Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 64, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

OPTIMAL DECAY RATES FOR HIGHER-ORDER DERIVATIVES

OF SOLUTIONS TO 3D COMPRESSIBLE

NAVIER-STOKES-POISSON EQUATIONS

WITH EXTERNAL FORCE

LIUNA QIN, CHANGGUO XIAO, YINGHUI ZHANG

Abstract. We investigate optimal decay rates for higher-order spatial deriva-
tives of solutions to the 3D compressible Navier-Stokes-Poisson equations with

external force. The main novelty of this article is twofold: First, we prove

the first and second order spatial derivatives of the solutions converge to zero
at the L2-rate (1 + t)−5/4, which is faster than the L2-rate (1 + t)−3/4 in

Li-Zhang [15]. Second, for well-chosen initial data, we show the lower optimal

decay rates of the first order spatial derivative of the solutions. Therefore, our
decay rates are optimal in this sense.

1. Introduction

This study concerns the initial value problem of the isentropic Navier-Stokes-
Poisson equations

∂tρ+∇ · (ρu) = 0,

ρ[∂tu+ (u · ∇)u] +∇P (ρ) = ρ∇φ+ µ∆u+ (µ+ ν)∇(∇ · u) + ρF,

∆φ = ρ− ρ̄, lim
|x|→∞

φ(x, t) = 0,

(ρ, u)(x, 0) = (ρ0, u0)(x).

(1.1)

Here the time variable is t ≥ 0, and the spatial coordinate is x is in R3. The
unknown functions are the density ρ > 0, the velocity u, and the electrostatic
potential φ. ρ̄ > 0 stands for the constant background doping profile. The constants
µ and ν are the viscosity coefficients satisfying µ > 0 and 2µ + 3ν ≥ 0. F (x) =
(F1(x), F2(x), F3(x)) is a given external force, here, for simplicity, we assume that
F = −∇ψ(x). P = P (ρ) is the pressure. In this paper, we always assume that
P = P (ρ) is a C2-function in the neighborhood of ρ̄ and satisfies P ′(ρ) > 0 for
ρ > 0. The typical example is P (ρ) = Aργ corresponding to polytropic (γ > 1)
and isothermal fluid (γ = 1). The main purpose of this article is to show optimal
decay rates for higher-order spatial derivatives of solutions for (1.1) for the initial

2020 Mathematics Subject Classification. 76B03, 35Q35.
Key words and phrases. Compressible Navier-Stokes-Poisson system; external force;

higher-order derivative.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted August 1, 2021. Published September 7, 2022.

1



2 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

data around stationary solutions. And the stationary problem takes the form

∇ · (ρ̃ũ) = 0,

ρ̃(ũ · ∇)ũ+∇P (ρ̃) = ρ̃∇φ̃− ρ̃∇ψ + µ∆ũ+ (µ+ ν)∇(∇ · ũ),

∆φ̃ = ρ̃− ρ̄,
ρ̃→ ρ̄, ũ→ 0 as |x| → ∞.

(1.2)

1.1. History of the problem. Before stating our main results, let’s briefly review
some former results which are closely related. Many mathematicians are interested
in studying the large time behavior of solutions and the global well-posedness for
the compressible Navier-Stokes-Poisson system, see, for example [2, 3, 4, 7, 9, 11,
13, 12, 10, 14, 15, 17, 19, 20, 23, 21, 26, 31, 30] and references therein. In the
following, we only review some results about the decay rates for the compressible
Navier-Stokes-Poisson system with and without the external potential force.

When there is no external force, Li-Matsumura-Zhang [11] proved that the den-
sity of the NSP system converges to its equilibrium state at the L2-rate (1+ t)−3/4,

but the momentum of the NSP system decays at the L2-rate (1 + t)−
1
4 . Later,

they also showed the similar results for the non-isentropic case [13]. Recently,
Wang [21] obtained the optimal decay rates of the higher-order spatial derivatives
of the solution via the pure energy method. Furthermore, Wang and Wang [26]
established global existence and decay estimates of classical solutions to the com-
pressible NSP system in three and higher dimensions, which is faster than ones of
[11, 25]. For the the bipolar Navier-Stokes-Poisson (BNSP) system in 3D, since it
has non-conservative structure and the interaction of two fluids through the elec-
tric field, up to now, there are very few results. By employing a detailed analysis
on Green’s function of the linearized system and some elaborate energy estimates,
Wang-Xu[27] obtained the global existence and the Hs decay rates of classical solu-
tions for the BNSP system. Besides, Wu-Zhang-Zhang [28] established the optimal
decay rates of solutions by a regularity interpolation trick and delicate energy meth-
ods. It should be noted that Chen-Wu-Zhang in [6] showed the explicit influences
of the electric field on the qualitative behaviors of solutions.

When there is an external force F = −∇ψ(x), the problem becomes much more
complicated due to the appearance of the non-trivial stationary solution. For the
BNSP system with external force, Zhao-Li [31] gave the optimal Lp-convergence
rates of the solutions towards the stationary solution. Recently, Li and Zhang [15]
proved the existence of the solution to the stationary problem (1.2) and long time
behavior of the Cauchy problem (1.1). Their main results can be outlined as follows:
There exists ε1 > 0 such that if

‖∆ψ‖H2 +

1∑
k=0

‖(1 + |x|)∇k∆ψ‖ ≤ ε1, (1.3)

then the stationary problem (1.2) has a unique solution (ρ̃, ũ, φ̃)(x) satisfying ũ = 0
and

‖ρ̃− ρ̄‖H4 + ‖∇φ̃‖H3 + ‖(1 + |x|)(ρ̃− ρ̄)‖H3 + ‖(1 + |x|)∇φ̃‖H2 ≤ Cε1. (1.4)

Moreover, if ‖(ρ0− ρ̄, u0)‖H2∩L1 is sufficiently small, then the Cauchy problem (1.1)
admits a unique global solution (ρ, u, φ) satisfying

‖∇(ρ− ρ̃, u,∇φ−∇φ̃)(t)‖H1 ≤ C(1 + t)−3/4. (1.5)



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 3

However, it is clear that the L2 decay rate of the second order spatial derivative
of the solution (ρ, u,∇φ) is (1+ t)−3/4 in (1.5), which is the same as that of its first
order spatial derivative and is slower than that of the heat equation. Therefore,
the decay rate of the second order spatial derivative of the solution (ρ, u,∇φ) is not
optimal in this sense. The main motivation of this paper is to provide a general
framework that can be used to extract the optimal decay rates of the solution to
the Cauchy problem (1.1). More precisely, we will modify the methods developed
in [6, 29] to derive the lower optimal decay rates of the first order spatial derivative
of the solutions.

