EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2247-2285
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Global existence of weak solutions to 3D compressible
primitive equations of atmospheric dynamics with

degenerate viscosity

Jules Ouya1, Arouna Ouédraogo1,∗

1 Département de Mathématiques, Laboratoire de Mathématiques, Informatique et
Applications (L@MIA), Université Norbert Zongo, Koudougou, Burkina Faso

Abstract. In this work, we show the existence of global weak solutions to the three-dimensional
compressible primitive equations of atmospheric dynamics with degenerate viscosity density-dependent
for large initial data. With a pressure law of the form ρ2, we represent the vertical velocity as a
function of the density and the horizontal one which will be important in using Faedo-Galerkin
method to obtain the global existence of the approximate solutions. In analogy with the cases
in [17–19], we prove that the weak solutions satisfy the basic energy inequality and the Bresch-
Desjardins entropy inequality. Based on these estimates and using compactness arguments, we
prove the global existence of weak solutions of (1) by vanishing the parameters in our approximate
system step by step.

2020 Mathematics Subject Classifications: 35Q30, 35B40, 76D05, 34C35

Key Words and Phrases: Compressible primitive equations, global weak solutions, degenerate
viscosity density-dependent

1. Introduction

We are interested in the study of equations of type Primitive Compressible Equations,
CPEs. These are the equations governing the motions of the dynamics of the atmosphere.
They belong to the class of geophysical fluid dynamics equations. More precisely, in the
hierarchy of models, the CPEs are situated between the non-hydrostatic equations and
the Saint-Venant equations. The primitive equations are derived from the said hydrostatic
approximation in which, the conservation of vertical momentum is replaced by the hydro-
static equation. In general, the CPEs are obtained from the full Navier-Stokes equations
for modeling the atmosphere with an anisotropic viscosity tensor. Taking advantage of
the difference between the horizontal and vertical dimensions of the atmosphere (10 to 20
kilometers for altitude versus thousands of kilometers for length), we obtain the following
hydrostatic model:

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4931

Email addresses: ouyajules6@gmail.com (J. Ouya), arounaoued2002@yahoo.fr (A. Ouédraogo)

https://www.ejpam.com 2247 © 2023 EJPAM All rights reserved.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2248



∂tρ+ divx(ρu) + ∂y(ρv) = 0,

∂t(ρu) + divx(ρu⊗ u) + ∂y(ρuv) +∇xp(ρ) + rρ|u|u
= divx

(
2ν1Dx(u)

)
+ ∂y(ν2∂yu),

∂yp(ρ) = −gρ.

(1)

Here, t > 0, x = (x1, x2) and y are respectively the temporal, horizontal and vertical
variables. (x, y) ∈ Ω = Ωx× (0, 1), with Ωx = T2 the two-dimensional torus.The functions
ρ and p represent respectively the density and the pressure of the medium. They each
depend on x, y and t. The vector U = (u = (u1, u2), v) is the velocity of the flow (where u
is the horizontal velocity and v the vertical velocity). u⊗u is the matrix with components
uiuj , divx = ∂x1 +∂x2 is the divergence operator and Dx is the strain tensor with Dx(u) =
∇xu+ (∇xu)

t

2
along the horizontal directions. The term rρ|u|u, with r > 0 a positive

constant, comes from the quadratic friction source and is useful for the mathematical
study (see [3]). ν1 and ν2 are the turbulence viscosities in the horizontal and vertical
direction respectively. They depend on the density ρ.
In (1), The first equation expresses the conservation of mass, the second, the evolution of
the momentum and the last equation, ∂yp(ρ) = −gρ, is from the hydrostatic approximation
with g > 0 the free fall acceleration. In order to close the system, we assume that the
fluid is Newtonian.
F. Wang et al., in [19], investigated the global existence of weak solutions to CPE (1)
in which the pressure is assumed to be P (ρ) = c2ρ. In [11], Liu and Titi have studied
problem (1) with P (ρ) = ργ , γ > 1, without buoyancy (no gravity).
The main aim of this paper is to extend the result of [11, 19] for a pressure law of the
form p(ρ) = aργ , with constant γ > 1 and a > 0 a given positive constant. This is one of
the perspectives put forward by Timak Ngom in his thesis defended in 2010 [3]. But we
will consider the particular case γ = 2, in which the density can not longer be written,

ρ(t, x, y) = ξ(t, x)e
−g
c2

y as in [19]. However, using the hydrostatic equation it can take the
following form

ρ(t, x, y) = ξ(t, x) + ϕ(y), (2)

where ϕ(y) =
g

2a
(1− y) and (t, x) ∈ [0; +∞[×T2, ξ(t, x) ≥ 0, the new densities.

For simplicity and without lost of generality, we take a = 1 and

ν1(ρ) = ν2(ρ) = ξ(t, x) + ϕ(y). (3)

Then, for ξ > 0 the system (1) becomes
∂t

(
ξ(1 + ϕ

ξ )
)
+ divx

(
ξ(1 + ϕ

ξ )u
)
+ ∂y

(
ξ(1 + ϕ

ξ )v
)
= 0,

∂t

(
ξ(1 + ϕ

ξ )u
)
+ divx

(
ξ(1 + ϕ

ξ )u⊗ u
)
+ ∂y

(
ξ(1 + ϕ

ξ )uv
)
+ rξ(1 + ϕ

ξ )|u|u

+∇x

(
ξ(1 + ϕ

ξ )
)2

= divx

(
2ξ(1 + ϕ

ξ )Dx(u)
)
+ ∂y

(
ξ(1 + ϕ

ξ )∂yu
)
,

(4)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2249

where (x, y) ∈ Ω and t ≥ 0.
We state the asymptotic regime

uj =
∑
i≥0

εiuij , j = 1, 2, v =
∑
i≥0

εivi, ρ =
∑
i≥0

εiρi =
∑
i≥0

εiξi
(
1 +

ϕi

ξi

)
,

where ε = ϕ0

ξ0
. We introduce the asymptotic development at the main order ε0 in (4) and

omit the powers ”0” to obtain

∂tξ + divx(ξu) + ∂y (ξv) = 0,

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u

= divx (2ξDx(u)) + ∂y (ξ∂yu) ,

∂yξ = 0.

(5)

The boundary conditions on ∂Ω are expressed as periodic conditions on ∂Ωx:v/y=0
= v/y=1

= 0,

∂yu/y=0
= ∂yu/y=1

= 0.
(6)

The initial data are written

ρ|t=0 = ξ0(x), ρu|t=0 = ξ0u0 = m0(x, y), (7)

with ξ0 ≥ 0 a.e in Ωx a bounded non-negative function, i.e., there exists a positive number
M such that

0 ≤ ξ0 ≤M < +∞. (8)

Furthermore, we assume that the initial data satisfies:
u0 =

m0

ξ0
if ξ0 ̸= 0 and u0 = 0 elsewhere,

|m0|2

ξ0
= 0, a.e on {(x, y) ∈ Ω : ξ0(x) = 0}

(9)

and 
ξ0 ∈ L1(Ω) ∩ L2(Ω), ∇x

√
ξ0 ∈ L2(Ω),

m0 ∈ L
4
3 (Ω), m0u0 =

m2
0

ξ0
∈ L1(Ω).

(10)

The factor 1 + ϕ
ξ is approximately equal to 1, meaning that the difference between the

vertical and horizontal components of the density is small. In general, in certain fluid dy-
namics problems, density can influence fluid velocity. The factor 1+ ϕ

ξ , close to 1, suggests



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2250

that the variation in vertical density has a negligible effect on fluid velocity, indicating that
other forces or factors are predominant in determining velocity. In particular, for a fluid in
hydrostatic equilibrium the vertical pressure difference is balanced by the vertical density
difference, so the factor 1 + ϕ

ξ close to 1, indicates that the vertical density variation has
a negligible effect on the pressure balance, suggesting that the fluid is mainly balanced by
the horizontal density variation.
In the case of the atmosphere, air density can influence fluid velocity. In the atmosphere,
vertical density variation due to temperature (and therefore pressure) variations is an im-
portant factor contributing to atmospheric circulation, including convective movements
and meteorological phenomena such as updrafts and downdrafts. However, in some sit-
uations where other forces or factors are predominant, the factor 1 + ϕ

ξ may be close to
1, suggesting that the variation in vertical density has a negligible effect on fluid velocity.
For example, in a situation where atmospheric pressure gradients are very steep, pressure
differences may be the main driver of atmospheric motions, and the vertical density vari-
ation may have a relatively small effect on fluid velocity. In such a case, the factor 1 + ϕ

ξ
may be close to 1.
Formally, multiplying the momentum equation (5)2 by horizontal velocity u, then inte-
grating by parts on Ω, we obtain the energy equality

d

dt

∫
Ω

(
1

2
ξ|u|2 + ξ2

)
dxdy + r

∫
Ω
ξ|u|3 dxdy +

∫
Ω
ξ
(
2|Dx(u)|2 + |∂yu|2

)
dxdy = 0.

In this paper, motivated by [17] and especially [19], we will investigate the global existence
of weak solutions to CPE (1) in which the pressure is assumed to be P (ρ) = aργ with
γ > 1 and a > 0 a constant (a = 1, γ = 2 for simplicity). The key issue in our proof is
to construct the approximate solutions satisfying lower bound of the density and Bresch-
Desjardins entropy. We will first face a new difficulty on how to estimate the vertical
velocity v since there is no equation on it. In order to overcome this difficulty, we represent
the vertical velocity v as a function of the density ξ and the horizontal velocity u and use
the Faedo-Galerkin method to prove the existence of the approximate solutions. What’s
more, similar to [10, 17–19], we construct the approximate solutions by adding viscosity
term in the continuity equation, adding drag, cold pressure, quantum and higher derivative
terms in the momentum equation (see (17) for details). Using compactness arguments, we
prove the global existence of weak solutions of CPE by vanishing the parameters in our
approximate system step by step.
The rest of the paper is organized as follows. In the next section , we present some
elementary inequality and compactness theorems which will be used frequently in the
whole proof. In section 3, we show the existence of global solutions to the approximate
system by using the Faedo-Galerkin method. In section 4, we deduce the Bresch-Desjardins
entropy estimates. In section 5, using the standard compactness arguments, we pass to
the limits as the parameters tend to zero, step by step.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2251

2. Preliminary and main results

We give here the basic inequalities which are useful for the next, the definition of
solution in our context and state the main theorems.

Lemma 1. (Aubin-Lions, see [16]). Let X0, X and X1 be three Banach spaces with
X0 ⊆ X ⊆ X1. Suppose that X0 is compactly embedded in X and X is continuously
embedded in X1. For 1 ≤ p, q ≤ +∞, let

W =
{
u ∈ Lp

(
[0, T ];X0

)
: ∂tu ∈ Lq

(
[0, T ];X1

)}
.

(I) If p < +∞, then the embedding of W into Lp
(
[0, T ];X

)
is compact;

(II) If p = +∞ and q > 1, then the embedding of W into C
(
[0, T ];X

)
is compact.

Lemma 2. (see [19] Lemma 2.3). Let Ω ⊂ Rn be a bounded measurable set. Suppose that
(fn)n∈N ⊂ Lp(Ω), ∥fn∥Lp(Ω)≤ C with C a positive constant and fn −→ f a.e in Ω. Then

(I) f ∈ Lp(Ω) and ∥f∥Lp(Ω)≤ C;

(II) fn −→ f strongly in Lp̄(Ω), for any p̄ ∈ [1, p).

Lemma 3. (see [14], theorem 9.3)(Gagliardo-Nirenberg interpolation inequality)
For a function u : Ω −→ R defined on a bounded Lipschitz domain Ω ⊂ Rn and for all
1 ≤ q, r ≤ ∞ and an integer m, suppose that a real number θ and a natural number j are
such that

1

p
=
j

n
+
(1
r
− m

n

)
θ +

1− θ

q
and

j

m
≤ θ ≤ 1.

Then we have
∥Dju∥p≤ C1∥Dmu∥θr∥u∥1−θ

q +C2∥u∥s, (11)

where s > 0 is an arbitrary constant. The constants C1 and C2 depend upon the domain
Ω as well as m, n, j, r, q et θ.

To construct the regular solutions of the approximation scheme, we represent the unknown
vertical velocity v as a function of the density ξ and the horizontal velocity u. To this
end, by differentiating (5)1 with respect to y, we obtain

−ξ∂2yv = ∂ydivx(ξu). (12)

Solving (12) yields

v(y) = −divx(ξũ(y))

ξ
+ y

divx(ξū)

ξ
. (13)

Then

∂yv(y) = −divx(ξu)

ξ
+

divx(ξū)

ξ
, (14)

where

ũ(y) =

∫ y

0
u(τ)dτ and ū =

∫ 1

0
u(τ)dτ.

We now give the definition of weak solutions in our context.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2252

Definition 2.1. A weak solution of the problem (5)-(7) is a triplet (ξ, u, v) satisfying:
(1) (7) holds in D′(Ω);
(2) ξ, u and v belong to the classes

ξ ∈ L∞(
[0, T ];L1(Ω) ∩ L2(Ω)

)
,

√
ξu ∈ L∞(

[0, T ];L2(Ω)
)
,

√
ξ ∈ L∞(

[0, T ];H1(Ω)
)
, ξ

1
3u ∈ L3

(
[0, T ];L3(Ω)

)
,

ξ∇xu ∈ L2
(
[0, T ];L2(Ω)

)
, ξ(∇xu)

t ∈ L2
(
[0, T ];L2(Ω)

)
,

√
ξv ∈ L2

(
[0, T ];L2(Ω)

)
,

√
ξ∂yv ∈ L2

(
[0, T ];L2(Ω)

)
,

√
ξ∂yu ∈ L2

(
[0, T ];L2(Ω)

)
, ∇xξ ∈ L2

(
[0, T ];L2(Ω)

)
,

∇x
√
ξ ∈ L∞ (

[0, T ];L2(Ω)
)
.

