Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

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J. Differential Equations 257 (2014) 4319–4368

www.elsevier.com/locate/jde

Asymptotic limits of solutions to the initial boundary

value problem for the relativistic Euler–Poisson

equations

La-Su Mai a, Hai-Liang Li b, Kaijun Zhang c,∗

a Department of Mathematics, Capital Normal University, Beijing 100048, PR Chinab Department of Mathematics and BCMIIS, Capital Normal University, Beijing 100048, PR China

c School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, PR China

Received 23 February 2014; revised 19 August 2014

Available online 22 September 2014

Abstract

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assump-tions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ satisfying c = τ−1/2 when the relaxation time τ tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solu-tion of the unipolar hydrodynamic model for semiconductors when the light speed c → ∞. In addition, the related convergence rate results are also obtained.© 2014 Elsevier Inc. All rights reserved.

Keywords: Relativistic Euler–Poisson equations; Global smooth solution; Combined zero-relaxation and non-relativistic limit; Non-relativistic limit

* Corresponding author.E-mail address: [email protected] (K. Zhang).

http://dx.doi.org/10.1016/j.jde.2014.08.0100022-0396/© 2014 Elsevier Inc. All rights reserved.

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4320 L.-S. Mai et al. / J. Differential Equations 257 (2014) 4319–4368

1. Introduction

We consider the relativistic Euler–Poisson equations which describe the motion of an isen-tropic relativistic electro-fluid [13]:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(ρ√

1 − υ2/c2

)t

+(

ρυ√1 − υ2/c2

)x

= 0,(n(ρ)c2 + p(n(ρ))

c2 − υ2· υ

)t

+(

n(ρ)c2 + p(n(ρ))

c2 − υ2· υ2 + p

(n(ρ)

))x

= ρφx√1 − υ2/c2

− ρυ/τ

√1 − υ2/c2,

φxx = ρ − C(x)√1 − υ2/c2

(1.1)

where ρ, υ , c, τ and φ represent the proper density of charge, the velocity of electric fluid, the speed of light, the relaxation time and the electrostatic potential, respectively. The pressure p(n)

is given by γ -law, namely,

p(n) = σ 2nγ for γ ≥ 1, (1.2)

where σ is taken to be a constant such that 0 < σ < c. The mass-energy density of electric fluid n(ρ) is a function of ρ. The relation between n(ρ) and ρ satisfies

dn

dρ= n + p(n)/c2

ρ(1.3)

due to the first law of thermal dynamic [13,24]. Note that (1.3) is the well-known Bernoulli equation when γ > 1 which can be solved exactly. We impose (1.3) on the initial value at a given point ρ = ρ as

n(ρ) = ρ, (1.4)

with ρ being a positive constant. Then, we solve the initial value problem (1.3)–(1.4) and obtain

n(ρ) =⎧⎨⎩ρ

− σ2

c2 ρc2+σ2

c2 , γ = 1,

ρ[1 − σ 2

c2 (ργ−1 − ργ−1)] 11−γ , γ > 1

(1.5)

for ρ ≥ ρ.

From the point of view of physics [1,8,13,30,31], relativistic electrodynamics uses the clas-sical method to study the interaction of relativistic charged particles and electromagnetic field, motion of a particle in electromagnetic field and the field excitation. Here the so called relativis-tic charged particle refers to its speed close to the speed of light in vacuum c, so its motion does not obey Newtonian equations must be used to deal with the relativistic equation of particle.

L.-S. Mai et al. / J. Differential Equations 257 (2014) 4319–4368 4321

Relativistic electrodynamics is mainly used in high-energy accelerator and high-energy electron-ics and high-energy astrophysics (such as the supernova explosions, the gravitational collapse, the galaxy formation and the accretion onto a neutron star or a black hole, etc.). However, the model (1.1) can be considered to describe the motion of isentropic relativistic electro-fluid when the speed of charged particles is very large but less than the speed of light and electric field effect is more strong than that of magnetic field, see [8,13]. For this system, the first equation describes the conservation of charge number, the second one is the momentum equation with relaxation term and the third equation indicates the effect of electric field. The energy equation can be derived from the momentum equation (1.1)2 and (1.3).

From the point of view of mathematics, one of the main motivations for studying system (1.1) is that in the non-relativistic limit c → ∞, this system formally reduces to the following classical non-relativistic Euler–Poisson system which is the mathematical simplified model for hydrodynamic model of semiconductors [4,16]:⎧⎪⎨⎪⎩

ρt + (ρυ)x = 0,

(ρυ)t + (ρυ2 + p(ρ)

)x

= ρφx − ρυ

τ,

φxx = ρ − C(x).

(1.6)

Note that the second equation of (1.6) contains the momentum relaxation term −ρuτ

, which is very important to obtain the exponential decay estimates of global smooth solutions for this system. Compared to the Euler–Poisson system (1.6), many mathematicians are also interested in the Euler–Poisson equation without relaxation. In this system, the Poisson equation describes the gravitational interaction, see [2,3,7,12,27].

The system (1.1) is supplemented with the initial data

ρ(0, x) = ρ(x), υ(0, x) = υ(x), x ∈ (0,1), (1.7)

and the boundary conditions

ρ(t,0) = ρl > 0, ρ(t,1) = ρr > 0, t ≥ 0, (1.8)

φ(t,0) = 0, φ(t,1) = φr, t ≥ 0, (1.9)

where ρl, ρr and φr are given constants. For simplicity, we assume that the relaxation time τsatisfies 0 < τ ≤ 1, the equation of state is given by (1.2) with γ = 1 and the background ion density C(x) ∈ C1[0, 1] satisfies

0 < C ≤ C(x) ≤ C and∣∣(C − Q)(x)

∣∣ ≤ C0δ, x ∈ [0,1], (1.10)

where C := infx∈[0,1] C(x), C := supx∈[0,1] C(x), Q := ρl + x(ρr − ρl), δ := |φr | + |ρr − ρl | is the boundary strength and C0 is a positive constant independent of the speed of light c, t and δ.

Denoting by jc the time dependent relativistic current density

jc := ρυ√1 − υ2/c2

, (1.11)

we can uniquely solve υ from (1.11) if infx∈(0,1) ρ(x) > 0. Then, the system (1.1) is equivalent to

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(√ρ2 + j2

c /c2)t+ (jc)x = 0,(

n(ρ)(c2 + σ 2)

c2ρ2·√

ρ2 + j2c /c2 · jc

)t

+(

n(ρ)(c2 + σ 2)

c2ρ2· j2

c + σ 2n(ρ)

)x

=√

1 + j2c /(cρ)2 · ρφx − jc,

φxx =√

1 + j2c /(cρ)2 · (ρ − C(x)

)(1.12)

with the boundary conditions (1.8) and (1.9) and initial data

ρ(0, x) := ρ(x), jc(0, x) = j(x) := ρ(x)υ(x)√1 − υ2(x)/c2

, x ∈ (0,1). (1.13)

For the smooth solutions with infx∈(0,1) ρ(x) > 0, the system (1.12) can be written as⎧⎪⎨⎪⎩a1 · ρt + c−2a2 · jc · (jc)t + (jc)x = 0,

c−2a3 · jcρt + a4 · (jc)t + a5 · ρx + a6 · jc · (jc)x = a7 · φx − jc,

φxx = a8 · (ρ − C(x)),

(1.14)

where ai = ai(ρ, j2c ) are given by

a1(ρ, j2

c

) := ργc, a2(ρ, j2

c

) := γc, (1.15)

a3(ρ, j2

c

) := n(ρ)ρ−3γc · (1 + σ 2/c2) · g1(ρ, j2

c

), (1.16)

a4(ρ, j2

c

) := n(ρ)ρ−2γc · (1 + σ 2/c2) · (ρ2 + 2j2c /c2), (1.17)

a5(ρ, j2

c

) := n(ρ)ρ−3 · (1 + σ 2/c2) · g1(ρ, j2

c

),

a6(ρ, j2

c

) := 2n(ρ)ρ−2 · (1 + σ 2/c2), (1.18)

a7(ρ, j2

c

) := γ −1c , a8

(ρ, j2

c

) := ρ−1γ −1c , γc := 1/

√ρ2 + j2

c /c2, (1.19)

g1(ρ, j2

c

) := σ 2ρ2 − (1 − σ 2/c2) · j2

c , n(ρ) = ρ− σ2

c2 ρc2+σ2

c2 . (1.20)

It is easy to see that if (ρ, jc, φ) is a unique solution to the IBVP (1.14), (1.13), (1.8) and (1.9)with infx∈(0,1) ρ(x) > 0, then υ can be uniquely determined by (1.11) and (ρ, υ, φ) is a unique solution of the IBVP (1.1), (1.7), (1.8) and (1.9). Thus, we will work with the IBVP (1.14), (1.13), (1.8) and (1.9).

