# Stability analysis of the incompressible porous media equation and the
Stokes transport system via energy structure

Jaemin Park  Department Mathematik Und Informatik, Universität Basel, CH-4051
Basel, Switzerland<EMAIL_ADDRESS>
###### Abstract.

In this paper, we revisit asymptotic stability for the two-dimensional
incompressible porous media equation and the Stokes transport system in a
periodic channel. It is well-known that a stratified density, which strictly
decreases in the vertical direction, is asymptotically stable under
sufficiently small and smooth perturbations. We provide improvements in the
regularity assumptions on the perturbation and in the convergence rate. Unlike
the standard approach for stability analysis relying on linearized equations,
we directly address the nonlinear problem by exploiting the energy structure
of each system. While it is widely known that the potential energy is a
Lyapunov functional in both systems, our key observation is that the second
derivative of the potential energy reveals a (degenerate) coercive structure,
which arises from the fact that the solution converges to the minimizer of the
energy.

###### Key words and phrases:

Asymptotic stability - incompressible fluds - porous media equation - Stokes
transport system

###### 2020 Mathematics Subject Classification:

76S05 - 35Q35 -34D05 - 76B03

Acknowledgements. The author acknowledges the support of the SNSF Ambizione
grant No. 216083. The author also extends gratitude to Luis Martínez–Zoroa and
Yao Yao for fruitful conversations during the course of this research.

###### Contents

  1. 1 Introduction
  2. 2 Preparation: Time-average decay and vertical rearrangement
  3. 3 Stability in the IPM equation
  4. 4 Stability in the Stokes transport system
  5. A Proof of Lemma 3.1

##  1\. Introduction

In this paper, we investigate asymptotic stability in the incompressible
porous media equation (IPM) and the Stokes transport system in a periodic
channel $\Omega:=\mathbb{T}^{2}\times(0,1)$. To introduce the models, let us
consider a continuity equation with a velocity field $u(t,x)$,

$\rho_{t}+\nabla\cdot(u\rho)=0,\text{ in $\Omega$ and
$\rho(0,x)=\rho_{0}(x)$,}$

for some nonnegative scalar-valued function $\rho_{0}$, which will be referred
to as density throughout the paper. Neglecting physical constants, the
velocity $u$ in each model is determined by $\rho$ via

$\begin{cases}u=-\nabla p-\begin{pmatrix}0\\\ \rho\end{pmatrix},\text{ with
}\nabla\cdot u=0\text{ in $\Omega$ and }u\cdot\vec{n}=0\text{ on
$\partial\Omega$,}&\text{(IPM) }\\\ \Delta u=-\nabla p-\begin{pmatrix}0\\\
\rho\end{pmatrix},\text{ with }\nabla\cdot u\text{ in $\Omega$ and }u=0\text{
on $\partial\Omega$},&\text{(Stokes) }\end{cases}$

where $\vec{n}$ denotes the unit normal vector on $\partial\Omega$. We note
that, given $\rho$, the above equations (referred to as Darcy’s law and the
steady Stokes equation for the IPM and the Stokes transport system,
respectively) uniquely determine the velocity $u$ and the pressure $p$ (up to
a constant) (See e.g. [constantin1988navier, Chapter 1] for the IPM and
[galdi2011introduction, Chapter IV] for the Stokes).

Both the IPM equation and the Stokes transport system describe the evolution
of transported density driven by an incompressible fluid subject to gravity.
Depending on the physical context, these equations can be interpreted in
various ways. However, as our primary focus is on their mathematical analysis,
interested readers are directed to [ingham1998transport,
hofer2018sedimentation, mecherbet2018sedimentation] for a more detailed
exploration of the physical motivations.

In both equations, it is well-known that any stratified density,
$\rho_{s}(x)=\rho_{s}(x_{1},x_{2})=\rho_{s}(x_{2})$, which is independent of
the horizontal variable $x_{1}\in\mathbb{T}$, is a steady state. Indeed, the
vector field $(0,\rho_{s}(x_{2}))^{T}$ can be easily represented as a gradient
field. Therefore the velocity determined by either Darcy’s law or the steady
Stokes equation corresponds to a trivial vector field $u\equiv 0$. In the
paper, our stability analysis will be focused on the stratified densities
satisfying the following additional condition:

$\displaystyle\partial_{2}\rho_{s}(x_{2})<0,\text{ for all $x\in\Omega$}.$
(1.1)

In the remainder of this section, we will provide a brief overview of relevant
background information and present our main results separately for each
equation.

###  1.1. The IPM equation

We recall that the IPM equation: For $(t,x)\in\mathbb{R}^{+}\times\Omega$

$\displaystyle\text{(IPM)}=\begin{cases}\rho_{t}+u\cdot\nabla\rho=0,\\\
u=-\nabla p-\begin{pmatrix}0\\\ \rho\end{pmatrix},\quad\nabla\cdot
u=0,\end{cases}$ (1.2)

with boundary condition $u\cdot\vec{n}=0$ on $\partial\Omega$ and
$\rho(0,x)=\rho_{0}(x)$. Thanks to the incompressibility, the velocity can be
recovered in terms of the stream function $\Psi$:

$\displaystyle u=\nabla^{\perp}\Psi,\text{ where $\Psi$ is a solution to
}\begin{cases}-\Delta\Psi=\partial_{1}\rho&\text{ in $\Omega$,}\\\
\Psi=0&\text{ on $\partial\Omega$.}\end{cases}$ (1.3)

We first review well-posedness theory. For strong solutions, the local well-
posedness with smooth initial data for the IPM equation (1.1) can be derived
by a standard energy estimate in the case where the spatial domain $\Omega$
does not have a boundary. More precisely, when $\Omega=\mathbb{T}^{2}$ or
$\mathbb{R}^{2}$, any initial data in $H^{k}$ for $k>2$ possesses a unique
local-in-time solution $\rho\in C([0,T];H^{k}(\Omega))$ (see
[cordoba2007analytical, kiselev2023small]). However, when the boundary is
present, the energy estimate becomes more involved. Especially it was proved
in [castro2019global] that the IPM equation in a periodic channel is locally
wellposed in a subspace of $H^{k}(\Omega)$ for $k\geq 3$ with an additional
conditions on the boundary behavior of the solution (See Theorem 3.3 for a
more precise statement). Unlike the local well-posedness, the problem of
global existence v.s. finite time blow-up for smooth initial data still
remains open, while several blow up criteria have been established in
[cordoba2007analytical]. For weak solutions, the global existence of the weak
solutions with $L^{p}$ initial data is currently not available to the best of
our knowledge. The nonuniqueness in the class
$L^{\infty}([0,T]\times\mathbb{T}^{2})$ by means of the convex integration
[szekelyhidi2012relaxation]. We also note that the global existence of patch-
type solutions (the so-called Muskat problem) with the presence of a surface
tension have been established in [jacobs2021weak] using the optimal transport
theory.

While the global well-posedness for general initial data (sufficiently smooth)
is still out of reach, there are several stability results near a particular
steady state $\rho_{s}(x_{2}):=1-x_{2}$, which, as a byproduct, ensures the
global existence. Considering the IPM equation in the spatial domain
$\Omega=\mathbb{R}^{2}$, it was proved in [elgindi2017asymptotic] that if
$\rVert\rho_{0}-\rho_{s}\rVert_{H^{k}(\mathbb{R}^{2})}\leq\epsilon$ with
$k\geq 20$, the solution converges eventually to $\rho_{s}$, satisfying
$\rVert\rho(t)-\rho_{s}\rVert_{H^{3}(\mathbb{R}^{2})}\leq
C\frac{\epsilon}{(1+t)^{1/4}}$. In the same paper, the author also established
asymptotic stability in $\Omega=\mathbb{T}^{2}$ in the class $H^{k}$ for
$k\geq 20$ and proved that such perturbed solutions eventually converge to a
stratified density, which might not be the same as $\rho_{s}$. Stability in a
periodic channel $\Omega=\mathbb{T}\times(0,1)$ was investigated in
[castro2019global], proving that in the class $H^{k}(\Omega)$ for $k\geq 10$ a
perturbed solution eventually converges to a stratified density, which again
might not be the same as $\rho_{s}$. While these results requires sufficiently
large regularity on the initial perturbation, the authors in
[bianchini2024relaxation] recently reduced such regularity assumptions,
establishing asymptotic stability in the class $H^{k}(\mathbb{R}^{2})\cap
H^{1-s}(\mathbb{R}^{2})$ for $k>3$ and $0<s<1$. We also mention the work
[jo2024quantitative] which proves asymptotic stability in a periodic channel
for a perturbation in $H^{k}$ for $3<k\in\mathbb{N}$, assuming that the
vertical derivative of the steady state $\rho_{s}$ is sufficiently large,
depending on the size of the perturbation.

These results concerning asymptotic stability reveal the difficulty to specify
a permanent description of the long time limit of the solution. The main
challenge stems from the fact that the equilibria of the IPM are not isolated;
given two stratified densities $\rho_{s}(x_{2})$ $f(x_{2})$, any function of
the form $\rho_{s}(x_{2})+\epsilon f(x_{2})$ for any $\epsilon>0$ is another
equilibrium. However, as noted in [dalibard2023long] the IPM equation is a
transport equation with an incompressible flow, therefore as long as such a
limit is achieved as a strong limit in $C^{1}$, each super-level set of the
solution $\rho(t)$ must preserve its topological properties and the area.
Given an initial data, a stratified density preserving such properties can be
uniquely determined as the so-called vertical (decreasing) rearrangement
$\rho^{*}$,

$\rho(x)\mapsto\rho^{*}(x):=\int_{0}^{\infty}1_{\left\\{0\leq
x_{2}\leq|\left\\{\rho>s\right\\}|\right\\}}ds.$

Note that such vertical rearrangement is invariant under any measure
preserving continuous diffeomorphism, especially, $\rho(t)^{*}=\rho_{0}^{*}$.
From this property, one can easily deduce that the long-time limit that was
not specified in the above earlier works must be indeed $\rho_{0}^{*}$, which
can be precisely specified from the initial data.

A somewhat trivial but crucial observation is that $\rho^{*}$ is a local
minimizer of the potential energy defined as

$E_{P}(\rho):=\int_{\Omega}\rho(x)x_{2}dx.$

More precisely, one can easily deduce that for any measure preserving
diffeomorphism $h:\Omega\mapsto\Omega$, it holds that

$E_{P}(\rho^{*}\circ h)\geq E_{P}(\rho^{*}),$

and the equality is obtained if and only if $h$ is the identity map. Moreover,
the potential energy is a Lyapunov functional to the IPM equation in the sense
that given a sufficiently smooth solution $\rho(t)$, namely,

$\frac{d}{dt}E_{P}(\rho(t))=\int_{\Omega}u_{2}\rho
dx=\int_{\Omega}u\cdot\begin{pmatrix}0\\\
\rho\end{pmatrix}dx=-\int_{\Omega}|u|^{2}dx,$

where the last equality is due to Darcy’s law and the incompressibility of the
flow. In this view, the solution to the IPM equation can be thought of as a
minimizing curve associated to the potential energy whose long-time limit is
the ground state of the energy. Hence the asymptotic stability can be obtained
by achieving a sufficiently fast decay of $E_{P}(\rho(t))$ towards
$E_{P}(\rho_{0}^{*})$. Our precise statement of the main theorem for the IPM
equation is as follows:

######  Theorem 1.1.

Let $k\in\mathbb{N}$ be such that $k\geq 3$ and $\rho_{s}(x_{2}):=1-x_{2}$.
There exist $\epsilon=\epsilon(k)>0$ and $C=C(k)>0$ such that if
$\rho_{0}-{\rho}_{s}\in H^{k}_{0}(\Omega)$ and
$\rVert\rho_{0}-{\rho}_{s}\rVert_{H^{k}(\Omega)}\leq\epsilon$, then there
exists a unique solution $\rho\in C([0,\infty);H^{k}(\Omega))$ to the IPM
equation (1.2) and it satisfies

$\rVert\rho(t)-{\rho}_{s}\rVert_{H^{k}(\Omega)}\leq C\epsilon,\text{ for all
$t>0$},$

Furthermore, the potential energy decays as

$E_{P}(\rho(t))-E_{P}(\rho_{0}^{*})\leq C\epsilon^{2}t^{-k}.$

Consequently, the solution $\rho(t)$ converges to the vertical rearrangement
of the initial data:

$\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}(\Omega)}\leq
C\frac{\epsilon}{t^{k/2}},\text{ for all $t>0$.}$

A few remarks are in order:

######  Remark 1.2.

As mentioned earlier, several results concerning asymptotic stability for the
IPM equation are available in the literature (e.g., [bianchini2024relaxation,
elgindi2017asymptotic, castro2019global]). Compared to these results, our
theorem requires slightly weaker regularity on the initial perturbation.
Furthermore, our poof in this paper is essentially different from all of these
results in that we do not rely on the linearized system but exploit the decay
of the potential energy. The key observation of this paper is that the
potential energy is not only a Lyapunov functional but also reveals another
(degenerate) coercive structure in its second derivative, i.e.,
$\left(\frac{d}{dt}\right)^{2}E_{P}(\rho(t))\geq\rVert
u_{2}\rVert_{L^{2}}^{2}$. See Subsection 1.3 for more detailed explanation how
such property can be used in the proof.

######  Remark 1.3.

The author expects that the regularity assumption $H^{k}$ for $k\geq 3$ can be
relaxed even further, and a similar strategy used this paper would work out
for initial perturbation $\rho_{0}-\rho_{s}$ sufficiently small $H^{k}$ for
$k>1+\sqrt{3}$. More concrete evidence of such an expectation will be
explained in Subsection 1.3. The main reasons why we do not included such a
stronger statement are because we do not want to introduce extra complications
of the proof by involving fractional Sobolev spaces, and the local well-
posedness of the equation is currently not directly available for lower
regularity spaces (see Theorem 3.3). We emphasize that the perturbation
regularity cannot be relaxed too much, considering the long time instability
result in $H^{2-\epsilon}$ for any $\epsilon>0$ [kiselev2023small, Theorem
1.5]. Noticing that $H^{2}$ barely fails to embed into $C^{1}$, it seems to be
an interesting question whether an asymptotic stability can be established for
a small perturbation in $H^{2+\epsilon}$.

######  Remark 1.4.

From the stream function formulation (1.3), it is straightforward to see the
velocity and the density formally stay in the same regularity class,
especially, $\rVert
u\rVert_{L^{2}}\leq\rVert\rho-\overline{\rho}\rVert_{L^{2}}$, where
$\overline{\rho}$ is the spatial average of $\rho$. However, compared to the
decay rate of the density stated in the above theorem, our argument reveals
slightly better decay rates of the kinetic energy and the anisotropic kinetic
energy in a time-average sense, that is,

$\frac{2}{t}\int_{t/2}^{t}\rVert u(s)\rVert_{L^{2}}^{2}ds\leq
C_{k}\frac{\epsilon^{2}}{t^{k+1}},\text{ and }\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq C_{k}\frac{\epsilon^{2}}{t^{k+2}},\text{ for
all $t>0$, $k\geq 3$.}$

######  Remark 1.5.

Theorem 1.1 only concerns perturbations near a specific steady state
$\rho_{s}(x_{2})=1-x_{2}$. However, the author expects that our result can be
generalized to perturbations near any sufficiently regular stratified density
$\rho_{s}$ such that $\inf_{x\in\Omega}\partial_{2}\rho_{s}(x_{2})>0$. Again
the reason why we do not pursue such a more general statement is due to a lack
of an exact statement in the literature about a local well-posedness theorem
near general steady states. Instead, we will establish asymptotic stability
near general stratified steady states for the Stokes transport system (see
Theorem 1.6), which could provide more evident ideas for the IPM equation as
well.

###  1.2. Stokes transport system

The Stokes transport system is another active scalar equation and it shares
several interesting properties with the IPM equation. We first recall the
system:

$\displaystyle\text{(Stokes)}=\begin{cases}\rho_{t}+u\cdot\nabla\rho=0,&\text{for
$(t,x)\in\mathbb{R}^{+}\times\Omega$}\\\ \nabla\cdot u=0,\\\ \Delta u=-\nabla
p-\begin{pmatrix}0\\\ \rho\end{pmatrix},\quad u=0&\text{ on
$\partial\Omega$}.\end{cases}$ (1.4)

As in the IPM, the incompressibility condition allows for a stream function
formulation for the velocity field:

$\displaystyle u=\nabla^{\perp}\Psi,\text{ where $\Psi$ solves
}\begin{cases}\Delta^{2}\Psi=\partial_{1}\rho&\text{ in $\Omega$,}\\\
\Psi=\nabla\Psi=0&\text{ on $\partial\Omega$.}\end{cases}$ (1.5)

From the stream function formulation, one can easily notice that the velocity
in the Stokes transport is much more regular than in the IPM equation. Indeed,
such regular structure enables a global well-posedness theorem in a standard
manner; if $\rho_{0}\in H^{k}(\Omega)$ for $k\geq 3$, then there exists a
unique solution $\rho\in C([0,\infty),H^{k}(\Omega))$ (see, [dalibard2023long,
Theorem A.1] or [leblond2022well, Theorem 1.1]). We also mention that the
regular of the velocity can yield quite robust structures for weak solutions,
for instance, $L^{3}$ initial data $\rho_{0}$ yields a unique regular
Lagrangian solution for three-dimensional model [inversi2023lagrangian,
Theorem 2.2, Theorem 2.4].

Various long time behaviors of the system (1.4) have been investigated in
[dalibard2023long], where the authors studied asymptotic stability and
boundary layer formation for initial data near $\rho_{s}(x_{2})=1-x_{2}$. We
also mention that the authors in [gancedo2022long, gancedo2024global] studied
the interface problem, where $\rho$ is given as a characteristic function
representing two different fluid densities, establishing global existence and
asymptotic stability/instability of the interface depending on the Reyleigh-
Taylor stability criterion.

In regard to the Stokes transport system, our main result in this paper is a
slight extension of the asymptotic stability obtained in [dalibard2023long],
especially concerning the regularity assumption on the initial perturbation.
This result will be established by adapting a similar strategy that we exploit
for the IPM equation based on the energy structure. More precise statement is
as follows:

######  Theorem 1.6.

Let $\rho_{s}(x_{2})$ be a stratified steady state such that

$\displaystyle\gamma:=\inf_{x\in\Omega}\left(-\partial_{2}\rho_{s}(x_{2})\right)>0,\quad\rVert\rho_{s}\rVert_{H^{4}(\Omega)}<\infty.$
(1.6)

Then there exist
$\epsilon=\epsilon(\gamma,\rVert\rho_{s}\rVert_{H^{4}(\Omega)})$ and
$C=C(\gamma,\rVert\rho_{s}\rVert_{H^{4}})$ such that if
$\rho_{0}-{\rho}_{s}\in H^{2}_{0}(\Omega)\cap H^{4}(\Omega)$ and
$\rVert\rho_{0}-{\rho}_{s}\rVert_{H^{4}}\leq\epsilon$, then the unique
solution $\rho\in C([0,\infty);H^{4}(\Omega))$ to the Stokes transport system
(1.4) satisfies

$\rVert\rho(t)-{\rho}_{s}\rVert_{H^{4}(\Omega)}\leq C\epsilon,\text{ for all
$t>0$}.$

Furthermore, the potential energy decays as

$E_{P}(\rho(t))-E_{P}(\rho_{0}^{*})\leq C\epsilon^{2}t^{-2}.$

Consequently, the solution $\rho(t)$ converges to its vertical rearrangement:

$\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}(\Omega)}\leq
C\frac{\epsilon}{t},\text{ for all $t>0$.}$

######  Remark 1.7.

A similar asymptotic stability result was already provided in
[dalibard2023long, Theorem 1.1], where the authors assumed that the initial
perturbation is small in $H^{6}(\Omega)$ near $\rho_{s}(x_{2})=1-x_{2}$. The
authors also provided clear evidence that such a result can be obtained near
more general stratified steady state which is sufficiently regular. As noted
earlier, the proof presented in this paper is different in that our analysis
does not use the linearized equation and it is an energy functional based
method. This approach requires a slightly weaker regularity assumption for the
initial perturbation. However, as in the IPM equation, there is a threshold of
the regularity for the stability. Indeed, [leblond2023well, Theorem 3.7.2]
provides an example of long time instability for small initial perturbation in
$H^{2-\epsilon}$ near any steady state.

######  Remark 1.8.

As in Remark 1.4, our proof reveals a slight faster decay estimate for the
velocity compared to that of the density. More precisely, we obtain

$\frac{2}{t}\int_{t/2}^{t}\rVert\nabla u(s)\rVert_{L^{2}}^{2}ds\leq
C\frac{\epsilon^{2}}{t^{3}},\text{ and }\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq C\frac{\epsilon^{2}}{t^{4}},\text{ for all
$t>0$.}$

###  1.3. A sketch of the proof of Theorem 1.1

Let us describe the structure of the proof for the IPM equation. A similar
strategy will be adapted to prove the stability for the Stokes transport
system.

