}\|_{L^{2}}^{2}$
$\displaystyle\leq-\frac{C_{2}}{\rho_{M}^{4}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{6}+C_{3}\left(\rho_{M}\|\tilde{\rho}\|_{L^{2}}^{2}+\|\tilde{\rho}\|_{L^{2}}^{4}\right),$
(4.4)

for any $t\in[t_{0},t_{0}+T]$.

The next proposition shows that one can upgrades the $L^{\infty}_{t}L^{2}_{x}$
control of $\rho$ to an $L^{\infty}_{t,x}$ control:

######  Proposition 4.3.

Let $\rho(t,x)$ be a regular, nonnegative solution to (4.1) on
$[t_{0},t_{0}+T]$ for some $t_{0}\geq 0$. Suppose that
$\|\rho(t,\cdot)-\rho_{M}\|_{L^{2}}\leq 2M$ for all $t\in[t_{0},t_{0}+T]$ and
some $M\geq\max(1,\rho_{M})$. Then we also have $\sup_{t_{0}\leq t\leq
t_{0}+T}\|\rho(t,\cdot)-\rho_{M}\|_{L^{\infty}}\leq CM^{2}$, where $C$ is a
universal constant.

The proofs of the propositions above are standard, and we refer interested
readers to [31] for related proofs of Proposition 4.1–4.3. We therefore omit
them here.

######  Remark 4.1.

We allow such generality in (4.1) that Propositions 4.1–4.3 hold for both the
static problem (2.7) and the full problem (5.1). This is because an
incompressible, viscous flow satisfying Lions boundary condition necessarily
satisfies all requirements prescribed in (4.1).

Finally, we establish a corollary that signifies a characteristic time for
which the $L^{2}$ norm $\|\rho-\rho_{M}\|_{L^{2}}^{2}$ doubles.

######  Corollary 4.1.

Suppose $\rho(t,x)$ is a regular solution to (4.1) on $[t_{0},t_{0}+T]$ for
some $t_{0}\geq 0$ and $T>0$. If $\|\rho(t_{0})-\rho_{M}\|_{L^{2}}^{2}\leq M$,
then there exists a universal constant $c_{0}>0$ and a time
$T_{*}(M,\rho_{M}):=c_{0}\min(M^{-1},\rho_{M}^{-1})$ such that

$\sup_{t_{0}\leq t\leq
t_{0}+T_{*}(M,\rho_{M})}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq 2M,$ (4.5)

and

$\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\|\nabla\rho(t)\|_{L^{2}}^{2}dt\leq
2M.$ (4.6)

###### Proof.

Without loss of generality, we only prove the case when $t_{0}=0.$ We also fix
$T_{*}(M,\rho_{M}):=c_{0}\min(M^{-1},\rho_{M}^{-1})$, where the exact size of
$c_{0}$ will be specified later. To show (4.5), we follow a standard barrier
argument: suppose that there exists a time $\tau\in[0,T_{*}(M,\rho_{M})]$ such
that $\|\rho(\tau)-\rho_{M}\|_{L^{2}}^{2}=2M$, and
$\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}<2M$ for any $t\in[0,\tau)$. Integrating
(4.2) from $0$ to $\tau$, we have

$\displaystyle 2M$ $\displaystyle=\|\rho(\tau)-\rho_{M}\|_{L^{2}}^{2}\leq
M+2\rho_{M}\int_{0}^{\tau}\|\rho(s)-\rho_{M}\|_{L^{2}}^{2}dx+C_{0}\int_{0}^{\tau}\|\rho(s)-\rho_{M}\|_{L^{2}}^{4}ds$
$\displaystyle\leq M+4\rho_{M}M\tau+4M^{2}C_{0}\tau<2M,$

if we choose $c_{0}<\frac{1}{4(1+C_{0})}$. This yields a contradiction.

To show (4.6), we integrate (4.3) from $0$ to $T_{*}(M,\rho_{M})$, which
yields:

$\displaystyle\|\rho(T_{*}(M,\rho_{M}))-\rho_{M}\|_{L^{2}}^{2}$
$\displaystyle+\int_{0}^{T_{*}(M,\rho_{M})}\|\nabla\rho(t)\|_{L^{2}}^{2}dt\leq
M+2\rho_{M}\int_{0}^{T_{*}(M,\rho_{M})}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}dt$
$\displaystyle+C_{0}\int_{0}^{T_{*}(M,\rho_{M})}\|\rho(t)-\rho_{0}\|_{L^{2}}^{4}dt$
$\displaystyle\leq M+2\rho_{M}(2M)c_{0}\rho_{M}^{-1}+C_{0}(4M^{2})c_{0}M^{-1}$
$\displaystyle<M+\frac{M}{1+C_{0}}+\frac{C_{0}}{1+C_{0}}M=2M.$

The proof is thus complete. ∎

For the rest of this work, $T_{*}(M,\rho_{M})$ is defined as in Corollary 4.1
after one fixes $M$ and $\rho_{M}$.

####  4.1.2 Fluid Flow Estimates

In this subsection, we restrict ourselves to the static problem (2.1) and
genuinely exploit the Biot-Savart-type relation (2.8). In particular, we show
that an appropriate mixing-type norm can be made sufficiently small in the
regime of large parameter $g$.

######  Proposition 4.4.

Let $(\rho,u)$ be the regular solution to equation (2.7) on
$[t_{0},t_{0}+T_{*}]$ for some $t_{0}\geq 0$. If
$\|\rho(t_{0})-\rho_{M}\|_{L^{2}}^{2}=M\geq 1$, the following bound holds:

$\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt\leq
C(\rho_{M})g^{-1}.$ (4.7)

###### Proof.

We compute that

$\displaystyle\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt$
$\displaystyle=\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}(-\Delta_{D})^{-1}\partial_{1}\rho(-\Delta_{D})^{-1}\partial_{1}\rho
dxdt$
$\displaystyle=\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}\partial_{1}\rho(-\Delta_{D})^{-2}\partial_{1}\rho
dxdt$
$\displaystyle=-\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}\rho\partial_{1}(-\Delta_{D})^{-2}\partial_{1}\rho
dxdt$
$\displaystyle=\frac{1}{g}\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}\rho
u_{2}dxdt.$ (4.8)

Note that we have used the self-adjointness of $(-\Delta_{D})^{-1}$ in the
second equality, integration by parts in the third equality, and (2.8) in the
last equality. On the other hand, we test the $\rho$-equation of (2.7) by
$x_{2}$, which yields:

$\displaystyle\int_{\Omega}\rho u_{2}dx$
$\displaystyle=-\int_{\Omega}x_{2}(u\cdot\nabla\rho)dx=\int_{\Omega}x_{2}\left(\partial_{t}\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))\right)dx,$
$\displaystyle=A_{1}(t)+A_{2}(t)+A_{3}(t)$

where the first equality follows from integration by parts and no-flux
boundary condition satisfied by $u$. Integrating in time from $t_{0}$ to
$t_{0}+T_{*}(M,\rho_{M})$, we note that the RHS contributes to the following
terms: using Proposition 4.1 and the bound (4.6), we first estimate
$A_{1},A_{2}$ by

$\displaystyle\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}A_{1}dt$
$\displaystyle=\int_{\Omega}x_{2}(\rho(t_{0}+T_{*}(M,\rho_{M}))-\rho(t_{0}))dx\leq
C\|\rho_{0}\|_{L^{1}}\leq C(\rho_{M})$
$\displaystyle\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}A_{2}dt$
$\displaystyle=-\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}\partial_{2}\rho
dxdt\leq
CT_{*}(M,\rho_{M})^{1/2}\left(\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\|\nabla\rho(t)\|_{L^{2}}^{2}dt\right)^{1/2}\leq
C,$

where we used the definition of $T_{*}(M,\rho_{M})$ in the final inequality of
$A_{2}$ estimate. To estimate $A_{3}$, we use Sobolev embedding, elliptic
estimate, and interpolation inequality to obtain that

$\displaystyle A_{3}$
$\displaystyle=\int_{\Omega}\rho\partial_{2}(-\Delta_{N})^{-1}(\rho-\rho_{M})\leq\|\rho\|_{L^{2}}\|\partial_{2}(-\Delta_{N})^{-1}(\rho-\rho_{M})\|_{L^{2}}$
$\displaystyle\leq(\|\rho-\rho_{M}\|_{L^{2}}+\rho_{M})\|\partial_{2}(-\Delta_{N})^{-1}(\rho-\rho_{M})\|_{W^{1,6/5}}$
$\displaystyle\leq(\|\rho-\rho_{M}\|_{L^{2}}+\rho_{M})\|\rho-\rho_{M}\|_{L^{6/5}}$
$\displaystyle\leq
C\rho_{M}^{2/3}\|\rho-\rho_{M}\|_{L^{2}}^{1/3}(\|\rho-\rho_{M}\|_{L^{2}}+\rho_{M})$
$\displaystyle\leq C(\rho_{M})M^{2/3}.$

Then we may estimate $A_{3}$ by

$\displaystyle\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}A_{3}dt$
$\displaystyle\leq C(\rho_{M})M^{-1}M^{2/3}=C(\rho_{M})M^{-1/3}\leq
C(\rho_{M}),$

where we used the assumption that $M>1$. Collecting the estimates above
yields:

$\left|\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\int_{\Omega}\rho
u_{2}dxdt\right|\leq C(\rho_{M}),$ (4.9)

Therefore by (4.1.2):

$\int_{t_{0}}^{t_{0}+T_{*}(M,\rho_{M})}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt\leq
C(\rho_{M})g^{-1},$

which concludes the proof. ∎

###  4.2 An Improved Nash-Type Inequality

This section is dedicated to proving a Nash’s inequality with improved
coefficient. Let us recall the following result from [31], which is valid for
any function $\rho\in L^{2}([s,r],H^{1}(\Omega))$, $0\leq s<r\leq\infty$:

######  Proposition 4.5 (Corollary 6.2 in [31]).

Assume that for some $N\geq 1$,

$\int_{s}^{r}\|\partial_{x_{1}}\rho\|_{H^{-1}_{0}}^{2}dt\leq
C_{1}N^{-1}\int_{s}^{r}\|\tilde{\rho}\|_{L^{2}}^{2}dt.$ (4.10)

Then there exists a constant $C(C_{1})>0$ such that

$\int_{s}^{r}\|\tilde{\rho}\|_{L^{2}}^{2}\,dt\leq
CN^{-1/4}\left(\int_{s}^{r}\|\rho\|_{L^{1}}^{2}\,dt\right)^{1/2}\left(\int_{s}^{r}\|\nabla\tilde{\rho}\|_{L^{2}}^{2}\,dt\right)^{1/2}.$
(4.11)

######  Lemma 4.1.

Fix $g>0$. Assume that there exists constants $C_{4}>0$ so that

$\int_{s}^{r}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt\leq
C_{4}g^{-9/10}\left(\int_{s}^{r}\|\tilde{\rho}\|_{L^{2}}^{2}dt\right)^{2},$
(4.12)

and

$\int_{s}^{r}\|\nabla\rho\|_{L^{2}}^{2}dt\leq\frac{g^{9/10}}{C_{4}},$

then there exists some constant $C=C(C_{4})$ such that the following estimate
holds:

$\int_{s}^{r}\|\tilde{\rho}\|_{L^{2}}^{2}dt\leq
Cg^{-9/80}\left(\int_{s}^{r}\|\rho-\rho_{M}\|_{L^{1}}^{2}dt\right)^{1/2}\left(\int_{s}^{r}\|\nabla\rho\|_{L^{2}}^{2}dt\right)^{5/8}.$
(4.13)

###### Proof.

Using definition of $H^{-1}_{0}$ norm and Cauchy-Schwarz inequality, we have

$\|\partial_{1}\tilde{\rho}\|_{H^{-1}_{0}}^{2}=\int_{\Omega}\partial_{1}\tilde{\rho}(-\Delta_{D})^{-1}\partial_{1}\tilde{\rho}dx\leq\|\partial_{1}\tilde{\rho}\|_{L^{2}}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}\leq\|\nabla\rho\|_{L^{2}}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}},$

where we used the fact that $\partial_{1}\rho=\partial_{1}\tilde{\rho}$.
Integrating the above inequality from $s$ to $r$ and applying Cauchy-Schwartz
inequality again, we have

$\displaystyle\int_{s}^{r}\|\partial_{1}\tilde{\rho}\|_{H^{-1}_{0}}^{2}dt$
$\displaystyle\leq\left(\int_{s}^{r}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt\right)^{1/2}\left(\int_{s}^{r}\|\nabla\rho\|_{L^{2}}^{2}dt\right)^{1/2}$
$\displaystyle\leq
C_{4}^{1/2}g^{-9/20}\left(\int_{s}^{r}\|\nabla\rho\|_{L^{2}}^{2}dt\right)^{1/2}\int_{s}^{r}\|\tilde{\rho}\|_{L^{2}}^{2}dt.$

Then an application of Proposition 4.5 via replacing $N$ by
$C_{4}^{-1/2}g^{9/20}\left(\int_{s}^{r}\|\nabla\rho\|_{L^{2}}^{2}dt\right)^{-1/2}\geq
1$ implies (4.13). ∎

###  4.3 Norm-Stabilizing Property and Global Well-Posedness

With the a priori estimates set up in the previous section, we are ready to
prove the norm-stabilizing property, from which the global existence of the
static problem (2.7) follows. Keeping our choice of time $T_{*}$ in mind, we
now prove Proposition 2.3.

###### Proof of Proposition 2.3.

Choose $M_{0},T_{*},\mathfrak{g}_{0}$ as in Remark 4.2. Then (2.9) follows
directly from Corollary 4.1. We thus focus on the proof of (2.10). Assume for
the sake of contradiction that $\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}>M_{0}/8$ for
any $t\in\left[t_{0}+\frac{T_{*}}{2},t_{0}+T_{*}\right]$. For
$g\geq\mathfrak{g}_{0}$, we discuss the following dichotomy:

  1. Case 1:

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\tilde{\rho}\|_{L^{2}}^{2}dt\leq
c_{0}g^{-1/20}.$ (4.14)

In this case, we will take advantage of the estimate (4.4). We first observe
that due to the contradiction assumption, we have a lower bound:

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\rho-\rho_{M}\|_{L^{2}}^{2}dt\geq\frac{T_{*}M_{0}}{16}=\frac{c_{0}}{16}.$

Combining with (4.14) and using orthogonality between $\bar{\rho}$ and
$\tilde{\rho}$, we have

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{2}dt\geq(\frac{1}{16}-g^{-1/20})c_{0}\geq\frac{c_{0}}{32},$
(4.15)

by choice of $\mathfrak{g}_{0}$ in the last inequality. On the other hand, an
application of Hölder’s inequality implies that

$\displaystyle\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{2}dt$
$\displaystyle\leq\left(\frac{T_{*}}{2}\right)^{2/3}\left(\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{6}dt\right)^{1/3}.$

Combining the inequality above with (4.15), we have

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{6}dt\geq\frac{4}{T_{*}^{2}}\left(\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{2}dt\right)^{3}\geq\frac{4c_{0}^{3}}{32^{3}T_{*}^{2}}=\frac{4c_{0}M_{0}^{2}}{32^{3}}.$
(4.16)

Now, we integrate (4.4) from $T_{*}/2$ to $T_{*}$, which yields:

$\displaystyle\|\rho(t_{0}+T_{*})-\rho_{M}\|_{L^{2}}^{2}$
$\displaystyle\leq\|\rho(t_{0}+T_{*}/2)-\rho_{M}\|_{L^{2}}^{2}-\frac{4}{\rho_{M}^{4}}\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\bar{\rho}-\rho_{M}\|_{L^{2}}^{6}dt$
$\displaystyle+C_{3}\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\left(\rho_{M}\|\tilde{\rho}\|_{L^{2}}^{2}+\|\tilde{\rho}\|_{L^{2}}^{4}\right)dt$
$\displaystyle\leq
2M_{0}-C(\rho_{M})M_{0}^{2}+C_{3}(\rho_{M}c_{0}g^{-1/20}+2M_{0}c_{0}g^{-1/20})$
$\displaystyle\leq 2M_{0}-C(\rho_{M})M_{0}^{2}+Cg^{-1/20}M_{0},$

where we used (4.14) and (4.16) in the second inequality. Choose $M_{0}$ so
that

$C(\rho_{M})M_{0}^{2}-(2C+\frac{15}{8})M_{0}>0,$ (4.17)

we have

$C(\rho_{M})M_{0}^{2}>(2C+\frac{15}{8})M_{0}\geq(Cg^{-1/20}+\frac{15}{8})M_{0},$

as $\mathfrak{g}_{0}\geq 1$. We thus conclude that

$\|\rho(t_{0}+T_{*})-\rho_{M}\|_{L^{2}}^{2}<\frac{M_{0}}{8},$

which yields a contradiction to our assumption.

  2. Case 2: $\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\tilde{\rho}\|_{L^{2}}^{2}dt>c_{0}g^{-1/20}$. In this case, since we choose $M_{0}$ such that $\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}>1$ on $[t_{0}+T_{*}/2,t_{0}+T_{*}]$, we can use estimate (4.7) to obtain that

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|(-\Delta_{D})^{-1}\partial_{1}\rho\|_{L^{2}}^{2}dt\leq
C(\rho_{M})g^{-1}<C(\rho_{M})g^{-9/10}\left(\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\tilde{\rho}\|_{L^{2}}^{2}dt\right)^{2}.$

If

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\nabla\rho\|_{L^{2}}^{2}dt>\frac{g^{9/10}}{C(\rho_{M})},$

Then we select $g$ large such that $g^{9/10}\geq 2C(\rho_{M})M_{0}$, which
yields
$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\nabla\rho\|_{L^{2}}^{2}dt>2M_{0}$. But
this contradicts with (4.6). Suppose otherwise, we can invoke Lemma 4.1 to
conclude that

$c_{0}g^{-1/20}<\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\tilde{\rho}\|_{L^{2}}^{2}dt\leq
C(\rho_{M})g^{-9/80}\left(\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\rho-\rho_{M}\|_{L^{1}}^{2}dt\right)^{1/2}\left(\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\nabla\rho\|_{L^{2}}^{2}dt\right)^{5/8}$
(4.18)

Rearranging (4.18) and invoking Proposition 4.1, we have

$\int_{t_{0}+T_{*}/2}^{t_{0}+T_{*}}\|\nabla\rho\|_{L^{2}}^{2}dt\geq
C(\rho_{M})\frac{g^{1/10}}{T_{*}^{4/5}}=C(\rho_{M})g^{1/10}M_{0}^{4/5}>2M_{0},$
(4.19)

where we used our choice of $\mathfrak{g}_{0}$ in the final inequality. Note
that we also used the relation $T_{*}=c_{0}M_{0}^{-1}$ above. However, (4.19)
contradicts the a priori estimate (4.6).

Thus, both cases in the dichotomy lead to a contradiction and we have
completed the proof. ∎

######  Remark 4.2.

The specific choice of parameters $M_{0},T_{*},$ and $\mathfrak{g}_{0}$ are
summarized as follows: first choose
$M_{0}=\max(8,\rho_{M},\rho_{M}^{2},2\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2},\mathcal{C}_{0}(\rho_{M}))$,
where $\mathcal{C}_{0}$ is fixed such that (4.3), (4.17), and Proposition 4.3
hold. Then we choose

$T_{*}=c_{0}\min(M_{0}^{-1},\rho_{M}^{-1})=c_{0}M_{0}^{-1},\quad\mathfrak{g}_{0}=\max\left\\{32^{20},C(\rho_{M})M_{0}^{10/9},\left(\frac{1+2c_{0}}{C(\rho_{M})}M_{0}^{1/5}\right)^{10}\right\\},$

where the universal constant $c_{0}$ is defined in Corollary 4.1.
$C(\rho_{M})$ is a constant that only depends on $\rho_{M}$.

Note that we can conclude global well-posedness of system (2.1) as a direct
consequence of Proposition 2.3.

######  Corollary 4.2.

For arbitrary nonnegative initial data $\rho_{0}\in C^{\infty}(\Omega)$, there
exists $mfg_{0}=\mathfrak{g}_{0}(\rho_{M})>0$ sufficiently large that the
solution of the static problem (2.7) is globally regular for all
$g\geq\mathfrak{g}_{0}$.

###### Proof.

Let $\mathfrak{g}_{0},M_{0}$ be chosen as in Proposition 2.3 (and therefore
$T_{*}$ is fixed). Let $t_{0}\geq 0$ be the first time such that
$\|\rho-\rho_{M}\|_{L^{2}}^{2}=M_{0}$. If $t_{0}=\infty$, then this means that
$\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq M_{0}$ for all $t\geq 0$ as
$M_{0}\geq\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2}$. Hence by the regularity
criterion stated in Theorem 3.1, we are already done. We may then suppose that
$t_{0}<\infty$. By Proposition 2.3, there exists
$T_{1}\in[t_{0}+T_{*}/2,t_{0}+T_{*}]$ such that
$\|\rho(t_{0}+T_{1})-\rho_{M}\|_{L^{2}}^{2}=M_{0}/8<M_{0}$. Moreover, we know
that $\sup_{0\leq t\leq t_{0}+T_{*}}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
2M_{0}$. Then let $t_{1}$ to be the first time after $t_{0}+T_{1}$ such that
$\|\rho-\rho_{M}\|_{L^{2}}^{2}=M_{0}$. Repeating the argument above, there
exists $T_{2}\in[t_{1}+T_{*}/2,t_{1}+T_{*}]$ such that
$\|\rho(t_{1}+T_{2})-\rho_{M}\|_{L^{2}}^{2}=M_{0}/8$ and
$\sup_{t_{0}+T_{1}\leq t\leq t_{1}+T_{*}}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
2M_{0}$. Repeating such steps indefinitely and by the fact that $T_{*}>0$, we
arrive at

$\sup_{t\geq 0}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq 2M_{0}<\infty,$

which implies global well-posedness by the $L^{2}$ criterion in Theorem 3.1. ∎

##  5 Analysis of the Full Stokes Problem

In this section, we study the full Keller-Segel-Stokes system i.e. (1.3) with
$\delta=0$, and we prove Theorem 1.1 in the $\delta=0$ case. For convenience,
let us recall the system here:

$\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,\quad\text{
in }\Omega$ (5.1a) $\partial_{t}u-B\Delta u+\nabla p=Bg\rho
e_{2},\quad\operatorname{div}(u)=0,\quad\text{ in }\Omega$ (5.1b)
$\partial_{2}\rho\big{|}_{\partial\Omega}=0,\;u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0,$
(5.1c) $\rho(0,x)=\rho_{0}(x)\geq 0,\;u(0,x)=u_{0}(x),$ (5.1d)

As pointed out in Section 2.3, we prove Proposition 2.4 in the $\delta=0$ case
with bootstrap assumption (A). Throughout this section, we assume $g\geq
g_{0}$ and $B\geq B_{0}$, where $g_{0}$ and $B_{0}$ are sufficiently large and
we refer the readers to Remark 5.4 for their exact values.

