Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 74, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.74

ZERO-VISCOSITY-CAPILLARITY LIMIT FOR THE CONTACT

DISCONTINUITY FOR THE 1-D FULL COMPRESSIBLE

NAVIER-STOKES-KORTEWEG EQUATIONS

JIAXUE CHEN, YEPING LI, RONG YIN

Abstract. In this article, we study the zero-viscosity-capillarity limit problem for the one-

dimensional full compressible Navier-Stokes-Korteweg equations. This equation models com-
pressible viscous fluids with internal capillarity and heat conductivity. We prove that if the

solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then

there exist smooth solutions to the one-dimensional full compressible Navier-Stokes-Korteweg
system which converge to the inviscid solution away from the contact discontinuity. It converges

a rate of ϵ1/4 as the the viscosity µ = ϵ, heat-conductivity coefficient α = νϵ and the capillarity

κ = λϵ2 and ϵ tends to zero. The proof is completed using the energy method and the scaling
technique.

1. Introduction

The purpose of this paper is to study the asymptotic equivalence between the solutions of the
one-dimensional full compressible Navier-Stokes-Korteweg equations and those of the compressible
full Euler system when the viscosity, the heat-conductivity coefficient and the capillarity satisfy
some conditions. The one-dimensional full compressible Navier-Stokes-Korteweg (denoted as NSK
in the sequel) equations in Lagrangian coordinates are expressed as

vt − ux = 0,

ut + px = µ
(ux
v

)
x
+
κ

v

(1
v

(1
v

(1
v

)
x

)
x

)
x
,(

e+
u2

2

)
t
+ (pu)x = α

(θx
v

)
x
+
(
µ
uux
v

+ κu
( 1

v2
(1
v

(1
v

)
x

)
x
− 1

2

(1
v

(1
v

)
x

)2))
x
,

(1.1)

where v, u, θ, p and e denote the specific volume, the velocity, the temperature, the pressure,
and the internal energy, respectively, and µ, α and κ are the viscosity and heat-conductivity and
capillary coefficients, respectively. Here x is the Lagrangian coordinate, so that x = constant
corresponds to a particle path. Here we only study the ideal polytropic gas, so that the pressure
p and the internal energy e are related with v and θ by the following equations of state

p = R
θ

v
, e =

R

γ − 1
θ + constant, (1.2)

where R > 0 is the gas constant and γ ∈ (1, 2] is the adiabatic exponent.
System (1.1) is known to be a model system for two phase flow with phase transition between

liquid and vapor in compressible fluid. Based on the works of Van der Waals [27] and Korteweg
[17], the rigorous derivation of the corresponding equations is due to Dunn and Serrin [8] and
Heida and Málek [13], respectively. Finally, one can see easily that when κ = 0, the system (1.1)
is reduced to the classical compressible Navier-Stokes equation.

2020 Mathematics Subject Classification. 76W05, 35B40.
Key words and phrases. Full compressible Navier-Stokes-Korteweg equation; compressible Euler system;

zero-viscosity-capillarity limit; contact discontinuity.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted May 19, 2025. Published July 16, 2025.

1



2 J. CHEN, Y. LI, R. YIN EJDE-2025/74

The compressible NSK equations have attracted a lot of attention of physicists and math-
ematicians because of its physical importance, complexity, rich phenomena, and mathematical
challenges. Here we only refer to some study results about the full compressible NSK equations
(1.1). Hattori and Li [12] considered the local existence and global existence of smooth solution
for three-dimensional non-isentropic compressible NSK equations in Sobolev space. Haspot [11]
showed the existence results of strong solutions to the nonisothermal compressible NSK equa-
tions in R3. Chen, He and Zhao [3] obtained the global classical solutions of the one dimensional
full compressible NSK equations with large initial data. Hou, Peng and Zhu [15] got the global
classical solutions for the three-dimensional non-isentropic compressible NSK equation with small
initial energy. Chen and Zhao [7] discussed the existence, uniqueness and nonlinear stability
of stationary solutions to the Cauchy problem of the three-dimensional full compressible NSK
equations. Cai, Tan and Xu [1] established the existence of the time periodic solution to the
three-dimensional full compressible NSK equations with a sufficiently small external force which
is periodic in the time variable. Zhang and Tan [31] showed decay estimates of smooth solutions
for the non-isentropic compressible fluid models of Korteweg type in R3. Kotschote [18, 19, 20]
established the local existence of strong solution, and global existence and time-asymptotics of
strong solution of the non-isentropic compressible NSK equations in a bounded domain with C3-
boundary. About the stability of basic nonlinear wave patterns such as the discontinuity wave,
viscous contact wave and the rarefaction wave of one dimensional compressible NSK equations,
we can refer to [4, 5, 6, 9, 24, 26] and the references therein.

Moreover, vanishing viscosity limit is one of the important problems in the theory of com-
pressible fluids. Goodman and Xin [10] and Hoff and Liu [14] pioneered, respectively, the study
on the viscous limit of piecewise smooth solutions for the hyperbolic conservation law, and on
the vanishing viscosity limit of the viscous compressible isentropic Navier-Stokes equations for
piecewise constant shock. After these work, the limit problem of the system of hyperbolic con-
servation laws and of the compressible Navier-Stokes equations is an important problem and has
been extensively investigated by many authors for the cases that the solution of the inviscid flows
is smooth, or contains singularities such as shocks and the vacuum state. Since the compressible
NSK equations are the capillarity approximation of the classical compressible Navier-Stokes equa-
tions (see [16]), one of the important topics about the compressible NSK equations is to study the
zero-viscosity-capillarity limit. Charve and Haspot [2] proved the existence of the global strong
solution of the one-dimensional isentropic NSK equations, and then showed that the global strong
solution converges to a weak-entropy solution of the compressible Euler equations. Li and Luo
[22], and Li and Zhu [23] showed zero-viscosity-capillarity limit towards rarefaction wave without
and with vacuum for one-dimensional the compressible NSK equations, respectively. Yin and Li
[29] also discussed the zero-viscosity-capillarity limit towards planar rarefaction wave for the two-
dimensional isentropic NSK equations. Nevertheless, it is more significant and difficult to study
the viscosity-vanishing limit for the non-isentropic (full) NSK equation (1.1) from both physical
and mathematical points of view. Lastly, Wang and Yao [28], and Yin, Li and Qian [30] discussed
zero-viscosity-capillarity limit towards rarefaction wave for one-dimensional full NSK system, re-
spectively. Here, we are going to investigate the zero-viscosity-capillarity limit problem for the
one-dimensional full compressible NSK equations (1.1). For this, we assume that the coefficients
of viscosity, capillary, heat-conductivity µ, κ and α satisfy

µ = ϵ, κ = λϵ2, α = νϵ. (1.3)

Then, formally as ϵ→ 0, (1.1) becomes the well-known compressible Euler system

vt − ux = 0,

ut + px = 0,(
e+

u2

2

)
t
+ (pu)x = 0.

(1.4)



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 3

Further, the Riemann problem to the corresponding Euler system (1.4) with Riemann initial data

(v, u, θ)(0, x) =

{
(v−, u−, θ−), if x < 0,

(v+, u+, θ+), if x > 0,
(1.5)

where v±(> 0), u±, s± are given constants, has a contact discontinuity (see [21]), which takes the
form

(V̄ , Ū , Θ̄)(t, x) =

{
(v−, u−, θ−), if x < 0,

(v+, u+, θ+), if x > 0,
(1.6)

provided that

u− = u+, p− =
Rθ−
v−

=
Rθ+
v+

= p+. (1.7)

As in [25], we first construct the viscous contact wave (V CD, UCD,ΘCD)(t, x) as follows. Let
the pressure of the profile (V CD, UCD,ΘCD)(t, x) be almost constant, that is,

PCD = R
ΘCD

V CD
≈ p+ = p−, (1.8)

it indicates that the energy equation (1.1)3 is

R

γ − 1
Θt + p+Ux = α

(Θx

V

)
x
. (1.9)

Substituting (1.8) into (1.9) and using (1.1)1 yield a nonlinear diffusion equation

Θt = aα(
Θx

Θ
)x, Θ(t,±∞) = θ±, a =

p+(γ − 1)

R2γ
> 0,

which admits a unique self-similar solution

Θ̂(t, x) = Θ̂
( x√

1 + t

)
.