1.2. Main results. In this article, we use Hk(R3) to denote the usual Sobolev
spaces with norm ‖ · ‖Hk and write ‖ · ‖k := ‖ · ‖Hk for convenience. Generally,
we use Lp (1 ≤ p ≤ ∞) to denote the usual Lp(R3) spaces with norm ‖ · ‖Lp .
The notation a . b means that a ≤ Cb for a universal positive constant which is
independent of time t. For simplicity, we write‖(A,B)‖X := ‖A‖X + ‖B‖X . For
a radial function φ ∈ C∞0 (R3

ξ) such that φ(ξ) = 1 when |ξ| ≤ 1 and φ(ξ) = 0

when |ξ| ≥ 2, we define the low–frequency part and the high-frequency part of f as
follows

fL = F−1[φ(ξ)f̂ ], and fH = F−1[(1− φ(ξ))f̂ ]

Before stating our main results, let us recall the following a priori estimates for
the Cauchy problem (1.1) in [15].

Proposition 1.1. For T > 0, let (ρ − ρ̃, u, φ − φ̃)(x, t) be a solution of (1.1) in
[0, T ] and introduce E(T ) = sup0≤t≤T ‖(ρ − ρ̃, u)(·, t)‖2H2 . Then there exists δ > 0
such that if

E(T ) + ε1 ≤ δ, (1.6)

then the following a-priori estimate holds

‖(ρ− ρ̃, u,∇φ−∇φ̃)(·, t)‖2H2 +

∫ t

0

‖(ρ− ρ̃,∇u,∇2φ−∇2φ̃)(·, s)‖2H2ds

≤ C‖(ρ0 − ρ̃, u0)‖2H2 ,

(1.7)

for any t ∈ [0, T ], where C is a positive constant independent of t.

Now, we are in a position to state our main results.

Theorem 1.2. Let (ρ0 − ρ̄, u0)(x) ∈ H2(R3) ∩ L1(R3), there exists 0 < δ0 < ε1
such that if

‖(ρ0 − ρ̄, u0)‖H2 + ‖∆ψ‖H2 +

1∑
k=0

‖(1 + |x|)∇k∆ψ‖L2 ≤ δ0, (1.8)

then the solution (ρ, u, φ)(x, t) of the Cauchy problem (1.1) has time decay estimate

‖∇(ρ− ρ̃, u,∇φ−∇φ̃)(t)‖H1 ≤ C(1 + t)−5/4, (1.9)

for any t ∈ [0,∞).

Theorem 1.3. Suppose that all the hypotheses of Theorem 1.2 are satisfied, and

the Fourier transform (ρ̂0 − ρ̄, û0, ∇̂φ0) satisfies

|ρ̂0 − ρ̃(ξ)| ≥ Cδ3/20 , |û0(ξ)| = 0, |∇̂φ0(ξ)| ≥ Cδ3/20 , for 0 ≤ |ξ| � 1,



4 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

where c0 is a positive constant. Then if

U0 = ‖(ρ0 − ρ̄, u0,∇φ0)‖L1 ≤ δ0 , (1.10)

then for any t ∈ [0,∞), it holds that

min
{
‖∇(ρ− ρ̃)(t)‖L2 , ‖∇u(t)‖L2 , ‖∇(∇φ−∇φ̃)(t)‖L2

}
≥ Cδ3/20 (1 + t)−5/4.

(1.11)

Remark 1.4. Compared to the decay rate (1.5) of [15], the decay rate (1.9) implies
that the first and second order spatial derivatives of the solution converge to zero
at the L2–rate (1 + t)−5/4, which is faster than the L2–rate (1 + t)−3/4 in (1.5).
Furthermore, the decay rate (1.11) also gives the lower optimal decay rate of the
first order spatial derivative of the solution. So, our decay rates are optimal in this
sense.

Now, let us sketch the strategy of proving Theorem 1.2-Theorem 1.3 and explain
some main difficulties and techniques involved in the process. Roughly speaking,
we will make full use of the benefit of a pure energy method, the low-frequency
and high-frequency decomposition f = fL + fH , where fL and fH stand for the
low-frequency part and high-frequency part of f , respectively. Our strategy can be
outlined as follows.

For the proof of Theorem 1.2, we hope to establish the optimal decay rate for
the first order spatial derivative of solution to the 3D compressible Navier-Stokes-
Poisson equations (1.1). Our strategy mainly involves the following three steps.
Firstly, we deduce the first order low frequency decay estimates including the lin-
ear decay estimates part and the nonlinear energy estimates part. Fortunately, we
can obtain the optimal decay rates on ‖∇k(n̄L, ūL,∇Φ̄L)‖L2 for the corresponding
linearized NSP system with an external force from Chen-Wu-Zhang [6] directly. Un-
fortunately, when the nonlinear terms in (2.2)-(2.5) of equations (2.1) are estimated,
the main difficulty comes from the term involving (ρ̃ − ρ̄), which however has no
specific time decay rate. Our key method to get over this difficulty is to introduce
a time-weighted energy functional M(t) = sup0≤s≤t(1 + s)5/2‖∇(n, u,∇Φ)(s)‖2H1 ,
thus we can deduce the first order low frequency decay estimates as follows (see the
proof of Lemma 3.2 for details).

‖∇(nL, uL, (∇Φ)L)(t)‖2L2 ≤ C(1 + t)−5/2(‖U0‖2L1 + δ20M(t)). (1.12)

Secondly, we deduce the first order and the second order high frequency decay
estimates. In this step, we establish the relevant energy estimates as follows:

1

2

d

dt

(P ′(ρ̄)

ρ̄
‖∇lnH‖2L2 + ρ̄‖∇luH‖2L2 + ‖∇l(∇Φ)H‖2L2

)
+ µ‖∇l+1uH‖2L2 + (µ+ ν)‖∇l∇ · uH‖2L2

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇u‖2H2 + ‖∇n‖2H1),

(1.13)

for l = 1, 2. Note that the energy inequality (1.13) only gives the dissipative
estimate for uH . To look for the dissipative estimates of nH and ∇ΦH , we will use
the benefit of the low-frequency and high-frequency decomposition to employ the
new interactive energy functional to get (see the proof of Lemma 3.4 for details):

d

dt
〈∇l−1uH ,∇lnH〉+

P ′(ρ̄)