We consider initial data $\rho_{0}$ such that
$\rVert\rho_{0}-\overline{\rho}\rVert_{H^{k}}\leq\epsilon$ for sufficiently
small $\epsilon>0$ and we will assume $k\geq 3$. For convenience, we denote

$\theta(t):=\rho(t)-{\rho}_{s},\quad\rho_{0}^{*}:=\text{ the vertical
(decreasing) rearrangement of $\rho_{0}$,}$

and

$E(t):=\int_{\Omega}(\rho(t)-\rho_{0}^{*})(x)x_{2}dx,\quad
K(t):=\rVert\nabla\Psi(t)\rVert_{L^{2}}^{2}.$

Thanks to the weight, $x_{2}$, in the integral expression for $E$, it is
evident that $\rho_{0}^{*}$ is the unique minimizer of $\rho\mapsto
E_{P}(\rho)$ among all the functions which can be obtained by a pushforward of
$\rho_{0}$ under a measure preserving map. Also if $\rho_{0}^{*}$ is a non-
degenerate minimizer, then $\rho\mapsto E_{P}(\rho)$ is expected to satisfy a
quadratic lower bound in a suitable space. In this paper, we will look for
such a lower bound in $L^{2}(\Omega)$ and establish in Proposition 2.5 that

$\displaystyle E(t)\sim\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}^{2}.$ (1.7)

Furthermore, as long as the solution $\rho(t)$ stays close to a stratified
density in the space $H^{3}(\Omega)$, the fact that $H^{3}(\Omega)$
continuously embeds into $C^{1}(\Omega)$ suggests that each level set of
$\rho(t)$ is also a graph of the horizontal variable $x_{1}$, from which one
can infer that

$\displaystyle\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}\leq
C\rVert\partial_{1}\rho(t)\rVert_{L^{2}}.$ (1.8)

On the other hand, the time derivative of $E(t)$ can be computed as

$\displaystyle\frac{d}{dt}E(t)=\frac{d}{dt}\int\rho(t,x)x_{2}dx=\int u_{2}\rho
dx=\int\partial_{1}\Psi\rho dx=-\int\Psi\partial_{1}\rho dx=\int\Psi\Delta\Psi
dx=-K(t).$ (1.9)

The Biot-Savart law in (1.3) and the Gargliado-Nirenberg inequality tell us
that

$\rVert\partial_{1}\rho\rVert_{L^{2}}=\rVert\Delta\Psi\rVert_{L^{2}}\leq
C_{k}\rVert\nabla\Psi\rVert_{H^{k}(\Omega)}^{1/k}\rVert\nabla\Psi\rVert_{L^{2}}^{(k-1)/k}.$

Combining this with (1.7) and (1.8), we get

$K(t)=\rVert\nabla\Psi\rVert_{L^{2}}^{2}\geq_{C}\rVert\partial_{1}\rho\rVert_{L^{2}}^{2k/(k-1)}\rVert\nabla\Psi\rVert_{H^{k}}^{-2/k}\geq_{C}E(t)^{k/(k-1)}\rVert\nabla\Psi\rVert_{H^{k}}^{-2/(k-1)}.$

Substituting this into (1.9), we obtain

$\displaystyle\frac{d}{dt}E(t)\leq_{C}-E(t)^{k/(k-1)}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{-2/(k-1)}.$
(1.10)

This inequality is the main source of the asymptotic stability. Let us use the
following notation which is slightly different from the usual convention: For
$\alpha>0$ and $f:\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$,

$\displaystyle f(t)=O(t^{-\alpha}),\text{ if \
$\frac{2}{t}\int_{t/2}^{t}f(s)ds\leq C(1+t)^{-\alpha}$ for some $C>0$}.$
(1.11)

Clearly $f(t)=O(t^{-\alpha})$ means that $f$ decays like $t^{-\alpha}$ in
average. Let us make an ansatz:

$\displaystyle\rVert\theta(t)\rVert_{H^{k}}\lesssim\epsilon,\quad\rVert\nabla\Psi(t)\rVert_{H^{k}}=O(t^{-\alpha}),\text{
for all $t>0$, for some $\alpha>0$}$ (1.12)

Under this ansatz, one can immediately deduce from the inequality (1.10) that

$\displaystyle E(t)=O({t^{-(k+2\alpha-1)}}).$ (1.13)

With this energy decay rate, the energy variation in time (1.9) suggests that
$K(t)$ decays faster than $E(t)$ by a factor of $t^{-1}$. Indeed, this
elementary heuristic can be made rigorous by measuring the decay rates as an
average (Lemma 2.2). Hence we can deduce

$K(t)=O(t^{-(k+2\alpha)}).$

Having such a decay rate for $K(t)$, we will proceed to look at a higher
derivative of the potential energy in time. A key observation is that the
second derivative of the energy $E(t)$ also exhibits a coercive structure,
namely,

$\left(\frac{d}{dt}\right)^{2}E(t)=-\frac{d}{dt}K(t)\geq C\rVert
u_{2}(t)\rVert_{L^{2}}^{2},$

which is the result of Proposition 3.7. We emphasize that such coercive
structure should not come as a surprise, because the solution is expected to
converge to a non-degenerate minimizer of the potential energy. Again, our
notation (1.11) allows us to postulate that $\rVert u_{2}\rVert_{L^{2}}^{2}$
will decay faster than $K(t)$ by a factor $t^{-1}$, that is,

$\displaystyle\rVert u_{2}(t)\rVert_{L^{2}}^{2}=O(t^{-(k+2\alpha+1)}).$ (1.14)

So far, the decay rates of the energies have been derived under the ansatz
(1.12), therefore it must be justified in order to close the argument. To this
end, in Proposition 3.6, we will derive the following estimate (an informal
estimate is presented at this point for simplicity):

$\displaystyle\frac{d}{dt}\rVert\theta(t)\rVert_{H^{k}}^{2}+\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}\leq
C\rVert u_{2}\rVert_{W^{1,\infty}}\rVert\theta(t)\rVert_{H^{k}}^{2}.$ (1.15)

Using again the Gagliardo-Nirenberg interpolation inequality and Young’s
inequality, we deduce

$\rVert u_{2}\rVert_{W^{1,\infty}}\leq C\rVert
u_{2}\rVert_{L^{2}}^{1-2/k}\rVert u_{2}\rVert_{H^{k}}^{2/k}\leq C\rVert
u_{2}\rVert_{L^{2}}^{1-2/k}\rVert\nabla\Psi\rVert_{H^{k}}^{2/k}\leq\eta\rVert\nabla\Psi\rVert_{H^{k}}^{2}+C_{\eta}\rVert
u_{2}\rVert_{L^{2}}^{(k-2)/(k-1)},$

for any $\eta\ll 1$. Noting that
$\rVert\theta(t)\rVert_{H^{k}}\lesssim\epsilon\ll 1$ as long as the ansatz
(1.12) is valid, we substitute this estimate into (1.15), yielding that

$\displaystyle\frac{d}{dt}\rVert\theta(t)\rVert_{H^{k}}^{2}+\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}\leq
C\rVert
u_{2}\rVert_{L^{2}}^{(k-2)/({k-1})}\rVert\theta(t)\rVert_{H^{k}}^{2}=\epsilon^{2}O(t^{-(k+2\alpha+1)(k-2)/(2k-2)}),$
(1.16)

where the last equality is due to (1.14). In this differential inequality, a
sufficient condition for the ansatz (1.12) to persist is that the right-hand
side should decay faster than $O(t^{-(1)})$, that is,

$\displaystyle\frac{(k+2\alpha+1)(k-2)}{2k-2}>1.$ (1.17)

Indeed, if this condition is satisfied, integrating the both sides of (1.16)
in time yields that

$\sup_{t>0}\rVert\theta(t)\rVert_{H^{k}}^{2}+\int_{0}^{\infty}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt\leq
C_{k,\alpha,\theta_{0}}\epsilon^{2}.$

In this case, $t\mapsto\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}$ is integrable in
time, which indicates that our ansatz (1.12) should hold at least for some
$\alpha\geq 1/2$;

$\frac{2}{t}\int_{t/2}^{t}\rVert\nabla\Psi(s)\rVert_{H^{k}}ds\leq\left(\frac{2}{t}\int_{t/2}^{t}\rVert\nabla\Psi(s)\rVert_{H^{k}}^{2}ds\right)^{1/2}\leq
C_{k,\alpha,\theta_{0}}\epsilon\sqrt{\frac{2}{t}}\text{ for all $t>0$.}$

For $\alpha\geq 1/2$, the minimum value of $k$ for the sufficient condition
(1.17) to hold can be directly computed:

$\frac{(k+2\alpha+1)(k-2)}{2k-2}>1\Longleftarrow\frac{(k+2)(k-2)}{2k-2}>1\Longleftarrow
k>1+\sqrt{3}.$

The range of $k$ stated in Theorem 1.1 is strong enough to satisfy the
sufficient condition for the above scheme. Especially, (1.13) with $\alpha\geq
1/2$ directly gives the decay rate of the potential energy stated in the
Theorem 1.1, resulting in the desired asymptotic stability.

###  1.4. Organization of the paper

In Section 2, we collect useful tools concerning simple ODE problems and
quantitative estimates for the potential energy. The stability analysis for
the IPM equation and the Stokes transport system will be separately
investigated in Section 3 and Section 4 respectively.

###  1.5. Conventional notations

Following the conventional practice, we denote by $C$ an implicit positive
constant that may vary from line to line. In the case where $C$ depends on a
quantity, say $A$, we will represent it as $C_{A}$ or $C(A)$. For two
quantities, $A$ and $B$, we will also use the notation $A\leq_{C}B$,
indicating that $A\leq CB$ for some constant $C>0$. We denote

$\displaystyle C^{\infty}_{0}(\Omega):=\left\\{f\in
C^{\infty}(\Omega):{\text{supp}(f)}\subset\Omega\right\\},$ (1.18)

where $\text{supp}(f)$ is the closed support of $f$.

##  2\. Preparation: Time-average decay and vertical rearrangement

###  2.1. Time-average decay rates in differential inequalities

In the proof of asymptotic stability, we will measure the decay rates of the
energy quantities in a time-average manner. To prepare for this analysis, we
will gather useful lemmas concerning simple differential inequalities. In what
follows $[0,T]$ will denote an arbitrary time interval for some $T>0$.

######  Lemma 2.1.

Let $\alpha>0,\ 1<n$. Let $a(t)$ and $f(t)$ be nonnegative functions on
$[0,T]$ such that

$\frac{d}{dt}f(t)\leq-a(t)^{-\alpha}f(t)^{n},\quad f(0)=f_{0}.$

Then, $f$ satisfies

$f(t)\leq_{\alpha,n}\frac{A^{\alpha/(n-1)}}{t^{(\alpha+1)/(n-1)}},\text{for
all $t\in[0,T]$, where $A:=\int_{0}^{t}a(s)ds.$}$

###### Proof.

Dividing the differential inequality by $f(t)^{n}$ and integrating it in time,
we find that

$\displaystyle\frac{1}{f^{n-1}(t)}-\frac{1}{f_{0}^{n-1}}\geq_{n}\int_{0}^{t}a(s)^{-\alpha}ds.$
(2.1)

Since $\alpha>0$, Jensen’s inequality yields that
$\int_{0}^{t}a(s)^{-\alpha}\frac{ds}{t}\geq\left(\int_{0}^{t}a(s)ds\right)^{-\alpha}t^{\alpha}=A^{-\alpha}t^{\alpha},$
which implies

$\int_{0}^{t}a(s)^{-\alpha}ds\geq_{n}A^{-\alpha}t^{\alpha+1}.$

Plugging this into (2.1), we get
$\frac{1}{f^{n-1}(t)}\geq_{n}A^{-\alpha}t^{\alpha+1}$, which immediately gives
the desired result. ∎

######  Lemma 2.2.

Let $n>0$ and $f(t),g(t),h(t)$ be nonnegative functions on $[0,T]$ such that

$\displaystyle\frac{d}{dt}f(t)\leq-g(t),\ \frac{d}{dt}g(t)\leq-h(t),\text{ and
}f(t)\leq\frac{C}{t^{n}},$ (2.2)

for some $C>0$. Then, it holds that

$\frac{2}{t}\int_{t/2}^{t}g(s)ds\leq_{n}\frac{C}{t^{n+1}},\text{ and
}\frac{2}{t}\int_{t/2}^{t}h(s)ds\leq_{n}\frac{C}{t^{n+2}},\text{ for all
$t\in[0,T]$.}$

###### Proof.

Let us choose $s,t\in[0,T]$ arbitrary so that $0\leq s\leq t\leq T$.
Integrating $f^{\prime}(t)\leq-g(t)$ over $[s,t]$ for $s\in(0,t)$, we get
$f(t)-f(s)+\int_{s}^{t}g(u)du\leq 0$. Hence, the upper bound of $f(t)$ tells
us that

$\displaystyle\int_{s}^{t}g(u)du\leq f(s)\leq\frac{C}{s^{n}},\text{ for
$0<s<t<T$.}$ (2.3)

Plugging in $s=t/2$, we see that

$\displaystyle\frac{2}{t}\int_{t/2}^{t}g(u)du\leq\frac{C}{t^{n+1}},$ (2.4)

and this is the desired estimate for $g$.

Similarly, we integrate $g^{\prime}\leq-h$ and observe that

$\displaystyle\int_{s}^{t}h(u)du\leq g(s),\text{ for $0<s<t<T$}.$ (2.5)

Integrating one more time in $s$ over $[t/4,t]$, we find that the left-hand
side must satisfy

$\int_{t/4}^{t}\int_{s}^{t}h(u)duds=\int_{t/4}^{t}h(u)\int_{t/4}^{u}dsdu=\int_{t/4}^{t}h(u)(u-t/4)du\geq\int_{t/2}^{t}h(u)(u-t/4)du\geq\frac{t}{4}\int_{t/2}^{t}h(u)du.$

On the other hand, integrating the right-hand side in (2.5) over $[t/4.t]$
yields

$\int_{t/4}^{t}g(s)ds=\int_{t/4}^{t/2}g(s)ds+\int_{t/2}^{t}g(s)ds\leq_{n}\frac{C}{(t/2)^{n}}+\frac{C}{t^{n}}\leq_{n}\frac{C}{t^{n}},$

where we used (2.4). Putting them together, we obtain

$t\int_{t/2}^{t}h(u)du\leq_{n}\frac{C}{t^{n}},$

Dividing the both sides by $t^{2}$, we derive the desired estimate for $h$,
finishing the proof. ∎

It is an elementary fact that if a bounded function $f$ exhibits a decay rate
$O(t^{-(1+\epsilon)})$, it is integrable over all time, i.e.,
$\int_{0}^{\infty}f(t)dt<C_{\epsilon}<\infty$. In the next lemma, we
demonstrate that if $f(t)$ decays like $O(t^{1+\epsilon})$ in a time-average
sense, the same conclusion holds.

######  Lemma 2.3.

Let $T>2$ and $n>1$. Let $f(t)$ be a nonnegative function on $[0,T]$ such that

$\frac{2}{t}\int_{t/2}^{t}f(s)ds\leq\frac{E}{t^{n}},\text{ for some $E>0$, for
all $t\in[2,T]$}.$

Then, for $\alpha\in(1/n,1]$, we have

$\int_{1}^{T}f(t)^{\alpha}ds\leq C_{\alpha,n}E^{\alpha},$

where $C_{\alpha,n}>0$ does not depend on $T$.

###### Proof.

We pick $N\in\mathbb{N}$ such that

$\displaystyle\frac{T}{2^{N+1}}\leq 1\leq\frac{T}{2^{N}}\leq 2,$ (2.6)

and define $T_{i}:={2^{-i}}T$, for $i=0,\ldots,N$. We decompose

$\int_{1}^{T}f(t)^{\alpha}ds=\int_{1}^{T_{N}}f(t)^{\alpha}dt+\sum_{i=1}^{N}\int_{T_{i}}^{T_{i-1}}f(t)^{\alpha}dt.$

Since $T_{N}\leq 2$ and $\alpha\leq 1$, applying Jensen’s inequality, we
obtain

$\displaystyle\int_{1}^{T_{N}}f(t)^{\alpha}dt\leq\int_{1}^{2}f(t)^{\alpha}dt\leq\left(\int_{1}^{2}f(t)dt\right)^{\alpha}\leq
E^{\alpha}.$ (2.7)

For $1\leq i\leq N$, again Jensen’s inequality gives us that

$\int_{T_{i}}^{T_{i-1}}f(t)^{\alpha}dt\leq\left(\frac{1}{T_{i-1}-T_{i}}\int_{T_{i}}^{T_{i-1}}f(t)dt\right)^{\alpha}|T_{i-1}-T_{i}|=\left(\frac{1}{T_{i}}\int_{T_{i}}^{T_{i-1}}f(t)dt\right)^{\alpha}T_{i}\leq_{\alpha}E^{\alpha}T_{i}^{1-\alpha
n},$

where the last inequality follows from the decay hypothesis for $f$. Summing
over $i=1,\ldots,N$, we get

$\sum_{i=1}^{N}\int_{T_{i}}^{T_{i-1}}f(t)^{\alpha}dt\leq_{\alpha}E^{\alpha}T^{1-\alpha
n}\sum_{i=1}^{N}\left({2^{\alpha
n-1}}\right)^{i}\leq_{\alpha,n}E^{\alpha}\left(\frac{T}{2^{N}}\right)^{1-\alpha
n}\leq C_{\alpha,n}E^{\alpha},$

where the last inequality follows from (2.6). Combining this with (2.7), we
finish the proof.∎

###  2.2. Vertical rearrangement

Given a Borel measurable function $f$ on $\Omega$, we define its vertical
(decreasing) rearrangement as

$\displaystyle f^{*}(x_{2}):=\int_{0}^{\infty}1_{\left\\{0\leq
x_{2}\leq|\left\\{f>s\right\\}|\right\\}}ds.$ (2.8)

By its definition, it is clear that $x_{2}\mapsto f^{*}(x_{2})$ is monotone
decreasing. In the rest of the section, we consider a stratified density
$\rho_{s}(x)=\rho_{s}(x_{2})$, a function $f$ that is close to $\rho_{s}$ in a
Sobolev space and its vertical rearrangement.

Before presenting the lemmas, let us collect some basic properties for
$\rho_{s}$. We will always assume that

$\displaystyle\gamma:=\inf_{\Omega}(-\partial_{2}\rho(x_{2}))>0,\quad\rVert\rho_{s}\rVert_{H^{4}(\Omega)}<\infty.$
(2.9)

By the monotonicity of $\rho_{s}$, we can describe the image of $\rho_{s}$ as

$\displaystyle I:=\rho_{s}(\Omega)=[\rho_{s}(1),\rho_{s}(0)].$ (2.10)

The inverse function theorem, combined with the assumption that $\gamma>0$,
guarantees the existence of the inverse of $\rho_{s}$, that is,
$\phi_{0}:=\rho_{s}^{-1}:I\mapsto[0,1]$ is well-defined. Moreover, since
$\rho_{s}$ depends on the single variable $x_{2}$, the regularity assumption
in (2.9), combined with the usual Sobolev embedding theorem, ensures that
$\rho_{s}\in C^{3}(\Omega)$, and
$\rVert\rho_{s}\rVert_{C^{3}}\leq_{C}\rVert\rho_{s}\rVert_{H^{4}}.$ With such
information, one can straightforwardly deduce the following estimates:

$\displaystyle\rVert\partial_{s}\phi_{0}\rVert_{L^{\infty}}+\rVert\partial_{ss}\phi_{0}\rVert_{L^{\infty}}+\rVert\partial_{2}\rho_{s}\rVert_{L^{\infty}}+\rVert\partial_{22}\rho_{s}\rVert_{L^{\infty}}+\rVert\partial_{222}\rho_{s}\rVert_{L^{\infty}}\leq
C(\gamma,\rVert\rho_{s}\rVert_{H^{4}}).$ (2.11)

Noting that $H^{3}(\Omega)$ continuously embeds into $C^{1}(\Omega)$, one can
infer that if a function $f$ is sufficiently close to $\rho_{s}$ in $H^{3}$,
similar properties of the level sets and the inverse function of $f$ can be
quantitatively estimated. This is will be the main implication of the next
lemma. In the rest of this section, the implicit constant $C$, that appears in
the proofs, may depend on $\gamma$ and $\rVert\rho_{s}\rVert_{H^{4}}$ but we
will omit its dependence in the notation for simplicity.