Furthermore, Θ̂(t, x) is a monotone function, increasing if θ+ > θ− and decreasing if θ+ < θ−. Let

δCD = |θ+ − θ−|, then Θ̂(t, x) satisfies

|(ϵ(1 + t))
k
2 ∂kxΘ̂|+ |Θ̂(t, x)− θ±| ≤ CδCDe−

c0x2

ϵ(1+t) , as |x| → ∞, k ≥ 1. (1.10)

With Θ̂(t, x) so determined, we can define the contact wave profile (V CD, UCD,ΘCD)(t, x) as
follows:

V CD =
RΘ̂

p+
, UCD = u− +

α(γ − 1)

Rγ

Θ̂x

Θ̂
, ΘCD = Θ̂. (1.11)

Then (V CD, UCD,ΘCD) satisfies

∥V CD − V̄ , UCD − Ū ,ΘCD − Θ̄∥Lp = O(ϵ
1
2p )(1 + t)

1
2p , p ≥ 1, (1.12)

and
V CD
t − UCD

x = 0,

UCD
t + PCD

x = ϵ(
UCD
x

V CD
)x +RCD

1 ,

R

γ − 1
ΘCD

t + PCDUCD
x = α(

ΘCD
x

V CD
)x + ϵ

(UCD
x )2

V CD
+RCD

2 ,

(1.13)

where

RCD
1 = O(δCD)ϵ1/2(1 + t)−

3
2 e−

c0x2

ϵ(1+t) ,

RCD
2 = O(δCD)ϵ(1 + t)−2e−

c0x2

ϵ(1+t) ,

as |x| → ∞. Now we state our main results in the following theorem.



4 J. CHEN, Y. LI, R. YIN EJDE-2025/74

Theorem 1.1. For a given (v−, u−, θ−), suppose that (v+, u+, θ+) satisfies (1.7). Let (V̄ , Ū , Θ̄)(t, x)
be a contact discontinuity solution of the form (1.6) with finite strength to the Euler system (1.4).
Then, there exists constant ϵ0 > 0, such that for each ϵ ∈ (0, ϵ0], there is a smooth solution (v, u, θ)
to (1.1) on R× R+, with the same initial data as (V CD, UCD,ΘCD), satisfying

(v − V̄ , u− Ū , θ − Θ̄) ∈ C0(0,+∞;L2),

(v, u, θ)x ∈ C0(0,+∞;L2), vxx ∈ C0(0,+∞;L2),

(u, θ)xx ∈ L2(0,+∞;L2), vxxx ∈ L2(0,+∞;L2).

Moreover, for each arbitrarily large T > 0 and small h > 0, it holds that

sup
0≤t≤T

∥(v − V̄ , u− Ū , θ − Θ̄)(t, ·)∥2 ≤ Cϵ1/2, (1.14)

sup
0≤t≤T, |x|≥h

|(v − V̄ , u− Ū , θ − Θ̄)(t, ·)| ≤ Cϵ1/4. (1.15)

Remark 1.2. Theorem 1.1 shows the zero-viscosity-capillarity limit of the one-dimensional full
NSK equations with the same order of the viscosity and the heat-conductivity, and the higher
order capillarity, as the corresponding full Euler equations have a contact discontinuity. It is more
interest to study the limit for the one-dimensional full NSK equations as the viscosity, the heat-
conductivity and the capillarity have same order. Further, it is also important that we investigate
the limit with the heat-conductivity and the capillarity and without the viscosity as in [25]. Finally,
the convergence rate in (1.15) may not be optimal. We conjecture that it can be improved to be
ϵ1/2. These will be left for future study.

The remaining part of this paper is organized as follows. In Section 2, we reformulate the
problem and give the proof of Theorem 1.1. We also collect the a priori estimates needed in the
proof of our main theorem, in Proposition 2.1. Then, we establish the a-priori estimates for the
reformulated problem in Section 3.
Notation. Throughout this paper, c and C denote two universal positive constant which is
independent of time t and may vary from line to line. Lp(R)(1 ≤ p < ∞) are the spaces of

measurable functions whose p-powers are integrable on R, with the norm ∥ · ∥Lp =
(∫

R | · |pdx
) 1

p .
For the case that p = 2, we simply denote ∥ · ∥L2 by ∥ · ∥. And L∞(R) is the space of bounded
measurable functions on R, with the norm ∥ ·∥L∞ = ess supx∈R| · |. Furthermore, for a nonnegative
integer k, Hk(R) denotes the usual L2-type Sobolev space of order k. We write ∥ · ∥k for the
standard norm of Hk(R). Finally, we denote by C([0, T ];Hk(R)) (resp. L2(0, T ;Hk(R))) the
space of continuous (resp. square integrable) functions on [0, T ] with values taken in a Banach
space Hk(R).

2. Reformulation of the Problem and Proof of Theorem 1.1

In this section, we reformulate the original problem (1.1) and (1.5) in terms of the perturbated
variables, then give the proof of Theorem 1.1. To begin with, suppose that U ≡ (v, u, θ) is the
exact solution to (1.1) with the initial data U(x, 0) = (V CD, UCD,ΘCD)(x, 0), and define the
perturbated variables:

ϕ = v − V CD, ψ = u− UCD, ζ = θ −ΘCD.

Let

y =
x

ϵ
, τ =

1 + t

ϵ
,

then from (1.1) and (1.13), one sees that

ϕτ − ψy = 0,

ψτ + (p− PCD)y =
(uy
v

−
UCD
y

V CD

)
y
+
λ

v

(1
v

(1
v

(1
v

)
y

)
y

)
y
−R1,

R

γ − 1
ζτ + (puy − PCDUCD

y ) = ν
(θy
v

−
ΘCD

y

V CD

)
y
+
(u2y
v

−
(UCD

y )2

V CD

)
+ λuy

(5v2y
2v6

− vyy
v5

)
−R2,

(2.1)



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 5

with the initial data

ϕ(τ0, y) = ψ(τ0, y) = ζ(τ0, y) = 0, (2.2)

where τ0 = 1
ϵ , R1 = ϵRCD

1 , R2 = ϵRCD
2 and

|∂kyV CD|+ |∂kyΘCD| ≤ Cϵ
k
2 e−c0y

2/τ , |∂k−1
y UCD| ≤ C|∂kyΘCD|, k ≥ 1

|R1| ≤ Cϵ3/2e−c0y
2/τ , |R2| ≤ Cϵ2e−c0y

2/τ .
(2.3)

Set τ1 = T+1
ϵ . Then we only need to show that for suitably small ϵ, the Cauchy problem

(2.1) and (2.2) has a unique ”small” smooth solution on R × [τ0, τ1]. By the standard existence
and uniqueness theory (cf. [12]), and the continuous induction argument, it suffices to show the
following a priori estimate.

Proposition 2.1. Suppose that the problem (2.1) and (2.2) has a solution (ϕ, ψ, ζ) ∈ C0(τ0, τ2;L
2)

for some τ2 ∈ (τ0, τ1]. Then there exist positive constants ϵ1, η1 and C, independent of ϵ, such
that if

0 < ϵ ≤ ϵ1, sup
τ0≤τ≤τ2

(∥ϕ∥2 + ∥(ψ, ζ)∥1) ≤ η1 (2.4)

for small ϵ1 and η1, then

sup
τ0≤τ≤τ2

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ2

τ0

∥(ϕy, ψy, ζy, ϕyy)∥2 dτ ≤ Cϵ1/2, (2.5)

and

sup
τ0≤τ≤τ2

∥(ϕy, ψy, ζy, ϕyy)(τ)∥2 +
∫ τ2

τ0

∥(ϕyy, ψyy, ζyy, ϕyyy)∥2 dτ ≤ Cϵ2/3. (2.6)

Suppose that Proposition 2.1 is true, we then are in a position to prove Theorem 1.1 as follows.

Proof of Theorem 1.1. For any T > 0, in view of (2.5), we have

sup
0≤t≤T

∥(v − V CD, u− UCD, θ −ΘCD)(t)∥2

= ϵ sup
τ0≤τ≤τ1

∥(v − V CD, u− UCD, θ −ΘCD)(τ)∥2

≤ Cϵ3/2.

Then it follows from this and (1.12) that

sup
0≤t≤T

∥(v − V̄ , u− Ū , θ − Θ̄)(t)∥2

≤ sup
0≤t≤T

∥(v − V CD, u− UCD, θ −ΘCD)(t)∥2 + sup
0≤t≤T

∥(V CD − V̄ , UCD − Ū ,ΘCD − Θ̄)(t)∥2

≤ Cϵ1/2,

which gives (1.14). Finally,

∥(v − V CD, u− UCD, θ −ΘCD)(t)∥L∞ ≤ C∥(ϕ, ψ, ζ)(t)∥1/2∥(ϕy, ψy, ζy)(t)∥1/2 ≤ Cϵ1/4.

This, together with (1.10), yields (1.15). Hence we have completed the proof of Theorem 1.1. □

3. A priori estimate

In this section, we shall prove Proposition 2.1. First, notice that the smallness of η1 in (2.4)
guarantees that

2v+ ≥ v = V CD + ϕ ≥ v−
2
, 2θ̄ ≥ θ = ΘCD + ζ ≥ θ

2
, (3.1)

where θ = inft≥0, x∈R ΘCD(t, x) and θ̄ = supt≥0, x∈R ΘCD(t, x). For the sake of clarity, we will
divide the proof of Proposition 2.1 into some Lemmas. That is, Proposition 2.1 can be obtained
by the following Lemmas 3.1-3.3.