2ρ̄
‖∇lnH‖2L2 + ‖∇l(∇Φ)H‖2L2

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇u‖2H2 + ‖∇n‖2H1) + C1‖∇2uH‖2H1 ,

(1.14)



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 5

for l = 1, 2. Thirdly, we prove the upper optimal decay rates of the solutions. We
choose two sufficiently large positive constants D0 and T0, and define the temporal
energy functional

E(t) = D0‖∇(nH , uH ,∇ΦH)‖2H1 +

2∑
l=1

〈∇l−1uH ,∇lnH〉, (1.15)

which is equivalent to ‖∇(nH , uH ,∇ΦH)‖2H1 . From (1.13) and (1.14), we can obtain

d

dt
E(t) + C3E(t) . (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇uL‖2L2 + ‖∇nL‖2L2). (1.16)

By using Gronwall’s inequality and noticing that T0 is large enough and δ0 is
sufficiently small, we can deduce that

E(t) ≤ C(1 + t)−5/2(E(0) + ‖U0‖2L1 + δ20M(t)), (1.17)

which together with low frequency decay rate in (1.12), the definition of M(t)
and the smallness of δ0 implies the key uniform time-independent bound on M(t).
Therefore, this proves (1.9) and completes the proof of Theorem 1.2.

For the proof of Theorem 1.3, we will employ Duhamel’s principle, the lower
decay rates of the linear system in (3.1) and (1.15), and Theorem 1.2 to obtain the
lower optimal decay rate of the solution.

2. Reformulation of original problem

In this section, we reformulate the Cauchy problem (1.1). Let (ρ, u, φ) = (n +

ρ̃, u,Φ + φ̃). Then (1.1) is equivalent to

∂tn+ ρ̄∇ · u = f11 + f12,

∂tu− µ1∆u− µ2∇(∇ · u) +
P ′(ρ̄)

ρ̄
∇n−∇Φ = f21 + f22,

∆Φ = n, lim
|x|→∞

Φ(x, t) = 0,

(n, u)(x, 0) = (n0, u0)(x) = (ρ0 − ρ̃, u0)(x),

(2.1)

where µ1 = µ/ρ̄, µ2 = (µ+ ν)/ρ̄,

f11 = −∇ · ((ρ̃− ρ̄)u), (2.2)

f12 = −∇ · (nu), (2.3)

f21 = −
(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇ρ̃−

(P ′(ρ̃)

ρ̃
− P ′(ρ̄)

ρ̄

)
∇n

+ (ρ̃− ρ̄)(
µ1

ρ̃
∆u+

µ2

ρ̃
∇div u),

(2.4)

f22 = −(u · ∇)u−
(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇n

+
( 1

n+ ρ̃
− 1

ρ̃

)
(µ∆u+ (µ+ ν)∇(∇ · u)),

(2.5)

Notice that

P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃
∼ n, P ′(ρ̃)

ρ̃
− P ′(ρ̄)

ρ̄
∼ ρ̃− ρ̄, 1

n+ ρ̃
− 1

ρ̃
∼ n.



6 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

3. Proof of Theorem 1.2

In this section we shall prove Theorem (1.2) by using good properties of the low-
frequency and high-frequency decomposition, and delicate energy estimates. First,
we recall the L2 time decay rates on the linearized system of (2.1).

Lemma 3.1. Let k be an integer. Assume that (n̄, ū,∇Φ̄) is the solution of the
linearized system of (2.1) with the initial data (n̄0, ū0,∇Φ̄0) ∈ H2(R3) ∩ L1(R3),
then for 1 ≤ k ≤ 2,, it holds that

‖∇k(n̄L, ūL,∇Φ̄L)‖L2 . (1 + t)−
3
4−

k
2 ‖(n̄0, ū0,∇Φ̄0)‖L1 . (3.1)

For a proof of the above lemma, see [6]. Before deriving the L2 time decay rates
on the nonlinear system (2.1), let us define the time-weighted energy functional

M(t) = sup
0≤s≤t

(1 + s)5/2‖∇(n, u,∇Φ)(s)‖2H1 . (3.2)

Notice that M(t) is non-decreasing, and we will deduce the L2 time decay rates on
the low-frequent part of the solution to the nonlinear system (2.1) as follows.

Lemma 3.2. Under the assumptions in Theorem 1.2, the solution U = (n, u,∇Φ)
of the nonlinear system (2.1) satisfies the decay estimate

‖∇(nL, uL, (∇Φ)L)(t)‖2L2 . (1 + t)−5/2(‖U0‖2L1 + δ20M(t)). (3.3)

Proof. To derive the decay on (∇nL,∇uL,∇(∇Φ)L), we need to estimate the non-
linear terms S(t) := (f11, f12, f21, f22)t as follows. By virtue of (1.4), (1.7) , (1.8),
(2.2) -(2.5) ,(3.2), Lemma 5.2, Lemma 5.3, Hölder’s inequality and Hardy inequality,
we have

‖S(t)L‖L1 .‖∇ · [(ρ̃− ρ̄)u]‖L1 + ‖∇ · (nu)‖L1 + ‖n∇(ρ̃− ρ̄)‖L1

+ ‖((ρ̃− ρ̄)∇n, (ρ̃− ρ̄)∆u, (ρ̃− ρ̄)∇(∇ · u))‖L1

+ ‖(u · ∇u, n∇n, n∆u, n∇(∇ · u))‖L1

.‖(1 + |x|)∇(ρ̃− ρ̄)‖L2

∥∥∥ u

1 + |x|

∥∥∥
L2

+ ‖ρ̃− ρ̄‖L2‖∇u‖L2

+
∥∥∥ n

1 + |x|

∥∥∥
L2
‖(1 + |x|)∇(ρ̃− ρ̄)‖L2 + ‖ρ̃− ρ̄‖L2‖(∇n,∆u)‖L2

+ ‖(u, n)‖L2‖(∇u,∇n,∆u)‖L2

.δ0‖(∇u,∇n,∆u)‖L2 + ‖(u, n)‖L2‖(∇u,∇n,∆u)‖L2

.δ0(1 + t)−5/4
√
M(t).