We organize this section in the following way: in Section 5.1, we prove
estimates of various derivatives of both $\rho$ and $u$ given bootstrap
assumption (A). In Section 5.2, we set up a comparison scheme in between the
full problem (5.1) and the static problem (2.7). Then we close the bootstrap
argument by utilizing estimates shown in Section 5.1.

###  5.1 Technical Controls

####  5.1.1 Control of $\nabla\rho$

The following estimate of $\nabla\rho$ in any time interval of length $T_{*}$
will be useful in the control of $\partial_{t}u,\partial_{t}\rho$ later on.

######  Lemma 5.1.

For arbitrary $t_{0}\in[0,\mathcal{T}_{0}]$ and given bootstrap assumption
(A), we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\nabla\rho\|_{L^{2}}^{2}dt\leq 18M_{0}.$ (5.2)

###### Proof.

This lemma follows directly from applying Corollary 4.1 with $M=9M_{0}$. ∎

####  5.1.2 Uniform Control of $u$

In this part, we give an estimate of $\|u\|_{L^{2}}$ that is uniform over the
slightly longer time interval $[0,\mathcal{T}_{0}+T_{*}/9]$.

######  Lemma 5.2.

Given bootstrap assumption (A), the following estimate holds:

$\sup_{0\leq t\leq\mathcal{T}_{0}+T_{*}/9}\|u(t)\|_{L^{2}}^{2}\leq
C(\rho_{0},u_{0})g^{2}M_{0}.$ (5.3)

###### Proof.

Testing (5.1b) by $u$, using $\operatorname{div}u=0$ and the boundary
condition, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u\|_{L^{2}}^{2}+B\|\nabla
u\|_{L^{2}}^{2}$ $\displaystyle\leq
Bg\|\rho\|_{L^{2}}\|u\|_{L^{2}}\leq\frac{B}{2C_{p}}\|u\|_{L^{2}}^{2}+2C_{p}Bg^{2}\|\rho\|_{L^{2}}^{2}.$

Note that we used the following computation to calculate the gradient term:

$\displaystyle\int_{\Omega}u\Delta udx$
$\displaystyle=\int_{\mathbb{T}}\int_{0}^{\pi}u_{j}\partial_{i}\partial_{i}u_{j}dx_{2}dx_{1}=-\|\nabla
u\|_{L^{2}}^{2}+\int_{\mathbb{T}}u_{j}\partial_{2}u_{j}\big{|}_{x_{2}=0}^{x_{2}=\pi}dx_{1}$
$\displaystyle=-\|\nabla
u\|_{L^{2}}^{2}+\int_{\mathbb{T}}(u_{1}\partial_{2}u_{1}+u_{2}\partial_{2}u_{2})\big{|}_{x_{2}=0}^{x_{2}=\pi}dx_{1}$
$\displaystyle=-\|\nabla u\|_{L^{2}}^{2},$

where we used the Lions boundary condition in the final equality.

On the other hand, Poincaré inequality implies that $\|u\|_{L^{2}}^{2}\leq
C_{p}\|\nabla u\|_{L^{2}}^{2}$ as $u\in W$. Combining the two estimates above
and rearranging, we obtain the following differential inequality:

$\frac{d}{dt}\|u\|_{L^{2}}^{2}\leq-\frac{B}{C_{p}}\|u\|_{L^{2}}^{2}+4C_{p}Bg^{2}\|\rho\|_{L^{2}}^{2}.$

Invoking Duhamel’s formula, we have the following integral form of the above
estimate: for $t\in[0,\mathcal{T}_{0}+T_{*}/9]$,

$\|u(t)\|_{L^{2}}^{2}\leq
e^{-\frac{B}{C_{p}}t}\|u_{0}\|_{L^{2}}^{2}+4C_{p}Bg^{2}\int_{0}^{t}e^{-\frac{B}{C_{p}}(t-s)}\|\rho(s)\|_{L^{2}}^{2}ds.$
(5.4)

To estimate (5.4), we use bootstrap assumption (A) and Corollary 4.1 to
conclude that

$\sup_{0\leq t\leq\mathcal{T}_{0}+T_{*}/9}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
18M_{0}.$

Thus, we deduce from (5.4) that:

$\displaystyle\|u(t)\|_{L^{2}}^{2}$
$\displaystyle\leq\|u_{0}\|_{L^{2}}^{2}+4C_{p}Bg^{2}(\sup_{0\leq
t\leq\mathcal{T}_{0}+T_{*}/9}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}+2\pi^{2}\rho_{M}^{2})\int_{0}^{t}e^{-\frac{B}{C_{p}}(t-s)}ds$
$\displaystyle\leq\|u_{0}\|_{L^{2}}^{2}+4C_{p}Bg^{2}(18M_{0}+2\pi^{2}\rho_{M}^{2})\frac{C_{p}}{B}(1-e^{-\frac{Bt}{C_{p}}})$
$\displaystyle\leq C(\rho_{0},u_{0})g^{2}M_{0},$

where the last inequality follows from our choice that
$M_{0}\geq\rho_{M}^{2}$. ∎

####  5.1.3 Control of $\partial_{t}\rho$

We will aim to control the time derivative $\partial_{t}\rho$ within a time
interval of length $T_{*}$ in this section. This estimate turns out to be
crucial in the control of time derivative of velocity later on.

######  Lemma 5.3.

Assuming the bootstrap assumption (A). For any $t_{0}\in[0,\mathcal{T}_{0}]$,
we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}M_{0}^{2}$ (5.5)

###### Proof.

Picking arbitrary $\varphi\in H^{1}_{0}(\Omega)$ and using (5.1a), we have

$\displaystyle\left|\int_{\Omega}\partial_{t}\rho\varphi dx\right|$
$\displaystyle\leq\left|\int_{\Omega}u\cdot\nabla\rho\varphi
dx\right|+\left|\int_{\Omega}\Delta\rho\varphi
dx\right|+\left|\int_{\Omega}\varphi\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))dx\right|$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

Using incompressibility of $u$ and integrating by parts, we first see that

$I_{1}=\left|\int_{\Omega}u\cdot\nabla\varphi(\rho-\rho_{M})dx\right|\leq\|\rho-\rho_{M}\|_{L^{\infty}}\|u\|_{L^{2}}\|\nabla\varphi\|_{L^{2}},$

Applying integration by parts in a similar fashion, we also have

$\displaystyle I_{2}$
$\displaystyle=\left|\int_{\Omega}\nabla\rho\cdot\nabla\varphi
dx\right|\leq\|\nabla\rho\|_{L^{2}}\|\nabla\varphi\|_{L^{2}},$ $\displaystyle
I_{3}$
$\displaystyle=\left|\int_{\Omega}\rho\nabla\varphi\cdot\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M})dx\right|\leq\|\rho\|_{L^{4}}\|\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M})\|_{L^{4}}\|\nabla\varphi\|_{L^{2}}$
$\displaystyle\leq\|\rho\|_{L^{4}}\|\rho-\rho_{M}\|_{L^{2}}\|\nabla\varphi\|_{L^{2}}\leq\|\rho\|_{L^{\infty}}^{1/2}\|\rho-\rho_{M}\|_{L^{2}}^{3/2}\|\nabla\varphi\|_{L^{2}}.$

Combining estimates of $I_{i}$ above and using duality, we conclude that for
fixed $t\in[t_{0},t_{0}+T_{*}/9]$,

$\displaystyle\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}$
$\displaystyle\lesssim\|\rho-\rho_{M}\|_{L^{\infty}}^{2}\|u\|_{L^{2}}^{2}+\|\nabla\rho\|_{L^{2}}^{2}+\|\rho\|_{L^{\infty}}\|\rho-\rho_{M}\|_{L^{2}}^{3}$
$\displaystyle\leq
C(\rho_{0},u_{0})(g^{2}M_{0}^{3}+M_{0}^{5/2})+\|\nabla\rho\|_{L^{2}}^{2}\leq
C(\rho_{0},u_{0})g^{2}M_{0}^{3}+\|\nabla\rho\|_{L^{2}}^{2},$

where we used the bootstrap assumption (A), Proposition 4.3, and Lemma 5.2 in
the second inequality. Finally, we integrate over $[t_{0},t_{0}+T_{*}/9]$ and
invoke Lemma 5.1 to conclude that

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}dt\leq\frac{c_{0}M_{0}^{-1}}{9}C(\rho_{0},u_{0})g^{2}M_{0}^{3}+CM_{0}\leq
C(\rho_{0},u_{0})g^{2}M_{0}^{2}$

since $g\geq 1$ and $M_{0}\geq 1$. ∎

####  5.1.4 Control of $\partial_{t}u$

In this section, we finish the sequence of preliminary estimates by showing
bounds on the time derivative $\partial_{t}u$. As the first step, we show an
estimate of $\partial_{t}u$ near time zero:

######  Lemma 5.4.

Consider time
$s_{0}=c_{0}\min(\|\rho_{0}-\rho_{M}\|_{L^{2}}^{-2},\rho_{M}^{-1})$, where
$c_{0}$ is chosen as in Corollary 4.1. Then the following estimate holds:

$\int_{0}^{s_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})B^{2}g^{2}.$ (5.6)

In particular, there exists $\tau_{0}\in(0,s_{0})$ such that

$\|\partial_{t}u(\tau_{0})\|_{L^{2}}^{2}\leq\frac{C(\rho_{0},u_{0})B^{2}g^{2}}{s_{0}}=:\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2}.$
(5.7)

###### Proof.

Note that given (5.6), (5.7) directly follows from Chebyshev’s inequality.
Hence, we focus on showing (5.6). Multiplying $\partial_{t}u$ on both sides of
(5.1b) and integrating in space, we have

$\|\partial_{t}u\|_{L^{2}}^{2}+B\int_{\Omega}\partial_{t}u\cdot(-\Delta
u)dx=Bg\int_{\Omega}\rho\partial_{t}u_{2}dx.$ (5.8)

Since $u\in W$, we can further compute that

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}$
$\displaystyle=\int_{\Omega}\partial_{t}\partial_{i}u_{j}\partial_{i}u_{j}dx=\int_{\Omega}\partial_{t}u\cdot(-\Delta
u)dx+\int_{\partial\Omega}\partial_{t}u_{j}\partial_{2}u_{j}dS$
$\displaystyle=\int_{\Omega}\partial_{t}u\cdot(-\Delta
u)dx+\int_{\partial\Omega}\partial_{t}u_{1}\cancelto{0}{\partial_{2}u_{1}}dS+\int_{\partial\Omega}\cancelto{0}{\partial_{t}u_{2}}\partial_{2}u_{2}dS$
$\displaystyle=\int_{\Omega}\partial_{t}u\cdot(-\Delta u)dx.$

