Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 107, pp. 1–14.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.107

LINEAR STABILITY OF THE COUETTE FLOW FOR NON-ISENTROPIC

COMPRESSIBLE FLUIDS

XIAOPING ZHAI

Abstract. In this article, we study the linear stability of a two-dimensional non-isentropic

compressible fluid with vanishing shear viscosity in the context of Couette flow on an infinitely

long flat torus T×R. By employing explicit weighted energy estimates and the Fourier multipliers
method, we first establish the inviscid damping of the incompressible component of the velocity.

Subsequently, we derive an upper bound which is superlinear in time for the compressible part

of the fluid. Furthermore, we demonstrate an enhanced dissipation phenomenon for the velocity
field under certain quality conditions pertaining to the initial density, initial temperature, and

incompressible component of the initial velocity field.

1. Introduction and main result

In this article, we study the long-time asymptotic behavior of the linearized two dimensional
non-isentropic compressible Navier-Stokes equations in a domain T×R. The governing equations
(in non-dimensional variables) are

ϱt + u · ∇ϱ+ ϱdivu = 0,

ϱ(ut + u · ∇u) +
1

γM2
∇P =

1

Re
(µ∆u+ (ν + µ)∇divu) ,

ϱ(ϑt + u · ∇ϑ) + (γ − 1)P divu =
γµ

σRe
∆ϑ+

γ(γ − 1)M2

Re

(µ
2
|∇u+∇u⊤|2 + ν|divu|2

)
.

(1.1)

Here t ≥ 0 is time, (x, y) ∈ T× R is the spatial coordinate and T = R/Z. The unknown u is the
velocity vector, ϱ is the density, ϑ is the temperature, P = ϱϑ is the pressure. γ > 1 is the ratio of
specific heats, M > 0 is the Mach number of the reference state, Re > 0 is the Reynolds number,
and σ > 0 is the Prandtl number. The two constant viscosity coefficients µ and ν are the shear
viscosity and the volume viscosity respectively. The equations (1.1) then express respectively the
conservation of mass, the balance of momentum, and the balance of energy under internal pressure,
viscosity forces, and the conduction of thermal energy.

A comprehensive understanding of the stability of compressible or incompressible shear flows
is a fundamental problem in fluid mechanics and has been the subject of both theoretical and
practical interest in astrophysics and engineering, see [1, 2, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18,
19, 20, 22, 24, 25, 26, 29, 30] for the compressible fluid and [3, 4, 5, 10, 11, 21, 23, 27, 28] for
incompressible fluid. The aim of the present paper is to study the long-time asymptotic behavior
of the linearized non-isentropic compressible Navier-Stokes equations around the Couette flow.
That is we seek a stationary solution of (1.1) with a constant mean pressure which have the form

ϱsh = ϱsh(y), ush =

(
y
0

)
, ϑsh = ϑsh(y), with ϱsh(y)ϑsh(y) = 1. (1.2)

2020 Mathematics Subject Classification. 76E05, 76E19.

Key words and phrases. Stability; compressible Navier-Stokes equations; Couette flow; enhanced dissipation.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted May 23, 2025. Published November 14, 2025.

1



2 X. ZHAI EJDE-2025/107

Obviously, when µ ̸= 0, because of the strong nonlinear term |∇u+∇u⊤|2 appeared in the third
equation of (1.1), it is straightforward to verify that ϑsh(y) must satisfy the restricted relation

γµ

σRe
∂yyϑsh(y) = −γµ(γ − 1)M2

Re
. (1.3)

To solve (1.3), we can choose ϑsh(y) as

ϑsh(y) = ϑr[r + (1− r)y − (1− 1

ϑr
)y2] (1.4)

where r > 0 is the temperature ratio and ϑr is the recovery temperature defined as

ϑr := 1 +
(γ − 1)σM2

2
.

Because of the complicated form of ϑsh(y), to study the long-time asymptotic behaviour of (1.1)
around the stationary solution defined in (1.2) and (1.4) is a very difficult problem. To the best
of our knowledge, there are only a few results in this direction, see [1, 2, 6, 7, 8, 13, 14, 15, 16, 17,
18, 19, 20, 25, 26].

Because of the mathematical challenges, to approach the problem, here, we only consider a
simple case of (1.1) with the shear viscosity coefficient µ = 0 and the volume viscosity ν ̸= 0. In
this case, system (1.1) can be rewritten as

ϱt + u · ∇ϱ+ ϱdivu = 0,

ϱ(ut + u · ∇u) +
1

γM2
∇(ϱϑ) =

ν

Re
∇divu,

ϱ(ϑt + u · ∇ϑ) + (γ − 1)ϱϑdivu =
νγ(γ − 1)M2

Re
|divu|2.

(1.5)

It is straightforward to verify that the Couette flow,

ϱsh = 1, ush =

(
y
0

)
, ϑsh = 1, (1.6)

is a stationary solution of (1.5). Our goal is to understand the stability and large-time behavior
of perturbations near this Couette flow.

Before presenting our main result, let us give a short review of the extensive mathematical
results about stability analysis on the compressible Navier-Stokes equations. Glatzel [14, 15]
studied the linear inviscid and viscous stability properties of the compressible Couette flow via a
normal mode analysis in simplified flow model with constant viscous coefficients and a constant
density profile. Duck et al. [13] proved the linear stability of the plane Couette flow for the non-
isentropic compressible Navier-Stokes equations. Chagelishvili et al. [8] considered the inviscid
stability of the 2D Couette flow. By means of some formal approximation, they showed that the
energy of acoustic perturbations grows linear in time due to the transfer of energy from the mean
flow to perturbations. Taking advantage of a fourth-order finite-difference method and a spectral
collocation method, Hu et al. [16] studied the viscous linear stability of supersonic Couette flow for
a perfect gas governed by Sutherland viscosity law. Kagei [17] proved that the plane Couette flow
in an infinite layer is asymptotically stable if the Reynolds and Mach numbers are sufficiently small.
Li et al. [20] investigated the stability analysis of the plane Couette flow for the 3D compressible
Navier-Stokes equations with Navier-slip boundary condition at the bottom boundary. They shown
that the plane Couette flow is asymptotically stable for small perturbation provided that the slip
length, Reynolds and Mach numbers satisfy some restricted relation. Recently, Antonelli et al. [1]
studied the linear stability properties of the 2D isentropic compressible Euler equations linearized
around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in
the domain T×R. Later then, they in [2] also studied the linear stability properties of perturbations
around the homogeneous Couette flow for a 2D isentropic inviscid or viscous compressible fluid.
Moreover, in the inviscid case, they proved the inviscid damping for the solenoidal component of
the velocity field and Lyapunov type instability for the density and the irrotational component
of the velocity field. In the viscous case, they obtained the enhanced dissipation phenomenon.
Zeng et al. [29] considered the linear stability of the three dimensional isentropic compressible