######  Lemma 2.4.

Suppose $\rho_{s}$ satisfies (2.9). There exists
$\delta=\delta(\gamma,\rVert\rho_{s}\rVert_{H^{4}})>0$, such that if

$f={\rho}_{s}\text{ on $\partial\Omega$ and }\rVert
f-{\rho}_{s}\rVert_{H^{3}(\Omega)}\leq\delta,$

then there exist $\phi_{1}:I\mapsto[0,1]$ and $h:\mathbb{T}\times I\to[0,1]$
such that

$\displaystyle\int_{\mathbb{T}}h(x_{1},s)dx_{1}=0\text{ and
}f(x_{1},\phi_{1}(s)+h(x_{1},s))=s=f^{*}(\phi_{1}(s)),\text{ for
$(x_{1},s)\in\mathbb{T}\times I$.}$

Furthermore, the following estimates hold:

$\displaystyle\rVert\partial_{ss}(\phi_{1}-\phi_{0})\rVert_{L^{\infty}}+\rVert\partial_{s}(\phi_{1}-\phi_{0})\rVert_{L^{\infty}}+\rVert
h\rVert_{L^{\infty}}+\rVert\partial_{s}h\rVert_{L^{\infty}}\leq C\rVert
f-{\rho}_{s}\rVert_{H^{3}}.$ (2.12)

where $C>0$ is a constant which depends on $\rVert\rho_{s}\rVert_{H^{4}}$ and
$\gamma$.

Figure 1. Illustration of the level sets of $\rho_{s}$ and $f$. For each $s\in
I$, $\phi_{1}$ is uniquely determined so that the deviation $x_{1}\mapsto
h(x_{1},s)$ has zero average in $x_{1}$.

###### Proof.

An illustration for the proof of the lemma is presented in Figure 1.

We notice that the image of $f$ is exactly equal to the interval $I$, which is
the image of ${\rho}_{s}$. Indeed, we have

$\displaystyle\partial_{2}f(x_{1},x_{2})$
$\displaystyle=\partial_{2}{\rho}_{s}(x_{2})+(\partial_{2}f(x_{2})-\partial_{2}{\rho}_{s}(x_{2}))\leq-\gamma+C\rVert
f-{\rho}_{s}\rVert_{H^{3}}\leq-\gamma+C\delta<0,$ (2.13)

for sufficiently small $\delta>0$. Hence, $x_{2}\mapsto f(x_{1},x_{2})$ is
strictly decreasing. Since $f={\rho}_{s}$ on $\partial\Omega$, it follows that
$f(\overline{\Omega})=I$.

Thanks to the monotonicity, the implicit function theorem tells us that there
is a parametrization of the level curves, $\left\\{f=s\right\\}$, which we
will denote by $\phi(\cdot,s)$, that is,

$\displaystyle f(x_{1},\phi(x_{1},s))=s\text{ for each
$(x_{1},s)\in\mathbb{T}\times I$.}$ (2.14)

Let us rewrite it as

$\displaystyle\phi(x_{1},s)=\phi_{0}(s)+g(x_{1},s),\text{ where
}g(x_{1},s):=\phi(x_{1},s)-\phi_{0}(s).$ (2.15)

Now, we will aim to derive necessary estimates for $g$. Writing

$\displaystyle
0=\rho_{s}(\phi_{0}(s))-f(x_{1},\phi(s))=\rho_{s}(\phi_{0}(s))-\rho_{s}(\phi(x_{1},s))+\left(\rho_{s}(\phi(x_{1},s))-f(x_{1},\phi(x_{1},s))\right),$

we notice that

$\displaystyle\int_{0}^{g(x_{1},s)}\partial_{2}\rho_{s}(y+\phi_{0}(s))dy=\rho_{s}(\phi(x_{1},s))-f(x_{1},\phi(x_{1},s)).$
(2.16)

Since $\partial_{2}\rho_{s}<-\gamma<0$ and $\rVert
f-\rho_{s}\rVert_{L^{\infty}}\leq_{C}\rVert f-\rho_{s}\rVert_{H^{3}}$, we find
that

$\displaystyle\rVert g\rVert_{L^{\infty}}\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}}.$ (2.17)

Differentiating (2.16) with respect to $s$, we obtain

$\displaystyle\partial_{s}g(x_{1},s)\partial_{2}\rho_{s}(\phi(x_{1},s))+\int_{0}^{g(x_{1},s)}\partial_{22}\rho_{s}(y+\phi_{0}(s))\partial_{s}\phi_{0}(s)dy=(\partial_{2}\rho_{s}-\partial_{2}f)(x_{1},\phi(x_{1},s))\partial_{s}\phi(x_{1},s).$
(2.18)

Again, using $\partial_{2}\rho_{s}<-\gamma<0$, we have
$|\partial_{s}g\partial_{2}\rho_{s}|\geq C|\partial_{s}g|$. On the other hand,
using (2.11) and (2.17), we can estimate the integral on the left-hand side as

$\displaystyle\left|\int_{0}^{g(x_{1},s)}\partial_{22}\rho_{s}(y+\phi_{0}(s))\partial_{s}\phi_{0}(s)dy\right|\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}}.$

For the right-hand side, at each point $(x_{1},s)\in\mathbb{T}\times I$, we
have

$|(\partial_{2}\rho_{s}-\partial_{2}f)\partial_{s}\phi|\leq_{C}\rVert\rho_{s}-f\rVert_{H^{3}}|\partial_{s}\phi|\leq_{C}\rVert\rho_{s}-f\rVert_{H^{3}}|\partial_{s}\phi_{0}|+\delta|\partial_{s}g|\leq_{C}\rVert\rho_{s}-f\rVert_{H^{3}}+\delta|\partial_{s}g|.$

Assuming $\delta$ is sufficiently small depending on $\gamma$, these estimates
and (2.18) give us

$\displaystyle\rVert\partial_{s}g\rVert_{L^{\infty}}\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}}.$ (2.19)

Next let us estimate $\partial_{ss}g$. Once again, differentiating (2.18) in
$s$ and using the chain rule, we get (at each point $(x_{1},s)$)

$\displaystyle\partial_{ss}g\partial_{2}\rho_{s}(\phi)$
$\displaystyle=A_{1}+A_{2}+A_{3}+A_{4}+A_{5}+A_{6},$ (2.20)

where

$\displaystyle A_{1}$
$\displaystyle:=-\partial_{s}g\partial_{22}\rho_{s}(\phi)\partial_{s}\phi,$
$\displaystyle A_{2}$
$\displaystyle:=-\partial_{s}g\partial_{22}\rho_{s}(\phi)\partial_{s}\phi_{0}$
$\displaystyle A_{3}$
$\displaystyle:=-\int_{0}^{g}\partial_{222}\rho_{s}(y+\phi_{0})(\partial_{s}\phi_{0})^{2}dy$
$\displaystyle A_{4}$
$\displaystyle:=-\int_{0}^{g}\partial_{22}\rho_{s}(y+\phi_{0})\partial_{ss}\phi_{0}(s)dy,$
$\displaystyle A_{5}$
$\displaystyle:=\partial_{22}(\rho_{s}-f)(\phi)(\partial_{s}\phi)^{2}$
$\displaystyle A_{6}$
$\displaystyle:=\partial_{2}(\rho_{s}-f)(\phi)\partial_{ss}\phi.$

Again, $\partial_{2}\rho_{s}\leq-\gamma<0$ gives us that the left-hand side of
(2.20) can be estimated as

$|\partial_{ss}g\partial_{2}\rho_{s}(\phi)|\geq C|\partial_{ss}g|.$

Using (2.11), (2.17) and (2.19), it is straightforward that

$\displaystyle|A_{1}|,|A_{2}|,|A_{3}|,|A_{4}|\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}},\quad|A_{5}|\leq_{C}|\partial_{22}(\rho_{s}-f)|,\quad|A_{6}|\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}}+\delta|\partial_{ss}g|.$

Plugging them into (2.20), we obtain a pointwise estimate:

$\displaystyle|\partial_{ss}g(x_{1},s)|$ $\displaystyle\leq_{C}\rVert
f-\rho_{s}\rVert_{H^{3}}+|\partial_{22}\rho_{s}(\phi(x_{1},s))-\partial_{22}f(x_{1},\phi(x_{1},s))|.$
(2.21)

Then for each fixed $x_{1}$, we treat the functions above as a function of
$s$. Applying the Sobolev embedding $W^{1,1}(I)\mapsto L^{\infty}(I)$, we have
that for each $x_{1}\in\mathbb{T}$,

$\displaystyle\sup_{s\in I}$
$\displaystyle|\partial_{22}\rho_{s}(\phi(s))-\partial_{22}f(\phi(s))|\leq_{C}\int_{I}|\partial_{s}(\partial_{22}(\rho_{s}-f)(\phi(s)))|ds+\int_{I}|\partial_{22}(\rho_{s}-f)(\phi(s))|ds$
$\displaystyle\leq_{C}\int_{I}|\partial_{222}(\rho_{s}-f)(\phi(s))\partial_{s}\phi(s)|ds+\int_{I}|\partial_{22}(\rho_{s}-f)(\phi(s))|ds$
$\displaystyle\leq_{C}\int_{0}^{1}|\partial_{222}(\rho_{s}-f)(x_{1},x_{2})|dx_{2}+\int_{0}^{1}|\partial_{22}(\rho_{s}-f)(x_{1},x_{2})|dx_{2}\left(\sup_{s}|\partial_{s}\phi(s)|^{-1}\right),$
(2.22)

where we used the change of variables $\phi(s)\to x_{2}$ in the last
inequality. Note that

$|\partial_{s}\phi(s)|\geq|\partial_{s}\phi_{0}(s)|-|\partial_{s}g(s)|\geq\gamma-\delta>C,$

where the second inequality is due to (2.19). Hence, integrating (2.2) in
$x_{1}$ over $\mathbb{T}$, we get

$\int_{\mathbb{T}}\sup_{s\in
I}|\partial_{22}\rho_{s}(\phi(s))-\partial_{22}f(\phi(s))|dx_{1}\leq_{C}\rVert\rho_{s}-f\rVert_{W^{3,1}(\Omega)}+\rVert\rho_{s}-f\rVert_{W^{2,1}(\Omega)}\leq_{C}\rVert\rho_{s}-f\rVert_{H^{3}(\Omega)}.$

Combining this with (2.21), we arrive at

$\displaystyle\sup_{s\in I}\int_{\mathbb{T}}|\partial_{ss}g(x,s)|dx\leq\rVert
f-\overline{\rho}\rVert_{H^{3}(\Omega)}.$ (2.23)

Towards the proof, we define $\phi_{1}$ and $h$ as follows:

$\phi_{1}:=\phi_{0}(s)+\frac{1}{2\pi}\int_{\mathbb{T}}g(x,s)dx,\quad
h(x,s):=g(x,s)-\int_{\mathbb{T}}g(x,s)dx.$

Let us check if $\phi_{1}$ and $h$ satisfy the desired properties. By its
definition, we have $\int_{\mathbb{T}}h(x,s)dx=0$ for each $s\in I$. Also,
(2.14) and (2.15) tells us that

$f(x,\phi_{1}(s)+h(x,s))=f(x,\phi_{0}(s)+g(x,s))=f(x,\phi(x,s))=s.$

Collecting the estimates for $g$ obtained in (2.17), (2.19) and (2.23), we see
that

$\displaystyle\rVert
h\rVert_{L^{\infty}},\rVert\partial_{s}h\rVert_{L^{\infty}(\mathbb{T}\times
I)},\ \rVert\partial_{s}(\phi_{1}-\phi_{0})\rVert_{L^{\infty}(\mathbb{T}\times
I)},\
\rVert\partial_{ss}(\phi_{1}-\phi_{0})\rVert_{L^{\infty}(\mathbb{T}\times
I)}\leq C\rVert f-\overline{\rho}\rVert_{H^{3}(\Omega)},$ (2.24)

which proves (2.12). To finish the proof, we have to prove
$f^{*}(\phi_{1}(s))=s.$ To this end, observe that for each $s\in I$, it holds
that

$\left|\left\\{(x_{1},x_{2})\in\Omega:\phi_{1}^{-1}>s\right\\}\right|=\int_{\mathbb{T}}\int_{0}^{1}1_{\left\\{\phi_{1}^{-1}(x_{2})\geq
s\right\\}}dx=\int_{\mathbb{T}}\int_{0}^{\phi_{1}(s)}dx=2\pi\phi_{1}(s).$

We also have

$\displaystyle\left|\left\\{(x_{1},x_{2})\in\Omega:{f}>s\right\\}\right|$
$\displaystyle=\int_{\mathbb{T}}\int_{0}^{1}1_{\left\\{f(x_{1},x_{2})>s\right\\}}dx=\int_{\mathbb{T}}\int_{0}^{1}1_{\left\\{x_{2}\leq\phi(x_{1},s)\right\\}}dx=\int_{\mathbb{T}}\phi(x_{1},s)dx_{1}$
$\displaystyle=2\pi\left(\phi_{0}(s)+\frac{1}{2\pi}\int_{\mathbb{T}}g(x_{1},s)dx_{1}\right)$
$\displaystyle=2\pi\phi_{1}(s).$

This implies that the areas of every super-level set of $\phi_{1}^{-1}$ and
$f$ are equal. Recalling the definition of the rearrangement in (2.8), we
arrive at

$f^{*}(x_{2})=\int_{0}^{\infty}1_{0\leq
x_{2}\leq|\left\\{f>s\right\\}|}ds=\int_{0}^{\infty}1_{0\leq
x_{2}\leq|\left\\{\phi_{1}^{-1}>s\right\\}|}ds=\phi^{-1}(x_{2}).$

This proves $f^{*}(\phi_{1}(s))=s.$ ∎

Recall that for $f\in L^{1}(\Omega)$, its potential energy is defined as

$\displaystyle E_{P}(f):=\int_{\Omega}f(x)x_{2}dx.$ (2.25)

It is well-known that for any stratified function $\rho_{s}$ is a critical
point of $E_{P}$ under any divergence-free perturbation. In the case where $f$
is sufficiently close to $\rho_{s}$, the next proposition will quantitatively
demonstrate that $f^{*}$ is the unique minimizer of $E_{P}$ among all
functions with the same area of super-level sets.

######  Proposition 2.5.

Suppose $\rho_{s}$ satisfies (2.9). There exists
$\delta=\delta(\gamma,\rVert\rho_{s}\rVert_{H^{4}})>0$ such that if
$f={\rho}_{s}$ on $\partial\Omega$ and $\rVert
f-{\rho}_{s}\rVert_{H^{3}(\Omega)}\leq\delta$, then

$\displaystyle C^{-1}\rVert f-f^{*}\rVert_{L^{2}(\Omega)}^{2}\leq
E_{P}(f)-E_{P}(f^{*})\leq C\rVert f-f^{*}\rVert_{L^{2}(\Omega)}^{2}.$ (2.26)

Moreover, we have

$\displaystyle\rVert\partial_{1}f\rVert_{L^{2}(\Omega)}\geq C\rVert
f-f^{*}\rVert_{L^{2}(\Omega)}.$ (2.27)

The constant $C$ depends only on $\gamma$ and $\rVert\rho_{s}\rVert_{H^{4}}$.

###### Proof.

Let us prove (2.26) first. For sufficiently small $\delta>0$, Lemma 2.4
ensures the existence of $\phi_{1},h$ such that

$\displaystyle\ \int_{\mathbb{T}}h(x,s)dx$ $\displaystyle=0,\quad
f(x,\phi_{1}(s)+h(x,s))=s=f^{*}(\phi_{1}(s)),$ (2.28)

and

$\displaystyle\rVert\partial_{ss}(\phi_{1}-\phi_{0})\rVert_{L^{\infty}}+\rVert\partial_{s}\phi_{1}-\partial_{s}\phi_{0}\rVert_{L^{\infty}}+\rVert
h\rVert_{L^{\infty}}+\rVert\partial_{s}h\rVert_{L^{\infty}}\leq C\delta,$
(2.29)

where $\phi_{0}$ is the inverse of $\rho_{s}$, which trivially implies
$\partial_{s}\phi_{0}(s)=\frac{1}{\partial_{2}\rho_{s}(\phi_{0}(s))}$. From
(2.9), it follows that

$\displaystyle 0<\frac{1}{C}\leq-\partial_{s}\phi_{0}<C.$ (2.30)

Since $f^{*}$ is the inverse of $\phi_{1}$, the inverse function theorem and
the estimates for $\phi_{1}$ in (2.29) tell us that

$\displaystyle\rVert\partial_{22}f^{*}-\partial_{22}\rho_{s}\rVert_{L^{\infty}(\Omega)}+\rVert\partial_{2}f^{*}-\partial_{2}\rho_{s}\rVert_{L^{\infty}(\Omega)}\leq
C\delta.$ (2.31)

Denoting $\phi(x,s):=\phi_{1}(s)+h(x_{1},s)$ and using the change of
variables, $x_{2}\to\phi(x_{1},s)$, we have

$\displaystyle\rVert f-f^{*}\rVert_{L^{2}(\Omega)}^{2}$
$\displaystyle=\int_{\mathbb{T}}\int_{I}|f(x,\phi(x,s))-f^{*}(\phi(x,s))|^{2}|\partial_{s}\phi(x,s)|dxds.$
(2.32)

Thanks to (2.28), we have

$\displaystyle f(x,\phi(x,s))$ $\displaystyle-f^{*}(\phi(x,s))$
$\displaystyle=s-f^{*}(\phi_{1}(s))+\left(f^{*}(\phi_{1}(s))-f^{*}(\phi_{1}(s)+h(x,s))\right)$
$\displaystyle=f^{*}(\phi_{1}(s))-f^{*}(\phi_{1}(s)+h(x,s))$
$\displaystyle=\partial_{2}f^{*}(\phi_{1}(s))h(x,s)+O\left(|\partial_{22}f^{*}||h(x_{1},s)|^{2}\right)$
$\displaystyle=\partial_{2}\rho_{s}h(x,s)+O\left(|\partial_{2}(\rho_{s}-f^{*})||h(x_{1},s)|\right)+O\left(|\partial_{22}f^{*}||h(x,s)|^{2}\right)$
$\displaystyle=\partial_{2}\rho_{s}h(x,s)+O(\delta|h(x,s)|),$ (2.33)

where the last equality follows from (2.31). Note that (2.29) also implies
$|\partial_{s}\phi-\partial_{s}\phi_{0}|\leq_{C}\delta$, thus (2.30) gives

$\displaystyle 0<\frac{1}{C}\leq-\partial_{s}\phi\leq C.$ (2.34)

Plugging this and (2.2) into (2.32), we obtain, for sufficiently small
$\delta>0$,

$\displaystyle C^{-1}\rVert h\rVert_{L^{2}(\mathbb{T}\times[0,1])}\leq\rVert
f-f^{*}\rVert_{L^{2}(\Omega)}\leq C\rVert
h\rVert_{L^{2}(\mathbb{T}\times[0,1])}.$ (2.35)

On the other hand, $E_{P}(f)$ can be written as

$\displaystyle E_{P}(f)$
$\displaystyle=\int_{\mathbb{T}}\int_{0}^{1}f(x_{1},x_{2})x_{2}dx_{2}$
$\displaystyle=-\int_{\mathbb{T}}\int_{I}f(x_{1},\phi(x_{1},s))\phi(x_{1},s)\partial_{s}\phi(x_{1},s)dsdx_{1}$
$\displaystyle=-\int_{\mathbb{T}}\int_{I}s\frac{1}{2}\partial_{s}\left(\phi(x_{1},s)\right)^{2}dsdx_{1}.$

Since $f=f^{*}=\rho_{s}$ on $\partial\Omega$ and $\rho_{s}$ is strictly
decreasing, we have

$\phi(x_{1},\rho_{s}(0))=\phi_{1}(\rho_{s}(0))=0,\quad\phi(x_{1},\rho_{s}(1))=\phi_{1}(\rho_{s}(1))=1.$

Using this, we can continue the computation above as

$\displaystyle E_{P}(f)$
$\displaystyle=-\frac{1}{2}\int_{\mathbb{T}}\phi(x_{1},\rho_{s}(0))^{2}-\phi(x_{1},\rho_{s}(1))^{2}dx_{1}+\frac{1}{2}\int_{\mathbb{T}}\int_{I}(\phi_{1}(s)+h(x_{1},s))^{2}dsdx_{1}$
$\displaystyle=\pi+\frac{1}{2}\int_{\mathbb{T}}\int_{0}^{1}\phi_{1}(s)^{2}+h(x_{1},s)^{2}dsdx_{1},$

where the last equality follows from that $h$ has zero average in $x$.
Similarly, we have

$\displaystyle E_{P}(f^{*})$
$\displaystyle=\int_{\mathbb{T}}\int_{0}^{1}f^{*}(x_{2})dx=-\int_{\mathbb{T}}\int_{I}f^{*}(\phi_{1}(s))\phi_{1}(s)\partial_{s}\phi_{1}(s)dsdx_{1}=\pi+\frac{1}{2}\int_{\mathbb{T}}\int_{I}\phi_{1}(s)^{2}ds.$

Consequently, we arrive at
$E(f)-E(f^{*})=\frac{1}{2}\int_{\mathbb{T}}\int_{I}|h(x_{1},s)|^{2}dsdx_{1}.$
Combining this with (2.35), the estimates in (2.26) is verified.