For smooth solutions, we additionally need for ρ the compatibility conditions of orders 0 to 1 at points (0, 0) and (0, 1) respectively. By solving (jc)t from (1.14)2 and substituting it into (1.14)1, if

g1(ρ, j2

c

)> 0, and inf

x∈(0,1)ρ > 0, (1.21)

then the compatibility conditions at points (0, 0) and (0, 1) respectively satisfy


ρ(0,0) = ρl, ρt (0,0) = Ψ(ρl, jx(0), ρx(0),φx(0,0), j(0)

) = 0, (1.22)

and

ρ(0,1) = ρr, ρt (0,1) = Ψ(ρr, jx(1), ρx(1),φx(0,1), j(1)

) = 0, (1.23)

where Ψ (ρ, (jc)x, ρx, φx, jc)(t, x) = Ψ (t, x) is denoted by

Ψ (t, x) := −Γ(ρ, j2

c

) · [ρ3 · (jc)x − c−2(σ 2ρ2 − (1 − σ 2/c2) · j2

c

) · jcρx

]− Γ

(ρ, j2

c

) · c−2n−1(ρ)ρ3jc · (1 + σ 2/c2)−1 · (√ρ2 + j2c /c2 · φx − jc

),

Γ(ρ, j2

c

) := (ρ2 + (

1 − σ 2/c2) · j2c /c2)−1 · 1/

√ρ2 + j2

c /c2 (1.24)

with φx being solved from (1.14)3 and (1.9)

φx(t, x) :=x∫

0

√1 + j2

c /(cρ)2 · (ρ − C)(t, z)dz + φr

−1∫

0

y∫0

√1 + j2

c /(cρ)2 · (ρ − C)(t, z)dzdy. (1.25)

Notice that the first inequality of (1.21) implies the uniform elliptic condition for the steady state system of (1.14) and the last inequality gives the positive property of ρ. In this paper, we study the asymptotic limits of the global smooth solutions for system (1.14) with the properties (1.21).

We first investigate asymptotic limits of the steady state system for (1.14) which satisfies⎧⎨⎩(jc)x = 0,

a5 · ρx = a7 · φx − jc,

φxx = a8 · (ρ − C),

(1.26)

where jc is the steady state relativistic current density

jc = ρυ√1 − υ2/c2

, (1.27)

and

a5(ρ, j2

c

) := n(ρ)ρ−3 · (1 + σ 2/c2) · g1(ρ, j2

c

),

g1(ρ, j2

c

) := σ 2ρ2 − (1 − σ 2/c2) · j2

c ,

a7(ρ, j2

c

) := γ −1c , a8

(ρ, j2

c

) := ρ−1c γ −1

c , γc = 1/

√ρ2 + j2

c /c2,

n(ρ) = ρ− σ2

c2 ρc2+σ2

c2


with ρ being positive constant ρ ≥ ρ > 0. Formally, this system reduces to the following steady state system of (1.6) as c → ∞

⎧⎪⎪⎪⎨⎪⎪⎪⎩(j )x = 0,(

j2

ρ+ σ 2ρ

)x

= ρφx − j

τ,

φxx = ρ − C(x),

(1.28)

with j = ρυ .In [13], we constructed the unique smooth solution to the BVP (1.26), (1.8) and (1.9) when the

boundary strength δ = |ρr − ρl | + |φr | is suitably small. The existence of smooth solution was obtained by the Schauder fixed point theorem and the uniqueness was obtained by the energy method. We also considered the zero-relaxation limit and non-relativistic limit of the smooth solution by the energy method. For the limit τ → 0, we introduced a new scaling and proved that at the rate of 1

c2 the smooth solution to the new scaled problem converges to the steady state solution of the relativistic drift-diffusion equation with the same boundary conditions (1.8) and (1.9). For the non-relativistic limit c → ∞, we obtained that the smooth solution to the BVP (1.26), (1.8) and (1.9) converges to the subsonic steady state solution for system (1.6) with the boundary conditions (1.8) and (1.9).

In [14], we proved the existence of unique global smooth solution to the IBVP (1.14), (1.13), (1.8)–(1.9) and (1.22)–(1.23) when the boundary strength δ is small and the initial date is the small perturbation of the smooth steady state solution. More precisely, there exist positive con-stants δ∗(c, τ), β(c, τ), M(c, τ) such that if

∥∥(ρ − ρ, j − jc)∥∥2

H 2(0,1)+ δ ≤ δ∗(c, τ ),

then, the IBVP (1.14), (1.13), (1.8)–(1.9) and (1.22)–(1.23) admits a unique global smooth solu-tion (ρ, jc, φ) satisfying

∥∥(ρ − ρ, jc − jc, φ − φ)(t, ·)∥∥2H 2(0,1)

≤ M(c, τ)∥∥(ρ − ρ, j − jc)

∥∥2H 2(0,1)

e−β(c,τ )t (1.29)

for any t ∈ [0, ∞).In this paper under the conditions of both the initial data being the small perturbation of given

steady state solution and the boundary strength being small, the following two main results are obtained.

(i) For suitably small τ and c = τ 1/2, the new scaled solution (ρτ , jτ , φτ ) (see (2.1)) to system (1.14) globally exists in time and tends exponentially to its steady state solution as t → ∞uniformly with respect to τ , see Theorem 2.2. Moreover, as τ → 0, the convergence of this solution to the solution of drift diffusion equations is also considered in Theorem 2.3.

(ii) For large enough c, similar to the result in (i), the global smooth solution to system (1.14)exists and goes to the steady state solution exponentially fast as t → ∞ uniformly with respect to c. Moreover, the global smooth solution converges to the subsonic smooth solution of the classical Euler–Poisson system (1.6), see Theorem 3.3 and Theorem 3.4.


There has been a lot of work to study the zero-relaxation limit for system (1.6). Marcati and Natalini [17] established the global weak solution to the Cauchy problem of this system for the equation of state given by (1.2) with 1 < γ ≤ 3

5 in terms of the compensated compactness method. [18] investigated the zero-relaxation limit of such a solution. [29,32] respectively dis-cussed the existence of weak solution and the zero-relaxation limit of the initial boundary value problem for the case γ > 5

3 . Recently, Nishibata and Suzuki [23] considered the zero-relaxation limit of the global smooth solution as well as its convergence rate to this system by using the en-ergy method. In their result, the initial data is not necessarily to be the small perturbation of some steady state solutions, but the equation of state was taken p(ρ) = σ 2ρ. As far as the relaxation limits for the steady state system and multidimensional system of (1.6) and more general system are concerned, we refer the readers to [5,19,25,26,28].

