# Vanishing Mach Number Limit of
Stochastic Compressible Flows

Gui-Qiang G. Chen  Mathematical Institute, University of Oxford, Oxford OX2
6GG, UK<EMAIL_ADDRESS>,  Michele Coti Zelati  Department of
Mathematics, Imperial College London, London, SW7 2AZ, UK  m.coti-
<EMAIL_ADDRESS>and  Chin Ching Yeung  Mathematical Institute,
University of Oxford, Oxford OX2 6GG, UK<EMAIL_ADDRESS>
###### Abstract.

We study the vanishing Mach number limit for the stochastic Navier-Stokes
equations with $\gamma$-type pressure laws, with focus on the one-dimensional
case. We prove that, if the stochastic term vanishes with respect to the Mach
number sufficiently fast, the deviation from the incompressible state of the
solutions (for $\gamma\geq 1$) and the invariant measures (for $\gamma=1$) is
governed by a linear stochastic acoustic system in the limit. In particular,
the critically sufficient decay rate for the stochastic term is slower than
the corresponding results with deterministic external forcing due to the
martingale structure of the noise term, and the blow-up of the noise term for
the fluctuation system can be allowed.

###### Key words and phrases:

Stochastic compressible Navier-Stokes equations, zero-Mach limit, acoustic
systems, invariant measures

###### 2020 Mathematics Subject Classification:

35R60, 76N06, 37L40, 35B25

###### Contents

  1. 1 Introduction
    1. 1.1 Stochastic Navier-Stokes equations
    2. 1.2 Vanishing Mach number limit
    3. 1.3 Main results
    4. 1.4 Techniques and key points of the analysis
    5. 1.5 Organization of the paper
  2. 2 Energy Structure for the Navier-Stokes Equations
    1. 2.1 The main energy functional and its properties
    2. 2.2 Energy estimates for the Navier-Stokes equations
    3. 2.3 Estimates for the fluctuation system
  3. 3 Analysis of the Acoustic System
    1. 3.1 Energy estimates
    2. 3.2 Exponential contraction and stability
    3. 3.3 Long-time behavior
  4. 4 Finite-Time Convergence: Proof of Theorem 1.1
  5. 5 Convergence of Invariant Measures
    1. 5.1 Invariant measures for the fluctuation system
    2. 5.2 Invariant measures for the acoustic system
    3. 5.3 Contraction property of the acoustic system semigroup
    4. 5.4 Estimates of the invariant measures
    5. 5.5 Proof of Theorem 1.2

##  1\. Introduction

We are concerned with the vanishing Mach number limit of the stochastic
Navier-Stokes equations with $\gamma$-type pressure laws:

$\begin{cases}\displaystyle\,\mathrm{d}\rho+(\rho
u)_{x}\,\mathrm{d}t=0,\\\\[2.84526pt] \displaystyle\,\mathrm{d}(\rho u)+(\rho
u^{2}+\frac{1}{\varepsilon^{2}}\rho^{\gamma})_{x}\,\mathrm{d}t=u_{xx}\,\mathrm{d}t+\rho\sigma_{\varepsilon}\,\mathrm{d}W_{t},\end{cases}$
(NS)

where $\rho=\rho(t,x)$ and $u=u(t,x)$ represent the density and velocity of
the fluid in the position $x\in(0,1)$ at time $t\geq 0$, respectively. The
system is complemented with homogeneous Dirichlet boundary conditions:

$\displaystyle u(t,0)=u(t,1)=0\qquad\mbox{ for all $t>0$},$ (1.1)

and initial data:

$\displaystyle\rho(0,x)=\rho_{0}(x),\quad u(0,x)=u_{0}(x)\qquad\text{ for all
$x\in(0,1)$}.$ (1.2)

The conservation of mass guaranteed by the continuity equation in system (NS)
allows to fix, without loss of generality, the initial mass to $1$, implying
that

$\int_{0}^{1}\rho(x,t)\,\mathrm{d}x=1\qquad\text{ for all $t>0$}.$ (1.3)

The constant $\varepsilon>0$ represents the Mach number of the flow, which is
(proportional to) the ratio of the typical flow speed and the speed of sound
propagating in the flow. Without forcing, in the limit $\varepsilon\to 0$, the
fluid behaves as an incompressible one. In the one-dimensional case, we expect
the velocity to vanish due to the boundary conditions, and the density to
approach a constant because of the continuity equation. In the presence of
forcing, this necessarily imposes the coefficient $\sigma_{\varepsilon}$ to
vanish as $\varepsilon\to 0$: system (NS) is therefore forced by a collection
of independent white noise processes, suitably colored in space in an
$\varepsilon$-dependent fashion through the coefficient
$\sigma_{\varepsilon}=\varepsilon^{\beta}\sigma$, with $\beta>0$ and $\sigma$
fixed and independent of $\varepsilon$.

The goal of this paper is to analyze the higher-order fluctuations in the
vanishing Mach number limit. For this purpose, we asymptotically expand the
solutions in the form111The first-order term of the density $\rho$ vanishes by
the momentum equation.:

$\begin{cases}\rho=1+\varepsilon^{2}\rho_{2}+\varepsilon^{3}\rho_{3}+\cdots=1+\varepsilon^{2}\bar{\rho}_{\varepsilon},\\\\[2.84526pt]
u=\varepsilon
u_{1}+\varepsilon^{2}u_{2}+\cdots=\varepsilon\bar{u}_{\varepsilon}.\end{cases}$
(1.4)

We can now rewrite the system in the new variables
$(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ as the following
_fluctuation system_ :

$\begin{cases}\displaystyle\,\mathrm{d}\bar{\rho}_{\varepsilon}+\big{(}\varepsilon(\bar{\rho}_{\varepsilon})_{x}\bar{u}_{\varepsilon}+\frac{1}{\varepsilon}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})(\bar{u}_{\varepsilon})_{x}\big{)}\,\mathrm{d}t=0,\\\\[4.2679pt]
\displaystyle\,\mathrm{d}\bar{u}_{\varepsilon}+\big{(}\varepsilon\bar{u}_{\varepsilon}(\bar{u}_{\varepsilon})_{x}+\frac{\gamma}{\varepsilon}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-2}(\bar{\rho}_{\varepsilon})_{x}\big{)}\,\mathrm{d}t=\frac{(\bar{u}_{\varepsilon})_{xx}}{1+\varepsilon^{2}\bar{\rho}}\,\mathrm{d}t+\varepsilon^{\beta-1}\sigma\,\mathrm{d}W_{t},\end{cases}$
(FS)

with initial data:

$\bar{\rho}(0,x)=\bar{\rho}_{0,\varepsilon}(x),\quad\bar{u}(0,x)=\bar{u}_{0,\varepsilon}(x)\qquad\text{
for all $x\in(0,1)$},$ (1.5)

and boundary conditions:

$\bar{u}(t,0)=\bar{u}(t,1)=0\qquad\text{ for all $t>0$}.$ (1.6)

As $\varepsilon\to 0$, system (FS) is formally well-approximated by the
following linear acoustic system:

$\begin{cases}\displaystyle\,\mathrm{d}r_{\varepsilon}+\frac{1}{\varepsilon}(v_{\varepsilon})_{x}\,\mathrm{d}t=0,\\\\[7.11317pt]
\displaystyle\,\mathrm{d}v_{\varepsilon}+\frac{\gamma}{\varepsilon}(r_{\varepsilon})_{x}\,\mathrm{d}t=(v_{\varepsilon})_{xx}\,\mathrm{d}t+\varepsilon^{\beta-1}\sigma\,\mathrm{d}W_{t},\end{cases}$
(AS)

equipped with initial data:

$r_{\varepsilon}(0,x)=r_{0,\varepsilon}(x),\quad
v_{\varepsilon}(0,x)=v_{0,\varepsilon}(x)\qquad\text{ for all $x\in(0,1)$},$
(1.7)

and boundary conditions:

$v_{\varepsilon}(t,0)=v_{\varepsilon}(t,1)=0\qquad\text{ for all }t>0.$ (1.8)

The acoustic system (AS) is formally a stochastically forced linear damped
wave equation:

$\,\mathrm{d}(r_{\varepsilon})_{t}-\frac{1}{\varepsilon^{2}}(r_{\varepsilon})_{xx}\,\mathrm{d}t=(r_{\varepsilon})_{txx}\,\mathrm{d}t+\varepsilon^{\beta-2}\sigma_{x}\,\mathrm{d}W_{t}.$

The acoustic system (AS) is mathematically easier to study, compared to the
original nonlinear system (FS), and it reveals the fine wave-like structure of
the system in the vanishing Mach number limit. Justifying this limit is the
main goal of this paper.

###  1.1. Stochastic Navier-Stokes equations

The mathematical literature concerning the stochastic
Navier-Stokes equations is vast, which has seen tremendous growth in the last
decades. We limit ourselves to recording a few relevant results concerning the
existence and long-time behavior of solutions. In the two-dimensional
incompressible setting, the existence and uniqueness of solutions have been
established in various instances (different domains, boundary conditions,
regularity of initial data, and structure of the noise term); see
[kuksin2012mathematics, da1996ergodicity] and the references therein.
Moreover, the ergodicity and uniqueness of an invariant measure for the
corresponding Markov semigroup are also by now well-established under minimal
requirements on the noise structure [hairer2006ergodicity,
flandoli1995ergodicity]. The picture in the three-dimensional case is
complicated by the well-known well-posedness issues. The existence of global
weak solutions is known (see [Flandoli08] and the references therein), and it
has also been shown that there exists a strong Feller Markov selection with a
unique invariant measure [DaPrato03, FlandoliRomito08].

In comparison, the theory for compressible fluids is much less developed. Most
of the results concern the barotropic cases. The existence of solutions has
been obtained by using a pathwise approach in [1d_98] in the one-dimensional
case, in [tornatore2000global] for a special two-dimensional case, and in
[feireisl2013compressible] for the three-dimensional case. Furthermore, the
existence of global weak solutions with more general noise terms and adiabatic
exponent $\gamma>\frac{3}{2}$ in the three-dimensional setting has been
obtained in [breit2016stochastic, smith2017random, wang2015global]
independently by using a several-layer approximation scheme, similar to the
one used in the deterministic case in [feireisl01]. In addition, the existence
and uniqueness of local-in-time (up to a positive stopping time) pathwise
strong solutions have been obtained in [breit2018local] for the barotropic
Navier-Stokes equations with multiplicative noise on a torus. Besides, the
incompressible limit of the stochastic isotropic Navier-Stokes has been
studied in [breit2016incompressible].

There are only a few results available in the study of invariant measures and
stationary solutions. In [gas2013mesure], the existence of invariant measures
has been proved for an approximated version of the one-dimensional stochastic
compressible Navier-Stokes equations in Lagrangian coordinates, while
[michele] provides the construction of an invariant measure for system (NS)
for the ideal gas pressure law, with $\gamma=1$. In the three-dimensional
case, stationary solutions to the isentropic Navier-Stokes equations have been
constructed in [breit2019stationary]. However, due to the uniqueness issues,
stationarity is not intended in terms of an invariant measure for a related
Markov semigroup. In this regard, a Markov selection for the three-dimensional
system has been constructed in [markov_selection_3d_comp].

###  1.2. Vanishing Mach number limit

Extensive research has been dedicated to the investigation of the low Mach
number limit within the realm of smooth solutions for the deterministic
compressible Euler system, even extending to multidimensional scenarios. This
rich body of work includes seminal contributions by Klainerman-Majda [A1],
Majda [A2], Metivier-Schochet [A3], Schochet [A4], and a plethora of related
references.

Chen-Christoforou-Zhang in [A5] provided a rigorous mathematical foundation
for the zero Mach limit, specifically focusing on global discontinuous
solutions, characterized by bounded variation, of the compressible Euler
equations. Their results illustrate the behavior of second-order coefficients
in the asymptotic expansion concerning the Mach number, highlighting the
relevance of the linear acoustic system within the Euler framework. In the
context of the stochastic compressible Navier-Stokes equations, the present
study builds upon this framework, further developing and extending it. Our
work entails the derivation of the corresponding stochastic linear acoustic
system, as described in (AS), and the subsequent validation of the vanishing
Mach limit.

For additional insights into the low Mach number limit for global weak
solutions pertaining to the isentropic Navier-Stokes equations, we refer to
the work of Lions-Masmoudi [VML5] for the deterministic setting and to the
work of Breit-Feireisl-Hofmanova [breit2016incompressible] for the stochastic
case.

###  1.3. Main results

We fix a filtered probability space
$(\Omega,\Sigma,\mathbb{P},(\mathcal{F}_{t})_{t\geq 0})$ satisfying the usual
conditions such that it supports a sequence of independent real-valued Wiener
processes $(W^{k})_{k=1}^{\infty}$. Let
$\sigma=(\sigma_{k})_{k=1}^{\infty}\subset H^{1}_{0}(0,1)\cap H^{2}(0,1)$ such
that

$\|\sigma_{xx}\|_{L^{2}}^{2}:=\sum_{k=1}^{\infty}\int_{0}^{1}|(\sigma_{k})_{xx}|^{2}\,\mathrm{d}x<\infty.$
(1.9)

This regularity assumption is tacitly assumed throughout this paper. The noise
term is defined as in the time-integrated sense as

$\displaystyle\int_{0}^{t}\varepsilon^{\beta}\rho\sigma\,\mathrm{d}W_{s}:=\sum_{k=1}^{\infty}\int_{0}^{t}\varepsilon^{\beta}\rho\sigma_{k}\,\mathrm{d}W^{k}_{s},$

where integrals are understood in the Itô’s sense.

The basic energy functional for the compressible Navier-Stokes equations (NS)
can be written as

$\displaystyle\mathcal{E}(\rho,u)=\int_{0}^{1}\Big{(}\frac{1}{2}\rho
u^{2}+H(\rho)+\frac{1}{2}\frac{\rho_{x}u}{\rho}+\frac{1}{4}\frac{\rho_{x}^{2}}{\rho^{3}}\Big{)}\,\mathrm{d}x,$
(1.10)

where $H(\rho)$ is the pressure potential given by

$\displaystyle
H(\rho)=\frac{1}{\varepsilon^{2}}\begin{cases}\rho\log\rho-\rho+1&\text{ if
}\gamma=1,\\\\[2.84526pt]
\frac{1}{\gamma-1}(\rho^{\gamma}-\gamma(\rho-1))&\text{ if
}\gamma>1.\end{cases}$ (1.11)

The energy structure revealed by $\mathcal{E}$ is crucial in our analysis. The
first main result is a finite-time convergence of solutions of system (FS) to
those of system (AS), in the whole range of pressure exponents $\gamma\geq 1$.

######  Theorem 1.1.

Let $\beta>\frac{1}{2}$ and $T>0$. Assume
$(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})_{\varepsilon\in(0,1]}$
and $(r_{0,\varepsilon},v_{0,\varepsilon})_{\varepsilon\in(0,1]}$ are
collections of $\mathcal{F}_{0}$-measurable random variables in
$H^{1}(0,1)\times H^{1}_{0}(0,1)$ such that

$\int_{0}^{1}\bar{\rho}_{0,\varepsilon}\,\mathrm{d}x=0,\quad
1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon}>0,\quad\int_{0}^{1}r_{0,\varepsilon}\,\mathrm{d}x=0\qquad\,\,\,\,\mbox{almost
surely}.$

Furthermore, assume that there exist $\alpha>1$ and $\lambda>0$ such that

$\mathbb{E}\left[\mathcal{E}(1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon},\varepsilon\bar{u}_{0,\varepsilon})^{p}\right]\lesssim
p!\,\lambda^{p}\varepsilon^{\alpha p}\qquad\text{ for any }p\geq 1\text{ and
}\ \varepsilon\in(0,1].$ (1.12)

Then there exists $\varepsilon_{0}\in(0,1]$ such that the corresponding
solutions $(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ of system (FS)
and $(r_{\varepsilon},v_{\varepsilon})$ of system (AS) satisfy the convergence
estimate:

$\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\big{(}\gamma\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}-v_{\varepsilon}\|_{L^{2}}^{2}\big{)}\Big{]}\lesssim\Big{(}\mathbb{E}\big{[}\gamma\|\bar{\rho}_{0,\varepsilon}-r_{0,\varepsilon}\|_{L^{2}}^{4}+\|\bar{u}_{0,\varepsilon}-v_{0,\varepsilon}\|_{L^{2}}^{4}\big{]}\Big{)}^{\frac{1}{2}}+\varepsilon^{\alpha-1}+\varepsilon^{2\beta-1}$
(1.13)

for all $\varepsilon\in(0,\varepsilon_{0}]$.

We also translate the above result to the convergence of the invariant
measures for $\gamma=1$, which is the second main result in this paper. The
phase space for system (FS) is denoted by

$\bar{\mathcal{X}}_{\varepsilon}=\Big{\\{}(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})\in
H^{1}(0,1)\times
H^{1}_{0}(0,1)\,:\,\,\int_{0}^{1}\bar{\rho}_{\varepsilon}\,\mathrm{d}x=0,\,\,1+\varepsilon^{2}\bar{\rho}_{\varepsilon}>0\Big{\\}}.$

Thanks to the results in [michele], the corresponding Markov semigroup has an
invariant probability measure $\bar{\mu}_{\varepsilon}$ on
$\bar{\mathcal{X}}_{\varepsilon}$; see Section 5. On the other hand, the
Markovian framework for system (AS) can be set up in the larger phase space:

$\displaystyle\bar{\mathcal{X}}=\Big{\\{}(r,v)\in H^{1}(0,1)\times
H^{1}_{0}(0,1)\,:\,\int_{0}^{1}r\,\mathrm{d}x=0\Big{\\}}.$ (1.14)

We denote by $\omega_{\varepsilon}$ the unique invariant probability measure
of system (AS) in $\bar{\mathcal{X}}$.

######  Theorem 1.2.

Let $\gamma=1$. There exists $\varepsilon_{0}>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$,

$\displaystyle\mathcal{W}_{L^{2}}(\bar{\mu}_{\varepsilon},\omega_{\varepsilon})\lesssim\varepsilon^{\beta-\frac{1}{2}},$

where $\mathcal{W}_{L^{2}}$ is the Wasserstein metric on $\bar{\mathcal{X}}$
defined by using the $L^{2}\times L^{2}$ norm222We have abused the notation
$\bar{\mu}_{\varepsilon}$ to mean the Borel probability measure on
$\bar{\mathcal{X}}$ that naturally extends $\bar{\mu}_{\varepsilon}$. See
(5.1) for the precise definition..