Combining with (5.8), we obtain the following energy inequality: for
$t\in[0,s_{0}]$

$\|\partial_{t}u\|_{L^{2}}^{2}+B\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}\leq
B^{2}g^{2}\|\rho\|_{L^{2}}^{2}\leq
B^{2}g^{2}(2\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2}+4\pi^{2}\rho_{M}^{2}),$ (5.9)

where we used our choice of $s_{0}$ and Corollary 4.1 in the final inequality.
Integrating (5.9) on interval $[0,s_{0}]$, we have

$\displaystyle\int_{0}^{s_{0}}\|\partial_{t}u\|_{L^{2}}^{2}dt\leq
B\|u_{0}\|_{1}^{2}+C(\rho_{0})B^{2}g^{2}\leq C(\rho_{0},u_{0})B^{2}g^{2},$

since $g,B>1$. This concludes our proof. ∎

With the $L^{2}$ estimate (5.7) of $\partial_{t}u$ near time zero above, we
can further derive estimates of $\partial_{t}u$ up to $\mathcal{T}_{0}$ thanks
to the bootstrap assumption (A). More precisely, we have:

######  Lemma 5.5.

Assume the bootstrap assumption (A) holds. There exists
$\tau_{0}=\tau_{0}(\rho_{0})<s_{0}$ such that

$\sup_{\tau_{0}\leq t\leq\mathcal{T}_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}\leq
2\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2}.$ (5.10)

Moreover, for any $t_{0}\in[\tau_{0},\mathcal{T}_{0}]$, we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})Bg^{2}.$ (5.11)

######  Remark 5.1.

We remark that (5.11) does not follow directly from integrating (5.10) in
time. In fact, doing so will only lead to an insufficient upper bound of size
$B^{2}g^{2}M_{0}^{-1}$.

###### Proof.

To prove both (5.10) and (5.11), we need to study the evolution of
$\partial_{t}u$. By differentiating (5.1b) in time, it is straightforward to
check that $\partial_{t}u$ satisfies the following equation:

$\partial_{t}^{2}u-B\Delta\partial_{t}u+\nabla\partial_{t}p=Bg(\partial_{t}\rho)e_{2}.$

Testing by $\partial_{t}u$, using $\operatorname{div}(\partial_{t}u)=0$ and
the fact that $\partial_{t}u$ still satisfies the Lions boundary condition, we
have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\partial_{t}u\|_{L^{2}}^{2}+B\|\nabla\partial_{t}u\|_{L^{2}}^{2}$
$\displaystyle=Bg\int_{\Omega}(\partial_{t}u_{2})(\partial_{t}\rho)dx\leq
Bg\|\partial_{t}u_{2}\|_{H^{1}_{0}}\|\partial_{t}\rho\|_{H^{-1}_{0}}$
$\displaystyle\leq\frac{B}{2}\|\partial_{t}u_{2}\|_{H^{1}_{0}}^{2}+2Bg^{2}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2},$

where we used the $H^{1}_{0}$–$H^{-1}_{0}$ duality in the first inequality.
The use of duality is indeed valid since the Lions boundary condition implies
that $u_{2}$, and thus $\partial_{t}u_{2}$, satisfies zero Dirichlet boundary
condition. Now rearranging the inequality above and applying Poincaré
inequality (2.2) to the diffusion term, we obtain the following differential
inequality:

$\frac{d}{dt}\|\partial_{t}u\|_{L^{2}}^{2}\leq-\frac{B}{C_{p}}\|\partial_{t}u\|_{L^{2}}^{2}+CBg^{2}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}.$
(5.12)

Let $\tau_{0}$ be chosen as in Lemma 5.4. Writing (5.12) in Duhamel form and
using (5.7), we obtain that for any $t\in[\tau_{0},\tau_{0}+T_{*}/9]$,

$\|\partial_{t}u(t)\|_{L^{2}}^{2}\leq\mathcal{C}_{1}(\rho_{0},u_{0})e^{-\frac{B(t-\tau_{0})}{C_{p}}}B^{2}g^{2}+CBg^{2}\int_{\tau_{0}}^{t}e^{-\frac{B(t-s)}{C_{p}}}\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}ds=J_{1}+J_{2}.$
(5.13)

The key point is that we have to estimate in two different time scales. Choose
$B$ such that

$B^{-1/2}<T_{*}/9.$ (5.14)

If $t\in[\tau_{0},\tau_{0}+B^{-1/2}]$, we simply have

$J_{1}\leq\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2},\;J_{2}\leq
CBg^{2}\int_{0}^{T_{*}/9}\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}ds\leq
C(\rho_{0},u_{0})Bg^{4}M_{0}^{2},$

where we used Lemma 5.3 in the last inequality. Choosing $B$ sufficiently
large that

$C(\rho_{0},u_{0})g^{2}M_{0}^{2}<\mathcal{C}_{1}(\rho_{0},u_{0})B,$ (5.15)

we have

$\|\partial_{t}u(t)\|_{L^{2}}^{2}\leq
2\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2}.$

If $t\in[\tau_{0}+B^{-1/2},\tau_{0}+T_{*}/9]$, we instead estimate that

$J_{1}\leq\mathcal{C}_{1}(\rho_{0},u_{0})e^{-\frac{B^{1/2}}{C_{p}}}B^{2}g^{2}\leq\frac{\mathcal{C}_{1}(\rho_{0},u_{0})}{2}B^{2}g^{2},$

if we choose $B$ such that

$e^{-\frac{B^{1/2}}{C_{p}}}\leq\frac{1}{2}.$ (5.16)

Moreover, $J_{2}\leq C(\rho_{0},u_{0})Bg^{4}M_{0}^{2}$ as in the first case.
But then we may choose $B$, if needed, even larger such that

$C(\rho_{0},u_{0})Bg^{4}M_{0}^{2}<\frac{\mathcal{C}_{1}(\rho_{0},u_{0})}{2}B^{2}g^{2}.$
(5.17)

This yields that

$\sup_{\tau_{0}+B^{-1/2}\leq
t\leq\tau_{0}+T_{*}/9}\|\partial_{t}u(t)\|_{L^{2}}^{2}<2\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2}$

with
$\|\partial_{t}u(\tau_{0}+T_{*}/9)\|_{L^{2}}^{2}\leq\mathcal{C}_{1}(\rho_{0},u_{0})B^{2}g^{2}$.
Now, we can iterate the above steps at $t_{i}=\tau_{0}+iT_{*}/9$. Since
$T_{*}>0$ is fixed, the argument terminates after finitely many iterations.
Thus, we conclude the proof of (5.10). To prove (5.11), we write (5.12) in
time-integrated Duhamel form starting at time $t_{0}\geq\tau_{0}$:

$\displaystyle\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt$
$\displaystyle\leq\|\partial_{t}u(t_{0})\|_{L^{2}}^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}e^{-\frac{B}{C_{p}}(t-t_{0})}dt$
$\displaystyle+CBg^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\int_{t_{0}}^{t}e^{-\frac{B(t-s)}{C_{p}}}\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}dsdt$
$\displaystyle\leq
2\mathcal{C}_{1}B^{2}g^{2}\frac{C_{p}}{B}(1-e^{\frac{-BT_{*}}{9C_{p}}})$
$\displaystyle+CBg^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\left(\int_{t_{0}}^{s}e^{-\frac{B(t-s)}{C_{p}}}dt\right)\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}dt$
$\displaystyle\leq
C(\rho_{0},u_{0})Bg^{2}+Cg^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}dt$
$\displaystyle\leq C(\rho_{0},u_{0})(Bg^{2}+g^{4}M_{0}^{2})\leq
C(\rho_{0},u_{0})Bg^{2},$

if we choose $B$ sufficiently large that

$B\geq g^{2}M_{0}^{2}.$ (5.18)

Note that we used (5.10) and Fubini theorem in the second inequality, and
Lemma 5.3 in the third inequality. This concludes the proof of (5.11). ∎

######  Remark 5.2.

The fundamental reason that we prove Lemma 5.4 and choose starting time
$\tau_{0}>0$ in Lemma 5.5 is due to the lack of compatibility condition
imposed on initial datum $u_{0}$. For such general datum, one generally cannot
obtain $\partial_{t}u|_{t=0}$ by taking the limit of $t\to 0$ of the equation.
We refer to [21] for a more detailed discussion on compatible data for
parabolic problems.

######  Remark 5.3.

If we additionally assume that $t_{0}\in[\tau_{0},\mathcal{T}_{0}-T_{*})$, we
may repetitively apply the arguments in proving (5.11) to obtain a bound in a
larger time interval:

$\int_{t_{0}}^{t_{0}+T_{*}}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})Bg^{2}.$

###  5.2 Closing the Bootstrap

As the final step, we show the improved estimate as stated in Proposition 2.4.
Consider time $\mathcal{T}_{1}$ to be the first time such that
$\|\rho-\rho_{M}\|_{L^{2}}^{2}$ achieves $M_{0}$. Note that $\mathcal{T}_{1}$
verifies the following two properties:

  1. 1.

$\mathcal{T}_{1}>s_{0}$ due to $M_{0}>2\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2}$ and
Corollary 4.1;

  2. 2.

By the definition of $T_{*}$, $\mathcal{T}_{1}+T_{*}<\mathcal{T}_{0}$.

Then we set up the following comparison scheme: let $\rho_{s},u_{s}$ be the
global regular solution to (2.7) with initial time $\mathcal{T}_{1}$ and
initial condition $\rho_{s}(\mathcal{T}_{1},x)=\rho(\mathcal{T}_{1},x)$.
Define $r=\rho-\rho_{s}$, $v=u-u_{s}$. A straightforward computation yields
the following equations for $r$ and $v$:

$\partial_{t}r-\Delta r+v\cdot\nabla\rho+u_{s}\cdot\nabla
r+\operatorname{div}(r\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))+\operatorname{div}(\rho_{s}\nabla(-\Delta_{N})^{-1}r)=0,$
(5.19a) $-\Delta
v+\nabla(B^{-1}p-p_{s})=-\frac{1}{B}\partial_{t}u+gre_{2},\;\operatorname{div}v=0,$
(5.19b)
$r(\mathcal{T}_{1},x)=0,\;v(\mathcal{T}_{1},x)=u(\mathcal{T}_{1},x)-u_{s}(\mathcal{T}_{1},x),$
(5.19c)
$\partial_{2}r|_{\partial\Omega}=0,\;v_{2}|_{\partial\Omega}=0,\;\partial_{2}v_{1}|_{\partial\Omega}=0.$
(5.19d)

Note that in view of local well-posedness of (5.1) and $L^{2}$ estimate (4.3),
the system (5.19) has a regular solution at least on
$[\mathcal{T}_{1},\mathcal{T}_{0}]$. Thus, we may test (5.19a) by $r$ and use
incompressibility of $u_{s}$ to deduce that for any
$t\in[\mathcal{T}_{1},\mathcal{T}_{1}+T_{*}]$:

$\displaystyle\frac{1}{2}\frac{d}{dt}\|r\|_{L^{2}}^{2}+\|\nabla
r\|_{L^{2}}^{2}$
$\displaystyle=-\int_{\Omega}rv\cdot\nabla\rho-\int_{\Omega}r\operatorname{div}(r\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))-\int_{\Omega}r\operatorname{div}(\rho_{s}\nabla(-\Delta_{N})^{-1}r)$
$\displaystyle=K_{1}+K_{2}+K_{3}.$