The most important model in special relativity is the relativistic Euler equations which reduceto the well-known classical non-relativistic compressible Euler equations in the Newton limit (|υ|/c → 0). Min and Ukai [15] investigated the non-relativistic global limit of BV weak solution of the relativistic Euler equations with the equation of state satisfying (1.2) and γ = 1. Later, Li and Ren [11] considered the global limit of entropy solution for the same problem as [15], but the equation of state is generalized to the case 1 < γ < 2. Li and Geng [10] investigated the relation between the relativistic Euler equations and the classical compressible Euler equations in the non-relativistic global limit of BV solution as c → ∞. Recently, Geng and Li [6] discussed the non-relativistic global limit of the entropy solution to the Cauchy problem of the 3 ×3 relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy. For more details on the related results on the non-relativistic limits for the local smooth solution of the three dimensional relativistic Euler equations see [20,21].

For the combined zero-relaxation and non-relativistic limit τ → 0 (c = 1√τ), we generalize

the result in [23] to (1.14) by using the energy method. The limit system is the well-known drift-diffusion model which is a coupled system of a uniformly parabolic equation and the Poisson equation. Thus, as τ → 0, the initial layer occurs between the initial function j0(x) of system (2.2) and the initial function j0(0, x) of system (2.9), if j0(x) = j0(0, x), x ∈ (0, 1). However, similar to results in [23], the initial layer vanishes for any t ∈ (0, ∞) as τ → 0 or t → ∞. Com-pared to system (1.6), the first equation of system (2.2) has an extra term of jτ

s , due to the effect of the Lorentz factor

√1 − υ2/c2 in system (1.1). This property makes our computations and

estimates to be more complicated and tedious. Thus, we have to estimate the time derivatives of jτ in order to obtain the uniform estimates for the local and global smooth solution for system (2.2). Due to the initial layer, it is difficult to estimate the time derivative of jτ . For system (1.6), it is enough to obtain the estimate of the space derivatives of j to obtain the similar results. We technically assume that c = 1√

τand make more precise estimates. In such assumptions, 1

2 is the

best order in c = 1τα (α > 0): if α > 1

2 , we can obtain the results like the case of 12 ; if 0 < α < 1

2 , it is hard to obtain the uniform estimates. For the non-relativistic limit c → ∞, the limit system is the classical Euler–Poisson system (1.6) which is a coupled system of the compressible Euler equations and the Poisson equation. Thus, the non-relativistic limit is a classical limit without singularity. The results for this limit can be proven by the method similar to the case of the combined zero-relaxation and non-relativistic limit.

This paper is arranged as follows. In Section 2, we study the combined zero-relaxation and non-relativistic limit for system (2.2). We first consider the limit for the steady state system (2.12). Then, the uniform asymptotic stability of steady state solution with respect to τ is con-structed in Section 2.1 and the combined zero-relaxation and non-relativistic limit of the global


smooth solution to system (2.2) as well as its convergence rate are made in Section 2.2. In Sec-tion 3, we consider the non-relativistic limit for system (1.14) and obtain the results as similar as that in Section 2.

Notation. Throughout this paper, δ := |ρr − ρl | + |φr | denotes the boundary strength. L2(0, 1)

is the space of square-integrable real valued functions defined on (0, 1) with the norm ‖ · ‖, and Hl(0, 1) denotes the usual Sobolev space with the norm ‖ · ‖l , especially, ‖ · ‖0 = ‖ · ‖. Ck([0, T ], Hl(0, 1)) denotes the space of the k-times continuously differentiable functions on the interval [0, T ] with values in Hl(0, 1) and

Xk

([0, T ]) :=k⋂

i=0

Ci([0, T ];Hk−i (0,1)

)(1.30)

for k = 0, 1, 2 with the norm

‖u‖ :=k∑

i=0

sup[0,T ]

∥∥∂it u(t)

∥∥Hk−i (0,1)

.

The space Ck[0, 1] denotes the space of the functions whose derivatives up to k-th order are continuous and bounded over [0, 1] with the norm

‖f ‖Ck[0,1] :=k∑

i=0

supx∈[0,1]

∣∣∂ixf (x)

∣∣. (1.31)

2. Combined zero-relaxation and non-relativistic limit

In this section, we consider the combined zero-relaxation and non-relativistic limit for the global smooth solution to the IBVP (1.14), (1.13), (1.8)–(1.9) and (1.22)–(1.23) with properties (1.21) when the light speed c and the relaxation time τ satisfy c = 1√

τ.

Introduce the new scaled variables:

t := s

τ, ρτ := ρ

(s

τ, x

), j τ (s, x) := j ( s

τ, x)

τ,

φτ (s, x) := φ

(s

τ, x

), c := 1√

τ; (2.1)

then (1.14) is transformed into⎧⎪⎨⎪⎩aτ

1 · ρτs + τ 3aτ

2 · jτ j τs + jτ

x = 0,

τ 3aτ3 · jτ ρτ

s + τ 2aτ4 · jτ

s + aτ5 · ρτ

x + τ 2aτ6 · jτ j τ

x = aτ7 · φτ

x − jτ ,

φτxx = aτ

8 · (ρτ − C(x)),

(2.2)

where aτi = ai(ρ

τ , (τjτ )2) are respectively determined by (1.15)–(1.19). Thanks to (1.13), the initial data for this system is


ρτ (0, x) = ρ(x), (2.3)

jτ (0, x) = j0(x) = j(x)

τ. (2.4)

Similar to (1.22) and (1.23), if

gτ1

(ρτ ,

(τjτ

)2)> 0 and inf

x∈(0,1)ρτ > 0, (2.5)

then the compatibility conditions at points (0, 0) and (0, 1) respectively are the following

ρτ (0,0) = ρl, ρτs (0,0) = Ψ τ

(ρl, (j0)x(0), ρx(0),φτ

x (0,0), j0(0)) = 0, (2.6)

and

ρτ (0,1) = ρl, ρτs (0,1) = Ψ τ

(ρl, (j0)x(1), ρx(1),φτ

x (0,1), j0(1)) = 0, (2.7)

where Ψ τ (ρτ , jτx , ρτ

x , φτx , jτ )(s, x) = Ψ τ (t, x) is given by (1.24) and φτ

x satisfies

φτx (s, x) :=

x∫0

√1 + τ 3

(jτ /ρτ

)2 · (ρτ − C)(s, z)dz + φr

−1∫

0

y∫0

√1 + τ 3

(jτ /ρτ

)2 · (ρτ − C)(s, z)dzdy. (2.8)

Formally, as τ → 0, system (2.2) reduces to the well-known drift-diffusion equations:

⎧⎪⎨⎪⎩ρ0

s + j0x = 0,(

σ 2ρ0)x

= ρ0φ0x − j0,

φ0xx = ρ0 − C(x).

(2.9)

It is easy to see that the initial function j0(0, x) can be determined by (2.9)2 as

j0(0, x) = ρφ0x(0, x) − σ 2ρx (2.10)

with

φ0x(0, x) :=

x∫0

(ρ − C)(z)dz + φr −1∫

0

y∫0

(ρ − C)(z)dzdy.


The steady state system of (2.2) reads⎧⎪⎨⎪⎩j τx = 0,

aτ5 · (ρτ

)x

= aτ7 · φτ

x − j τ ,

φτxx = aτ

8 · ρτ − C(x),

(2.11)

where

aτ5

(ρτ ,

(τ j τ

)2) := n(ρτ

) · (1 + τσ 2)/(ρτ)3 · gτ

1

(ρτ ,

(τ j τ

)2), (2.12)

aτ7

(ρτ ,

(τ j τ

)2) :=√(

ρτ)2 + τ 3

(j τ

)2, (2.13)

aτ8

(ρτ ,

(τ j τ

)2) :=√

1 + τ 3(j τ /ρτ

)2, n

(ρτ

) := (ρτ /ρ

)τσ 2 · ρτ , (2.14)

gτ1

(ρτ ,

(τjτ

)2) := σ 2(ρτ)2 − (

1 − τσ 2) · (τ j τ)2

. (2.15)

For the limit τ → 0, it reduces to the steady state system of (2.9)⎧⎪⎨⎪⎩j0x = 0,(σ 2ρ0)

x= ρ0φ0

x − j0,

φ0xx = ρ0 − C(x).