######  Remark 1.3.

Concerning our main results, we have several remarks in order:

  1. (i)

Assumption (1.9) implies in particular that

$\|\sigma\|_{L^{\infty}}^{2}:=\sum_{k=1}^{\infty}\|\sigma_{k}\|_{L^{\infty}}^{2}\leq\sum_{k=1}^{\infty}\|(\sigma_{k})_{x}\|_{L^{2}}^{2}<\infty.$

This covers the natural choice that $\sigma_{k}=\alpha_{k}\sin(k\pi x)$ with
$(\alpha_{k})_{k=1}^{\infty}\subset\mathbb{R}$ and
$\sum_{k=1}^{\infty}k^{4}\alpha_{k}^{2}<\infty$.

  2. (ii)

The results apply to the case: $\beta\in(\frac{1}{2},1)$, where the noise term
in the fluctuation system (FS) blows up as $\varepsilon\to 0$.

  3. (iii)

The results above are better than the corresponding situation with a
deterministic external forcing, in which case $\beta>1$ is required, namely
the external forcing must vanish with the Mach number. The improvement in our
cases is due to the martingale structure of the stochastic integrals.

###  1.4. Techniques and key points of the analysis

Before presenting the detailed analysis in the subsequent sections, we
summarize here the novel ingredients and key observations that constitute the
foundation of our approach.

$\diamond$ Polynomial and exponential moment estimates for the energy for all
$\gamma\geq 1$. These generalize the results obtained in [michele] where only
the isothermal pressure law ($\gamma=1$) is considered (see Proposition 2.3).
Instead of trying to control the source terms from the random forcing by using
the dissipative terms (which are degenerate for $\gamma>1$) in the energy
balance (see (2.2) below) as in [michele], we directly use $\mathcal{E}$ to
estimate them, resulting in the bounds that grow exponentially in time.

$\diamond$ Polynomial moment estimates on the upper and lower bounds of the
density at small Mach numbers. These estimates are a consequence of the
exponential moment estimates for $\mathcal{E}$. The estimates allow us to
obtain fundamental controls on the fluctuation variables, which play a
critical role on the finite-time convergence analysis (see Proposition 2.7).

$\diamond$ Exponential contraction property of the acoustic system due to the
viscosity. We prove that the acoustic system has a fine structure of global
exponential contraction. This property allows the analysis of the large-time
behavior of the system and is of paramount importance in translating the
finite-time convergence analysis to the study of the convergence of invariant
measures (which could informally be considered as capturing the information on
the behavior of the system when $t=\infty$). There is a mismatch between the
topology used to prove the finite-time convergence and the contraction
property: In order to overcome this issue, we employ an $\varepsilon$-weighted
$H^{1}$ norm that is stronger than the $L^{2}$ norm, but coincides with the
$L^{2}$ norm in the limit $\varepsilon\to 0$.

$\diamond$ Novel estimates of the invariant measures. We prove that
statistically stationary solutions distributed as $\bar{\mu}_{\varepsilon}$
satisfy the assumption of Theorem 1.1. These are some novel estimates on the
invariant measures of the compressible Navier-Stokes equations. In particular,
we obtain the finite exponential moments of $\mathcal{E}$. We believe these
estimates are of independent interest.

###  1.5. Organization of the paper

In Section 2, we make the estimates of solutions of the Navier-Stokes
equations (NS) and use them as a starting point for the analysis of system
(FS) in Section 2.3. Next, we study the properties of the acoustic system (AS)
in Section 3. All these estimates are gathered together in Section 4 to give a
complete proof of Theorem 1.1. The study of the properties and the convergence
of invariant measures is left for Section 5.

Throughout the paper, we adopt the convention to denote $C$ as the constants
whose exact values are irrelevant and may be different at each occurrence. We
also use the subscripts to emphasize the dependence of the constants on
various parameters. In all cases, $C$ is independent of $\varepsilon$.

##  2\. Energy Structure for the Navier-Stokes Equations

In this section, we study the properties of solutions of problem (1.1)–(1.2)
for system (NS) in detail. Considering the pathwise transformation

$\hat{u}_{\varepsilon}=u_{\varepsilon}-\sum_{k=1}^{\infty}\int_{0}^{t}\varepsilon^{\beta}\sigma_{k}\,\mathrm{d}W^{k}_{s},$

we can transform system (NS) into a system with random coefficients but
without stochastic integrals. Then, using the ideas in [onedns], we can obtain
the existence and uniqueness of strong pathwise solutions of problem
(1.1)–(1.2) for system (NS) (see also [1d_96, 1d_98, Hoff00, michele]):

######  Proposition 2.1.

Assume $(\rho_{0,\varepsilon},u_{0,\varepsilon})_{\varepsilon\in(0,1]}$ is a
collection of $\mathcal{F}_{0}$-measurable random variables in $H^{1}\times
H^{1}_{0}$ such that

$\int_{0}^{1}\rho_{0,\varepsilon}\,\mathrm{d}x=1,\quad\rho_{0,\varepsilon}>0\qquad\,\,\mbox{almost
surely}.$

Then there exists a unique adapted strong pathwise solution
$(\rho_{\varepsilon},u_{\varepsilon})$ of system (NS) such that

$\rho_{\varepsilon}\in C([0,\infty)\times[0,1])\cap
L^{\infty}_{\mathrm{loc}}(0,\infty;H^{1}),\,\,\,\,u_{\varepsilon}\in
C([0,\infty);H^{1}_{0})\cap
L^{2}_{\mathrm{loc}}(0,\infty;H^{2})\qquad\,\mbox{almost surely}.$

Moreover, the map $t\mapsto\rho(t,\cdot)\in H^{1}$ is weakly continuous and,
for every $t>0$, the density is bounded away from vacuum:

$\rho(t,\cdot)^{-1}\in L^{\infty}(0,1)\qquad\,\mbox{almost surely}.$

We skip the proof of this classical result and limit ourselves to derive some
_a priori_ estimates for the system.

###  2.1. The main energy functional and its properties

The classical a priori estimates of solutions of the compressible Navier-
Stokes equations usually begin with the physical energy. However, in the one-
dimensional case, we can use the energy $\mathcal{E}$ defined in (1.10), which
is more powerful as it provides additional estimates on the $H^{1}$-norm of
the density. A similar structure in Lagrangian coordinates appeared in
[Kanel]. The energy has been used in [Hoff99, Hoff00] for deterministic
isothermal flows to study the large-time behavior of solutions. Its
generalization for the density-dependent viscosities, which even applies to
some multidimensional cases, is sometimes known as the BD-entropy that was
introduced in Bresch-Desjardins [bresch2003existence, bresch2004some,
bresch2007existence, bresch2003some]. The entropy has then been used
critically to prove various a priori estimates, particularly on the lower
bound of the density in the one-dimensional case (see e.g. [onedns,
li2008vanishing]). We now record some properties of $\mathcal{E}(\rho,u)$.

######  Proposition 2.2.

Let $(\rho,u)\in H^{1}\times H^{1}_{0}$ such that
$\int_{0}^{1}\rho(x)\,\mathrm{d}x=1$ and $\rho(x)>0$. Then

  1. (i)

$\displaystyle\int_{0}^{1}\rho u^{2}\,\mathrm{d}x\leq 4\mathcal{E}(\rho,u)$,

  2. (ii)

$\displaystyle\int_{0}^{1}\frac{|\rho_{x}|^{2}}{\rho^{3}}\,\mathrm{d}x\leq
8\mathcal{E}(\rho,u)$,

  3. (iii)

$\displaystyle\int_{0}^{1}H(\rho)\,\mathrm{d}x\leq\mathcal{E}(\rho,u)$,

  4. (iv)

$\displaystyle\|\rho^{-1}\|_{L^{\infty}}\leq
2\big{(}1+2\mathcal{E}(\rho,u)\big{)}$,

  5. (v)

$\displaystyle\max\\{\|\rho^{-1}\|_{L^{\infty}},\|\rho\|_{L^{\infty}}\\}\leq\exp(\sqrt{8\mathcal{E}(\rho,u)})$.

###### Proof.

The first two estimates follow respectively from the pointwise inequalities

$\left|\frac{\rho_{x}u}{\rho}\right|\leq\frac{1}{2}\rho
u^{2}+\frac{1}{2}\frac{|\rho_{x}|^{2}}{\rho^{3}},\qquad\left|\frac{\rho_{x}u}{\rho}\right|\leq\rho
u^{2}+\frac{1}{4}\frac{|\rho_{x}|^{2}}{\rho^{3}}.$

The third estimate follows from the fact that

$\frac{1}{2}\rho
u^{2}+\frac{1}{2}\frac{\rho_{x}u}{\rho}+\frac{1}{4}\frac{|\rho_{x}|^{2}}{\rho^{3}}\geq
0,$

which can be seen from the above two inequalities.

To prove the last two estimates, we note that there exists $x_{0}\in[0,1]$
such that $\rho(x_{0})=1$, since $\int_{0}^{1}\rho(x)\,\mathrm{d}x=1$ and
$\rho(x)$ is continuous. Hence, for all $x\in[0,1]$,

$\displaystyle\big{|}\rho^{-\frac{1}{2}}(x)-1\big{|}$
$\displaystyle=\big{|}\rho^{-\frac{1}{2}}(x)-\rho^{-\frac{1}{2}}(x_{0})\big{|}=\Big{|}\int_{x_{0}}^{x}\big{(}\rho^{-\frac{1}{2}}(y)\big{)}_{y}\,\mathrm{d}y\Big{|}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{1}\big{|}\rho^{-\frac{3}{2}}\rho_{x}\big{|}\,\mathrm{d}x\leq\frac{1}{2}\Big{(}\int_{0}^{1}\frac{|\rho_{x}|^{2}}{\rho^{3}}\,\mathrm{d}x\Big{)}^{\frac{1}{2}}\leq\sqrt{2\mathcal{E}(\rho,u)}.$

The triangle inequality then implies the desired estimate. Likewise, we have

$\displaystyle|\log\rho(x)|$
$\displaystyle=|\log\rho(x)-\log\rho(x_{0})|=\Big{|}\int_{x_{0}}^{x}\big{(}\log\rho(y)\big{)}_{y}\,\mathrm{d}y\Big{|}\leq\Big{(}\int_{0}^{1}\rho(y)\,\mathrm{d}y\Big{)}^{\frac{1}{2}}\Big{(}\int_{0}^{1}\frac{\rho_{x}^{2}}{\rho^{3}}\,\mathrm{d}y\Big{)}^{\frac{1}{2}}.$

The first integral in the inequality above is $1$ due to the mass condition,
while the second integral is bounded by $8\mathcal{E}(\rho,u)$. Exponentiating
both sides gives the last estimate. ∎

###  2.2. Energy estimates for the Navier-Stokes equations

For solutions of system (NS), we can now derive the following estimates:

######  Proposition 2.3.

Assume $(\rho_{0,\varepsilon},u_{0,\varepsilon})_{\varepsilon\in(0,1]}$ is a
collection of $\mathcal{F}_{0}$-measurable random variables in $H^{1}\times
H^{1}_{0}$ such that

$\int_{0}^{1}\rho_{0,\varepsilon}\,\mathrm{d}x=1,\quad\,\rho_{0,\varepsilon}>0\qquad\mbox{almost
surely}.$

Then the corresponding solution $(\rho_{\varepsilon},u_{\varepsilon})$ of
system (NS) satisfies the following properties:

  1. (i)

For all $p\geq 2$, there exists $C_{p}>0$ such that, for all
$\varepsilon\in(0,1]$ and $T\geq 1$,

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\Big{(}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\|(u_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s+\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\|\rho_{\varepsilon}^{\frac{\gamma-3}{2}}(\rho_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s\Big{)}^{p}\Big{]}$
$\displaystyle\,\leq 3^{p-1}{\rm
e}^{T}\,\mathbb{E}\big{[}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{p}\big{]}+C_{p}\,{\rm
e}^{T}\,T^{p+1}\varepsilon^{2\beta p}\|\sigma\|^{2p}_{L^{\infty}},$ (2.1)

with $C_{p}=O(1296^{p}p^{p})$ as $p\to\infty$.

  2. (ii)

For

$\eta_{0}(\varepsilon,T):=(4000T\varepsilon^{2\beta}\|\sigma\|^{2}_{L^{\infty}})^{-1},$
(2.2)

there exists $C>0$ such that, for all $\varepsilon\in(0,1]$, $T\geq 1$, and
$0<\eta<\eta_{0}(\varepsilon,T)$,

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\eta\sup_{0\leq t\leq
T}\big{(}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\|(u_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t+\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\|\rho_{\varepsilon}^{(\gamma-3)/2}(\rho_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\big{)}\Big{)}\Big{]}$
$\displaystyle\leq\frac{{\rm
e}^{T}}{3}\mathbb{E}\big{[}\exp\big{(}3\eta\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})\big{)}\big{]}+\frac{CT{\rm
e}^{T}}{1-\eta/\eta_{0}(\varepsilon,T)}.$ (2.3)

###### Proof.

For the first estimate, we employ the Itô’s formula to perform a direct
computation to obtain the energy balance:

$\displaystyle\,\mathrm{d}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\Big{(}\int_{0}^{1}|(u_{\varepsilon})_{x}|^{2}\,\mathrm{d}x+\frac{1}{\varepsilon}\int_{0}^{1}\rho_{\varepsilon}^{\gamma-3}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t$
$\displaystyle=\frac{1}{2}\varepsilon^{2\beta}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}t+\varepsilon^{\beta}\int_{0}^{1}\Big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\Big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{t}.$
(2.4)

Note that the last two terms are interpreted as

$\displaystyle\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}t=\sum_{k=1}^{\infty}\int_{0}^{1}\rho_{\varepsilon}\sigma_{k}^{2}\,\mathrm{d}x\,\mathrm{d}t,$
$\displaystyle\int_{0}^{1}\Big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\Big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{t}=\sum_{k=1}^{\infty}\int_{0}^{1}\Big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\Big{)}\sigma_{k}(x)\,\mathrm{d}x\,\mathrm{d}W^{k}_{t};$

also see [onedns] for the similar computation in the deterministic case
without the Itô’s correction term and [michele] for the computation in the
stochastic case with $\gamma=1$. Therefore, for all $p\in[2,\infty)$ and
$T\geq 1$, it holds almost surely that

$\displaystyle\bigg{(}\sup_{0\leq t\leq
T}\Big{(}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\int_{0}^{1}|(u_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\int_{0}^{1}\rho_{\varepsilon}^{\gamma-3}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}s\Big{)}\bigg{)}^{p}$
$\displaystyle=\bigg{(}\sup_{0\leq t\leq
T}\Big{(}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})+\frac{\varepsilon^{2\beta}}{2}\int_{0}^{t}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}s+\varepsilon^{\beta}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{s}\Big{)}\bigg{)}^{p}$
$\displaystyle\leq
3^{p-1}\bigg{(}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{p}+\sup_{0\leq
t\leq
T}\Big{(}\frac{\varepsilon^{2\beta}}{2}\int_{0}^{t}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}s\Big{)}^{p}$
$\displaystyle\qquad\qquad+\sup_{0\leq t\leq
T}\Big{(}\varepsilon^{\beta}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{s}\Big{)}^{p}\bigg{)}.$

We now estimate the last two terms in the bracket on the right-hand side. For
the first term, we have

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\Big{(}\frac{\varepsilon^{2\beta}}{2}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}\Big{]}$
$\displaystyle=\frac{\varepsilon^{2{p\beta}}}{2^{p}}\,\sup_{0\leq t\leq
T}\Big{(}\sum_{k=1}^{\infty}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}\sigma_{k}^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}$
$\displaystyle\leq\frac{\varepsilon^{2{p\beta}}}{2^{p}}\,\sup_{0\leq t\leq
T}\Big{(}\sum_{k=1}^{\infty}\|\sigma_{k}\|_{L^{\infty}}^{2}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}$
$\displaystyle\leq\frac{\varepsilon^{2{p\beta}}}{2^{p}}\,\sum_{k=1}^{\infty}\|\sigma_{k}\|_{L^{\infty}}^{2p}T^{p}\leq\frac{\varepsilon^{2{p\beta}}}{2^{p}}\|\sigma\|_{L^{\infty}}^{2p}T^{p}.$

Next, we estimate the second term involving a (local) martingale. For a local
martingale $(N_{t})_{t\geq 0}$, we write $(\langle N\rangle_{t})_{t\geq 0}$
for its quadratic variation process. Now, by the Burkholder-Davis-Gundy
inequality, there exists $G_{p}>0$ such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\Big{(}\varepsilon^{\beta}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{s}\Big{)}^{p}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\Big{(}\sum_{k=1}^{\infty}\varepsilon^{\beta}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\,\mathrm{d}W^{k}_{s}\Big{)}^{p}\bigg{]}$
$\displaystyle\leq G_{p}\,\varepsilon^{p\beta}\,\mathbb{E}\bigg{[}\sup_{0\leq
t\leq
T}\Big{\langle}\sum_{k=1}^{\infty}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\,\mathrm{d}W^{k}_{s}\Big{\rangle}^{\frac{p}{2}}\bigg{]}$
$\displaystyle=G_{p}\,\varepsilon^{p\beta}\,\mathbb{E}\bigg{[}\Big{(}\sum_{k=1}^{\infty}\int_{0}^{T}\Big{(}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\Big{)}^{2}\,\mathrm{d}s\Big{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
G_{p}\,\varepsilon^{p\beta}\,\|\sigma\|_{L^{\infty}}^{p}\,\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}u_{\varepsilon}^{2}\,\mathrm{d}x\,\mathrm{d}t+\int_{0}^{T}\int_{0}^{1}\frac{|(\rho_{\varepsilon})_{x}|^{2}}{\rho_{\varepsilon}^{3}}\,\mathrm{d}x\Big{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
12G_{p}\,\varepsilon^{p\beta}\,\|\sigma\|_{L^{\infty}}^{p}\,\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}t\Big{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
12G_{p}\,\varepsilon^{p\beta}\,T^{\frac{p}{2}-1}\|\sigma\|_{L^{\infty}}^{p}\,\mathbb{E}\Big{[}\int_{0}^{T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})^{\frac{p}{2}}\,\mathrm{d}t\Big{]}$
$\displaystyle\leq
3^{-(p-1)}\mathbb{E}\Big{[}\int_{0}^{T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})^{p}\,\mathrm{d}t\Big{]}+\frac{144^{p}}{4}3^{p-1}G_{p}^{2}T^{p-1}\varepsilon^{2p\beta}\|\sigma\|_{L^{\infty}}^{2p}.$

As a result, we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{0\leq t\leq
T}\Big{(}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\int_{0}^{1}|(u_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}s+\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\int_{0}^{1}\rho_{\varepsilon}^{\frac{\gamma-3}{2}}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}s\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\int_{0}^{T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})^{p}\,\mathrm{d}t\Big{]}+3^{p-1}\mathbb{E}\Big{[}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{p}\Big{]}+\Big{(}\frac{1296^{p}}{36}G_{p}^{2}T^{p-1}+\frac{3^{p-1}}{2^{p}}T\Big{)}T^{p-1}\varepsilon^{2\beta
p}\|\sigma\|_{L^{\infty}}^{2p}.$

The assertion now follows from the Gronwall inequality.