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 3

Navier-Stokes equations on T × R × T. They proved the enhanced dissipation phenomenon and
the lift-up phenomenon around the Couette flow (y, 0, 0)⊤. The motivation of the present paper
is to generalize the results obtained by Antonelli et al. [1, 2] to the non-isentropic compressible
Navier-Stokes equations with vanished shear viscosity.

We denote

ρ = ϱ− ϱsh, v = u− ush, θ = ϑ− ϑsh.

The linearized system of (1.5) around the Couette flow (1.6) reads as follows

∂tρ+ y∂xρ+ divv = 0,

∂tv + y∂xv +

(
vy

0

)
+

1

γM2
(∇ρ+∇θ) = ν∇divv,

∂tθ + y∂xθ + (γ − 1) divv = 0.

(1.7)

The above linear system is very different from the isentropic compressible Euler equation in [1, 30]
and the isentropic compressible Navier-Stokes equation in [2]. On the one hand, as we only have
dissipation on the compressible part of v, the behavior of (ρ,∇divv, θ) is similar to the isentropic
compressible Navier-Stokes equation discussed in [2], i.e., the density, the compressible part of the
velocity field, and the temperature experience a Lyapunov type instability. In contrast, because
of the lack of dissipation on the incompressible part of v, the incompressible part of the velocity
experiences an inviscid damping just like the classical incompressible Euler equation. On the other
hand, we define

ω = ∇⊥ · v, with ∇⊥ = (−∂y, ∂x)
⊤,

we can find that the equation of the incompressible part of the velocity connects with the com-
pressible part of the velocity through the equation

∂tω + y∂xω − divv = 0. (1.8)

Hence, assuming the initial data satisfies some quality relation (see (1.17) in the following) and
exploiting the special linear structure of (1.7), we can transfer the dissipation of ∇ divv to the
incompressible part of the velocity field ω. As a result, we can prove the enhanced dissipation
phenomenon of the the velocity field. The phenomenon of enhanced dissipation has been widely
studied in the physics literature [2, 3, 4, 11, 23], and has recently received a lot of attention from
the mathematical community. Here, we give more explanation about the enhanced dissipation
phenomenon. For example, considering the equation

∂tF + y∂xF = ν(∂xx + ∂yy)F, F (x, y, 0) = F0(x, y). (1.9)

Applying the Fourier transform to the governing equation and using the coordinate transformation,
we obtain the evolution equation in the frequency domain,

∂tF̂ − k∂ηF̂ = −ν(k2 + η2)F̂ , F̂ (k, η, 0) = F̂0(k, η), (1.10)

where (k, η) denotes the dual variables in Fourier space corresponding to the spatial coordinates.
The Fourier transform convention used in this derivation, along with its fundamental properties,
will be specified in the subsequent section.

Making the natural change of variables

ξ := η + kt, H(k, ξ, t) := F̂ (k, η, t),

we find that

∂tH(k, ξ, t) = −ν(k2 + (ξ − kt)2)H(k, ξ, t), H(k, ξ, 0) = F̂0(k, ξ).

Integrating in time yields

H(k, ξ, t) = F̂0(k, ξ)e
−ν

∫ t
0
(k2+(ξ−kτ)2) dτ .

Therefore,

F̂ (k, η, t) = H(k, ξ, t) = F̂0(k, η + kt)e−ν
∫ t
0
k2+(η+k(t−τ))2 dτ

= F̂0(k, η + kt)e−ν(k2+η2)te−
1
3νk

2t3−νkηt2 .
(1.11)



4 X. ZHAI EJDE-2025/107

This explicit representation reflects the enhanced dissipation. The dissipation time scale isO(ν−
1
3 ),

which is much faster than the standard dissipation time scale O(ν−1). Clearly the dissipation rate
is inhomogeneous and depends on the frequencies k.

Before going into details of our theorem, we introduce some notation. We define

α = divv, ω = ∇⊥ · v, with ∇⊥ = (−∂y, ∂x)
⊤,

according to the Helmholtz projection operators, we have

v = (vx, vy)⊤ := P[v] +Q[v] (1.12)

with
P[v] := ∇⊥∆−1ω, Q[v] := ∇∆−1α. (1.13)

From the above definition, one infers that

vy = ∂y(∆
−1)α+ ∂x(∆

−1)ω, (1.14)

from which, we can rewrite (1.7) in terms of (ρ, α, ω, θ) that

∂tρ+ y∂xρ+ α = 0,

∂tα+ y∂xα+ 2∂x(∂y(∆
−1)α+ ∂x(∆

−1)ω) +
1

γM2
(∆ρ+∆θ) = ν∆α,

∂tω + y∂xω − α = 0,

∂tθ + y∂xθ + (γ − 1)α = 0.

(1.15)

Obviously, the above system is a closed system regarding of (ρ, α, ω, θ). Let

f̂(k, η) =
1

2π

∫∫
T×R

e−i(kx+ηy)f(x, y) dx dy,

f(x, y) =
1

2π

∑
k

∫
R
ei(kx+ηy)f̂(k, η) dη,

then we define f ∈ Hs(T× R) if

∥f∥2Hs =
∑
k

∫
⟨k, η⟩2s|f̂ |2(k, η) dη < +∞, with ⟨k, η⟩ =

√
1 + k2 + η2.