Now, let us prove (2.27). Differentiating (2.28) in $x_{1}$, we see that

$0=\partial_{1}(f(x_{1},\phi(x_{1},s)))=\partial_{1}f(x_{1},\phi(x,s))+\partial_{2}f(x_{1},\phi(x_{1},s))\partial_{1}h(x_{1},s).$

Similarly, diffierentiating (2.28) in $s$ yields
$1=\partial_{2}f(x_{1},\phi(x_{1},s))\partial_{s}\phi(s)$, thus

$\partial_{1}f(x_{1},\phi(x_{1},s))=-\frac{\partial_{1}h(x_{1},s)}{\partial_{s}\phi(x_{1},s)}.$

Then, using the change of variables $x_{2}\to\phi(x_{1},s)$ and also using
(2.34), we obtain

$\rVert\partial_{1}f\rVert_{L^{2}(\Omega)}\geq
C\rVert\partial_{1}h\rVert_{L^{2}(\mathbb{T}\times I)}\geq C\rVert
h\rVert_{L^{2}(\mathbb{T}\times I)},$

where the last inequality follows from the zero-average in $x$ of $h$ in
(2.28) and the Poincaré inequality. Therefore, combining this with (2.35), we
conclude that (2.27) holds. ∎

##  3\. Stability in the IPM equation

In this section, we aim to prove the asymptotic stability for the
incompressible porous media equation (1.2). Throughout the section, we will
fix

$\displaystyle\rho_{s}(x_{2}):=1-x_{2}.$ (3.1)

###  3.1. Preliminaries for the IPM

Let us review important previous results concerning the local existence of the
IPM equation (1.2) in the domain $\Omega=\mathbb{T}\times(0,1)$. For further
details, we refer readers to the paper by Castro–Córdoba–Lear
[castro2019global].

We recall the following spaces from [castro2019global, Section 1]: For
$k\in\mathbb{N}$,

$\displaystyle X^{k}(\Omega)$ $\displaystyle:=\left\\{f\in
H^{k}(\Omega):\partial_{2}^{n}f|_{\partial\Omega}=0,\text{ for
}n=0,2,4,.\ldots k^{\star}\right\\},\text{ where
}k^{\star}:=\begin{cases}k-2,&\text{ if $k$ is even},\\\ k-1&\text{ if $k$ is
odd}.\end{cases}$ (3.2)

That is, $X^{k}(\Omega)$ is the closure of $\left\\{f\in
C^{\infty}({\Omega}):\partial_{2}^{n}f|_{\partial\Omega}=0,\text{ for
}n=0,2,4,.\ldots k^{\star}\right\\}$ in the norm $H^{k}$. It is worth noting
that the usual trace theorem, $H^{k}(\Omega)\hookrightarrow
H^{k-1}(\partial\Omega)$, ensures that the vanishing normal derivatives in the
above definition is well-defined. For convenience, we denote

$X^{\infty}(\Omega):=\cap_{k\in\mathbb{N}}X^{k}(\Omega).$

By definition, it is clear from (1.18) that

$\displaystyle C^{\infty}_{0}(\Omega)\subset X^{\infty}(\Omega).$ (3.3)

If $f\in X^{\infty}(\Omega)$, it holds that

$\displaystyle\partial_{1}^{n_{1}}\partial_{2}^{2n_{2}}f\in
X^{\infty}(\Omega),\text{ for any
$n_{1},n_{2}\in\mathbb{N}\cup\left\\{0\right\\}$, }$ (3.4)

where $\partial_{n}$ denotes the normal derivative of $f$ on
$\partial_{\Omega}$. Especially, we can use the integration by parts in the
vertical variable without a boundary integral:

$\displaystyle\int_{\Omega}\partial_{2}^{k+1}f(x)\partial_{2}^{k+1}g(x)dx=-\int_{\Omega}\partial_{2}^{k}f(x)\partial_{2}^{k+2}g(x)dx,\text{
for any $k\in\mathbb{N}\cup\left\\{0\right\\}$ and $f,g\in
X^{\infty}(\Omega)$.}$ (3.5)

In a usual domain without boundary, for instance $\mathbb{R}^{2}$, it is
trivial that the Sobolev norms can be bounded by looking at only each single
directional derivatives, that is,

$\rVert f\rVert_{\dot{H}^{k}(\mathbb{R}^{2})}\leq
C\left(\rVert\partial_{1}^{k}f\rVert_{L^{2}(\mathbb{R}^{2})}+\rVert\partial_{2}^{k}f\rVert_{L^{2}(\mathbb{R}^{2})}\right).$

In a bounded domain, this property may depend on the boundary condition, since
a mixed derivative might not be well controlled. While the next lemma seems
intuitively obvious, we will give a proof for the sake of completeness,
although the proof will be postponed to Appendix A.

######  Lemma 3.1.

Let $f\in X^{\infty}(\Omega)$. For any
$n,k\in\mathbb{N}\cup\left\\{0\right\\}$, we have

$\displaystyle\rVert\partial_{1}^{n}\partial_{2}^{k}f\rVert_{L^{2}(\Omega)}\leq
C_{n,k}\left(\rVert\partial_{1}^{n+k}f\rVert_{L^{2}(\Omega)}+\rVert\partial_{2}^{n+k}f\rVert_{L^{2}(\Omega)}\right).$
(3.6)

Consequently, we have

$\displaystyle\rVert f\rVert_{{H}^{k}}\leq
C_{k}\left(\rVert\partial_{1}^{k}f\rVert_{L^{2}}+\rVert\partial_{2}^{k}f\rVert_{L^{2}}\right)\text{
for all $k\geq 0$}.$ (3.7)

Let us consider a solution $\rho(t)$ to the IPM equation. We denote

$\displaystyle\theta(t):=\rho(t)-\rho_{s}.$ (3.8)

Substituting $\rho=\theta+{\rho}_{s}$ in (1.2), one can easily see that
$\theta(t)$ solves

$\displaystyle\begin{cases}\theta_{t}+u\cdot\nabla\theta=u_{2},\\\
u=\nabla^{\perp}\Psi,\end{cases}\text{ with
}\begin{cases}-\Delta\Psi=\partial_{1}\theta,&\text{ in $\Omega$},\\\
\Psi=0,&\text{ on $\partial\Omega$.}\end{cases}$ (3.9)

The next lemma tells us that if the solution $\theta(t)\in X^{k}$, then the
stream function $\Psi(t)$ behaves in a similar manner.

######  Lemma 3.2.

[castro2019global, Lemma 3.1] Let $f\in X^{k}(\Omega)$ and let $\Psi$ be a
solution to

$\begin{cases}\Delta\Psi=-\partial_{1}f&\text{ in $\Omega$},\\\ \Psi=0&\text{
on $\partial\Omega$.}\end{cases}$

Then $\Psi\in X^{k+1}(\Omega)$ and it satisfies
$\rVert\Psi\rVert_{H^{k+1}}\leq C_{k}\rVert f\rVert_{H^{k}}$.

The local well-posedness to the equation (3.9) was established in
[castro2019global]:

######  Theorem 3.3.

[castro2019global, Theorem 4.1] Let $k\in\mathbb{N}$ with $k\geq 3$. For any
$\theta_{0}\in X^{k}(\Omega)$, there exists a time
$T=T(\rVert\theta_{0}\rVert_{H^{3}})>0$ and a unique solution $\theta\in
C(0,T;X^{k}(\Omega))$ for the equation (3.9).

Thanks to the local well-posedness theorem, we will assume that the initial
data is smooth, that is, $\theta_{0}\in X^{\infty}$. In view of the statement
of Theorem 1.1, the general case where $\theta_{0}\in H^{k}_{0}(\Omega)$ will
be managed by usual compactness argument in the end of the section.

###  3.2. Energy estimates

In this section, we aim to derive an a priori energy estimate. The main result
will be given in Proposition 3.6.

Let us recall the basic tame estimates concerning the Sobolev spaces. In the
next lemma, we use the following notations: For
$\alpha\in(\mathbb{N}\cup\left\\{0\right\\})^{2}$,

$\alpha=(\alpha_{1},\alpha_{2}),\quad|\alpha|:=\alpha_{1}+\alpha_{2},\quad\partial^{\alpha}:=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}.$

######  Lemma 3.4.

[castro2019global, Lemma 4.2] Let $f,g\in C^{\infty}(\Omega)$. Then, for
$\alpha,\beta\in(\mathbb{N}\cup\left\\{0\right\\})^{2}$, we have

$\displaystyle\rVert\partial^{\alpha}f\partial^{\beta}g\rVert_{L^{2}}$
$\displaystyle\leq_{\alpha,\beta}\rVert f\rVert_{H^{|\alpha|+|\beta|}}\rVert
g\rVert_{L^{\infty}}+\rVert g\rVert_{H^{|\alpha|+|\beta|}}\rVert
f\rVert_{L^{\infty}},$
$\displaystyle\rVert\partial^{\alpha}(fg)-f\partial^{\alpha}g\rVert_{L^{2}}$
$\displaystyle\leq_{\alpha,\beta}\rVert f\rVert_{H^{|\alpha|}}\rVert
g\rVert_{L^{\infty}}+\rVert f\rVert_{W^{1,\infty}}\rVert
g\rVert_{H^{|\alpha|-1}}.$

######  Lemma 3.5.

For $f\in H^{1}(\Omega)$ and
$\overline{f}(x_{2}):=\frac{1}{2\pi}\int_{\mathbb{T}}f(x_{1},x_{2})dx_{1}$, it
holds that

$\rVert
f-\overline{f}\rVert_{L^{\infty}}\leq_{C}\rVert\partial_{1}f\rVert_{H^{1}}.$

###### Proof.

We notice the following pointwise estimate:

$\displaystyle|(f-\overline{f})(x)|^{2}=\left(\frac{1}{2\pi}\int_{\mathbb{T}}f(x_{1},x_{2})-f(z,x_{2})dz\right)^{2}\leq
C\int_{\mathbb{T}}(f(x_{1},x_{2})-f(z,x_{2}))^{2}dz.$ (3.10)

The integrand in the right-hand side can be written as
$f(x_{1},x_{2})-f(z,x_{2})=\int_{x_{1}}^{z}\partial_{1}f(a,x_{2})da.$ For each
fixed $x_{1}$, we apply the Sobolev embedding $H^{1}([0,1])\hookrightarrow
L^{\infty}([0,1])$ to the map
$x_{2}\mapsto\int_{x_{1}}^{z}\partial_{1}f(a,x_{2})da$, yielding that

$\sup_{x_{2}\in[0,1]}\left|\int_{x_{1}}^{z}\partial_{1}f(a,x_{2})da\right|^{2}\leq
C\int_{0}^{1}\int_{x_{1}}^{z}|\partial_{12}f(a,y)|^{2}+|\partial_{1}f(a,y)|^{2}dady\leq\rVert\partial_{1}f\rVert_{H^{1}}^{2}.$

Therefore, taking the supremum over $x\in\Omega$ in (3.10), the desired result
follows. ∎

######  Proposition 3.6.

Let $\theta_{0}\in C^{\infty}_{0}(\Omega)$ and $\theta(t)\in
C(0,T;X^{\infty}(\Omega))$ be the unique smooth solution to (3.9) for some
$T>0$. For $k\geq 3$, it holds that

$\displaystyle\frac{d}{dt}\left(\rVert\partial_{1}^{k}\theta\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{k}\theta\rVert_{L^{2}}^{2}\right)$
$\displaystyle\leq-
C_{k}(1-C_{k}\rVert\theta\rVert_{H^{k}})\rVert\nabla\Psi\rVert_{{H}^{k}}^{2}+C_{k}\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}.$

###### Proof.

In the proof, the implicit constant $C$ may depend on $k$, but its dependence
will be omitted for simplicity. In what follows $\partial_{i}$ will denote
either $\partial_{1}$ or $\partial_{2}$. Using (3.9), we compute

$\displaystyle\frac{1}{2}\frac{d}{dt}\rVert\partial_{i}^{k}\theta\rVert_{L^{2}}^{2}=-\int\partial_{i}^{k}(u\cdot\nabla\theta)\partial_{i}^{k}\theta
dx+\int\partial_{i}^{k}u_{2}\partial_{i}^{k}\theta dx$ (3.11)

We simplify the linear term first. Recalling from (3.9) that
$u_{2}=\partial_{1}\Psi$ and $\partial_{1}\theta=-\Delta\Psi$, we have

$\displaystyle\int\partial_{i}^{k}u_{2}\partial_{i}^{k}\theta dx$
$\displaystyle=\int\partial_{1}\partial_{i}^{k}\Psi\partial_{i}^{k}\theta
dx=-\int\partial_{i}^{k}\Psi\partial_{i}^{k}\partial_{1}\theta
dx=\int\partial_{i}^{k}\Psi\partial_{i}^{k}\Delta\Psi dx$
$\displaystyle=\int_{\partial\Omega}\partial_{i}^{k}\Psi\nabla(\partial_{i}^{k}\Psi)\cdot\vec{n}(x)d\sigma(x)-\int_{\Omega}|\nabla\partial_{i}^{k}\Psi|^{2}dx=-\int_{\Omega}|\nabla\partial_{i}^{k}\Psi|^{2}dx,$

where the last equality follows from (3.5), which ensures that the integral
over $\partial\Omega$ vanishes. Since $\theta\in X^{\infty}(\Omega)$, it
follows from Lemma 3.2 and Lemma 3.1 that
$\sum_{i=1,2}\int_{\Omega}|\nabla\partial_{i}^{k}\Psi|^{2}dx\geq_{C}\rVert\nabla\Psi\rVert_{\dot{H}^{k}}^{2}$.
Since $\Psi=0$ on $\partial\Omega$, the Poincaré inequality gives us
$\rVert\nabla\Psi\rVert_{\dot{H}^{k}}^{2}\geq_{C}\rVert\nabla\Psi\rVert_{H^{k}}$,
consequently,

$\displaystyle\sum_{i=1,2}\int\partial_{i}^{k}u_{2}\partial_{i}^{k}\theta
dx\leq_{C}-\rVert\nabla\Psi\rVert_{{H}^{k}}^{2}.$ (3.12)

Now, we move on to the nonlinear term. We write

$\displaystyle\int\partial_{i}^{k}(u\cdot\nabla\theta)\partial_{i}^{k}\theta
dx$
$\displaystyle=\int\left(\partial_{i}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{i}^{k}\theta\right)\partial_{i}^{k}\theta
dx+\int u\cdot\nabla\partial_{i}^{k}\theta\partial_{i}^{k}\theta dx$
$\displaystyle=\int\left(\partial_{i}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{i}^{k}\theta\right)\partial_{i}^{k}\theta
dx+\int u\cdot\nabla\left(\frac{1}{2}(\partial_{i}^{k}\theta)^{2}\right)dx$
$\displaystyle=\int\left(\partial_{i}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{i}^{k}\theta\right)\partial_{i}^{k}\theta
dx,$ (3.13)

where the last equality follows from the integration by parts and
$u\cdot\vec{n}=0$ on the boundary. We claim that

$\displaystyle\left|\int\left(\partial_{i}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{i}^{k}\theta\right)\partial_{i}^{k}\theta
dx\right|\leq C\left(\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert\nabla\Psi\rVert_{{H}^{k}}^{2}\rVert\theta\rVert_{H^{k}}\right),$
(3.14)

either $i=1$ or $i=2$. Once we have the above claim, combining it with (3.12)
yields the desired energy estimate. Thus, in the rest of the proof, we will
aim to prove the claim (3.14). We consider two cases, $i=1$ and $i=2$,
separately.

Case $i=1$. The Cauchy-Schwarz inequality gives us

$\displaystyle\left|\int\left(\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\right)\partial_{1}^{k}\theta
dx\right|\leq_{C}\rVert\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\rVert_{L^{2}}\rVert\partial_{1}^{k}\theta\rVert_{L^{2}},$
(3.15)

while Lemma 3.4 tells us that

$\rVert\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\rVert_{L^{2}}\leq
C\left(\rVert u\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}+\rVert
u\rVert_{H^{k}}\rVert\nabla\theta\rVert_{L^{\infty}}\right).$

Since $k\geq 3$, the Sobolev inequality gives us

$\displaystyle\rVert
u\rVert_{W^{1,\infty}}=\rVert\nabla\Psi\rVert_{W^{1,\infty}}\leq
C\rVert\nabla\Psi\rVert_{H^{k}},\text{ and
}\rVert\nabla\theta\rVert_{L^{\infty}}\leq C\rVert\theta\rVert_{H^{k}}.$
(3.16)

Hence we have
$\rVert\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\rVert_{L^{2}}\leq
C\rVert\nabla\Psi\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}}$. Plugging this into
(3.15), we obtain

$\left|\int\left(\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\right)\partial_{1}^{k}\theta
dx\right|\leq
C\rVert\nabla\Psi\rVert_{H^{k}}\rVert\partial_{1}^{k}\theta\rVert_{L^{2}}\rVert\theta\rVert_{H^{k}}.$

Furthermore, from the Poisson equation in (3.9), we find that

$\rVert\partial_{1}^{k}\theta\rVert_{L^{2}}=\rVert\partial_{1}^{k-1}\Delta\Psi\rVert_{L^{2}}\leq
C\rVert\nabla\Psi\rVert_{H^{k}},$

therefore, we conclude

$\displaystyle\left|\int\left(\partial_{1}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{1}^{k}\theta\right)\partial_{1}^{k}\theta
dx\right|\leq C\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}},$
(3.17)

which verifies (3.14).