In order to show the asymptotic estimate of the constant in (2.1), we first
observe that

$N_{t}:=\varepsilon^{\beta}\int_{0}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{s}$

is a continuous martingale. To see this, we compute similar to the above that,
for all $t\geq\tau\geq 0$ and $m\geq 2$,

$\displaystyle\mathbb{E}\big{[}|N_{t}-N_{\tau}|^{m}\big{]}$
$\displaystyle=\varepsilon^{\beta
m}\mathbb{E}\bigg{[}\Big{(}\sum_{k=1}^{\infty}\int_{\tau}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\,\mathrm{d}W^{k}_{s}\Big{)}^{m}\bigg{]}$
$\displaystyle\leq G_{m}\,\varepsilon^{\beta
m}\,\mathbb{E}\bigg{[}\Big{(}\sum_{k=1}^{\infty}\int_{\tau}^{t}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\,\mathrm{d}W^{k}_{s}\bigg{)}^{\frac{m}{2}}\Bigg{]}$
$\displaystyle\leq 12G_{m}\,\varepsilon^{\beta
m}\,\|\sigma\|_{L^{\infty}}^{m}\,\mathbb{E}\bigg{[}\Big{(}\int_{\tau}^{t}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}r\Big{)}^{\frac{m}{2}}\bigg{]}$
$\displaystyle\leq 12G_{m}\,\varepsilon^{\beta
m}\,\|\sigma\|_{L^{\infty}}^{m}\,(t-\tau)^{\frac{m}{2}}\mathbb{E}\bigg{[}\Big{(}\sup_{s\leq
r\leq
t}\mathcal{E}(\rho_{\varepsilon}(t,\cdot),u_{\varepsilon}(t,\cdot))\Big{)}^{\frac{m}{2}}\bigg{]}$
$\displaystyle=C_{m}\varepsilon^{\beta m}(t-\tau)^{\frac{m}{2}}.$

Since $m\geq 2$ can be arbitrary large, by the Kolmogorov continuity theorem,
$N_{t}$ is almost surely Hölder continuous with Hölder exponent as any number
less than $\frac{1}{2}$. In particular, since $N_{t}$ is continuous, the
asymptotic estimate of the constant in (2.1) follows from the fact that
$G_{p}=O(p^{\frac{p}{2}})$ as $p\to\infty$, thanks to [BDG].

We now show ((ii)). Define

$Y_{t}=\sup_{0\leq t\leq
T}\Big{(}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\|(u_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s+\frac{1}{2\varepsilon^{2}}\int_{0}^{t}\|\rho_{\varepsilon}^{\frac{\gamma-3}{2}}(\rho_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s\Big{)}.$

Then

$\displaystyle\exp(\eta Y_{t})$
$\displaystyle=\sum_{k=0}^{\infty}\frac{\eta^{k}}{k!}\mathbb{E}\big{[}Y_{t}^{k}\big{]}$
$\displaystyle\leq\sum_{k=0}^{\infty}\frac{\eta^{k}}{k!}\Big{(}3^{k-1}{\rm
e}^{t}\,\mathbb{E}\big{[}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{k}\big{]}+C_{k}{\rm
e}^{t}t^{k+1}\varepsilon^{2\beta k}\|\sigma\|^{2k}_{L^{\infty}}\Big{)}$
$\displaystyle\leq\frac{{\rm
e}^{t}}{3}\sum_{k=0}^{\infty}\frac{(3\eta)^{k}}{k!}\mathbb{E}\big{[}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{k}\big{]}+Ct{\rm
e}^{t}\sum_{k=0}^{\infty}\frac{\big{(}1296t\eta\varepsilon^{2\beta}\|\sigma\|^{2}_{L^{\infty}}\big{)}^{k}}{k!}k^{k}$
$\displaystyle\leq\frac{{\rm
e}^{t}}{3}\mathbb{E}\big{[}\exp\big{(}3\eta\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})\big{)}\big{]}+Ct{\rm
e}^{t}\sum_{k=0}^{\infty}\big{(}4000t\eta\varepsilon^{2\beta}\|\sigma\|^{2}_{L^{\infty}}\big{)}^{k}$
$\displaystyle\leq\frac{{\rm
e}^{t}}{3}\mathbb{E}\big{[}\exp\big{(}3\eta\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})\big{)}\big{]}+\frac{Ct{\rm
e}^{t}}{1-4000t\eta\varepsilon^{2\beta}\|\sigma\|^{2}_{L^{\infty}}},$

if $0<\eta<\min\\{\frac{\eta(\varepsilon)}{3},\eta_{0}(\varepsilon,T)\\}$.
Note that we have used the Stirling’s approximation $k^{k}\sim k!\,{\rm
e}^{k}(2\pi k)^{-\frac{1}{2}}$ for large $k$ in the penultimate inequality.
This completes the proof. ∎

######  Remark 2.4.

In the isothermal case $\gamma=1$, the estimates in Proposition 2.3 can be
improved. In particular, the factor ${\rm e}^{T}$ in the first estimate can be
replaced by a $T$-independent constant and the smallness of $\eta$ needed in
the second estimate can also be chosen to be independent of $T$; see
[michele].

###  2.3. Estimates for the fluctuation system

The energy estimates established for system (NS) in §2.2 constitute the basis
for the analysis of the fluctuation system (FS). The estimates here will be
critical to establish the finite-time convergence results in Theorem 1.1. We
first restate Proposition 2.1 in the new variables
$(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ defined in (1.4).

######  Proposition 2.5.

Assume that
$(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})_{\varepsilon\in(0,1]}$
is a collection of $\mathcal{F}_{0}$-measurable random variables in
$H^{1}\times H^{1}_{0}$ such that

$\int_{0}^{1}\bar{\rho}_{0,\varepsilon}\,\mathrm{d}x=0,\quad
1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon}>0\qquad\,\,\,\mbox{almost
surely}.$

Then there exists a unique adapted strong pathwise solution
$(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ to (FS) such that

$\bar{\rho}_{\varepsilon}\in C([0,\infty)\times[0,1])\cap
L^{\infty}_{\mathrm{loc}}(0,\infty;H^{1}),\,\,\,\,\bar{u}_{\varepsilon}\in
C([0,\infty);H^{1}_{0})\cap
L^{2}_{\mathrm{loc}}(0,\infty;H^{2})\qquad\mbox{almost surely}.$

Moreover, the map: $t\mapsto\bar{\rho}_{\varepsilon}(t,\cdot)\in H^{1}$ is
weakly continuous and, for each $t>0$,

$(1+\varepsilon^{2}\bar{\rho}_{\varepsilon}(t,\cdot))^{-1}\in
L^{\infty}(0,1)\qquad\,\,\mbox{almost surely}.$

We now further assume that (1.12) holds. In particular, this implies that
there exists $C>0$ such that, for all $p\geq 1$ and $\varepsilon\in(0,1]$, for
all $0\leq\eta<\lambda^{-1}\varepsilon^{-\alpha}$,

$\displaystyle\mathbb{E}\left[\exp\big{(}\eta\mathcal{E}(1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon},\varepsilon\bar{u}_{0,\varepsilon})\big{)}\right]\leq\frac{C}{1-\eta\lambda\varepsilon^{\alpha}}\qquad\mbox{for
any $p\geq
1,\varepsilon\in(0,1],\eta\in(0,\lambda^{-1}\varepsilon^{-\alpha})$}.$ (2.5)

In fact, the assumption is equivalent to asserting that
$\mathcal{E}(1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon},\varepsilon\bar{u}_{0,\varepsilon})$
has an exponential tail. That is, there exist $\tilde{C},\tilde{\lambda}>0$
such that, for all $R>0$,

$\mathbb{P}\left[\mathcal{E}(1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon},\varepsilon\bar{u}_{0,\varepsilon})\geq
R\right]\leq\tilde{C}{\rm
e}^{-\frac{R}{\tilde{\lambda}\varepsilon^{\alpha}}}.$

######  Remark 2.6.

The assumption on the initial data covers some important cases of interest.
For example, if the initial data are deterministic and bounded in $H^{1}\times
H^{1}_{0}$ uniformly in $\varepsilon\in(0,1]$, then (1.12) is satisfied with
$\alpha=2$. Also, as we will see later, for $\gamma=1$, if
$(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})$ is a statistically
stationary solution of the fluctuation system (FS), then the assumption is
satisfied with $\alpha=2\beta$.

We now quantify better the estimates of solutions of system (FS).

######  Proposition 2.7.

Let $T\geq 1$, and let $(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})$
be satisfying (1.12) as above. Then

  1. (i)

For all $p\geq 1$, there exists $C>0$, independent of
$(\varepsilon,\alpha,\beta)$, such that

$\displaystyle\mathbb{E}\Big{[}\max\big{\\{}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{p}\|_{L^{\infty}},\sup_{0\leq
t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-p}\|_{L^{\infty}}\big{\\}}\Big{]}\leq
C$ (2.6)

for all $\varepsilon\in(0,\varepsilon_{0}]$ for some $\varepsilon_{0}>0$
depending only on $p,T,\alpha,\beta$, and $\|\sigma\|_{L^{\infty}}$.

  2. (ii)

For all $p\geq 1$, there exists $C_{p}=C_{p}(p,\gamma,\lambda,T)>0$,
independent of $(\alpha,\beta,\varepsilon)$, such that, for all
$\varepsilon\in(0,\varepsilon_{0}]$ $($ for $\varepsilon_{0}>0$ depending on
$p,T,\alpha,\beta$, and $\|\sigma\|_{L^{\infty}}$),

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\left(\|\bar{\rho}_{\varepsilon}\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}\|_{L^{2}}^{2}+\varepsilon^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\right)\,\mathrm{d}x+\int_{0}^{T}\big{(}\|(\bar{\rho}_{\varepsilon})_{x}\|^{2}_{L^{2}}+\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\big{)}\,\mathrm{d}t\Big{)}^{p}\bigg{]}$
$\displaystyle\leq
C_{p}\big{(}\varepsilon^{(\alpha-2)p}+\varepsilon^{(2\beta-2)p}\big{)},$ (2.7)

  3. (iii)

There exists $C=C(T)$ such that, for all $\varepsilon\in(0,1]$ and $\delta>0$
with

$\displaystyle\delta<\min\big{\\{}\frac{1}{6\lambda\varepsilon^{\alpha-2}},\,\frac{\eta_{0}(\varepsilon,T)\varepsilon^{2}}{2}\big{\\}}$
(2.8)

and $\eta_{0}$ defined in Proposition 2.3,

$\mathbb{E}\Big{[}\exp\Big{(}\delta\int_{0}^{T}\int_{0}^{1}\bar{u}_{\varepsilon}^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}\Big{]}\leq
C.$ (2.9)

###### Proof.

We divide the proof into three steps.

1\. We first go back to the original variables
$\rho_{\varepsilon}=1+\varepsilon^{2}\bar{\rho}_{\varepsilon}$ and
$u_{\varepsilon}=\varepsilon\bar{u}_{\varepsilon}$, which solve (NS) with the
initial data
$(1+\varepsilon^{2}\bar{\rho}_{0,\varepsilon},\varepsilon\bar{u}_{0,\varepsilon})$.

Fix $p\geq 1$. To prove (2.6), since $\alpha,\beta>0$, there exists
$\varepsilon_{0}>0$ such that, for all $\varepsilon\in(0,\varepsilon_{0})$,
$8p^{2}\leq\min\\{(6\lambda\varepsilon^{\alpha})^{-1},\frac{\eta_{0}(\varepsilon,T)}{2}\\}$,
where $\eta_{0}(\varepsilon,T)$ is defined in (2.2). For all such
$\varepsilon$, by Proposition 2.2, we have

$\displaystyle\max\Big{\\{}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}^{p}\|_{L^{\infty}},\,\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}^{-p}\|_{L^{\infty}}\Big{\\}}\leq\exp\Big{(}\big{(}\sup_{0\leq
t\leq
T}8p^{2}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\big{)}^{\frac{1}{2}}\Big{)}.$

We note that the function $\mu:[1,\infty)\to[1,\infty)$, defined by
$\mu(x)=\exp((\log x)^{\frac{1}{2}})$ for all $x\geq 1$, is concave and
strictly increasing. Therefore, we have

$\displaystyle\max\Big{\\{}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}^{p}\|_{L^{\infty}},\,\sup_{0\leq t\leq
T}\|\rho^{-p}\|_{L^{\infty}}\Big{\\}}$
$\displaystyle\leq\mathbb{E}\Big{[}\exp\Big{(}\big{(}\sup_{0\leq t\leq
T}8p^{2}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\big{)}^{\frac{1}{2}}\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}\big{[}\mu(\exp(\sup_{0\leq t\leq
T}8p^{2}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})))\big{]}$
$\displaystyle\leq\mu(\mathbb{E}[\exp(\sup_{0\leq t\leq
T}8p^{2}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon}))])$
$\displaystyle\leq\mu(\frac{{\rm
e}^{T}}{3}\mathbb{E}[\exp(\frac{3}{6\lambda\varepsilon^{\alpha}}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon}))]+\frac{CT{\rm
e}^{T}}{1-(\eta_{0}(\varepsilon,T)/2)/\eta_{0}(\varepsilon,T)})$
$\displaystyle=\mu(\frac{{\rm
e}^{T}}{3}\mathbb{E}[\exp(\frac{1}{2\lambda\varepsilon^{\alpha}}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon}))]+2CT{\rm
e}^{T})\leq C,$

by means of ((ii)) and (2.5).

2\. We now tackle (2.7) and start by estimating
$\mathbb{E}[\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2})}^{2p}]$. By
Taylor’s theorem, for all $z\in\mathbb{R}$, there exists $\xi$ between $1$ and
$1+z$ such that

$\displaystyle
H(1+z)=H(1)+H^{\prime}(1)z+\frac{H^{\prime\prime}(\xi)}{2}z^{2}.$

It follows from the fact that $H^{\prime\prime}$ is monotone with
$H^{\prime\prime}(1)=\frac{\gamma}{\varepsilon^{2}}$ that

$\displaystyle\int_{0}^{1}H(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})\,\mathrm{d}x\geq
b(t)\int_{0}^{1}\varepsilon^{4}\bar{\rho}_{\varepsilon}^{2}\,\mathrm{d}x$
(2.10)

with

$\displaystyle
b(t):=\frac{1}{2}\min\big{\\{}\frac{\gamma}{\varepsilon^{2}},\,\inf_{x\in[0,1]}H^{\prime\prime}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})\big{\\}}.$
(2.11)

Note that, by (2.6), there exists $C_{p,\gamma}>0$ such that, for all
sufficiently small $\varepsilon>0$,

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}b(t)^{-2p}\Big{)}^{\frac{1}{2}}\bigg{]}\leq C_{p,\gamma}\varepsilon^{2p}.$

We use (2.10) to compute that

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\int_{0}^{1}\varepsilon^{4}\bar{\rho}_{\varepsilon}^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}b(t)\int_{0}^{1}\varepsilon^{4}\bar{\rho}_{\varepsilon}^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq
t\leq T}b(t)^{-2p}\Big{)}^{\frac{1}{2}}\bigg{]}$ $\displaystyle\leq
C\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\Big{)}^{p}\bigg{]}\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq
t\leq T}b(t)^{-2p}\Big{)}^{\frac{1}{2}}\bigg{]}$ $\displaystyle\leq
C_{p,\gamma}\varepsilon^{2}\left(3^{2p-1}{\rm
e}^{T}\,\mathbb{E}[\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon})^{2p}]+C_{2p}\,{\rm
e}^{T}\,T^{2p+1}\varepsilon^{4\beta
p}\|\sigma\|^{4p}_{L^{\infty}}\right)^{\frac{1}{2}}$ $\displaystyle\leq
C_{p,\gamma}\varepsilon^{2p}\big{(}\varepsilon^{2p\alpha}+\varepsilon^{4p\beta}\big{)}^{\frac{1}{2}}$
$\displaystyle\leq
C_{p,\gamma}\varepsilon^{2p}\big{(}\varepsilon^{p\alpha}+\varepsilon^{2p\beta}\big{)},$

from which the bound on
$\mathbb{E}[\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2})}^{2p}]$
follows. Regarding $\bar{u}_{\varepsilon}$, we similarly have

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\varepsilon^{2}\int_{0}^{1}\bar{u}_{\varepsilon}^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\inf_{x\in[0,1]}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})\int_{0}^{1}\varepsilon^{2}\bar{u}_{\varepsilon}^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq
t\leq
T}\big{(}\displaystyle\inf_{x\in[0,1]}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})\big{)}^{-1}\Big{)}^{p}\bigg{]}$
$\displaystyle\leq C_{p}\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq t\leq
T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\Big{)}^{\\!2p}\Big{]}\bigg{)}^{\frac{1}{2}}\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq
t\leq
T}\big{(}\displaystyle\inf_{x\in[0,1]}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon}))^{-1}\Big{)}^{\\!2p}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C_{p}\big{(}\varepsilon^{p\alpha}+\varepsilon^{2p\beta}\big{)},$

giving the required bound on
$\mathbb{E}\|\bar{u}_{\varepsilon}\|_{L^{\infty}(0,T;L^{2})}^{2p}$. To
complete the proof of (2.7), we need two more estimates on
$(\bar{\rho}_{\varepsilon})_{x}$. The first is

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\int_{0}^{1}\varepsilon^{4}|(\bar{\rho}_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}^{3-\gamma}\|_{L^{\infty}}\Big{)}^{p}\Big{(}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}^{\gamma-3}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}^{3-\gamma}\|_{L^{\infty}}\Big{)}^{2p}\Big{]}\bigg{)}^{\frac{1}{2}}\bigg{(}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}\int_{0}^{1}\rho_{\varepsilon}^{\gamma-3}(\rho_{\varepsilon})_{x}^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{2p}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C_{p}\big{(}\varepsilon^{p\alpha}+\varepsilon^{2p\beta}\big{)},$

while the second is

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\int_{0}^{1}\varepsilon^{4}|(\bar{\rho}_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}$
$\displaystyle=\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\int_{0}^{1}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\Big{)}^{p}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}\|_{L^{\infty}}^{3}\int_{0}^{1}\frac{|(\rho_{\varepsilon})_{x}|^{2}}{\rho_{\varepsilon}^{3}}\,\mathrm{d}x\Big{)}^{p}\bigg{]}$
$\displaystyle\leq 8\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}\|_{L^{\infty}}^{3}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})\Big{)}^{p}\bigg{]}$
$\displaystyle\leq 8\bigg{(}\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\|\rho_{\varepsilon}\|_{L^{\infty}}^{6p}\Big{]}\bigg{)}^{\frac{1}{2}}\bigg{(}\mathbb{E}\Big{[}\sup_{0\leq
t\leq
T}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})^{2p}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C\big{(}\varepsilon^{p\alpha}+\varepsilon^{2p\beta}\big{)}.$

We observe that, using estimate (2.1) again, we have

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\int_{0}^{1}\varepsilon^{2}|(\bar{u}_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}\bigg{]}=\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\int_{0}^{1}|(u_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{p}\bigg{]}\leq
C_{p}\big{(}\varepsilon^{p\alpha}+\varepsilon^{2p\beta}\big{)},$

which gives the desired bound on $(\bar{u}_{\varepsilon})_{x}$.