Now, we can state the main result of the present paper.

Theorem 1.1. Let γ > 1, 0 < ν < 1 and 0 < M ≤ ν−1. Assume that (ρin, αin, ωin, θin) ∈
H

3
2 (T× R) is the initial data of (1.15) with∫

T
ρin dx =

∫
T
αin dx =

∫
T
ωin dx =

∫
T
θin dx = 0. (1.16)

Then, there exists a positive constant C independent of γ, ν,M such that∥∥P[v]x(t)∥∥
L2

≤ C⟨t⟩−1/2γ−1 exp(CM(M+ 1))
( 1

M

(
∥ρin∥H3/2 + ∥θin∥H3/2

)
+ ∥αin∥H3/2 + γ∥ωin∥H3/2

)
,∥∥P[v]y(t)∥∥

L2

≤ C⟨t⟩−3/2γ−1 exp(CM(M+ 1))
( 1

M

(
∥ρin∥H3/2 + ∥θin∥H3/2

)
+ ∥αin∥H3/2 + γ∥ωin∥H3/2

)
,∥∥Q[v](t)

∥∥L2 +
γ

M
∥ρ(t)∥ − L2 +

γ

M
∥θ(t)∥L2

≤ C⟨t⟩1/2
{∥∥ (γ − 1)ρin − θin

M

∥∥
L2 + (γ + 1) exp(CM(M + 1))

×
( 1

M
∥ρin∥H1 +

1

M
∥θin∥H1 + ∥αin∥H1 + γ∥ωin∥H1

)}
.

Moreover, if ρin, θin, and ωin additionally satisfy the relation

ρin + γωin + θin = 0, (1.17)



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 5

we can obtain the enhanced dissipation of the velocity field,∥∥P[v]x(t)∥∥
L2 ≤ C⟨t⟩−1/2e−

1
16ν

1/3t exp(CM(M + 1))
(
∥αin∥H3/2 +

1

M
∥ωin∥H3/2

)
,∥∥P[v]y(t)∥∥

L2 ≤ C⟨t⟩−3/2
e−

1
16ν

1/3t exp(CM(M + 1))
(
∥αin∥H3/2 +

1

M
∥ωin∥H3/2

)
,∥∥Q[v](t)

∥∥
L2 +

1

M
∥ρ(t) + θ(t)∥L2

≤ C⟨t⟩1/2e− 1
32ν

1/3t(1 + γ) exp(CM(M + 1))
(
∥αin∥H1 +

1

M
∥ωin∥H1

)
.

At first glance, the enhanced dissipation phenomenon of the velocity field is some surprising
because of there is only dissipation for the compressible part of the velocity. This mainly benefits
from the relation (1.17) which gives rise to ω = − 1

γ (ρ+θ). The special relation connects compress-

ible and incompressible phenomena. Namely, an increase of the vorticity need to be compensated
by a decrease for the density and the temperature.

In [1], Antonelli et al. studied the linear stability properties of the 2D isentropic compressible
Euler equations linearized around a shear flow given by a monotone profile, close to the Couette
flow, with constant density, in the domain T× R. For the non-isentropic compressible fluid, how
to obtain a similar result is an interesting problem. This is left in the future work.

Remark 1.2. For 0 < ν < 1, Theorem 1.1 holds for the whole subsonic regime. Formally
let M → 0, the behavior of fluid subsonic regime may be very similar to the incompressible
case. Physically, when M → 0+, our system indeed formally converges to an incompressible
regime, as the dominant pressure term 1

γM2∇P enforces divv ≈ 0. However, the convergence

is subtle because: (i) The temperature equation remains active in our non-isentropic model. (ii)
The volume viscosity ν maintains dissipation even as M → 0. We recognize that this transition
between compressible and incompressible regimes presents important theoretical questions. A
rigorous examination of this asymptotic limit will be the focus of our ongoing research efforts.

2. Proof of the main theorem

2.1. Preliminary and a priori estimates. First of all, we are concerned with the dynamics of
the x-averages of the perturbations. To reveal the distinction between the zero mode case k = 0
and the nonzero modes k ̸= 0. We define

f0(y) :=
1

2π

∫
T
f(x, y) dx, f̸=(x, y) := f(x, y)− f0(y),

which represents the projection onto 0 frequency and the projection onto non-zero frequencies.
Because the structure of the Couetee flow and that the equations are linear, it is clear that

the zero mode in x has an independent dynamics with respect to other modes. Consequently, in
our analysis we can decouple the evolution of the k = 0 mode from the rest of the perturbation.
Integration in x equations in (1.15), one infer that

∂tρ0 = −α0,

∂tα0 = − 1

γM2
∂yyρ0 −

1

γM2
∂yyθ0 + ν∂yyα0,

∂tω0 = α0,

∂tθ0 = −(γ − 1)α0.

(2.1)

From (2.1), we can further get α0 satisfies the damped wave equations

∂ttα0 − ν∂t∂yyα0 −
1

M2
∂yyα0 = 0, in R, (2.2)

and ρ0 + θ0 satisfies the wave equation

∂tt(ρ0 + θ0)−
1

M2
∂yy(ρ0 + θ0) = 0, in R. (2.3)



6 X. ZHAI EJDE-2025/107

Hence, given ρin0 = αin
0 = θin0 = ωin

0 = 0, we can get that for all t ≥ 0,

ρ0(t) = α0(t) = θ0(t) = ω0(t) = 0.

Consequently, in our analysis we can decouple the evolution of the k = 0 mode from the rest of
the perturbation. Let us consider the coordinate transform(

x
y

)
7→

(
X
Y

)
=

(
x− yt

y

)
.

Under the new coordinate transform, the differential operators change as follows

∂x = ∂X , ∂y = ∂Y − t∂X , ∆ = ∆L := ∂XX + (∂Y − t∂X)2.