6 J. CHEN, Y. LI, R. YIN EJDE-2025/74

First, we establish the first energy estimate for the unknown variable (ϕ, ψ, ζ)(t, x) to problem
(2.1)-(2.2). For this, let us introduce the function Φ(s) = s− ln s−1, which is the convex function
for any s > 0. Then from (2.1), after a direct and tedious computation, we arrive at(1

2
ψ2 + λ

ϕ2y
2v5

+RΘCDΦ
( v

V CD

)
+
RΘCD

γ − 1
Φ
( θ

ΘCD

))
τ
+
ψ2
yΘ

CD

vθ
+ ν

ζ2yΘ
CD

vθ2

+
(
(p− PCD)ψ −

(uy
v

−
UCD
y

V CD

)
ψ − λ

1

v2
(1
v

(1
v

)
y

)
y
ψ − λ

ϕyψy

v5
+ λ

v2yψ

2v6

− ν
(θy
v

−
ΘCD

y

V CD

)ζ
θ

)
y
+ PCDUCD

y

(
γΦ

( v

V CD

)
+Φ

(θV CD

vΘCD

))
=

(
ν
(ΘCD

y

V CD

)
y
+

(UCD
y )2

V CD
+R2

)(
(γ − 1)Φ

( v

V CD

)
− Φ

(ΘCD

θ

))
+

UCD
y

vV CD
ϕψy + ν

ΘCD
y

vθ2
ζζy + ν

ΘCDΘCD
y

vθ2V CD
ϕζy − ν

(ΘCD
y )2

vθ2V CD
ϕζ

+
2UCD

y

vθ
ζψy −

(UCD
y )2

vθV CD
ϕζ − λ

5UCD
y

2v6
ϕ2y + λ

V CD
yy

v5
ψy − λ

5(V CD
y )2

2v6
ψy

+ λ(ψy + Uy)
(5(ϕy + Vy)

2

2v6
− ϕyy + Vyy

v5
)ζ
θ
−R1ψ −R2

ζ

θ
.

(3.2)

Here we used

1

v

(1
v

(1
v

(1
v

)
y

)
y

)
y
ψ =

( 1

v2
(1
v

(1
v

)
y

)
y
ψ
)
y
+
(vy
v3

)
y

ψy

v2
− 2

(vy
v3

)
y

ψvy
v3

=
( 1

v2
(1
v

(1
v

)
y

)
y
ψ +

vyψy

v5
− 2

ψv2y
v6

)
y
−

(
ϕ+ V CD

)
y
ψyy

v5

+
4(ϕy + V CD

y )2ψy

v6
+ 2

ψvyvyy
v6

−
6ψv3y
v7

=
( 1

v2
(1
v

(1
v

)
y

)
y
ψ +

vyψy

v5
− 2

ψv2y
v6

−
V CD
y ψy

v5
+
v2yψ

v6
)
y
−

( ϕ2y
2v5

)
τ

−
5ϕ2yU

CD
y

2v6
+
V CD
yy ψy

v5
−

5ψy(V
CD
y )2

2v6

with the help of (2.1)1. Moreover, there exists a positive constant C1 and C2 such that

C1(s− t)2 ≤ Φ
(s
t

)
≤ C2(s− t)2. (3.3)

Lemma 3.1. Suppose that the assumptions in Proposition 2.1 hold. Then it holds that

sup
τ0≤τ≤τ2

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ2

τ0

∥(ϕy, ψy, ζy, ϕyy)∥2 dτ ≤ Cϵ1/2. (3.4)

Proof. Integrating (3.2) with respect to τ and y over [τ0, τ ] × R (τ ≤ τ2) and using (1.3), (3.1)
and (3.3) shows that

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ

τ0

∥∥∥√UCD
y (ϕ, ζ)

∥∥∥2 dτ + ∫ τ

τ0

∥(ψy, ζy)∥2 dτ ≤ C

6∑
i=1

∫ τ

τ0

∫
R
Hidydτ, (3.5)

where

H1 = |UCD
y V CD

yy ζ|+ |UCD
y (V CD

y )2ζ|+ |R1ψ|+ |R2ζ|,
H2 = (|ΘCD

yy |+ |ΘCD
y V CD

y |+ (UCD
y )2 + |UCD

y |)(ϕ2 + ζ2)

+ |(ΘCD
y )2ϕζ|+ |(UCD

y )2ϕζ|+ |R2|(ϕ2 + ζ2),

H3 = |UCD
y ϕψy|+ |ΘCD

y ζζy|+ |ΘCD
y ϕζy|+ |UCD

y ζψy|+ |(V CD
y )2ζψy|

+ |UCD
y V CD

y ζϕy|+ |V CD
yy ζψy|+ |UCD

y ζϕyy|,
H4 = |V CD

yy ψy|+ |(V CD
y )2ψy|+ |UCD

y ϕ2y|,



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 7

H5 = |V CD
y ζϕyψy|+ |UCD

y ζϕ2y|,
H6 = |ζψyϕyy|+ |ζψyϕ

2
y|.

Now let us estimate each term on the right hand side of (3.5). First, we employ Young inequality
and (2.3) to obtain

∫ τ

τ0

∫
R
|UCD

y V CD
yy ζ|dydτ ≤ ϵ

∫ τ

τ0

∫
R
|ζ|2 dydτ + ϵ−1

∫ τ

τ0

∫
R
|UCD

y V CD
yy |2 dydτ

≤ ϵ

∫ τ

τ0

∫
R
|ζ|2 dydτ + Cϵ−1

∫ τ

τ0

∫
R

(
ϵe−c0y

2/τ · ϵe−c0y
2/τ

)2
dydτ

≤ ϵ

∫ τ

τ0

∥ζ∥2 dτ + Cϵ3
∫ τ

τ0

∫
R
e−

4c0y2

τ dydτ

≤ ϵ

∫ τ

τ0

∥ζ∥2 dτ + Cϵ3
∫ τ

τ0

τ1/2 dτ

≤ ϵ

∫ τ

τ0

∥ζ∥2 dτ + Cϵ3/2.

(3.6)

Similarly, estimating the rest terms in
∫ τ

τ0

∫
RH1dydτ , we have

∫ τ

τ0

∫
R
H1dydτ ≤ ϵ

∫ τ

τ0

(∥ψ∥2 + ∥ζ∥2) dτ + C
(
ϵ1/2 + ϵ3/2

)
. (3.7)

Next, utilizing Young inequality, (2.3) and e−c0y
2/τ ≤ 1, one gets

∫ τ

τ0

∫
R
|(ΘCD

y )2ϕζ|dydτ ≤ C

∫ τ

τ0

∫
R

(
ϵ1/2e−c0y

2/τ
)2
(ϕ2+ζ2) dydτ ≤ Cϵ

∫ τ

τ0

(∥ϕ∥2+∥ζ∥2) dτ. (3.8)

In the same way, we can deal with the remainder terms in
∫ τ

τ0

∫
RH2dydτ and the terms in∫ τ

τ0

∫
RH3dydτ to obtain

∫ τ

τ0

∫
R
H2dydτ ≤ C(ϵ+ ϵ2)

∫ τ

τ0

(∥ϕ∥2 + ∥ζ∥2) dτ, (3.9)

and ∫ τ

τ0

∫
R
H3dydτ ≤ C

(
ϵ1/2 + ϵ

) ∫ τ

τ0

∥ϕ∥2 dτ + C
(
ϵ1/2 + ϵ+ ϵ3/2

) ∫ τ

τ0

∥ζ∥2 dτ

+ Cϵ

∫ τ

τ0

∥ψy∥2 dτ + Cϵ1/2
∫ τ

τ0

∥ζy∥2 dτ

+ Cϵ3/2
∫ τ

τ0

∥ϕy∥2 dτ + Cϵ

∫ τ

τ0

∥ϕyy∥2 dτ.

(3.10)

With (3.6) and (3.8) in hand, it is easy to obtain

∫ τ

τ0

∫
R
H4dydτ ≤ 1

8

∫ τ

τ0

∥ψy∥2 dτ + Cϵ

∫ τ

τ0

∥ϕy∥2 dτ + Cϵ1/2. (3.11)



8 J. CHEN, Y. LI, R. YIN EJDE-2025/74

Moreover, using (2.3), Hölder inequality, Sobolev inequality, (2.4) and Young inequality, and noting

e−c0y
2/τ ≤ 1, one obtains∫ τ

τ0

∫
R
H5dydτ ≤ C

∫ τ

τ0

∫
R

(∣∣ϵ1/2e−c0y
2/τζϕyψy

∣∣+ ∣∣ϵe−c0y
2/τζϕ2y

∣∣) dydτ
≤ C

∫ τ

τ0

(
ϵ1/2∥ζ∥L∞∥ϕy∥∥ψy∥+ ϵ∥ζ∥L∞∥ϕy∥2

)
dτ

≤ C

∫ τ

τ0

(
ϵ1/2∥ζ∥1∥ϕy∥∥ψy∥+ ϵ∥ζ∥1∥ϕy∥2

)
dτ

≤ C

∫ τ

τ0

(
ϵ1/2η1∥ϕy∥∥ψy∥+ ϵη1∥ϕy∥2

)
dτ

≤ C
(
ϵ1/2 + ϵ

)
η1

∫ τ

τ0

∥ϕy∥2 dτ + Cϵ1/2η1

∫ τ

τ0

∥ψy∥2 dτ.