3\. Finally, we turn to (2.9), and fix any $\delta>0$ complying with (2.8).
Then, by ((ii)), we have

$\displaystyle\mathbb{E}\Big{[}\exp(\delta\int_{0}^{t}\int_{0}^{1}\bar{u}_{\varepsilon}^{2}\,\mathrm{d}x\,\mathrm{d}t)\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\exp(\frac{\delta}{\varepsilon^{2}}\int_{0}^{t}\int_{0}^{1}u^{2}\,\mathrm{d}x\,\mathrm{d}t)\Big{]}$
$\displaystyle\leq\frac{{\rm
e}^{T}}{3}\mathbb{E}\Big{[}\exp(\frac{3\delta}{\varepsilon^{2}}\mathcal{E}(\rho_{0,\varepsilon},u_{0,\varepsilon}))\Big{]}+\frac{CT{\rm
e}^{T}}{1-\delta/(\varepsilon^{2}\eta_{0}(\varepsilon,T))}$
$\displaystyle\leq\frac{C{\rm
e}^{T}}{3(1-3\delta\lambda\varepsilon^{\alpha-2})}+\frac{CT{\rm
e}^{T}}{1-\delta/(\varepsilon^{2}\eta_{0}(\varepsilon,T))}\leq C.$

This completes the proof. ∎

##  3\. Analysis of the Acoustic System

The starting point of our analysis of the acoustic system (AS) is the
existence and uniqueness of solutions with the initial data
$(r_{0,\varepsilon},v_{0,\varepsilon})$, which are proved directly by using
again a pathwise transformation:

$\hat{r}_{\varepsilon}=r_{\varepsilon}-\sum_{k=1}^{\infty}\int_{0}^{t}\varepsilon^{\beta-2}\sigma_{k}\,\,\mathrm{d}W^{k}_{s}.$

######  Proposition 3.1.

Assume that $(r_{0,\varepsilon},v_{0,\varepsilon})_{\varepsilon\in(0,1]}$ is a
collection of $\mathcal{F}_{0}$-measurable random variables in $H^{1}\times
H^{1}_{0}$ such that

$\int_{0}^{1}r_{0,\varepsilon}\,\mathrm{d}x=0\qquad\mbox{almost surely}.$

Then there exists a unique adapted strong pathwise solution
$(r_{\varepsilon},v_{\varepsilon})$ of system (AS) such that

$r_{\varepsilon}\in C([0,\infty);H^{1}),\quad v_{\varepsilon}\in
C([0,\infty);H^{1}_{0})\cap
L^{2}_{\mathrm{loc}}(0,\infty;H^{2})\qquad\mbox{almost surely}.$

The energy structure of (AS) is captured by the energy functional:

$\mathcal{G}(r,v)=\int_{0}^{1}\Big{(}\frac{1}{2}(\gamma
r^{2}+v^{2})+\frac{1}{2}\varepsilon
vr_{x}+\frac{\varepsilon^{2}}{4}|r_{x}|^{2}\Big{)}\,\mathrm{d}x,$

which can be viewed as analogous to $\mathcal{E}$ in (1.10).

###  3.1. Energy estimates

We start by a series of the bounds on the solutions of (AS).

######  Proposition 3.2.

Let $(r_{\varepsilon},v_{\varepsilon})$ be the solution of system (AS) with
initial data $(r_{0,\varepsilon},v_{0,\varepsilon})$ as in Proposition 3.1,
and let

$\kappa_{\varepsilon}:=\frac{1}{4\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}}.$

For all $R>0$, the following exponential martingale estimate holds:

$\displaystyle\mathbb{P}\Big{[}\sup_{t\geq
0}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\frac{1}{4}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s-\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}t\Big{)}-\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\geq
R\Big{]}$ $\displaystyle\,\,\leq{\rm e}^{-\kappa_{\varepsilon}R}.$ (3.1)

As a consequence, for any $m\geq 1$, there exists $c_{m}>0$ independent of
$\varepsilon\in(0,1]$ such that, for all $T\geq 1$,

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\frac{1}{4}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s\Big{)}^{m}\Big{]}$
$\displaystyle\leq
c_{m}\left(\kappa_{\varepsilon}^{-m}+\varepsilon^{2m\beta-2m}\|\sigma\|_{L^{\infty}}^{2m}T^{m}+\mathbb{E}\left[\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})^{m}\right]\right).$
(3.2)

Moreover, for any $\delta\in(0,\frac{\kappa_{\varepsilon}}{2})$, there exist a
constant $C_{\delta}>0$ independent of $\varepsilon\in(0,1]$ such that, for
all $T\geq 1$,

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\delta\sup_{0\leq t\leq
T}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\frac{1}{4}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s\Big{)}\Big{)}\Big{]}$
$\displaystyle\leq
C_{\delta}\exp\big{(}\frac{\delta}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}T\big{)}\sqrt{\mathbb{E}\big{[}\exp\big{(}2\delta\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\big{)}\big{]}}.$
(3.3)

###### Proof.

By Itô’s formula, the two equations in (AS) imply that

$\,\mathrm{d}\Big{(}\int_{0}^{1}\frac{1}{2}(\gamma
r_{\varepsilon}^{2}+v_{\varepsilon}^{2})\,\mathrm{d}x\Big{)}+\Big{(}\int_{0}^{1}|(v_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t=\Big{(}\int_{0}^{1}\frac{1}{2}\varepsilon^{2\beta-2}\sigma^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t+\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma
v_{\varepsilon}\,\mathrm{d}x\Big{)}\,\mathrm{d}W_{t}.$

Again, the first integral on the right-hand-side is

$\int_{0}^{1}\frac{1}{2}\varepsilon^{2\beta-2}\sigma^{2}\,\mathrm{d}x=\sum_{k=1}^{\infty}\int_{0}^{1}\frac{1}{2}\varepsilon^{2\beta-2}\sigma_{k}^{2}\,\mathrm{d}x,$

and the last integral is a short form representing

$\sum_{k=1}^{\infty}\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma_{k}v_{\varepsilon}\,\mathrm{d}x\Big{)}\,\mathrm{d}W^{k}_{t}.$

This equality, together with the observation that

$\,\mathrm{d}\Big{(}\varepsilon
v_{\varepsilon}(r_{\varepsilon})_{x}+\frac{\varepsilon^{2}}{2}|(r_{\varepsilon})_{x}|^{2}\Big{)}=-vv_{xx}\,\mathrm{d}t-\gamma|(r_{\varepsilon})_{x}|^{2}\,\mathrm{d}t+\varepsilon^{\beta}(r_{\varepsilon})_{x}\sigma\,\mathrm{d}W_{t},$

gives

$\displaystyle\,\mathrm{d}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\frac{1}{2}\Big{(}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}t$
$\displaystyle=\Big{(}\int_{0}^{1}\frac{1}{2}\varepsilon^{2\beta-2}\sigma^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t+\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}W_{t}.$
(3.4)

Define

$\displaystyle Z(t)$
$\displaystyle:=\int_{0}^{t}\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}W_{s}$
$\displaystyle=\sum_{k=1}^{\infty}\int_{0}^{t}\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma_{k}\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}W^{k}_{s}.$

Then $(Z(t))_{t\geq 0}$ is a (local) martingale. It follows from (3.4) that,
for all $t\geq 0$,

$\displaystyle\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\frac{1}{2}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle=\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})+\frac{1}{2}\varepsilon^{2\beta-2}\int_{0}^{t}\int_{0}^{1}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}s+Z(t).$
(3.5)

Also, the quadratic variation of $Z(t)$ satisfies

$\displaystyle\langle Z\rangle(t)$
$\displaystyle=\sum_{k=1}^{\infty}\int_{0}^{t}\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma_{k}\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}^{2}\,\mathrm{d}s$
$\displaystyle\leq\varepsilon^{2(\beta-1)}\sum_{k=1}^{\infty}\int_{0}^{t}\|\sigma_{k}\|_{L^{2}}^{2}\big{\|}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{\|}_{L^{2}}^{2}\,\mathrm{d}s$
$\displaystyle\leq
2\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2}\int_{0}^{t}\|v_{\varepsilon}\|_{L^{2}}^{2}\,\mathrm{d}s+\frac{1}{2}\varepsilon^{2\beta}\|\sigma\|_{L^{\infty}}^{2}\int_{0}^{t}\int_{0}^{1}|(r_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\leq\frac{1}{2\kappa_{\varepsilon}}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s.$

Let

$\Phi(t):=\mathcal{G}(r_{\varepsilon},v_{\varepsilon})(t)+\frac{1}{4}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s.$

Then

$\displaystyle\Phi(t)-\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}t-\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})$
$\displaystyle\leq Z(t)-\frac{\kappa_{\varepsilon}}{2}\langle
Z\rangle(t)+\Big{(}\frac{\kappa_{\varepsilon}}{2}\langle
Z\rangle(t)-\frac{1}{4}\int_{0}^{t}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s\Big{)}$
$\displaystyle\leq Z(t)-\frac{\kappa_{\varepsilon}}{2}\langle Z\rangle(t).$

Thus, it follows from the exponential martingale estimates that, for all
$R>0$,

$\displaystyle\mathbb{P}\Big{[}\sup_{t\geq
0}\big{(}\Phi(t)-\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}t\big{)}-\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\geq
R\Big{]}$ $\displaystyle\leq\mathbb{P}\Big{[}\sup_{t\geq
0}\big{(}Z(t)-\frac{\kappa_{\varepsilon}}{2}\langle Z\rangle(t)(t)\big{)}\geq
R\Big{]}$ $\displaystyle\leq{\rm e}^{-\kappa_{\varepsilon}R},$

which is (3.1). To prove (3.2), fix $T>0$ and denote

$Y:=\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}T+\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon}).$

We also denote by $\chi$ the characteristic function of the set:
$\\{\sup_{0\leq t\leq T}\Phi(t)>Y\\}$. Then we have

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq T}\Phi(t)^{m}\Big{]}$
$\displaystyle\leq 2^{m-1}\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}(\Phi(t)-Y)^{m}\chi\Big{]}+2^{m-1}\mathbb{E}\big{[}Y^{m}\chi\big{]}+\mathbb{E}\Big{[}\sup_{0\leq
t\leq T}\Phi(t)^{m}(1-\chi)\Big{]}$ $\displaystyle\leq
2^{m-1}\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}(\Phi(t)-Y)^{m}\chi\Big{]}+2^{m-1}\mathbb{E}[Y^{m}]$
$\displaystyle=2^{m-1}\int_{0}^{\infty}\mathbb{P}\Big{[}\sup_{0\leq t\leq
T}(\Phi(t)-Y)^{m}\chi\geq\lambda\Big{]}\,\mathrm{d}\lambda+2^{m-1}\mathbb{E}\big{[}Y^{m}\big{]}$
$\displaystyle\leq 2^{m-1}\int_{0}^{\infty}{\rm
e}^{-\kappa_{\varepsilon}\lambda^{1/m}}\,\mathrm{d}\lambda+2^{m-1}\mathbb{E}\Big{[}\Big{(}\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}T+\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\Big{)}^{m}\Big{]}$
$\displaystyle\leq
c_{m}\Big{(}\kappa_{\varepsilon}^{-m}+\varepsilon^{2m\beta-2m}\|\sigma\|_{L^{\infty}}^{2m}T^{m}+\mathbb{E}\big{[}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})^{m}\big{]}\Big{)},$

as we want. Turning to (3.3), we fix $0<\delta<\frac{\kappa_{\varepsilon}}{2}$
and argue similarly to obtain

$\displaystyle\mathbb{E}\Big{[}\exp\big{(}\delta\sup_{0\leq t\leq
T}\Phi(t)\big{)}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\exp\big{(}\delta\sup_{0\leq t\leq
T}(\Phi(t)-Y)\chi\big{)}\exp(\delta Y\chi)\exp\big{(}\delta(1-\chi)\sup_{0\leq
t\leq T}\Phi(t)\big{)}\Big{]}$ $\displaystyle\leq\mathbb{E}\Big{[}\exp(\delta
Y)\exp\big{(}\delta\sup_{0\leq t\leq T}(\Phi(t)-Y)\chi\big{)}\Big{]}$
$\displaystyle\leq\sqrt{\mathbb{E}\big{[}\exp(2\delta
Y)\big{]}}\sqrt{\mathbb{E}\Big{[}\exp\big{(}2\delta\sup_{0\leq t\leq
T}(\Phi(t)-Y)\chi\big{)}\Big{]}}$
$\displaystyle\leq\sqrt{\mathbb{E}\big{[}\exp(2\delta
Y)\big{]}}\sqrt{\int_{1}^{\infty}\mathbb{P}\Big{[}\sup_{0\leq t\leq
T}(\Phi(t)-Y)\chi\geq\frac{1}{2\delta}\log\lambda\Big{]}\,\mathrm{d}\lambda}$
$\displaystyle\leq\sqrt{\mathbb{E}\Big{[}\exp\big{(}2\delta(\frac{1}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}T+\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon}))\big{)}\Big{]}}\sqrt{\int_{1}^{\infty}\lambda^{-\kappa_{\varepsilon}/(2\delta)}\,\mathrm{d}\lambda}$
$\displaystyle=C_{\delta}\exp\big{(}\frac{\delta}{2}\varepsilon^{2\beta-2}\|\sigma\|_{L^{\infty}}^{2}T\big{)}\sqrt{\mathbb{E}\big{[}\exp\big{(}2\delta\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\big{)}\big{]}}.$

This concludes the proof. ∎

###  3.2. Exponential contraction and stability

Due to the linearity of system (AS) and the additivity of the noise term, the
stability of the acoustic system can be analyzed through the corresponding
deterministic system. The main result of this section is the following:

######  Proposition 3.3.

Let $\varepsilon\in(0,1]$. Let $(r_{\varepsilon},v_{\varepsilon})\in
C([0,\infty);H^{1})\times C([0,\infty;H^{1}_{0})\cap L^{2}_{\text{\rm
loc}}(0,\infty;H^{2})$ be a solution to the deterministic system:

$\begin{cases}(r_{\varepsilon})_{t}+\frac{1}{\varepsilon}(v_{\varepsilon})_{x}=0,\\\
(v_{\varepsilon})_{t}+\frac{\gamma}{\varepsilon}(r_{\varepsilon})_{x}=(v_{\varepsilon})_{xx}.\end{cases}$
(3.6)

Then the following exponential stability estimates hold: For every $t\geq 0$,

$\displaystyle\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))\leq\mathcal{G}(r_{\varepsilon}(0,\cdot),v_{\varepsilon}(0,\cdot)){\rm
e}^{-\frac{t}{2}},$ (3.7)
$\displaystyle\|(r_{\varepsilon},\,v_{\varepsilon})(t,\cdot)\|_{H^{1}}^{2}\leq
8\gamma\|(r_{\varepsilon},\,v_{\varepsilon})(0,\cdot)\|_{H^{1}}^{2}{\rm
e}^{-\frac{t}{4}},$ (3.8)
$\displaystyle\|(r_{\varepsilon},\,v_{\varepsilon})(t,\cdot)\|_{L^{2}}^{2}+\varepsilon^{2}\|((r_{\varepsilon})_{x},(v_{\varepsilon})_{x})(t,\cdot)\|_{L^{2}}^{2}$
$\displaystyle\quad\leq
6\gamma\Big{(}\|(r_{\varepsilon},\,v_{\varepsilon})(0,\cdot)\|_{L^{2}}^{2}+\varepsilon^{2}\|((r_{\varepsilon})_{x},(v_{\varepsilon})_{x})(0,\cdot)\|_{L^{2}}^{2}\Big{)}{\rm
e}^{-\frac{t}{2}}.$ (3.9)

###### Proof.

First, notice the following pointwise inequality:

$\displaystyle\frac{1}{2}(\gamma
r_{\varepsilon}^{2}+v_{\varepsilon}^{2})+\frac{1}{2}\varepsilon
v_{\varepsilon}(r_{\varepsilon})_{x}+\frac{\varepsilon^{2}}{4}|(r_{\varepsilon})_{x}|^{2}\leq\frac{\gamma}{2}r_{\varepsilon}^{2}+\frac{3}{4}v_{\varepsilon}^{2}+\frac{\varepsilon^{2}}{2}|(r_{\varepsilon})_{x}|^{2}.$

Then, applying the Poincaré inequality for both $r_{\varepsilon}$ and
$v_{\varepsilon}$ in the second inequality, we have

$\displaystyle\mathcal{G}(r_{\varepsilon},v_{\varepsilon})$
$\displaystyle\leq\frac{\gamma+\varepsilon^{2}}{2}\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}+\frac{3}{4}\|v_{\varepsilon}\|_{L^{2}}^{2}\leq\gamma\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}+\|(v_{\varepsilon})_{x}\|_{L^{2}}^{2}$
(3.10)

for all $\varepsilon\in(0,1]$. By the equations, we have the following
equality (see (3.4)):

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})=-\frac{\gamma}{2}\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}-\frac{1}{2}\|(v_{\varepsilon})_{x}\|_{L^{2}}^{2},$
(3.11)

which implies

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})\leq-\frac{1}{2}\mathcal{G}(r_{\varepsilon},v_{\varepsilon}).$

The first estimate (3.7) then follows from the Gronwall inequality.