We define

R(t,X, Y ) = ρ(t,X + tY, Y ), A(t,X, Y ) = α(t,X + tY, Y ),

Ω(t,X, Y ) = ω(t,X + tY, Y ), Θ(t,X, Y ) = θ(t,X + tY, Y ).

Then, the linear system (1.15) reduces to the following system in the new coordinates,

∂tR = −A,

∂tA = ν∆LA− 2∂X(∂Y − t∂X)(∆−1
L )A− 2∂XX(∆−1

L )Ω− 1

γM2
(∆LR+∆LΘ),

∂tΩ = A,

∂tΘ = −(γ − 1)A.

(2.4)

We want to analyze the system (2.4) on the frequency space, in analogy with respect to the
incompressible Couette flow. So we define the symbol associated with −∆L as

p(t, k, η) = k2 + (η − kt)2,

and denote the symbol associated to the operator 2∂X(∂Y − t∂X) as

(∂tp)(t, k, η) = −2k(η − kt).

In the moving frame, for the Laplacian operator, there hold the following inequalities.

Lemma 2.1. Let p = −∆̂L = k2 + (η − kt)2, then for any function f ∈ Hs+2β(T × R), it holds
that

∥p−βf∥Hs ≤ C
1

⟨t⟩2β
∥f∥ −Hs+2β , ∥pβf∥Hs ≤ C⟨t⟩2β∥f∥Hs+2β , (2.5)

for any β > 0.

Proof. The bound (2.5) follows just by Plancherel Theorem and the basic inequalities for japanese
brackets ⟨k, η⟩ ≤ C⟨η − ξ⟩⟨k, ξ⟩. □

In proving Theorem 1.1, we use some main ideas from [1] and [2] but we are faced with a
number of technical difficulties because of a more complicated system. We first get by taking the
Fourier transform of (2.4) that

∂tR̂ = −Â,

∂t = −νpÂ+
∂tp

p
Â− 2k2

p
Ω̂ +

p

γM2 (R̂+ Θ̂),

∂tΩ̂ = Â,

∂tΘ̂ = −(γ − 1)Â.

(2.6)

To exploit the special structure of the system (2.6), we introduce an unknown good function Φ as

Φ =
R+Θ

γ
(2.7)



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 7

from which we can rewrite (2.6) as

∂tΦ̂ = −Â,

∂t = −νpÂ+
∂tp

p
Â− 2k2

p
Ω̂ +

p

M2
Φ̂.

(2.8)

To break through the barrier involved in the term Ω in (2.8), we deduce from

∂t(R+ γΩ+Θ) = 0

that it holds

R+ γΩ+Θ = Rin + γΩin +Θin

which combining with (2.7) leads to

Ω = Φin +Ωin − Φ. (2.9)

Hence, substituting (2.9) into (2.8), we obtain a closed system only involved in Φ̂, and  other
than the initial data

∂tΦ̂ = −Â,

∂t = −νpÂ+
∂tp

p
Â+ (

p

M2
+

2k2

p
)Φ̂− 2k2

p
(Φ̂in + Ω̂in).

(2.10)

In the following, to obtain the enhanced dissipation, we introduce the “ghost multiplier” which
has been used in [3, 4].

Let multiplier m solve the linear ODE for k ̸= 0,

∂tm

m
= − ν1/3

[ν1/3|t− η
k |]2 + 1

,

m(0, k, η) = 1.

Notice that there is a constant c (independent of k, η, t, and ν) such that c < m(t, k, η) ≤ 1. In
particular, its presence does not change a norm

∥m(t,∇)⟨∇⟩σf∥L2 ≈ ∥⟨∇⟩σf∥L2 , with ⟨̂∇⟩σf(k, η) :=
(
1 + (k2 + η2)

)σ/2
f̂(k, η).

The crucial property that m satisfies is

1 ≲ ν−1/6
(√

−∂tm

m
(t, k, η) + ν1/2|k, η − kt|

)
for k ̸= 0, (2.11)

which implies that

∥f ̸=∥2L2 ≲ ν−1/3
(∥∥∥√−∂tm

m
f̸=

∥∥∥2
L2

+ ν∥∇Lf̸=∥2L2

)
. (2.12)

The following lemma plays a crucial role in our subsequent analysis.

Lemma 2.2. For any (ρin, αin, ωin, θin) ∈ Hs(T×R) with s ≥ 0. Assume that γ > 1, 0 < ν < 1,
and 0 < M ≤ ν−1. Then there exists a positive constant C independent of γ, ν,M such that

1

M
∥(p−1/4Φ̂)(t)∥Hs + ∥(p−3/4Â)(t)∥Hs

≤ C exp(CM(M + 1))
( 1

M
∥Φ̂in∥Hs + ∥Âin∥Hs + ∥Φ̂in + Ω̂in∥Hs

)
.

(2.13)

Proof. For any s ≥ 0, we define two weighted functions involved in Φ̂ and  as

Z1(t) :=
1

M
⟨k, η⟩s(m−1p−1/4Φ̂)(t), (2.14)

Z2(t) := ⟨k, η⟩s(m−1p−3/4Â)(t). (2.15)



8 X. ZHAI EJDE-2025/107

From the equations in (2.10) and definitions of Z1 and Z2, a simple computations gives

∂tZ1 = −∂tm

m
Z1 −

1

4

∂tp

p
Z1 −

1

M
p1/2Z2,

∂tZ2 = −
(∂tm

m
+ νp

)
Z2 +

1

4

∂tp

p
Z2

( 1

M
p1/2 + 2M

k2

p3/2

)
Z1 − ⟨k, η⟩s 2m

−1k2

p
7
4

(Φ̂in + Ω̂in).