(2.16)

Before stating the main results of this section, we first investigate the combined relaxation and non-relativistic limit of the steady state system (2.11) as follows.

Lemma 2.1. Assume that (1.8), (1.9) and (1.10) hold. Then there exist positive constants JR , δR and KR such that for all 0 ≤ δ < δR , the system (2.11) with (1.8) and (1.9) admits a unique solution (ρτ , jτ , φτ ) ∈ C1[0, 1] × [−JR,JR] × C2(0, 1) satisfying

ρ∗ = min{ρr, ρl,C} ≤ ρτ ≤ max{ρr, ρl,C},|JR| ≤

√σ 2ρ2∗/2,∥∥(

ρτx , ρτ

xx, jτ , φτ

x , φτxx

)∥∥C0[0,1] ≤ KRδ,

and the following estimate holds∥∥ρτ − ρ0∥∥

1 + ∣∣j τ − j0∣∣ + ∥∥φτ − φ0

∥∥2 = O

(τ−1), (2.17)

where (ρ, φ) is the unique solution of the system (2.16) with (1.8) and (1.9) corresponding to j0

with |j0| ≤ JR .

Proof. This lemma can be proved by the argument similar to Theorem 3 in [13]. �Now we give the main results of this section as follows.


Theorem 2.2. Let (ρ, j , φ) be the solution of the BVP (2.11), (1.8) and (1.9) in Lemma 2.1. Assume that the initial data (ρ, j0) ∈ H 2(0, 1) satisfies (2.3) and (2.4). The boundary conditions(1.8) and (1.9) hold. In addition, the compatibility conditions (2.5) and (2.6) are assumed to be valid. Then there exist positive constants τ , δ, M and β independent of τ such that if∥∥ρ − ρτ

∥∥22 + ∥∥j0 − j τ

∥∥21 + ∥∥τ(j0)xx

∥∥2 + δ ≤ δ and 0 < τ ≤ τ , (2.18)

the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) admits a unique global smooth solution (ρτ , jτ , φτ ) ∈ X2[0, ∞) satisfying∥∥(

ρτ − ρτ)(s, ·)∥∥2

1 + ∥∥(jτ − j τ

)(s, ·)∥∥2

1 + ∥∥τjτxx(s, ·)

∥∥2 + ∥∥(φτ − φτ

)(s, ·)∥∥2

2

≤ M · (∥∥ρ − ρτ∥∥2

2 + ∥∥j0 − j τ∥∥2

1 + ∥∥τ(j0)xx

∥∥2)e−βs . (2.19)

Proof. Based on the a priori estimates in Lemma 2.7 and Lemma 2.8 of Section 2.1, we can expand the local smooth solution in Lemma 2.6 to the global smooth solution of the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) by the standard continuation argument. The estimate in (2.19) can be obtained by the energy method if δ and τ are suitably small. �Theorem 2.3. Let (ρτ , jτ , φτ ) be the solution of the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and(2.6)–(2.7) in Theorem 2.2. Then there exist positive constants m1, m2, M∗, and α independent of τ such that if δ and τ in (2.18) are sufficiently small, the following estimates hold for any s ∈ [0, ∞)

m1 ≤ infx∈[0,1]ρ

τ , supx∈[0,1]

ρτ ≤ m2, (2.20)

∣∣τjτ∣∣ ≤

√σ 2(m1)2/2, (2.21)∥∥ρτ (s, ·)∥∥2

2 + ∥∥jτ (s, ·)∥∥21 + ∥∥(

ρτs , τρτ

xs, τ2ρτ

ss, τ2jτ

s , τ 2jτxs, τ

4jτss, τj

τxx

)(s, ·)∥∥2 ≤ M∗,

(2.22)s∫

0

∥∥(ρτ

xs, τρτss, τj

τs , τ 3jτ

ss, jτxx

)(t, ·)∥∥2

dt ≤ M∗ · (1 + s). (2.23)

Moreover, ∥∥(ρτ − ρ0)(s, ·)∥∥2

1 ≤ M∗τα,∥∥(

ρτ − ρ0)s(s, ·)∥∥2 ≤ M∗s−1τα, (2.24)∥∥(

jτ − j0)(s, ·)∥∥2 ≤ M∗τα + M∗∥∥(

jτ − j0)(0, ·)∥∥2e− s

τ2 , (2.25)∥∥(jτ − j0)

x(s, ·)∥∥2 ≤ M∗s−1τα, (2.26)∥∥τ 2jτ

s (s, ·)∥∥2 ≤ M∗τα + M∗∥∥(

jτ − j0)(0, ·)∥∥2e− s

τ2 , (2.27)

where (ρ0, j0, φ0) is a unique solution of the IBVP (2.9), (2.10), (2.3), (1.8) and (1.9) which is given by Lemma 2.9 in Section 2.2.


Proof. Due to Lemma 2.1 and Theorem 2.2, the estimates in (2.20)–(2.23) are easily obtained. The convergence rate estimate in (2.24)–(2.26) will be proved in Section 2.2. �2.1. Uniform asymptotic stability of steady state solutions with respect to τ

We consider the asymptotic stability of the steady state solution obtained in Theorem 2.2. To achieve this, we divide our work into two steps. The well-posedness of local smooth solution of the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) is established in Step 1 and the a priori estimate for asymptotic stability is constructed in Step 2.

Step 1. Uniform estimates for local smooth solutions. We construct the uniform estimate with respect to τ and find a local existence time T independent of τ for the local smooth solution to the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) with properties (2.5). We, [14], have investigated the well-posedness of local smooth solution for this problem by the standard itera-tion methods as in [22]. Since the a priori estimate and local existence time of our result in [14]may depend on τ , we have to modify our result to the following Lemma 2.6 in order to take the limit τ → 0.

For reader’s convenience, the result in [14] is stated as follows. We, [14], assumed that

0 < mτ1 ≤ inf

x∈(0,1)ρ, sup

x∈(0,1)

ρ ≤ mτ2, (2.28)

|j0| ≤√

σ 2(mτ

1

)2/2, (2.29)

where mτ1 and mτ

2 are positive constants depending on τ .

Lemma 2.4. Let the initial data (ρ, j0) ∈ H 2(0, 1) satisfy (2.3), (2.4), (2.28) and (2.29) and the boundary conditions (1.8) and (1.9) hold. In addition, the compatibility conditions (2.6) and (2.7) are assumed to hold. Then, there exist positive constants mτ

1 , mτ2 , Mτ

1 and T τ depending on τ such that the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) admits a unique local smooth solution (ρτ , jτ , φτ ) ∈ X2[0, T τ ] satisfying (2.28), (2.29) and∥∥(

ρτ , jτ)(s, ·)∥∥2

2 + ∥∥(ρτ

s , j τs

)(s, ·)∥∥2

1 + ∥∥(ρτ

ss, jτss

)(s, ·)∥∥2 ≤ Mτ, for 0 ≤ t ≤ T τ , (2.30)

where the positive constant Mτ also depends on mτ1, mτ

2 , ‖(ρ, j0)‖22, φr and C.

We assume that (ρτ , jτ , φτ ) ∈ X2[0, T ] is a set of functions satisfying

0 <m1

2≤ inf

x∈(0,1)ρ, sup

x∈(0,1)

ρ ≤ 2m2, (2.31)

∣∣τjτ∣∣ ≤

√σ 2(m1)2/2, (2.32)∥∥ρτ (s, ·)∥∥2

2 + ∥∥jτ (s, ·)∥∥21 + ∥∥(

ρτs , τρτ

xs, τ2ρτ

ss, τ2jτ

s , τ 2jτxs, τ

4jτss, τj

τxx

)(s, ·)∥∥2

+s∫

0

∥∥(ρτ

xs, τρτss, τj

τs , τ 3jτ

ss, jτxx

)(t, ·)∥∥2

dt ≤ 2M, (2.33)

for some positive constants m1, m2 and M .