Case $i=2$. Splitting $u\cdot\nabla=u_{1}\partial_{1}+u_{2}\partial_{2}$, we
have

$\displaystyle\int\left(\partial_{2}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{2}^{k}\theta\right)\partial_{2}^{k}\theta
dx$
$\displaystyle=\int\left(\partial_{2}^{k}(u_{1}\partial_{1}\theta)-u_{1}\partial_{2}^{k}\partial_{1}\theta\right)\partial_{2}^{k}\theta
dx+\int\left(\partial_{2}^{k}(u_{2}\partial_{2}\theta)-u_{2}\partial_{2}^{k}\partial_{2}\theta\right)\partial_{2}^{k}\theta
dx$ $\displaystyle=:I_{1}+I_{2}.$ (3.18)

The first integral $I_{1}$ can be estimated as before; applying the Cauchy-
Schwarz inequality, we get

$\displaystyle|I_{1}|\leq
C\rVert\partial_{2}^{k}(u_{1}\partial_{1}\theta)-u_{1}\partial_{2}^{k}\partial_{1}\theta\rVert_{L^{2}}\rVert\partial_{2}^{k}\theta\rVert_{L^{2}},$
(3.19)

while Lemma 3.4 gives us

$\displaystyle\rVert\partial_{2}^{k}(u_{1}\partial_{1}\theta)-u_{1}\partial_{2}^{k}\partial_{1}\theta\rVert_{L^{2}}$
$\displaystyle\leq C\left(\rVert
u_{1}\rVert_{W^{1,\infty}}\rVert\partial_{1}\theta\rVert_{H^{k-1}}+\rVert
u_{1}\rVert_{H^{k}}\rVert\partial_{1}\theta\rVert_{L^{\infty}}\right)$
$\displaystyle\leq
C\rVert\nabla\Psi\rVert_{H^{k}}\left(\rVert\partial_{1}\theta\rVert_{H^{k-1}}+\rVert\partial_{1}\theta\rVert_{L^{\infty}}\right),$

where we used the estimate for $u$ in (3.16) to get the second inequality.
Again, using the Sobolev inequality and the Poisson equation in (3.9), we
estimate

$\rVert\partial_{1}\theta\rVert_{L^{\infty}}\leq_{C}\rVert\partial_{1}\theta\rVert_{H^{k-1}}=\rVert\Delta\Psi\rVert_{H^{k-1}}\leq\rVert\nabla\Psi\rVert_{H^{k}},$

which gives us
$\rVert\partial_{2}^{k}(u_{1}\partial_{1}\theta)-u_{1}\partial_{2}^{k}\partial_{1}\theta\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}^{2}$.
Plugging this into (3.19), we conclude

$\displaystyle|I_{1}|\leq
C\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\partial_{2}^{k}\theta\rVert_{L^{2}}\leq
C\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}.$ (3.20)

Now, let us estimate $I_{2}$ in (3.2). We denote

$\displaystyle\overline{\theta}(x_{2}):=\frac{1}{2\pi}\int_{\mathbb{T}}\theta(x_{1},x_{2})dx_{1}.$
(3.21)

We split $I_{2}$ as

$\displaystyle
I_{2}=\int\left(\partial_{2}^{k}(u_{2}\partial_{2}(\theta-\overline{\theta}))-u_{2}\partial_{2}^{k}\partial_{2}(\theta-\overline{\theta})\right)\partial_{2}^{k}\theta
dx+\int\left(\partial_{2}^{k}(u_{2}\partial_{2}\overline{\theta})-u_{2}\partial_{2}^{k}\partial_{2}\overline{\theta}\right)\partial_{2}^{k}\theta
dx=:I_{21}+I_{22}$ (3.22)

We estimate $I_{21}$ first. In a similar manner as above, the Cauchy-Schwarz
inequality and the Sobolev inequalities yield

$\displaystyle|I_{21}|$ $\displaystyle\leq_{C}\left(\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\partial_{2}(\theta-\overline{\theta})\rVert_{H^{k-1}}+\rVert
u_{2}\rVert_{H^{k}}\rVert\partial_{2}(\theta-\overline{\theta})\rVert_{L^{\infty}}\right)\rVert\theta\rVert_{H^{k}}$
$\displaystyle\leq_{C}\left(\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert(\theta-\overline{\theta})\rVert_{H^{k}}+\rVert
u_{2}\rVert_{H^{k}}\rVert\partial_{2}(\theta-\overline{\theta})\rVert_{L^{\infty}}\right)\rVert\theta\rVert_{H^{k}}$
$\displaystyle\leq_{C}\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert
u_{2}\rVert_{H^{k}}\rVert\partial_{2}(\theta-\overline{\theta})\rVert_{L^{\infty}}\rVert\theta\rVert_{H^{k}}.$
(3.23)

Moreover, Lemma 3.5 implies

$\rVert\partial_{2}(\theta-\overline{\theta})\rVert_{L^{\infty}}\leq_{C}\rVert\partial_{12}\theta\rVert_{H^{1}}\leq_{C}\rVert\partial_{2}\Delta\Psi\rVert_{L^{2}}\leq_{C}\rVert\nabla\Psi\rVert_{H^{k}},$

where the last inequality follows from $k\geq 3$. Plugging this and $\rVert
u\rVert_{H^{k}}\leq\rVert\nabla\Psi\rVert_{H^{k}}$ into (3.2), we obtain

$\displaystyle|I_{21}|\leq C\left(\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}\right).$
(3.24)

Next, we estimate $I_{22}$ in (3.22). By expanding $I_{22}$ using the product
rule, we have

$\displaystyle
I_{22}=\sum_{j=1}^{k}C_{k,j}\int\partial_{2}^{j}u_{2}\partial_{2}^{k-j+1}\overline{\theta}\partial_{2}^{k}\theta
dx.$ (3.25)

When $j=1$, we have

$\displaystyle\int\partial_{2}u_{2}\partial_{2}^{k}\overline{\theta}\partial_{2}^{k}\theta
dx\leq_{C}\rVert\partial_{2}u_{2}\rVert_{L^{\infty}}\rVert\overline{\theta}\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}}\leq_{C}\rVert\partial_{2}u_{2}\rVert_{L^{\infty}}\rVert\theta\rVert_{H^{k}}^{2}.$
(3.26)

For $j\geq 2$, noting that $u_{2}=\partial_{1}\Psi$, we can apply the
integration by parts in each integral as

$\displaystyle\int\partial_{2}^{j}u_{2}\partial_{2}^{k-j+1}\overline{\theta}\partial_{2}^{k}\theta
dx$
$\displaystyle=-\int\partial_{2}^{j}\Psi\partial_{2}^{k-j+1}\overline{\theta}dx\partial_{1}\partial_{2}^{k}\theta
dx$
$\displaystyle=-\int_{\partial\Omega}\partial_{2}^{j}\Psi\partial_{2}^{k-j+1}\overline{\theta}\partial_{1}\partial_{2}^{k-1}\theta
d\sigma(x)+\int_{\Omega}\partial_{2}\left(\partial_{2}^{j}\Psi\partial_{2}^{k-j+1}\overline{\theta}\right)\partial_{1}\partial_{2}^{k-1}\theta
dx$
$\displaystyle=\int_{\Omega}\partial_{2}\left(\partial_{2}^{j}\Psi\partial_{2}^{k-j+1}\overline{\theta}\right)\partial_{1}\partial_{2}^{k-1}\theta
dx.$

To see the last inequality, note that by Lemma 3.2, we have that
$\Psi,\overline{\theta},\partial_{1}\theta$ are all in $X^{\infty}$. Since at
least one of $j,k-j+1,k-1$ must be even, the definition of the space
$X^{k}(\Omega)$ in (3.2) tells us that the boundary integral must vanish.

To continue, we apply the Cauchy-Schwarz inequality to get

$\displaystyle\left|\int\partial_{2}^{j}u_{2}\partial_{2}^{k-j+1}\overline{\theta}\partial_{2}^{k}\theta
dx\right|$
$\displaystyle=\left|\int_{\Omega}\partial_{2}\left(\partial_{2}^{j}\Psi\partial_{2}^{k-j+1}\overline{\theta}\right)\partial_{1}\partial_{2}^{k-1}\theta
dx\right|$
$\displaystyle\leq\left(\rVert\partial_{2}^{j+1}\Psi\partial_{2}^{k-j+1}\overline{\theta}\rVert_{L^{2}}+\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}\right)\rVert\partial_{1}\partial_{2}^{k-1}\theta\rVert_{L^{2}}$
$\displaystyle\leq\left(\rVert\partial_{2}^{j+1}\Psi\partial_{2}^{k-j+1}\overline{\theta}\rVert_{L^{2}}+\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}\right)\rVert\nabla\Psi\rVert_{H^{k}},\text{
for $j\geq 2$},$ (3.27)

where we used
$\rVert\partial_{1}\partial_{2}^{k-1}\theta\rVert_{L^{2}}=\rVert\partial_{2}^{k-1}\Delta\Psi\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}$
to get the last inequality.

Let us estimate
$\rVert\partial_{2}^{j+1}\Psi\partial_{2}^{k-j+1}\overline{\theta}\rVert_{L^{2}}$
in (3.2). Since $2\leq j\leq k$, we have

$\displaystyle\rVert\partial_{2}^{j+1}\Psi\partial_{2}^{k-j+1}\overline{\theta}\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}\rVert\partial_{2}^{k-j+1}\overline{\theta}\rVert_{L^{\infty}}\leq\rVert\nabla\Psi\rVert_{H^{k}(\Omega)}\rVert\overline{\theta}\rVert_{H^{k-j+2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}},$
(3.28)

where the second last inequality follows from the Sobolev inequality, noticing
that $\overline{\theta}$ depends only on the variable $x_{2}$.

Let us estimate
$\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}$
in (3.2). When $j=k$, we have

$\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}\leq_{C}\rVert\Psi\rVert_{H^{k}}\rVert\partial_{22}\overline{\theta}\rVert_{L^{\infty}}\leq_{C}\rVert\nabla\Psi\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}}.$

When $j\leq k-1$, we have

$\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}\leq\rVert\partial_{2}^{j}\Psi\rVert_{L^{\infty}}\rVert\partial_{2}^{k-j+2}\theta\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}}.$

where in the the last inequality, we used the Sobolev inequality and $j\geq
2$. Thus, we obtain

$\rVert\partial_{2}^{j}\Psi\partial_{2}^{k-j+2}\overline{\theta}\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{k}}\rVert\theta\rVert_{H^{k}},\text{
for all $j\geq 2$.}$

Plugging this and (3.28) into (3.2), we get

$\left|\int\partial_{2}^{j}u_{2}\partial_{2}^{k-j+1}\overline{\theta}\partial_{2}^{k}\theta
dx\right|\leq
C\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}\text{ for $j\geq
2$.}$

Combining this with (3.26) and plugging them into (3.25), we see that

$|I_{22}|\leq_{C}\rVert\partial_{2}u_{2}\rVert_{L^{\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}.$

Plugging this and (3.24) into (3.22), we get

$|I_{2}|\leq_{C}\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}.$

Plugging this and (3.20) into (3.2), we conclude

$\left|\int\left(\partial_{2}^{k}(u\cdot\nabla\theta)-u\nabla\partial_{2}^{k}\theta\right)\partial_{2}^{k}\theta
dx\right|\leq_{C}\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\theta\rVert_{H^{k}}^{2}+\rVert\nabla\Psi\rVert_{H^{k}}^{2}\rVert\theta\rVert_{H^{k}}.$

Combining this with (3.17), we obtain (3.14). ∎

###  3.3. Analysis of the energy structure

The main objective in this subsection is to derive sufficient convergence rate
of $\rho(t)$ to the equilibrium $\rho_{0}^{*}$, while $\rho$ stays close to
${\rho}_{s}$. We emphasize that $\theta(t)=\rho(t)-\rho_{s}$ will stay small
but not necessarily decay. Thus it is important to distinguish the roles of
$\theta(t)$ and $\rho(t)-\rho_{0}^{*}$. We will consider the potential energy
and the kinetic energy defined as

$\displaystyle E(t):=\int_{\Omega}(\rho(t)-\rho_{0}^{*})x_{2}dx,\quad
K(t):=\rVert u(t)\rVert_{L^{2}}^{2}.$ (3.29)

Since we are interested in a solution that is close to $\rho_{s}$, we will
assume, throughout this subsection, that

$\displaystyle\rVert\theta(t)\rVert_{H^{3}}^{2}+\int_{0}^{T}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt\leq\delta\leq\delta_{0},\text{
for $t\in[0,T]$ for some $T>0$ and $\delta_{0}\ll 1$.}$ (3.30)

######  Proposition 3.7.

Let $k\geq 3$. There exists $\delta_{0}=\delta_{0}(k)>0$ such that if
$\rho(t)$ satisfies (3.30), then

$\displaystyle\frac{d}{dt}E(t)=-K(t),\quad\frac{d}{dt}K(t)\leq-C_{k}\rVert
u_{2}\rVert_{L^{2}}^{2}.$ (3.31)

###### Proof.

Differentiating $E(t)$, we see that

$\displaystyle\frac{d}{dt}E(t)=\int_{\Omega}\rho_{t}x_{2}dx=-\int_{\Omega}u\cdot\nabla\rho
x_{2}dx=\int_{\Omega}u_{2}\rho dx.$ (3.32)

Using (1.3), we can further simplify the last expression as

$\int_{\Omega}u_{2}\rho dx=\int_{\Omega}\partial_{1}\Psi\rho
dx=-\int_{\Omega}\Psi\partial_{1}\rho dx=\int_{\Omega}\Psi\Delta\Psi
dx=-\int_{\Omega}|\nabla\Psi|^{2}dx,$

therefore, we get

$\displaystyle\frac{d}{dt}E(t)=-\int_{\Omega}|\nabla\Psi|^{2}dx=-K(t).$ (3.33)

In order to estimate $\frac{d}{dt}K(t)$, we differentiate (3.32) and obtain

$\frac{1}{2}\left(\frac{d}{dt}\right)^{2}E(t)=\frac{1}{2}\frac{d}{dt}\int_{\Omega}u_{2}\rho
dx=\frac{1}{2}\int_{\Omega}u_{2}\rho_{t}dx+\frac{1}{2}\int_{\Omega}\partial_{t}u_{2}\rho
dx.$

Using $u_{2}=\partial_{1}\Psi$ and $\partial_{1}\rho=\Delta\Psi$, we have

$\int_{\Omega}\partial_{t}u_{2}\rho
dx=-\int_{\Omega}\partial_{t}\Psi\partial_{1}\rho
dx=\int_{\Omega}\partial_{t}\Psi\Delta\Psi
dx=\int_{\Omega}\partial_{t}\Delta\Psi\Psi
dx=-\int_{\Omega}{\partial_{t}}\partial_{1}\rho\Psi
dx=\int_{\Omega}\partial_{t}\rho u_{2}dx$

Therefore, we get

$\displaystyle\frac{1}{2}\left(\frac{d}{dt}\right)^{2}E(t)$
$\displaystyle=\int_{\Omega}u_{2}\rho_{t}=-\int_{\Omega}u_{2}u\cdot\nabla\rho
dx=-\int_{\Omega}u_{2}^{2}\partial_{2}\rho
dx-\int_{\Omega}u_{2}u_{1}\partial_{1}\rho dx$
$\displaystyle\geq-\int_{\Omega}u_{2}^{2}\partial_{2}\rho dx-\rVert
u_{2}\rVert_{L^{2}}\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}}.$

Using the assumption on $\rVert\theta\rVert_{H^{3}}$ in (3.30) and (3.1), we
have

$-\int_{\Omega}u_{2}^{2}\partial_{2}\rho dx=-\int
u_{2}^{2}\partial_{2}\rho_{s}dx-\int u_{2}^{2}\partial_{2}\theta dx\geq\rVert
u_{2}\rVert_{L^{2}}^{2}-\sqrt{\delta_{0}}\rVert u_{2}\rVert_{L^{2}}^{2}\geq
C\rVert u_{2}\rVert_{L^{2}}^{2}.$

Using (3.33), we see that the above inequality implies

$\displaystyle C\frac{d}{dt}K(t)+\rVert u_{2}\rVert_{L^{2}}^{2}\leq C\rVert
u_{2}\rVert_{L^{2}}\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}}.$ (3.34)

Let us estimate $\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}(\Omega)}$. Using
(1.3), we rewrite

$\displaystyle\rVert
u_{1}\partial_{1}\rho\rVert_{L^{2}(\Omega)}=\rVert\partial_{2}\Psi\Delta\Psi\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{L^{4}}\rVert\Delta\Psi\rVert_{L^{4}}\leq\rVert\nabla\Psi\rVert_{H^{3}}\rVert\Psi\rVert_{L^{2}},$
(3.35)

where the last inequality is due to the Gagliardo-Nirenberg interpolation
theorem. Moreover, we notice from (1.3) that
$g(x_{2}):=\int_{\mathbb{T}}\Psi(x_{1},x_{2})dx_{1}$ satisfies

$\partial_{22}g(x_{2})=\int_{\mathbb{T}}\partial_{22}\Psi(x_{1},x_{2})dx_{1}=\int_{\mathbb{T}}\Delta\Psi(x_{1},x_{2})-\partial_{11}\Psi(x_{1},x_{2})dx_{1}=\int_{\mathbb{T}}\partial_{1}(-\rho+\partial_{1}\Psi)dx=0,$

with the boundary condition, $g(0)=g(1)=0$. Therefore, $g=0$ for all
$x_{2}\in[0,1]$. In other words, the map $x_{1}\to\Psi(x_{1},x_{2})$ has zero
average for each fixed $x_{2}$. Then, the Poincaré inequality tells us

$\rVert\Psi\rVert_{L^{2}}\leq\rVert\partial_{1}\Psi\rVert_{L^{2}}=\rVert
u_{2}\rVert_{L^{2}}.$

Plugging this into (3.35), we get

$\rVert
u_{1}\partial_{1}\rho\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{H^{3}}\rVert
u_{2}\rVert_{L^{2}}\leq\rVert\theta\rVert_{H^{3}}\rVert
u_{2}\rVert_{L^{2}}\leq\sqrt{\delta_{0}}\rVert u_{2}\rVert_{L^{2}},$

where the second inequality follows from Lemma 3.2 and the last inequality
follows from (3.30). Plugging this into (3.34), we obtain that for
sufficiently small $\delta_{0}>0$, $\frac{d}{dt}K(t)\leq-C\rVert
u_{2}\rVert_{L^{2}}^{2}.$ Combining this with (3.33), we finish the proof of
the proposition. ∎

######  Corollary 3.8.

Let $k\geq 3$. There exists $\delta_{0}=\delta_{0}(k)>0$ such that if (3.30)
holds, then,

$\displaystyle E(t)$ $\displaystyle\leq\frac{C\delta}{t^{k}},$
$\displaystyle\frac{2}{t}\int_{t/2}^{t}K(s)ds$
$\displaystyle\leq\frac{C\delta}{t^{k+1}},$
$\displaystyle\frac{2}{t}\int_{t/2}^{t}\rVert u_{2}(s)\rVert_{L^{2}}^{2}ds$
$\displaystyle\leq\frac{C\delta}{t^{k+2}},$

for all $t\in[0,T]$.

###### Proof.

Thanks to Proposition 2.5, it holds that

$\displaystyle C^{-1}\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}^{2}\leq E(t)\leq
C\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}^{2}.$ (3.36)

Now, using the Gagliardo-Nirenberg interpolation theorem, we observe that

$\rVert\partial_{1}\rho\rVert_{L^{2}(\Omega)}=\rVert\Delta\Psi\rVert_{L^{2}}\leq
C\rVert\nabla\Psi\rVert_{H^{k}}^{1/k}\rVert\nabla\Psi\rVert_{L^{2}}^{(k-1)/k}.$

On the other hand, applying (2.27), we get
$\rVert\partial_{1}\rho\rVert_{L^{2}}\geq
C\rVert\rho-\rho_{0}^{*}\rVert_{L^{2}}$. Combining this with the above
estimate, we find

$\rVert\nabla\Psi\rVert_{L^{2}}^{2}\geq
C\left(\rVert\nabla\Psi\rVert_{H^{k}}^{-1/k}\rVert\rho-\rho_{0}^{*}\rVert_{L^{2}}\right)^{2k/(k-1)}\geq
C\rVert\nabla\Psi\rVert_{H^{k}}^{-2/(k-1)}E(t)^{k/(k-1)}.$

where the last inequality follows from (3.36). Hence, the variation of the
potential energy in (3.31) must satisfy

$\displaystyle\frac{d}{dt}E(t)\leq-C\left(\rVert\nabla\Psi\rVert_{H^{k}}^{2}\right)^{-1/(k-1)}E(t)^{k/(k-1)}.$
(3.37)

Applying Lemma 2.1 with $\alpha=1/(k-1)>0$ and $n=k/(k-1)$ and
$a(t)=\rVert\nabla\Psi\rVert_{H^{k}}^{2}$, we get

$\displaystyle E(t)$ $\displaystyle\leq\frac{C\delta}{t^{k}}.$ (3.38)

Applying Lemma 2.2 to (3.31) with $f(t)=E(t),\ g(t)=K(t),\ h(t)=C\rVert
u_{2}(t)\rVert_{L^{2}}^{2}$, we get

$\frac{2}{t}\int_{t/2}^{t}K(s)ds\leq\frac{C}{t^{k+1}},\text{ and
}\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq\frac{C\delta}{t^{k+2}}.$

Together with (3.38), we obtain the desired estimates. ∎

###  3.4. Proof of Theorem 1.1

Let $k\geq 3$ be fixed. Let $\delta_{0}$ be be fixed as in Proposition 3.7 .
We claim that there exists $\epsilon_{0}(k)>0$ such that if
$\theta_{0}:=\rho_{0}-{\rho}_{s}\in C^{\infty}_{0}(\Omega)$ and

$\displaystyle\rVert\rho_{0}-{\rho}_{s}\rVert_{H^{k}}\leq\epsilon\leq\epsilon_{0},$

then for all $t>0$, it holds that

$\displaystyle\rVert\theta(t)\rVert_{H^{k}}^{2}+\int_{0}^{t}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt<C\epsilon^{2},\text{
for some $C=C(k)>0$.}$ (3.39)

Let us suppose for the moment that the claim is true. From Theorem 3.3, we
know that the maximal existence time depends only on
$\rVert\theta(t)\rVert_{H^{3}}$, thus, the solution $\rho$ exists globally in
time. Also, if necessary, we can further assume that $\epsilon$ is small
enough to ensure Corollary 3.8 is applicable, that is, the solution $\rho(t)$
satisfies

$\displaystyle
E(t)\leq\frac{C\epsilon^{2}}{t^{k}},\quad\frac{2}{t}\int_{t/2}^{t}K(s)ds\leq\frac{C\epsilon^{2}}{t^{k+1}},\quad\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq\frac{C\epsilon^{2}}{t^{k+2}},\text{ for all
$t>0$}.$ (3.40)

In order to prove the theorem for $\rho_{0}\in H^{k}_{0}$ without assuming the
smoothness, we can simply find an approximation $\rho_{0,n}$ such that
$\rho_{0,n}-\overline{\rho}\in C^{\infty}_{0}(\Omega)$ such that
$\rho_{0,n}\to\rho_{0}$ in $H^{k}$. Then (3.40) and (3.39) are satisfied by
the global solutions $\rho_{n}(t)$, with initial data $\rho_{0,n}$. Since all
estimates are uniform in $n$, the unique limit
$\rho(t):=\lim_{n\to\infty}\rho_{n}(t)$ is the global in time solution
satisfying the same estimates, that is,

$\rVert\rho(t)-\rho_{s}\rVert_{H^{k}}^{2}\leq\epsilon^{2}\quad\int_{\Omega}(\rho(t)-\rho_{0}^{*})(x)x_{2}dx\leq
C\epsilon^{2}t^{-k},\text{ for all $t>0$}.$

Together with (2.26), we obtain all the desired estimates to establish Theorem
1.1.