(15)

(3) The mass equation is valid in the sense of the distributions and the following equality
holds:∫

Ω
m0Φ(0, x, y)dxdy −

∫
Ω
ξuΦ(T, x, y)dxdy +

∫ T

0

∫
Ω
(ξu∂tΦ+ ξu⊗ u : ∇xΦ) dxdydt

+

∫ T

0

∫
Ω

(
ξuv∂yΦ+ ξ2divxΦ

)
dxdydt−

∫ T

0

∫
Ω
rξ|u|uΦdxdydt

−
∫ T

0

∫
Ω
(2ξDxu : ∇xΦ+ ξ∂yu∂yΦ) dxdydt = 0, (16)

for any regular test function Φ(t, x, y) ∈ C∞
c ([0, T ]× Ω).

We now state our main results.

Theorem 1. Suppose that the initial data satisfies (10).
Then, ∀ T > 0, (5)-(7) has a weak solution (ξ, u, v) in the sense of Definition 2.1.

Theorem 2. Under an assumption similar to the Theorem 1, (1)-(3) and (6) − (7) has
a weak solution (ρ, u, v) in the sense of Definition 2.1, if we replace (ξ, u, v) by (ρ, u, v)
accordingly.

3. Approximation problem

In this section, we construct in a similar way to [9, 18, 19] and chapter 7 of [4] the Faedo-
Galerkin scheme of the approximate system (17) below, and prove the global existence of
solutions to this system. Then, we will show the global existence of weak solutions to the
problem (1)-(3) (6)− (7) by making the parameters of our approximate system tend to 0
step by step.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2253

3.1. Faedo-Galerkin scheme

In order to prove the global existence of weak solutions for the compressible primitive
equations, we consider the following approximate system :

∂tξ + divx(ξu) + ∂y (ξv) = ϵ∆xξ,

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u+ r1u

+α∆2
xu = divx (2ξD(u)) + ∂y (ξ∂yu) + ϵ∇xξ · ∇xu

+r2∇xξ
−β + k1ξ∇x

(
∆x

√
ξ√

ξ

)
+ δξ∇x∆

5
xξ,

∂yξ = 0,

(17)

where (x, y) ∈ Ω, t ≥ 0, β ≥ 10 and the vertical velocity v can be expressed as

v(y) = −divx(ξũ(y))

ξ
+ y

divx(ξū)

ξ
.

In order to keep the density bounded, we add the extra terms ϵ∇xξ
−β and δξ∇x∆

5
xξ.

This allows us to take ∇ (ln ξ) as a test function to derive the Bresch-Desjardins entropy.
Moreover, the term r1u is used to control the density near the vacuum, and rξ|u|u is used

to make sure that
√
ξu is strong convergence in L2

(
[0, T ];L2(Ω)

)
. The term ∆x

√
ξ√

ξ
is called

Bohm potential which can be interpreted as a quantum potential.
For T > 0, We introduce a finite dimensional space Xn = span{ψ1, . . . , ψn}, n ∈ N, where
ψi is the eigenfunction of the Laplacian: −∆ψi = λiψi in Ω with λi the eigenvalue for ψi.
We have the periodic conditions on ∂Ωx: ∂yψi/y=0 = ∂yψi/y=1 = 0.
{ψi} is an orthonormal basis of L2(Ω) which is also an orthogonal basis of H2(Ω). Let
(ξ0, u0) ∈ C∞(Ω) be some initial data satisfying ξ0 ≥ λ for some λ > 0. We notice that
the velocity u ∈ C

(
[0, T ];Xn

)
is given by

u(x, y, t) =
n∑

i=1

λi(t)ψi(x, y), (t, x, y) ∈ [0, T ]× Ω

for some functions λi(t), and the norm of u in C ([0, T ];Xn) can be written as

∥u∥C([0,T ];Xn(Ω))= sup
t∈[0,T ]

n∑
i=1

|λi(t)|.

Since Xn is a finite-dimensional space, all the norms are equivalent on Xn. Hence, u can be
bounded in C

(
[0, T ];Ck(Ω)

)
for all k ∈ N, then there exists a constant C > 0 dependent

on k such that
∥u∥C([0,T ];Ck(Ω))≤ C∥u∥C([0,T ];L2(Ω)). (18)

Now, we approach the continuity equation by adding a viscosity term, ϵ∆xξ :∂tξ + divx(ξu) + ∂y(ξv) = ∂tξ + divx(ξū) = ϵ∆xξ,

ξ0 ∈ C∞(Ω), ξ0 ≥ λ > 0,
(19)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2254

where (t, x, y) ∈ [0, T ]× Ω with T > 0.
For any given U = (u, v) in C ([0, T ];Xn), by the classical theory of parabolic equations,
there exists a classical solution ξ(t, x) ∈ C1

(
[0, T ];C3(Ω)

)
to the approximated system

(19). (see [12], [Lemma 3.1, [19]]) for details and proofs.
We show that this solution is continuously dependent on u and by the comparison principle,
we also show that it satisfies the following inequality

0 < ξ(x)e−
∫ T
0 ∥divxu∥L∞dt ≤ ξ(t, x) ≤ ξ(x)e

∫ T
0 ∥divxu∥L∞dt, ∀x ∈ Ωx = T2, t ≥ 0, (20)

where

ξ(x) = inf
x∈T2

ξ0(x), ξ(x) = sup
x∈T2

ξ0(x).

Moreover, from (20) and the initial conditions of (19), there exists a positive constant η
such that

0 < η ≤ ξ(t, x) ≤ η−1, for (t, x) ∈ [0, T ]× Ωx. (21)

Thus, using the mass equation, we construct the continuous linear operator
S : C0 ([0, T ];Xn) −→ C0

(
[0, T ];Ck(Ω)

)
by S(u) = ξ, and there exists a constant Cn,k

dependent on k ≥ 1 and on n such that

∥S(u1)− S(u2)∥C([0,T ];Ck(Ω))≤ Cn,k∥u1 − u2∥C([0,T ];L2(Ω)). (22)

3.2. Faedo-Galerkin approximation for the weak formulation of the mo-
mentum

We now want to solve the momentum equation (17)2 on the space Xn. For any test
function Φ ∈ Xn, the approximate solution un ∈ C0

(
[0, T ];Xn

)
satisfies:

∫
Ω
ξnun(T )ΦdX −

∫
Ω
m0ΦdX −

∫ T

0

∫
Ω
(ξnun ⊗ un) : ∇xΦdXdt

−
∫ T

0

∫
Ω
ξnunvn∂yΦdXdt+ r

∫ T

0

∫
Ω
ξn|un|unΦdXdt+

∫ T

0

∫
Ω
∇xξ

2
n · ΦdXdt

+ r1

∫ T

0

∫
Ω
unΦdXdt+ α

∫ T

0

∫
Ω
∆xun ·∆xΦdXdt+

∫ T

0

∫
Ω
ξn∂yun∂yΦdXdt (23)

= −2

∫ T

0

∫
Ω
ξnDx(un) : ∇xΦdXdt+ ϵ

∫ T

0

∫
Ω
(∇xξn · ∇xun) ΦdXdt

− r2

∫ T

0

∫
Ω
ξ−β
n divxΦdXdt− 2k1

∫ T

0

∫
Ω
Φ∆x

√
ξn∇x

√
ξndXdt

− k1

∫ T

0

∫
Ω
divxΦ

(
∆x

√
ξn

)√
ξndXdt+ δ

∫ T

0

∫
Ω
Φξn∇x∆

5
xξndXdt,



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2255

where m0 = ξn(0, x, y)un(0, x, y) and dX = dxdy.
As in [4, 5, 9, 19], to solve (23) we introduce the linear operator

M[ξ] : Xn −→ X
′
n, <M[ξ]u, v >=

∫
Ω
ξu.vdx, u, v ∈ Xn.

Using the Lax-Milgram theorem, we show that this operator is invertible and M−1 is
Lipschitz continuous. Let us reformulate equation (23) as follows:

un(t) = M−1[S(un(t))]
(
M[ξ0](u0) +

∫ T

0
N (S(un), un) (s)ds

)
, (24)

where

S(un) = ξn,

N (S(un), un) (s) = divx (2ξnDx(un)) + ∂y (ξn∂yun)− divx(ξun ⊗ un)− ∂y(ξnunvn)

−α∆2
xun − ϵ∇xξn · ∇xun + k1ξ∇x

(
∆x

√
ξn√

ξn

)
−∇xξ

2
n

+r2∇xξ
−β
n − r1un − rξ|un|un + δξn∇x∆

5
xξn.

For more details, we refer the readers to [4–9, 13, 15, 19].
In view of the Lipschitz continuous estimates for S and M−1, the nonlinear equation (24)
can be solved on a short time interval [0, τ ], where τ ≤ T , using a fixed point theorem on
the Banach space C ([0, T ];Xn). We thus obtain a unique local solution in time (ξn, un, vn)
to problems (19) and (24).
Next we will extend this obtained local solution to be a global one. Differentiating (23)
with respect to time t, taking Φ = un and integrating by parts with respect to x over Ω,
we get

∫
Ω

d

dt

(
ξn
u2n
2

)
dX +

∫
Ω
un · ∇xξ

2dX + r

∫
Ω
ξn|un|3dX +

∫
Ω
ξn|∂yun|2dX

+ r1

∫
Ω
u2ndX − r2

∫
Ω
un · ∇xξ

−β
n dX +

∫
Ω
2ξn|Dx(un)|2dX + α

∫
Ω
|∆un|2dX

+

∫
Ω
ξn|∂yun|2dX + k1

∫
Ω

∆x
√
ξn√

ξn
divx (ξnun) dX + δ

∫
Ω
divx (ξnun)∆

5
xξndX = 0. (25)

Furthermore, we estimate the terms of the left hand side in (25) one by one:

∫
Ω
un · ∇xξ

2
ndX = −2

∫
Ω
ξndivx(ξnun)dX

= −2

∫
Ω
ξn (ϵ∆xξn − ∂tξn − ∂y(ξnvn)) dX



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2256

=
d

dt

∫
Ω
ξ2ndX + 2ϵ

∫
Ω
|∇xξn|2 dX, (26)

where we used the fact that ∂yξn = 0 and integration by parts.
Next we deal with the cold pressure and high order derivative of the density terms as
follows

−r2
∫
Ω
un · ∇xξ

−β
n dX = −r2

β

β + 1

∫
Ω
ξ−β−1
n (ϵ∆xξn − ∂tξn − ∂y(ξnvn)) dX

=
r2

β + 1

d

dt

∫
Ω
ξ−β
n dX +

4ϵr2
β

∫
Ω
|∇xξ

−β
2

n |2dX, (27)

δ

∫
Ω
divx(ξnun)∆

5
xξndX = δ

∫
Ω
(ϵ∆xξn − ∂tξn − ∂y(ξnvn))∆

5
xξndX

=
δ

2

d

dt

∫
Ω
|∇x∆

2
xξn|2dX + δϵ

∫
Ω
|∆3

xξn|2dX. (28)

Finally, we estimate the quantum term

k1

∫
Ω

∆x
√
ξn√

ξn
divx(ξnun)dX = k1

∫
Ω

∆x
√
ξn√

ξn
(ϵ∆xξn − ∂tξn − ∂y(ξnvn)) dX

= k1
d

dt

∫
Ω
|∇x

√
ξn|2dX +

k1ϵ

2

∫
Ω
ξn|∇2

x ln ξn|2dX, (29)

where we used

2ξn∇x

(
∆x

√
ξn√

ξn

)
= 2ξn∇x

(
divx

(
∇x

√
ξn√

ξn

)
−∇x

√
ξn · ∇x

1√
ξn

)
= ξndivx

(
∇2

x ln ξn
)
+

1

2
ξn∇x (∇x ln ξn)

2

= divx
(
ξn∇2

x ln ξn
)
.

Substituting (26)-(29) in (25), we obtain the following energy equality:

d

dt
E(ξn, un) + 2ϵ

∫
Ω
|∇xξn|2dX + r1

∫
Ω
u2ndX + r

∫
Ω
ξn|un|3dX + α

∫
Ω
|∆un|2dX

+

∫
Ω
2ξn|Dx(un)|2dX +

4ϵr2
β

∫
Ω
|∇xξ

−β
2

n |2dX +

∫
Ω
ξn|∂yun|2dX

+
k1ϵ

2

∫
Ω
ξn|∇2

x ln ξn|2dX + δϵ

∫
Ω
|∆3

xξn|2dX = 0 (30)

on [0, τ ] where,

E(ξn, un) =

∫
Ω

(
1

2
ξnu

2
n + ξ2n +

r2
β + 1

ξ−β
n + k1|∇x

√
ξn|2 +

δ

2
|∇x∆

2
xξn|2

)
dX,



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2257

and

E0(ξn, un) =

∫
Ω

(
1

2
ξ0u

2
0 + ξ20 +

r2
β + 1

ξ−β
0 + k1|∇x

√
ξ0|2 +

δ

2
|∇x∆

2
xξ0|2

)
dX.