(3.12)

Finally, it follows from Hölder inequality, Sobolev inequality, Young inequality and (2.4), that∫ τ

τ0

∫
R
H6dydτ ≤ C

∫ τ

τ0

(∥ϕy∥2L∞∥ψy∥∥ζ∥+ ∥ζ∥L∞∥ϕyy∥∥ψy∥) dτ

≤ C

∫ τ

τ0

(∥ϕy∥∥ϕyy∥∥ψy∥∥ζ∥+ ∥ζ∥1∥ϕyy∥∥ψy∥) dτ

≤ C(η1 + η21)
(∫ τ

τ0

∥ψy∥2 dτ +
∫ τ

τ0

∥ϕyy∥2 dτ
)
.

(3.13)

Therefore, substituting the estimates of (3.7), (3.9), (3.10), (3.11), (3.12) and (3.13) into (3.5)
and noting that ϵ and η1 are suitably small, we have

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ

τ0

∥∥∥√UCD
y (ϕ, ζ)

∥∥∥2 dτ + ∫ τ

τ0

∥(ψy, ζy)∥2 dτ

≤ Cϵ

∫ τ

τ0

∥ψ∥2 dτ + C
(
ϵ1/2 + ϵ+ ϵ3/2 + ϵ2

) ∫ τ

τ0

∥ζ∥2 dτ

+ C
(
ϵ1/2 + ϵ+ ϵ2

) ∫ τ

τ0

∥ϕ∥2 dτ + C
(
ϵ+ η1 + η21

) ∫ τ

τ0

∥ϕyy∥2 dτ

+ C
(
ϵ+ ϵ3/2 +

(
ϵ1/2 + ϵ

)
η1
) ∫ τ

τ0

∥ϕy∥2 dτ + C
(
ϵ1/2 + ϵ3/2

)
.

(3.14)

Next, we deal with the double integral of ϕ2y and ϕ2yy. We multiply (1.13)2 by
ϕy

v to obtain( ϕ2y
2v2

− ψ
ϕy
v

)
τ
+
(
λ
ϕy
v3

(1
v

(1
v

)
y

)
y
+
ψψy

v

)
y
+
Rθ

v3
ϕ2y + λ

ϕ2yy
v6

=
ψ2
y

v
+R

ζyϕy
v2

−
ψψyV

CD
y

v2
+
ψϕyU

CD
y

v2
+

(R
v
− R

V CD

)ϕy
v
ΘCD

y

−
(Rθ
v2

− RΘCD

(V CD)2
)ϕy
v
V CD
y +

UCD
yy ϕϕy

v2V CD
+
V CD
y ψyϕy

v3

−
UCD
y V CD

y (v + V CD)ϕϕy

v3(V CD)2
+ λ

(5ϕ2yϕyy
v7

−
6ϕ4y
v8

−
V CD
yy ϕyy

v6

+
2ϕ2yV

CD
yy

v7
+

8ϕyϕyyV
CD
y

v7
+

2ϕyV
CD
y V CD

yy

v7
+

3ϕyy(V
CD
y )2

v7

−
18ϕ3yV

CD
y

v8
−

18ϕ2y(V
CD
y )2

v8
−

6ϕy(V
CD
y )3

v8
)
+R1

ϕy
v
.

(3.15)

Here we also use that(1
v

(1
v

(1
v

)
y

)
y

)
y

ϕy
v2

=
(ϕy
v3

(1
v

(1
v

)
y

)
y

)
y
− 1

v

(1
v

(1
v

)
y

)
y

(ϕy
v2

)
y



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 9

=
(ϕy
v3

(1
v

(1
v

)
y

)
y

)
y
+
(vy
v3

)
y

(ϕyy
v3

− 2ϕyvy
v4

)
=

(ϕy
v3

(1
v

(1
v

)
y

)
y

)
y
+
ϕ2yy
v6

+
V CD
yy ϕyy

v6
−

5ϕ2yϕyy

v7
−

2ϕ2yV
CD
yy

v7

−
8ϕyϕyyV

CD
y

v7
−

2ϕyV
CD
y V CD

yy

v7
−

3ϕyy(V
CD
y )2

v7
+

6ϕ4y
v8

+
18ϕ3yV

CD
y

v8
+

18ϕ2y(V
CD
y )2

v8
+

6ϕy(V
CD
y )3

v8
.

Then integrating (3.15) with respect to τ and y over [τ0, τ ]×R (τ ≤ τ2) and using (1.3) and (3.1)
yield

∥ϕy(τ)∥2 +
∫ τ

τ0

(∥ϕy∥2 + ∥ϕyy∥2)dτ

≤ C
(
∥ψ(τ)∥2 +

∫ τ

τ0

∥ψy∥2dτ +
5∑

i=1

∫ τ

τ0

∫
R
Iidydτ

)
.

(3.16)

Here we used the inequality∫
R
(ψ
ϕy
v
)(τ) dy ≤ 1

2

∫
R
ϕ2y(τ) dy + C

∫
R
ψ2(τ) dy,

and Ii(i = 1, . . . , 5) are given by

I1 = |ζyϕy|, I2 = ϕ4y + ϕ2y|ϕyy|,
I3 = |V CD

yy ϕyy|+ |V CD
y V CD

yy ϕy|+ |(V CD
y )2ϕyy|+ |(V CD

y )3ϕy|+ |R1ϕy|,
I4 = |V CD

y ψψy|+ |UCD
y ψϕy|+ |ΘCD

y ϕϕy|+ |V CD
y (ζ + ϕ)ϕy|+ |UCD

yy ϕϕy|+ |UCD
y V CD

y ϕϕy|,
I5 = |V CD

y ψyϕy|+ |V CD
yy ϕ2y|+ |V CD

y ϕyϕyy|+ |(V CD
y )2ϕ2y|+ |V CD

y ϕ3y|.

Now we estimate each term on the right-hand side of (3.16). First, using Hölder inequality and
Young inequality, it is easy to obtain∫ τ

τ0

∫
R
I1dydτ ≤ 1

8

∫ τ

τ0

∥ϕy∥2 dτ + C

∫ τ

τ0

∥ζy∥2 dτ. (3.17)

By Hölder inequality, Sobolev inequality, Young inequality and (2.4), it holds that∫ τ

τ0

∫
R
I2dydτ ≤ C

∫ τ

τ0

∥ϕy∥2L∞∥ϕy∥2 dτ + C

∫ τ

τ0

∥ϕy∥L∞∥ϕy∥∥ϕyy∥dτ

≤ C(η1 + η21)

∫ τ

τ0

∥ϕy∥2 dτ + Cη1

∫ τ

τ0

∥ϕyy∥2 dτ.
(3.18)

Moreover, as (3.6) and (3.8), we have∫ τ

τ0

∫
R
I3dydτ ≤ 1

8

∫ τ

τ0

∥ϕyy∥2 dτ + ϵ

∫ τ

τ0

∥ϕy∥2 dτ + Cϵ1/2, (3.19)

and ∫ τ

τ0

∫
R
I4dydτ ≤ C

(
ϵ1/2 + ϵ+ ϵ3/2

) ∫ τ

τ0

(∥ϕ∥2 + ∥ϕy∥2) dτ + C
(
ϵ1/2 + ϵ

) ∫ τ

τ0

∥ψ∥2 dτ

+ Cϵ1/2
∫ τ

τ0

(∥ζ∥2 + ∥ψy∥2) dτ.
(3.20)

Finally, similar to (3.8) and (3.12), one obtains∫ τ

τ0

∫
R
I5dydτ

≤ Cϵ1/2
∫ τ

τ0

∥ψy∥2 dτ + C
(
ϵ1/2 + ϵ+ ϵ1/2η1

) ∫ τ

τ0

∥ϕy∥2 ,dτ + Cϵ1/2
∫ τ

τ0

∥ϕyy∥2 dτ.
(3.21)



10 J. CHEN, Y. LI, R. YIN EJDE-2025/74

Then substituting estimates (3.17)-(3.21) into (3.16) and noting that ϵ and η1 are suitably small,
we have

∥ϕy(τ)∥2 +
∫ τ

τ0

(∥ϕy∥2 + ∥ϕyy∥2)dτ

≤ C
(
∥ψ(τ)∥2 +

∫ τ

τ0

(∥ψy∥2 + ∥ζy∥2)dτ + (ϵ1/2 + ϵ+ ϵ3/2)

∫ τ

τ0

∥ϕ∥2 dτ

+
(
ϵ1/2 + ϵ

) ∫ τ

τ0

∥ψ∥2 dτ + ϵ1/2
∫ τ

τ0

∥ζ∥2 dτ + ϵ1/2
)
,

which together with (3.14) and the small behavior of ϵ and η1 yields

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ

τ0

∥(ϕy, ψy, ζy, ϕyy)∥2 dτ

≤ C
(
ϵ1/2 + ϵ+ ϵ3/2 + ϵ2

) ∫ τ

τ0

∥(ϕ, ζ)∥2 dτ + C
(
ϵ1/2 + ϵ

) ∫ τ

τ0

∥ψ∥2 dτ + C
(
ϵ1/2 + ϵ3/2

)
.