Now, differentiating the first equation in (3.6) with respect to $x$ gives

$\displaystyle((r_{\varepsilon})_{x})_{t}+\frac{1}{\varepsilon}(v_{\varepsilon})_{xx}=0.$

With this, multiplying the second equation of (3.6) by
$(v_{\varepsilon})_{xx}$ and integrating in $x$, we have

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{0}^{1}\frac{1}{2}|(v_{\varepsilon})_{x}|^{2}\,\mathrm{d}x+\int_{0}^{1}|(v_{\varepsilon})_{xx}|^{2}\,\mathrm{d}x=\int_{0}^{1}\frac{\gamma}{\varepsilon}(r_{\varepsilon})_{x}(v_{\varepsilon})_{xx}\,\mathrm{d}x=-\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{0}^{1}\frac{\gamma}{2}|(r_{\varepsilon})_{x}|^{2}\,\mathrm{d}x,$

which implies

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\Big{(}\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}$
$\displaystyle=-2\int_{0}^{1}|(v_{\varepsilon})_{xx}|^{2}\,\mathrm{d}x\leq-2\int_{0}^{1}|(v_{\varepsilon})_{x}|^{2}\,\mathrm{d}x.$
(3.12)

The last inequality is due to the Poincaré inequality applied to
$(v_{\varepsilon})_{x}$ that is of zero mean. Combining this with (3.11), we
have

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}$
$\displaystyle\leq-\frac{\gamma}{2}\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}-\frac{5}{2}\|(v_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\leq-\frac{1}{4}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}.$

The second estimate (3.8) now follows from the Gronwall inequality and the
bounds:

$\displaystyle\frac{1}{4}\int_{0}^{1}\big{(}\gamma
r_{\varepsilon}^{2}+v_{\varepsilon}^{2}+\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x$
$\displaystyle\leq\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x$
$\displaystyle\leq
2\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x.$

Similarly, (3.11)–(3.12) imply

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\varepsilon^{2}\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}$
$\displaystyle\leq-\frac{\gamma}{2}\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}-\frac{1}{2}\|(v_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\leq-\frac{1}{2}\Big{(}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\varepsilon^{2}\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}.$

The last estimate (3.9) now follows from the Gronwall inequality and the
bounds:

$\displaystyle\frac{1}{4}\int_{0}^{1}\big{(}r_{\varepsilon}^{2}+v_{\varepsilon}^{2}+\varepsilon|(r_{\varepsilon})_{x}|^{2}+\varepsilon|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x$
$\displaystyle\leq\mathcal{G}(r_{\varepsilon},v_{\varepsilon})+\varepsilon^{2}\int_{0}^{1}\big{(}\gamma|(r_{\varepsilon})_{x}|^{2}+|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x$
$\displaystyle\leq\frac{3\gamma}{2}\int_{0}^{1}\big{(}r_{\varepsilon}^{2}+v_{\varepsilon}^{2}+\varepsilon|(r_{\varepsilon})_{x}|^{2}+\varepsilon|(v_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x.$

This concludes the proof. ∎

A straightforward consequence of the above result is the stability and
continuous dependence of solutions of problem (1.7)–(1.8) for system (AS),
which follows from the observation that the difference of any two solutions of
system (AS) is a solution to the deterministic system (3.6).

######  Corollary 3.4.

Let $\varepsilon\in(0,1]$. Let $(r_{1,\varepsilon},v_{1,\varepsilon})$ and
$(r_{2,\varepsilon},v_{2,\varepsilon})$ be two solutions of problem
(1.7)–(1.8) for the stochastic system (AS). Then, for all $t\geq 0$,

$\displaystyle\big{\|}(r_{1,\varepsilon}-r_{2,\varepsilon},\,v_{1,\varepsilon}-v_{2,\varepsilon})(t,\cdot)\big{\|}_{H^{1}}^{2}\leq
8\gamma\big{\|}(r_{1,\varepsilon}-r_{2,\varepsilon},\,v_{1,\varepsilon}-v_{2,\varepsilon})(0,\cdot)\big{\|}_{H^{1}}^{2}{\rm
e}^{-\frac{t}{4}},$
$\displaystyle\big{\|}(r_{1,\varepsilon}-r_{2,\varepsilon},\,v_{1,\varepsilon}-v_{2,\varepsilon})(t,\cdot)\big{\|}_{L^{2}}^{2}+\varepsilon^{2}\big{\|}((r_{1,\varepsilon})_{x}-(r_{2,\varepsilon})_{x},\,(v_{1,\varepsilon})_{x}-(v_{2,\varepsilon})_{x})(t,\cdot)\big{\|}_{L^{2}}^{2}$
$\displaystyle\,\,\,\leq
6\gamma\Big{(}\big{\|}(r_{1,\varepsilon}-r_{2,\varepsilon},\,v_{1,\varepsilon}-v_{2,\varepsilon})(0,\cdot)\big{\|}_{L^{2}}^{2}+\varepsilon^{2}\big{\|}((r_{1,\varepsilon})_{x}-(r_{2,\varepsilon})_{x},\,(v_{1,\varepsilon})_{x}-(v_{2,\varepsilon})_{x})(0,\cdot)\big{\|}_{L^{2}}^{2}\Big{)}{\rm
e}^{-\frac{t}{2}},$

almost surely.

###  3.3. Long-time behavior

We can further explore the dissipative property of system (AS) to establish
the following energy dissipative estimates.

######  Proposition 3.5.

Let $(r_{\varepsilon},v_{\varepsilon})$ be the solution of system (AS) with
initial data $(r_{0,\varepsilon},v_{0,\varepsilon})$. Then, for all $p\geq 1$,
there exists $C_{p}>0$ independent of $\varepsilon$ such that, for all $t\geq
0$,

$\displaystyle\mathbb{E}\big{[}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))^{p}\big{]}\leq
C_{p}\Big{(}\mathbb{E}\big{[}{\rm
e}^{-\frac{pt}{2}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})^{p}\big{]}+\varepsilon^{2(\beta-1)p}\|\sigma\|_{L^{\infty}}^{2p}\Big{)}.$
(3.13)

Moreover, for all
$0<\eta<(8\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2})^{-1}$, there
exists $C_{\eta}>0$ independent of $\varepsilon$ such that, for all $t\geq 0$,

$\displaystyle\mathbb{E}\big{[}\exp\big{(}\eta\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot)\big{)}\big{]}\leq
C_{\eta}\exp\big{(}2\eta\varepsilon^{2(\beta-1)}\big{)}\sqrt{\mathbb{E}\big{[}\exp\big{(}2{\rm
e}^{-\frac{t}{4}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\big{)}\big{]}}.$
(3.14)

###### Proof.

From (3.4) and (3.10), we obtain

$\displaystyle\,\mathrm{d}\big{(}{\rm
e}^{\frac{t}{4}}\mathcal{G}(r_{\varepsilon},v_{\varepsilon})\big{)}+\frac{1}{4}\Big{(}{\rm
e}^{\frac{t}{4}}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}t$
$\displaystyle\leq\Big{(}{\rm
e}^{\frac{t}{4}}\int_{0}^{1}\frac{1}{2}\varepsilon^{2(\beta-1)}\sigma^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t+\Big{(}{\rm
e}^{\frac{t}{4}}\int_{0}^{1}\varepsilon^{\beta-1}\sigma\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}W_{t}.$

Hence, for all $t\geq 0$, we have

$\displaystyle\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))$
$\displaystyle\leq{\rm
e}^{-\frac{t}{4}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})+2\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\|\sigma\|_{L^{\infty}}^{2}$
$\displaystyle\quad+\varepsilon^{\beta-1}\int_{0}^{t}{\rm
e}^{-\frac{t-s}{4}}\int_{0}^{1}\sigma\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\,\mathrm{d}W_{s}$
$\displaystyle\quad-\frac{1}{4}\Big{(}\int_{0}^{t}{\rm
e}^{-\frac{t-s}{4}}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\Big{)}\,\mathrm{d}s.$

Therefore, for all $t\geq 0$ and $R>0$, we deduce

$\displaystyle\mathbb{P}\Big{[}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))-{\rm
e}^{-\frac{t}{2}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})-2\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\|\sigma\|_{L^{\infty}}^{2}\geq R\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}\varepsilon^{\beta-1}\int_{0}^{t}{\rm
e}^{-\frac{t-s}{2}}\int_{0}^{1}\sigma\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\,\mathrm{d}W_{s}-\frac{1}{4}\int_{0}^{t}{\rm
e}^{-\frac{t-s}{4}}\int_{0}^{1}\big{(}|(v_{\varepsilon})_{x}|^{2}+|(r_{\varepsilon})_{x}|^{2}\big{)}\,\mathrm{d}x\,\mathrm{d}s\geq
R\Big{]}$ $\displaystyle\leq{\rm e}^{-\kappa_{\varepsilon}R},$

where
$\kappa_{\varepsilon}=(4\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2})^{-1}$
is the same in Proposition 3.2. The last inequality is due to
[Mattingly02]*Lemma A.1 and the fact that

$\displaystyle\sum_{k=1}^{\infty}\Big{(}\int_{0}^{1}\varepsilon^{\beta-1}\sigma_{k}\big{(}v_{\varepsilon}+\frac{1}{2}\varepsilon(r_{\varepsilon})_{x}\big{)}\,\mathrm{d}x\Big{)}^{2}\leq
2\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2}\Big{(}\|(v_{\varepsilon})_{x}\|_{L^{2}}^{2}+\frac{1}{4}\varepsilon^{2}\|(r_{\varepsilon})_{x}\|_{L^{2}}^{2}\Big{)}.$

As a result, we use a computation similar to the proof of Proposition 3.2 to
obtain that, for all $p\geq 1$,

$\displaystyle\mathbb{E}\Big{[}\big{(}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))-{\rm
e}^{-\frac{t}{2}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})-2\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\|\sigma\|_{L^{\infty}}^{2}\big{)}^{p}\Big{]}$
$\displaystyle\leq p\int_{0}^{\infty}\lambda^{p-1}{\rm
e}^{-\kappa_{\varepsilon}\lambda}\,\mathrm{d}\lambda=\kappa_{\varepsilon}^{-p}p!\,p.$

Consequently, we have

$\displaystyle\mathbb{E}\big{[}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))^{p}\big{]}$
$\displaystyle\qquad\leq 3^{p-1}\mathbb{E}\Big{[}{\rm
e}^{-\frac{pt}{2}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})^{p}\Big{]}+3^{p-1}\,2^{p}\,\varepsilon^{2(\beta-1)p}(1-{\rm
e}^{-\frac{t}{4}})^{p}\|\sigma\|_{L^{\infty}}^{2p}$
$\displaystyle\qquad\quad+3^{p-1}\mathbb{E}\Big{[}\big{(}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))-{\rm
e}^{-\frac{t}{4}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})-2\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\|\sigma\|_{L^{\infty}}^{2}\big{)}^{p}\Big{]}$
$\displaystyle\qquad=3^{p-1}\mathbb{E}\Big{[}{\rm
e}^{-\frac{pt}{2}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})^{p}\Big{]}+3^{p-1}\,2^{p}\,\varepsilon^{2(\beta-1)p}(1-{\rm
e}^{-\frac{t}{4}})^{p}\|\sigma\|_{L^{\infty}}^{2p}+\kappa_{\varepsilon}^{-p}p!\,p,$

which leads to (3.13). Likewise, we can deduce that, for all
$\eta\in(0,\kappa_{\varepsilon})$,

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\eta\big{(}\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))-{\rm
e}^{-\frac{t}{4}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})-2\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\|\sigma\|_{L^{\infty}}^{2}\big{)}\Big{)}\Big{]}$
$\displaystyle\leq
1+\int_{1}^{\infty}\lambda^{-\frac{\kappa_{\varepsilon}}{\eta}}\,\mathrm{d}\lambda=\frac{\kappa_{\varepsilon}}{\kappa_{\varepsilon}-\eta}.$

Hence, for all $t\geq 0$ and $\eta\in(0,\frac{\kappa_{\varepsilon}}{2})$,

$\mathbb{E}\big{[}\exp\big{(}\eta\mathcal{G}(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot)\big{)}\big{]}\leq\exp\big{(}2\eta\varepsilon^{2(\beta-1)}(1-{\rm
e}^{-\frac{t}{4}})\big{)}\sqrt{\frac{\kappa_{\varepsilon}}{\kappa_{\varepsilon}-2\eta}\mathbb{E}\big{[}\exp\big{(}2{\rm
e}^{-\frac{t}{4}}\mathcal{G}(r_{0,\varepsilon},v_{0,\varepsilon})\big{)}\big{]}},$

from which (3.14) follows. This completes the proof. ∎

##  4\. Finite-Time Convergence: Proof of Theorem 1.1

In this section, we gather the estimates established in the previous sections
to provide a proof of Theorem 1.1. We first estimate the difference between
$(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ and
$(r_{\varepsilon},v_{\varepsilon})$.

######  Proposition 4.1.

There exists a generic constant $C>0$, independent of any parameters, such
that, for all $t\geq 0$, the following estimate holds almost surely:

$\displaystyle\gamma\|\bar{\rho}_{\varepsilon}(t)-r_{\varepsilon}(t)\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}(t)-v_{\varepsilon}(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s$
$\displaystyle\leq\exp\Big{(}\int_{0}^{t}\frac{\varepsilon}{\gamma}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s\Big{)}$
$\displaystyle\quad\times\Big{(}\gamma\|\bar{\rho}_{\varepsilon}(0)-r_{\varepsilon}(0)\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}(0)-v_{\varepsilon}(0)\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{t}\int_{0}^{1}\bar{u}_{\varepsilon}^{4}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\qquad\quad+C\varepsilon^{4}\int_{0}^{t}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}x\,\mathrm{d}s$
$\displaystyle\qquad\quad+C|\gamma-2|\varepsilon^{2}\int_{0}^{t}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x\,\mathrm{d}s+C\gamma^{2}\varepsilon\int_{0}^{t}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s\Big{)}.$
(4.1)

###### Proof.

Taking the difference of the first equation of (FS) and the first equation of
(AS), and multiplying it by $(\bar{\rho}_{\varepsilon}-r_{\varepsilon})$, we
have

$\displaystyle\frac{1}{2}\big{(}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})^{2}\big{)}_{t}+\frac{1}{\varepsilon}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})=-\varepsilon(\bar{\rho}_{\varepsilon}\bar{u}_{\varepsilon})_{x}(\bar{\rho}_{\varepsilon}-r_{\varepsilon}).$

Likewise, taking the difference of the second equation of (FS) and the second
equation of (AS), and multiplying it with
$(\bar{u}_{\varepsilon}-v_{\varepsilon})$, we obtain

$\displaystyle\frac{1}{2}\big{(}(\bar{u}_{\varepsilon}-v_{\varepsilon})^{2}\big{)}_{t}+\frac{\gamma}{\varepsilon}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})$
$\displaystyle=(\bar{u}_{\varepsilon}-v_{\varepsilon})_{xx}(\bar{u}_{\varepsilon}-v_{\varepsilon})+\frac{\gamma}{\varepsilon}\big{(}1-(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-2}\big{)}(\bar{\rho}_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})$
$\displaystyle\quad-\varepsilon^{2}\frac{\bar{\rho}_{\varepsilon}(\bar{u}_{\varepsilon})_{xx}}{1+\varepsilon^{2}\bar{\rho}_{\varepsilon}}(\bar{u}_{\varepsilon}-v_{\varepsilon})-\varepsilon
uu_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon}).$

Summing the two equations and integrating in $x\in(0,1)$, we obtain

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\big{(}\gamma\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}-v_{\varepsilon}\|_{L^{2}}^{2}\big{)}+2\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle=-2\gamma\varepsilon\int_{0}^{1}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})(\bar{\rho}_{\varepsilon})_{x}\bar{u}_{\varepsilon}\,\mathrm{d}x-2\gamma\varepsilon\int_{0}^{1}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})\bar{\rho}_{\varepsilon}(\bar{u}_{\varepsilon})_{x}\,\mathrm{d}x$
$\displaystyle\quad+\frac{2\gamma}{\varepsilon}\int_{0}^{1}\big{(}1-(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-2}\big{)}(\bar{\rho}_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x$
$\displaystyle\quad+\varepsilon\int_{0}^{1}\bar{u}_{\varepsilon}^{2}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\,\mathrm{d}x-2\varepsilon^{2}\int_{0}^{1}\frac{\bar{\rho}_{\varepsilon}}{1+\varepsilon^{2}\bar{\rho}_{\varepsilon}}(\bar{u}_{\varepsilon})_{xx}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x$
$\displaystyle=:\sum_{i=1}^{5}I_{i}.$

Notice that

$\displaystyle I_{1}+I_{2}$
$\displaystyle=-2\gamma\varepsilon\int_{0}^{1}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})(\bar{\rho}_{\varepsilon})_{x}\bar{u}_{\varepsilon}\,\mathrm{d}x-2\gamma\varepsilon\int_{0}^{1}(\bar{\rho}_{\varepsilon}-r_{\varepsilon})\bar{\rho}_{\varepsilon}(\bar{u}_{\varepsilon})_{x}\,\mathrm{d}x$
$\displaystyle\leq
2\gamma\varepsilon\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}\|\bar{u}_{\varepsilon}\|_{L^{\infty}}+2\gamma\varepsilon\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}$
$\displaystyle\leq
4\gamma\varepsilon\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}$
$\displaystyle\leq\varepsilon\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+C\gamma^{2}\varepsilon\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}.$

Next, we estimate $I_{3}$. For $\gamma=1$,

$\displaystyle I_{3}$ $\displaystyle\leq
2\varepsilon\left|\int_{0}^{1}\frac{\bar{\rho}_{\varepsilon}(\bar{\rho}_{\varepsilon})_{x}}{1+\varepsilon^{2}\bar{\rho}_{\varepsilon}}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x\right|$
$\displaystyle=2\varepsilon\left|\int_{0}^{1}\Big{(}\frac{1}{\varepsilon^{2}}\bar{\rho}_{\varepsilon}-\frac{1}{\varepsilon^{4}}\log(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})\Big{)}_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x\right|$
$\displaystyle=2\varepsilon\left|\int_{0}^{1}\Big{(}\frac{1}{\varepsilon^{2}}\bar{\rho}_{\varepsilon}-\frac{1}{\varepsilon^{4}}\log(1+\varepsilon^{2}\rho)\Big{)}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\,\mathrm{d}x\right|$
$\displaystyle\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{1}\Big{(}\frac{1}{\varepsilon^{2}}\bar{\rho}_{\varepsilon}-\frac{1}{\varepsilon^{4}}\log(1+\varepsilon^{2}\rho)\Big{)}^{2}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{1}\frac{|\bar{\rho}_{\varepsilon}|^{4}}{(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{2}}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-1}\|_{L^{\infty}}^{2}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x,$

where we used here the inequality:
$\frac{1}{\varepsilon^{2}}x-\frac{1}{\varepsilon^{4}}\log(1+\varepsilon^{2}x)\leq\frac{x^{2}}{1+\varepsilon^{2}x}$,
which is true for all $x$ with $1+\varepsilon^{2}x>0$.