(3.4)

By using equation (2.1)1-(2.1)2, Lemma 3.1, Duhamel’s principle, and Hölder’s
inequality, we have

‖∇(nL, uL, (∇Φ)L)(t)‖L2

≤ C(1 + t)−5/4‖(n0, u0,∇Φ0)‖L1 +

∫ t

0

(1 + t− s)−5/4‖S(s)‖L1ds

≤ C(1 + t)−5/4‖U0‖L1 + Cδ0
√
M(t)

∫ t

0

(1 + t− s)−5/4(1 + s)−5/4ds

≤ C(1 + t)−5/4(‖U0‖L1 + δ0
√
M(t)),

(3.5)

which implies (3.3). �



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 7

Next, we turn to derive the first and second order high-frequency decay estimates
of the solution, which are stated in the following Lemma.

Lemma 3.3. Under the assumption of Theorem 1.2, for l = 1, 2, it holds that

1

2

d

dt

(P ′(ρ̄)

ρ̄
‖∇lnH‖2L2 + ρ̄‖∇luH‖2L2 + ‖∇l(∇Φ)H‖2L2

)
+ µ‖∇l+1uH‖2L2 + (µ+ ν)‖∇l∇ · uH‖2L2

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇u‖2H2 + ‖∇n‖2H1),

(3.6)

for any t ∈ [0,∞).

Proof. For l = 1, 2, by taking

〈∇lF−1[(1− φ(ξ))F(2.1)1],
P ′(ρ̄)

ρ̄
∇lnH〉+ 〈∇lF−1[(1− φ(ξ))F(2.1)2], ρ̄∇luH〉,

and using integration by parts, we obtain

1

2

d

dt

(P ′(ρ̄)

ρ̄
‖∇lnH‖2L2 + ρ̄‖∇luH‖2L2 + ‖∇l(∇Φ)H‖2L2

)
+ µ‖∇l+1uH‖2L2 + (µ+ ν)‖∇l∇ · uH‖2L2

=
P ′(ρ̄)

ρ̄
〈∇lfH11 +∇lfH12,∇lnH〉+ ρ̄〈∇lfH21 +∇lfH22,∇luH〉

− 〈∇lfH11 +∇lfH12,∇lΦH〉 :=

6∑
i=1

Ii.

(3.7)

Next, we shall estimate each term at the right-hand side of (3.7). Firstly, for
term I1, by using (1.4), (2.2), Lemmas 5.2–5.4, Hölder’s and Young inequalities, we
obtain

|I1| = |
P ′(ρ̄)

ρ̄
〈∇l∇ · [(ρ̃− ρ̄)u]H ,∇lnH〉|

. ‖∇l+1[(ρ̃− ρ̄)u]‖L2‖∇lnH‖L2

. (‖ρ̃− ρ̄‖L∞‖∇l+1u‖L2 + ‖∇l+1(ρ̃− ρ̄)‖L2‖u‖L∞)‖∇lnH‖L2

. (‖∇(ρ̃− ρ̄)‖H1‖∇l+1u‖L2 + ‖∇l+1(ρ̃− ρ̄)‖L2‖∇u‖H1)‖∇lnH‖L2

. δ0(‖∇u‖2H2 + ‖∇nH‖2H1).

(3.8)

For the terms I2 , from (2.3), we have

I2 = −P
′(ρ̄)

ρ̄
〈∇l[∇ · (nu)]H ,∇lnH〉

= −P
′(ρ̄)

ρ̄
〈∇l(n∇ · u)H +∇l(u · ∇n)H ,∇lnH〉

:= I2,1 + I2,2.

(3.9)



8 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

For the term I2,1, it follows from the decay rate of (1.5), Lemmas 5.1 and 5.4,
Hölder’s and Young inequalities that

|I2,1|

. ‖∇l(n∇ · u)‖L2‖∇lnH‖L2

. (‖n‖L∞‖∇l+1u‖L2 + ‖∇u‖L∞‖∇ln‖L2)‖∇lnH‖L2

. ‖∇n‖1/2L2 ‖∇2n‖1/2L2 ‖∇l+1u‖L2‖∇ln‖L2

+ ‖∇2u‖1/2L2 ‖∇ln‖1/2L2 ‖∇ln‖3/2L2 ‖∇3u‖1/2L2

. (1 + t)−
3
2 ‖∇l+1u‖L2 + (1 + t)−3/4‖∇ln‖3/2L2 ‖∇3u‖1/2L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇l+1u‖2L2 + (1 + t)−3/4(‖∇ln‖2L2 + ‖∇3u‖2L2)

. (1 + t)−5/2 + (1 + t)−1/2(‖∇n‖2H1 + ‖∇u‖2H2).

(3.10)

For the term I2,2, we first rewrite it as follows

I2,2 = 〈∇l(u · ∇n)H ,∇lnH〉

= 〈∇l(u · ∇n)−∇l(u · ∇n)L,∇lnH〉

= 〈∇l(u · ∇nH) +∇l(u · ∇nL)−∇l(u · ∇n)L,∇lnH〉

:=

3∑
i=1

I2,2,i.

(3.11)

For the term I2,2,1, if l = 1, we have

|I2,2,1| = |〈∇(u · ∇nH),∇nH〉|
. (‖u‖L∞‖∇2nH‖L2 + ‖∇nH‖L6‖∇u‖L3)‖∇nH‖L2

. ‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇2nH‖L2‖∇nH‖L2

. (1 + t)−
3
2 ‖∇nH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇n‖2L2 ;

(3.12)

if l = 2, it is easy to see that

I2,2,1 = 〈∇2(u · ∇nH),∇2nH〉
= 〈u · ∇(∇2nH) + 2∇u · ∇2nH +∇2u · ∇nH ,∇2nH〉.

By using integration by parts, we have

∫
R3

u · ∇(∇2nH) · ∇2nHdx = −1

2

∫
R3

divu|∇2nH |2dx,



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 9

and hence we obtain

|I2,2,1|
. (‖∇u‖L∞‖∇2nH‖L2 + ‖∇nH‖L3‖∇2u‖L6)‖∇2nH‖L2

. (‖∇2u‖1/2L2 ‖∇3u‖1/2L2 ‖∇2n‖L2 + ‖∇nH‖1/2L2 ‖∇2nH‖1/2L2 ‖∇3u‖L2)‖∇2n‖L2

. ‖∇2u‖1/2L2 ‖∇2n‖1/2L2 ‖∇3u‖1/2L2 ‖∇2n‖3/2L2 + ‖∇n‖1/2L2 ‖∇2n‖3/2L2 ‖∇3u‖L2

. (1 + t)−3/4‖∇3u‖1/2L2 ‖∇2n‖3/2L2 + (1 + t)−
3
2 ‖∇3u‖L2

. (1 + t)−5/2 + (1 + t)−1/2(‖∇2n‖2L2 + ‖∇3u‖2L2).