We first treat $K_{2},K_{3}$. Integrating by parts, we have

$\displaystyle K_{2}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\nabla(r^{2})\cdot\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M})=\frac{1}{2}\int_{\Omega}r^{2}(\rho-\rho_{M})dx\leq\frac{1}{2}\|\rho-\rho_{M}\|_{L^{\infty}}\|r\|_{L^{2}}^{2}.$

To estimate $K_{3}$, we integrate by parts and obtain

$\displaystyle K_{3}$ $\displaystyle=\int_{\Omega}\rho_{s}\nabla
r\cdot\nabla(-\Delta_{N})^{-1}rdx\leq\|\nabla
r\|_{L^{2}}\|\rho_{s}\|_{L^{4}}\|\nabla(-\Delta_{N})^{-1}r\|_{L^{4}}$
$\displaystyle\leq\frac{1}{4}\|\nabla
r\|_{L^{2}}^{2}+\frac{3}{4}\|\rho_{s}\|_{L^{4}}^{2}\|\nabla(-\Delta_{N})^{-1}r\|_{L^{4}}^{2}$
$\displaystyle\leq\frac{1}{4}\|\nabla
r\|_{L^{2}}^{2}+\frac{3}{4}\|\rho_{s}\|_{L^{4}}^{2}\|r\|_{L^{2}}^{2},$

where we used the Sobolev embedding $H^{1}\subset L^{4}$ and the standard
elliptic estimate. Finally, we treat $K_{1}$, which potentially contributes
the growth of $\|r\|_{L^{2}}$ due to the existence of $v$ in the term. In
fact, we have

$\displaystyle K_{1}$ $\displaystyle=-\int_{\Omega}rv\cdot\nabla\rho
dx=\int_{\Omega}\operatorname{div}(rv)\rho
dx-\underbrace{\int_{\partial\Omega}rv_{2}\rho d\sigma}_{=0\text{ by no-flux
boundary condition of }v}$ $\displaystyle=\int_{\Omega}\nabla r\cdot v\rho
dx\leq\|\nabla
r\|_{L^{2}}\|v\|_{L^{\infty}}\|\rho\|_{L^{2}}\leq\frac{1}{4}\|\nabla
r\|_{L^{2}}^{2}+\frac{3}{8}\|v\|_{2}^{2}\|\rho\|_{L^{2}}^{2},$

where we used the Sobolev embedding $H^{2}\subset L^{\infty}$ in the final
inequality.

Observe now that $-B^{-1}\partial_{t}u(t,\cdot)+gr(t,\cdot)e_{2}\in H$.
Indeed, as $u(t,\cdot)\in W$, we in particular have
$\int_{\Omega}u_{1}(x)dx=0$. This leads to

$\int_{\Omega}\left(-B^{-1}\partial_{t}u+gre_{2}\right)_{1}dx=\int_{\Omega}-B^{-1}\partial_{t}u_{1}dx=-B^{-1}\partial_{t}\int_{\Omega}u_{1}(x)dx=0,$

and this verifies the observation. Hence, we are eligible to apply the
elliptic estimate (2.3) and obtain

$\|v\|_{2}^{2}\lesssim\frac{1}{B^{2}}\|\partial_{t}u\|_{L^{2}}^{2}+g^{2}\|r\|_{L^{2}}^{2}.$

Thus collecting the estimates above, we have the following differential
inequality:

$\displaystyle\frac{d}{dt}\|r\|_{L^{2}}^{2}+\|\nabla r\|_{L^{2}}^{2}$
$\displaystyle\leq
C\left(g^{2}\|\rho\|_{L^{2}}^{2}+\|\rho-\rho_{M}\|_{L^{\infty}}+\|\rho_{s}\|_{L^{4}}^{2}\right)\|r\|_{L^{2}}^{2}+\frac{1}{B^{2}}\|\rho\|_{L^{2}}^{2}\|\partial_{t}u\|_{L^{2}}^{2}$
$\displaystyle\leq
F(t)\|r\|_{L^{2}}^{2}+\frac{2}{B^{2}}(\|\rho-\rho_{M}\|_{L^{2}}^{2}+2\pi^{2}\rho_{M}^{2})\|\partial_{t}u\|_{L^{2}}^{2},$

where we set
$F(t)=C(g^{2}\|\rho\|_{L^{2}}^{2}+\|\rho-\rho_{M}\|_{L^{\infty}}+\|\rho_{s}\|_{L^{4}}^{2})$.
Note that by the bootstrap assumption, Proposition 4.3, and Proposition 2.3,
we have

$\int_{\mathcal{T}_{1}}^{\mathcal{T}_{1}+T_{*}}F(s)ds\leq
C(\rho_{0})(g^{2}+1).$

Now we would like to estimate $\|r(t)\|_{L^{2}}^{2}$ on
$[\mathcal{T}_{1},\mathcal{T}_{1}+T_{*}]$. Since $r(\mathcal{T}_{1},x)=0$, an
application of Grönwall inequality and bootstrap assumption (A) yields:

$\displaystyle\|r(t)\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{1}{B^{2}}\int_{\mathcal{T}_{1}}^{t}(\|\rho(s)-\rho_{M}\|_{L^{2}}^{2}+2\pi^{2}\rho_{M}^{2})\|\partial_{t}u(s)\|_{L^{2}}^{2}ds\left[\exp\left(\int_{\mathcal{T}_{1}}^{t}F(s)ds\right)\right]$
$\displaystyle\leq\frac{C(\rho_{0})(M_{0}+1)}{B^{2}}\int_{\mathcal{T}_{1}}^{\mathcal{T}_{1}+T_{*}}\|\partial_{t}u(s)\|_{L^{2}}^{2}ds\cdot
e^{g^{2}+1}$
$\displaystyle\leq\frac{C(\rho_{0},u_{0})g^{2}(M_{0}+1)e^{g^{2}+1}}{B},$
(5.20)

where we used Remark 5.3 due to the two properties satisfied by
$\mathcal{T}_{1}$.

Now, we choose $B$ sufficiently large that

$\sup_{\mathcal{T}_{1}\leq
t\leq\mathcal{T}_{1}+T_{*}}\|r(t)\|_{L^{2}}^{2}\leq\frac{C(\rho_{0},u_{0})g^{2}(M_{0}+1)e^{g^{2}+1}}{B}<\frac{M_{0}}{4}.$
(5.21)

On the other hand, we note that
$\|\rho_{s}(t)\|_{L^{1}}=\|\rho(t)\|_{L^{1}}=\|\rho_{0}\|_{L^{1}}$ for all
$t\geq\mathcal{T}_{1}$, and
$\|\rho_{s}(\mathcal{T}_{1})-\rho_{M}\|_{L^{2}}^{2}=M_{0}$ by the definition
of $\mathcal{T}_{1}$. Then by Remark 4.2, there exists
$\tilde{M_{0}}=\max(8,\rho_{M},2M_{0},\mathcal{C}_{0}(\rho_{M}))$,
$\tilde{\mathfrak{g}_{0}}=\tilde{\mathfrak{g}_{0}}(\tilde{M_{0}},\rho_{M})$,
$\tilde{T_{*}}=c_{0}\tilde{M_{0}}^{-1}$ such that for any
$g\geq\tilde{\mathfrak{g}_{0}}$, we have

$\sup_{t_{0}\leq t\leq
t_{0}+\tilde{T_{*}}}\|\rho_{s}(t)-\rho_{M}\|_{L^{2}}^{2}\leq 2\tilde{M_{0}},$

and there exists $\tilde{T}\in(\tilde{T_{*}}/2,\tilde{T_{*}})$ such that

$\|\rho_{s}(t_{0}+\tilde{T})-\rho_{M}\|_{L^{2}}^{2}\leq\frac{\tilde{M_{0}}}{8}.$

Observe that $\tilde{M_{0}}=2M_{0}$: by definition it is clear that
$\tilde{M_{0}}\geq 2M_{0}$. Conversely, by definition of $M_{0}$ we know that
$2M_{0}>M_{0}\geq\max(8,\rho_{M},\mathcal{C}_{0}(\rho_{M}))$, which implies
that $\tilde{M_{0}}\leq 2M_{0}$.

Thus particularly, we have $\tilde{T_{*}}=\frac{T_{*}}{2}$, and we choose
$g_{0}:=4{\mathfrak{g}}_{0}\geq\tilde{\mathfrak{g}}_{0}$ in view of Remark
4.2. Moreover, for $g\geq g_{0}$, we know that
$\|\rho_{s}(t)-\rho_{M}\|_{L^{2}}^{2}\leq 2\tilde{M_{0}}=4M_{0}$ for
$t\in[\mathcal{T}_{1},\mathcal{T}_{1}+T_{*}/2]$, and there exists
$T\in[T_{*}/4,T_{*}/2]$ such that
$\|\rho_{s}(\mathcal{T}_{1}+T)-\rho_{M}\|_{L^{2}}^{2}\leq\frac{\tilde{M_{0}}}{8}=\frac{M_{0}}{4}$.
By Cauchy-Schwartz inequality and (5.21), we conclude that

$\displaystyle\sup_{\mathcal{T}_{1}\leq
t\leq\mathcal{T}_{1}+T_{*}/2}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}$
$\displaystyle\leq 2\sup_{\mathcal{T}_{1}\leq
t\leq\mathcal{T}_{1}+T_{*}/2}\|\rho_{s}(t)-\rho_{M}\|_{L^{2}}^{2}+2\sup_{\mathcal{T}_{1}\leq
t\leq\mathcal{T}_{1}+T_{*}/2}\|r(t)\|_{L^{2}}^{2}$ $\displaystyle\leq
4M_{0}+\frac{M_{0}}{4}<5M_{0},$

and

$\|\rho(\mathcal{T}_{1}+T)-\rho_{M}\|_{L^{2}}^{2}<M_{0}.$

Now, we may iterate this argument starting from time $\mathcal{T}_{1}+T$.
Since $T\geq\frac{T_{*}}{4}>0$, we may repeat the above argument finitely many
times to conclude the improved bound

$\sup_{0\leq t\leq\mathcal{T}_{0}}\|\rho(t)-\rho_{M}\|_{L^{2}}^{2}\leq
5M_{0}.$

This concludes the proof of Proposition 2.4.

######  Remark 5.4.

We summarize the choice of parameters $g_{0}$ and $B_{0}$ as follows. We first
choose $g_{0}=4\mathfrak{g}_{0}$, where $\mathfrak{g}_{0}$ is selected in
Remark 4.2. We then choose $B_{0}>1$ such that (5.14), (5.15), (5.16), (5.17),
(5.18), and (5.21) hold.