Lemma 2.5. There exist positive constants m1, m2, τ and M independent of τ such that if (ρτ , jτ , φτ ) ∈ X2[0, T ] is a solution of the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7)with properties (2.31)–(2.33), then the following estimates hold for all 0 < τ ≤ τ

m1 − C(m1,m2,M) · √s ≤ infx∈(0,1)

ρ, supx∈(0,1)

ρ ≤ m2 + C(m1,m2,M) · √s, (2.34)

∣∣τjτ∣∣ ≤

√σ 2(m1)2/2 + C(m1,m2,M) · s, (2.35)

∥∥ρτ (s, ·)∥∥22 + ∥∥jτ (s, ·)∥∥2

1 + ∥∥(ρτ

s , τρτxs, τ

2ρτss, τ

2jτs , τ 2jτ

xs, τ4jτ

ss, τjτxx

)(s, ·)∥∥2

+s∫

0

∥∥(ρτ

xs, τρτss, τj

τs , τ 3jτ

ss, jτxx

)(t, ·)∥∥2

dt ≤ M + C(m1,m2,M) · s, (2.36)

where C(m1,m2,M) is the positive constant only depending on m1, m2 and M , but independent of τ .

Proof. Determine m1, m2 by

m1 = infx∈(0,1)

ρ(x) and m2 = supx∈(0,1)

ρ(x).

Since

ρτ (s, x) = ρ(x) +s∫

0

ρτs (t, x)dt,

the Sobolev and Schwarz inequalities show (2.34), with the help of (2.31)–(2.33). Similarly, if

|j0| ≤√

σ 2(m1)2/2

holds, then (2.35) is easily obtained.We now construct the estimate in (2.36). By (2.2)1 × aτ

4 − (2.2)2 × (τaτ2 · jτ ), we have

Aτ · ρτs + Bτ · jτ

x − τaτ2 aτ

5 · jτ ρτx + τaτ

2 aτ7 · jτφτ

x − τaτ2 · (jτ

)2 = 0, (2.37)

where

Aτ := aτ2 aτ

4 − τ 4aτ2 aτ

3 · (jτ)2

= n(ρτ

) · (1 + τσ 2)(ρτ)−3 · ((ρτ

)2 + τ 3(1 − τσ 2) · (jτ)2)

, (2.38)

Bτ := aτ4 − τ 3aτ

2 aτ6 · (jτ

)2

= n(ρτ

) · (1 + τσ 2)/√(ρτ

)2 + τ 3(jτ

)2. (2.39)


Differentiating (2.37) with respect to s gives

jτxs = −Aτ

(Bτ

)−1 · ρτss + τaτ

2 aτ5

(Bτ

)−1jτ ρτ

xs − τaτ2 aτ

7

(Bτ

)−1 · jτφτxs + 2τaτ

2

(Bτ

)−1 · jτ j τs

− (Aτ

(Bτ

)−1)s· ρτ

s + (τaτ

2 aτ5

(Bτ

)−1 · jτ)s· ρτ

x + (τaτ

2 aτ7

(Bτ

)−1 · jτ)s· φτ

x

+ (τaτ

2

(Bτ

)−1)s· (jτ

)2. (2.40)

Differentiating (2.2)2 with respect to x, we obtain

(τ 3aτ

3 · jτ ρτs

)x

+ aτ4 · τ 2jτ

xs + (aτ

5 · ρτx

)x

+ (τ 2aτ

6 · jτ j τx

)x

+ (τ 2aτ

4

)x

· jτs

= aτ7 · φτ

xx + (aτ

7

)x

· φτx − jτ

x . (2.41)

Substituting (2.2)1, (2.2)3 and (2.40) into (2.41) yields

Gτ1 · τ 2ρτ

ss + τ 2(Gτ2 · jτ ρτ

s

)x

− (Gτ

3 · ρτx

)x

+ aτ1 · ρτ

s + Gτ4 · ρτ + τ 3Gτ

5 · jτφτxs + Gτ

6 · φτx

+ Gτ7 · τ 2ρτ

s + Gτ8 · τ 3ρτ

x + Gτ9 · τ 2jτ

s + Gτ10 · τ 3jτ + Gτ

11 · τ 3jτx − Gτ

4 · C(x) = 0, (2.42)

where

Gτ1 := n

(ρτ

)(ρτ

)−5 · (1 + τσ 2) · ((ρτ)2 + τ 3 · (jτ

)2) · ((ρτ)2 + (

1 − τσ 2) · τ 3(jτ)2)

,

(2.43)

Gτ2 := τ

(aτ

3 + aτ2 aτ

4 aτ5

(Bτ

)−1) + Aτaτ6

(Bτ

)−1, (2.44)

Gτ3 := n

(ρτ

)(ρτ

)−5(1 + τσ 2) · ((ρτ)2 + 2τ 3(jτ

)2) · ((σρτ)2 + (

1 − τσ 2) · (τjτ)2)

,

(2.45)

Gτ4 := (

ρτ)−3 · ((ρτ

)2 + 2τ 3 · (jτ)) · ((ρτ

)2 + τ 3 · (jτ))

,

Gτ5 := aτ

2 aτ4 aτ

7

(Bτ

)−1, (2.46)

Gτ6 := (

aτ7

)x

+ τ 3((aτ2 aτ

4 aτ5

(Bτ

)−1 · jτ)x

+ aτ4 · (aτ

2 aτ7

(Bτ

)−1 · jτ)s

), (2.47)

Gτ7 := aτ

4 · (Aτ(Bτ

)−1)s+ (

τ 3aτ2 aτ

4 aτ5

(Bτ

)−1jτ

)x,

Gτ8 := −aτ

4 · (aτ2 aτ

5

(Bτ

)−1jτ

)s, (2.48)

Gτ9 := τ

(1 − 2aτ

4

(Bτ

)−1) · aτ2 jτ − (

aτ4

)x, (2.49)

Gτ10 := −aτ

4 · (aτ2

(Bτ

)−1)sj τ − (

aτ2 aτ

6

(Bτ

)−1)x

· (jτ)2

,

Gτ11 := 3aτ

2 aτ6

(Bτ

)−1(jτ

)2. (2.50)

Multiplying (2.42) by ρτs , integrating it over (0, s) × (0, 1) and integrating by parts, we get


1

2

1∫0

(Gτ

1 · (τρτs

)2 + Gτ3 · (ρτ

x

)2 + Gτ4 · (ρτ

)2)dx

−1∫

0

(G1 · (τρτ

s (0, x))2 + G3 · ρ2

x + G4 · ρ2)dx

+s∫

0

1∫0

aτ1 · (ρτ

s

)2dxdt +

3∑i=1

Eτi = 0, (2.51)

where

Eτ1 := 1

2

s∫0

1∫0

(∂sG

τ1 · (τρτ

s

)2 + ∂x

(Gτ

3 · jτ) · (τρτ

s

)2 + ∂sGτ3 · (ρτ

x

)2)dxdt

+ 1

2

s∫0

1∫0

∂sGτ4 · (ρτ

)2dxdt, (2.52)

Eτ2 :=

s∫0

1∫0

(τ 3Gτ

5 · jτφτxs + Gτ

6 · φτx

) · ρτs dxdt, (2.53)

Eτ3 :=

s∫0

1∫0

(Gτ

7 · τ 2ρτs + Gτ

8 · τ 3ρτx + Gτ

9 · τ 2jτs + Gτ

10 · τ 3jτ) · ρτ

s dxdt

+ 1

2

s∫0

1∫0

(Gτ

11 · τ 3jτx − Gτ

4 · C(x)) · ρτ

s dxdt. (2.54)

Due to (2.33), the Sobolev inequality shows∣∣(ρτx , τρτ

s , j τ , τjτx , τ 2jτ

s

)∣∣ ≤ C(M). (2.55)

From now on, C(m1,m2,M) are the generic positive constants depending on m1,m2, and M , but independent of s, τ and δ. By (2.31)–(2.32), (2.43)–(2.46) and the chain rule, we obtain∣∣Eτ