(2.16)

Now, by multiplying the first equation by Z̄1 and the second equation by Z̄2 in (2.16) respectively,
we obtain

1

2

d

dt
|Z1|2 = −∂tm

m
|Z1|2 −

1

4

∂tp

p
|Z1|2 −

1

M
p1/2Re(Z̄1Z2), (2.17)

1

2

d

dt
|Z2|2 = −

(∂tm
m

+ νp
)
|Z2|2 +

1

4

∂tp

p
|Z2|2 +

1

M
p1/2Re(Z1Z̄2)

+ 2M
k2

p3/2
Re(Z1Z̄2)− ⟨k, η⟩s 2m

−1k2

p
7
4

Re((Φ̂in + Ω̂in)Z̄2).

(2.18)

From p(t, k, η) = k2 + (η − kt)2 > 0, one has for any t > η/k it holds ∂tp/p > 0, the third term
on the right-hand side of the first equation in (2.16) acts as a damping term for Z1. Instead,
∂tp/p < 0 for t < η/k, hence it induces a growth on Z1. However, the situation is opposite for the
second equation involved in Z2. That is to say, for t > η/k, the term (∂tp/p)Z2 induces a growth,
for t < η/k, the term (∂tp/p)Z2 acts as a damping term. Thus, there is a competition between
Z1 and Z2. To balance this relation, we have to consider the time derivative of the mixed terms
involved in Z1, Z2:

d

dt

( ∂tp

p3/2
Z1

)
= −∂tm

m

∂tp

p3/2
Z1 +

( 2k2

p3/2
− 7

4

(∂tp)
2

p
5
2

)
Z1 −

1

M

∂tp

p
Z2 (2.19)

from this and the second equation in (2.16), we obtain

M

4

d

dt

( ∂tp

p3/2
Re(Z̄1Z2)

)
= −1

4

∂tp

p
(|Z2|2 − |Z1|2) +

M

4

(2k2
p

3
2

− 3

2

(∂tp)
2

p5/2

)
Re(Z̄1Z2)

− M

2

∂tm

m

∂tp

p3/2
Re(Z̄1Z2)−ν

M

4

∂tp

p1/2
Re(Z̄1Z2) +M2 k

2∂tp

2p3
|Z1|2

− ⟨k, η⟩sm−1Mk2∂tp

2p
13
4

Re
(
(Φ̂in + Ω̂in)Z̄2

)
.

(2.20)

It is obvious that the first term on the right=hand side of (2.20) could cancel two bad terms

− 1
4
∂tp
p |Z1|2 appeared in (2.17) and 1

4
∂tp
p |Z1|2 appeared in (2.18).

Because of the lack of a diffusive term in the equation of Φ̂, we have to exploit the special
structural characteristics (wave structure) of (2.16) to find hidden dissipation for Z1. So, we also
need to consider the time derivative of the mixed terms involved in Z1, Z2 with different weight
as

d

dt

(
p−1/2Z1

)
= −∂tm

m
p−1/2Z1 −

3

4

∂tp

p3/2
Z1 −

1

M
Z2 (2.21)

which combined with the second equation in (2.16) give rise to

− d

dt

(
p−1/2Re(Z̄1Z2)

)
= − 1

M

(
1 + 2M2 k

2

p2

)
|Z1|2 +

1

2

∂tp

p3/2
Re(Z̄1Z2)

+ 2
∂tm

mp1/2
Re(Z̄1Z2) +

1

M
|Z2|2 + νp1/2Re(Z̄1Z2)

+ ⟨k, η⟩s 2m
−1k2

p9/4
Re((Φ̂in + Ω̂in)Z̄1).

(2.22)



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 9

Finally, to define a coercive energy functional, we need to consider the time derivative of the term
p−3/2∂tpZ1 or p−3/2∂tpZ2. Here, we choose the former

M2

2

d

dt

∣∣ ∂tp
p3/2

Z1

∣∣2
= M2

(2k2∂tp
p3

− 7

4

(∂tp)
3

p4

)
|Z1|2−M2 ∂tm

m

(∂tp)
2

p3
|Z1|2−M

(∂tp)
2

p
5
2

Re(Z̄1Z2).

(2.23)

Now, we define the energy functional

E(t) = 1

2

(
1 +M2 (∂tp)

2

p3

)
|Z1|2(t) +

1

2
|Z2|2(t) +

(M
4

∂tp

p3/2
Re(Z̄1Z2)

)
(t)

−
(Mν1/3

4
p−1/2Re(Z̄1Z2)

)
(t).

(2.24)

Multiplying by Mν1/3

4 on both hand side of (2.22) then summing (2.17), (2.18), (2.20), and (2.23)
gives

d

dt
E(t) +

(∂tm
m

+
ν1/3

4

(
1 + 2M2 k

2

p2
)
+M2 ∂tm

m

(∂tp)
2

p3

)
|Z1|2 +

(∂tm
m

+ νp
)
|Z2|2

=
ν1/3

4
|Z2|2 +

Mν
4
3

4
p1/2Re(Z̄1Z2)−

νM

4

∂tp

p1/2
Re(Z̄1Z2) +M2

(5k2∂tp
2p3

− 7

4

(∂tp)
3

p4

)
|Z1|2

+M
(ν1/3

8

∂tp

p
3
2

+
5

2

k2

p3/2
− 11

8

(∂tp)
2

p
5
2

+
ν1/3

2

∂tm

mp1/2

)
Re(Z̄1Z2)

+ ⟨k, η⟩sMν1/3m−1k2

2p9/4
Re((Φ̂in + Ω̂in)Z̄1)− ⟨k, η⟩sMm−1k2∂tp

2p
13
4

Re((Φ̂in + Ω̂in)Z̄2)

:= I1 + I2 + I3 + I4 + I5 + I6 + I7.