In the rest of the proof, we aim to prove (3.39). Towards a contradiction, let
$T^{*}>0$ be the first time that (3.39) breaks down, that is,

$\displaystyle\rVert\theta(T^{*})\rVert_{H^{k}(\Omega)}^{2}+\int_{0}^{T^{*}}\rVert\nabla\Psi(T^{*})\rVert_{H^{k}(\Omega)}^{2}dt=M\epsilon^{2}\ll
1,$ (3.41)

for some $M>0$, which will be chosen later. Towards a contradiction, let us
denote

$f(t):=\rVert\partial_{1}^{k}\theta\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{k}\theta\rVert_{L^{2}}^{2}.$

Since $\theta_{0}=\rho_{0}-\overline{\rho}\in C^{\infty}_{0}(\Omega)$, we have
$\theta_{0}\in X^{\infty}(\Omega)$. Therefore it follows from Theorem 3.3 that
$\theta(t)\in X^{\infty}(\Omega)$ for $t\in[0,T^{*}]$. Since $\theta(t)$
vanishes on the boundary, Lemma 3.1 tells us that

$\displaystyle C^{-1}\rVert\theta(t)\rVert_{H^{k}}^{2}\leq f(t)\leq
C\rVert\theta(t)\rVert_{H^{k}}^{2}\leq CM\epsilon^{2},\quad f(0)\leq
C\epsilon^{2}.$ (3.42)

Using the Sobolev embedding $W^{1,\infty}(\Omega)\hookrightarrow
H^{k}(\Omega)$ for $k\geq 3$, we have

$\displaystyle\rVert u_{2}\rVert_{W^{1,\infty}}\leq\rVert
u_{2}\rVert_{H^{k}}\leq\rVert\nabla\Psi\rVert_{H^{k}}\rVert\leq\rVert\theta\rVert_{H^{k}}\leq
f(t)^{1/2},$ (3.43)

where the second last inequality follows from (3.2). Using this and
Proposition 3.6, we observe that $f(t)$ satisfies

$\displaystyle\frac{d}{dt}f(t)\leq-C\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}+Cf(t)^{3/2},\text{
for $t\in[0,T^{*}]$.}$ (3.44)

From this we see that $f$ satisfies $\frac{d}{dt}f(t)\leq Cf^{3/2}$. Since
$f(0)\leq C\epsilon^{2}$, one can easily find that

$\displaystyle f(t)\leq\frac{C}{(\epsilon^{-1}-t)^{2}},\text{ for
$t\in[0,T^{*}]$.}$ (3.45)

This implies that for sufficiently small $\epsilon>0$, it holds that

$\displaystyle f(t)\leq C\epsilon^{2},\text{ for $t\in[0,2]$.}$ (3.46)

In other words, for (3.41) to happen, we must have

$\displaystyle T^{*}>2.$ (3.47)

Now we analyze the energy inequality in a more careful manner. Using the
Gagliardo-Nirenberg interpolation theorem, we see that

$\rVert u_{2}\rVert_{W^{1,\infty}}\leq\rVert u_{2}\rVert_{H^{k}}^{2/k}\rVert
u_{2}\rVert_{L^{2}}^{1-2/k}\leq\rVert\nabla\Psi\rVert_{H^{k}}^{2/k}\rVert
u_{2}\rVert_{L^{2}}^{1-2/k}.$

Hence, using Young’s inequality, we derive

$\displaystyle\rVert u_{2}\rVert_{W^{1,\infty}}f(t)$
$\displaystyle\leq\rVert\nabla\Psi\rVert_{H^{k}}^{2/k}\rVert
u_{2}\rVert_{L^{2}}^{1-2/k}f(t)$
$\displaystyle\leq\eta\rVert\nabla\Psi\rVert_{H^{k}}^{2}+C_{\eta}f(t)^{k/(k-1)}\rVert
u_{2}\rVert_{L^{2}}^{(k-2)/(k-1)}$
$\displaystyle\leq\eta\rVert\nabla\Psi\rVert_{H^{k}}^{2}+C_{\eta}(M\epsilon^{2})^{k/(k-1)}\rVert
u_{2}\rVert_{L^{2}}^{(k-2)/(k-1)},\text{ for any $\eta>0$}.$

Substituting this into (3.44) with sufficiently small $\eta$ depending only on
the implicit constant $C$, we arrive at

$f^{\prime}(t)\leq-C\rVert\nabla\Psi\rVert_{H^{k}}^{2}+C(M\epsilon^{2})^{k/(k-1)}\rVert
u_{2}\rVert_{L^{2}}^{(k-2)/(k-1)},$

Integrating the above in $t$ over $[1,T^{*}]$, we obtain

$\displaystyle
f(T^{*})-f(1)+C_{k}\int_{1}^{T^{*}}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt\leq
C_{k}(M\epsilon^{2})^{k/(k-1)}\int_{1}^{T^{*}}\left(\rVert
u_{2}(t)\rVert_{L^{2}}^{2}\right)^{\frac{k-2}{2(k-1)}}dt.$ (3.48)

Thanks to $M\epsilon^{2}\ll 1$ in (3.41), we apply Corollary 3.8 and obtain

$\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq\frac{CM\epsilon^{2}}{t^{k+2}},\text{ for all
$t\in[0,T^{*}]$.}$

Since $k\geq 3$, we have $n:=k+2>1$ and
$\alpha:=\frac{k-2}{2(k-1)}\in(1/n,1)$. Thus applying Lemma 2.3 with $f=\rVert
u_{2}\rVert_{L^{2}}^{2}$ and $E=CM\epsilon^{2}$, we get

$\int_{1}^{T^{*}}\left(\rVert
u_{2}(t)\rVert_{L^{2}}^{2}\right)^{\frac{k-2}{2(k-1)}}dt\leq
C(M\epsilon^{2})^{\frac{k-2}{2(k-1)}}.$

Plugging this and (3.46) into (3.48), we obtain

$f(T^{*})+C\int_{0}^{T^{*}}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt\leq
C\left(\int_{0}^{1}\rVert\nabla\Psi\rVert_{H^{k}}^{2}dt+\epsilon^{2}+(M\epsilon^{2})^{\frac{3k-2}{2(k-1)}}\right)\leq
C\epsilon^{2}+C(M\epsilon^{2})^{\frac{3k-2}{2(k-1)}},$

where we used (3.43) to justify the last inequality. Finally using
$f(T^{*})\geq C\rVert\theta\rVert_{H^{k}}^{2}$, which is due to the last
inequality in (3.43), we notice that for (3.41) to hold, we must have

$M\epsilon^{2}=\rVert\theta(T^{*})\rVert_{H^{k}}^{2}+\int_{0}^{T^{*}}\rVert\nabla\Psi(T^{*})\rVert_{H^{k}(\Omega)}^{2}dt\leq_{C}f(T^{*})+\int_{0}^{T^{*}}\rVert\nabla\Psi(t)\rVert_{H^{k}}^{2}dt\leq
C\epsilon^{2}+C(M\epsilon^{2})^{\frac{3k-2}{2(k-1)}}.$

Since $k\geq 3$, we have $\frac{3k-2}{2(k-1)}>1$. Therefore our assumption
$M\epsilon^{2}\ll 1$ in (3.41) and the above estimate give us

$M\epsilon^{2}\leq C\epsilon^{2}.$

This leads to a contradiction with (3.41), if $M$ is chosen strictly larger
than some implicit constant $C$, which depends only on $k$. This finishes the
proof of Theorem 1.1.

##  4\. Stability in the Stokes transport system

In this section, our goal is to prove the asymptotic stability for the Stokes
transport system (1.4). Throughout the section, we will assume that
$\rho_{s}=\rho_{s}(x_{2})$ is a function that is independent of $x_{1}$, and
satisfies

$\displaystyle\gamma:=\inf_{x\in\Omega}\left(-\partial_{2}\rho_{s}(x_{2})\right)>0,\quad\rVert\rho_{s}\rVert_{H^{4}(\Omega)}<\infty.$
(4.1)

In the rest of the paper, we allow the implicit constant $C$ to depend on
$\gamma$ and $\rVert\rho_{s}\rVert_{H^{4}}$, but we omit its dependence in the
notation for simplicity. While the proof will exhibit a structure quite
similar to that of the previous section, we will furnish enough details for
the sake of completeness of the paper.

###  4.1. Preliminaries for the Stokes transport system

Let us briefly review relevant results concerning the Stokes transport system
in the periodic channel For more details, we refer readers to the paper by
Dalibard–Guillod–Leblond [dalibard2023long] and additional references therein.

Let us consider a solution $\rho(t)$ to the Stokes transport system. As in the
previous section we denote

$\displaystyle\theta(t):=\rho(t)-\rho_{s}.$ (4.2)

Substituting $\rho=\theta+{\rho}_{s}$ in (1.4), it immediately follows that

$\displaystyle\begin{cases}\theta_{t}+u\cdot\nabla\theta=-\partial_{2}\rho_{s}u_{2},\\\
\nabla\cdot u=0,\\\ \end{cases}with\
\begin{cases}\Delta^{2}\Psi=\partial_{1}\theta,&\text{ in $\Omega$,}\\\
\Psi=\nabla\Psi=0,&\text{ on $\partial\Omega$}.\end{cases}$ (4.3)

Thanks to the Bilaplacian operator in the equation for the stream function,
the velocity in the Stokes transport system is much regular compared to that
of the IPM equation. More quantitative estimate can be found in the next
lemma.

######  Lemma 4.1.

[dalibard2023long, Lemma B.1] Let $f\in H^{k}(\Omega)$ for $k\geq-2$ and
$\Psi$ be a solution to

$\begin{cases}\Delta^{2}\Psi=f&\text{ in $\Omega$},\\\
\Psi=\nabla\Psi=0&\text{ on $\partial\Omega$.}\end{cases}$

Then $\Psi\in H_{0}^{2}(\Omega)\cap H^{k+4}(\Omega)$ and
$\rVert\Psi\rVert_{H^{k+4}}\leq C_{k}\rVert f\rVert_{H^{k}}$.

Thanks to more regular nature of the velocity, the global in time existence of
the solution follows in a standard way. More precise statement can be
formulated as below.

######  Theorem 4.2.

[dalibard2023long, Lemma 2.1, Theorem A.1] Let $\theta_{0}\in H^{k}(\Omega)$
with $k\geq 3$. Then there exists a unique global solution $\rho(t)$ to (1.4)
such that $\rho\in C(\mathbb{R}^{+};H^{m}(\Omega))$. Furthermore, denoting
$\theta:=\rho-{\rho}_{s}$ and
$\overline{\theta}(x_{2}):=\frac{1}{2\pi}\int_{\mathbb{T}}\theta(x_{1},x_{2})dx_{1}$,
it holds that if
$\theta_{0}=\partial_{n}\theta=\partial_{n}^{2}\overline{\theta}=0$ on
$\partial\Omega$, then

$\theta(t)=\partial_{2}\theta(t)=\partial_{2}^{2}\overline{\theta}(t)=0,\text{
on $\partial\Omega$.}$

In view of Theorem 1.6, we will consider initial data $\theta_{0}\in
H^{2}_{0}(\Omega)\cap H^{4}(\Omega)$. The usual trace theorem ensures that
$\theta,\partial_{2}\theta$ and $\partial_{22}\overline{\theta}$ are indeed
well-defined on the boundary $\partial\Omega$ and vanish identically as stated
in the above theorem.

###  4.2. Energy estimates

In this subsection, our objective is to derive an a priori energy estimate.
The main result is presented in Proposition 4.4. As one can notice from the
proposition, we will focus on estimating the evolution of
$\rVert\Delta^{2}\theta\rVert_{L^{2}}$, rather than directly examining
$\rVert\theta\rVert_{H^{4}}$. The reason is to mitigate potential
complications in the finer analysis of the anisotropic nature of the nonlinear
term in the equation. As expected, the norm
$\rVert\Delta^{2}\theta\rVert_{L^{2}}$ is equivalent to
$\rVert\theta\rVert_{H^{4}}$ for solutions vanishing on the boundary. Although
the immediate equivalence between these two norms may not be apparent through
elementary integration by parts, Lemma 4.1 provides a rigorous justification

######  Lemma 4.3.

If $f\in H^{4}(\Omega)$ satisfies $f=\partial_{2}f=0$ on $\partial\Omega$ and
$\int_{\Omega}f(x)dx=0$, then

$\rVert f\rVert_{{H}^{2}(\Omega)}\leq C\rVert\Delta
f\rVert_{L^{2}(\Omega)},\quad\rVert f\rVert_{H^{4}(\Omega)}\leq
C\rVert\Delta^{2}f\rVert_{L^{2}(\Omega)}.$

###### Proof.

The only problematic part is to ensure that the mixed derivatives can be
estimated by the Laplacian/Bilaplacian operator. The first inequality is
trivial since $\rVert\Delta
f\rVert_{L^{2}}^{2}=\rVert\partial_{11}f\rVert_{L^{2}}^{2}+\rVert\partial_{22}f\rVert_{L^{2}}^{2}+2\rVert\partial_{12}f\rVert_{L^{2}}^{2}$,
which is strong enough to control all the second derivatives of $f$. For the
second inequality, we simply apply Lemma 4.1 with $k=0$, yielding the desired
result. ∎

######  Proposition 4.4.

Let $\theta_{0}\in H^{2}_{0}(\Omega)\cap H^{4}(\Omega)$ and $\theta(t)$ be the
unique smooth solution to (4.3). It holds that

$\displaystyle\frac{d}{dt}\left(\rVert\Delta^{2}\theta\rVert_{L^{2}}^{2}\right)$
$\displaystyle\leq_{C}-C(1-C\rVert\Delta^{2}\theta\rVert_{L^{2}})\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{2}+\left(\rVert
u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}\theta\rVert_{L^{2}}^{2}$
$\displaystyle\ +\rVert
u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}+\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}.$

###### Proof.

Using (4.3), we compute

$\displaystyle\frac{1}{2}\frac{d}{dt}\rVert\Delta^{2}\theta\rVert_{L^{2}}^{2}=-\int\Delta^{2}(u\cdot\nabla\theta)\Delta^{2}\theta
dx+\int\Delta^{2}(-\partial_{2}\rho_{s}u_{2})\Delta^{2}\theta dx$ (4.4)

We simplify the linear term first. Let us write

$\int\Delta^{2}(-\partial_{2}\rho_{s}u_{2})\Delta^{2}\theta
dx=\int\left(\Delta^{2}(-\partial_{2}\rho_{s}u_{2})-(-\partial_{2}\rho_{s})\Delta^{2}u_{2}\right)\Delta^{2}\theta
dx+\int(-\partial_{2}\rho_{s})\Delta^{2}u_{2}\Delta^{2}\theta
dx=:A_{1}+A_{2}.$

For $A_{1}$, the usual tame estimate yields

$\displaystyle|A_{1}|$
$\displaystyle\leq_{C}\rVert\Delta^{2}(-\partial_{2}\rho_{s}u_{2})-(-\partial_{2}\rho_{s})\Delta^{2}u_{2}\rVert_{L^{2}}\rVert\Delta^{2}\theta\rVert_{L^{2}}$
$\displaystyle\leq_{C}\left(\rVert\partial_{2}\rho_{s}\rVert_{L^{\infty}}\rVert
u_{2}\rVert_{H^{3}}+\rVert\partial_{2}\rho_{s}\rVert_{H^{4}}\rVert
u_{2}\rVert_{L^{\infty}}\right)\rVert\Delta^{2}\theta\rVert_{L^{2}}$
$\displaystyle\leq_{C}\rVert
u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}.$ (4.5)

For $A_{2}$, we can write

$\displaystyle A_{2}$
$\displaystyle=\int-\partial_{2}\rho_{s}\partial_{1}\Delta^{2}\Psi\Delta^{2}\theta
dx=\int\partial_{2}\rho_{s}\partial_{1}\theta\partial_{1}\Delta^{2}\theta
dx=\int\Delta(\partial_{2}\rho_{s}\partial_{1}\theta)\partial_{1}\Delta\theta
dx$
$\displaystyle=\int\partial_{2}\rho_{s}|\partial_{1}\Delta\theta|^{2}dx+\int(\partial_{222}\rho_{s}\partial_{1}\theta+2\partial_{22}\rho_{s}\partial_{12}\theta)\partial_{1}\Delta\theta
dx=:A_{21}+A_{22},$ (4.6)

while $A_{22}$ can be estimated as

$\displaystyle|A_{22}|$
$\displaystyle\leq_{C}\rVert\partial_{222}\rho_{s}\rVert_{L^{\infty}}\rVert\partial_{1}\theta\rVert_{H^{1}}\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}\leq
C\rVert\partial_{1}\theta\rVert_{H^{1}}\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}$
$\displaystyle\leq_{C}\rVert\partial_{1}\theta\rVert_{L^{2}}^{1/2}\rVert\partial_{1}\theta\rVert_{H^{2}}^{3/2}\leq\rVert\partial_{1}\theta\rVert_{L^{2}}^{1/2}\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{3/2}\leq\frac{\inf(-\partial_{2}\rho_{s})}{4}A_{21}+C\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}$

where the last inequality follows from Young’s inequality. Thus, using (4.1),
we observe that (4.2) can be written as

$A_{2}\leq-\frac{3}{4}\inf(-\partial_{2}\rho_{s})\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{2}+C\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}\leq-C\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{2}+C\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}.$

Combining this with (4.2), we get

$\displaystyle\int\Delta^{2}(-\partial_{2}\rho_{s}u_{2})\Delta^{2}\theta dx$
$\displaystyle\leq-C\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{2}+C\left(\rVert
u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}+\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}\right).$
(4.7)

where the last equality follows from Lemma 4.2, which ensures that the
integral over $\partial\Omega$ appearing in the integration by parts vanishes.