1

∣∣ ≤ C(m1,m2,M) · s. (2.56)

From (2.2)2 and (1.9), it follows that

φτ (s, x) =x∫

0

y∫0

aτ8 · (ρτ − C

)(s, z)dzdy + x ·

(φr −

1∫0

y∫0

aτ8 · (ρτ − C

)(s, z)dzdy

),

(2.57)


which in combination with (2.31) and (2.32) yields

∥∥φτ (s, ·)∥∥C2[0,1] ≤ C(m1,m2,C). (2.58)

Differentiating (2.2)3 with respect to s gives

φτxxs = ∂sa

τ8 · (ρτ − C(x)

) + aτ8 · ρτ

s . (2.59)

Multiplying this equation by φτs ,

1∫0

(φτ

xs

)2 ≤ C(m1,m2)

1∫0

((ρτ

s

)2 + (τjτ

s

)2)dx, (2.60)

with the help of φτs (s, 0) = φτ

s (s, 1) = 0 and the Poincaré inequality. In virtue of (2.47)–(2.50), we also get

∣∣Eτ2

∣∣ + ∣∣Eτ3

∣∣ ≤ C(m1,m2,M) · s. (2.61)

Substituting (2.56), (2.61) into (2.51), we have, with the help of (2.31)–(2.33), that

∥∥(ρτ

s , ρτx , ρτ

)(s, ·)∥∥2 +

s∫0

1∫0

(ρτ

)2dxdt

≤ M1(m1,m2,‖ρ‖2

1,‖j0‖21

) + C(m1,m2,M) · s, (2.62)

which in combination with (2.2)2 yields

∥∥jτ (s, ·)∥∥2 +s∫

0

1∫0

(τjτ

s

)2dxdt ≤ M3 + C(m1,m2,M) · s. (2.63)

Here and in the sequel, Mj = Mj(m1,m2, ‖ρ‖2i , ‖j0‖2

i , φr,C) for j = 1, 2, ... , and i = 0, 1, 2, denote the positive constants depending only on m1, m2, φr , C, ‖ρ‖2

i and ‖j0‖2i (i = 0, 1, 2),

but independent of s, δ, τ and M .The next step is to get the estimates for the second order derivatives. Differentiating (2.42)

with respect to s, we have

Gτ1 · τ 2ρτ

sss − (Gτ

2 · τ 2jτ ρτss

)x

− (Gτ

3 · ρτxs

)x

+ aτ1 · ρτ

ss + Gτ4 · ρτ

s +4∑

i=1

Rτi = 0, (2.64)

where


Rτ1 := ∂sG

τ1 · τ 2ρτ

ss − τ 2(∂s

(Gτ

2 · jτ)ρτ

s

)x

− (∂sG

τ3 · ρτ

x

)x

+ ∂saτ1 · ρτ

s + ∂sGτ4 · ρτ , (2.65)

Rτ2 := Gτ

5 · τ 3jτφτxss + Gτ

6 · φτxs + τ 3∂s

(Gτ

5 · jτ)φτ

xs + ∂sGτ6 · φτ

x , (2.66)

Rτ3 := Gτ

7 · τ 2ρτss + Gτ

8 · τ 3ρτxs + Gτ

9 · τ 2jτss + Gτ

10 · τ 3jτs + Gτ

11 · τ 3jτxs, (2.67)

Rτ4 := ∂sG

τ7 · τ 2ρτ

s + ∂sGτ8 · τ 3ρτ

x + ∂sGτ9 · τ 2jτ

s + ∂sGτ10 · τ 3jτ + ∂sG

τ11 · τ 3jτ

x

− ∂sGτ4 · C(x). (2.68)

Multiplying (2.64) by ρτs , similar to (2.51), we have

1∫0

(Gτ

1 · τ 2ρτssρ

τs + aτ

1 · (ρτs )2

2

)dx −

1∫0

(G1 · τ 2ρτ

ss(0, x)ρτs (0, x) + a1 · (ρτ

s (0, x))2

2

)dx

+s∫

0

1∫0

(Gτ

4 · (ρτs

)2 + Gτ3 · (ρτ

xs

)2 − Gτ1 · (τρτ

ss

)2)dxdt

+ Rτ +3∑1

s∫0

1∫0

Rτi ρτ

s dxdt = 0, (2.69)

where

Rτ := −s∫

0

1∫0

(∂sG

τ1 · τ 2ρτ

ssρτs − Gτ

2 · τ 2jτ ρτssρ

τxs + ∂sa

τ1 · (ρτ

s )2

2

)dx. (2.70)

It is easy to obtain

∣∣Rτ∣∣ +

s∫0

1∫0

Rτ1 · ρτ

s dxdt

≤ C(m1,m2,M) · s + 1

8

s∫0

1∫0

aτ1 · (τρτ

ss

)2dxdt + ε

s∫0

1∫0

(ρτ

xs

)2dxdt, (2.71)

for the arbitrarily positive constant ε independent of s, δ and τ . From (2.57), we have

1∫0

(φτ

xss

)2dx ≤ C(m1,m2)

1∫0

(∣∣ρτs

∣∣4 + τ 6∣∣jτ

s

∣∣2 + ∣∣ρτss

∣∣2 + τ 4∣∣jτ

ss

∣∣2)dx. (2.72)

A straightforward calculation shows


∣∣∂sGτ6

∣∣ ≤ C(m1,m2) · (∣∣ρτs

∣∣ + ∣∣jτs

∣∣) · (∣∣ρτx

∣∣ + τ 2∣∣jτ

x

∣∣ + τ 2∣∣ρτ

s

∣∣)+ C(m1,m2) · ((∣∣ρτ

s

∣∣ + ∣∣jτx

∣∣ + τ 2∣∣jτ

s

∣∣) · τ 3∣∣jτ

s

∣∣ + τ 2∣∣jτ

xs

∣∣)+ C(m1,m2) · (τ 2

∣∣ρτss

∣∣ + τ 3∣∣jτ

ss

∣∣ + ∣∣ρτxs

∣∣). (2.73)

Thus, it holds that

s∫0

1∫0

∂sGτ6 · φτ

xρτs dxdt

≤ C(m1,m2,M) · s + ε

s∫0

1∫0

(ρτ

xs

)2dxdt + 1

3

s∫0

1∫0

τ 6(jτss

)2dxdt. (2.74)

Differentiating (2.2)2 with respect to s,

aτ4 · τ 2jτ

ss + jτs + ∂sa

τ4 · τ 2jτ

s + (τ 3aτ

3 · jτ ρτs + aτ

5 · ρτx + aτ

7 · φτx + τ 2aτ

6 · jτ j τx

)s= 0.