(2.25)

Now we are in a position to bound the terms on the right-hand side of (2.25). First, from
(2.11), it holds

∂tm

m
+ νp ≥ ν1/3. (2.26)

Hence, the first term I1 can be absorbed directly in the left. We next consider I2. With the aid
of the Cauchy-Schwarz inequality, one has

|I2| ≤
Mν1/3

8
(ν|Z1|2 + νp|Z2|2) ≤

Mν

8
(ν1/3|Z1|2) +

Mν1/3

8
(νp|Z2|2). (2.27)

As a result, to absorb I2 by the left, we need the assumption Mν ≤ 1.
Since |∂tp| ≤ 2|k|p1/2, we can bound the terms I3 as follows

|I3| ≤
ν

4

(2M |k|p1/2

p1/2
Re(Z̄1Z2)

)
≤ ν

4

(
4
M2k2

p
|Z1|2 +

1

4
(p|Z2|2)

)
≤ νM2k2

p
|Z1|2 +

1

16
νp|Z2|2.

(2.28)

In the same manner, from |∂tp| ≤ 2|k|p1/2 and the fact that |k|p−3/2 ≤ 1, we have

|I4| ≤ 19M2 |k|3

p5/2
|Z1|2 ≤ CM2 k

2

p
|Z1|2. (2.29)

In the following, we bound the terms in I5. Thanks to |∂tp| ≤ 2|k|p1/2 again, we have

|I5| ≤
Mν1/3

4

|k|
p2

Re(Z̄1Z2) +
M

4

|k|2

p3/2
Re(Z̄1Z2) +

Mν1/3

2

∂tm

mp1/2
Re(Z̄1Z2)

:= I5,1 + I5,2 + I5,3.
(2.30)



10 X. ZHAI EJDE-2025/107

From ν ≤ 1 and p−
3
2 ≤ 1, we can bound I5,1 as

|I5,1| ≤ CM
k2

p
(|Z1|2 + |Z2|2). (2.31)

The term I5,2 can be controlled similarly if noticing the fact that |k|p−1 ≤ 1.

For the last term I5,3, we can use Mν1/3 ≤ 1 and p−1/2 ≤ 1 to obtain

|I5,3| ≤ C
∂tm

m
(|Z1|2 + |Z2|2). (2.32)

Substituting the above estimates involved in I5,1, I5,2, I5,3 into (2.30), we obtain

|I5| ≤ CM
k2

p
(|Z1|2 + |Z2|2) + C

∂tm

m
(|Z1|2 + |Z2|2). (2.33)

From ν ≤ 1, p−1 ≤ 1, |∂tp| ≤ 2|k|p1/2 and the multiplier m−1 is a bound Fourier multiplier, we
obtain

Mm−1k2∂tp

2p13/4
+

Mν1/3m−1k2

2p9/4
≤ CM

|k|2

p
;

this and the Young inequality give rise to

|I6|+ |I7| ≤ CM
k2

p

(
⟨k, η⟩2s|Φ̂in + Ω̂in|2 + (|Z1|2 + |Z2|2)

)
. (2.34)

Noticing that

M2 ∂tm

m

(∂tp)
2

p3
> 0,

and then inserting (2.27), (2.28), (2.29), (2.30), (2.33), (2.34) into (2.25), we obtain

d

dt
E(t) + ν1/3

16

(
(1 + 4M2 k

2

p2
)|Z1|2 + |Z2|2

)
≤ CM

k2

p
⟨k, η⟩2s|Φ̂in + Ω̂in|2 + C

(
M(M + 1)

k2

p
+

∂tm

m

)
E(t).

(2.35)

As

4M2 k
2

p2
≥ M2 (∂tp)

2

p3
,

we obtain

d

dt
E(t) + ν1/3

16
E(t) ≤ CM

k2

p
⟨k, η⟩2s|Φ̂in + Ω̂in|2 + C

(
M(M + 1)

k2

p
+ 2

∂tm

m

)
E(t). (2.36)

It is easy to check that∫ t

0

k2

p(τ)
dτ =

∫ t

0

dτ

(ηk − τ)2 + 1
= arctan(

η

k
− t)− arctan(

η

k
).

As a result, applying Gronwall’s inequality to (2.36) we have

E(t) ≤ C
(
E(0) + ⟨k, η⟩2s|Φ̂in + Ω̂in|2

)
exp(CM(M + 1). (2.37)

From |∂tp| < p, it’s not hard to check that

E(t) ≈ 1

4

(
(1 +M2 (∂tp)

2

p3
)|Z1|2 + |Z2|2

)
(t)

which combines with the fact that m is a bounded Fourier multiplier and the definitions of Z1, Z2,
we can obtain ∑

k

∫
E(t) dη ≈ 1

M2
∥p−1/4Φ̂(t)∥2Hs + ∥p−3/4Â(t)∥2Hs (2.38)



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 11

which implies that

1

M
∥(p−1/4Φ̂)(t)∥Hs + ∥(p−3/4Â)(t)∥Hs

≤ C exp(CM(M + 1))
( 1

M
∥Φ̂in∥Hs + ∥Âin∥Hs + ∥Φ̂in + Ω̂in∥Hs

)
.

(2.39)

This completes the proof.. □

2.2. Proof of Theorem 1.1 for general ρin, θin, ωin. Thanks to the previous Lemma, we are
ready to conclude the proof of Theorem 1.1. First, from (2.9) and Lemma 2.2, we have

∥Ω(t)∥Hs = ∥Φ(t)− Φin − Ωin∥Hs

= M∥p1/4(M−1p−1/4Φ̂)(t)∥Hs + ∥Φin +Ωin∥Hs

≤ CM⟨t⟩1/2∥M−1p−1/4Φ̂(t)∥
Hs+1

2
+ ∥Φin∥Hs + ∥Ωin∥Hs

≤ Cγ−1 exp(CM(M + 1))⟨t⟩1/2Cin,s+ 1
2

(2.40)

with

Cin,s+ 1
2
:=

1

M
∥ρin + θin∥

Hs+1
2
+ ∥αin∥

Hs+1
2
+ γ∥ωin∥

Hs+1
2
.

Recall the definition of P[v] in (1.13), it holds

∥P[v]x(t)∥L2 = ∥∂y∆−1ω(t)∥L2 = ∥(∂Y − t∂X)(∆−1
L Ω)(t)∥L2 ≤ C∥((−∆L)

−1/2Ω)(t)∥L2 .