(3.13)

Consequently, (3.12) together with (3.13) implies that for 1 ≤ l ≤ 2,

|I2,2,1| . (1 + t)−5/2 + (1 + t)−1/2(‖∇ln‖2L2 + ‖∇3u‖2L2). (3.14)

The term I2,2,2 can be estimated as follows

|I2,2,2|

. ‖∇l(u · ∇nL)‖L2‖∇lnH‖L2

. (‖u‖L∞‖∇l+1nL‖L2 + ‖∇nL‖L∞‖∇lu‖L2)‖∇lnH‖L2

. (‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇ln‖L2 + ‖∇2nL‖1/2L2 ‖∇3nL‖1/2L2 ‖∇lu‖L2)‖∇lnH‖L2

. (‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇ln‖L2 + ‖∇n‖1/2L2 ‖∇2n‖1/2L2 ‖∇lu‖L2)‖∇lnH‖L2 (3.15)

. (1 + t)−
3
2 ‖∇lnH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇lnH‖2L2 ,

where we have used (5.2). Similar to the proof of (3.12), we also have

|I2,2,3|

. ‖∇l(u · ∇n)L‖L2‖∇lnH‖L2

. ‖∇l−1(u · ∇n)‖L2‖∇lnH‖L2

. (‖u‖L∞‖∇ln‖L2 + ‖∇l−1u‖L6‖∇n‖L3)‖∇lnH‖L2

. (‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇ln‖L2 + ‖∇n‖1/2L2 ‖∇2n‖1/2L2 ‖∇lu‖L2)‖∇ln‖L2

. (1 + t)−
3
2 ‖∇ln‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇ln‖2L2 .

(3.16)

Substituting (3.14)–(3.16) into (3.11), we arrive at

|I2,2| . (1 + t)−5/2 + (1 + t)−1/2(‖∇n‖2H1 + ‖∇3u‖2L2), (3.17)

Thus, combining (3.10) with (3.17) gives rise to

|I2| . (1 + t)−5/2 + (1 + t)−1/2(‖∇n‖2H1 + ‖∇u‖2H2). (3.18)



10 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

For the term I3, from (2.4), one has

I3

= ρ̄〈∇lfH21,∇luH〉

= ρ̄
〈
∇l
[
−
(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇(ρ̃− ρ̄)

]H
,∇luH

〉
+ ρ̄
〈
∇l
[
−
(P ′(ρ̃)

ρ̃
− P ′(ρ̄)

ρ̄

)
∇n
]H
,∇luH

〉
+ ρ̄
〈
∇l
[µ1

ρ̃
(ρ̃− ρ̄)∆u

]H
,∇luH

〉
+ ρ̄
〈
∇l
[µ2

ρ̃
(ρ̃− ρ̄)∇ div u

]H
,∇luH

〉
: =

4∑
i=1

I3,i.

(3.19)

By employing a similar argument as in the proof of (3.8), we have

|I3,1| . ‖∇l[n∇(ρ̃− ρ̄)]‖L2‖∇luH‖L2

. (‖n‖L∞‖∇l+1(ρ̃− ρ̄)‖L2 + ‖∇ln‖L2‖∇(ρ̃− ρ̄)‖L∞)‖∇luH‖L2

. (δ0‖∇n‖H1 + δ0‖∇ln‖L2)‖∇luH‖L2

. δ0(‖∇l+1uH‖2L2 + ‖∇n‖2H1).

(3.20)

For the term I3,2, by using integration by parts, we have

|I3,2| . |〈∇l[(ρ̃− ρ̄)∇n]H ,∇luH〉|

. |〈∇l−1[(ρ̃− ρ̄)∇n]H ,∇l∇ · uH〉|

. (‖∇l−1(ρ̃− ρ̄)‖L∞‖∇n‖L2 + ‖ρ̃− ρ̄|L∞‖∇ln‖L2)‖∇l∇ · uH‖L2

. (‖∇l(ρ̃− ρ̄)‖H1‖∇n‖L2 + ‖∇(ρ̃− ρ̄)‖H1‖∇ln‖L2)‖∇l∇ · uH‖L2

. δ0(‖∇l+1uH‖2L2 + ‖∇n‖2H1).

(3.21)

Similarly, for the term I3,3, we have

|I3,3| . |〈∇l[(ρ̃− ρ̄)∆u]H ,∇l∇uH〉|

. |〈∇l−1[(ρ̃− ρ̄)∆u]H ,∇l∇ · uH〉|

. (‖∇l−1(ρ̃− ρ̄)‖L∞‖∆u‖L2 + ‖ρ̃− ρ̄‖L∞‖∇l+1u‖L2)‖∇l∇ · uH‖L2

. (‖∇l(ρ̃− ρ̄)‖H1‖∆u‖L2 + ‖∇(ρ̃− ρ̄)‖H1‖∇l+1u‖L2)‖∇l∇ · u‖L2

. δ0‖∇2u‖2H1 .

(3.22)

The term I3,4 can be estimated in the same way, and it holds that

|I3,4| . δ0‖∇2u‖2H1 . (3.23)

Putting the above inequalities (3.20)-(3.23) into (3.19), we have

|I3| . δ0(‖∇2u‖2H1 + ‖∇n‖2H1). (3.24)



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 11

For the term I4, using (2.5) and integration by parts, it holds that

I4 = ρ̄〈∇lfH22,∇luH〉

= ρ̄〈∇l−1fH22,∇l∇ · uH〉

= ρ̄〈∇l−1(−u · ∇u)H ,∇l∇ · uH〉

− ρ̄
〈
∇l−1

[(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇n
]H
,∇l∇ · uH

〉
+ ρ̄
〈
∇l−1

[
µ
( 1

n+ ρ̃
− 1

ρ̃

)
∆u
]H
,∇l∇ · uH

〉
+ ρ̄
〈
∇l−1

[
(µ+ ν)

( 1

n+ ρ̃
− 1

ρ̃

)
∆u
]H
,∇l∇ · uH

〉
: =

4∑
i=1

I4,i.