Now we estimate the integral coming from the nonlinear term in (4.4). We
derive a necessary estimate in a separate lemma. Indeed, Lemma 4.5 gives us
that

$\displaystyle\left|\int\Delta^{2}(u\cdot\nabla\theta)\Delta^{2}\theta
dx\right|\leq_{C}\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}\theta\rVert_{L^{2}}^{2}+\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}\rVert\Delta^{2}\theta\rVert_{L^{2}}.$
(4.8)

Using Lemma 4.1,

$\rVert
u\rVert_{H^{5}}\leq_{C}\rVert\Psi\rVert_{H^{5}}\leq_{C}\rVert\partial_{1}\theta\rVert_{H^{1}}\leq_{C}\rVert\partial_{1}\Delta\theta\rVert_{L^{2}},$

where the last inequality follows from Lemma 4.3. Plugging this into (4.8),
and combining it with (4.7), we conclude the proposition. ∎

######  Lemma 4.5.

Let $u$ be a smooth divergence free vector field such that $u=0$ on
$\partial\Omega$ and $f$ be a smooth scalar-valued function such that
$f=\partial_{2}f=\partial_{2}^{2}\overline{f}=0$ on $\partial\Omega$ where
$\overline{f}:=\frac{1}{2\pi}\int_{\mathbb{T}}f(x_{1},x_{2})dx_{1}$. Then, we
have

$\displaystyle\left|\int_{\Omega}\Delta^{2}(u\cdot\nabla
f)\Delta^{2}fdx\right|\leq_{C}\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}^{2}+\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}}$

###### Proof.

We split the integral $\int_{\Omega}\Delta^{2}(u\cdot\nabla f)\Delta^{2}fdx$
as

$\displaystyle\int_{\Omega}\Delta^{2}(u\cdot\nabla f)\Delta^{2}fdx$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u\cdot\nabla
f)-u\cdot\nabla\Delta^{2}f\right)\Delta^{2}\theta+\int_{\Omega}u\cdot\nabla\left(\frac{1}{2}(\Delta^{2}f)^{2}\right)dx$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u\cdot\nabla
f)-u\cdot\nabla\Delta^{2}\theta\right)\Delta^{2}fdx,$

where the last integral vanishes due to the incompressibility of $u$ and the
boundary condition $u=0$ on $\Omega$. Next, we further decompose the last
integral as

$\displaystyle\int_{\Omega}\Delta^{2}(u\cdot\nabla f)\Delta^{2}fdx$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u\cdot\nabla
f)-u\cdot\nabla\Delta^{2}f\right)\Delta^{2}fdx$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u_{1}\partial_{1}f)-u_{1}\Delta^{2}\partial_{1}f\right)\Delta^{2}fdx+\int_{\Omega}\left(\Delta^{2}(u_{2}\partial_{2}f)-u_{2}\Delta^{2}\partial_{2}f\right)\Delta^{2}fdx$
$\displaystyle=:I_{1}+I_{2}.$ (4.9)

Estimate for $I_{1}$. Using the Cauchy-Schwarz inequality, we estimate

$|I_{1}|\leq\rVert\Delta^{2}(u_{1}\partial_{1}f)-u_{1}\Delta^{2}\partial_{1}f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}}.$

Using Lemma 3.4, we have

$\displaystyle\rVert\Delta^{2}(u_{1}\partial_{1}f)-u_{1}\Delta^{2}\partial_{1}f\rVert_{L^{2}}$
$\displaystyle\leq\rVert
u_{1}\rVert_{H^{4}}\rVert\partial_{1}f\rVert_{L^{\infty}}+\rVert
u_{1}\rVert_{W^{1,\infty}}\rVert\partial_{1}f\rVert_{H^{3}}$
$\displaystyle\leq\rVert
u_{1}\rVert_{H^{4}}\rVert\partial_{1}f\rVert_{H^{2}}+\rVert
u_{1}\rVert_{W^{1,\infty}}\rVert\partial_{1}f\rVert_{H^{3}}.$

where the second inequality is due to the Sobolev embedding
$H^{2}(\Omega)\hookrightarrow L^{\infty}(\Omega)$. Then, using Lemma 4.3, we
can replace $\rVert\partial_{1}f\rVert_{H^{2}}$ and
$\rVert\partial_{1}f\rVert_{H^{3}}$ in the above estimate by
$\rVert\partial_{1}\Delta f\rVert_{L^{2}}$ and
$\rVert\Delta^{2}f\rVert_{L^{2}}$, respectively. Hence, we obtain

$\displaystyle|I_{1}|\leq\left(\rVert
u_{1}\rVert_{H^{4}}\rVert\partial_{1}\Delta f\rVert_{L^{2}}+\rVert
u_{1}\rVert_{W^{1,\infty}}\rVert\Delta^{2}f\rVert_{L^{2}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}.$
(4.10)

Before we start estimating $I_{2}$, we consider the structure of the bi-
Laplacian acting on a product of two functions. Expanding the bi-Laplacian
using the formula $\Delta(fg)=\Delta gh+2\nabla g\cdot\nabla h+g\Delta h$, we
have

$\displaystyle\Delta^{2}(gh)-f\Delta^{2}h$ $\displaystyle=\Delta(\Delta
gh+2\nabla g\cdot\nabla h+g\Delta h)-g\Delta^{2}h$
$\displaystyle=\Delta^{2}gh+2\nabla\Delta g\cdot\nabla h+\Delta g\Delta h$
$\displaystyle\ +2\left(\nabla\Delta g\cdot\nabla
h+\nabla\partial_{1}g\cdot\nabla\partial_{1}h+\nabla\partial_{2}g\cdot\nabla\partial_{2}h+\nabla
g\cdot\nabla\Delta h\right)$ $\displaystyle\ +\Delta g\Delta h+\nabla
g\cdot\nabla\Delta h$ $\displaystyle=\Delta^{2}gh+4\nabla\Delta g\cdot\nabla
h+2{\left(\nabla\partial_{1}g\cdot\nabla\partial_{1}h+\nabla\partial_{2}g\cdot\nabla\partial_{2}h+\Delta
g\cdot\Delta h\right)}+\nabla g\cdot\nabla\Delta h.$ (4.11)

Estimate for $I_{2}$. Next, we move on to estimate $I_{2}$ in (4.2). Recalling
our notation that
$\overline{f}(x_{2}):=\frac{1}{2\pi}\int_{\mathbb{T}}f(x_{1},x_{2})dx_{2},$ we
split $I_{2}$ as

$\displaystyle I_{2}$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u_{2}\partial_{2}f)-u_{2}\Delta^{2}\partial_{2}f\right)\Delta^{2}fdx$
$\displaystyle=\int_{\Omega}\left(\Delta^{2}(u_{2}\partial_{2}(f-\overline{f}))-u_{2}\Delta^{2}\partial_{2}(f-\overline{f})\right)\Delta^{2}fdx+\int_{\Omega}\left(\Delta^{2}(u_{2}\partial_{2}\overline{f})-u_{2}\Delta^{2}\partial_{2}\overline{f}\right)\Delta^{2}{f}dx$
$\displaystyle=:I_{21}+I_{22}.$ (4.12)

Again, the Cauchy-Schwarz inequality gives us

$|I_{21}|\leq\rVert\Delta^{2}(u_{2}(f-\overline{f}))-u_{2}\Delta^{2}\partial_{2}(f-\overline{f})\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}},$

while Lemma 3.4 yields

$\displaystyle\rVert\Delta^{2}(u_{2}(f-\overline{f}))-u_{2}\Delta^{2}\partial_{2}(f-\overline{f})\rVert_{L^{2}}$
$\displaystyle\leq_{C}\rVert
u_{2}\rVert_{H^{4}}\rVert\partial_{2}(f-\overline{f})\rVert_{L^{\infty}}+\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert f-\overline{f}\rVert_{H^{4}}$
$\displaystyle\leq_{C}\rVert
u_{2}\rVert_{H^{4}}\rVert\partial_{12}f\rVert_{{H}^{1}}+\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert f\rVert_{H^{4}}$ $\displaystyle\leq_{C}\rVert
u_{2}\rVert_{H^{4}}\rVert\partial_{1}\Delta f\rVert_{L^{2}}+\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\Delta^{2}f\rVert_{L^{2}}.$

where the second inequality is due to
$\rVert\overline{f}\rVert_{H^{4}(\Omega)}\leq\rVert f\rVert_{H^{4}(\Omega)}$
and Lemma 3.5, and the last inequality follows from Lemma 4.3. Hence, we get

$\displaystyle|I_{21}|\leq\left(\rVert
u_{2}\rVert_{H^{4}}\rVert\partial_{1}\Delta f\rVert_{L^{2}}+\rVert
u_{2}\rVert_{W^{1,\infty}}\rVert\Delta^{2}f\rVert_{L^{2}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}.$
(4.13)

Next, we estimate $I_{22}$. Recalling $I_{22}$ from (4.2) and applying (4.2)
with $g=u_{2},\ h=\partial_{2}\overline{f}$, we have

$\displaystyle I_{22}$
$\displaystyle=\int_{\Omega}\Delta^{2}u_{2}\partial_{2}\overline{f}\Delta^{2}fdx$
$\displaystyle+\int_{\Omega}(4\nabla\Delta
u_{2}\cdot\nabla\partial_{2}\overline{f}+2\left(\nabla\partial_{1}u_{2}\cdot\nabla\partial_{12}\overline{f}+\nabla\partial_{2}u_{2}\cdot\nabla\partial_{22}\overline{f}+\Delta
u_{2}\Delta\partial_{2}\overline{f}+\nabla
u_{2}\cdot\nabla\Delta\partial_{2}\overline{f}\right))\Delta^{2}fdx$
$\displaystyle=\int_{\Omega}\Delta^{2}u_{2}\partial_{2}\overline{f}\Delta^{2}fdx+\int_{\Omega}(4\partial_{2}\Delta
u_{2}\cdot\partial_{22}\overline{f}+2\left((\partial_{22}u_{2}+\Delta
u_{2})\cdot\partial_{222}\overline{f}+\partial_{2}u_{2}\cdot\partial_{2222}\overline{f}\right))\Delta^{2}fdx$
$\displaystyle=:I_{221}+I_{222},$ (4.14)

where the second last equality is due to the fact that $\overline{f}$ does not
depend on the variable $x_{1}$. For $I_{222}$, using the Cauchy-Schwarz
inequality, we just need a crude estimate,

$\displaystyle|I_{222}|$ $\displaystyle\leq_{C}\left(\rVert
u_{2}\rVert_{H^{3}(\Omega)}\rVert\partial_{22}\overline{f}\rVert_{L^{\infty}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\rVert
f\rVert_{H^{4}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}$
$\displaystyle\leq_{C}\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert
f\rVert_{H^{4}}\rVert\Delta^{2}f\rVert_{L^{2}}$
$\displaystyle\leq_{C}\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}^{2},$ (4.15)

where we used the Sobolev embedding $L^{\infty}([0,1])\hookrightarrow
H^{1}([0,1])$ to have
$\rVert\partial_{22}\overline{f}\rVert_{L^{\infty}}\leq\rVert{f}\rVert_{H^{4}}$
in the the second inequality, and we used Lemma 4.3 in the last inequality.

Lastly, we estimate $I_{221}$. Let $\Psi$ be the stream function of $u$ such
that

$\int_{\Omega}\Psi dx=0,\quad u=\nabla^{\perp}\Psi\text{ with
}\Psi=\nabla\Psi=0\text{ on $\partial\Omega$}.$

Indeed, such a stream function exists since $u$ is divergence free and,
especially, $u=0$ on $\partial\Omega$. Using the integration by parts, we get

$\displaystyle I_{221}$
$\displaystyle=\int_{\Omega}\partial_{1}\Delta^{2}\Psi\partial_{2}\overline{f}\Delta^{2}fdx=-\int\left(\Delta^{2}\Psi\partial_{2}\overline{f}\right)\Delta^{2}\partial_{1}fdx=-\int\Delta\left(\Delta^{2}\Psi\partial_{2}\overline{f}\right)\Delta\partial_{1}fdx,$

where the integration by parts in the last equality is justified by the
assumption that $\partial_{2}\theta=\partial_{22}\overline{f}=0$ on
$\partial\Omega$. Therefore, expanding
$\Delta(\Delta^{2}\Psi\partial_{2}\overline{f})$, we get

$\displaystyle I_{221}$
$\displaystyle=-\int(\Delta^{3}\Psi\partial_{2}\overline{f}+2\partial_{2}\Delta^{2}\Psi\cdot\partial_{22}\overline{f}+\Delta^{2}\Psi\partial_{222}\overline{f})\partial_{1}\Delta\theta
dx.$

Then, we obtain

$\displaystyle|I_{221}|$
$\displaystyle\leq\rVert\Psi\rVert_{H^{6}}\rVert\overline{f}\rVert_{W^{3,\infty}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\leq\rVert u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\overline{f}\rVert_{H^{4}}\leq\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}},$

where we used Lemma 4.3 in the last inequality. Combining this with (4.2) and
(4.2), we get

$|I_{22}|\leq\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}^{2}+\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}}.$

Combining this with (4.13) and (4.2), we have

$|I_{2}|\leq\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}^{2}+\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}}.$

Combining this with (4.10) and (4.2), we get

$\left|\int_{\Omega}\Delta^{2}(u\cdot\nabla\theta)\Delta^{2}\theta
dx\right|\leq\left(\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\right)\rVert\Delta^{2}f\rVert_{L^{2}}^{2}+\rVert
u\rVert_{H^{5}}\rVert\partial_{1}\Delta
f\rVert_{L^{2}}\rVert\Delta^{2}f\rVert_{L^{2}}.$

This finishes the proof. ∎

###  4.3. Analysis of the energy structure

As in the previous section for the IPM equation, we define

$\displaystyle E(t):=\int_{\Omega}(\rho(t)-\rho_{0}^{*})x_{2}dx,\quad
K(t):=\rVert\Delta\Psi(t)\rVert_{L^{2}}^{2},$ (4.16)

where $\rho_{0}^{*}$ is the vertical rearrangement of the initial density
$\rho_{0}$. Throughout this subsection, we will assume that

$\displaystyle\rVert\theta(t)\rVert_{H^{3}}^{2}+\int_{0}^{T}\rVert\Delta^{2}\Psi(t)\rVert_{H^{2}}^{2}dt\leq\delta\leq\delta_{0}\ll
1,\text{ for $t\in[0,T]$ for some $T>0$ and $\delta_{0}>0$}.$ (4.17)

######  Proposition 4.6.

There exists $\delta_{0}=\delta_{0}(\gamma,\rVert\rho_{s}\rVert_{H^{4}})>0$
such that if (4.17) holds, then

$\displaystyle\frac{d}{dt}E(t)=-K(t),\quad\frac{d}{dt}K(t)\leq-C\rVert
u_{2}\rVert_{L^{2}}^{2}.$ (4.18)

###### Proof.

Differentiating the energy difference $E(t)$ in time, we see that

$\displaystyle\frac{d}{dt}E(t)=\int_{\Omega}\rho_{t}x_{2}dx=-\int_{\Omega}u\cdot\nabla\rho
x_{2}dx=\int_{\Omega}u_{2}\rho dx.$ (4.19)

Using (1.5), we can further simplify the last expression as

$\int_{\Omega}u_{2}\rho dx=\int_{\Omega}\partial_{1}\Psi\rho
dx=-\int_{\Omega}\Psi\partial_{1}\rho dx=-\int_{\Omega}\Psi\Delta^{2}\Psi
dx=-\int_{\Omega}|\Delta\Psi|^{2}dx,$

therefore, we get

$\displaystyle\frac{d}{dt}E(t)=-\int_{\Omega}|\Delta\Psi|^{2}dx=-K(t).$ (4.20)

In order to estimate $\frac{d}{dt}K(t)$, taking a derivative in (4.19), we
compute

$\frac{1}{2}\left(\frac{d}{dt}\right)^{2}E(t)=\frac{1}{2}\frac{d}{dt}\int_{\Omega}u_{2}\rho
dx=\frac{1}{2}\int_{\Omega}u_{2}\rho_{t}dx+\frac{1}{2}\int_{\Omega}\partial_{t}u_{2}\rho
dx.$

Using $u_{2}=\partial_{1}\Psi$ and $\partial_{1}\rho=\Delta^{2}\Psi$, we have

$\int_{\Omega}\partial_{t}u_{2}\rho
dx=-\int_{\Omega}\partial_{t}\Psi\partial_{1}\rho
dx=-\int_{\Omega}\partial_{t}\Psi\Delta^{2}\Psi
dx=-\int_{\Omega}\partial_{t}\Delta^{2}\Psi\Psi
dx=-\int_{\Omega}{\partial_{t}}\partial_{1}\rho\Psi
dx=\int_{\Omega}\partial_{t}\rho u_{2}dx$

Therefore, we get

$\displaystyle\frac{1}{2}\left(\frac{d}{dt}\right)^{2}E(t)$
$\displaystyle=\int_{\Omega}u_{2}\rho_{t}=-\int_{\Omega}u_{2}u\cdot\nabla\rho
dx=-\int_{\Omega}u_{2}^{2}\partial_{2}\rho
dx-\int_{\Omega}u_{2}u_{1}\partial_{1}\rho dx$
$\displaystyle\geq-\int_{\Omega}u_{2}^{2}\partial_{2}\rho dx-\rVert
u_{2}\rVert_{L^{2}}\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}}.$

Using the assumption on $\rVert\theta\rVert_{H^{3}}$ in (4.17), we have

$-\int_{\Omega}u_{2}^{2}\partial_{2}\rho dx=-\int
u_{2}^{2}\partial_{2}\rho_{s}dx-\int
u_{2}^{2}\partial_{2}(\rho-\rho_{s})dx\geq\gamma\rVert
u_{2}\rVert_{L^{2}}^{2}-\sqrt{\delta_{0}}\rVert u_{2}\rVert_{L^{2}}^{2}\geq
C\rVert u_{2}\rVert_{L^{2}}^{2},$

where we used the strict positivity of $\gamma$ which is assumed in (4.1).
Using (4.20), we see that the above inequality implies

$\displaystyle\frac{d}{dt}K(t)+C\rVert u_{2}\rVert_{L^{2}}^{2}\leq C\rVert
u_{2}\rVert_{L^{2}}\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}}.$ (4.21)

Let us estimate $\rVert u_{1}\partial_{1}\rho\rVert_{L^{2}}^{2}$. Using (1.5),
we rewrite

$\displaystyle\rVert
u_{1}\partial_{1}\rho\rVert_{L^{2}(\Omega)}=\rVert\partial_{2}\Psi\Delta^{2}\Psi\rVert_{L^{2}}\leq\rVert\nabla\Psi\rVert_{L^{4}}\rVert\Delta^{2}\Psi\rVert_{L^{4}}\leq\rVert\Delta^{2}\Psi\rVert_{H^{2}}\rVert\Psi\rVert_{L^{2}},$
(4.22)

where the last inequality is due to the Gagliardo-Nirenberg interpolation
theorem. Moreover, we notice from (1.5) that
$g(x_{2}):=\int_{\mathbb{T}}\Psi(x_{1},x_{2})dx_{1}$ satisfies

$\displaystyle\partial_{2222}g(x_{2})$
$\displaystyle=\int_{\mathbb{T}}\partial_{2222}\Psi(x_{1},x_{2})dx_{1}=\int_{\mathbb{T}}\Delta^{2}\Psi(x_{1},x_{2})-(\partial_{1111}+2\partial_{1122})\Psi(x_{1},x_{2})dx_{1}=\int_{\mathbb{T}}\partial_{1}\rho
dx=0,$

with the boundary condition, $g=\partial_{2}g=0$ on $\partial\Omega$.
Therefore, $g=0$ for all $x_{2}\in[0,1]$. In other words, the map
$x_{1}\to\Psi(x_{1},x_{2})$ has zero average for each fixed $x_{2}$. Then, the
Poincaré inequality tells us

$\rVert\Psi\rVert_{L^{2}}\leq\rVert\partial_{1}\Psi\rVert_{L^{2}}=\rVert
u_{2}\rVert_{L^{2}}.$

Substituting this into (4.22), we get

$\rVert
u_{1}\partial_{1}\rho\rVert_{L^{2}}\leq\rVert\Delta^{2}\Psi\rVert_{H^{2}}\rVert
u_{2}\rVert_{L^{2}}\leq\rVert\rho-{\rho}_{s}\rVert_{H^{3}}\rVert
u_{2}\rVert_{L^{2}}\leq\sqrt{\delta_{0}}\rVert u_{2}\rVert_{L^{2}},$

where the second inequality follows from Lemma 4.1 (noting that
$\Delta^{2}\Psi=\partial_{1}(\rho-\rho_{s})$ and $\rho-\rho_{s}=0$ on
$\partial\Omega$) and the last inequality follows from (4.17). Plugging this
into (4.21), we obtain that for sufficiently small $\delta>0$,
$\frac{d}{dt}K(t)\leq-C\rVert u_{2}\rVert_{L^{2}}^{2}.$ Combining this with
(4.20), we finish the proof of the proposition. ∎

######  Corollary 4.7.

There exists $\delta_{0}=\delta_{0}(\gamma,\rVert\rho_{s}\rVert_{H^{4}})>0$
such that if (4.17) holds, then,

$\displaystyle E(t)$ $\displaystyle\leq\frac{C\delta}{t^{2}},$
$\displaystyle\frac{2}{t}\int_{t/2}^{t}K(s)ds$
$\displaystyle\leq\frac{C\delta}{t^{3}},$
$\displaystyle\frac{2}{t}\int_{t/2}^{t}\rVert u_{2}(s)\rVert_{L^{2}}^{2}ds$
$\displaystyle\leq\frac{C\delta}{t^{4}},$

for all $t\in[0,T]$.