For $\gamma>1$ and $\gamma\neq 2$, we proceed as

$\displaystyle I_{3}$
$\displaystyle\leq\frac{2\gamma}{\varepsilon}\left|\int_{0}^{1}\big{(}1-(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-2}\big{)}(\bar{\rho}_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x\right|$
$\displaystyle=\frac{2\gamma}{\varepsilon^{3}}\left|\int_{0}^{1}\Big{(}1+\varepsilon^{2}\bar{\rho}_{\varepsilon}-\frac{1}{\gamma-1}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-1}\Big{)}_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x\right|$
$\displaystyle=\frac{2\gamma}{\varepsilon^{3}}\left|\int_{0}^{1}\Big{(}1+\varepsilon^{2}\bar{\rho}_{\varepsilon}-\frac{1}{\gamma-1}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-1}\Big{)}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\,\mathrm{d}x\right|$
$\displaystyle\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+\frac{C}{\varepsilon^{6}}\int_{0}^{1}\Big{(}1+\varepsilon^{2}\bar{\rho}_{\varepsilon}-\frac{1}{\gamma-1}(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-1}\Big{)}^{2}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C|\gamma-2|\varepsilon^{2}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x.$

Lastly, for $\gamma=2$, $I_{3}=0$.

In all cases, we have

$I_{3}\leq\frac{1}{4}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C|\gamma-2|\varepsilon^{2}\,\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x.$

The fourth integral can be bounded as

$I_{4}=\varepsilon\int_{0}^{1}\bar{u}_{\varepsilon}^{2}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\,\mathrm{d}x\leq\frac{1}{8}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{1}|\bar{u}_{\varepsilon}|^{4}\,\mathrm{d}x.$
(4.2)

Finally, we have

$\displaystyle I_{5}$
$\displaystyle=-2\varepsilon^{2}\int_{0}^{1}\frac{\bar{\rho}_{\varepsilon}}{1+\varepsilon^{2}\bar{\rho}_{\varepsilon}}(\bar{u}_{\varepsilon})_{xx}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x$
$\displaystyle=2\varepsilon^{2}\int_{0}^{1}\frac{(\bar{\rho}_{\varepsilon})_{x}}{(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{2}}(\bar{u}_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})\,\mathrm{d}x+2\varepsilon^{2}\int_{0}^{1}\frac{\bar{\rho}_{\varepsilon}}{1+\varepsilon^{2}\bar{\rho}_{\varepsilon}}(\bar{u}_{\varepsilon})_{x}(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{16}\|\bar{u}_{\varepsilon}-v_{\varepsilon}\|_{L^{\infty}}^{2}+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\quad\,+\frac{1}{16}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-1}\|_{L^{\infty}}^{2}\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\leq\frac{1}{8}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-1}\|_{L^{\infty}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\big{(}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}+\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}}^{2}\big{)}$
$\displaystyle\leq\frac{1}{8}\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}.$

Combining all the above, we conclude the following inequality almost surely:

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\Big{(}\gamma\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}-v_{\varepsilon}\|_{L^{2}}^{2}\Big{)}+\|(\bar{u}_{\varepsilon}-v_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\leq\varepsilon\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\quad+C|\gamma-2|\varepsilon^{2}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x+C\gamma^{2}\varepsilon\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{1}|\bar{u}_{\varepsilon}|^{4}\,\mathrm{d}x$
$\displaystyle\leq\frac{\varepsilon}{\gamma}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\big{(}\gamma\|\bar{\rho}_{\varepsilon}-r_{\varepsilon}\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}-v_{\varepsilon}\|_{L^{2}}^{2}\big{)}$
$\displaystyle\quad+C\gamma^{2}\varepsilon\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}+C\varepsilon^{2}\int_{0}^{1}|\bar{u}_{\varepsilon}|^{4}\,\mathrm{d}x+C\varepsilon^{4}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}$
$\displaystyle\quad+C|\gamma-2|\varepsilon^{2}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x.$

Note that, by estimate (2.8) in Proposition 2.7, there exists $C>0$ such that,
for any sufficiently small $\varepsilon>0$ and all $t>0$,

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{t}\frac{\varepsilon}{\gamma}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}s\Big{]}\leq
C\big{(}\varepsilon^{\alpha-2}+\varepsilon^{2\beta-2}\big{)}<\infty.$

This implies that
$\frac{\varepsilon}{\gamma}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}$ is
locally integrable in time almost surely for sufficiently small
$\varepsilon>0$. The Gronwall’s inequality applied to (4) gives the desired
result. ∎

We now complete the proof of Theorem 1.1. Fix $T>0$. Since (4.1) is valid for
all $0\leq t\leq T$, we take the expectation and use the Cauchy-Schwartz
inequality to obtain

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\big{(}\gamma\|\bar{\rho}_{\varepsilon}(t)-r_{\varepsilon}(t)\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}(t)-v(t)\|_{L^{2}}^{2}\big{)}\Big{]}$
$\displaystyle\leq\bigg{(}\mathbb{E}\Big{[}\exp\Big{(}\frac{2}{\gamma}\int_{0}^{T}\varepsilon\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}\Big{]}\bigg{)}^{\frac{1}{2}}\bigg{(}\mathbb{E}\Big{[}\big{(}\gamma\|\bar{\rho}_{\varepsilon}(0)-r_{\varepsilon}(0)\|_{L^{2}}^{2}+\|\bar{u}_{\varepsilon}(0)-v(0)\|_{L^{2}}^{2}\big{)}^{2}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\quad+\bigg{(}\mathbb{E}\Big{[}\exp\Big{(}\frac{2}{\gamma}\int_{0}^{T}\varepsilon\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\qquad\,\times\bigg{(}\mathbb{E}\Big{[}\Big{(}C\varepsilon^{2}\int_{0}^{T}\int_{0}^{1}|\bar{u}_{\varepsilon}|^{4}\,\mathrm{d}x\,\mathrm{d}t+C\varepsilon^{4}\int_{0}^{T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle\qquad\qquad\qquad+C|\gamma-2|\varepsilon^{2}\int_{0}^{T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x\,\mathrm{d}t+C\gamma^{2}\varepsilon\int_{0}^{T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\Big{]}\bigg{)}^{\frac{1}{2}}.$
(4.3)

We can control the expression on the right-hand side by using the estimates on
the derivation variables obtained in Proposition 2.7. To begin with, since
$\alpha>1$ and $\beta>\frac{1}{2}$, one may take $\varepsilon$ to be
sufficiently small such that the choice $\delta=\frac{2\varepsilon}{\gamma}$
satisfies constraint (2.8). Hence, Proposition 2.7 implies that, for all
sufficiently small $\varepsilon>0$,

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{2}{\gamma}\int_{0}^{T}\varepsilon\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}\Big{]}\leq
C.$

Next, using also Proposition 2.7, we have

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\int_{0}^{1}\bar{u}_{\varepsilon}^{4}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\|\bar{u}_{\varepsilon}\|_{L^{\infty}}^{2}\|\bar{u}_{\varepsilon}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|\bar{u}_{\varepsilon}\|_{L^{2}}^{2}\int_{0}^{T}\|(\bar{u}_{\varepsilon})_{x}\|^{2}_{L^{2}}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq t\leq
T}\|\bar{u}_{\varepsilon}\|_{L^{2}}^{2}\Big{)}^{4}\Big{]}\bigg{)}^{\frac{1}{2}}\bigg{(}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}\|(\bar{u}_{\varepsilon})_{x}\|^{2}_{L^{2}}\,\mathrm{d}t\Big{)}^{4}\Big{]}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C\big{(}\varepsilon^{\alpha-2}+\varepsilon^{2\beta-2}\big{)}^{4}.$

Likewise, Proposition 2.7 also gives

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\sup_{0\leq
t\leq
T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\int_{0}^{T}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{-2}\|_{L^{\infty}}^{2}\Big{)}^{6}\Big{]}\bigg{)}^{\frac{1}{3}}\bigg{(}\mathbb{E}\Big{[}\Big{(}\sup_{0\leq
t\leq
T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\Big{)}^{6}\Big{]}\bigg{)}^{\frac{1}{3}}\bigg{(}\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}\|(\bar{u}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{6}\Big{]}\bigg{)}^{\frac{1}{3}}$
$\displaystyle\leq
C\varepsilon^{-4}\big{(}\varepsilon^{\alpha-2}+\varepsilon^{2\beta-2}\big{)}^{4}.$

By Proposition 2.7 again, we have

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{1}|\bar{\rho}_{\varepsilon}|^{4}\,\mathrm{d}x\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\int_{0}^{T}\|\bar{\rho}_{\varepsilon}\|_{L^{\infty}}^{2}\|\bar{\rho}_{\varepsilon}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\Big{(}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}\sup_{0\leq
t\leq
T}\|\bar{\rho}_{\varepsilon}\|_{L^{2}}^{2}\int_{0}^{T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\bigg{(}\\!\mathbb{E}\Big{[}\sup_{0\leq t\leq
T}\|(1+\varepsilon^{2}\bar{\rho}_{\varepsilon})^{\gamma-3}\|_{L^{\infty}}^{6}\Big{]}\bigg{)}^{\frac{1}{3}}\\!\bigg{(}\\!\mathbb{E}\Big{[}\Big{(}\sup_{0\leq
t\leq
T}\|\bar{\rho}_{\varepsilon}\|_{L^{2}}^{2}\,\mathrm{d}x\Big{)}^{\\!6}\Big{]}\bigg{)}^{\frac{1}{3}}\\!\bigg{(}\\!\mathbb{E}\Big{[}\Big{(}\int_{0}^{T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{\\!\\!6}\Big{]}\bigg{)}^{\frac{1}{3}}$
$\displaystyle\leq
C\big{(}\varepsilon^{\alpha-2}+\varepsilon^{2\beta-2}\big{)}^{4}.$

Finally, a further application of Proposition 2.7 entails

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int_{0}^{T}\|(\bar{\rho}_{\varepsilon})_{x}\|_{L^{2}}^{2}\,\mathrm{d}t\Big{)}^{2}\bigg{]}\leq
C\big{(}\varepsilon^{\alpha-2}+\varepsilon^{2\beta-2}\big{)}^{2}.$

Combining all of the above shows that the right-hand side of
(LABEL:eq:L^2_diff_exp_control) can be controlled as stated in (1.13), so that
the proof of Theorem 1.1 is now completed.

##  5\. Convergence of Invariant Measures

In this section, we consider the convergence of invariant measures of system
(FS) to the corresponding invariant measures of system (AS) as $\varepsilon\to
0$, which leads to the complete proof of Theorem 1.2. We restrict to the case
that $\gamma=1$, the only case that the existence of invariant measures for
the Navier-Stokes Equations (NS) (and hence (FS)) has been known so far.

###  5.1. Invariant measures for the fluctuation system

Let

$\bar{\mathcal{X}}_{\varepsilon}=\Big{\\{}(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})\in
H^{1}(0,1)\times
H^{1}_{0}(0,1)\,:\,\int_{0}^{1}\bar{\rho}_{\varepsilon}\,\mathrm{d}x=0,\,\,1+\varepsilon^{2}\bar{\rho}_{\varepsilon}>0\Big{\\}}$

be the state space for system (FS). Following [michele], we equip
$\bar{\mathcal{X}}_{\varepsilon}$ with the $L^{2}\times L^{2}$ metric (see
Remark 5.1). The Markov semigroup
$(\bar{\mathcal{P}}_{t}^{\varepsilon})_{t\geq 0}$ associated to system (FS) is
a collection of operators
$\bar{\mathcal{P}}_{t}^{\varepsilon}:\mathcal{M}_{b}(\bar{\mathcal{X}}_{\varepsilon})\to\mathcal{M}_{b}(\bar{\mathcal{X}}_{\varepsilon})$
on the set of all bounded Borel measurable functions on
$\bar{\mathcal{X}}_{\varepsilon}$ given by

$\displaystyle\bar{\mathcal{P}}_{t}^{\varepsilon}(\phi)(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})=\mathbb{E}\big{[}\phi(\bar{\rho}_{\varepsilon}(t,\cdot),\bar{u}_{\varepsilon}(t,\cdot))\big{]}$

for all $\phi\in\mathcal{M}_{b}(\bar{\mathcal{X}}_{\varepsilon})$ and
$(\rho_{0,\varepsilon},u_{0,\varepsilon})\in\bar{\mathcal{X}}_{\varepsilon}$,
where $(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})$ is the solution of
problem (1.5)–(1.6) for system (FS) with initial data
$(\bar{\rho}_{0,\varepsilon},\bar{u}_{0,\varepsilon})\in\bar{\mathcal{X}}_{\varepsilon}$.
A Borel probability measure $\bar{\mu}_{\varepsilon}$ on
$\bar{\mathcal{X}}_{\varepsilon}$ is said to be an invariant measure of
semigroup $(\bar{\mathcal{P}}_{t}^{\varepsilon})_{t\geq 0}$ if

$(\bar{\mathcal{P}}_{t}^{\varepsilon})^{*}\bar{\mu}_{\varepsilon}=\bar{\mu}_{\varepsilon}\qquad\text{for
all }t\geq 0,$

where $(\bar{\mathcal{P}}_{t}^{\varepsilon})^{*}\bar{\mu}_{\varepsilon}$ is
the push-forward measure of $\bar{\mu}_{\varepsilon}$ by
$(\bar{\mathcal{P}}_{t}^{\varepsilon})^{*}$. Equivalently, the probability
measure $\bar{\mu}_{\varepsilon}$ satisfies

$\int_{\bar{\mathcal{X}}_{\varepsilon}}\bar{\mathcal{P}}_{t}^{\varepsilon}(\phi)\,\mathrm{d}\bar{\mu}_{\varepsilon}=\int_{\bar{\mathcal{X}}_{\varepsilon}}\phi\,\mathrm{d}\bar{\mu}_{\varepsilon}\qquad\,\,\mbox{for
all $t\geq 0$ and $\phi\in\mathcal{M}_{b}(\bar{\mathcal{X}}_{\varepsilon})$}.$

An invariant measure represents the law of a statistically stationary
solution.

We briefly describe the main result in [michele], which implies the existence
of an invariant probability measure $\bar{\mu}_{\varepsilon}$ for
$(\bar{\mathcal{P}}_{t}^{\varepsilon})_{t\geq 0}$. In [michele], the Naiver-
Stokes equations (NS) are considered in the phase space

$\displaystyle\mathcal{X}=\Big{\\{}(\rho,u)\in H^{1}(0,1)\times
H^{1}_{0}(0,1)\,:\,\int_{0}^{1}\rho\,\mathrm{d}x=1,\,\,\rho>0\Big{\\}}$

equipped with the $L^{2}\times L^{2}$ norm. For each $\varepsilon>0$, it has
been proved that there exists an invariant measure $\mu_{\varepsilon}$ for the
corresponding Markov semigroup $(\mathcal{P}_{t}^{\varepsilon})_{t\geq 0}$.
Now we can define a bijection
$\mathcal{H}_{\varepsilon}:\bar{\mathcal{X}}_{\varepsilon}\to\mathcal{X}$ by

$\mathcal{H}_{\varepsilon}(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})=(1+\varepsilon^{2}\bar{\rho}_{\varepsilon},\,\varepsilon\bar{u}_{\varepsilon}).$

Then
$\bar{\mu}_{\varepsilon}:=(\mathcal{H}_{\varepsilon}^{-1})^{*}\mu_{\varepsilon}$
is an invariant measure for semigroup
$(\bar{\mathcal{P}}_{t}^{\varepsilon})_{t\geq 0}$. Indeed, for all
$\phi\in\mathcal{M}_{b}(\bar{\mathcal{X}}_{\varepsilon})$,

$\displaystyle\int_{\bar{\mathcal{X}}_{\varepsilon}}\bar{\mathcal{P}}_{t}^{\varepsilon}\phi(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})\,\mathrm{d}\bar{\mu}_{\varepsilon}$
$\displaystyle=\int_{\mathcal{X}}\bar{\mathcal{P}}_{t}^{\varepsilon}\phi\big{(}\mathcal{H}_{\varepsilon}^{-1}(\rho_{\varepsilon},u_{\varepsilon})\big{)}\,\mathrm{d}\mu_{\varepsilon}=\int_{\mathcal{X}}\mathcal{P}_{t}^{\varepsilon}\big{(}\phi\circ\mathcal{H}_{\varepsilon}^{-1}\big{)}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}\mu_{\varepsilon}$
$\displaystyle=\int_{\mathcal{X}}\phi\circ\mathcal{H}_{\varepsilon}^{-1}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}\mu_{\varepsilon}=\int_{\bar{\mathcal{X}}_{\varepsilon}}\phi(\bar{\rho}_{\varepsilon},\bar{u}_{\varepsilon})\,\mathrm{d}\bar{\mu}_{\varepsilon}.$

######  Remark 5.1.

The state space $\mathcal{X}$ considered in [michele], equipped with the
$L^{2}\times L^{2}$ norm, is a non-complete metric space. Nonetheless, one can
establish an _approximate Feller_ property for
$\mathcal{P}_{t}^{\varepsilon}$, which in turn facilitates the construction of
an invariant measure. However, since the $L^{2}\times L^{2}$ topology and the
$H^{1}\times H^{1}_{0}$ topology generate the same Borel $\sigma$-algebra on
$\mathcal{X}$, the invariance of a Borel probability measure with respect to
$\mathcal{P}_{t}^{\varepsilon}$ is not affected by the norm chosen.