(3.22)

Then we can conclude from (3.22) by using the classical Gronwall inequality for τ ∈ [τ0, τ2] and
the smallness of ϵ,

∥(ϕ, ψ, ζ, ϕy)(τ)∥2 +
∫ τ

τ0

∥(ϕy, ψy, ζy, ϕyy)∥2 dτ ≤ Cϵ1/2,

which is (3.4) in Lemma 3.1. This completes the proof. □

Next, we derive the derivative estimate. For this, let us multiply (1.13)2 by −ψyy and ∂y(1.13)3
by

ζy
ΘCD . We have(1
2
ψ2
y +

RΘCD

2(V CD)2
ϕ2y +

λ

2v5
ϕ2yy

)
τ
+
ψ2
yy

v
+
(UCD

yy ψy

v
− ψτψy −

(
PCD
vy ϕ

+ PCD
θy ζ

)
ψy −

UCD
y V CD

y ψy

v2
−
UCD
yy ψy

V CD
+
UCD
y V CD

y ψy

(V CD)2
+ λ

(ψyy

v2
(
1

v
(
1

v
)y)y

+
ψyyV

CD
yy

v5
−

3ψyyv
2
y

v6
+

10V CD
y V CD

yy ψy

v6
−

15(V CD
y )3ψy

v7
−
V CD
yyy ψy

v5
))

y

=
(Rθ
v

− RΘCD

V CD
+

RΘCD

(V CD)2
ϕ− R

V CD
ζ
)
y
ψyy +

(( RΘCD

(V CD)2
)
y
ϕ−

( R

V CD

)
y
ζ
)
y
ψy

+
1

2

( RΘCD

(V CD)2
)
τ
ϕ2y +

R

V CD
ζyψyy +

ψyϕyψyy

v2
+
ψyψyyV

CD
y

v2

+
ϕyψyyU

CD
y

v2
+
(UCD

yy

v
−
UCD
y V CD

y

v2
)
y
ψy −

5λ

2v6
ϕ2yy(ψy + UCD

y )

− λ
(5(ϕyϕyy + 2ϕyV

CD
yy + ϕyyV

CD
y )

v6
−

15(ϕ3y + 3ϕ2yV
CD
y + 3ϕy(V

CD
y )2)

v7
)
ψyy

+ λ
(10V CD

y V CD
yy

v6
−

15(V CD
y )3

v7
−
V CD
yyy

v5
)
y
ψy −

(UCD
yy

V CD
−
UCD
y V CD

y

(V CD)2
)
y
ψy +R1ψyy,

(3.23)

and ( ζ2y
2ΘCD

)
τ
+

ν

vΘCD
ζ2yy +

( Rθ

vΘCD
ψyζy −

R

v
ψyζy −

νζy
ΘCD

(
θy
v
)y

+
ν(ΘCD

yy v −ΘCD
y vy)

v2ΘCD
ζy −

u2yζy

vΘCD
+

(UCD
y )2ζy

vΘCD
− λζyuy

ΘCD

(5v2y
2v6

− vyy
v5

)
+
λζyU

CD
y

ΘCD

(5(V CD
y )2

2v6
−
V CD
yy

v5
)
+R2

ζy
ΘCD

)
y

=
1

2

( 1

ΘCD

)
τ
ζ2y +

( 1

ΘCD

)
y
ψyζy

(Rθ
v

− RΘCD

V CD

)
+

1

ΘCD
ψyζyy

(Rθ
v

− RΘCD

V CD

)



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 11

− 1

ΘCD
ψyζy

(RΘCD

V CD

)
y
−
ζyU

CD
y

ΘCD

(Rθ
v

− RΘCD

V CD

)
y
−

(Rθ
v

− RΘCD

V CD

)ζyUCD
yy

ΘCD

+
ν

ΘCD
ζyy

ζyϕy + ζyV
CD
y + ϕyΘ

CD
y

v2
+

(
ν

1

ΘCD

ΘCD
yy v −ΘCD

y V CD
y

v2
)
y
ζy

− ν
( 1

ΘCD

)
y
ζy

(ζyyv − ζyϕy)− (ζyV
CD
y + ϕyΘ

CD
y ) + (ΘCD

yy v −ΘCD
y V CD

y )

v2

+ ν
( 1

ΘCD

)
y
ζy

ΘCD
yy V CD −ΘCD

y V CD
y

(V CD)2
−
(
ν

1

ΘCD

ΘCD
yy V CD −ΘCD

y V CD
y

(V CD)2
)
y
ζy

−
(ψy + UCD

y )2

v

( 1

ΘCD

)
y
ζy +

(UCD
y )2

V CD

( 1

ΘCD

)
y
ζy −

ψ2
y + 2ψyU

CD
y

vΘCD
ζyy

− λ(ψy + UCD
y )

(5(ϕy + V CD
y )2

2v6
−
ϕyy + V CD

yy

v5
)( 1

ΘCD

)
y
ζy

−
( (UCD

y )2

V CDΘCD

)
y
ζy −

λζyyψy

ΘCD

(5(ϕy + V CD
y )2

2v6
−
ϕyy + V CD

yy

v5
)

+ λ
((5(V CD

y )2

2v6
−
V CD
yy

v5
)UCD

y

ΘCD

)
y
ζy −

R

V CD
ψyyζy +

( (UCD
y )2

vΘCD

)
y
ζy

−
λζyyU

CD
y

ΘCD

(5(ϕ2y + 2ϕyV
CD
y )

2v6
− ϕyy

v5
)
+R2

( ζy
ΘCD

)
y
. (3.24)

Lemma 3.2. Suppose that the assumptions in Proposition 2.1 hold. Then it holds that

sup
τ0≤τ≤τ2

∥(ϕy, ψy, ζy, ϕyy)(τ)∥2 +
∫ τ2

τ0

∥(ψyy, ζyy)∥2 dτ

≤ C
(
ϵ1/2 + ϵ

) ∫ τ2

τ0

∥ϕyy∥2 dτ + C
(
η21 + ϵ

1
4 + ϵ

) ∫ τ2

τ0

∥ϕyyy∥2 dτ + Cϵ
2
3 .

(3.25)

Proof. Taking a suitable linear combination of (3.23)-(3.24) and integrating the resultant equation
with respect to τ and y over [τ0, τ ]× R (τ ≤ τ2), we obtain

∥(ϕy, ψy, ζy, ϕyy)(τ)∥2 +
∫ τ

τ0

∥(ψyy, ζyy)∥2 dτ ≤ C

6∑
i=1

∫ τ

τ0

∫
R
Ji dydτ, (3.26)

where

J1 = |ϕϕyψyy|+ |ζζyψyy|+ |ζψyζyy|+ |ϕψyζyy|+ |ζyϕyζyy|+ |ϕyψyψyy|
+ |ψ2

yζyy|+ |ψyϕyyζyy|+ |ϕ3yψyy|+ |ψyϕ
2
yζyy|,

J2 = |ϕyψy(Θ
CD
y + V CD

y )|+ |ζyψy(V
CD
y +ΘCD

y )|+ |ϕ2yUCD
y |+ |ζ2yUCD

y |
+ |ζyϕyUCD

y |+ |ϕyψy(U
CD
y V CD

y + UCD
yy )|+ |ζ2yΘCD

y V CD
y |

+ |ζyϕy((ΘCD
y )2 +ΘCD

yy +ΘCD
y V CD

y + (UCD
y )2)|+ |ψyζyU

CD
y ΘCD

y |
+ |ϕyψy(V

CD
y V CD

yy + (V CD
y )3 + V CD

yyy )|+ |ψyζyΘ
CD
y ((V CD

y )2 + V CD
yy )|

+ |ϕyζy(UCD
y V CD

y ΘCD
y + UCD

y (V CD
y )2 + UCD

y V CD
yy )|,

J3 = |ϕψy(V
CD
y ΘCD

y +ΘCD
yy + (V CD

y )2 + V CD
yy )|+ |ζψy((V

CD
y )2 + V CD

yy )|
+ |(ϕ+ ζ)ζyU

CD
yy |+ |ϕψy(V

CD
y UCD

yy + UCD
yyy + (V CD

y )2UCD
y + V CD

yy UCD
y )|

+ |ϕζy(ΘCD
y (ΘCD

yy + V CD
yy + (V CD

y )2 + (UCD
y )2) + (ΘCD

y )2V CD
y +ΘCD

yyy

+ΘCD
yy V CD

y + V CD
y (UCD

y )2 + UCD
y UCD

yy )|,
J4 = |ϕyψyyU

CD
y |+ |ϕyψyy(V

CD
yy + (V CD

y )2)|+ |ζyζyy(ΘCD
y + V CD

y )|
+ |ϕyζyyΘCD

y |+ |ψyζyyU
CD
y |+ |ϕyyζyUCD

y ΘCD
y |+ |ζyyψy((V

CD
y )2 + V CD

yy )|
+ |ϕyζyyUCD

y V CD
y |+ |ζyϕyyyV CD

y |+ |ϕyyζyyUCD
y |,



12 J. CHEN, Y. LI, R. YIN EJDE-2025/74

J5 = |ψy((V
CD
y )2V CD

yy + (V CD
yy )2 + V CD

y V CD
yyy + (V CD

y )4 + V CD
yyyy)|+ |R1ψyy|

+ |ζy(UCD
y ((V CD

y )2ΘCD
y + V CD

yy ΘCD
y + (V CD

y )3 + V CD
y V CD

yy + V CD
yyy )