###### Proof.

Thanks to Proposition 2.5, we know that there is a constant $C>0$ such that

$\displaystyle C^{-1}\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}^{2}\leq E(t)\leq
C\rVert\rho(t)-\rho_{0}^{*}\rVert_{L^{2}}^{2}.$ (4.23)

Now, using the Gagliardo-Nirenberg interpolation theorem, we estimate

$\rVert\partial_{1}\rho\rVert_{L^{2}(\Omega)}=\rVert\Delta^{2}\Psi\rVert_{L^{2}}\leq
C\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{1/2}\rVert\Delta\Psi\rVert_{L^{2}(\Omega)}^{1/2}.$

On the other hand, applying (2.27), we observe that
$\rVert\partial_{1}\rho\rVert_{L^{2}}\geq
C\rVert\rho-\rho_{0}^{*}\rVert_{L^{2}}$. Combining this with the above
estimate, we get

$\rVert\Delta\Psi\rVert_{L^{2}}^{2}\geq\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{-2}\rVert\rho-\rho_{0}^{*}\rVert_{L^{2}}^{4}\geq
C\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{-2}E(t)^{2}.$

where the last inequality follows from (4.23). Hence, the variation of the
potential energy in (4.18) yields

$\displaystyle\frac{d}{dt}E(t)\leq-C\left(\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}\right)^{-1}E(t)^{2}.$
(4.24)

Applying Lemma 2.1 with $\alpha=1>0$ and $n=2$ and
$a(t)=\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}$, we get

$E(t)\leq\frac{CA}{t^{2}},\text{ where
$A=\int_{0}^{t}\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}ds$.}$

Under the assumptions in (3.30), we have $A\leq C\delta$. Therefore,

$\displaystyle E(t)\leq\frac{C\delta}{t^{2}}.$ (4.25)

Applying Lemma 2.2 with $f(t)=E(t),\ g(t)=K(t),\ h(t)=C\rVert
u_{2}(t)\rVert_{L^{2}}^{2}$, it follows from (4.18) that

$\frac{2}{t}\int_{t/2}^{t}K(s)ds\leq\frac{C}{t^{3}},\text{ and
}\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq\frac{C\delta}{t^{4}}.$

Together with (4.25), we obtain the desired estimates. ∎

###  4.4. Proof of Theorem 1.6

Let $\delta_{0}$ be fixed so that Proposition 4.6 and Corollary 4.7 are
applicable. We claim that there exists
$\epsilon=\epsilon(\gamma,\rVert\rVert\rho_{s}\rVert_{H^{4}})>0$ such that if
$\rho_{0}-{\rho}_{s}\in H^{2}_{0}(\Omega)\cap H^{4}(\Omega)$ and

$\displaystyle\rVert\rho_{0}-{\rho}_{s}\rVert_{H^{4}}\leq\epsilon\ll 1,$
(4.26)

then for all $t>0$, it holds that

$\displaystyle\rVert\Delta^{2}\theta(t)\rVert_{L^{2}(\Omega)}^{2}+\int_{0}^{t}\rVert\Delta^{2}\Psi(t)\rVert_{H^{2}(\Omega)}^{2}dt<C\epsilon^{2},\text{
for some $C>0$.}$ (4.27)

Let us suppose for the moment that the claim is true. Then by Corollary 4.7,
the solution $\rho(t)$ satisfies

$\displaystyle
E(t)\leq\frac{C\epsilon^{2}}{t^{2}},\quad\frac{2}{t}\int_{t/2}^{t}K(s)ds\leq\frac{C\epsilon^{2}}{t^{3}},\quad\frac{2}{t}\int_{t/2}^{t}\rVert
u_{2}(s)\rVert_{L^{2}}^{2}ds\leq\frac{C\epsilon^{2}}{t^{4}},\text{ for all
$t>0$}.$ (4.28)

Especially, the potential energy estimate in (4.28) and (2.26) give us all the
necessary properties to establish Theorem 1.6.

In the rest of the proof, we aim to prove (4.27). Towards a contradiction, let
$T^{*}>0$ be the first time that (4.27) breaks down, that is,

$\displaystyle\rVert\Delta^{2}\theta(T^{*})\rVert_{L^{2}(\Omega)}^{2}+\int_{0}^{T^{*}}\rVert\Delta^{2}\Psi(t)\rVert_{H^{2}(\Omega)}^{2}dt=M\epsilon^{2}\ll
1,$ (4.29)

for some $M\gg 1$, which will be chosen later. Towards a contradiction, let us
denote

$f(t):=\rVert\Delta^{2}\theta(t)\rVert_{L^{2}}^{2},\quad
g(t):=\rVert\Delta^{2}\Psi(t)\rVert_{H^{2}}^{2}.$

Since $\theta_{0}:=\rho_{0}-\overline{\rho}\in H^{2}_{0}\cap H^{4}$, it
follows from Theorem 4.2 that $\theta(t)$ satisfies
$\theta=\partial_{2}\theta=\partial_{22}\overline{\theta}=0$ on
$\partial\Omega$. Then, Lemma 4.3 and our assumption on $T^{*}$ tell us that

$\displaystyle C^{-1}g(t)\leq{C}^{-1}\rVert\theta(t)\rVert_{H^{4}}^{2}\leq
f(t)\leq C\rVert\theta(t)\rVert_{H^{4}}^{2}\leq CM\epsilon^{2},\text{ for all
$t\in[0,T^{*}]$},\quad f(0)\leq C\epsilon^{2},$ (4.30)

where the initial condition is due to (4.26). In particular, we have

$\rVert\partial_{1}\Delta\theta\rVert_{L^{2}}^{2}=\rVert\Delta\partial_{1}\theta\rVert_{L^{2}}^{2}\geq\rVert\partial_{1}\theta\rVert_{H^{2}}^{2}=\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}=g(t).$

Using this and (4.30), we derive from Proposition 4.4 that

$\displaystyle\frac{d}{dt}f(t)\leq-Cg(t)+C\left(\rVert
u_{2}\rVert_{H^{3}}+\rVert u_{2}\rVert_{W^{2,\infty}}\right)f(t)+C\rVert
u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}+\rVert\partial_{1}\theta\rVert_{L^{2}}^{2},\text{
for $t\in[0,T^{*}]$.}$ (4.31)

With this differential inequality, we first derive a crude estimate for ”short
time” first and more careful analysis will follows afterwards.

Using the Sobolev embedding $W^{2,\infty}(\Omega)\hookrightarrow
H^{4}(\Omega)$, we have

$\displaystyle\rVert u_{2}\rVert_{H^{3}}+\rVert
u_{2}\rVert_{W^{2,\infty}}\leq\rVert
u_{2}\rVert_{H^{4}}\leq\rVert\nabla\Psi\rVert_{H^{4}}\leq\rVert\partial_{1}\theta\rVert_{H^{1}}\leq\sqrt{f(t)},$
(4.32)

where the last inequality follows from (4.30). From this and (4.31) we see
that $f$ satisfies

$\frac{d}{dt}f(t)\leq C(f(t)^{3/2}+f(t))\leq Cf(t),\quad\text{ for
$t\in[0,T^{*}]$}.$

Using $f(0)\leq C\epsilon^{2}$, we can immediately deduce from this
differential inequality that $f(t)\leq C\epsilon^{2}e^{Ct},$ for
$t\in[0,T^{*}]$. Since $g(t)\leq Cf(t)$, we can easily deduce that

$\displaystyle f(t)+\int_{0}^{t}g(s)ds\leq C\epsilon^{2}e^{Ct}\text{ for
$t\in[0,T^{*}]$}.$ (4.33)

Thus, for (4.29) to occur, we must have
$M\epsilon^{2}\leq_{C}\epsilon^{2}e^{CT^{*}}$. This gives us a lower bound of
$T^{*}$,

$T^{*}\geq C\log M.$

Let us pick

$\displaystyle T^{*}_{M}:=\log\log M\gg 1.$ (4.34)

Without loss of generality, we can assume $M$ is sufficiently large to ensure
that

$T^{*}_{M}<T^{*}.$

Then , it follows from (4.33) that

$\displaystyle f(T^{*}_{M})+\int_{0}^{T^{*}_{M}}g(t)dt\leq C\epsilon^{2}(\log
M)^{C}.$ (4.35)

Now, we will estimate $f(t)$ more carefully for $t\geq T^{*}_{M}$. Using the
Gagliardo-Nirenberg interpolation theorem, we see that

$\displaystyle\rVert u_{2}\rVert_{W^{2,\infty}}+\rVert
u_{2}\rVert_{H^{3}}\leq_{C}\rVert u_{2}\rVert_{L^{2}}^{2/5}\rVert
u_{2}\rVert_{H^{5}}^{3/5}\leq_{C}\rVert
u_{2}\rVert_{L^{2}}^{2/5}\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{3/5}.$ (4.36)

Hence, Young’s inequality can give us

$\displaystyle\left(\rVert u_{2}\rVert_{W^{2,\infty}}+\rVert
u_{2}\rVert_{H^{3}}\right)f(t)$ $\displaystyle\leq_{C}\rVert
u_{2}\rVert_{L^{2}}^{2/5}\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{3/5}f(t)$
$\displaystyle\leq_{C}\eta\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}+C_{\eta}f(t)^{10/7}\rVert
u_{2}\rVert_{L^{2}}^{4/7}$ $\displaystyle\leq_{C}\eta
g(t)+C_{\eta}(M\epsilon^{2})^{10/7}\rVert u_{2}\rVert_{L^{2}}^{4/7},\text{ for
any $\eta>0$}.$ (4.37)

Again using (4.36) and the estimate $\rVert\theta(t)\rVert_{H^{4}}\leq
C\sqrt{M\epsilon^{2}}$ in (4.30), we get

$\displaystyle\rVert u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}$
$\displaystyle\leq_{C}\rVert
u_{2}\rVert_{L^{2}}^{2/5}\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{3/5}\sqrt{M\epsilon^{2}}$
$\displaystyle=\rVert
u_{2}\rVert_{L^{2}}^{2/5}\sqrt{M\epsilon^{2}}g(t)^{3/10}$
$\displaystyle\leq_{C}\eta
g(t)+C_{\eta}\left(\sqrt{M\epsilon^{2}}\right)^{10/7}\rVert
u_{2}\rVert_{L^{2}}^{4/7},\text{ for any $\eta>0$,}$ (4.38)

where the last inequality follows from Young’s inequality. Similarly, the
Gagliardo-Nirenberg interpolation theorem and the definition of $K(t)$ in
(4.16) give us

$\displaystyle\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}=\rVert\Delta^{2}\Psi\rVert_{L^{2}}^{2}\leq
C\rVert\Delta^{2}\Psi\rVert_{H^{2}}\rVert\Delta\Psi\rVert_{L^{2}}\leq\eta\rVert\Delta^{2}\Psi\rVert_{H^{2}}^{2}+C_{\eta}\rVert\Delta\Psi\rVert_{L^{2}}^{2}\leq_{C}\eta
g(t)+C_{\eta}K(t).$ (4.39)

Summing up this, (4.4) and (4.4), we obtain

$\displaystyle\left(\rVert u_{2}\rVert_{W^{2,\infty}}+\rVert
u_{2}\rVert_{H^{3}}\right)f(t)$ $\displaystyle+\rVert
u_{2}\rVert_{H^{3}}\rVert\Delta^{2}\theta\rVert_{L^{2}}+\rVert\partial_{1}\theta\rVert_{L^{2}}^{2}$
$\displaystyle\leq C\eta g(t)+C_{\eta}(\sqrt{M\epsilon^{2}})^{10/7}\rVert
u_{2}\rVert_{L^{2}}^{4/7}+K(t).$

Substituting this into (4.31) and choosing $\eta$ small enough, we arrive at

$\displaystyle\frac{d}{dt}f(t)+Cg(t)\leq_{C}\left(\sqrt{M\epsilon^{2}}\right)^{10/7}\rVert
u_{2}\rVert_{L^{2}}^{4/7}+K(t),\text{ for $t\in[0,T^{*}]$}.$ (4.40)

The estimate for $\rVert u_{2}\rVert_{L^{2}}$ in Corollary 4.7 tells us that

$\frac{2}{t}\int_{t/2}^{t}s^{1/4}\rVert u_{2}(s)\rVert_{L^{2}}^{2}ds\leq
Ct^{1/4}\frac{2}{t}\int_{t/2}^{t}\rVert u_{2}(s)\rVert_{L^{2}}^{2}ds\leq
C\frac{M\epsilon^{2}}{t^{15/4}},\text{ for all $t\in[0,T^{*}]$.}$

Hence applying Lemma 2.3 with $\alpha=2/7$, $n=15/4$, we obtain

$\int_{1}^{T^{*}}t^{1/14}\rVert
u_{2}(t)\rVert_{L^{2}}^{4/7}dt=\int_{1}^{T^{*}}\left(t^{1/4}\rVert
u_{2}(t)\rVert_{L^{2}}^{2}\right)^{2/7}dt\leq
C\left(M\epsilon^{2}\right)^{2/7}.$

Thanks to (4.34), this implies

$\displaystyle\int_{T_{M}^{*}}^{T^{*}}\left(\sqrt{M\epsilon^{2}}\right)^{10/7}\rVert
u_{2}(t)\rVert_{L^{2}}^{4/7}dt\leq\left(\sqrt{M\epsilon^{2}}\right)^{10/7}(T_{M}^{*})^{-1/14}\int_{T^{*}_{M}}^{T^{*}}t^{1/14}\rVert
u_{2}(t)\rVert_{L^{2}}^{4/7}dt\leq CM\epsilon^{2}(T^{*}_{M})^{-1/14}.$ (4.41)

Similarly, for the term $K(t)$ in (4.40), we use Corollary 4.7 to see

$\frac{2}{t}\int_{t/2}^{t}s^{1/14}K(s)ds\leq
Ct^{1/14}\frac{2}{t}\int_{t/2}^{t}K(s)ds\leq
C\frac{M\epsilon^{2}}{t^{41/14}}.$

Applying Lemma 2.3 with $\alpha=1$ and $n=41/14$, we get
$\int_{1}^{T^{*}}t^{1/14}K(t)dt\leq CM\epsilon^{2}$. Therefore

$\int_{T^{*}_{M}}^{T^{*}}K(s)ds\leq\left(T^{*}_{M}\right)^{-1/14}\int_{T^{*}_{M}}^{T^{*}}s^{1/14}K(s)ds\leq
C(T^{*}_{M})^{-1/14}M\epsilon^{2}.$

Combining this with (4.41), integrating (4.40) over $t\in[T_{M}^{*},T^{*}]$
gives

$f(T^{*})+\int_{T_{M}^{*}}^{T^{*}}g(s)ds\leq_{C}f(T^{*}_{M})+\left(T^{*}_{M}\right)^{-1/14}M\epsilon^{2}$

Together with (4.35) and (4.34), this estimate finally gives us

$f(T^{*})+\int_{0}^{T^{*}}g(t)dt\leq_{C}M\epsilon^{2}\left(M^{-1}(\log
M)^{C}+(\log\log M)^{-1/14}\right).$

Comparing this to (4.29), we must have

$1\leq C\left(M^{-1}(\log M)^{C}+(\log\log M)^{-1/14}\right),$

which leads to a contradiction if $M$ is chosen sufficiently large depending
only on the implicit constant $C$. This finishes the proof.

##  Appendix A Proof of Lemma 3.1

We argue by induction on $k$, that is, we will prove that for every
$k\in\mathbb{N}$,

$\displaystyle\rVert\partial_{1}^{n}\partial_{2}^{k}f\rVert_{L^{2}}\leq
C_{n,k}\left(\rVert\partial_{1}^{n+k}f\rVert_{L^{2}}+\rVert\partial_{2}^{n+k}f\rVert_{L^{2}}\right),\text{
for all $n\in\mathbb{N}\cup\left\\{0\right\\}$ and $f\in X^{\infty}$.}$ (A.1)

When $k=0$, the above estimate holds true trivially. Now, let us aim to prove
(A.1) for $k\to k+1$, assuming that the statement is true for some $k\geq 0$.
We split the cases $n=0,1$ and $n\geq 2$.

Case: $n=0$. It is trivial that

$\displaystyle\rVert\partial_{2}^{k+1}f\rVert_{L^{2}}\leq
C_{k}\left(\rVert\partial_{1}^{k+1}f\rVert_{L^{2}}+\rVert\partial_{2}^{k+1}f\rVert_{L^{2}}\right).$
(A.2)

Case: $n=1$. We have

$\displaystyle\rVert\partial_{1}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}=\int_{\Omega}\partial_{1}\partial_{2}^{k+1}f(x)\partial_{1}\partial_{2}^{k+1}f(x)dx$
$\displaystyle=-\int_{\Omega}\partial_{11}\partial_{2}^{k+1}f(x)\partial_{2}^{k+1}f(x)dx=\int_{\Omega}\partial_{11}\partial_{2}^{k}f(x)\partial_{2}^{k+2}f(x)dx$
$\displaystyle\leq\rVert\partial_{11}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{k+2}f\rVert_{L^{2}}^{2},$

where the integration by parts to get the second equality is justified by
(3.5). Using the induction hypothesis (A.1) for $k$, we have
$\rVert\partial_{11}\partial_{2}^{k}f\rVert_{L^{2}}^{2}\leq
C_{k}\left(\rVert\partial_{1}^{k+2}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{k+2}f\rVert_{L^{2}}^{2}\right)$,
hence

$\displaystyle\rVert\partial_{1}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}\leq
C_{k}\left(\rVert\partial_{1}^{k+2}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{k+2}f\rVert_{L^{2}}^{2}\right).$
(A.3)

Case: $n\geq 2$. We claim that

$\displaystyle\rVert\partial_{1}^{n-j}\partial_{2}^{k+1+j}f\rVert^{2}_{L^{2}(\Omega)}\leq\frac{1}{2}\left(\rVert\partial_{1}^{n-j+1}\partial_{2}^{k+j}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}^{n-j-1}\partial_{2}^{k+j+2}f\rVert_{L^{2}}^{2}\right),\text{
for all $0\leq j\leq n-1$.}$ (A.4)

Suppose (A.4) holds true for the moment. Then we have

$\displaystyle\sum_{j=0}^{n-1}\rVert\partial_{1}^{n-j}\partial_{2}^{k+1+j}f\rVert^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq\frac{1}{2}\sum_{j=0}^{n-1}\rVert\partial_{1}^{n-j+1}\partial_{2}^{k+j}f\rVert_{L^{2}}^{2}+\frac{1}{2}\sum_{j=0}^{n-1}\rVert\partial_{1}^{n-j-1}\partial_{2}^{k+j+2}f\rVert_{L^{2}}^{2}=:S_{1}+S_{2}$
(A.5)

The first summation in the right-hand side can be written as

$\displaystyle S_{1}$
$\displaystyle=\frac{1}{2}\left(\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\sum_{j=0}^{n-2}\rVert\partial_{1}^{n-j}\partial_{2}^{k+j+1}f\rVert_{L^{2}}^{2}\right)$
$\displaystyle=\frac{1}{2}\left(\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}+\sum_{j=1}^{n-2}\rVert\partial_{1}^{n-j}\partial_{2}^{k+j+1}f\rVert_{L^{2}}^{2}\right)$

and the second summation can be written as

$\displaystyle S_{2}$
$\displaystyle=\frac{1}{2}\left(\sum_{j=1}^{n-1}\rVert\partial_{1}^{n-j}\partial_{2}^{k+j+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\right)$
$\displaystyle=\frac{1}{2}\left(\sum_{j=1}^{n-2}\rVert\partial_{1}^{n-j}\partial_{2}^{k+j+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}^{1}\partial_{2}^{k+n}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\right)$

Summing them up, we can write the right-hand side of (A.5) as

$\displaystyle
S_{1}+S_{2}=\sum_{j=1}^{n-2}\rVert\partial_{1}^{n-j}\partial_{2}^{k+1+j}f\rVert^{2}_{L^{2}(\Omega)}+\frac{1}{2}\left(\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}\partial_{2}^{k+n}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\right)$

Plugging this into (A.5), and after the cancellations of the summation in $j$
over $1$ to $n-2$, we obtain

$\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}\partial_{2}^{k+n}f\rVert_{L^{2}}^{2}\leq\frac{1}{2}\left(\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}\partial_{2}^{k+n}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\right),$

which is equivalent to

$\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{1}\partial_{2}^{k+n}f\rVert_{L^{2}}^{2}\leq\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}.$

Hence, dropping the second term in the left-hand side, we get

$\rVert\partial_{1}^{n}\partial_{2}^{k+1}f\rVert_{L^{2}}^{2}\leq\rVert\partial_{1}^{n+1}\partial_{2}^{k}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\leq
C_{n,k}\left(\rVert\partial_{1}^{n+k+1}f\rVert_{L^{2}}^{2}+\rVert\partial_{2}^{n+k+1}f\rVert_{L^{2}}^{2}\right),$

where we used the induction hypothesis (A.1) for $k$. Combining this with
(A.2) and (A.3), we see that the statement (A.1) holds true for all $n$, for
$k+1$. The only missing part is a proof of (A.4).

Proof of (A.4). It is straightforward that

$\displaystyle\rVert\partial_{1}^{n-j}\partial_{2}^{k+1+j}f\rVert^{2}_{L^{2}(\Omega)}$
$\displaystyle=\int_{\Omega}\partial_{1}^{n-j}\partial_{2}^{k+1+j}f(x)\partial_{1}^{n-j}\partial_{2}^{k+1+j}f(x)dx$
$\displaystyle=-\int_{\Omega}\partial_{1}\partial_{1}^{n-j}\partial_{2}^{k+1+j}f(x)\partial_{1}^{n-j-1}\partial_{2}^{k+1+j}f(x)dx$
$\displaystyle=\int_{\Omega}\partial_{1}^{n-j+1}\partial_{2}^{k+j}f(x)\partial_{1}^{n-j-1}\partial_{2}^{k+2+j}f(x)dx$
$\displaystyle\leq\frac{1}{2}\rVert\partial_{1}^{n-j+1}\partial_{2}^{k+j}f\rVert_{L^{2}}^{2}+\frac{1}{2}\rVert\partial_{1}^{n-j-1}\partial_{2}^{k+2+j}f\rVert_{L^{2}}^{2},$

where the second and the third equality follow from the integration by parts
which is valid due to (3.5) and the last inequality follows from Young’s
inequality. This proves the claim (A.4).

## References