###  5.2. Invariant measures for the acoustic system

The state space for system (AS) is given in (1.14) by

$\displaystyle\bar{\mathcal{X}}=\Big{\\{}(r,v)\in H^{1}(0,1)\times
H^{1}_{0}(0,1)\,:\,\int_{0}^{1}r\,\mathrm{d}x=0\Big{\\}}.$

We may equip the space, $\bar{\mathcal{X}}$, with either the $L^{2}\times
L^{2}$ norm or the $H^{1}\times H^{1}_{0}$ norm333For the same reason as in
Remark 5.1, the choice of the norm does not affect the invariance of a Borel
probability measure., and we denote them by $\bar{\mathcal{X}}_{L^{2}}$ and
$\bar{\mathcal{X}}_{H^{1}}$ respectively when we want to emphasis the choice
of the metric. Similar to the above case, given the existence and uniqueness
of solutions to system (AS), there is a well-defined semigroup
$(\mathcal{Q}_{t}^{\varepsilon})_{t\geq 0}$ associated to (AS) in
$\bar{\mathcal{X}}$ for all $\varepsilon>0$. More specifically, for all $t\geq
0$, map
$\mathcal{Q}_{t}^{\varepsilon}:\mathcal{M}_{b}(\bar{\mathcal{X}})\to\mathcal{M}_{b}(\bar{\mathcal{X}})$
is given by

$\displaystyle\mathcal{Q}_{t}^{\varepsilon}(\phi)(r_{0,\varepsilon},v_{0,\varepsilon})=\mathbb{E}\phi(r_{\varepsilon}(t,\cdot),v_{\varepsilon}(t,\cdot))$

for all $\phi\in\mathcal{M}_{b}(\bar{\mathcal{X}})$ and
$(r_{0,\varepsilon},v_{0,\varepsilon})\in\bar{\mathcal{X}}$, where
$(r_{\varepsilon},v_{\varepsilon})$ is the solution of problem (1.7)–(1.8) for
system (AS) with initial data $(r_{0,\varepsilon},v_{0,\varepsilon})$.
Moreover, there is a unique invariant measure $\omega_{\varepsilon}$ for
$(\mathcal{Q}_{t}^{\varepsilon})_{t\geq 0}$ (see Corollary 5.3 below).

In order to compare the invariant measures between systems (FS) and (AS), we
want to embed $\bar{\mathcal{X}}_{\varepsilon}$ into $\bar{\mathcal{X}}$. Note
that
$\bar{\mathcal{X}}_{\varepsilon_{2}}\subset\bar{\mathcal{X}}_{\varepsilon_{1}}$
for $0<\varepsilon_{1}<\varepsilon_{2}$. In fact, we have the following
relation:

$\displaystyle\bar{\mathcal{X}}=\lim_{\varepsilon\to
0}\bar{\mathcal{X}}_{\varepsilon}:=\bigcup_{\varepsilon>0}\bar{\mathcal{X}}_{\varepsilon}.$

As a result, for any probability measure $\nu$ on $\bar{\mathcal{X}}$ for
$\varepsilon>0$, we may extend it to be a probability measure $\tilde{\nu}$ on
$\bar{\mathcal{X}}$ by defining for all Borel measurable set
$A\subseteq\bar{\mathcal{X}}$,

$\displaystyle\tilde{\nu}(A)=\nu(A\cap\bar{\mathcal{X}}_{\varepsilon}).$ (5.1)

In the sequel, we omit the tilde sign and abuse the notation $\nu$ to mean its
natural extension $\tilde{\nu}$.

###  5.3. Contraction property of the acoustic system semigroup

In this section, we prove a contraction property of semigroup
$(\mathcal{Q}_{t}^{\varepsilon})_{t\geq 0}$, which will be crucial to
translate the finite-time convergence of solutions to the convergence of
invariant measures.

Since the solutions are not known to be pathwise contractive in the
$L^{2}\times L^{2}$ norm, we define an $\varepsilon$-dependent norm that
coincides with the $L^{2}\times L^{2}$ norm in the limit $\varepsilon\to 0$
and has the pathwise contraction property. For all $\varepsilon\in(0,1]$,
define the $\varepsilon$-norm $\|\cdot\|_{\varepsilon}$ on $\bar{\mathcal{X}}$
by

$\displaystyle\|(r,v)\|_{\varepsilon}^{2}=\|(r,\,v)\|_{L^{2}}^{2}+\varepsilon^{2}\|(r_{x},\,v_{x})\|_{L^{2}}^{2}.$
(5.2)

We now define the Wasserstein metrics for probability measures on
$\bar{\mathcal{X}}$ by using the metrics on $\bar{\mathcal{X}}$. For a metric
$d$ on $\bar{\mathcal{X}}$, let444Here, we denote the set of all Borel
probability measures on $\bar{\mathcal{X}}$ by
$\mathfrak{P}(\bar{\mathcal{X}})$.

$\mathfrak{P}_{d}=\Big{\\{}\bar{\eta}\in\mathfrak{P}(\bar{\mathcal{X}})\,:\,\int_{\bar{\mathcal{X}}}d((0,0),(r,v))\,\mathrm{d}\bar{\eta}(r,v)<\infty\Big{\\}}.$

be the collection of Borel probability measures on $\bar{\mathcal{X}}$ with
finite metric moment. Then the Wasserstein metric $\mathcal{W}_{d}$
associating with $d$ is defined for all $\eta_{1},\eta_{2}\in\mathfrak{P}_{d}$
as

$\displaystyle\mathcal{W}_{d}(\eta_{1},\eta_{2})=\inf_{\bar{\eta}\in\mathcal{C}(\eta_{1},\eta_{2})}\int_{\bar{\mathcal{X}}\times\bar{\mathcal{X}}}d\big{(}(r_{1},v_{1}),(r_{2},v_{2})\big{)}\,\mathrm{d}\bar{\eta}((r_{1},v_{1}),(r_{2},v_{2})),$
(5.3)

where the infimum is taken over the set, $\mathcal{C}(\eta_{1},\eta_{2})$, of
all couplings of $\eta_{1}$ and $\eta_{2}$, i.e., all measures on
$\mathcal{X}\times\mathcal{X}$ with marginals $\eta_{1}$ and $\eta_{2}$. Note
that $\mathcal{W}_{d}$ turns $\mathfrak{P}_{d}$ into a metric space. If, in
addition, $d$ is complete, then so is $\mathcal{W}_{d}$; see for instance
[Villani09]. We denote by $\mathcal{W}_{L^{2}},\mathcal{W}_{H^{1}}$, and
$\mathcal{W}_{\varepsilon}$ the Wasserstein metrics corresponding to the
$L^{2}\times L^{2}$ metric, the $H^{1}\times H^{1}_{0}$ metric, and the
$\varepsilon$-norm (5.2), respectively. We also write $\mathfrak{P}_{L^{2}}$,
$\mathfrak{P}_{H^{1}}$, and $\mathfrak{P}_{\varepsilon}$ for the corresponding
collections of probability measures. Moreover, we remark that the infimum in
(5.3) can be attained for all the three metrics
$\mathcal{W}_{L^{2}},\mathcal{W}_{H^{1}}$, and $\mathcal{W}_{\varepsilon}$
(cf. [Villani09]*Theorem 4.1). Moreover, it follows from the Monge–Kantorovich
duality that

$\displaystyle\mathcal{W}_{d}(\eta_{1},\eta_{2})=\sup_{\textrm{Lip}_{d}(\phi)\leq
1}\left|\int_{\bar{\mathcal{X}}}\phi\,\mathrm{d}\eta_{1}-\int_{\bar{\mathcal{X}}}\phi\,\mathrm{d}\eta_{2}\right|,$
(5.4)

where the supremum is taken over the set of all Lipschitz functions
$\phi:\bar{\mathcal{X}}\to\mathbb{R}$ with Lipschitz constants less than $1$
when $\bar{\mathcal{X}}$ is equipped with metric $d$.

We can establish the following contraction property for semigroup
$\mathcal{Q}_{t}^{\varepsilon}$ on $\bar{\mathcal{X}}_{H^{1}}$, as well as on
$\bar{\mathcal{X}}$ equipped with the $\varepsilon$-norm, which may be seen as
a simple version of [hairer2008spectral]*Theorem 2.5. The main ingredient is
the exponential contraction properties of solutions expressed in Corollary
3.4.

######  Proposition 5.2.

Let $\varepsilon\in(0,1]$. Then, for all $t\geq 0$ and for probability
measures $\eta_{1}$ and $\eta_{2}$ on $\bar{\mathcal{X}}$,

$\displaystyle\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{1},(\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{2})\leq
2\sqrt{2}{\rm e}^{-\frac{t}{8}}\mathcal{W}_{H^{1}}(\eta_{1},\eta_{2}),$ (5.5)
$\displaystyle\mathcal{W}_{\varepsilon}((\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{1},(\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{2})\leq\sqrt{6}{\rm
e}^{-\frac{t}{4}}\mathcal{W}_{\varepsilon}(\eta_{1},\eta_{2}),$ (5.6)

where $(\mathcal{Q}_{t}^{\varepsilon})^{*}\bar{\eta}$ is the push-forward
measure of $\bar{\eta}$ by $\mathcal{Q}_{t}^{\varepsilon}$.

###### Proof.

We first show that the conclusion is true when $\eta_{1}$ and $\eta_{2}$ are
the Dirac measures. Indeed, let $g_{1},g_{2}\in\bar{\mathcal{X}}$. Then, by
the Monge–Kantorovich duality (5.4), we have

$\displaystyle\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{g_{1}},(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{g_{2}})=\sup_{\textrm{Lip}_{H^{1}}(\phi)\leq
1}\left|\mathcal{Q}_{t}^{\varepsilon}\phi(g_{1})-\mathcal{Q}_{t}^{\varepsilon}\phi(g_{2})\right|.$

Denote by $(r_{\varepsilon}(t;g),v_{\varepsilon}(t,g))$ the solution of system
(AS) at time $t\geq 0$ stating at the initial condition:
$(r_{0,\varepsilon},v_{0,\varepsilon})=g$. Taking any $\phi\in
C(\bar{\mathcal{X}}_{H_{1}})$ with $\textrm{Lip}_{H_{1}}(\phi)\leq 1$, we have

$\displaystyle\left|\mathcal{Q}_{t}^{\varepsilon}\phi(g_{1})-\mathcal{Q}_{t}^{\varepsilon}\phi(g_{2})\right|$
$\displaystyle=\left|\mathbb{E}\big{[}\phi(r_{\varepsilon}(t;g_{1}),v_{\varepsilon}(t,g_{1}))-\phi(r_{\varepsilon}(t;g_{2}),v_{\varepsilon}(t,g_{2}))\big{]}\right|$
$\displaystyle\leq\mathbb{E}\Big{[}\left(\|r_{\varepsilon}(t;g_{1})-r_{\varepsilon}(t;g_{2})\|^{2}_{H^{1}}+\|v_{\varepsilon}(t;g_{1})-v_{\varepsilon}(t;g_{2})\|^{2}_{H^{1}}\right)^{\frac{1}{2}}\Big{]}$
$\displaystyle\leq 2\sqrt{2}{\rm e}^{-\frac{t}{8}}\|g_{1}-g_{2}\|_{H^{1}\times
H^{1}_{0}}$ $\displaystyle=2\sqrt{2}{\rm
e}^{-\frac{t}{8}}\mathcal{W}_{H^{1}}(\delta_{g_{1}},\delta_{g_{2}}).$

The second inequality is due to (3.8) in Corollary 3.4.

Next, we prove the general case. Let $\eta_{1}$ and $\eta_{2}$ be two
probability measures on the $\bar{\mathcal{X}}$. Take
$\bar{\eta}\in\mathcal{C}(\eta_{1},\eta_{2})$ such that the infimum in (5.3)
is attained. Then

$\displaystyle\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{1},(\mathcal{Q}_{t}^{\varepsilon})^{*}\eta_{2})$
$\displaystyle\leq\int_{\mathcal{X}\times\mathcal{X}}\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{g_{1}},(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{g_{2}})\,\mathrm{d}\bar{\eta}(g_{1},g_{2})$
$\displaystyle\leq 2\sqrt{2}{\rm
e}^{-\frac{t}{8}}\int_{\mathcal{X}\times\mathcal{X}}\mathcal{W}_{H^{1}}(\delta_{g_{1}},\delta_{g_{2}})\,\mathrm{d}\bar{\eta}(g_{1},g_{2})$
$\displaystyle=2\sqrt{2}{\rm
e}^{-\frac{t}{8}}\mathcal{W}_{H^{1}}(\eta_{1},\eta_{2}).$

This proves (5.5). The proof of (5.6) is the same, except (3.9) is used in
place of (3.8). ∎

The contraction property implies the existence and uniqueness of the invariant
probability measure for system (AS).

######  Corollary 5.3.

The semigroup $(\mathcal{Q}_{t}^{\varepsilon})_{t\geq 0}$ admits the unique
invariant measure $\omega_{\varepsilon}$ on $\bar{\mathcal{X}}_{H^{1}}$ such
that $\omega_{\varepsilon}\in\mathfrak{P}_{H^{1}}$ and, for all
$\lambda\in\mathfrak{P}_{H^{1}}$,

$\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\lambda,\omega_{\varepsilon})\leq
2\sqrt{2}{\rm
e}^{-\frac{t}{8}}\mathcal{W}_{H^{1}}(\lambda,\omega_{\varepsilon})\qquad\,\,\mbox{for
every $t\geq 0$}.$ (5.7)

###### Proof.

Note that, by Proposition 5.2, map $(\mathcal{Q}_{t}^{\varepsilon})^{*}$ is a
contraction on $\mathfrak{P}_{H^{1}}$ for all sufficiently large $t\geq 0$.
Since $H^{1}\times H^{1}_{0}$ is a complete norm on $\bar{\mathcal{X}}$,
$\mathfrak{P}_{H^{1}}$ is a complete space. By the contraction mapping
principle, for all such $t$, there exists a probability measure
$\omega_{\varepsilon}^{t}\in\mathfrak{P}_{H^{1}}$ such that
$(\mathcal{Q}_{t}^{\varepsilon})^{*}\omega_{\varepsilon}^{t}=\omega_{\varepsilon}^{t}$.
The time-continuity of $(\mathcal{Q}_{t}^{\varepsilon})$ implies that
$\omega_{\varepsilon}^{t}=\omega_{\varepsilon}^{s}$ for all these sufficiently
large $t$ and $s$. By the semigroup property,
$\omega_{\varepsilon}=\omega_{\varepsilon}^{t}$ is invariant for all $t\geq
0$. The attraction property (5.7) is a consequence of Proposition 5.2.

To complete the proof, we still have to show that the uniqueness holds in the
entire space $\mathfrak{P}(\bar{\mathcal{X}})$, not just in
$\mathfrak{P}_{H^{1}}$. To see this, let
$\bar{\omega}_{\varepsilon}\in\mathfrak{P}(\bar{\mathcal{X}})$ be an invariant
measure for $(\mathcal{Q}_{t}^{\varepsilon})_{t\geq 0}$. Note that, for all
$\delta>0$, there exists $M_{\delta}>0$ such that

$m_{\varepsilon}^{\delta}:=\bar{\omega}_{\varepsilon}\big{(}\big{\\{}(r,v)\in\bar{\mathcal{X}}:\|r_{x}\|_{L^{2}}+\|v_{x}\|_{L^{2}}>M_{\delta}\big{\\}}\big{)}<\delta.$

Define $\bar{\omega}_{\varepsilon}^{\delta}\in\mathfrak{P}_{H^{1}}$ by

$\bar{\omega}_{\varepsilon}^{\delta}(A)=\frac{1}{1-m_{\varepsilon}^{\delta}}\int_{A}\mathbbm{1}_{\\{\|r_{x}\|_{L^{2}}+\|v_{x}\|_{L^{2}}\leq
M_{\delta}\\}}\,\mathrm{d}\bar{\omega}_{\varepsilon}$

for all Borel measurable set $A\subseteq\bar{\mathcal{X}}$. On one hand, since
$\bar{\omega}_{\varepsilon}^{\delta}$ converges to
$\bar{\omega}_{\varepsilon}$ setwise as $\delta\to 0$,
$(\mathcal{Q}_{t}^{\varepsilon})^{*}\bar{\omega}_{\varepsilon}^{\delta}$
converges to
$(\mathcal{Q}_{t}^{\varepsilon})^{*}\bar{\omega}_{\varepsilon}=\bar{\omega}_{\varepsilon}$
setwise and hence weakly as $\delta\to 0$. On the other hand, since
$\bar{\omega}_{\varepsilon}^{\delta}\in\mathfrak{P}_{H^{1}}$, (5.7) implies
that $(\mathcal{Q}_{t}^{\varepsilon})^{*}\bar{\omega}_{\varepsilon}^{\delta}$
converges weakly to $\omega$ as $t\to\infty$. This yields
$\bar{\omega}_{\varepsilon}=\omega_{\varepsilon}$ by exchanging the order of
the two limits, which is valid since the setwise (and hence weak) convergence
of $(\mathcal{Q}_{t}^{\varepsilon})^{*}\bar{\omega}_{\varepsilon}^{\delta}$ to
$\bar{\omega}_{\varepsilon}$ as $\delta\to 0$ is uniform in $t\geq 0$. This
completes the proof. ∎

We can also deduce further estimates for the invariant measure
$\omega_{\varepsilon}$, which can be seen as moment bounds for the
corresponding statistically stationary solutions.