(2.75)

Multiplying this equation by τ 4jτss ,

s∫0

1∫0

τ 6(jτss

)2dxdt +

1∫0

τ 4 (j τs )2

2dx ≤ M4 + C(m1,m2,M) · s, (2.76)

which in combination with (2.74) gives

s∫0

1∫0

∂sGτ6 · φτ

xρτs dxdt ≤ 1

3M4 + C(m1,m2,M) · s + ε

s∫0

1∫0

(ρτ

xs

)2dxdt. (2.77)

Thus,

s∫0

1∫0

Rτ2 · ρsdxdt ≤ 1

3M4 + C(m1,m2,M) · s + ε

s∫0

1∫0

(ρτ

xs

)2dxdt. (2.78)

Similarly, the Cauchy–Schwarz inequality applied to (2.67) and (2.68) shows

s∫0

1∫0

(Rτ

3 + Rτ4

) · ρτs dxdt

≤ 1

3M4 + C(m1,m2,M) · s + ε

s∫ 1∫ ((ρτ

xs

)2 + (τρτ

ss

)2)dxdt. (2.79)

0 0


Substituting (2.71), (2.78) and (2.79) into (2.69), we have

1∫0

(Gτ

1 · τ 2ρτssρ

τs + aτ

1 · (ρτs )2

2

)dx

+ 1

2

s∫0

1∫0

(Gτ

4 · (ρτs

)2 + Gτ3 · (ρτ

xs

)2 − (Gτ

1 + aτ1

) · τ 2(ρτss

)2)dxdt

≤ M8 + C(m1,m2,M) · s (2.80)

for suitably small ε and 0 < τ ≤ τ0 with τ0 being a positive constant independent of s, δ and τ , where we have used∣∣∣∣∣

1∫0

(G1 · τ 2ρτ

ss(0, x)ρτs (0, x) + a1 · (ρτ

s (0, x))2

2

)dx

∣∣∣∣∣ ≤ M7 (2.81)

for any 0 < τ ≤ τ0.On the other hand, multiplying (2.64) by τ 2ρτ

ss , similar to (2.80), there exists a positive con-stant τ1 independent of s, δ and τ , such that for any 0 < τ ≤ τ1

1

2

1∫0

(Gτ

1 · τ 4(ρτss

)2 + Gτ3 · (ρτ

xs

)2 + Gτ4 · (τρτ

s

)2)dx + 1

2

s∫0

1∫0

aτ1 · (τρτ

ss

)2dxdt

≤ M9 + C(m1,m2,M) · s. (2.82)

Multiplying the two sides of this estimate by a positive constant K independent of s, δ and τ and combining the resulting estimate with (2.80), using (2.31)–(2.33), it follows that

∥∥(τ 2ρτ

ss, ρτs , τρτ

xs

)(s, ·)∥∥2 +

s∫0

∥∥(τρτ

ss, ρτxs, ρ

τs

)(t, ·)∥∥2

dt ≤ M11 + C(m1,m2,M) · s,

(2.83)

for the sufficiently large K .In virtue of (2.31)–(2.33), (2.58), (2.63) and (2.83), we get from (2.37)∥∥jτ

x (s, ·)∥∥2 ≤ M12 + C(m1,m2,M) · s. (2.84)

Similarly, it follows from (2.40)

1∫τ 4(jτ

xs

)2dx ≤ M13 + τ 2M14

1∫τ 4(jτ

xs

)2dx + C(m1,m2,M) · s. (2.85)

0 0


We take 0 < τ ≤ 12M14

and obtain

1∫0

τ 4(jτxs

)2dx ≤ 2M13 + C(m1,m2,M) · s. (2.86)

From (2.41),

1∫0

(ρτ

xx

)2dx ≤ M15 + τ 2M16

1∫0

(ρτ

xx

)2dx + M17

1∫0

(τjτ

xx

)2dx + C(m1,m2,M) · s. (2.87)

We take 0 < τ ≤ min{ 12M14

, 12M16

, τ0, τ1} and obtain

1∫0

(ρτ

xx

)2dx ≤ 2M15 + 2M17

1∫0

(τjτ

xx

)2dx + C(m1,m2,M) · s. (2.88)

Differentiating (2.37) with respect to x and multiplying it by τ 2jτxx ,

1∫0

(τjτ

xx

)2dx ≤ M18 + τ 2M19

1∫0

(ρτ

xx

)2dx + C(m1,m2,M) · s. (2.89)

Thus, we take 0 < τ ≤ min{ 12M14

, 12M16

, 14M17M19

, τ0, τ1} and obtain from (2.88)

1∫0

(ρτ

xx

)2dx ≤ 2M20 + C(m1,m2,M) · s, (2.90)

which in combination with (2.89) yields

1∫0

(τjτ

xx

)2dx ≤ M21 + C(m1,m2,M) · s. (2.91)

By analogy with (2.76),

1∫0

τ 8(jτss

)2dx ≤ M22 + C(m1,m2,M) · s. (2.92)

Finally, we chose τ∗ and M as


τ ∗ = min

{1

2M14,

1

2M16,

1

4M17M19, τ0, τ1

},

M = M1 + M3 + M4 + M12 + 2M13 + 2M20 + M21 + M22. (2.93)

Then, summing up (2.62), (2.63), (2.76), (2.84), (2.86), (2.90)–(2.92) gives (2.36). �Lemma 2.6. Suppose that the assumptions in Lemma 2.4 hold. Then, there exist positive con-stants m1, m2, δ, τ and T independent of τ such that if 0 ≤ δ < δ and 0 < τ ≤ τ , then the IBVP (2.2), (2.3)–(2.4), (1.8)–(1.9) and (2.6)–(2.7) admits a unique local smooth solution (ρτ , jτ , φτ ) ∈ X[0, T ] satisfying (2.31)–(2.33).

Proof. This lemma is easily proven by a contradiction argument with the help of estimates in Lemma 2.5 and we omit it here. �

Step 2. Uniform a priori estimates with respect to τ . As mentioned in introduction, we, [14], have constructed the a priori estimate in (1.29) with the positive constants Mτ and βτ

depending on τ . In this step, we obtain the uniform a priori estimate (2.19) with respect to τ , where the positive constants M and β are independent of τ .

Set

ψτ (s, x) := ρτ (s, x) − ρτ (x), J τ (s, ·) := jτ (s, x) − j τ (x),

eτ (s, x) := φτ (s, x) − φτ (x).

Subtracting (2.11) from (2.2), we obtain the system of (ψτ , ητ , eτ ) for [0, ∞) × (0, 1):

aτ1 · ψτ

s + τ 3aτ2 · (ητ + j τ

) · ητs + ητ

x = 0, (2.94)

τ 3aτ3 · (ητ + j τ

) · ψτs + aτ

4 · τ 2ητs + aτ

5 · ψτx + (

aτ5 − aτ

5

) · (ψτx + ρτ

x

) + aτ6 · (ητ + j τ

) · ητx

= aτ7 · eτ

x + (aτ

7 − aτ7

) · (eτx + φτ

x

) − ητ , (2.95)

eτxx = aτ

8 · ψτ + (aτ

8 − aτ8

) · (ψτ + ρτ − C(x)), (2.96)

where aτi = ai(ψ + ρ, τ 2(η + j )2) are given by (1.15)–(1.19), respectively. This system is sup-

plemented with the initial and boundary conditions:

ψτ (0, x) = ψ(x) := ρ(x) − ρτ (x), ητ (0, x) = η(x) := j0(x) − j τ , x ∈ [0,1], (2.97)

ψτ (s,0) = ψτ (s,1) = 0, s ≥ 0, (2.98)

eτ (s,0) = eτ (s,1) = 0, s ≥ 0. (2.99)

Since (ρτ , j τ , φτ ) ∈ H 2(0, 1), the unique existence of local smooth solution to the IBVP (2.94)–(2.99) follows from Lemma 2.1 and Lemma 2.6. We assume that a solution to the IBVP (2.94)–(2.99) exists for s ∈ [0, T ] with any fixed T > 0 and define

N(T ) := sup(∥∥ψτ (s, ·)∥∥2 + ∥∥ητ (s, ·)∥∥1 + ∥∥τjτ

xx

∥∥) � 1.
s∈[0,T ]


Thus there exist positive constants m1 and m2 independent of t such that for any t ∈ [0, T ],

m1 ≤ ψτ + ρτ ≤ m2, (2.100)∣∣τ(ητ + j τ

)∣∣ ≤√

σ 2(m1)2/2, (2.101)∣∣(ψτ ,ψτx , ητ , ητ

x

)∣∣ ≤ O(1)N(T ), (2.102)

where we have used the Sobolev inequality in order to obtain (2.102).We first write (2.95) and (2.96) into the more explicit forms. A straightforward computation

shows

aτ5 − aτ

5 = aτ5

(ρτ ,

(τjτ

)2) − aτ5

(ρτ ,

(τjτ

)2) + a5(ρτ ,

(τjτ

)2) − a5(ρτ ,

(τ j τ

)2)= hτ

1 · ψτ − τ 2hτ2

(ρτ

) · (ητ + 2j τ) · ητ , (2.103)

where

hτ1 = hτ

1

(ρτ ,ψτ , τ 2(ητ + j τ

)2) :=1∫

0

aτ5ρτ

(ρτ + μψτ , τ 2(ητ + j τ

)2)dμ,

a5ρτ

(ρτ ,

(τjτ

)2) := n(ρτ

)(ρτ

)−4 · (1 + τσ 2)[(σρτ)2 − (

2 − τσ 2) · (1 − τσ 2) · (τjτ)2]

,

h2 = h2(ρτ

) := n(ρτ

)(ρ)−4 · (1 + τσ 2).