Thus the energy equality (30) gives∫ τ

0
∥∆xun∥2L2(Ω)dt ≤ E0(ξn, un) < +∞. (31)

From dimXn <∞ and (20), the density is bounded with a positive constant, which means
that there exists a constant η > 0 such that

0 < η ≤ ξn(t, x) ≤
1

η
, (32)

for any t ∈ [0, τ ] and (x, y) ∈ Ω. Furthermore, from the basic energy equality (30) and
using (32), we also obtain

sup
t∈[0,τ ]

∫
Ω
ξnu

2
ndX ≤ E0(ξn, un) ≤ C <∞. (33)

From (30), we get

sup
t∈[0,τ ]

∫
Ω
ξn|Dx(un)|2dX ≤ E0(ξn, un). (34)

As all norms are equivalent on Xn, from (31) -(33), we obtain

sup
t∈[0,τ ]

∥un∥L∞(Ω)≤ C < +∞, (35)

sup
t∈[0,τ ]

∥∇xun∥L∞(Ω)≤ C and sup
t∈[0,τ ]

∥∆xun∥L∞(Ω)≤ C. (36)

Then, we can extend τ to T by repeating the above argument several times and obtain
un ∈ C ([0, T ];Xn) . In other words, we obtain a global solution (ξn, un, vn) of (19) and
(24) for any T > 0. Moreover, from (30), we have

sup
t∈[0,T ]

∫
Ω

√
ξnu

2
ndX ≤ E0(ξn, un). (37)

Also,
E(ξn, un) ≤ E0(ξn, un), (38)

which gives

∥ξn∥L∞(0,T ;H5(Ω))≤ C (E0(ξn, un), δ) and ∥ξ2n∥L∞(0,T ;L2(Ω))≤ C. (39)

Using Hölder’s inequality we have:∫
Ω
ξnū

2
ndxdy =

∫
Ω
ξn

(∫ 1

0
un(τ)dτ

)2
dxdy



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2258

≤
∫
Ω
ξn

(∫ 1

0
|un(τ)|2dτ

)
dxdy

≤
∫ 1

0

(
sup

t∈[0,T ]

∫
Ω
ξnu

2
ndxdy

)
dτ

≤
∫ 1

0
E0(ξn, un)dτ ≤ E0(ξn, un) (40)

and ∫
Ω
ξnũ

2
ndxdy =

∫
Ω
ξn

(∫ y

0
un(τ)dτ

)2
dxdy

≤
∫
Ω
ξn

(∫ y

0
|un(τ)|2dτ

)
dxdy

≤
∫
Ω
ξn

(∫ 1

0
|un(τ)|2dτ

)
dxdy

≤
∫
Ω
ξn

(∫ 1

0
12dτ

)(∫ 1

0
u2n(τ)dτ

)
dxdz

≤ E0(ξn, un). (41)

In addition, the basic energy (30) also gives∫ T

0

∫
Ω
ξn|∇2

x ln ξn|2dXdt ≤ E0(ξn, un) (42)

and

sup
t∈[0,T ]

∫
Ω
δ|∇x∆

2
xξn|2dX ≤ E0(ξn, un), (43)

which, with (32), implies that ξn is a positive regular function.
We need the following Lemma from Jüngel [9].

Lemma 4. For any positive regular function ξ(x, y), we have∫
Ω
|∇2

√
ξ|2dxdy ≤ 7

∫
Ω
ξ|∇2 ln ξ|2dxdy (44)

and ∫
Ω
|∇ξ

1
4 |4dxdy ≤ 8

∫
Ω
ξ|∇2 ln ξ|2dxdy. (45)

Therefore, the energy equality (30) and the Lemma (4) allow us to state the following
lemma.

Lemma 5. For any smooth positive function ξ(t, x, y), we have the following estimate:

(kϵ)
1
2 ∥
√
ξn∥L2([0,T ];H2(Ω))+(kϵ)

1
4 ∥∇xξ

1
4
n ∥L4([0,T ];L4(Ω))≤ C, (46)

for a constant C > 0 independent on n.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2259

To close this section, we give a summary of the approximate solution (ξn, un, vn) as
follows.

Proposition 3.1. Assume that (ξn, un, vn) is the solution of (19) and (24) on [0, T ]×Ω
constructed above. Then the solution satisfies the energy inequality

E(ξn, un) + 2ϵ

∫ T

0

∫
Ω
|∇xξn|2dXdt+ r1

∫ T

0

∫
Ω
u2ndXdt+ r

∫ T

0

∫
Ω
ξn|un|3dXdt

+

∫ T

0

∫
Ω
2ξn|Dx(un)|2dX + α

∫ T

0

∫
Ω
|∆xun|2dXdt+

4ϵr2
β

∫ T

0

∫
Ω
|∇xξ

−β
2

n |2dXdt

+ δϵ

∫ T

0

∫
Ω
|∆3

xξn|2dXdt+
∫ T

0

∫
Ω
ξn|∂yun|2dXdt+

ϵk1
2

∫ T

0

∫
Ω
ξn|∇2

x ln ξn|2dXdt

≤ E(ξ0, u0), (47)

with

E(ξn, un) =

∫
Ω

(
1

2
ξnu

2
n + ξ2n +

r2
β + 1

ξ−β
n + k1|∇x

√
ξn|2 +

δ

2
|∇x∆

2
xξn|2

)
dX.

In particular, we have the following estimates

√
ξnun ∈ L∞ (

[0, T ];L2(Ω)
)
, ξn ∈ L2

(
[0, T ];L2(Ω)

)
, r2ξ

−β
n ∈ L∞ (

[0, T ];L1(Ω)
)
,

√
ξnDxun ∈ L2

(
[0, T ];L2(Ω)

)
,
√
α∆xun ∈ L2

(
[0, T ];L2(Ω)

)
,
√
k1
√
ξn ∈ L∞(

[0, T ];H1(Ω)
)
,

√
δξn ∈ L∞(

[0, T ];H5(Ω)
)
,
√
ϵ∇x

√
ξn ∈ L2

(
[0, T ];L2(Ω)

)
;
√
ξn∂yun ∈ L2

(
[0, T ];L2(Ω)

)
,

√
ϵr2∇xξ

−β
2

n ∈ L2
(
[0, T ];L2(Ω)

)
,
√
δϵξn ∈ L2

(
[0, T ];H6(Ω)

)
,
√
r1un ∈ L2

(
[0, T ];L2(Ω)

)
,

ξ
1
3
n un ∈ L3

(
[0, T ];L3(Ω)

)
,
√
ϵk1

√
ξn ∈ L2

(
[0, T ];H2(Ω)

)
, 4
√
k1ϵ∇xξ

1
4
n ∈ L4

(
[0, T ];L4(Ω)

)
,

ξ2 ∈ L∞(
[0, T ];L1(Ω)

)
,
√
ξnūn ∈ L∞(

[0, T ];L2(Ω)
)
,
√
ξnũn ∈ L∞(

[0, T ];L2(Ω)
)
.

(48)
Now, we want to pass to the limit in (47) when n −→ +∞. We then state a series of
convergence lemmas.

3.3. Convergence results

We fix ϵ, r, r1, k1, α, δ, r2 > 0 and let us first take the limits when n −→ +∞.

Lemma 6. (Convergence of ξn). For any fixed positive constants ϵ, r1, k1, α, δ and r,



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2260

the following estimates hold:

∥∂t
√
ξn∥

L∞
(
[0,T ];W−1, 32 (Ω)

)+∥
√
ξn∥L2

(
[0,T ];H2(Ω)

)≤ K,

∥ξn∥L∞
(
[0,T ];H5(Ω)

)+∥∂tξn∥
L∞

(
[0,T ];W−1, 32 (Ω)

)≤ K,

∥ξ−β
n ∥

L
5
3

(
[0,T ];L

5
3 (Ω)

)≤ K,

∥ξ2n∥L 5
3

(
[0,T ];L

5
3 (Ω)

)≤ K,

(49)

where K is independent of n, depends on ϵ, r2, δ, r1, initial data and T .
Moreover, up to an extracted subsequence ξn, when n −→ +∞ we have:

√
ξn −→

√
ξ strongly in L2

(
[0, T ];H1(Ω)

)
and

√
ξn −→

√
ξ a.e,

ξn −→ ξ strongly in C
(
[0, T ];H5(Ω)

)
and ξn −→ ξ a.e,

ξ−β
n −→ ξ−β strongly in L1

(
[0, T ];L1(Ω)

)
and ξ−β

n −→ ξ−β a.e.

(50)

Proof. The proof of (49) is done in Lemma 2.2 of [18].
It remains to prove (50). Since

√
ξn ∈ L∞(

[0, T ];H1(Ω)
)
, using Sobolev embedding theo-

rem (since k = 1, d = 3, p = 2 < d, then W 1,2(Ω) = H1(Ω) ↪→ Lp∗(Ω) with p∗ = dp
d−p = 6

) we deduce that
√
ξn ∈ L∞(

[0, T ];L6(Ω)
)
. Then, since

√
ξn ∈ L∞(

[0, T ];L6(Ω)
)
and√

ξnūn ∈ L∞(
[0, T ];L2(Ω)

)
, the Hölder inequality allows us to conclude that ξnūn =

√
ξn
√
ξnūn ∈ L∞(

[0, T ];L
3
2 (Ω)

)
. By (19), we have

∂tξn = ϵ∆xξn − divx(ξnun)− ∂y(ξnvn)

= ϵ∆xξn − divx(
√
ξn
√
ξnūn) ∈ L∞(

[0, T ];W−1, 3
2 (Ω)

)
. (51)

Using (51), we get ξn ∈ L∞(
[0, T ];H5(Ω)

)
∩L2

(
[0, T ];H6(Ω)

)
. From Aubin-Lions Lemma

1, we deduce that ξn ∈ C
(
[0, T ];H5(Ω)

)
. So, up to a subsequence, we have

ξn −→ ξ strongly in C
(
[0, T ];H5(Ω)

)
and ξn −→ ξ a.e.

Next, we claim that ξ−β
n is bounded in L

5
3

(
[0, T ];L

5
3 (Ω)

)
. Indeed, using the fact that

∇xξ
−β/2
n is bounded in L2

(
[0, T ];L2(Ω)

)
and the Sobolev embedding theorem, we deduce

that ξ−β
n is bounded in L1

(
[0, T ];L3(Ω)

)
. Then we apply Hölder inequality to get

∥ξ−β
n ∥

L
5
3

(
[0,T ];L

5
3 (Ω)

)≤ ∥ξ−β
n ∥

2
5

L∞
(
[0,T ];L1(Ω)

)∥ξ−β
n ∥

3
5

L1
(
[0,T ];L3(Ω)

)≤ K. (52)

Now, using the following Sobolev inequality, ∥ξ−1
n ∥L∞(Ω)≤ C

(
1+∥ξ−1

n ∥L3(Ω)

)3(
1+∥ξn∥Hk+2(Ω)

)2
for k ≥ 3

2 (see [[1], Lemma 2.1]), and the estimates of density in (48), we get

∥ξ−β
n ∥L∞(Ω)≤ C(r2, δ) a.e on [0, T ]. (53)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2261

Thus, we have ξ−β
n converges a.e to ξ−β. Thanks to (52) and Lemma 2, we have ξ−β

n −→
ξ−β strongly in L1

(
[0, T ];L1(Ω)

)
.

Using again the mass equation (19), we have

∂t
√
ξn =

1

2

1√
ξn
∂tξn =

1

2

1√
ξn

(
ϵ∆xξn − divx(

√
ξn
√
ξnūn)

)
.

By using (53) and the fact that ∂tξn ∈ L∞(
[0, T ];W−1, 3

2

)
, we have ∂t

√
ξn ∈ L∞(

[0, T ];W−1, 3
2

)
.

Note that
√
ξn ∈ L2

(
[0, T ];H2(Ω)

)
which is reflexive, so using the Aubin-Lions Lemma,

we obtain√
ξn −→

√
ξ strongly in L2

(
[0, T ];H1(Ω)

)
and

√
ξn −→

√
ξ a.e in [0, T ]× Ω.

Furthermore, as in [5] we show that
∥∇xξn∥L2

(
[0,T ]×Ω

)≤ K,

∥ξ2n∥L 5
3

(
[0,T ];L

5
3 (Ω)

)≤ ∥ξ2n∥
2
5

L∞
(
[0,T ];L1(Ω)

)∥ξ2n∥ 3
5

L1
(
[0,T ];L3(Ω)

)≤ K.
(54)

Using (49) and the fact that ξ2n converges almost everywhere to ξ2, we obtain

ξ2n −→ ξ2 strongly in L1
(
[0, T ];L1(Ω)

)
.

Lemma 7. (Convergence of momentum ξnun). Up to an extracted subsequence, we have
ξnun −→ ξu strongly in L2

(
[0, T ];L2(Ω)

)
and ξnun −→ ξu a.e in [0, T ]× Ω.

Proof. According to estimates (48), we know that un is bounded in L2
(
[0, T ];L2(Ω)

)
which is reflexive. So, up to a subsequence, we have

un ⇀ u weakly in L2
(
[0, T ];L2(Ω)

)
.

Recalling that ξn −→ ξ strongly in C
(
[0, T ];H5(Ω)

)
, we have

ξnun −→ ξu strongly in L1
(
[0, T ];L1(Ω)

)
.

Moreover, since ξn ∈ L∞(
[0, T ];H5(Ω)

)
, un ∈ L2

(
[0, T ];H2(Ω)

)
,
√
ξn∂yun ∈ L2

(
[0, T ];L2(Ω)

)
,

we deduce

∇(ξnun) = un.∇xξn + ξn∇xun + ξn∂yun ∈ L2
(
[0, T ];L2(Ω)

)
.

This last identity and the fact that ξnun ∈ L2
(
[0, T ];L2(Ω)

)
give ξnun ∈ L2

(
[0, T ];H1(Ω)

)
.

Next, we claim that ∂y(ξnun) ∈ L2
(
[0, T ];H−s(Ω)

)
, for some s > 0.