(3.25)

The four terms on the right-hand side of (3.25) can be estimated as follows. Similar
to the proof of (3.12), we have

|I4,1| . ‖∇l−1(u∇u)‖L2‖∇l∇ · uH‖L2

. (‖∇l−1u‖L6‖∇u‖L3 + ‖u‖L∞‖∇lu‖L2)‖∇l∇ · uH‖L2

. ‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇lu‖L2‖∇l∇ · uH‖L2

. (1 + t)−
3
2 ‖∇l∇ · uH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇l∇ · uH‖2L2

(3.26)

Similarly, for the term (3.26), we have

|I4,2| . (1 + t)−5/2 + (1 + t)−1/2‖∇l∇ · uH‖2L2 . (3.27)

and

|I4,3|+ |I4,4| . ‖∇l−1(n∆u)‖L2‖∇l∇ · uH‖L2

. (‖n‖L∞‖∇l+1u‖L2 + ‖∇l−1n‖L6‖∆u‖L3)‖∇l∇ · uH‖L2

. ‖∇n‖H1‖∇l+1u‖2L2 + ‖∇ln‖L2‖∆u‖H1‖∇l∇ · uH‖L2

. (1 + t)−3/4‖∇l+1u‖2L2 + (1 + t)−3/4‖∇∇ · u‖2H1

. (1 + t)−3/4‖∇∇ · u‖2H1 .

(3.28)

Substituting (3.26) - (3.28) into (3.25) gives

|I4| . (1 + t)−5/2 + (1 + t)−1/2‖∇∇ · u‖2H1 . (3.29)

For the term I5, we have

|I5| . |〈∇l∇ · [(ρ̃− ρ̄)u]H ,∇lΦH〉|

. |〈∇l[(ρ̃− ρ̄)u]H ,∇l∇ΦH〉|

. (‖∇l(ρ̃− ρ̄)‖L2‖u‖L∞ + ‖ρ̃− ρ̄‖L3‖∇lu‖L6)‖∇l−1nH‖L2

. (δ0‖∇u‖H1 + δ0‖∇l+1u‖L2)‖∇lnH‖L2

. δ0(‖∇u‖2H2 + ‖∇lnH‖2L2).

(3.30)



12 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

For the last term I6, we obtain

|I6| . |〈∇l∇ · (nu)H ,∇lΦH〉|

. |〈∇l(nu)H ,∇l∇ΦH〉|

. ‖∇l(nu)‖L2‖∇l−1nH‖L2

. ‖(n, u)‖L∞‖∇l(n, u)‖L2‖∇lnH‖L2

. (‖∇(n, u)‖1/2L2 ‖∇2(n, u)‖1/2L2 ‖∇l(n, u)‖L2‖∇lnH‖L2

. (1 + t)−
3
2 ‖∇lnH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇lnH‖2L2 .

(3.31)

Substituting (3.8),(3.18),(3.24) and (3.29)–(3.31) into (3.7), and using the smallness
of δ0, we obtain the estimate (3.6), and thus completes the proof. �

Notice that the energy inequality (3.6) only gives the dissipative estimate for
uH . Next, we will establish dissipation estimates for nH and ∇ΦH .

Lemma 3.4. Under the assumptions of Theorem 1.2, then for l = 1, 2, it holds
that

d

dt
〈∇l−1uH ,∇lnH〉+

P ′(ρ̄)

2ρ̄
‖∇lnH‖2L2 + ‖∇l(∇Φ)H‖2L2

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇u‖2H2 + ‖∇n‖2H1) + C1‖∇2uH‖2H1 ,

(3.32)

for any t ∈ [0,∞).

Proof. For l = 1, 2, by taking

〈∇lF−1[(1− φ(ξ))F(2.1)1],∇l−1uH〉+ 〈∇l−1F−1[(1− φ(ξ))F(2.1)2],∇lnH〉,

using integration by parts, Hölder’s inequality and Young’s inequality, we have

d

dt
〈∇l−1uH ,∇lnH〉+

P ′(ρ̄)

ρ̄
‖∇lnH‖2L2 + ‖∇l(∇Φ)H‖2L2

= 〈∇l(−ρ̄∇ · uH + fH11 + fH12),∇l−1uH〉

+ 〈∇l−1(µ1∆uH + µ2∇∇ · uH + fH21 + fH22),∇lnH〉

= 〈ρ̄∇luH ,∇luH〉+ 〈∇l[(ρ̃− ρ̄)u]H +∇l(nu)H ,∇luH〉

+ (µ1 + µ2)〈∇l+1uH ,∇lnH〉+ 〈∇l−1(fH21 + fH22),∇lnH〉

≤ C1‖∇l+1uH‖2L2 +
P ′(ρ̄)

2ρ̄
‖∇lnH‖2L2 + 〈∇l[(ρ̃− ρ̄)u]H ,∇luH〉

+ 〈∇l(nu)H ,∇luH〉+ 〈∇l−1(fH21 + fH22),∇lnH〉,

(3.33)

and for simplicity, we define

〈∇l[(ρ̃− ρ̄)u]H ,∇luH〉+ 〈∇l(nu)H ,∇luH〉+ 〈∇l−1(fH21 + fH22),∇lnH〉

:=

4∑
i=1

Ji.
(3.34)



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 13

Next, we shall estimate the terms on the right-hand side of (3.34) one by one. For
the terms J1 and J2, we have

|J1| = |〈∇l[(ρ̃− ρ̄)u]H ,∇luH〉|

. ‖∇l[(ρ̃− ρ̄)u]‖L2‖∇luH‖L2

. (‖∇l(ρ̃− ρ̄)‖L2‖u‖L∞ + ‖ρ̃− ρ̄‖L∞‖∇lu‖L2)‖∇luH‖L2

. (‖∇l(ρ̃− ρ̄)‖L2‖∇u‖H1 + ‖∇(ρ̃− ρ̄)‖H1‖∇lu‖L2)‖∇luH‖L2

. δ0‖∇u‖2H1 ,

(3.35)

and

|J2| = |〈∇l(nu)H ,∇luH〉|

. ‖∇l(nu)‖L2‖∇lnH‖L2

. ‖(n, u)‖L∞‖∇l(n, u)‖L2‖∇lnH‖L2

. (‖∇(n, u)‖1/2L2 ‖∇2(n, u)‖1/2L2 ‖∇l(n, u)‖L2‖∇lnH‖L2

. (1 + t)−
3
2 ‖∇lnH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇lnH‖2L2 .

(3.36)

For the term J3, it holds that

J3 = 〈∇l−1fH21,∇lnH〉

=
〈
∇l−1

[
−
(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇(ρ̃− ρ̄)

]H
,∇lnH

〉
+
〈
∇l−1

[
−
(P ′(ρ̃)

ρ̃
− P ′(ρ̄)

ρ̄

)
∇n
]H
,∇lnH

〉
+
〈
∇l−1

[µ1

ρ̃
(ρ̃− ρ̄)∆u

]H
,∇lnH

〉
+
〈
∇l−1

[µ2

ρ̃
(ρ̃− ρ̄)∇ div u

]H
,∇lnH〉

: =

4∑
i=1

J3,i.