Therefore, from p1/2⟨kt⟩ ≥ C⟨k, η⟩⟨kt⟩ ≥ C⟨η⟩ and estimate (2.40) we obtain

∥P[v]x(t)∥L2 ≤ C
1

⟨t⟩
∥Ω(t)∥H1 ≤ C⟨t⟩−1/2γ−1 exp(CM(M + 1))Cin, 32

. (2.41)

In the same manner, we can deal with the second component of P[v]y,

∥P[v]y(t)∥L2 = ∥∂x∆−1ω∥L2 = ∥∂X(∆−1
L Ω)(t)∥L2

= ∥k
p
Ω(t)∥L2 ≤ C⟨t⟩−2∥Ω(t)∥H1

≤ C⟨t⟩−3/2γ−1 exp(CM(M + 1))Cin,3/2.

(2.42)

Finally, we estimate the compressible part of the velocity. On the one hand, from the Helmholtz
decomposition and the change of coordinates, we obtain

∥Q[v](t)∥L2 +
1

M
∥ρ(t) + θ(t)∥L2 = ∥(−∆)−1/2α(t)∥L2 +

1

M
∥ρ(t) + θ(t)∥L2

= ∥(−∆L)
−1/2A(t)∥L2 +

1

M
∥R(t) + Θ(t)∥L2

= ∥(−∆L)
−1/2A(t)∥L2 +

γ

M
∥Φ(t)∥L2 .

(2.43)

As a result, from (2.43), Lemma 2.2 and that p ≤ ⟨t⟩2⟨k, η⟩2, we deduce that

∥Q[v](t)∥L2 +
1

M
∥ρ(t) + θ(t)∥L2 = ∥p1/4(p−3/4Â)(t)∥L2 +

γ

M
∥p1/4(p−1/4Φ̂)(t)∥L2

≤ C⟨t⟩1/2
(
∥(p−3/4Â)(t)∥H1 +

γ

M
∥(p−1/4Φ̂)(t)∥H1

)
≤ C⟨t⟩1/2(1 + γ)γ−1 exp(CM(M + 1))Cin,1.

(2.44)

On the other hand, by (1.15), we have (∂t + y∂x)((γ − 1)ρ− θ) = 0 which implies that

(γ − 1)ρ− θ = (γ − 1)ρin − θin.

Moreover, we have ∥∥ (γ − 1)ρ(t)− θ(t)

M

∥∥2
L2 =

∥∥ (γ − 1)ρin − θin

M

∥∥2
L2 .



12 X. ZHAI EJDE-2025/107

A simple computation gives

γ

M
ρ =

(γ − 1)ρ− θ

M
+

ρ+ θ

M
,

γ

M
θ = − (γ − 1)ρ− θ

M
+ (γ − 1)

ρ+ θ

M
.

Hence

γ

M
∥ρ(t)∥L2 ≤ C

∥∥ (γ − 1)ρ− θ

M

∥∥
L2 + C

∥∥ρ+ θ

M

∥∥
L2

≤ C⟨t⟩1/2
{∥∥ (γ − 1)ρin − θin

M

∥∥
L2 + (1 + γ)γ−1 exp(CM(M + 1))Cin,1

}
,

and

γ

M
∥θ(t)∥L2 ≤ C

∥∥ (γ − 1)ρ− θ

M

∥∥
L2 + C(γ − 1)

∥∥ρ+ θ

M

∥∥
L2

≤ C(γ − 1)
∥∥ (γ − 1)ρin − θin

M

∥∥
L2 + C⟨t⟩1/2(γ − γ−1) exp(CM(M + 1))Cin,1.

This proves the first case for general ρin, θin, ωin.

2.3. Proof of Theorem 1.1 with special ρin, θin, ωin satisfying (1.17). If

ρin + γωin + θin = 0,

from ∂t(R+ γΩ+Θ) = 0, we can infer that

Ω = −R+Θ

γ
= −Φ.

Thus, we obtain a closed system only involved in Φ̂ and Â

∂tΦ̂ = −Â,

∂t = −νpÂ+
∂tp

p
Â+ (

p

M2
+

2k2

p
)Φ̂.

(2.45)

For the above system, we can make a similar argument as in the proof of Lemma 2.2 to obtain

another version of (2.36) which do not involve in Φ̂in, Ω̂in that

d

dt
E(t) + ν1/3

16
E(t) ≤ C

(
M(M + 1)

k2

p
+

∂tm

m

)
E(t), (2.46)

Consequently, applying Gronwall’s inequality to (2.46), we have

E(t) ≤ C exp(CM(M + 1))e−
ν1/3

16 tE(0)
which combines with (2.38) to give

1

M
∥(p−1/4Φ̂)(t)∥Hs + ∥(p−3/4Â)(t)∥Hs

≤ C exp(CM(M + 1))e−
1
32ν

1/3t
( 1

γM
∥Φin∥Hs + ∥αin∥Hs

)
.

(2.47)

With this inequality in hand, we can follow the same argument as the derivation of (2.41), (2.42),
and (2.44) to obtain

∥P[v]x(t)∥L2 ≤ C exp(CM(M + 1))⟨t⟩−1/2e−
1
16ν

1/3t
( 1

γM
∥ρin + θin∥H3/2 + ∥αin∥H3/2

)
,

∥P[v]y(t)∥L2 ≤ C exp(CM(M + 1))⟨t⟩−3/2
e−

1
16ν

1/3t
( 1

γM
∥ρin + θin∥H3/2 + ∥αin∥H3/2

)
,

∥Q[v](t)∥L2 +
1

M
∥ρ(t) + θ(t)∥L2

≤ C(1 + γ) exp(CM(M + 1))⟨t⟩1/2e− 1
32ν

1/3t
( 1

γM
∥ρin + θin∥H1 + ∥αin∥H1

)
.