Indeed,

∂t(ξnun) =− divx(ξnun ⊗ un)− ∂y(ξnunvn)−∇xξ
2
n − r1un

− rξ|un|un − α∆2un + 2divx
(
ξnDx(un)

)
+ ∂y(ξn∂yun)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2262

+ ε∇xξn · ∇xun + r2∇xξ
−β
n + k1ξn∇x

(∆x
√
ξn√

ξn

)
+ δξn∇x∆

5
xξn. (55)

where

∂y(ξnunvn) = ∂y

(
ξnun

(
− divx(ξnũn)

ξn
+ y

divx(ξnūn)

ξn

))
= ∂y

(
− divx(ξnũn ⊗ un) + ξnũn.∇xun + ydivx(ξnūn ⊗ un)− yξnūn.∇xun

)
.

(56)

Based on the energy estimates (48), we get ∂y(ξnun) ∈ L2
(
[0, T ];H−5(Ω)

)
. Then, thanks

to the Aubin-Lions Lemma 1, ξnun converges strongly in L2
(
[0, T ];L2(Ω)

)
to a function

f ∈ L2
(
[0, T ];L2(Ω)

)
.

Also, since ξnun −→ ξu strongly in L1
(
[0, T ];L1(Ω)

)
, we have ξnun −→ ξu strongly in

L2
(
[0, T ];L2(Ω)

)
.

We have the following result.

Lemma 8. (see [19], Lemma 3.5). Up to an extracted subsequence, we have

√
ξnun −→

√
ξu strongly in L2

(
[0, T ];L2(Ω)

)
,

√
ξnũn −→

√
ξũ strongly in L2

(
[0, T ];L2(Ω)

)
,

√
ξnūn −→

√
ξū strongly in L2

(
[0, T ];L2(Ω)

)
.

By Lemma 8, we conclude that
√
ξnun −→

√
ξu,

√
ξnũn −→

√
ξũ and√

ξnūn −→
√
ξū almost everywhere in [0, T ]× Ω.

Lemma 9. (Convergence of (∂y(ξnunvn))n). Let Φ ∈ C∞
c

(
[0, T ] × Ω

)
be a regular test

function, then∫ T

0

∫
Ω
∂y(ξnunvn).Φdxdydt −→

∫ T

0

∫
Ω
∂y(ξuv).Φdxdydt as n −→ +∞.

Proof. Let Φ ∈ C∞
c

(
[0, T ]× Ω

)
be a smooth function. From (56), we have∫ T

0

∫
Ω
∂y(ξnunvn).Φdxdydt

= −
∫ T

0

∫
Ω
ξnunvn · ∂yΦdxdydt

= −
∫ T

0

∫
Ω
un

(
− divx(ξnũn) + ydivx(ξnūn)

)
· ∂yΦdxdydt

= −
∫ T

0

∫
Ω

(
− divx(ξnũn ⊗ un) + ξnũn · ∇xun + ydivx(ξnūn ⊗ un)− yξnūn · ∇xun

)
.∂yΦdxdydt



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2263

= −
∫ T

0

∫
Ω
ξnũn ⊗ un : ∂y∇xΦdxdydt+

∫ T

0

∫
Ω
ξnūn ⊗ un : y∂y∇xΦdxdydt

−
∫ T

0

∫
Ω
ξnũn · ∇xun · ∂yΦdxdydt+

∫ T

0

∫
Ω
ξnūn · ∇xun · y∂yΦdxdydt. (57)

By direct computation we have ∇x(
√
ξnun) =

√
ξn∇xun +∇x

√
ξn ⊗ un, thus∫ T

0

∫
Ω

√
ξn∇xun : Φdxdydt =

∫ T

0

∫
Ω

(
∇x(

√
ξnun)−∇x

√
ξn ⊗ un

)
: Φdxdydt

= −
∫ T

0

∫
Ω
(
√
ξnun) · divxΦdxdydt−

∫ T

0

∫
Ω
∇x

√
ξn ⊗ un : Φdxdydt

−→
n→+∞

−
∫ T

0

∫
Ω
(
√
ξu) · divxΦdxdydt−

∫ T

0

∫
Ω
∇x

√
ξ ⊗ u : Φdxdydt

=

∫ T

0

∫
Ω

√
ξ∇xu : Φdxdydt.

Hence, √
ξn∇xun ⇀

√
ξ∇xu weakly in L2

(
[0, T ];L2(Ω)

)
.

Combining this last weak convergence with (57), we get, after replacing ξn by
√
ξn
√
ξn

and using the previous Lemmas,∫ T

0

∫
Ω
∂y(ξnunvn).Φdxdydt −→

∫ T

0

∫
Ω
∂y(ξuv).Φdxdydt as n −→ +∞,

where ξv = −divx(ξũ) + ydivx(ξū).

Lemma 10. (see Subsection 3.3.5 of [19]) (Convergence of nonlinear diffusion terms).
For any smooth function Φ ∈ C∞

c

(
[0, T ]× Ω

)
, we have∫ T

0

∫
Ω
divx(ξnDx(un)) · Φdxdydt −→

∫ T

0

∫
Ω
divx(ξDx(u)) · Φdxdydt as n −→ +∞,∫ T

0

∫
Ω
ξn∇x∆

5
xξn · Φdxdydt −→

∫ T

0

∫
Ω
ξ∇x∆

5
xξ · Φdxdydt as n −→ +∞,∫ T

0

∫
Ω
ξn∇x

(∆x
√
ξn√

ξn

)
Φdxdydt −→

∫ T

0

∫
Ω
ξ∇x

(∆x
√
ξ√

ξ

)
Φdxdydt as n −→ +∞.

Due to the compactness above, taking the limits in the approximate system of (19)
and (24), then (ξ, u, v) satisfies

∂tξ + divx(ξu) + ∂y(ξv) = ϵ∆xξ on [0, T ]× Ω

and the following identity holds:∫
Ω
ξu(T )ΦdX −

∫
Ω
m0ΦdXdt−

∫ T

0

∫
Ω
(ξu⊗ u) : ∇xΦdX +

∫ T

0

∫
Ω
ξ∂yu∂yΦdXdt



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2264

−
∫ T

0

∫
Ω
ξuv∂yΦdXdt+

∫ T

0

∫
Ω
∇xξ

2 · ΦdXdt+ r

∫ T

0

∫
Ω
ξ|u|uΦdXdt

+ r1

∫ T

0

∫
Ω
uΦdXdt+ α

∫ T

0

∫
Ω
∆xu ·∆xΦdXdt

= −
∫ T

0

∫
Ω
2ξDx(u) : ∇xΦdXdt+ ϵ

∫ T

0

∫
Ω
(∇xξ · ∇xu) ΦdX − r2

∫ T

0

∫
Ω
ξ−βdivxΦdXdt

− 2k1

∫ T

0

∫
Ω
Φ∆x

√
ξ∇x

√
ξdXdt− k1

∫ T

0

∫
Ω
divxΦ

(
∆x

√
ξ
)√

ξdXdt+ δ

∫ T

0

∫
Ω
Φξ∇x∆

5
xξdXdt,

(58)

for any smooth function Φ ∈ C∞
c

(
[0, T ] × Ω

)
, where m0 = ξ(0, x, y)u(0, x, y) and dX =

dxdy.
Using the lower semi-continuity of convex functions, we take the limits in the energy
estimate (47) to obtain the following energy inequality:

sup
t∈[0,T ]

E(ξ, u) + 2ϵ

∫ T

0

∫
Ω
|∇xξ|2dXdt+ r1

∫ T

0

∫
Ω
u2dXdt+ r

∫ T

0

∫
Ω
ξ|u|3dXdt

+

∫ T

0

∫
Ω
2ξ|Dx(u)|2dXdt+ α

∫ T

0

∫
Ω
|∆xu|2dXdt+

4ϵr2
β

∫ T

0

∫
Ω
|∇xξ

−β
2 |2dXdt

+
ϵk1
2

∫ T

0

∫
Ω
ξ|∇2

x ln ξ|2dXdt+ δϵ

∫ T

0

∫
Ω
|∆3

xξ|2dXdt+
∫ T

0

∫
Ω
ξ|∂yu|2dXdt ≤ E(ξ0, u0),

(59)

with

E(ξ, u) =

∫
Ω

(
1

2
ξu2 + ξ2 +

r2
β + 1

ξ−β + k|∇x

√
ξ|2 + δ

2
|∇x∆

2
xξ|2

)
dX.

Consequently, we obtain the existence of weak solutions of the approximated system (17),
through the following.

Proposition 3.2. For any T > 0, the following system

∂tξ + divx(ξu) + ∂y (ξv) = ϵ∆xξ,

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u

+r1u+ α∆2
xu = 2divx (ξD(u)) + ∂y (ξ∂yu) + ϵ∇xξ · ∇xu

+r2∇xξ
−β + k1ξ∇x

(
∆x

√
ξ√

ξ

)
+ δξ∇x∆

5
xξ,

∂yξ = 0,

(60)

admits a weak solution (ξ, u, v) with continuous initial data. In particular, the weak solu-
tion satisfies the energy inequality (59).



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2265

4. Bresch-Desjardins Entropy

In this section, we deal with the Bresch-Desjardins (B-D) estimate for the approximate
system of the Proposition 3.2, which was first introduced by D. Bresch and B. Desjardins
in [2]. By (48) and (53), we have

ξ(t, x) ≥ C(δ, r2) > 0

ξ ∈ L∞(
[0, T ];H5(Ω)

)
∩ L2

(
[0, T ];H6(Ω)

)
.

(61)

As many authors have pointed out, a main difficulty in the proof in this type of model
is to pass to the limit in the nonlinear term ξu ⊗ u which requires a strong convergence
of

√
ξu. It seems necessary to obtain additional information on the density ξ. In this

perspective, we use a mathematical entropy, called B-D entropy. Thanks to (61), we

can take
∇ξ
ξ

= ∇ ln ξ as a test function to derive the B-D entropy from the momentum

equation. To this end, we first take the gradient of the mass equation with respect to x,
we obtain

∂t∇xξ +∇x (ξdivx(u)) +∇x(u · ∇xξ) + ∂y∇x(ξv) = ϵ∇x∆xξ. (62)

Multiplying (62) by 2 and writing the ∇xξ terms as ξ∇x ln ξ, we get

∂t (2ξ∇x ln ξ) + divx
(
2ξ∇t

xu
)
+ divx (2ξ∇x ln ξ ⊗ u) + 2∂y∇x(ξv) = 2ϵ∇x∆xξ. (63)

Then, we add (63) with that of the conservation of moments to obtain

∂t (ξ(u+ 2∇x ln ξ)) + divx (ξ(u+ 2∇x ln ξ)⊗ u) + ∂y (2∇x(ξv)) + ∂y(ξuv)

+∇xξ
2 + r1u+ rξ|u|u+ α∆2

xu = divx (2ξAx(u)) + ∂y(ξ∂yu)

+ r2∇xξ
−β − ϵ∇xξ · ∇xu+ 2ϵ∇x∆xξ + k1ξ∇x

(
∆x

√
ξ√

ξ

)
+ δξ∇x∆

5
xξ, (64)

where Ax(u) =
∇xu−∇t

xu

2
=

(
∂xiuj − ∂xjui

2

)
1≤i,j≤2

is the vorticity rate tensor.

Lemma 11. : Under the assumption (61), we have the following B-D entropy:∫
Ω

(1
2
ξ|u+ 2∇x ln ξ|2 − 2r1 ln ξ

)
dxdy + 2

∫ T

0

∫
Ω
ξ|∂yv|2dxdydt+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt

+
16r2
β

∫ T

0

∫
Ω
|∇xξ

−β/2|2dxdydt+ α

∫ T

0

∫
Ω
|∆xu|2dxdydt+ r1

∫ T

0

∫
Ω
u2dxdydt

+ k1

∫ T

0

∫
Ω
ξ|∇2

x ln ξ|2dxdydt+ 2δ

∫ T

0

∫
Ω
|∆3

xξ|2dxdydt+ 2

∫ T

0

∫
Ω
ξ|Ax(u)|2dxdydt

+ 8

∫ T

0

∫
Ω
ξ|∇x

√
ξ|2dxdydt+ 2r1ϵ

∫ T

0

∫
Ω

|∇xξ|2

ξ2
dxdydt+

∫ T

0

∫
Ω
ξ|∂yu|2dxdy



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2266

≤
∫
Ω

(
ξ0u

2
0 + 10(∇x

√
ξ0)

2 − 2r1 ln ξ0

)
dxdy + E0 + C + ϵC(δ, r2) +

√
αC(δ, r2),

(65)

where C is a generic positive constant depending on the initial data and other constants
but independent of ϵ, δ, r1, r2, α, and C(δ, r2) is a generic positive constant only depending
on δ and r2.