(3.37)

For J3,1 and J3,2, we obtain

|J3,1| . |〈∇l−1[n∇(ρ̃− ρ̄)]H ,∇lnH〉|

. ‖∇l−1[n∇(ρ̃− ρ̄)]‖L2‖∇lnH‖L2

. (‖∇(ρ̃− ρ̄)‖L3‖∇l−1n‖L6 + ‖∇l(ρ̃− ρ̄)‖L2‖n‖L∞)‖∇lnH‖L2

. (‖∇(ρ̃− ρ̄)‖H1‖∇2n‖L2 + ‖∇l(ρ̃− ρ̄)‖L2‖∇n‖H1)‖∇lnH‖L2

. δ0‖∇n‖2H1 .

(3.38)

|J3,2| . |〈∇l−1[(ρ̃− ρ̄)∇n]H ,∇lnH〉|

. ‖∇l−1[(ρ̃− ρ̄)∇n]‖L2‖∇lnH‖L2

. (‖ρ̃− ρ̄‖L∞‖∇ln‖L2 + ‖∇l−1(ρ̃− ρ̄)‖L3‖∇n‖L6)‖∇lnH‖L2

. (‖∇(ρ̃− ρ̄)‖H1‖∇ln‖L2 + ‖∇l−1(ρ̃− ρ̄)‖H1‖∇2n‖L2)‖∇lnH‖L2

. δ0‖∇n‖2H1 .

(3.39)



14 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

Similarly, for the term J3,3 and J3,4, we have

|J3,3|+ |J3,4|

. |〈∇l−1[(ρ̃− ρ̄)∆u]H ,∇lnH〉|

. ‖∇l−1[(ρ̃− ρ̄)∆u]‖L2‖∇lnH‖L2

. (‖ρ̃− ρ̄‖L∞‖∇l+1u‖L2 + ‖∇l−1(ρ̃− ρ̄)‖L∞‖∆u‖L2)‖∇lnH‖L2

. (‖∇(ρ̃− ρ̄)‖H1‖∇l+1u‖L2 + ‖∇l(ρ̃− ρ̄)‖H1‖∆u‖L2)‖∇lnH‖L2

. δ0(‖∇lnH‖2L2 + ‖∇2u‖2H1).

(3.40)

Substituting (3.38)–(3.40) into (3.37) yields

|J3| . δ0(‖∇n‖2H1 + ‖∇2u‖2H1). (3.41)

For the term J4, we have

J4

= 〈∇l−1fH22,∇lnH〉

= 〈∇l−1(−u · ∇u)H ,∇lnH〉 −
〈
∇l−1

[(P ′(n+ ρ̃)

n+ ρ̃
− P ′(ρ̃)

ρ̃

)
∇n
]H
,∇lnH

〉
+
〈
∇l−1

[
µ
( 1

n+ ρ̃
− 1

ρ̃

)
∆u
]H
,∇lnH

〉
+
〈
∇l−1

[
(µ+ ν)

( 1

n+ ρ̃
− 1

ρ̃

)
∇(∇ · u)

]H
,∇lnH

〉
:=

4∑
i=1

J4,i.

(3.42)

The four terms on the right-hand side of (3.42) can be estimated as follows. Firstly,
similar to the proof of (3.26), we have

|J4,1| . ‖∇l−1(u · ∇u)‖L2‖∇lnH‖L2

. (‖∇l−1u‖L6‖∇u‖L3 + ‖u‖L∞‖∇lu‖L2)‖∇lnH‖L2

. ‖∇u‖1/2L2 ‖∇2u‖1/2L2 ‖∇lu‖L2‖∇lnH‖L2

. (1 + t)−
3
2 ‖∇lnH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇lnH‖2L2 .

(3.43)

Similar to the proof of (3.43), we have

|J4,2| . ‖∇l−1(n · ∇n)‖L2‖∇lnH‖L2

. (1 + t)−5/2 + (1 + t)−1/2‖∇lnH‖2L2 ,
(3.44)

and

|J4,3|+ |J4,4| . ‖∇l−1(n∆u)‖L2‖∇lnH‖L2

. (‖n‖L∞‖∇l+1u‖L2 + ‖∇l−1n‖L6‖∆u‖L3)‖∇lnH‖L2

. ‖∇n‖1/2L2 ‖∇2n‖1/2L2 ‖∇ln‖L2‖∇l+1u‖L2 + ‖∇ln‖2L2‖∆u‖H1

. (1 + t)−
3
2 ‖∇l+1u‖L2 + (1 + t)−

3
2 ‖∆u‖H1

. (1 + t)−5/2 + (1 + t)−1/2‖∆u‖2H1 .

(3.45)



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 15

Substituting (3.43)–(3.45) into (3.42) yields

|J4| . (1 + t)−5/2 + (1 + t)−1/2(‖∇lnH‖2L2 + ‖∆u‖2H1). (3.46)

Substituting (3.35), (3.36), (3.41) and (3.46) into (3.34) and using (3.33), we can
prove (3.32) and thus completes the proof. �

Now, we are ready to prove Theorem 1.2. Now, multiplying (3.32) with some
positive number C2 = 2µ+ν

2C1
, and summing (3.6) from l = 1 to 2 leads to

d

dt

(P ′(ρ̄)

2ρ̄
‖∇nH‖2H1 +

ρ̄

2
‖∇uH‖2H1 +

1

2
‖∇(∇Φ)H‖2H1 +

2∑
l=1

C2〈∇l−1uH ,∇lnH〉
)

+
P ′(ρ̄)

2ρ̄
C2‖∇nH‖2H1 +

2µ+ ν

2
‖∇2uH‖2H1 + C2‖∇(∇Φ)H‖2H1 (3.47)

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇u‖2H2 + ‖∇n‖2H1)).