+ UCD
yy (V CD

y )2 + UCD
yy V CD

yy )|+ |R2ζyy|+ |R2Θ
CD
y ζy|,

J6 = |ϕ2yψyyV
CD
y |+ |ζ2yϕyΘCD

y |+ |ψ2
yζyΘ

CD
y |+ |ϕ2yζyyUCD

y |+ |ϕ2yζyUCD
y ΘCD

y |
+ |ϕ2yψyζyΘ

CD
y |+ |ψyϕyyζyΘ

CD
y |+ |ψyϕyζyV

CD
y ΘCD

y |+ |ζyyψyϕyV
CD
y |

+ |(ζy + ϕy)ψyζyΘ
CD
y |.

Now let us estimate each term on the right-hand side of (3.26). First, using Young inequality,
Sobolev inequality (2.4) and (3.4), one obtains∫ τ

τ0

∫
R
(|ζψyζyy|+ |ζyϕyζyy|) dydτ

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ + C

∫ τ

τ0

(∥ζ∥2L∞∥ψy∥2 + ∥ϕy∥2L∞∥ζy∥2) dτ

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ + C

∫ τ

τ0

(∥ζ∥∥ζy∥∥ψy∥2 + ∥ϕy∥∥ϕyy∥∥ζy∥2) dτ

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ + C sup

τ0≤τ≤τ
∥ζ∥∥ζy∥

∫ τ

τ0

∥ψy∥2 dτ

+ C sup
τ0≤τ≤τ

∥ϕy∥∥ϕyy∥
∫ τ

τ0

∥ζy∥2 dτ

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ +

1

16
sup

τ0≤τ≤τ
∥ζy∥2 + C sup

τ0≤τ≤τ
∥ζ∥2

( ∫ τ

τ0

∥ψy∥2 dτ
)2

+
1

48
sup

τ0≤τ≤τ
∥ϕyy∥2 + C sup

τ0≤τ≤τ
∥ϕy∥2

( ∫ τ

τ0

∥ζy∥2 dτ
)2

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ +

1

16
sup

τ0≤τ≤τ
∥ζy∥2 +

1

48
sup

τ0≤τ≤τ
∥ϕyy∥2 + Cϵ3/2,

(3.27)

and ∫ τ

τ0

∫
R
(|ψ2

yζyy|+ |ψyϕyyζyy|) dydτ

≤ 1

24

∫ τ

τ0

∫
R
ζ2yy dydτ + C

∫ τ

τ0

(∥ψy∥2L∞ + ∥ϕyy∥2L∞)∥ψy∥2 dτ

≤ 1

24

∫ τ

τ0

∥ζyy∥2 dτ + C

∫ τ

τ0

(∥ψy∥∥ψyy∥+ ∥ϕyy∥∥ϕyyy∥)∥ψy∥2 dτ

≤ 1

24

∫ τ

τ0

∥ζyy∥2 dτ +
1

88

∫ τ

τ0

∥ψyy∥2 dτ + C

∫ τ

τ0

∥ψy∥2∥ψy∥4 dτ

+ C

∫ τ

τ0

(∥ϕyy∥2 + ∥ϕyyy∥2)∥ψy∥2 dτ

≤ 1

24

∫ τ

τ0

∥ζyy∥2 dτ +
1

88

∫ τ

τ0

∥ψyy∥2 dτ + C sup
τ0≤τ≤τ

∥ψy∥2η21
∫ τ

τ0

∥ψy∥2 dτ

+ C sup
τ0≤τ≤τ

∥ϕyy∥2
∫ τ

τ0

∥ψy∥2 dτ + C sup
τ0≤τ≤τ

∥ψy∥2
∫ τ

τ0

∥ϕyyy∥2 dτ

≤ 1

24

∫ τ

τ0

∥ζyy∥2 dτ +
1

88

∫ τ

τ0

∥ψyy∥2 dτ + Cη21ϵ
1/2 sup

τ0≤τ≤τ
∥ψy∥2

+ Cϵ1/2 sup
τ0≤τ≤τ

∥ϕyy∥2 + Cη21

∫ τ

τ0

∥ϕyyy∥2 dτ.



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 13

In the same way, we can deal with the remainder terms in
∫ τ

τ0

∫
R J1 dydτ . Then we have∫ τ

τ0

∫
R
J1 dydτ ≤ 1

8

∫ τ

τ0

(∥ψyy∥2 + ∥ζyy∥2) dτ + Cη21

∫ τ

τ0

∥ϕyyy∥2 dτ

+
(1
8
+ Cη21ϵ

1/2
)

sup
τ0≤τ≤τ

∥ϕy∥2 +
1

8
sup

τ0≤τ≤τ
∥ζy∥2

+
(1
8
+ Cϵ1/2

)
sup

τ0≤τ≤τ
∥ϕyy∥2 + Cη21ϵ

1/2 sup
τ0≤τ≤τ

∥ψy∥2 + Cϵ3/2.

(3.28)

Next, employing (3.4) and using a similar process to (3.8), we have∫ τ

τ0

∫
R
J2 dydτ ≤ C

(
ϵ+ ϵ3/2 + ϵ2 + ϵ5/2

)
. (3.29)

By (2.3), Hölder inequality, Sobolev inequality, Young inequality and (3.4), we obtain∫ τ

τ0

∫
R
|V CD

y ΘCD
y ϕψy|dydτ

≤ C

∫ τ

τ0

∫
R
|ϵe−

2c0y2

τ ϕψy|dydτ

≤ C

∫ τ

τ0

ϵ∥ϕ∥∥ψy∥L4τ1/8 dτ

≤ C

∫ τ

τ0

ϵ∥ϕ∥∥ψy∥1/4∥ψyy∥3/4τ1/8 dτ

≤ 1

24

∫ τ

τ0

∥ψyy∥2 dτ + C

∫ τ

τ0

∥ϕ∥8/5∥ψy∥2/5ϵ8/5τ1/5 dτ

≤ 1

24

∫ τ

τ0

∥ψyy∥2 dτ + C

∫ τ

τ0

∥ϕ∥2ϵ11/8τ5/24 dτ + C

∫ τ

τ0

∥ψy∥2ϵ5/2τ1/6 dτ

≤ 1

24

∫ τ

τ0

∥ψyy∥2 dτ + C sup
τ0≤τ≤τ

∥ϕ∥2ϵ 11
8 ϵ−29/24 + C sup

τ0≤τ≤τ
∥ψy∥2ϵ5/2ϵ−7/6

≤ 1

24

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ1/6 sup
τ0≤τ≤τ

∥ϕ∥2 + Cϵ4/3 sup
τ0≤τ≤τ

∥ψy∥2

≤ 1

24

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ4/3 sup
τ0≤τ≤τ

∥ψy∥2 + Cϵ2/3.

We can similarly estimate the other terms in
∫ τ

τ0

∫
R J3 dydτ to obtain∫ τ

τ0

∫
R
J3 dydτ ≤ 1

8

∫ τ

τ0

(∥ψyy∥2 + ∥ζyy∥2) dτ + Cϵ

∫ τ

τ0

∥ϕyy∥2 dτ

+ Cϵ17/6 sup
τ0≤τ≤τ

∥ζy∥2 + C
(
ϵ4/3 + ϵ13/3

)
sup

τ0≤τ≤τ
∥ψy∥2

+ Cϵ17/6 sup
τ0≤τ≤τ

∥ϕy∥2 + C
(
ϵ2/3 + ϵ

31
24 + ϵ23/12 + ϵ61/24

)
.