Proof. Multiplying (64) by u+ 2∇x ln ξ = ψ and integrating over Ω, we obtain:∫
Ω
∂t(ξψ)ψdxdy +

∫
Ω
divx(ξψ ⊗ u)ψdxdy +

∫
Ω
∂y(ξuv)ψdxdy + 2

∫
Ω
∂y∇x(ξv)ψdxdy

+

∫
Ω
∇xξ

2ψdxdy +

∫
Ω

(
r1u+ rξ|u|u+ α∆2u

)
ψdxdy − 2

∫
Ω
divx(ξAx(u))ψdxdy

−
∫
Ω
∂y(ξ∂yu)ψdxdy − r2

∫
Ω
∇xξ

−βψdxdy + ϵ

∫
Ω
(∇xξ · ∇xu)ψdxdy − 2ϵ

∫
Ω

(
∇x∆xξ

)
ψdxdy

− k1

∫
Ω
ξ∇x

(∆x
√
ξ√

ξ

)
ψdxdy − δ

∫
Ω
(ξ∇x∆

5
xξ)ψdxdy = 0. (66)

The two first terms of (66) give:

•
∫
Ω
∂t(ξψ)ψdxdy +

∫
Ω
divx(ξψ ⊗ u)ψdxdy

=

∫
Ω
ψ2∂tξdxdy +

∫
Ω
ξ∂t

ψ2

2
dxdy +

∫
Ω

(
ξψ divx(u) + u · ∇x(ξψ)

)
ψdxdy

=

∫
Ω
ψ2∂tξdxdy +

∫
Ω
ξ∂t

ψ2

2
dxdy +

∫
Ω

(
ξψ2divx(u) +

1

2
ξu∇xψ

2 + uψ2∇xξ
)
dxdy

=

∫
Ω
ψ2∂tξdxdy +

∫
Ω
ξ∂t

ψ2

2
dxdy +

∫
Ω

[
ψ2

(
ξdivx(u) + u∇xξ

)
− 1

2
divx(ξu)ψ

2
]
dxdy

=

∫
Ω
ψ2∂tξdxdy +

1

2

∫
Ω
ξ∂tψ

2dxdy +
1

2

∫
Ω
ψ2divx(ξu)dxdy. (67)

Remark that ∂y(ξuv) = ∂y(ξvψ)− 2∂y(v∇xξ), then the third term of (66) becomes

•
∫
Ω
∂y(ξuv)ψdxdy =

∫
Ω
∂y(ξvψ)ψdxdy − 2

∫
Ω
∂y(v∇xξ)ψdxdy

=

∫
Ω
ψ2∂y(ξv)dxdy +

∫
Ω
ξv∂y

ψ2

2
dxdy − 2

∫
Ω
ψ∇xξ∂y(v)dxdy

=

∫
Ω
ψ2∂y(ξv)dxdy −

1

2

∫
Ω
ψ2∂y(ξv)dxdy + 2

∫
Ω
v∇xξ∂y(ψ)dxdy

=
1

2

∫
Ω
ψ2∂y(ξv)dxdy + 2

∫
Ω
v∇xξ∂yudxdy. (68)

Adding the last term of (67) and the first term in the right hand side of (68), we obtain

1

2

∫
Ω
ψ2divx(ξu)dxdy +

1

2

∫
Ω
ψ2∂y(ξv)dxdy =

1

2
ϵ

∫
Ω
ψ2∆xξdxdy −

1

2

∫
Ω
ψ2∂tξdxdy. (69)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2267

Thus, the first three terms of (66) give∫
Ω
(∂t(ξψ) + divx(ξψ ⊗ u) + ∂z(ξuv))ψdxdy

=
1

2

d

dt

∫
Ω
ξψ2dxdy + 2

∫
Ω
v∇xξ∂yudxdy +

1

2
ϵ

∫
Ω
ψ2∆xξdxdy. (70)

The fourth term of (66) gives

• 2

∫
Ω
∂y∇x(ξv)ψdxdy = −

∫
Ω
2∇x(ξv)∂yudxdy −

∫
Ω
4∇x(ξv)∂y(∇x ln ξ)dxdy

=

∫
Ω
2ξv∂ydivx(u)dxdy

=

∫
Ω
2
(
v∂ydivx(ξu)− v∇xξ∂yu

)
dxdy

=

∫
Ω
2
(
− ξv∂2yv − v∇xξ∂yu

)
dxdy

=

∫
Ω
2
(
ξ|∂yv|2 − v∇xξ∂yu

)
dxdy. (71)

Thus, the first four terms of (66) are exactly

1

2

d

dt

∫
Ω
ξψ2dxdy +

1

2
ϵ

∫
Ω
ψ2∆xξdxdy +

∫
Ω
2ξ|∂yv|2dxdy. (72)

Moreover, by summing the second term of (72) with ϵ
∫
Ω(∇xξ ·∇xu)ψdxdz, we obtain, by

replacing ψ by u+ 2∇x ln ξ,

ϵ

∫
Ω
(∇xξ · ∇xu)ψdxdy +

1

2
ϵ

∫
Ω
ψ2∆xξdxdy

= −2ϵ

∫
Ω
∇xξ · ∇2

x ln ξ · udxdy − 4ϵ

∫
Ω
∇xξ · ∇2

x ln ξ · ∇x ln ξdxdy. (73)

The term −2
∫
Ω divx(ξAx(u))ψdxdy in (66) can be written:

• − 2

∫
Ω
divx(ξAx(u))ψdxdy = −2

∫
Ω
divx(ξAx(u))udxdy − 2

∫
Ω
divx(2ξAx(u))∇x ln ξdxdy

= 2

∫
Ω
ξ|Ax(u)|2dxdy − 2

∫
Ω
divx(2ξAx(u))∇x ln ξdxdy.

(74)

The other terms in (66) yield :

•
∫
Ω
rξ|u|uψdxdy =

∫
Ω
rξ|u|3dxdy + 2r

∫
Ω
|u|u∇xξdxdy. (75)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2268

• −
∫
Ω
∂y(ξ∂yu)ψdxdy = −

∫
Ω
∂y(ξ∂yu)udxdy =

∫
Ω
ξ|∂yu|2dxdy, (76)

since
∫
Ω ∂y(ξ∂yu)∇x ln ξdxdy = 0.

Using (26), we get

•
∫
Ω
∇xξ

2ψdxdy =

∫
Ω
∇xξ

2udxdy + 2

∫
Ω
∇xξ

2∇x ln ξdxdy

=
d

dt

∫
Ω
ξ2dxdy + 2ϵ

∫
Ω
|∇xξ|2 dxdy + 2

∫
Ω
2ξ
√
ξ∇x

√
ξ.
2
√
ξ∇x

√
ξ

ξ
dxdy

=
d

dt

∫
Ω
ξ2dxdy + 2ϵ

∫
Ω
|∇xξ|2 dxdy + 8

∫
Ω
ξ|∇x

√
ξ|2dxdy (77)

On another note,

•
∫
Ω
r1uψdxdy = r1

∫
Ω
u2dxdy + 2r1

∫
Ω
u∇x ln ξdxdy

= r1

∫
Ω
u2dxdy + 2r1

∫
Ω
u
∇xξ

ξ
dxdy (78)

and

•
∫
Ω
α∆2uψdxdz = α

∫
Ω
∆2

xu.udxdy + 2α

∫
Ω
∆2

xu∇x ln ξdxdy

= α

∫
Ω
|∆xu|2dxdy + 2α

∫
Ω
∆xu∇x∆x ln ξdxdy. (79)

Referring to (27), we obtain

• − r2

∫
Ω
∇xξ

−βψdxdy = −r2
∫
Ω
∇xξ

−βudxdy − 2r2

∫
Ω
∇xξ

−β∇x ln ξdxdy

=
r2

β + 1

d

dt

∫
Ω
ξ−β
n dxdy +

4r2
β

(ϵ+ 4)

∫
Ω
|∇xξ

−β
2

n |2dxdy. (80)

As in (28), we get

• − δ

∫
Ω
(ξ∇x∆

5
xξ)ψdxdy = δ

∫
Ω
divx(ξu)∆

5
xξdxdy + 2δ

∫
Ω
∆xξ∆

5
xξdxdy

=
δ

2

d

dt

∫
Ω
|∇x∆

2
xξ|2dxdy + δϵ

∫
Ω
|∆3

xξ|2dxdy + 2δ

∫
Ω
|∆3

xξ|2dxdy.

(81)

Similarly, by exploiting (29), we finally obtain

• − k1

∫
Ω
ξ∇x

(∆x
√
ξ√

ξ

)
ψdxdy = k1

∫
Ω
divx(ξu)

(∆x
√
ξ√

ξ

)
dxdy + 2k1

∫
Ω

(∆x
√
ξ√

ξ

)
∆xξdxdy



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2269

= k1
d

dt

∫
Ω
|∇x

√
ξ|2 + k1ϵ

2

∫
Ω
ξ|∇2

x ln ξ|2dxdy + k1

∫
Ω
ξ|∇2

x ln ξ|2dxdy. (82)

Finally, putting the above results together, the equation (66) is written:

d

dt

∫
Ω

(1
2
ξψ2 + ξ2 +

r2
β + 1

ξ−β + k1|∇x

√
ξ|2 + δ

2
|∇x∆

2
xξ|2

)
dxdy

+ 2

∫
Ω
ξ|∂yv|2dxdy + r

∫
Ω
ξ|u|3dxdy +

∫
Ω
ξ|∂yu|2dxdy + r1

∫
Ω
u2dxdy

+
4r2
β

(ϵ+ 4)

∫
Ω
|∇xξ

−β/2|2dxdy + α

∫
Ω
|∆xu|2dxdy +

∫
Ω
2ξ|Ax(u)|2dxdy

+
k1(ϵ+ 2)

2

∫
Ω
ξ|∇2

x ln ξ|2 + δ(ϵ+ 2)

∫
Ω
|∆3

xξ|2dxdy +
∫
Ω
2ϵ|∇xξ|2dxdy + 8

∫
Ω
ξ|∇x

√
ξ|2dxdy

= 2

∫
Ω
divx(2ξAx(u))∇x ln ξdxdy − 2α

∫
Ω
∆xu∇x∆x ln ξdxdy + 2ϵ

∫
Ω
∇xξ · ∇2

x ln ξ · udxdy

+ 4ϵ

∫
Ω
∇xξ · ∇2

x ln ξ · ∇x ln ξdxdy + 2ϵ

∫
Ω
∇x∆xξψdxdy

− 2r

∫
Ω
|u|u∇xξdxdy − 2r1

∫
Ω
u
∇xξ

ξ
dxdy =

7∑
i=1

Ii. (83)

We have I1 = 0 due to the periodic conditions on Ωx, that is,

2

∫
Ω
divx(2ξAx(u))∇x ln ξdxdy =

∫
Ω
∂i
(
ξ(∂iuj − ∂jui)

)∂jξ
ξ
dxdy

=

∫
Ω

(∂iξ∂jξ
ξ

(∂iuj − ∂j) + (∂i∂iuj − ∂i∂jui)∂jξ
)
dxdy

=

∫
Ω
(∂i∂iuj∂jξ − ∂i∂jui∂jξ)dxdy

=

∫
Ω
(∂i∂iuj∂jξ − ∂j∂jui∂iξ)dxdy = 0. (84)

For I7, we have,

I7 = −2r1

∫
Ω
u
∇xξ

ξ
dxdy

= −2r1

∫
Ω

divx(ξu)

ξ
dxdy

= 2r1

∫
Ω

∂tξ + ∂y(ξv)− ε∆xξ

ξ
dxdy

= 2r1

∫
Ω
∂t ln ξdxdy − 2r1ϵ

∫
Ω

∆xξ

ξ
dxdy

= 2r1
d

dt

∫
Ω
ln ξdxdy − 2r1ϵ

∫
Ω

|∇xξ|2

ξ2
dxdy. (85)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2270

Substituting (84) and (85) into (83) and integrating with respect to time t, we obtain∫
Ω

(1
2
ξ|ψ|2 − 2r1 ln ξ

)
dxdy + 2

∫ T

0

∫
Ω
ξ|∂yv|2dxdydt+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt

+
16r2
β

∫ T

0

∫
Ω
|∇xξ

−β/2|2dxdydt+ α

∫ T

0

∫
Ω
|∆xu|2dxdydt+ r1

∫ T

0

∫
Ω
u2dxdydt

+ k1

∫ T

0

∫
Ω
ξ|∇2

x ln ξ|2 + 2δ

∫ T

0

∫
Ω
|∆3

xξ|2dxdydt+ 2

∫ T

0

∫
Ω
ξ|Ax(u)|2dxdydt

+ 8

∫ T

0

∫
Ω
ξ|∇x

√
ξ|2dxdydt+ 2r1ϵ

∫ T

0

∫
Ω

|∇xξ|2

ξ2
dxdydt+

∫ T

0

∫
Ω
ξ|∂yu|2dxdydt

≤
∫
Ω

(
ξ0u

2
0 + 10(∇x

√
ξ0)

2 − 2r1 ln ξ0
)
dxdydt+

∫ T

0

( 6∑
i=2

Ii
)
dt+ E0, (86)

where we used the energy inequality (59).
Now, we control the other terms Ii in (86).

•
∫ T

0
I2dt = −2α

∫ T

0

∫
Ω
∆xu · ∇x∆x ln ξdxdydt

= −2α

∫ T

0

∫
Ω
∆xu ·

(∇x∆xξ

ξ
− ∆xξ∇xξ

ξ2
− 2

(∇xξ.∇xξ)∇xξ

ξ2
+ 2

|∇xξ|2∇xξ

ξ3

)
dxdydt

≤ Cα

∫ T

0

∫
Ω
|∆xu|

(
ξ−1|∇3

xξ|+ ξ−2|∇xξ||∇2
xξ|+ ξ−3|∇xξ|3

)
dxdzdt

≤ C
√
α∥

√
α∆xu∥L2(L2(Ω))

[
∥ξ−1∥L∞(L∞(Ω))∥∇3

xξ∥L2(L2(Ω))

+ ∥ξ−1∥2L∞(L∞(Ω))∥∇xξ∥L∞(L∞(Ω))∥∇2
xξ∥L2(L2(Ω))+∥ξ−1∥3L∞(L∞(Ω))∥∇xξ∥3L6(L6(Ω))

]
≤ C

√
α∥

√
α∆xu∥L2(L2(Ω))

[
∥ξ−1∥3L∞(L∞(Ω))∥∇

3
xξ∥L2(L2(Ω))+∥ξ−1∥L∞(L∞(Ω))

]
≤ C(δ, r2)

√
α. (87)

•
∫ T

0
I3dt = 2ϵ

∫ T

0

∫
Ω
∇xξ · ∇2

x ln ξ · udxdzdt

= −ϵ
∫ T

0

∫
Ω
|∇xξ

ξ
|2divx(ξu)dxdzdt

= −ϵ
∫ T

0

∫
Ω

( |∇xξ|2

ξ
∇x · u+

|∇xξ|2

ξ2
u∇xξ

)
dxdzdt

≤ ϵ

∫ T

0

∫
Ω

(∇xξ)
2

ξ
3
2

√
ξ∇x · u+

(∇xξ)
2

ξ
5
2

√
ξu · ∇xξdxdzdt

≤ ϵ∥1
ξ
∥
3
2
L∞(L∞(Ω))∥∇xξ∥L∞(L∞(Ω))∥∇xξ∥L2(L2(Ω))∥

√
ξ∇xu∥L2(L2(Ω))



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2271

+ ϵ∥1
ξ
∥
5
2
L∞(L∞(Ω))∥∇xξ∥2L∞(L∞(Ω))∥∇xξ∥L2(L2(Ω))∥

√
ξu∥L2(L2(Ω))

≤ ϵC(δ, r2)
(
∥
√
ξ∇xu∥L2(L2(Ω))+1

)
≤ 1

2
∥
√
ξAx(u)∥2L2(L2(Ω))+ϵC(δ, r2), (88)

where we used,

∥
√
ξAx(u)∥2L2(L2(Ω))+∥

√
ξDx(u)∥2L2(L2(Ω))= ∥

√
ξ∇xu∥2L2(L2(Ω)).