EJDE-2025/107 STABILITY ANALYSIS FOR COMPRESSIBLE FLUIDS 13

The proof of Theorem 1.1 is complete.

Acknowledgments. We thank the referees for their comments that have helped to improve our
manuscript. This research was supported by the Foundation for Basic and Applied Basic Research
of Guangdong under grant number 2024A1515030115.

References

[1] P. Antonelli, M. Dolce, P. Marcati; Linear stability analysis for 2D shear flows near Couette in the isentropic

compressible Euler equations, arXiv : 2003.01694.
[2] P. Antonelli, M. Dolce, P. Marcati; Linear stability analysis of the homogeneous Couette flow in a 2D isentropic

compressible fluid, Ann. PDE, 7 (2021), Paper No. 24, 53 pp.

[3] J. Bedrossian, P. Germain, N. Masmoudi; On the stability threshold for the 3D Couette flow in Sobolev
regularity, Ann. of Math., 185 (2017), 541–608.

[4] J. Bedrossian, P. Germain, N. Masmoudi; Stability of the Couette flow at high reynolds numbers in two

dimensions and three dimensions, Bull. Amer. Math. Soc., 56 (2019), 373–414.
[5] R. Bianchini, M. Coti Zelati, M. Dolce; Linear inviscid damping for shear flows near Couette in the 2D stably

stratified regime, Indiana Univ. Math. J., 71 (2022), 1467–1504.

[6] W. Blumen, P. Drazin, D. Billings; Shear layer instability of an inviscid compressible fluid, part 2, J. Fluid
Mech., 71 (1975), 305–316.

[7] W. Blumen; Shear layer instability of an inviscid compressible fluid, J. Fluid Mech., 40 (1970), 769–781.

[8] G. Chagelishvili, A. Rogava, I. Segal; Hydrodynamic stability of compressible plane Couette flow, Physical
Review E, 50 (1994), R4283.

[9] G. Chagelishvili, A. Tevzadze, G. Bodo, S. Moiseev; Linear mechanism of wave emergence from vortices in
smooth shear flows, Physical review letters, 79 (1997), 3178.

[10] Q. Chen, T. Li, D. Wei, Z. Zhang; Transition threshold for the 2D Couette flow in a finite channel, Arch.

Ration. Mech. Anal., 238 (2020), 125–183.
[11] W. Deng, J. Wu, P. Zhang; Stability of Couette flow for 2D Boussinesq system with vertical dissipation, J.

Funct. Anal., 281 (2021), Paper No. 109255, 40 pp.

[12] P. Drazin, A Davey; Shear layer instability of an inviscid compressible fluid, part 3, J. Fluid Mech., 82 (1977),
255–260.

[13] P. W Duck, G. Erlebacher, M. Y. Hussaini; On the linear stability of compressible plane Couette flow, J. Fluid

Mech., 258 (1994), 131–165.
[14] W. Glatzel; Sonic instability in supersonic shear flows, Mon. Not. R. Astron. Soc., 233 (1988), 795–821.

[15] W. Glatzel; The linear stability of viscous compressible plane Couette flow, J. Fluid Mech., 202 (1989),

515–541.
[16] S. Hu, X. Zhong; Linear stability of viscous supersonic plane Couette flow, Phys. Fluids, 10 (1998), 709–729.

[17] Y. Kagei; Asymptotic behavior of solutions of the compressible Navier-Stokes equation around the plane
Couette flow, J. Math. Fluid Mech., 13 (2011), 1–31.

[18] Y. Kagei; Global existence of solutions to the compressible Navier-Stokes equation around parallel flows, J.

Differential Equations, 251 (2011), 3248–3295.
[19] Y. Kagei; Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a parallel flow,

Arch. Ration. Mech. Anal., 205 (2012), 585–650.
[20] H. Li, X. Zhang; Stability of plane couette flow for the compressible Navier-Stokes equations with Navier-slip

boundary, J. Differential Equations, 263 (2017), 1160–1187.
[21] Z. Lin, C. Zeng; Inviscid dynamical structures near Couette flow, Arch. Ration. Mech. Anal., 200 (2011),

1075–1097.
[22] M. Malik, J. Dey, M. Alam; Linear stability, transient energy growth, and the role of viscosity stratification in

compressible plane Couette flow, Physical Review E, 77 (2008), 036322.
[23] N. Masmoudi, W. Zhao; Enhanced dissipation for the 2D Couette flow in critical space, Comm. Partial

Differential Equations, 45 (2020), 1682–1701.
[24] P. Marcus, W. Press; On Green’s functions for small disturbances of plane Couette flow, J. Fluid Mech., 79

(1977), 525–534.

[25] M. Padmini, M. Subbiah; Stability of non-homentropic, inviscid, compressible shear flows, J. Math. Anal.

Appl., 241 (2000), 56–72.
[26] M. Subbiah, R. Jain; Stability of compressible shear flows, J. Math. Anal. Appl, 151 (1990), 34–41.

[27] D. Wei, Z. Zhang, W. Zhao; Linear inviscid damping for a class of monotone shear flow in sobolev spaces,
Comm. Pure. Appl. Math., 71 (2018), 617–687.

[28] J. Yang, Z. Lin; Linear inviscid damping for Couette flow in stratified fluid, J. Math. Fluid Mech., 20 (2018),

445–472.
[29] L. Zeng, Z. Zhang, R. Zi; Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes

equations, SIAM J. Math. Anal., 54 (2022), 5698–5741.
[30] X. Zhai; Linear stability analysis of the Couette flow for the two dimensional non-isentropic compressible Euler

equations, J. Differential Equations, 369 (2023), 215–228.



14 X. ZHAI EJDE-2025/107

Xiaoping Zhai

School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, China

Email address: pingxiaozhai@163.com


	1. Introduction and main result
	2. Proof of the main theorem
	2.1. Preliminary and a priori estimates
	2.2. Proof of Theorem 1.1 for general in, in,in
	2.3. Proof of Theorem 1.1 with special in, in,in satisfying (1.17)
	Acknowledgments

	References