##  6 Analysis of the Navier-Stokes Problem

In this section, we follow the recipe of the previous section to further
investigate the Keller-Segel-Navier-Stokes system i.e. (1.3) with $\delta=1$.
That is, we study

$\partial_{t}\rho+u\cdot\nabla\rho-\Delta\rho+\operatorname{div}(\rho\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))=0,\quad\text{in
}\Omega$ (6.1a) $\partial_{t}u+u\cdot\nabla u-B\Delta u+\nabla p=Bg\rho
e_{2},\quad\operatorname{div}(u)=0,\quad\text{in }\Omega$ (6.1b)
$\partial_{2}\rho\big{|}_{\partial\Omega}=0,\;u_{2}|_{\partial\Omega}=0,\;\partial_{2}u_{1}|_{\partial\Omega}=0,$
(6.1c) $\rho(0,x)=\rho_{0}(x)\geq 0,\;u(0,x)=u_{0}(x).$ (6.1d)

Indeed, the extra advection term $u\cdot\nabla u$ appearing in (6.1b) seems to
be a potentially dangerous term. As we will see later, this term will be of
lower order in the regime of $B$ sufficiently large. We thus can still close
the bootstrap argument. As in Sect. 5, we assume the bootstrap assumption (A),
and we aim to prove Proposition 2.4 in the case of $\delta=1$.

The section is organized as follows. In Section 6.1, we summarize several
technical controls that echo with those in Section 5.1. We also include
additional controls which treat the advection term $u\dot{\nabla}u$. In
Section 6.2, we close the bootstrapping using an analysis of the difference
equation that is similar to that in Section 5.2. Throughout this section, we
assume $g\geq g_{1}>1$, $B\geq B_{1}>1$, where $g_{1},B_{1}$ are sufficiently
large. The exact values of $g_{1},B_{1}$ can be seen in Remark 6.1.

###  6.1 Technical Controls

Same as our program of treating (5.1), we need to control terms involving
$u,\partial_{t}\rho$, and $\partial_{t}u$ given the bootstrap assumption (A).
We first remark that Lemma 5.1, 5.2, and 5.3 still hold in the scenario of
(6.1). For the sake of clarity, we reiterate the statements of those lemmas
here.

######  Lemma 6.1.

For arbitrary $t_{0}\in[0,\mathcal{T}_{0}]$ and given bootstrap assumption
(A), we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\nabla\rho\|_{L^{2}}^{2}dt\leq 18M_{0}.$ (6.2)

######  Lemma 6.2.

Given bootstrap assumption (A), the following estimate holds:

$\sup_{0\leq t\leq\mathcal{T}_{0}+T_{*}/9}\|u(t)\|_{L^{2}}^{2}\leq
C(\rho_{0},u_{0})g^{2}M_{0}.$ (6.3)

######  Lemma 6.3.

Assuming the bootstrap assumption (A). For any $t_{0}\in[0,\mathcal{T}_{0}]$,
we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}M_{0}^{2}$ (6.4)

Instead of giving detailed proofs to the statements above, We briefly point
out the differences of the proofs from their counterparts in Sect. 5. Lemma
6.1 follows directly from Corollary 4.1. To prove Lemma 6.2, we perform the
standard $L^{2}$ estimate on (6.1b). The only extra term appearing in the
estimate is

$\int_{\Omega}u(u\cdot\nabla u)dx,$

which vanishes due to incompressibility and no-flux boundary condition
satisfied by $u$. Finally, the derivation of Lemma 6.3 only utilizes the
divergence-free property of fluid flow $u$, and thus trivially applies to the
Navier-Stokes case.

####  6.1.1 Control of the $\partial_{t}u$ Near Time 0

In this subsection, we establish a counterpart of Lemma 5.4 that concerns the
size of $\|\partial_{t}u\|_{L^{2}}$ near time zero. To achieve this goal, we
first need to establish the following bound for the advection term
$u\cdot\nabla u$ near time zero:

######  Lemma 6.4.

Consider time
$s_{0}=c_{0}\min(\|\rho_{0}-\rho_{M}\|_{L^{2}}^{-2},\rho_{M}^{-1})$, where
$c_{0}$ is as in Corollary 4.1. There exists a constant $C=C(\rho_{0},u_{0})$
such that for $B\geq C(\rho_{0},u_{0})gM_{0}^{1/2}$, the following estimate
holds:

$\int_{0}^{s_{0}}\|u\cdot\nabla u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{4}M_{0}.$ (6.5)

###### Proof.

First, we obtain a bound for advection term $u\cdot\nabla u$: using Hölder
inequality and Ladyzhenskaya’s inequality in 2D, we observe that

$\displaystyle\|u\cdot\nabla u\|_{L^{2}}$
$\displaystyle\leq\|u\|_{L^{4}}\|\nabla
u\|_{L^{4}}\lesssim\|u\|_{L^{2}}^{1/2}\|u\|_{1}\|\nabla u\|_{1}^{1/2}$
$\displaystyle\leq\|u\|_{L^{2}}^{1/2}(\|u\|_{L^{2}}+\|\nabla
u\|_{L^{2}})(\|\nabla u\|_{L^{2}}^{1/2}+\|\nabla^{2}u\|_{L^{2}}^{1/2})$
$\displaystyle=\|u\|_{L^{2}}^{3/2}\|\nabla
u\|_{L^{2}}^{1/2}+\|u\|_{L^{2}}^{3/2}\|\nabla^{2}u\|_{L^{2}}^{1/2}+\|u\|_{L^{2}}^{1/2}\|\nabla
u\|_{L^{2}}^{3/2}+\|u\|_{L^{2}}^{1/2}\|\nabla
u\|_{L^{2}}\|\nabla^{2}u\|_{L^{2}}^{1/2}$

We show how to estimate the most singular term $\|u\|_{L^{2}}^{1/2}\|\nabla
u\|_{L^{2}}\|\nabla^{2}u\|_{L^{2}}^{1/2}$, and the rest of the terms follow
from a similar argument. Using the interpolation inequality $\|\nabla
u\|_{L^{2}}^{2}\leq\|u\|_{L^{2}}\|\Delta u\|_{L^{2}}$ for $u\in W$, and the
calculus lemma A.1, we have

$\|u\|_{L^{2}}^{1/2}\|\nabla
u\|_{L^{2}}\|\nabla^{2}u\|_{L^{2}}^{1/2}\leq\|u\|_{L^{2}}\|\Delta
u\|_{L^{2}}.$ (6.6)

Integrating in time from $0$ to $s_{0}$ and using (6.3), we have

$\int_{0}^{s_{0}}\|u\|_{L^{2}}\|\Delta u\|_{L^{2}}dt\leq\sup_{0\leq t\leq
s_{0}}\|u(t)\|_{L^{2}}^{2}\int_{0}^{s_{0}}\|\Delta u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}M_{0}\int_{0}^{s_{0}}\|\Delta u(t)\|_{L^{2}}^{2}dt.$

Proceeding similarly with other terms, we can conclude the following bound:

$\int_{0}^{s_{0}}\|u\cdot\nabla u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}M_{0}\int_{0}^{s_{0}}\|\Delta u(t)\|_{L^{2}}^{2}dt.$
(6.7)

Thus to close the estimate, we need to obtain a control of $\Delta u$ near
time zero. Testing (6.1b) by $-\Delta u$, we obtain that for any
$t\in(0,s_{0})$:

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}+B\|\Delta
u\|_{L^{2}}^{2}$ $\displaystyle=\int_{\Omega}(\Delta u)(u\cdot\nabla
u)dx+Bg\int_{\Omega}\rho\Delta u_{2}dx$ $\displaystyle\leq\|\Delta
u\|_{L^{2}}\|u\cdot\nabla u\|_{L^{2}}+\frac{B}{4}\|\Delta
u\|_{L^{2}}^{2}+Bg^{2}\|\rho\|_{L^{2}}^{2}$ $\displaystyle\leq
C\|u\|_{L^{2}}\|\Delta u\|_{L^{2}}^{2}+\frac{B}{4}\|\Delta
u\|_{L^{2}}^{2}+Bg^{2}\|\rho\|_{L^{2}}^{2}$ $\displaystyle\leq
C(\rho_{0},u_{0})gM_{0}^{1/2}\|\Delta u\|_{L^{2}}^{2}+\frac{B}{4}\|\Delta
u\|_{L^{2}}^{2}+Bg^{2}\|\rho\|_{L^{2}}^{2},$ (6.8)

where we used (6.6) in the second inequality and (6.3) in the last inequality
above. Choosing

$B\geq 4C(\rho_{0},u_{0})gM_{0}^{1/2}$ (6.9)

and rearraging the estimate above, we have

$\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}+B\|\Delta u\|_{L^{2}}^{2}\leq
Bg^{2}\|\rho\|_{L^{2}}^{2}\leq
C(\rho_{0})Bg^{2}\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2},$

where we used the definition of $s_{0}$ in the last inequality. Integrating
from $0$ to $s_{0}$:

$\displaystyle B\int_{0}^{s_{0}}\|\Delta u(t)\|_{L^{2}}^{2}dt$
$\displaystyle\leq\|\nabla
u_{0}\|_{L^{2}}^{2}+C(\rho_{0})Bg^{2}\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2}s_{0}\leq
C(\rho_{0},u_{0})Bg^{2}.$

Thus, we have

$\int_{0}^{s_{0}}\|\Delta u(t)\|_{L^{2}}^{2}dt\leq C(\rho_{0},u_{0})g^{2}.$

Combining with (6.7), we obtain the desired result. ∎

Thanks to Lemma 6.4, we are ready to prove the following result that parallels
Lemma 5.4:

######  Lemma 6.5.

Consider time $s_{0}$ defined in Lemma 6.4. There exists a constant
$C=C(\rho_{0},u_{0})\geq 1$ such that for $B\geq C(\rho_{0},u_{0})g^{2}M_{0}$,
the following estimate holds:

$\int_{0}^{s_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})B^{2}g^{2}.$ (6.10)

In particular, there exists $\tau_{0}\in(0,s_{0})$ such that

$\|\partial_{t}u(\tau_{0})\|_{L^{2}}^{2}\leq\frac{C(\rho_{0},u_{0})B^{2}g^{2}}{s_{0}}=:\mathcal{C}_{2}(\rho_{0},u_{0})B^{2}g^{2}.$
(6.11)

###### Proof.

Similar to the proof of Lemma 5.4, it suffices to prove (6.10). Testing (6.1b)
by $\partial_{t}u$, we obtain

$\displaystyle\|\partial_{t}u\|_{L^{2}}^{2}+\frac{B}{2}\frac{d}{dt}\|\nabla
u\|_{L^{2}}^{2}$
$\displaystyle=Bg\int_{\Omega}\rho\partial_{t}u_{2}-\int_{\Omega}\partial_{t}u(u\cdot\nabla
u)dx$
$\displaystyle\leq\frac{1}{2}\|\partial_{t}u\|_{L^{2}}^{2}+\frac{3}{4}\left(B^{2}g^{2}\|\rho\|_{L^{2}}^{2}+\|u\cdot\nabla
u\|_{L^{2}}^{2}\right).$

Rearranging and integrating in time from $0$ to $s_{0}$, we have

$B\|\nabla
u(s_{0})\|_{L^{2}}^{2}+\int_{0}^{s_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
B\|\nabla
u_{0}\|_{L^{2}}^{2}+\frac{3}{2}\int_{0}^{s_{0}}\left(B^{2}g^{2}\|\rho(t)\|_{L^{2}}^{2}+\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}\right)dt.$

Choosing $B$ so large that it satisfies the assumption of Lemma 6.4, we invoke
(6.5) and the definition of $s_{0}$ to conclude that:

$\displaystyle\int_{0}^{s_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt$
$\displaystyle\leq B\|\nabla
u_{0}\|_{L^{2}}^{2}+C(\rho_{0},u_{0})(B^{2}g^{2}\|\rho_{0}-\rho_{M}\|_{L^{2}}^{2}\cdot
s_{0}+g^{4}M_{0})$ $\displaystyle\leq
C(\rho_{0},u_{0})(B+B^{2}g^{2}+g^{4}M_{0})\leq C(\rho_{0},u_{0})B^{2}g^{2},$

where the last inequality follows from the assumption $B^{2}\geq g^{2}M_{0}$.
Hence, we have showed (6.10). ∎

####  6.1.2 Control of $\partial_{t}u$

We now prove a lemma analogous to Lemma 5.5, which motivates us to study the
time-differentiated equation for (6.1b). Similar to the approach deployed in
Section 5, we apply $\partial_{t}$ to equation (6.1b) and obtain:

$\partial_{t}u+\partial_{t}u\cdot\nabla
u+u\cdot\nabla\partial_{t}u-B\Delta\partial_{t}u+\nabla\partial_{t}p=Bg(\partial_{t}\rho)e_{2}.$

Testing by $\partial_{t}u$, using $\operatorname{div}\partial_{t}u=0$ and the
fact that $\partial_{t}u$ still satisfies the Lions boundary condition, we
obtain that:

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\partial_{t}u\|_{L^{2}}^{2}+B\|\nabla\partial_{t}u\|_{L^{2}}^{2}$
$\displaystyle=-\int_{\Omega}(\partial_{t}u_{j})(\partial_{j}u_{i})(\partial_{t}u_{i})dx+Bg\int_{\Omega}\partial_{t}\rho\partial_{t}u_{2}dx$

We estimate the second term on the RHS above in a similar fashion to Lemma
5.5, namely:

$Bg\int_{\Omega}\partial_{t}\rho\partial_{t}u_{2}dx\leq\frac{B}{4}\|\nabla\partial_{t}u\|_{L^{2}}^{2}+CBg^{2}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}$

To estimate the first term, an application of Ladyzhenskaya’s inequality in 2D
and Cauchy-Schwartz inequality gives:

$\displaystyle\int_{\Omega}(\partial_{t}u_{j})(\partial_{j}u_{i})(\partial_{t}u_{i})dx$
$\displaystyle\leq\|\partial_{t}u\|_{L^{4}}^{2}\|\nabla u\|_{L^{2}}\leq
C\|\partial_{t}u\|_{L^{2}}\|\nabla\partial_{t}u\|_{L^{2}}\|\nabla u\|_{L^{2}}$
$\displaystyle\leq\frac{B}{4}\|\nabla\partial_{t}u\|_{L^{2}}^{2}+\frac{C}{B}\|\partial_{t}u\|_{L^{2}}^{2}\|\nabla
u\|_{L^{2}}^{2}.$

Collecting the estimates above and invoking Poincaré inequality, we obtain the
following differential inequality:

$\frac{d}{dt}\|\partial_{t}u\|_{L^{2}}^{2}\leq\left(\frac{C}{B}\|\nabla
u\|_{L^{2}}^{2}-\frac{B}{C_{p}}\right)\|\partial_{t}u\|_{L^{2}}^{2}+CBg^{2}\|\partial_{t}\rho\|_{H^{-1}_{0}}^{2}$
(6.12)

By observing (6.12) and recalling our proof of Lemma 5.5, we need to control
the size of $\|\nabla u\|_{L^{2}}^{2}$ on a time interval of length $T_{*}/9$
so that $\left(\frac{C}{B}\|\nabla
u\|_{L^{2}}^{2}-\frac{B}{C_{p}}\right)\|\partial_{t}u\|_{L^{2}}^{2}$ is still
a damping term. The following lemma precisely addresses this issue.

######  Lemma 6.6.

Assume the bootstrap assumption (A). Then for any
$t_{0}\in[0,\mathcal{T}_{0}]$, $B\geq M_{0}$, we have the following bound:

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\nabla u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}$ (6.13)

###### Proof.

We test (6.1b) by $u$. Using incompressibility of $u$ and integration by
parts, we have for $t\in[0,\mathcal{T}_{0}+T_{*}/9]$:

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u\|_{L^{2}}^{2}+B\|\nabla
u\|_{L^{2}}^{2}$ $\displaystyle\leq gB\|\rho\|_{L^{2}}\|u\|_{L^{2}}\leq
C(\rho_{0},u_{0})gB(18M_{0}+2\pi^{2}\rho_{M}^{2})^{1/2}gM_{0}^{1/2}$
$\displaystyle\leq C(\rho_{0},u_{0})Bg^{2}M_{0},$

where we invoked the bootstrap assumption and (6.3) in the second inequality.
Fixing $t_{0}\in[0,\mathcal{T}_{0}]$, and integrating the above from $t_{0}$
to $t_{0}+T_{*}/9$, we obtain that

$\displaystyle B\int_{t_{0}}^{t_{0}+T_{*}/9}\|\nabla u(t)\|_{L^{2}}^{2}dt$
$\displaystyle\leq\sup_{0\leq
t\leq\mathcal{T}_{0}}\|u(t)\|_{L^{2}}^{2}+C(\rho_{0})Bg^{2}M_{0}\cdot
c_{0}M_{0}^{-1}$ $\displaystyle\leq C(\rho_{0},u_{0})(g^{2}M_{0}+Bg^{2})$
$\displaystyle\leq C(\rho_{0},u_{0})Bg^{2}.$

where used the choice

$B\geq M_{0}$ (6.14)

in the final inequality. Now, the desired estimate follows from dividing on
both sides by $B$. ∎

We are therefore ready to prove the following statement analogous to Lemma
5.5:

######  Lemma 6.7.

Assume the bootstrap assumption (A) and $B\geq C(\rho_{0},u_{0})gM_{0}^{1/4}$.
Then there exists $\tau_{0}=\tau_{0}(\rho_{0})\in(0,s_{0})$ such that we have

$\sup_{\tau_{0}\leq t\leq\mathcal{T}_{0}}\|\partial_{t}u(t)\|_{L^{2}}^{2}\leq
2\mathcal{C}_{2}(\rho_{0},u_{0})g^{2}B^{2}.$ (6.15)

Moreover, for any $t_{0}\in[\tau_{0},\mathcal{T}_{0}]$, we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{2}B.$ (6.16)

###### Proof.

Following the same approach as that in Lemma 5.5, we write (6.12) in Duhamel
form. Fixing any $t\in[\tau_{0},\tau_{0}+T_{*}/9]$, we have

$\displaystyle\|\partial_{t}u(t)\|_{L^{2}}^{2}$
$\displaystyle\leq\mathcal{C}_{2}(\rho_{0},u_{0})g^{2}B^{2}\exp\left(-\frac{B(t-\tau_{0})}{C_{p}}+\frac{C}{B}\int_{\tau_{0}}^{t}\|\nabla
u(s)\|_{L^{2}}^{2}ds\right)$
$\displaystyle+CBg^{2}\int_{\tau_{0}}^{t}\exp\left(-\frac{B(t-s)}{C_{p}}+\frac{C}{B}\int_{s}^{t}\|\nabla
u(\tau)\|_{L^{2}}^{2}d\tau\right)\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}ds$
(6.17)

Using (6.13), we choose $B$ sufficiently large such that

$\exp\left(\frac{C}{B}\int_{t_{0}}^{t}\|\nabla
u(s)\|_{L^{2}}^{2}ds\right)\leq\exp\left(C(\rho_{0},u_{0})gM_{0}^{1/4}B^{-1}\right)<\ln
2,$ (6.18)

from which we can reduce (6.1.2) to:

$\displaystyle\|\partial_{t}u(t)\|_{L^{2}}^{2}$ $\displaystyle\leq
2\mathcal{C}_{2}(\rho_{0},u_{0})g^{2}B^{2}e^{-\frac{B(t-t_{0})}{C_{p}}}+CBg^{2}\int_{t_{0}}^{t}e^{-\frac{B(t-s)}{C_{p}}}\|\partial_{t}\rho(s)\|_{H^{-1}_{0}}^{2}ds.$
(6.19)

The rest of the proof follows verbatim to that of Lemma 5.5, and we omit the
details here. ∎

####  6.1.3 Control of the Advection Term

In this subsection, we estimate the additional advection term $u\cdot\nabla u$
on time interval of length $T_{*}/9$ in the bootstrap regime.

######  Lemma 6.8.

Assuming the bootstrap assumption (A) and fixing
$t_{0}\in[0,\mathcal{T}_{0}]$, we have

$\int_{t_{0}}^{t_{0}+T_{*}/9}\|u\cdot\nabla u(t)\|_{L^{2}}^{2}dt\leq
C(\rho_{0},u_{0})g^{4}M_{0}$ (6.20)

###### Proof.

By the equation (6.1b), we have

$\displaystyle B^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\|\Delta u\|_{L^{2}}^{2}dt$
$\displaystyle\lesssim\int_{t_{0}}^{t_{0}+T_{*}/9}\|\partial_{t}u(t)\|_{L^{2}}^{2}+\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}+B^{2}g^{2}\|\rho(t)\|_{L^{2}}^{2}dt$ $\displaystyle\leq
C(\rho_{0},u_{0})(Bg^{2}+B^{2}g^{2})+C\int_{t_{0}}^{t_{0}+T_{*}/9}\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}dt,$

where we used the estimate (6.16) and the bootstrap assumption in the second
inequality. Combining the inequality above with (6.6), we obtain that

$\displaystyle\int_{t_{0}}^{t_{0}+\frac{T_{*}}{9}}\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}dt$ $\displaystyle\lesssim\sup_{0\leq
t\leq\mathcal{T}_{0}+T_{*}/9}\|u(t)\|_{L^{2}}^{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\|\Delta
u(t)\|_{L^{2}}^{2}dt$
$\displaystyle\leq\frac{C(\rho_{0},u_{0})g^{2}M_{0}}{B^{2}}\left(C(\rho_{0},u_{0})(Bg^{2}+B^{2}g^{2})+C\int_{t_{0}}^{t_{0}+T_{*}/9}\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}dt\right)$ $\displaystyle\leq
C(\rho_{0},u_{0})g^{4}M_{0}+\frac{1}{2}\int_{t_{0}}^{t_{0}+T_{*}/9}\|u\cdot\nabla
u(t)\|_{L^{2}}^{2}dt,$

after we chose

$B\geq\max(g^{4}M_{0},C(\rho_{0},u_{0})g^{2}M_{0}).$ (6.21)

Rearranging, we conclude (6.20). ∎

###  6.2 Closing the Bootstrap

We follow the plan of the Stokes case. Consider $\mathcal{T}_{1}\geq 0$ to be
the first time when $\|\rho-\rho_{M}\|_{L^{2}}^{2}$ achieves $M_{0}$. Let
$\rho_{s},u_{s}$ be the global solution to (2.7) with initial time
$\mathcal{T}_{1}$ and initial condition
$\rho_{s}(\mathcal{T}_{1},x)=\rho(\mathcal{T}_{1},x)$. Defining the difference
variables $r=\rho-\rho_{s}$, $v=u-u_{s}$, we note that $r,v$ satisfy the
following system (which is locally well-posed on
$[\mathcal{T}_{1},\mathcal{T}_{1}+T_{*}]$):