In addition, we also have

•
∫ T

0
I4dt = 4ϵ

∫ T

0

∫
Ω
∇xξ · ∇2

x ln ξ · ∇x ln ξdxdydt

= −2ϵ

∫ T

0

∫
Ω
∆xξ · |∇x ln ξ|2dxdydt

≤ 2ϵ∥1
ξ
∥2L∞(L∞(Ω))∥∇xξ∥L∞(L∞(Ω))∥∇xξ∥L2(L2(Ω))∥∆xξ∥L2(L2(Ω))

≤ ϵC(δ, r2). (89)

Furthermore, we have

•
∫ T

0
I5dt = 2ϵ

∫ T

0

∫
Ω
∇x∆xξ(u+ 2∇x ln ξ)dxdydt

≤ 2ϵ∥∇x∆xξ∥L2(L2(Ω))∥
√
ξu∥L2(L2(Ω))∥

√
ξ∥L∞(L∞(Ω))

+ 4ϵ∥∇x∆xξ∥L2(L2(Ω))∥∇xξ∥L2(L2(Ω))∥ξ−1∥L∞(L∞(Ω))

≤ ϵC(δ, r2) (90)

and

•
∫ T

0
I6dt = −2rϵ

∫ T

0

∫
Ω
|u|u∇xξdxdydt

= 2rϵ

∫ T

0

∫
Ω
divx(|u|u)ξdxdydt

= 2rϵ

∫ T

0

∫
Ω
ξ
(
|u|divxu+ u

u

|u|
∇xu

)
dxdydt

≤ C∥
√
ξu∥L2(L2(Ω))∥

√
ξ∇xu∥L2(L2(Ω))

≤ C +
1

2
∥
√
ξAx(u)∥2L2(L2(Ω)). (91)

So by replacing (87)-(91) in (86), we get the B-D entropy (65).

In the next section, we make the parameters of our approximate system tend to 0.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2272

5. Proof of mains results

5.1. Proof of Theorem 1

We make the proof of Theorem 1 in several steps.

5.1.1. Passing to the limits as ϵ, α −→ 0

We denote the solution to (60) at this level of approximation as (ξα,ϵ, uα,ϵ, vα,ϵ). From the
energy inequality (59), we obtain the following uniform regularities.

√
ξα,ϵuα,ϵ ∈ L∞(

[0, T ];L2(Ω)
)
, r2ξ

−β
α,ϵ ∈ L∞(

[0, T ];L1(Ω)
)
,√

ξα,ϵDx(uα,ϵ) ∈ L2
(
[0, T ];L2(Ω)

)
,

√
ξα,ϵ∂yuα,ϵ ∈ L2

(
[0, T ];L2(Ω)

)
,

√
δξα,ϵ ∈ L∞(

[0, T ];H5(Ω)
)
,

√
r1uα,ϵ ∈ L2

(
[0, T ];L2(Ω)

)
,

√
α∆uα,ϵ ∈ L2

(
[0, T ];L2(Ω)

)
, ξ

1
3
α,ϵuα,ϵ ∈ L3

(
[0, T ];L3(Ω)

)
.

(92)

The B-D entropy (65) gives the following additional uniform regularities:

√
δξα,ϵ ∈ L2

(
[0, T ];H6(Ω)

)
, ∇xξα,ϵ ∈ L2

(
[0, T ];L2(Ω)

)
√
r2∇xξ

−β/2
α,ϵ ∈ L2

(
[0, T ];L2(Ω)

)
,

√
ξα,ϵAx(uα,ϵ) ∈ L2

(
[0, T ];L2(Ω)

)
,√

ξα,ϵ∂yuα,ϵ ∈ L2
(
[0, T ];L2(Ω)

)
, ξ2α,ϵ ∈ L∞(

[0, T ];L1(Ω)
)
,√

ξα,ϵ∇x

√
ξα,ϵ ∈ L∞(

[0, T ];L2(Ω)
)
.

(93)

Similar to the proof of Lemma 5, we have the following uniform boundedness:√
k1
√
ξα,ϵ ∈ L2

(
[0, T ];H2(Ω)

)
; k

1
4
1 ∇xξ

1
4
α,ϵ ∈ L4

(
[0, T ];L4(Ω)

)
.

With the above regularities, we can show the following uniform compactness results.

Lemma 12. Let (ξα,ϵ, uα,ϵ, vα,ϵ) the weak solution of (60) satisfying (92) and (93), then
the following estimates hold:

∥∂t
√
ξα,ϵ∥

L∞
(
[0,T ];W−1, 32 (Ω)

)+∥
√
ξα,ϵ∥L2

(
[0,T ];H2(Ω)

)≤ K

∥ξα,ϵ∥L2
(
[0,T ];H6(Ω)

)+∥∂tξα,ϵ∥L2
(
[0,T ];H−1(Ω)

)≤ K

∥ξ−β
α,ϵ∥

L
5
3

(
[0,T ];L

5
3 (Ω)

)≤ K

∥ξα,ϵuα,ϵ∥
L2
(
[0,T ];W 1, 32 (Ω)

)+∥∂t(ξα,ϵuα,ϵ)∥L2
(
[0,T ];H−5(Ω)

)≤ K,

(94)

where K is independent of ϵ, α.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2273

Proof. The proof of Lemma 12 follows the same lines as the proof of Lemma 6.

Thanks to Lemmas 1 and 12, when α→ 0 and ϵ→ 0, we have the following compact-
ness results

√
ξα,ϵ −→

√
ξ strongly in L2

(
[0, T ];H1(Ω)

)
,

ξα,ϵ −→ ξ strongly in C
(
[0, T ];H5(Ω)

)
,

ξα,ϵ ⇀ ξ weakly in L2
(
[0, T ];H6(Ω)

)
,

uα,ϵ ⇀ u weakly in L2
(
[0, T ];L2(Ω)

)
,√

ξα,ϵ ⇀
√
ξ weakly in L2

(
[0, T ];H2(Ω)

)
,

ξα,ϵuα,ϵ −→ ξu strongly in L2
(
[0, T ];Lp(Ω)

)
, ∀1 ≤ p < 3,

ξ−β
α,ϵ −→ ξ−β strongly in L1

(
[0, T ];L1(Ω)

)
,√

ξα,ϵuα,ϵ −→
√
ξu strongly in L2

(
[0, T ];L2(Ω)

)
.

(95)

Hence,
√
ξα,ϵ −→

√
ξ, ξα,ϵ −→ ξ and ξ−β

α,ϵ −→ ξ−β almost everywhere (t, x).
By the results of regularities above, we can pass to the limits in (60) as α→ 0 and ϵ→ 0.
For any test function Φ ∈ C∞

c

(
[0, T ]× Ω

)
, we have∫ T

0

∫
Ω
∂y(ξα,ϵuα,ϵvα,ϵ)Φdxdydt = −

∫ T

0

∫
Ω
ξα,ϵuα,ϵvα,ϵ∂yΦdxdydt

−→ −
∫ T

0

∫
Ω
ξuv∂yΦdxdydt. (96)

Moreover, when ϵ, α −→ 0, we have

ϵ

∫ T

0

∫
Ω
∇xξα,ϵ · ∇xuα,ϵΦdxdydt

= −ϵ
∫ T

0

∫
Ω
∇x(∇xξα,ϵΦ)uα,ϵdxdydt

= −ϵ
∫ T

0

∫
Ω
∆xξα,ϵ · uα,ϵΦdxdydt− ϵ

∫ T

0

∫
Ω
∇xξα,ϵuα,ϵdivxΦdxdydt

≤ ϵ∥∆xξα,ϵ∥L2
(
[0,T ];L2(Ω)

)∥uα,ϵ∥L2
(
[0,T ];L2(Ω)

)∥Φ∥
L∞

(
[0,T ];L∞(Ω)

)
+ ϵ∥∇xξα,ϵ∥L2

(
[0,T ];L2(Ω)

)∥uα,ϵ∥L2
(
[0,T ];L2(Ω)

)∥divxΦ∥L∞
(
[0,T ];L∞(Ω)

)−→ 0 (97)

and

α

∫ T

0

∫
Ω
∆2uα,ϵ · Φdxdydt = α

∫ T

0

∫
Ω
∆uα,ϵ ·∆ϕdxdydt



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2274

≤
√
α∥

√
α∆uα,ϵ∥L2

(
[0,T ];L2(Ω)

)∥∆Φ∥
L2
(
[0,T ];L2(Ω)

)−→ 0. (98)

So, taking α, ϵ −→ 0 in (60), the system

∂tξ + divx(ξu) + ∂y(ξv) = 0,

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + r1u+ rξ|u|u

= 2divx
(
ξDx(u)

)
+ ∂y(ξ∂yu) + r2∇xξ

−β + k1ξ∇x

(∆x
√
ξ√

ξ

)
+ δξ∇x∆

5
xξ,

∂yξ = 0

(99)

holds in the sense of distribution on [0, T ]× Ω.
Moreover, thanks to the lower semi-continuity of convex functions and the strong conver-
gence of (ξα,ϵ, uα,ϵ, vα,ϵ), we can pass to the limits in the energy inequality (59) and the
B-D entropy (65) as α = ϵ −→ 0 with δ, r2, r0, k1 being fixed, to obtain

E(ξ, u) + r1

∫ T

0

∫
Ω
u2dXdt+ r

∫ T

0

∫
Ω
ξ|u|3dXdt

+

∫ T

0

∫
Ω
2ξ|Dx(u)|2dXdt+

∫ T

0

∫
Ω
ξ|∂yu|2dXdt ≤ E(ξ0, u0) (100)

and∫
Ω

(1
2
ξ|u+ 2∇x ln ξ|2 − 2r1 ln ξ

)
dxdy + 2

∫ T

0

∫
Ω
ξ|∂yv|2dxdydt+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt

+
16r2
β

∫ T

0

∫
Ω
|∇xξ

−β/2|2dxdydt+ r1

∫ T

0

∫
Ω
u2dxdydt+

∫ T

0

∫
Ω
ξ|∂yu|2dxdy

+ k1

∫ T

0

∫
Ω
ξ|∇2

x ln ξ|2dxdydt+ 2δ

∫ T

0

∫
Ω
|∆3

xξ|2dxdydt+ 2

∫ T

0

∫
Ω
ξ|Ax(u)|2dxdydt

+ 8

∫ T

0

∫
Ω
ξ|∇x

√
ξ|2dxdydt ≤

∫
Ω

(
ξ0u

2
0 + 10(∇x

√
ξ0)

2 − 2r1 ln ξ0

)
dxdy + E0 + C.

(101)

Thus, to conclude this part, we give an existence result of weak solutions to the system
(99).

Proposition 5.1. For any T > 0, the system

∂tξ + divx(ξu) + ∂y(ξv) = 0,

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u+ r1u

= 2divx
(
ξDx(u)

)
+ ∂y(ξ∂yu) + r2∇xξ

−β + k1ξ∇x

(∆x
√
ξ√

ξ

)
+ δξ∇x∆

5
xξ,

∂yξ = 0,

(102)

with (t, x, y) ∈ [0, T ]×Ω, admits a weak solution with appropriate initial data. In particular
the weak solution (ξ, u, v) satisfies the energy inequality (100) and the B-D entropy (101).



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2275

5.1.2. Passing to the limits as r2, δ −→ 0

In this step, we pass to the limit as r2 −→ 0, δ −→ 0 with k1, r1 fixed. We denote
by (ξr2,δ, ur2,δ, vr2,δ) the weak solution at this level. From Proposition 5.1, we have the
following regularity results:

√
ξr2,δur2,δ ∈ L∞(

[0, T ];L2(Ω)
)
,
√
ξr2,δDx(ur2,,δ) ∈ L2

(
[0, T ];L2(Ω)

)
,
√
ξr2,δ∂yur2,δ ∈ L2

(
[0, T ];L2(Ω)

)
,√

ξr2,δ∇x

√
ξr2,δ ∈ L∞(

[0, T ];L2(Ω)
)
,

√
δξr2,δ ∈ L∞(

[0, T ];H5(Ω)
)
,

√
δξr2,δ ∈ L2

(
[0, T ];H6(Ω)

)
,

ur2,δ ∈ L2
(
[0, T ];L2(Ω)

)
, ξ

1
3
r2,δ

ur2,δ ∈ L3
(
[0, T ];L3(Ω)

)
,

√
ξr2,δ∇xur2,δ ∈ L2

(
[0, T ];L2(Ω)

)
,

√
k1
√
ξr2,δ ∈ L2

(
[0, T ];H2(Ω)

)
, 4

√
k1∇xξ

1
4
r2,δ

∈ L4
(
[0, T ];L4(Ω)

)
,
√
δ∆3

xξr2,δ ∈ L2
(
[0, T ];L2(Ω)

)
,

√
r2∇xξ

−β/2
r2,δ

∈ L2
(
[0, T ];L2(Ω)

)
,

√
ξr2,δ∂yvr2,δ ∈ L2

(
[0, T ];L2(Ω)

)
, r2ξ

−β
r2,δ

∈ L∞(
[0, T ];L1(Ω)

)
,

r1 ln ξr2,δ ∈ L∞(
[0, T ];L1(Ω)

)
.