(3.30)

Moreover, utilizing (2.3), Hölder inequality, Sobolev inequality and Young inequality, one obtains∫ τ

τ0

∫
R
|UCD

y ϕyψyy|dydτ ≤ C

∫ τ

τ0

∫
R
|ϵe−c0y

2/τϕyψyy|dydτ

≤ C

∫ τ

τ0

ϵ∥ψyy∥∥ϕy∥L4τ1/8 dτ

≤ C

∫ τ

τ0

ϵ∥ψyy∥∥ϕy∥1/4∥ϕyy∥3/4τ1/8 dτ

≤ 1

64

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ2
∫ τ

τ0

∥ϕy∥1/2∥ϕyy∥3/2τ1/4 dτ



14 J. CHEN, Y. LI, R. YIN EJDE-2025/74

≤ 1

64

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ

∫ τ

τ0

∥ϕyy∥2 dτ + C

∫ τ

τ0

ϵ5∥ϕy∥2τ dτ

≤ 1

64

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ

∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ3 sup
τ0≤τ≤τ

∥ϕy∥2.

Similarly, we estimate the other terms in
∫ τ

τ0

∫
R J4 dydτ to obtain∫ τ

τ0

∫
R
J4 dydτ ≤ 1

8

∫ τ

τ0

(∥ψyy∥2 + ∥ζyy∥2) dτ + C(ϵ1/2 + ϵ)

∫ τ

τ0

∥ϕyy∥2 dτ

+ C(ϵ
1
4 + ϵ)

∫ τ

τ0

∥ϕyyy∥2 dτ + C
(
ϵ1/2 + ϵ3 + ϵ7

)
sup

τ0≤τ≤τ
∥ϕy∥2

+ Cϵ6 sup
τ0≤τ≤τ

∥ψy∥2 + Cϵ3 sup
τ0≤τ≤τ

∥ϕyy∥2

+ C
(
ϵ+ ϵ2 + ϵ6

)
sup

τ0≤τ≤τ
∥ζy∥2.

(3.31)

With (2.3), Hölder inequality and Young inequality in hand, we have∫ τ

τ0

∫
R
(|(V CD

y )2V CD
yy ψy|+ |R1ψyy|) dydτ

≤ C

∫ τ

τ0

∫
R
(|ϵ2e−

3c0y2

τ ψy|+ |ϵ3/2e−c0y
2/τψyy|) dydτ

≤ C

∫ τ

τ0

ϵ2∥ψy∥τ1/4 dτ +
1

8

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ3
∫ τ

τ0

∫
R
e−

2c0y2

τ dydτ

≤ Cϵ2 sup
τ0≤τ≤τ

∥ψy∥
∫ τ

τ0

τ1/4 dτ +
1

8

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ3
∫ τ

τ0

τ1/2 dτ

≤ Cϵ3/4 sup
τ0≤τ≤τ

∥ψy∥+
1

8

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ
3/2

≤ 1

8
sup

τ0≤τ≤τ
∥ψy∥2 +

1

8

∫ τ

τ0

∥ψyy∥2 dτ + Cϵ3/2.

Hence, we obtain∫ τ

τ0

∫
R
J5 dydτ

≤ 1

8

∫ τ

τ0

(∥ψyy∥2 + ∥ζyy∥2) dτ +
1

8
sup

τ0≤τ≤τ
∥ψy∥2 +

1

8
sup

τ0≤τ≤τ
∥ζy∥2 + C(ϵ3/2 + ϵ5/2).

(3.32)

By (2.3), Hölder inequality, Sobolev inequality, Young inequality and (3.4), we have∫ τ

τ0

∫
R
|UCD

y ϕ2yζyy|dydτ ≤ C

∫ τ

τ0

∫
R

∣∣ϵe−c0y
2/τϕ2yζyy

∣∣dydτ
≤ Cϵ

∫ τ

τ0

∥ϕy∥2L∞∥ζyy∥τ1/4 dτ

≤ Cϵ

∫ τ

τ0

∥ϕy∥∥ϕyy∥∥ζyy∥τ1/4 dτ

≤ ϵ1/2
∫ τ

τ0

∥ζyy∥2 dτ + Cϵ3/2
∫ τ

τ0

∥ϕy∥2∥ϕyy∥2τ1/2 dτ

≤ ϵ1/2
∫ τ

τ0

∥ζyy∥2 dτ + C sup
τ0≤τ≤τ

∥ϕy∥2∥ϕyy∥2

≤ ϵ1/2
∫ τ

τ0

∥ζyy∥2 dτ + Cϵ1/2 sup
τ0≤τ≤τ

∥ϕyy∥2.

(3.33)



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 15

Noting e−
nc0y2

τ ≤ 1, for n ∈ N+, similar to (3.28) and (3.33), one obtains∫ τ

τ0

∫
R
J6 dydτ ≤ C

(
ϵ1/2 + ϵ

) ∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ1/2
∫ τ

τ0

∥ζyy∥2 dτ + C
(
ϵ+ ϵ2

)
+
(1
8
+ Cϵ1/2

) ∫ τ

τ0

∥ψyy∥2 dτ + C
(
ϵ1/2 + ϵ+ ϵ3/2

)
sup

τ0≤τ≤τ
∥ϕyy∥2

+ C
(
ϵ1/2 + ϵ1/4η21 + ϵ

)
sup

τ0≤τ≤τ
∥ζy∥2 + C

(
ϵ1/2 + ϵ+ ϵ3/2

)
sup

τ0≤τ≤τ
∥ψy∥2.

(3.34)

Substituting these estimates (3.28)-(3.32) and (3.34) into (3.26) and noting that ϵ and η1 are
suitably small that we can prove (3.25) for τ ∈ [τ0, τ2] in Lemma 3.2. This completes the proof. □

Finally, we deal with the estimations for
∫ τ

τ0
∥ϕyy∥2 dτ and

∫ τ

τ0
∥ϕyyy∥2 dτ . To this end, we

multiply ∂y (1.13)2 by
ϕyy

v to have

(ϕ2yy
2v2

− ψyϕyy
v

)
τ
+

λ

v6
ϕ2yyy −

1

v
ψ2
yy +

Rθ

v3
ϕ2yy −

R

v2
ζyyϕyy

+
(UCD

yyy ϕy

v2
−

2UCD
yy V CD

y ϕy

v3
−
UCD
y V CD

yy ϕy

v3
+

2UCD
y (V CD

y )2ϕy

v4

−
UCD
yyy ϕy

vV CD
+

2UCD
yy V CD

y ϕy

v(V CD)2
+
UCD
y V CD

yy ϕy

v(V CD)2
−

2UCD
y (V CD

y )2ϕy

v(V CD)3

+ λ
(ϕyy
v2

(
1

v
(
1

v
(
1

v
)y)y)y +

V CD
yyy ϕyy

v6
−

10V CD
y V CD

yy ϕyy

v7
+

15(V CD
y )3ϕyy

v8

−
V CD
y V CD

yyy ϕy

v7
+

10(V CD
y )2V CD

yy ϕy

v8
−

15(V CD
y )4ϕy

v9
)
+
ψyψyy

v

−
(Rθ
v

− RΘCD

V CD

)
y

ϕyy
v

− Rθϕyϕyy
v3

+
Rζyϕyy
v2

−R1
ϕyy
v

)
y

=
1

2

( 1

v2
)
τ
ϕ2yy +

(UCD
yyy

v2
−

2UCD
yy V CD

y

v3
−
UCD
y V CD

yy

v3
+

2UCD
y (V CD

y )2

v4
)
y
ϕy

−
( UCD

yyy

vV CD
−

2UCD
yy V CD

y

v(V CD)2
−
UCD
y V CD

yy

v(V CD)2
+

2UCD
y (V CD

y )2

v(V CD)3
)
y
ϕy

+
2ϕyy(ϕy(ψyy + UCD

yy ) + ψyyV
CD
y )

v3
+

(ψy + UCD
y )ϕ2yy + ψyϕyyV

CD
yy

v3

−
2ψy(ϕy + V CD

y )2ϕyy + 2UCD
y (ϕ2y + 2ϕyV

CD
y )ϕyy

v4

+ λ
(V CD

yyy

v6
−

10V CD
y V CD

yy

v7
+

15(V CD
y )3

v8
)
y
ϕyy

− λ
(V CD

y V CD
yyy

v7
−

10(V CD
y )2V CD

yy

v8
+

15(V CD
y )4

v9
)
y
ϕy

+
λ

v

(10(V CD
y + ϕy)ϕyy + 10ϕyV

CD
yy

v6

−
15(ϕ3y + 3ϕ2yV

CD
y + 3ϕy(V

CD
y )2)

v7
)
ϕyyy

+
λ

v2

(ϕyyy + V CD
yyy

v5
−

10(ϕy + V CD
y )(ϕyy + V CD

yy )

v6

+
15(ϕy + V CD

y )3

v7

)
ϕyϕyy −

(1
v

)
τ
ψyϕyy +

(1
v

)
y
ψyψyy

+
λV CD

y

v2
(ϕyyy
v5

−
10(ϕyϕyy + ϕyV

CD
yy + ϕyy)

v6



16 J. CHEN, Y. LI, R. YIN EJDE-2025/74

+
15(ϕ3y + 3ϕ2yV

CD
y + 3ϕy(V

CD
y )2)

v7
)
ϕyy +

R

v2V CD
ϕΘCD

y ϕyyy

−
(Rθ
v

− RΘCD

V CD

)
y

(1
v

)
y
ϕyy −

((Rθ
v3

)
y
ϕy −

(R
v2

)
y
ζy
)
ϕyy

+
R

v3
ζϕyyyV

CD
y − RΘCD(v + V CD)

v3(V CD)2
ϕϕyyyV

CD
y −R1

(ϕyy
v

)
y
. (3.35)

Lemma 3.3. Suppose that the assumptions in Proposition 2.1 hold. Then it holds that

sup
τ0≤τ≤τ2

∥ϕyy(τ)∥2 +
∫ τ2

τ0

∥(ϕyy, ϕyyy)∥2 dτ

≤ C sup
τ0≤τ≤τ2

∥(ϕy, ψy, ζy)(τ)∥2 + C

∫ τ2

τ0

∥(ψyy, ζyy)∥2 dτ + Cϵ.