Next, choosing sufficiently large time T0 and positive constant D0, for t ≥ T0, we
define the temporary energy functional

E(t) = D0‖∇(nH , uH ,∇ΦH)‖2H1 +

2∑
l=1

〈∇l−1uH ,∇lnH〉, (3.48)

which is equivalent to ‖∇(nH , uH ,∇ΦH)‖2H1 since D0 is large enough. Using the
smallness of δ0, (3.47) and Lemma 5.4, for t ≥ T0, it holds that

d

dt
E(t) + ‖∇nH‖2H1 + ‖∇2uH‖2H1 + ‖∇(∇Φ)H‖2H1

. (1 + t)−5/2 + (δ0 + (1 + t)−1/2)(‖∇uL‖2L2 + ‖∇nL‖2L2),
(3.49)

where we have used the fact that T0 is large enough. On the other hand, it is clear
that

‖∇nH‖2H1 + ‖∇2uH‖2H1 + ‖∇(∇Φ)H‖2H1 ≥ C3E(t). (3.50)

Then by the Gronwall inequality and Lemma 3.2, we arrive at

E(t) ≤ E(0)e−C3t + C4

∫ t

0

e−C3(t−s)[(1 + s)−5/2 + (δ0 + (1 + s)−1/2)

× (‖∇u(s)L‖2L2 + ‖∇n(s)L‖2LL)]ds

≤ E(0)e−C3t + C4

∫ t

0

e−C3(t−s)[(1 + s)−5/2 + (δ0 + (1 + s)−1/2)]

× C(1 + s)−5/2(‖U0‖2L1 + δ20M(s))ds

≤ E(0)e−C3t + C5(‖U0‖2L1 + δ20M(t))

∫ t

0

(1 + t− s)−5/2(1 + s)−5/2ds

≤ C(1 + t)−5/2(E(0) + ‖U0‖2L1 + δ20M(t)).

(3.51)

Combining with Lemma 3.2 and (3.51), we deduce that

‖∇(n, u,∇Φ)‖2H1 ≤ ‖∇(nH , uH ,∇ΦH)‖2H1 + ‖∇(nL, uL,∇ΦL)‖2H1

≤ C(1 + t)−5/2(E(0) + ‖U0‖2L1 + δ20M(t)),
(3.52)

which together with the definition (3.2) of M(t) and the smallness of δ0 implies
that

M(t) ≤ C(E(0) + ‖U0‖2L1).



16 L. QIN, C. XIAO, Y. ZHANG EJDE-2022/64

This gives (1.9) and thus completes the proof.

4. Proof of Theorem 1.3

In this section, we devote ourselves to proving the lower decay estimates of the
global solution of Cauchy problem (2.1). By virtue of Duhamel’s principle, (1.9),
(3.3), (3.4) and Lemma(5.4), we have

min
{
‖∇n‖L2 , ‖∇u‖L2 , ‖∇Φ‖L2

}
≥ min

{
‖∇nL‖L2 , ‖∇uL‖L2 , ‖∇ΦL‖L2

}
≥ Cδ3/20 (1 + t)−5/4 − C

∣∣∣ ∫ t

0

(1 + t− s)− 5
4 ‖S(s)L‖L1

∣∣∣ds
≥ Cδ3/20 (1 + t)−5/4 − Cδ0

√
M(t)

∫ t

0

(1 + t− s)− 5
4 (1 + s)−5/4ds

≥ (Cδ
3/2
0 − Cδ0(E(0) + ‖U0‖2L1)

1
2 )(1 + t)−5/4.

(4.1)

This together with the fact that E(0), ‖U0‖L1 < δ0 implies (1.11), and thus com-
pletes the proof of Theorem 1.3.

5. Analytic tools

Now, we recall the Sobolev interpolation of Gagliardo-Nirenberg’s inequality.

Lemma 5.1 ([16]). Given 2 ≤ p ≤ +∞ and 0 ≤ k,m ≤ `, then for any f ∈ H`(R3),
we have

‖∇kf‖Lp . ‖∇mf‖αL2‖∇`f‖1−αL2 ,

where α ∈ [0, 1] satisfies

k

3
− 1

p
=
(m

3
− 1

2

)
α+

( `
3
− 1

2

)
(1− α).

Next, to estimate the L2-norm of the spatial derivatives of the product of two
functions, we shall record the following estimate.

Lemma 5.2 ([8]). Let k ≥ 1 be an integer, then it holds that

‖∇k(fg)‖Lp . ‖f‖Lp1 ‖∇kg‖Lp2 + ‖g‖Lp3 ‖∇kf‖Lp4 ,

where p, p2, p3 ∈ (1,+∞) and

1

p
=

1

p1
+

1

p2
=

1

p3
+

1

p4
.

Next, we recall Hardy inequality and Sobolev embedding estimates.

Lemma 5.3 ([18]). (i) If u(x) ∈ H1(R3), then the following inequalities hold:∥∥ u
|x|
∥∥
L2 ≤ C‖∇u‖L2 ,

‖u‖L6 ≤ C‖∇u‖L2 ,

‖u‖L3 ≤ C(‖u‖L2 + ‖u‖L6) ≤ C‖u‖H1 .

(ii) If u(x) ∈ H2(R3), then ‖u‖L∞ ≤ C‖∇u‖H1

We also have the following lemma concerning the estimate for the low–frequent
part and the high–frequent part of f .



EJDE-2022/64 3D COMPRESSIBLE NAVIER-STOKES-POISSON EQUATIONS 17

Lemma 5.4. If f ∈ Lk(R3) for any 2 ≤ r ≤ ∞, then

‖∇kf‖ ≤ ‖∇kfL‖+ ‖∇kfH‖ (5.1)

‖∇kfL‖ ≤ ‖∇k−1f‖, k ≥ 1, (5.2)

‖∇kfH‖ ≤ ‖∇k+1f‖, k ≥ 1, (5.3)

Lemma 5.5 ([22, 24]). If r1, r2 > 0, then∫ t

0

(1 + t− τ)−r1(1 + τ)−r2dτ ≤ C(r1, r2)(1 + t)−min{r1,r2,r1+r2−1−η}, (5.4)

for an arbitrarily small η > 0.

Acknowledgments. This work was partially supported by the Guangxi Natural
Science Foundation #2019JJG110003, #2019AC20214, and by the Key Laboratory
of Mathematical and Statistical Model (Guangxi Normal University), Education
Department of Guangxi Zhuang Autonomous Region.

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Liuna Qin
School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi

541004, China

Email address: liunaqin321@stu.gxnu.edu.cn

Changguo Xiao

School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi
541004, China

Email address: changguoxiao@mailbox.gxnu.edu.cn

Yinghui Zhang (corresponding author)
School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi

541004, China
Email address: yinghuizhang@mailbox.gxnu.edu.cn


	1. Introduction
	1.1. History of the problem
	1.2. Main results

	2. Reformulation of original problem
	3. Proof of Theorem ??
	4. Proof of Theorem ??
	5. Analytic tools
	Acknowledgments

	References