(103)
The inequality (46) remains true for r2, δ −→ 0. Thus, we have the uniform estimates
similar to the Lemma 12 with δ and r2. Moreover, we deduce the same compactness for
(ξr2 , ur2 , vr2) and for (ξδ, uδ, vδ). Therefore at this level of approximation, we focus only
on the convergence of r2∇ξ−β and that of δξ∇x∆

5
xξ. Here we state the following two

Lemmas.

Lemma 13. : For any ξr2 defined as in Proposition 5.1, we have

r2

∫ T

0

∫
Ω
ξ−β
r2 dxdydt −→ 0 as r2 −→ 0.

Proof. The proof is motivated by the method in [18, 19]. From the entropy B-D (101),
we have:

sup
t∈[0,T ]

∫
Ω

(
ln(

1

ξr2
)
)
+
dxdy ≤ C(r1) < +∞. (104)

Note that y ∈ R+ 7−→ ln( 1y )+ is a continuous convex function. Moreover, in combination
with the convex function property and Fatou’s Lemma, it follows that:∫

Ω

(
ln(

1

ξ
)
)
+
dxdy ≤

∫
Ω
lim inf
r2→0

(
ln(

1

ξr2
)
)
+
dxdy

≤ lim inf
r2→0

∫
Ω

(
ln(

1

ξr2
)
)
+
dxdy

≤ C(r1) < +∞. (105)

This means that
(
ln(1ξ )

)
+
is bounded in L∞(

[0, T ];L1(Ω)
)
. This allows us to deduce that∣∣{x/ξ(t, x) = 0}

∣∣ = 0 for almost every t ∈ [0, T ], (106)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2276

where |B| denotes the measure of set B.
Due to ξr2 −→ ξ strongly in C

(
[0, T ];H5(Ω)

)
, hence ξr2 −→ ξ a.e. Thus the above limits

and (106) deduce

r2ξ
−β
r2 −→ 0 a.e as r2 −→ 0. (107)

Moreover, using the Lemma 3 (interpolation inequality), we have

∥r2ξ−β
r2 ∥

L
5
3
(
[0,T ];L

5
3 (Ω)

)≤ ∥r2ξ−β
r2 ∥

2
5

L∞
(
[0,T ];L1(Ω)

)∥r2ξ−β
r2 ∥

3
5

L1
(
[0,T ];L3(Ω)

)≤ C, (108)

which combines with (107) and Lemma 2, and we have

r2ξ
−β
r2 −→ 0 strongly in L1

(
[0, T ];L1(Ω)

)
.

Lemma 14. For any ξδ defined as in the proposition 5.1, we have, for any test function
Φ

δ

∫ T

0

∫
Ω
ξδ∇x∆

5
xξδΦdxdydt −→ 0 as δ −→ 0. (109)

Proof. By (103) (where δ appears) and using the Gagliardo-Nirenberg interpolation
inequality, we have

∥∇5
xξδ∥L2≤ C∥∇6

xξδ∥
9
11
L2 ∥ξδ∥

2
11
L3 . (110)

Thus, ∫ T

0
δ
(∫

Ω
|∇5

xξδ|2dxdy
) 11

9
dt

≤ C
(

sup
t∈[0,T ]

∥ξδ∥L3

) 4
9

∫ T

0
δ

∫
Ω
|∇6

xξδ|2dxdydt. (111)

This implies

δ
9
22∇5

xξδ ∈ L
22
9
(
[0, T ];L2(Ω)

)
.

For any test function Φ ∈ C∞
c

(
[0, T ]; Ω

)
, we have

δ

∫ T

0

∫
Ω
ξδ∇x∆

5
xξδΦdxdydt

= −δ
∫ T

0

∫
Ω
∆2

xdivx(ξδΦ)∆
3
xξδdxdydt

= −δ
∫ T

0

∫
Ω
∆2

x

(
Φ∇xξδ + ξδdivxΦ

)
∆3

xξδdxdydt. (112)



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2277

Now we treat the term∣∣∣δ ∫ T

0

∫
Ω
∆2

x(∇xξδ)∆
3
xξδΦdxdydt

∣∣∣
≤ Cδ

1
11 ∥

√
δ∇6

xξδ∥L2(L2)∥δ
9
22∇5

xξδ∥
L
22
9 (L2)

∥Φ∥L11(L∞)−→ 0, (113)

as δ −→ 0.
As before, we can control the other term of

δ

∫ T

0

∫
Ω
ξδ∇x∆

5
xξδΦdxdydt.

Thus, we have:

δ

∫ T

0

∫
Ω
ξδ∇x∆

5
xξδΦdxdydt −→ 0 when δ −→ 0. (114)

So, we can pass to the limit r2, δ −→ 0 in (102)). Hence

∂tξ + divx(ξu) + ∂y(ξv) = 0

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u+ r1u

= 2divx
(
ξDx(u)

)
+ ∂y(ξ∂yu) + k1ξ∇x

(∆x
√
ξ√

ξ

)
∂yξ = 0,

(115)

holds in the sense of distribution on [0, T ]× Ω.
Due to the lower semi-continuity of convex functions, we can obtain the following energy
inequality (116) and B-D entropy (117) by passing to the limits in (100) and (101) as
r2, δ −→ 0,∫

Ω

(
1

2
ξu2 + ξ2 + k1|∇x

√
ξ|2

)
dxdy + r1

∫ T

0

∫
Ω
u2dxdydt

+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt+

∫ T

0

∫
Ω
2ξ|Dx(u)|2dxdydt+ g

∫ T

0

∫
Ω
vξ ln ξdxdydt

+

∫ T

0

∫
Ω
ξ|∂yu|2dxdydt ≤

∫
Ω

(
1

2
ξ0u

2
0 + ξ20 + k1|∇x

√
ξ0|2

)
ddxdy (116)

and∫
Ω

(1
2
ξ|u+ 2∇x ln ξ|2 − 2r1 ln ξ

)
dxdy + 2

∫ T

0

∫
Ω
ξ|∂yv|2dxdydt+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt

+ r1

∫ T

0

∫
Ω
u2dxdydt+

∫ T

0

∫
Ω
ξ|∂yu|2dxdy + k1

∫ T

0

∫
Ω
ξ|∇2

x ln ξ|2dxdydt



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2278

+ 2

∫ T

0

∫
Ω
ξ|Ax(u)|2dxdydt+ 8

∫ T

0

∫
Ω
ξ|∇x

√
ξ|2dxdydt

≤
∫
Ω

(
ξ0u

2
0 + 10(∇x

√
ξ0)

2 − 2r1 ln ξ0

)
dxdy + E0 + C. (117)

We obtain the existence of the weak solution (ξ, u, v) at this level of approximation given
in the following proposition.

Proposition 5.2. For any T > 0, the system (115) admits a weak solution with appro-
priate initial data. In particular, the weak solution (ξ, u, w) satisfies the energy inequality
(116) and the B-D entropy (117).

5.1.3. Passing to the limits as k1, r1 −→ 0

In this step, we pass to the limits as k1, r1 −→ 0. We denote by (ξk1,r1 , uk1,r1 , vk1,r1) the
weak solution at this level. Here, the weak solution satisfies the energy inequality (116)
and the B-D entropy (117), then, we have the following regularities

√
ξk1,r1uk1,r1 ∈ L∞(

[0, T ];L2(Ω)
)
,

√
ξk1,r1Dx(uk1,r1) ∈ L2

(
[0, T ];L2(Ω)

)
,

∇x

√
ξk1,r1 ∈ L∞(

[0, T ];L2(Ω)
)
,

√
ξk1,r1∂yuk1,r1 ∈ L2

(
[0, T ];L2(Ω)

)
,

ξ
1
3
k1,r1

uk1,r1 ∈ L3
(
[0, T ];L3(Ω)

)
,

√
ξk1,r1∇xuk1,r1 ∈ L2

(
[0, T ];L2(Ω)

)
,

uk1,r1 ∈ L2
(
[0, T ];L2(Ω)

)
, r1 ln ξk1,r1 ∈ L∞(

[0, T ];L1(Ω)
)
,√

ξk1,r1∂yvk1,r1 ∈ L2
(
[0, T ];L2(Ω)

)
,

√
k1
√
ξk1,r1 ∈ L2

(
[0, T ];H2(Ω)

)
,

√
ξk1,r1∇x

√
ξk1,r1 ∈ L2

(
[0, T ];L2(Ω)

)
, 4

√
k1∇xξ

1
4
k1,r1

∈ L4
(
[0, T ];L4(Ω)

)
.

(118)

Lemma 15. (Convergence of (
√
ξk1,r1)k1,r1). For

√
ξk1,r1 satisfying Proposition 5.2, we

deduce that (
√
ξk1,r1)k1,r1 is bounded in L∞(

[0, T ];H1(Ω)
)
and (∂t

√
ξk1,r1)k1,r1 is bounded

in L2
(
[0, T ];H−1(Ω)

)
.

Then, up to a subsequence, we have√
ξk1,r1 −→

√
ξ a.e. and strongly in L2

(
[0, T ];L2(Ω).

Furthermore, we have

ξk1,r1 −→ ξ a.e. and strongly in C
(
[0, T ];Lp(Ω), ∀ 1 ≤ p < 3.

Proof. Since ∥
√
ξk1,r1∥2L2(Ω)= ∥ξ0∥L1(Ω) and ∇x

√
ξk1,r1 ∈ L∞(

[0, T ];L2(Ω), we have√
ξk1,r1 ∈ L∞(

[0, T ];H1(Ω). Then, we claim that

∂t
√
ξk1,r1 = −1

2

√
ξk1,r1divx(uk1,r1)− uk1,r1 .∇x

√
ξk1,r1 −

1

2

√
ξk1,r1∂yvk1,r1



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2279

=
1

2

√
ξk1,r1divx(uk1,r1)− divx(

√
ξk1,r1uk1,r1)−

1

2

√
ξk1,r1∂yvk1,r1 . (119)

Indeed, we have
∂tξk1,r1 + ∂y(ξk1,r1vk1,r1) = −divx(ξk1,r1uk1,r1).

Furthermore, ∂t
√
ξk1,r1 =

1

2

1√
ξk1,r1

∂tξk1,r1 , hence

∂t
√
ξk1,r1 = −1

2

1√
ξk1,r1

(
divx(ξk1,r1uk1,r1) + ∂y(ξk1,r1vk1,r1)

)
.

By developing the divergence part and replacing ξk1,r1 by
√
ξk1,r1 .

√
ξk1,r1 , we obtain

∂t
√
ξk1,r1 = −1

2

√
ξk1,r1divx(uk1,r1)− uk1,r1∇x

√
ξk1,r1 −

1

2

√
ξk1,r1∂yvk1,r1 .

Adding and deducting
√
ξk1,r1divx(uk1,r1) in the above equality, we obtain (119).

This gives ∂t
√
ξk1,r1 ∈ L2

(
[0, T ];H−1(Ω)

)
. Using Lemma 1 we have

√
ξk1,r1 −→

√
ξ

strongly in L2
(
[0, T ];H1(Ω)

)
, thus in L2

(
[0, T ];L2(Ω)

)
and this gives that

√
ξk1,r1 −→

√
ξ

a.e.
Using the Sobolev injection theorem, we deduce that

√
ξk1,r1 is bounded in L∞(

[0, T ];L6(Ω)
)
,

so ξk1,r1 ∈ L∞(
[0, T ];L3(Ω)

)
. Then, we deduce by using the Hölder inequality that

ξk1,r1uk1,r1 =
√
ξk1,r1

√
ξk1,r1uk1,r1 ∈ L∞(

[0, T ];L
3
2 (Ω)

)
. (120)

This gives divx(ξk1,r1uk1,r1) ∈ L∞(
[0, T ];W−1,

3
2 (Ω)

)
. This last result, combined with√

ξk1,r1∂yvk1,r1 ∈ L2
(
[0, T ];L2(Ω)

)
and the continuity equation, give ∂t

√
ξk1,r1 ∈ L2

(
[0, T ];W−1,

3
2 (Ω)

)
.

Moreover, we have

∇xξk1,r1 = 2
√
ξk1,r1∇x

√
ξk1,r1 ∈ L∞(

[0, T ];L
3
2 (Ω)

)
. (121)

Then, we conclude that ξk1,r1 is bounded in L∞(
[0, T ];W 1,

3
2 (Ω)

)
. Now, we use Lemma 1

to get
ξk1,r1 −→ ξ strongly in C

(
[0, T ];Lp(Ω)

)
, ∀1 ≤ p < 3. (122)

Therefore, we get ξk1,r1 −→ ξ a.e.

Lemma 16. (Convergence of the momentum).
Up to a subsequence, the momentum mk1,r1 = ξk1,r1uk1,r1 satisfies

mk1,r1 −→ m strongly in L2
(
[0, T ];Lq(Ω)

)
, ∀1 ≤ q <

3

2
.