(3.36)

Proof. Integrating (3.35) with respect to τ and y over [τ0, τ ]× R (τ ≤ τ2), we have

∥ϕyy(τ)∥2 +
∫ τ

τ0

∥(ϕyy, ϕyyy)∥2 dτ

≤ C
(
∥ψy(τ)∥2 +

∫ τ

τ0

∥(ψyy, ζyy)∥2 dτ +
7∑

i=1

∫ τ

τ0

∫
R
Ki dydτ

)
,

(3.37)

where

K1 = |ϕyϕyyψyy|+ |ϕyϕyyϕyyy|+ |ϕ2yϕyy|+ |ϕyζyϕyy|+ |ϕyψyψyy|
+ |ϕ2yyψy|+ |ψ2

yϕyy|+ |ϕ3yϕyyy|+ |ψyϕ
2
yϕyy|+ |ϕ2yϕ2yy|+ |ϕ4yϕyy|,

K2 = |ϕ2y(UCD
yyy + UCD

yy V CD
y + UCD

y V CD
yy + UCD

y (V CD
y )2)|

+ |ϕ2y(V CD
y V CD

yyy + (V CD
y )2V CD

yy + (V CD
y )4)|,

K3 = |ϕϕy(V CD
y (UCD

yyy + UCD
y V CD

yy + UCD
y (V CD

y )2) + (V CD
y )2UCD

yy + UCD
yyyy

+ UCD
yy V CD

yy + UCD
y V CD

yyy )|,
K4 = |ψyψyyV

CD
y |+ |ψyϕyy(V

CD
yy + (V CD

y )2)|+ |ϕyϕyy(UCD
y V CD

y + UCD
yy )|

+ |ϕyϕyy(V CD
yyy + V CD

y V CD
yy + (V CD

y )3)|+ |ϕyϕyyy(V CD
yy + (V CD

y )2)|
+ |ϕyyϕyyyV CD

y |+ |ψyϕyyU
CD
y |+ |ϕyϕyy(V CD

y +ΘCD
y )|

+ |ζyϕyyV CD
y |+ |ϕyyψyyV

CD
y |+ |ϕ2yyUCD

y |+ |ϕ2yy(V CD
y )2|

+ |ϕyϕyyy(V CD
y +ΘCD

y )|+ |R1ϕyϕyy|,
K5 = |ϕyy(V CD

y V CD
yyy + (V CD

y )2V CD
yy + (V CD

y )4 + V CD
yyyy + (V CD

yy )2)|
+ |ϕy((V CD

y )2V CD
yyy + V CD

y V CD
yyyy + V CD

yy V CD
yyy + (V CD

y )3V CD
yy

+ V CD
y (V CD

yy )2 + (V CD
y )5)|+ |R1ϕyyy|+ |R1V

CD
y ϕyy|,

K6 = |ϕ2yϕyyyV CD
y |+ |ϕyϕ2yyV CD

y |+ |ϕ2yϕyy(V CD
yy + (V CD

y )2)|
+ |ϕ2yϕyyUCD

y |+ |ψyϕyϕyyV
CD
y |+ |ϕ3yϕyyV CD

y |,
K7 = |ϕϕ2y(UCD

yyy + UCD
yy V CD

y + UCD
y V CD

yy + UCD
y (V CD

y )2)|.

Firstly, similar to (3.28)-(3.32) and (3.34), one obtains∫ τ

τ0

∫
R
K1 dydτ ≤ 1

8

∫ τ

τ0

(∥ϕyy∥2 + ∥ψyy∥2 + ∥ϕyyy∥2) dτ

+
(1
8
+ Cη41ϵ

1/2
)

sup
τ0≤τ≤τ

∥ϕyy∥2

+ Cη21ϵ
1/2

(
sup

τ0≤τ≤τ
∥ϕy∥2 + sup

τ0≤τ≤τ
∥ψy∥2

)
+ Cϵ3/2,

(3.38)



EJDE-2025/74 SHORT ZERO-VISCOSITY-CAPILLARITY LIMIT 17∫ τ

τ0

∫
R
K2 dydτ ≤ Cϵ5/2, (3.39)∫ τ

τ0

∫
R
K3 dydτ ≤ 1

8

∫ τ

τ0

∥ζyy∥2 dτ + C
(
ϵ

17
6 + ϵ

35
6

)
sup

τ0≤τ≤τ
∥ζy∥2

+ C
(
ϵ

31
24 + ϵ

61
24

)
,

(3.40)

∫ τ

τ0

∫
R
K4 dydτ ≤ 1

8

∫ τ

τ0

(∥ϕyy∥2 + ∥ψyy∥2 + ∥ζyy∥2 + ∥ϕyyy∥2) dτ

+ Cϵ2 sup
τ0≤τ≤τ

∥ζy∥2 + C
(
ϵ2 + ϵ6 + ϵ10

)
sup

τ0≤τ≤τ
∥ϕy∥2

+ C
(
ϵ2 + ϵ6

)
( sup
τ0≤τ≤τ

∥ψy∥2 + sup
τ0≤τ≤τ

∥ϕyy∥2),

(3.41)

∫ τ

τ0

∫
R
K5 dydτ ≤ 1

8

∫ τ

τ0

∥ϕyyy∥2 dτ +
1

8
sup

τ0≤τ≤τ
∥(ϕy, ϕyy)∥2

+ C
(
ϵ3/2 + ϵ5/2

)
,

(3.42)

∫ τ

τ0

∫
R
K6 dydτ ≤ C

(
ϵ1/2 + ϵ+ ϵ1/2η1

) ∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ1/2
∫ τ

τ0

∥ψyy∥2 dτ

+ Cϵ1/2
∫ τ

τ0

∥ϕyyy∥2 dτ + C
(
ϵ1/2 + ϵ

)
sup

τ0≤τ≤τ
∥ϕyy∥2 + C

(
ϵ+ ϵ2

)
.

(3.43)

It follows from (2.3), Hölder inequality, Sobolev inequality, Young inequality and (3.4) that∫ τ

τ0

∫
R
K7 dydτ ≤ C

∫ τ

τ0

∫
R

∣∣ϵ2(e−c0y
2/τ + e−

2c0y2

τ + e−
3c0y2

τ

)
ϕϕ2y

∣∣dydτ
≤ Cϵ2

∫ τ

τ0

∥ϕ∥τ1/4∥ϕy∥2L∞ dτ

≤ Cϵ2
∫ τ

τ0

∥ϕ∥τ1/4∥ϕy∥∥ϕyy∥ dτ

≤ Cϵ2
∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ2
∫ τ

τ0

∥ϕ∥2τ1/2∥ϕy∥2 dτ

≤ Cϵ2
∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ2 sup
τ0≤τ≤τ

∥ϕ∥2 sup
τ0≤τ≤τ

∥ϕy∥2ϵ−
3
2

≤ Cϵ2
∫ τ

τ0

∥ϕyy∥2 dτ + Cϵ3/2. (3.44)

Substituting estimates (3.38)-(3.44) into (3.37) and noting that ϵ and η1 are suitably small, we
can prove (3.36) for τ ∈ [τ0, τ2] in Lemma 3.3. This completes the proof. □

Acknowledgements. The research was supported in part by the National Natural Science Foun-
dation of China (Grant No. 12171258).

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Jiaxue Chen

School of Mathematics and statistics, Nantong University, Nantong 226019, China
Email address: jxchen2025@163.com

Yeping Li

School of Mathematics and statistics, Nantong University, Nantong 226019, China
Email address: ypleemei@aliyun.com

Rong Yin (corresponding author)
School of Mathematics and statistics, Nantong University, Nantong 226019, China

Email address: yin.r@ntu.edu.cn


	1. Introduction
	2. Reformulation of the Problem and Proof of Theorem 1.1
	3. A priori estimate
	Acknowledgements

	References