######  Proposition 5.4.

Let $\varepsilon\in(0,1]$. Let $\omega_{\varepsilon}$ be the invariant measure
of $(\mathcal{Q}_{t}^{\varepsilon})$. Then

$\displaystyle\int_{\bar{\mathcal{X}}}\|(r_{x},\,v_{x})\|_{L^{2}}^{2}\,\mathrm{d}\omega_{\varepsilon}\leq\|\sigma\|_{L^{\infty}}^{2}\varepsilon^{2(\beta-1)}.$
(5.8)

Moreover, for all $p\geq 1$, there exists $C_{p}>0$, independent of
$\varepsilon$, such that

$\displaystyle\int_{\bar{\mathcal{X}}}\mathcal{G}(r,v)^{p}\,\mathrm{d}\omega_{\varepsilon}\leq
C_{p}\varepsilon^{2(\beta-1)p}\|\sigma\|_{L^{\infty}}^{2p}.$ (5.9)

Furthermore, for all
$\eta\in(0,\,(8\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2})^{-1})$,
there exists $C_{\eta}>0$, independent of $\varepsilon$, such that

$\int_{\bar{\mathcal{X}}}\exp(\eta\mathcal{G}(r,v))\,\mathrm{d}\omega_{\varepsilon}\leq
C_{\eta}\exp(2\eta\varepsilon^{2(\beta-1)}).$ (5.10)

###### Proof.

First, we integrate (3.5) with respect to measure $\omega_{\varepsilon}$,
which implies that, for all $t\geq 0$,

$\int_{\bar{\mathcal{X}}}\mathcal{Q}_{t}^{\varepsilon}\mathcal{G}(r,v)\,\mathrm{d}\omega_{\varepsilon}+\frac{1}{2}\int_{0}^{t}\int_{\bar{\mathcal{X}}}\mathcal{Q}_{s}^{\varepsilon}(\|(r_{x},\,v_{x})\|_{L^{2}}^{2})\,\mathrm{d}\omega_{\varepsilon}\,\mathrm{d}s=\int_{\bar{\mathcal{X}}}\mathcal{G}(r,v)\,\mathrm{d}\omega_{\varepsilon}+\frac{1}{2}\varepsilon^{2(\beta-1)}t\int_{0}^{1}\sigma^{2}\,\mathrm{d}x.$
(5.11)

Using the invariance of $\omega_{\varepsilon}$ and the fact that
$\omega_{\varepsilon}\in\mathfrak{P}_{H^{1}}$, we have

$\displaystyle\int_{\bar{\mathcal{X}}}\mathcal{Q}_{t}^{\varepsilon}\mathcal{G}(r,v)\,\mathrm{d}\omega_{\varepsilon}=\int_{\bar{\mathcal{X}}}\mathcal{G}(r,v)\,\mathrm{d}\omega_{\varepsilon}<\infty.$

Therefore, (5.11) implies

$\displaystyle\frac{1}{2}t\int_{\bar{\mathcal{X}}}\|(r_{x},\,v_{x})\|_{L^{2}}^{2}\,\mathrm{d}\omega_{\varepsilon}=\frac{1}{2}\varepsilon^{2(\beta-1)}t\int_{0}^{1}\sigma^{2}\,\mathrm{d}x,$

from which (5.8) follows. Next, let $\delta_{(0,0)}$ be the delta measure on
$\bar{\mathcal{X}}$ centred at $(0,0)\in\bar{\mathcal{X}}$. By Proposition
3.5, we see that, for all $p\geq 1$, there exists $C_{p}>0$, independent of
$\varepsilon$, such that

$\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\mathcal{G}(r,v)^{p}\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}=C_{p}\varepsilon^{2(\beta-1)p}\|\sigma\|_{L^{\infty}}^{2p}.$

Likewise, for all
$\eta\in(0,\,(8\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2})^{-1})$,
there exists $C_{\eta}>0$, independent of $\varepsilon$, such that

$\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\exp(\eta\mathcal{G}(r,v)\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}=C_{\eta}\exp\big{(}2\eta\varepsilon^{2(\beta-1)}\big{)}.$

By Corollary 5.3, we have

$\displaystyle\lim_{t\to\infty}\mathcal{W}_{H^{1}}((\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)},\omega_{\varepsilon})=0.$

This can be rephrased in terms of the Monge–Kantorovich duality (5.4). For all
globally Lipschitz function $f:\bar{\mathcal{X}}\to\mathbb{R}$,

$\displaystyle\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}f\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}=\int_{\bar{\mathcal{X}}}f\,\mathrm{d}\omega_{\varepsilon}.$
(5.12)

Let $p\geq 1$ and $N>0$. Choosing $f(r,v)=\min\\{\mathcal{G}(r,v)^{p},N\\}$ in
the above gives

$\displaystyle\int_{\bar{\mathcal{X}}}\min\\{\mathcal{G}(r,v)^{p},N\\}\,\mathrm{d}\omega_{\varepsilon}$
$\displaystyle=\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\min\\{\mathcal{G}(r,v)^{p},N\\}\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}$
$\displaystyle\leq\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\mathcal{G}(r,v)^{p}\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}=C_{p}\varepsilon^{2(\beta-1)p}\|\sigma\|_{L^{\infty}}^{2p}.$

Now, letting $N\to\infty$ gives the first estimate (5.9). Similarly, let
$\eta\in(0,(8\varepsilon^{2(\beta-1)}\|\sigma\|_{L^{\infty}}^{2})^{-1})$ and
$M>0$. Choosing $f(r,v)=\min\\{\exp(\eta\mathcal{G}(r,v)),M\\}$ in (5.12)
gives

$\displaystyle\int_{\bar{\mathcal{X}}}\min\\{\exp(\eta\mathcal{G}(r,v)),M\\}\,\mathrm{d}\omega_{\varepsilon}$
$\displaystyle=\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\min\\{\exp(\eta\mathcal{G}(r,v)),M\\}\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}$
$\displaystyle\leq\lim_{t\to\infty}\int_{\bar{\mathcal{X}}}\exp(\eta\mathcal{G}(r,v))\,\mathrm{d}(\mathcal{Q}_{t}^{\varepsilon})^{*}\delta_{(0,0)}=C_{\eta}\exp\big{(}2\eta\varepsilon^{2(\beta-1)}\big{)}.$

Now, letting $M\to\infty$ gives the second estimate (5.10). ∎

###  5.4. Estimates of the invariant measures

In order to apply the finite-time convergence result to study the convergence
of invariant measures, we have to estimate the statistically stationary
solutions distributed as invariant measures. In particular, we want to show
that they satisfy (1.12), which is the main goal of this section.

Since there is a natural correspondence between the invariant measures of
systems (FS) and (NS), we can confine our analysis to the latter system.
Recall that the energy balance (2.2) holds for system (NS):

$\displaystyle\,\mathrm{d}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\frac{1}{2}\Big{(}\int_{0}^{1}|(u_{\varepsilon})_{x}|^{2}\,\mathrm{d}x+\frac{1}{\varepsilon}\int_{0}^{1}\rho_{\varepsilon}^{\gamma-3}|(\rho_{\varepsilon})_{x}|^{2}\,\mathrm{d}x\Big{)}\,\mathrm{d}t$
(5.13)
$\displaystyle=\frac{1}{2}\varepsilon^{2\beta}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x\,\mathrm{d}t+\varepsilon^{\beta}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x\,\mathrm{d}W_{t}.$
(5.14)

To simplify the notation, we write that, for all
$(\rho_{\varepsilon},u_{\varepsilon})\in\mathcal{X}$,

$\displaystyle\mathcal{K}(\rho_{\varepsilon},u_{\varepsilon})=\frac{1}{2}\|(u_{\varepsilon})_{x}\|_{L^{2}}^{2}+\frac{1}{2\varepsilon^{2}}\|(\log\rho_{\varepsilon})_{x}\|_{L^{2}}^{2},$
$\displaystyle\mathcal{M}(\rho_{\varepsilon},u_{\varepsilon})=\varepsilon^{\beta}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma\,\mathrm{d}x,$
$\displaystyle\mathcal{R}(\rho_{\varepsilon},u_{\varepsilon})=\frac{\varepsilon^{2\beta}}{2}\int_{0}^{1}\rho_{\varepsilon}\sigma^{2}\,\mathrm{d}x,$

so that the above equality reads

$\,\mathrm{d}\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})+\mathcal{K}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}t=\mathcal{M}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}W_{t}+\mathcal{R}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}t.$

Note here that the terms
$\mathcal{M}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}W_{t}$ and
$\mathcal{R}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}t$ should be
interpreted as

$\displaystyle\mathcal{M}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}W_{t}=\sum_{k=1}^{\infty}\varepsilon^{\beta}\int_{0}^{1}\big{(}\rho_{\varepsilon}u_{\varepsilon}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}}{\rho_{\varepsilon}}\big{)}\sigma_{k}\,\mathrm{d}x\,\mathrm{d}W^{k}_{t},$
$\displaystyle\mathcal{R}(\rho_{\varepsilon},u_{\varepsilon})\,\mathrm{d}t=\sum_{k=1}^{\infty}\frac{\varepsilon^{2\beta}}{2}\int_{0}^{1}\rho_{\varepsilon}\sigma_{k}^{2}\,\mathrm{d}x\,\mathrm{d}t.$

We also define

$\displaystyle\mathcal{M}(\rho_{\varepsilon},u_{\varepsilon})^{2}:=\sum_{k=1}^{\infty}\Big{(}\varepsilon^{\beta}\int_{0}^{1}\big{(}\rho
u+\frac{1}{2}\frac{\rho_{x}}{\rho}\big{)}\sigma_{k}\,\mathrm{d}x\Big{)}^{2}$

for later use.

As a first step of the estimates, we show that
$\mathcal{E}(\rho_{\varepsilon},u_{\varepsilon})$ can be controlled by
$\mathcal{K}(\rho_{\varepsilon},u_{\varepsilon})$, a distinctive feature that
is true only for the isothermal pressure law and has been observed and used in
[Hoff99, Hoff00] to study the long-time behavior of the solutions in the
deterministic case.

######  Proposition 5.5.

Let $\varepsilon\in(0,1]$. Then, for all $(\rho,u)\in\mathcal{X}$,

$\displaystyle\mathcal{E}(\rho,u)\leq\big{(}\,\frac{3}{2}+2\|\rho\|_{L^{\infty}}^{2}+\varepsilon^{2}\|\rho^{-1}\|_{L^{\infty}}\big{)}\mathcal{K}(\rho,u),$
(5.15)

which implies

$\displaystyle\mathcal{E}(\rho,u)$
$\displaystyle\leq\big{(}\frac{3}{2}+2\exp(2\varepsilon\sqrt{2\mathcal{K}(\rho,u)})+\varepsilon^{2}\exp(\varepsilon\sqrt{2\mathcal{K}(\rho,u)})\big{)}\mathcal{K}(\rho,u).$
(5.16)

###### Proof.

It suffices to control each term in the definition of $\mathcal{E}(\rho,u)$ in
(1.10) separately. The first two terms are easily bounded by

$\frac{1}{2}\int_{0}^{1}\rho
u^{2}\,\mathrm{d}x\leq\frac{1}{2}\|u\|_{L^{\infty}}^{2}\int_{0}^{1}\rho\,\mathrm{d}x\leq\frac{1}{2}\|u_{x}\|_{L^{2}}^{2}\leq\mathcal{K}(\rho,u),$

and

$\displaystyle\int_{0}^{1}\frac{1}{\varepsilon^{2}}(\rho\log\rho-p+1)\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{\varepsilon^{2}}\int_{0}^{1}\rho(\rho-1)\,\mathrm{d}x=\frac{1}{\varepsilon^{2}}\int_{0}^{1}(\rho^{2}-1)\,\mathrm{d}x\leq\frac{1}{\varepsilon^{2}}\int_{0}^{1}\rho_{x}^{2}\,\mathrm{d}x$
$\displaystyle\leq\frac{1}{\varepsilon^{2}}\|\rho\|_{L^{\infty}}^{2}\|(\log\rho)_{x}\|_{L^{2}}^{2}\leq
2\|\rho\|_{L^{\infty}}^{2}\mathcal{K}(\rho,u).$

Moreover, we have

$\displaystyle\frac{1}{4}\int_{0}^{1}\frac{\rho_{x}^{2}}{\rho^{3}}\,\mathrm{d}x\leq\frac{\varepsilon^{2}}{2}\|\rho^{-1}\|_{L^{\infty}}\,\frac{1}{2\varepsilon^{2}}\|(\log\rho)_{x}\|_{L^{2}}^{2}\leq\frac{\varepsilon^{2}}{2}\|\rho^{-1}\|_{L^{\infty}}\mathcal{K}(\rho,u),$
$\displaystyle\frac{1}{2}\int_{0}^{1}\frac{\rho_{x}u}{\rho}\leq\frac{1}{4}\int_{0}^{1}\rho
u^{2}\,\mathrm{d}x+\frac{1}{4}\int_{0}^{1}\frac{\rho_{x}^{2}}{\rho^{3}}\,\mathrm{d}x\leq\frac{1}{2}\mathcal{K}(\rho,u)+\frac{\varepsilon^{2}}{2}\|\rho^{-1}\|_{L^{\infty}}\mathcal{K}(\rho,u).$

Combining the above, we obtain the first estimate (5.15).

The second estimate (5.16) follows from the first one and the fact that

$\displaystyle\|\log\rho\|_{L^{\infty}}\leq\|(\log\rho)_{x}\|_{L^{2}}\leq\varepsilon^{-1}\sqrt{2\mathcal{K}(\rho,u)}$

for all $\rho\in H^{1}$ with mass equal to 1. Indeed, there exists
$x_{0}\in[0,1]$ such that $\rho(x_{0})=1$ so that, for all $x\in[0,1]$,

$\displaystyle|\log\rho(x)|=|\log\rho(x)-\log\rho(x_{0})|=\Big{|}\int_{x_{0}}^{x}\big{(}\log\rho(y)\big{)}_{y}\,\mathrm{d}y\Big{|}\leq\|(\log\rho)_{y}\|_{L^{2}}.$

∎

In light of the above proposition, we define $h:[0,\infty)\to[0,\infty)$ given
by

$h(x):=\big{(}\frac{3}{2}+2\exp(2\varepsilon\sqrt{2x})+\varepsilon^{2}\exp(\varepsilon\sqrt{2x})\big{)}x.$
(5.17)

Note that $h$ is strictly increasing, bijective and convex. Using (5.14), we
can obtain a first bound for the invariant measures. The computation here is
similar to the one in the incompressible case; see [Flandoli08].

######  Proposition 5.6.

Let $F:[0,\infty)\to[0,\infty)$ be of class $C^{1}$ and increasing. Let
$\mu_{\varepsilon}$ be an invariant measure of system (NS) defined on the
state space $\mathcal{X}$. Then

$\int_{\mathcal{X}}F(\mathcal{E}(\rho,u))\mathcal{K}(\rho,u)\>\,\mathrm{d}\mu_{\varepsilon}\leq\int_{\mathcal{X}}F(\mathcal{E}(\rho,u))\mathcal{R}(\rho,u)\,\mathrm{d}\mu_{\varepsilon}+\frac{1}{2}\int_{\mathcal{X}}F^{\prime}(\mathcal{E}(\rho,u))(\mathcal{M}(\rho,u))^{2}\,\mathrm{d}\mu_{\varepsilon},$
(5.18)

where the last integral is to be understood as

$\int_{\mathcal{X}}F^{\prime}(\mathcal{E}(\rho,u))(\mathcal{M}(\rho,u))^{2}\,\mathrm{d}\mu_{\varepsilon}=\sum_{k=1}^{\infty}\int_{\mathcal{X}}F^{\prime}(\mathcal{E}(\rho,u))\varepsilon^{2\beta}\Big{(}\int_{0}^{1}\big{(}\rho
u+\frac{1}{2}\frac{\rho_{x}}{\rho}\big{)}\sigma_{k}\,\mathrm{d}x\Big{)}^{2}\,\mathrm{d}\mu_{\varepsilon}.$

###### Proof.

Note that, for each $\delta>0$, there exists $N_{\delta}>0$ such that

$\displaystyle
q_{\varepsilon}^{\delta}:=\mu_{\varepsilon}\left(\\{(\rho,u)\in\mathcal{X}\,:\,\mathcal{E}(\rho,u)>N_{\delta}\\}\right)<\delta.$

Define a probability measure $\mu_{\varepsilon}^{\delta}$ on $\mathcal{X}$ by

$\mu_{\varepsilon}^{\delta}(A)=\frac{1}{1-q_{\varepsilon}^{\delta}}\int_{A}\mathbbm{1}_{\\{\mathcal{E}(\rho,u)\leq
N_{\delta}\\}}\,\mathrm{d}\mu_{\varepsilon}$

for all Borel measurable set $A\subseteq\mathcal{X}$. Let
$(\rho_{0,\varepsilon}^{\delta},u_{0,\varepsilon}^{\delta})$ be an
$\mathcal{F}_{0}$-measurable random variable distributed as
$\mu_{\varepsilon}^{\delta}$. Let
$(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})$ be a solution of the
fluctuation system (FS) with initial data
$(\rho_{0,\varepsilon}^{\delta},u_{0,\varepsilon}^{\delta})$.

Define the function $\tilde{F}:[0,\infty)\to[0,\infty)$ by

$\displaystyle\tilde{F}(r)=\int_{0}^{r}F(y)\,\mathrm{d}y$

so that $\tilde{F}^{\prime}=F$. By Itô’ s formula, we have

$\displaystyle\,\mathrm{d}\tilde{F}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})=F(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))\,\mathrm{d}\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})+\frac{1}{2}F^{\prime}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))\,\mathrm{d}\langle\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}),\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\rangle,$

so that

$\displaystyle\,\mathrm{d}\tilde{F}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))+F(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))K(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\,\mathrm{d}t$
$\displaystyle=F(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))M(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\,\mathrm{d}W_{t}+F(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))R(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\,\mathrm{d}t$
$\displaystyle\quad+\frac{1}{2}F^{\prime}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))\big{(}M(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\big{)}^{2}\,\mathrm{d}t,$

where

$\displaystyle
F^{\prime}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))\big{(}M(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta})\big{)}^{2}\,\mathrm{d}t=\sum_{k=1}^{\infty}F^{\prime}(\mathcal{E}(\rho_{\varepsilon}^{\delta},u_{\varepsilon}^{\delta}))\Big{(}\varepsilon^{\beta}\int_{0}^{1}\big{(}\rho_{\varepsilon}^{\delta}u_{\varepsilon}^{\delta}+\frac{1}{2}\frac{(\rho_{\varepsilon})_{x}^{\delta}}{\rho_{\varepsilon}^{\delta}}\big{)}\sigma_{k}\,\mathrm{d}x\Big{)}^{2}\,\mathrm{d}t.$

Thus, for all $T>0$, we obtain

$\displaystyle\mathbb{E}\big{[}\tilde{F}(\mathcal{E}(\rho_{\varepsilon}^{\delta}(T,\cdot),u_{\varepsilon}^{\delta}(T,\cdot)))\big{]}+\mathbb{E}\Big{[}\int_{0}^{T}F(\mathcal{E}(\rho_{\varepsilon}^{\delta}(t,\cdot),u_{\varepsilon}^{\delta}(t,\cdot)))K(\rho_{\varepsilon}^{\delta}(t,\cdot),u_{\varepsilon}^{\delta}(T,\cdot))\,\mathrm{d}t\Big{]}$