In particular, mk1,r1 −→ m for (t, x) ∈ [0, T ]× Ω.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2280

Proof. Since

∇x(ξk1,r1uk1,r1) = ∇xξk1,r1 ⊗ uk1,r1 + ξk1,r1∇xuk1,r1

= 2∇x

√
ξk1,r1 ⊗

√
ξk1,r1uk1,r1 +

√
ξk1,r1

√
ξk1,r1∇xuk1,r1 ∈ L2

(
[0, T ];L1(Ω)

)
and

∂y(ξk1,r1uk1,r1) =
√
ξk1,r1

√
ξk1,r1∂yuk1,r1 ∈ L2

(
[0, T ];L

3
2 (Ω)

)
,

we use (120) to deduce that

ξk1,r1uk1,r1 ∈ L2
(
[0, T ];W 1,1(Ω)

)
.

Now, we claim that ∂y(ξk1,r1uk1,r1) is bounded in L2
(
[0, T ];H−s(Ω)

)
for some constant

s > 0.
Indeed,

∂y(ξk1,r1uk1,r1) = 2divx (ξk1,r1Dx(uk1,r1)) + ∂y(ξk1,r1∂yuk1,r1)

+ k1ξk1,r1∇x

(∆x

√
ξk1,r1√

ξk1,r1

)
+ δξk1,r1∇x∆

5
xξk1,r1 − divx(ξk1,r1uk1,r1 ⊗ uk1,r1)

− ∂y(ξk1,r1uk1,r1vk1,r1)−∇xξ
2
k1,r1 − r1uk1,r1 − rξk1,r1uk1,r1 |uk1,r1 |. (123)

With the estimates of (118), we deduce that

ξk1,r1uk1,r1 ⊗ uk1,r1 ∈ L∞(
[0, T ];L1(Ω)

)
, ξk1,r1uk1,r1vk1,r1 ∈ L2

(
[0, T ];L1(Ω)

)
and

ξk1,r1∂yuk1,r1 ∈ L2
(
[0, T ];L

3
2 (Ω)

)
, ξk1,r1Dx(uk1,r1) ∈ L2

(
[0, T ];L

3
2 (Ω)

)
.

In particular, thanks to the Sobolev injection theorem, we have

divx(ξk1,r1uk1,r1 ⊗ uk1,r1) ∈ L∞(
[0, T ];W−2,2(Ω)

)
, ∂y(ξk1,r1uk1,r1vk1,r1) ∈ L2

(
[0, T ];W−2,2(Ω)

)
,

∂y(ξk1,r1∂yuk1,r1) ∈ L2
(
[0, T ];W−2,2(Ω)

)
, divx(ξk1,r1Dx(uk1,r1)) ∈ L2

(
[0, T ];W−2,2(Ω)

)
,

k1ξk1,r1∇x

(∆x

√
ξk1,r1√

ξk1,r1

)
= k1∇x(

√
ξk1,r1∆x

√
ξk1,r1)− 2k1∆x

√
ξk1,r1∇x

√
ξk1,r1 ∈ L2

(
[0, T ];W−3,2(Ω)

)
.

Then, we obtain the boundedness of ∂t(ξk1,r1uk1,r1) in L
2
(
[0, T ];H−5(Ω)

)
.

Therefore, we use Lemma 1 to conclude the proof of Lemma 16.

Remark 1. We can define u(t, x, y) =
m(t, x, y)

ξ(t, x, y)
outside the vacuum set {x|ξ(t, x) = 0}.

Then, we obtain ξk1,r1uk1,r1 −→ ξu strongly in L2
(
[0, T ];Lq(Ω)

)
, ∀1 ≤ q < 1.5.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2281

With Lemmas 14 and 16, in a similar way to the proof of Lemma 8, we can deduce
that when k1, r1 −→ 0,√

ξk1,r1uk1,r1 −→
√
ξu strongly in L2

(
[0, T ];L2(Ω)

)
. (124)

Lemma 17. (Convergence of terms divx(ξk1,r1Dx(uk1,r1)), k1ξk1,r1∇x

(∆x

√
ξk1,r1√

ξk1,r1

)
and

r1uk1,r1).
For any test function Φ ∈ C∞

c

(
[0, T ]; Ω

)
, we have

r1

∫ T

0

∫
Ω
uk1,r1Φdxdydt −→ 0 as r1 −→ 0, (125)∫ T

0

∫
Ω
divx(ξk1,r1Dx(uk1,r1))Φdxdydt −→

∫ T

0

∫
Ω
divx(ξDx(u))Φdxdydt as r1, k1 −→ 0,(126)

k1

∫ T

0

∫
Ω
ξk1,r1∇x

(∆x

√
ξk1,r1√

ξk1,r1

)
Φdxdydt −→ 0 as k1 −→ 0. (127)

Proof. We take Φ ∈ C∞
c

(
[0, T ]; Ω

)
as a test function. Firstly, we prove (125). Using

Hölder’s inequality, we get

r1

∫ T

0

∫
Ω
uk1,r1Φdxdydt ≤

√
r1∥

√
r1uk1,r1∥L2

(
[0,T ];L2(Ω)

)∥Φ∥
L2
(
[0,T ];L2(Ω)

) −→
r1→0

0.

Secondly, we deal with (126). Recalling (57), we have∫ T

0

∫
Ω
divx(ξk1,r1Dx(uk1,r1))Φdxdydt

=
1

2

∫ T

0

∫
Ω

(
ξk1,r1uk1,r1 ·∆xΦ+ 2∇xΦ · ∇x

√
ξk1,r1 ·

√
ξk1,r1uk1,r1

)
dxdydt

+
1

2

∫ T

0

∫
Ω

(
ξk1,r1uk1,r1 · divx(∇t

xΦ) + 2∇t
xΦ · ∇x

√
ξk1,r1 ·

√
ξk1,r1uk1,r1

)
dxdydt.

Since ∇x

√
ξk1,r1 ∈ L∞(

[0, T ];L2(Ω)
)
, then the sequence ∇x

√
ξk1,r1 is weakly converges.

Now, using Lemmas 14 and 16, and the fact that
√
ξk1,r1uk1,r1 −→

√
ξu strongly in

L2
(
[0, T ];L2(Ω)

)
, we obtain

1

2

∫ T

0

∫
Ω

(
ξk1,r1uk1,r1 ·∆xΦ+ 2∇xΦ · ∇x

√
ξk1,r1 ·

√
ξk1,r1uk1,r1

)
dxdydt

+
1

2

∫ T

0

∫
Ω

(
ξk1,r1uk1,r1 · divx(∇t

xΦ) + 2∇t
xΦ · ∇x

√
ξk1,r1 ·

√
ξk1,r1uk1,r1

)
dxdydt

−→
k1,r1−→0

1

2

∫ T

0

∫
Ω

(
ξu ·∆xΦ+ 2∇xΦ · ∇x

√
ξ ·

√
ξu

)
dxdydt

+
1

2

∫ T

0

∫
Ω

(
ξu · divx(∇t

xΦ) + 2∇t
xΦ · ∇x

√
ξ ·

√
ξu

)
dxdydt,



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2282

which implies that∫ T

0

∫
Ω
divx(ξk1,r1Dx(uk1,r1))Φdxdydt −→

k1,r1−→0

∫ T

0

∫
Ω
divx(ξDx(u))Φdxdydt.

At last, we prove (127). Recalling (118), we have

k1

∫ T

0

∫
Ω
ξk1,r1∇x

(∆x

√
ξk1,r1√

ξk1,r1

)
Φdxdydt

= −2k1

∫ T

0

∫
Ω
∆x

√
ξk1,r1∇x

√
ξk1,r1Φdxdydt− k1

∫ T

0

∫
Ω
∆x

√
ξk1,r1

√
ξk1,r1divxΦdxdydt

≤ 2
√
k1∥

√
k1∆x

√
ξk1,r1∥L2

(
[0,T ];L2(Ω)

)∥∇x

√
ξk1,r1∥L2

(
[0,T ];L2(Ω)

)∥Φ∥
L∞

(
[0,T ];L∞(Ω)

)
+
√
k1∥

√
k1∆x

√
ξk1,r1∥L2

(
[0,T ];L2(Ω)

)∥√ξk1,r1∥L2
(
[0,T ];L2(Ω)

)∥divxΦ∥L∞
(
[0,T ];L∞(Ω)

)
−→ 0 as k1 −→ 0.

Passing to the limits in (116) and (117) the energy inequality and the B-D entropy
give respectively∫

Ω

(
1

2
ξu2 + ξ2

)
dxdy + r

∫ T

0

∫
Ω
ξ|u|3dxdydt+

∫ T

0

∫
Ω
ξ|∂yu|2dxdydt

+

∫ T

0

∫
Ω
2ξ|Dx(u)|2dxdydt ≤

∫
Ω

(
1

2
ξ0u

2
0 + ξ20

)
dxdy (128)

and ∫
Ω

(1
2
ξ|u+ 2∇x ln ξ|2

)
dxdy + 2

∫ T

0

∫
Ω
ξ|∂yv|2dxdydt+ r

∫ T

0

∫
Ω
ξ|u|3dxdydt

+

∫ T

0

∫
Ω
ξ|∂yu|2dxdy + 2

∫ T

0

∫
Ω
ξ|Ax(u)|2dxdydt+ 8

∫ T

0

∫
Ω
ξ|∇x

√
ξ|2dxdydt

≤
∫
Ω

(
ξ0u

2
0 + 10(∇x

√
ξ0)

2
)
dxdy +

∫
Ω

(
1

2
ξ0u

2
0 + ξ20

)
dxdy + C. (129)

Passing to the limits in (115) when k1 −→ 0 and r1 −→ 0, the system

∂tξ + divx(ξu) + ∂y(ξv) = 0

∂t(ξu) + divx(ξu⊗ u) + ∂y(ξuv) +∇xξ
2 + rξ|u|u

= 2divx
(
ξDx(u)

)
+ ∂y(ξ∂yu)

∂yξ = 0

(130)

holds in the sense of distribution on[0, T ]× Ω. Therefore Theorem 1 is proved.



J. Ouya, A. Ouédraogo / Eur. J. Pure Appl. Math, 16 (4) (2023), 2247-2285 2283

5.2. Proof of Theorem 2

Let us now consider the weak solution (ξ, u, v) of the system (7). From Theorem 1, we
have the following regularities:

ξ ∈ L∞(
[0, T ];H1(Ω)

)
,

√
ξu ∈ L∞(

[0, T ];L2(Ω)
)
,

ξ∇xu ∈ L2
(
[0, T ];L2(Ω)

)
, ξ(∇xu)

t ∈ L2
(
[0, T ];L2(Ω)

)
,

√
ξ∂yv ∈ L2

(
[0, T ];L2(Ω)

)
,

√
ξ∂yu ∈ L2

(
[0, T ];L2(Ω)

)
,

ξ
1
3u ∈ L3

(
[0, T ];L3(Ω)

)
,

√
ξv ∈ L2

(
[0, T ];L2(Ω)

)
.

(131)

Recall that ρ(t, x, y) = ξ(t, x) + ϕ(y), so we obtain the following properties:

∥√ρ∥
L∞

(
[0,T ];H1(Ω)

)≥ ∥
√
ξ∥

L∞
(
[0,T ];H1(Ω)

),
∥√ρu∥

L∞
(
[0,T ];L2(Ω)

)≥ ∥
√
ξu∥

L∞
(
[0,T ];L2(Ω)

),
∥ 3
√
ρu∥

L3
(
[0,T ];L3(Ω)

)≥ ∥ 3
√
ξu∥

L3
(
[0,T ];L3(Ω)

),
∥ρDxu∥L2

(
[0,T ];L2(Ω)

)≥ ∥ξDxu∥L2
(
[0,T ];L2(Ω)

),
∥√ρ∂yu∥L2

(
[0,T ];L2(Ω)

)≥ ∥
√
ξ∂yu∥L2

(
[0,T ];L2(Ω)

),
∥∂yρ∥L∞

(
[0,T ];L2(Ω)

)= 1

2
g|Ωx|.

(132)

In addition,

∥√ρv∥2
L2
(
[0,T ];L2(Ω)

) =

∫ T

0

∫ 1

0

∫
Ωx

|√ρv|2dxdydt

=

∫ T

0

∫ 1

0

∫
Ωx

(
ξv2 + ϕv2

)
dxdydt

≤ 2

∫ T

0

∫ 1

0

∫
Ωx

ξv2dxdydt+ g

∫ T

0

∫ 1

0

∫
Ωx

v2dxdydt (133)

and

∥√ρ∂yv∥2
L2
(
[0,T ];L2(Ω)

) =

∫ T

0

∫ 1

0

∫
Ωx

|√ρ∂yv|2dxdydt

≤ 2

∫ T

0

∫ 1

0

∫
Ωx

(
ξ(∂yv)

2 + ϕ(∂yv)
2
)
dxdydt

≤ C

∫ T

0

∫ 1

0

∫
Ωx

(
√
ξ∂yv)

2dxdydt+ g

∫ T

0

∫ 1

0

∫
Ωx

(∂yv)
2dxdydt.

(134)



REFERENCES 2284

With all of the above estimates, the Theorem 2 is proved, since (ρ, u, v) satisfies the
conditions of Definition 2.1. Thus our initial system (1) admits a global weak solution.

6. Conclusion

In this paper, we discussed in dimension d = 3, the existence of global weak solutions
to the three-dimensional compressible primitive equations of atmospheric dynamics with
degenerate viscosity density-dependent for large initial data. We have proven that the weak
solutions satisfy the basic energy inequality and the Bresch-Desjardins entropy inequality.
We have obtained the global existence of weak solutions of (1) by vanishing the parameters
in our approximate system step by step.

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