$\partial_{t}r-\Delta r+v\cdot\nabla\rho+u_{s}\cdot\nabla
r+\operatorname{div}(r\nabla(-\Delta_{N})^{-1}(\rho-\rho_{M}))+\operatorname{div}(\rho_{s}\nabla(-\Delta_{N})^{-1}r)=0,$
(6.22a) $-\Delta
v+\nabla(B^{-1}p-p_{s})=-\frac{1}{B}\partial_{t}u-\frac{1}{B}(u\cdot\nabla
u)+gre_{2},$ (6.22b)
$r(\mathcal{T}_{1},x)=0,\;v(\mathcal{T}_{1},x)=u(\mathcal{T}_{1},x)-u_{s}(\mathcal{T}_{1},x),$
(6.22c)
$\partial_{2}r|_{\partial\Omega}=0,\;v_{2}|_{\partial\Omega}=0,\;\partial_{2}v_{1}|_{\partial\Omega}=0.$
(6.22d)

Testing (6.22a) by $r$, we note that the only difference between the Navier-
Stokes case and the Stokes case is the estimate of $\|v\|_{2}^{2}$. We first
verify that the RHS of (6.22b) belongs to class $H$. Given $u\in W$, it
suffices for us to check $u\cdot\nabla u\in H$. Indeed:

$\int_{\Omega}u_{j}\partial_{j}u_{1}dx=-\int_{\Omega}\operatorname{div}(u)u_{1}dx+\int_{\partial\Omega}u_{2}u_{1}dS(x)=0,$

where we used that $u$ is incompressible and $u_{2}|_{\partial\Omega}=0$.
Thus, we are eligible to use (2.3):

$\|v\|_{2}^{2}\lesssim\frac{1}{B^{2}}\|\partial_{t}u\|_{L^{2}}^{2}+\frac{1}{B^{2}}\|u\cdot\nabla
u\|_{L^{2}}^{2}+g^{2}\|r\|_{L^{2}}^{2}.$

Using this estimate and by an identical computation to that in the Stokes
case, we derive the following energy inequality for $r$:

$\displaystyle\frac{d}{dt}\|r\|_{L^{2}}^{2}+\|\nabla r\|_{L^{2}}^{2}$
$\displaystyle\leq
F(t)\|r\|_{L^{2}}^{2}+\frac{1}{B^{2}}(\|\rho-\rho_{M}\|_{L^{2}}^{2}+2\pi^{2}\rho_{M}^{2})(\|\partial_{t}u\|_{L^{2}}^{2}+\|u\cdot\nabla
u\|_{L^{2}}^{2}),$

where
$F(t)=C(g^{2}\|\rho\|_{L^{2}}^{2}+\|\rho-\rho_{M}\|_{L^{\infty}}+\|\rho_{s}\|_{L^{4}}^{2})$.
Moreover, $F(t)$ satisfies the following estimate:

$\int_{\mathcal{T}_{1}}^{\mathcal{T}_{1}+T_{*}}F(s)ds\leq
C(\rho_{0})(g^{2}+1).$

Using Grönwall inequality, for $t\in[\mathcal{T}_{1},\mathcal{T}_{1}+T_{*}]$
we have

$\displaystyle\|r(t)\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{1}{B^{2}}\int_{\mathcal{T}_{1}}^{t}(\|\rho(s)-\rho_{M}\|_{L^{2}}^{2}+2\pi^{2}\rho_{M}^{2})(\|\partial_{t}u(s)\|_{L^{2}}^{2}+\|u\cdot\nabla
u(s)\|_{L^{2}}^{2})ds\left[\exp\left(\int_{\mathcal{T}_{1}}^{t}F(s)ds\right)\right]$
$\displaystyle\leq\frac{C(\rho_{0},u_{0})(M_{0}+1)}{B^{2}}(g^{2}B+g^{4}M_{0})\cdot
e^{g^{2}+1}\leq\frac{M_{0}}{4}$ (6.23)

if we choose $B$ sufficiently large. Note that we used Lemma 6.7 and 6.8 in
the second inequality. The rest of the proof follows from an identical
argument to that in the Stokes case, and we omit it here.

######  Remark 6.1.

We summarize the choice of parameters $g_{1}$ and $B_{1}$ as follows. We first
choose $g_{1}=4\mathfrak{g}_{0}$, where $\mathfrak{g}_{0}$ is selected in
Remark 4.2. We then choose $B_{1}>1$ such that (6.9), (6.14), (6.18), (6.21),
and (6.23) hold.

##  Appendix A Appendix

In this appendix, we first rigorously prove Proposition 2.1 and Proposition
2.2. For the convenience of the readers, we reiterate the statements of the
propositions as follows:

######  Proposition A.1.

Let $u\in V$. Then the following estimate holds:

$\|u\|_{L^{2}}\leq C_{p}\|\nabla u\|_{L^{2}},$ (A.1)

where $C_{p}$ denotes the Poincaré constant that only depends on the domain
$\Omega$.

###### Proof.

Since $\|u\|_{L^{2}}^{2}=\|u_{1}\|_{L^{2}}^{2}+\|u_{2}\|_{L^{2}}^{2}$, we will
estimate each velocity component separately. Since $u\in V$, then in
particular $u_{1}$ has zero spatial mean and $u_{2}\in H^{1}_{0}(\Omega)$.
Thus, (A.1) follows from applying Poincaré inequality to $u_{1}$ and $u_{2}$
separately. ∎

######  Proposition A.2.

Assuming $f\in H$, then (2.1) admits a unique solution $u\in W$ with estimate

$\|u\|_{2}\leq C(\Omega)\|f\|_{L^{2}}.$ (A.2)

In fact, $u$ is given by the following explicit formula:

$u=\nabla^{\perp}(-\Delta_{D})^{-2}(\partial_{2}f_{1}-\partial_{1}f_{2}).$
(A.3)

More generally, if $f\in H^{s}\cap H$, $s\geq 1$, we have the following
improved regularity estimate:

$\|u\|_{s+2}\leq C(\Omega)\|f\|_{s}.$ (A.4)

###### Proof.

To show well-posedness of problem (2.1), it is convenient to show on the level
of stream function. Since we would find $u$ such that $\int_{\Omega}u_{1}dx=0$
and $u_{2}|_{\partial\Omega}=0$, there exists a unique stream function $\psi$
such that $u=\nabla^{\perp}\psi$ by Hodge decomposition. Moreover, $\psi$
satisfies the following:

$\Delta\psi=\omega,\quad\psi|_{\partial\Omega}=0.$

Taking $\nabla^{\perp}\cdot$ on (2.1) and denoting $\omega=\nabla^{\perp}\cdot
u$, we must have

$-\Delta\omega=\partial_{1}f_{2}-\partial_{2}f_{1},\quad\omega|_{\partial\Omega}=0.$

Therefore, the discussion above motivates us to study the following auxiliary
coupled system:

$\begin{cases}\Delta\psi=\omega,\quad\psi|_{\partial\Omega}=0,\\\
-\Delta\omega=\partial_{1}f_{2}-\partial_{2}f_{1},\quad\omega|_{\partial\Omega}=0.\end{cases}$
(A.5)

First, note that since $f\in L^{2}$, we have
$\partial_{1}f_{2}-\partial_{2}f_{1}\in H^{-1}_{0}$ in the sense of
distribution. Then by standard elliptic theory, there exists a unique
$\omega\in H^{1}_{0}$ such that the $\omega$ equation holds, with estimate
$\|\omega\|_{1}\leq C(\Omega)\|f\|_{L^{2}}$ . Applying standard elliptic
theory again to $\psi$ equation, we have a unique solution $\psi\in H^{3}\cap
H^{1}_{0}$, with estimate $\|\psi\|_{3}\leq C(\Omega)\|\omega\|_{1}$. Define
$u=\nabla^{\perp}\psi$. Then $u$ strongly solves (2.1) with boundary
conditions $u_{2}|_{\partial\Omega}=\partial_{2}u_{1}|_{\partial\Omega}=0$
satisfied in the trace sense. Moreover, we have the estimate

$\|u\|_{2}\leq\|\psi\|_{3}\leq C(\Omega)\|f\|_{L^{2}}.$

Moreover from the discussion above, we obtain formula (A.3). Finally, to show
the higher regularity estimates (A.4), we note that given $f\in H^{s}\cap H$,
$s\geq 1$, $\nabla f\in H^{s-1}$. Using the explicit formula (A.3) and
standard elliptic estimate, we conclude (A.4). ∎

Finally, we present the following calculus lemma that would be useful in
Sect.6.

######  Lemma A.1.

$\|\nabla^{2}u\|_{L^{2}}=\|\Delta u\|_{L^{2}}$ for $u\in W\cap
C^{\infty}(\Omega)$.

###### Proof.

Integrating by parts twice, we obtain that

$\displaystyle\|\Delta u\|_{L^{2}}^{2}$
$\displaystyle=\int_{\Omega}\partial_{i}\partial_{i}u_{j}\partial_{k}\partial_{k}u_{j}dx$
$\displaystyle=\int_{\Omega}\partial_{i}\partial_{k}u_{j}\partial_{i}\partial_{k}u_{j}dx+\int_{\partial\Omega}\Delta
u_{j}\partial_{2}u_{j}dx_{1}-\int_{\partial\Omega}\partial_{k}\partial_{2}u_{j}\partial_{k}u_{j}dx_{1}$
$\displaystyle=:\|\nabla^{2}u\|_{L^{2}}^{2}+BT_{1}+BT_{2}.$

We moreover notice that

$\displaystyle BT_{1}$ $\displaystyle=\int_{\partial\Omega}\Delta
u_{1}\partial_{2}u_{1}dx_{1}+\Delta u_{2}\partial_{2}u_{2}dx_{1}=0,$

Now we observe that for any $x\in\partial\Omega$, since u is smooth and
belongs to the class $W$, we have

$\Delta
u_{2}=\partial_{1}^{2}u_{2}+\partial_{2}^{2}u_{2}=\partial_{2}^{2}u_{2}=-\partial_{1}\partial_{2}u_{1}=0,$

where we used $u_{2}|_{\partial\Omega}=0$ in the second equality,
$\operatorname{div}(u)=0$ in the third equality, and
$\partial_{2}u_{1}|_{\partial\Omega}=0$ in the last equality. Using this fact
and $\partial_{2}u_{1}=0$ for $x\in\partial\Omega$, we expand the sum and
obtain:

$\displaystyle BT_{2}$
$\displaystyle=\int_{\partial\Omega}\partial_{1}\partial_{2}u_{1}\partial_{1}u_{1}dx_{1}+\int_{\partial\Omega}\partial_{1}\partial_{2}u_{2}\partial_{1}u_{2}dx_{1}$
$\displaystyle+\int_{\partial\Omega}\partial_{2}\partial_{2}u_{1}\partial_{2}u_{1}dx_{1}+\int_{\partial\Omega}\partial_{2}\partial_{2}u_{2}\partial_{2}u_{2}dx_{1}$
$\displaystyle=\int_{\partial\Omega}\partial_{2}\partial_{2}u_{2}\partial_{2}u_{2}dx_{1},$

where we used the boundary conditions of $u$ in the last equality. Since
$\partial_{2}^{2}u_{2}=-\partial_{1}\partial_{2}u_{1}=0$ for
$x\in\partial\Omega$, we finally conclude that $BT_{2}=0$, and thus
$\|\nabla^{2}u\|_{L^{2}}=\|\Delta u\|_{L^{2}}$. ∎

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