Similarly,

aτ7 − aτ

7 = hτ3 · ψτ + τ 3hτ

4 · (ητ + 2j τc

) · ητ , (2.104)

aτ8 − aτ

8 = −τ 3hτ5 · (ητ + j τ

)2 · ψτ + τ 3hτ6 · (ητ + 2j τ

) · ητ , (2.105)

where

hτ3

(ψτ + ρτ , τ 2(ητ + j τ

)2, ρτ

) := ((ψτ + ρτ

) + ρτ) · (ωτ

1 + ωτ2

)−1,

h4(ψτ + ρτ , τ 2(ητ + j τ

)2, ρτ

) := (ωτ

1 + ωτ2

)−1,


)2, ρτ

) := (ρτ

)−1(ψτ + ρτ

)−1(ρτωτ

1 + (ψτ + ρτ

)ωτ

2

)−1

× ((ψτ + ρτ

) + ρτ),


)2, ρτ

) := (ρτ

)−1 · (ρτωτ1 + (

ψτ + ρτ)ωτ

2

)−1 · (ψτ + ρτ),

ωτ1 :=

√(ψτ + ρτ

)2 + τ 3(ητ + j τ

)2, ωτ

2 :=√(

ρτ)2 + τ 3

(j τ

)2.

Substituting (2.103) and (2.104) into (2.95) and (2.105) into (2.96), we respectively have


τ 3aτ3 · (ητ + j τ

) · ψτs + τ 2aτ

4 · ητs + ητ + τ 2aτ

6 · (ητ + j τ) · ητ

x + aτ5 · ψτ

x − aτ7 · eτ

x = Fτ ,

(2.106)

eτxx = aτ

8 · ψτ + τ 3Fτ5 · ψτ + τ 3Fτ

6 · ητ , (2.107)

where

Fτ := Fτ1 · ψτ

x + Fτ2 · ψτ + Fτ

3 · ητ + Fτ4 · eτ

x , (2.108)

Fτ1 := τ 2hτ

2

(ρτ

) · (ητ + 2j τ) · ητ − hτ

1 · ψτ ,

F τ2 := hτ

3 · φτx − hτ

1 · ρτx , (2.109)

Fτ3 := (

τ 2hτ2

(ρτ

) · ρτx + τ 3hτ

4 · φτx

) · (ητ + 2j τ),

F τ4 := hτ

3 · ψτ + τ 3hτ4 · (ητ + 2j τ

), (2.110)

Fτ5 := −hτ

5 · (ητ + j τ)2 · (ψτ + ρτ − C(x)

),

F6 := hτ6 · (ητ + 2j τ

) · (ψτ + ρτ − C(x)). (2.111)

Lemma 2.7. There exist positive constants δ0∗ , β0∗ , τ 0∗ and M independent of s, δ and τ , such that for any fixed T > 0, if

N(T ) + δ ≤ δ0∗ and 0 < τ ≤ τ 0∗ ,

then the following estimates hold for any s ∈ [0, T ]∥∥eτx(s, ·)∥∥2

1 + ∥∥eτxs(s, ·)

∥∥21 ≤ M

∥∥(ψτ ,ψτ

s

)(s, ·)∥∥2 + M · (N(T ) + δ

)∥∥ητ (s, ·)∥∥2, (2.112)∣∣(eτ

x , eτxx, τψτ

s , τ 2ητs

)∣∣ ≤ MN(T ), (2.113)∥∥ητx(s, ·)∥∥2 ≤ M

∥∥(ψτ

x ,ψτs ,ψτ

)(s, ·)∥∥2 + M · (N(T ) + δ

)∥∥ητ (s, ·)∥∥2, (2.114)∥∥ητ

s (s, ·)∥∥2 ≤ M∥∥(

ψτx ,ψτ

s ,ψτ , ητ)(s, ·)∥∥2

, (2.115)

d

ds

1∫0

((ητ )2

2+ (

aτ5 · ψτ − aτ

7 · eτx

) · ητ

)dx

+ d

ds

1∫0

(hτ

1 · (ψτ + ρτx

) − hτ3 · (φτ + eτ

x

)) · ψτητ dx +1∫

0

((ητ

)2 + (τητ

s

)2)dx

≤ M∥∥(

ψτs ,ψτ

x ,ψτ)(s, ·)∥∥2

. (2.116)

Proof. Due to (2.99), (2.102) and (2.107), we easily obtain

1∫ ((eτx

)2 + (eτxx

)2)dx ≤ M

1∫ (ψτ

)2dx + M · (N(T ) + δ

) 1∫ (ητ

)2dx. (2.117)

0 0 0


Differentiating (2.107) with respect to s gives

eτxxs = aτ

8 · ψs + τ 3(Fτ5 · ψτ + Fτ

6 · ητ)s. (2.118)

Multiplying this equation by eτs and integrating it over (0, 1), we get

1∫0

(eτxs

)2 ≤ M

1∫0

ψ2s dx + M · (N(T ) + δ

) 1∫0

(τητ

s

)2dx, (2.119)

with the help of eτs (s, 0) = eτ

s (s, 1) = 0, integration by parts and the Poincaré inequality. Taking L2-norm over (2.118) with respect to x and combining the resulting equation with (2.117) and (2.119) yields (2.112). By (2.107) and (2.112), it follows that∣∣(eτ

x , eτxx

)∣∣ ≤ O(1)N(T ). (2.120)

Similar to (2.37), we have

Aτ · ψτs + Bτ · ητ

x − τaτ2 · (ητ + j τ

) · ητ − τaτ2 aτ

5 · (ητ + j τ) · ψτ

x

+ τaτ2 aτ

7 · (ητ + j τ) · eτ

x + τaτ2 · (ητ + j τ

) · Fτ = 0, (2.121)

where Aτ = Aτ (ψτ + ρτ , τ 2(ητ + j τ )2) and Bτ = Bτ (ψτ + ρτ , τ 2(ητ + j τ )2) are given by (2.38) and (2.39), respectively. In virtue of (2.100) and (2.101), there exists a positive constant m3 such that

Aτ ≥ m3 and Bτ ≥ m3. (2.122)

Then, we obtain from (2.106) and (2.121) that∣∣(τψτs , τ 2ητ

s

)∣∣ ≤ MN(T ), (2.123)

which in combination with (2.120) leads to (2.113). We obtain (2.114) from (2.112) and (2.121). Similarly, multiply (2.106) by 1

aτ4

· τ 2ητs and integrate it over (0, 1) to get (2.115).

Multiplying (2.106) by ητ and integrating it over (0, 1), the Cauchy–Schwarz inequality shows

1∫0

(ητ

)2dx ≤ M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

x

)2 + τ 4(ητs

)2 + (eτx

)2)dx

+ M · (N(T ) + δ) 1∫

0

(ητ

)2dx, (2.124)

which in combination with (2.112) and (2.114) yields


1∫0

(ητ

)2dx ≤ M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx + τ 2Λ1∗

1∫0

(τητ

s

)2dx (2.125)

for sufficiently small N(T ) + δ and some positive constant Λ1∗ independent of s, δ and τ . Mul-tiplying (2.106) by ητ

s and integrating it over (0, 1), we have

d

ds

1∫0

(ητ )2

2dx +

1∫0

aτ4 · (τητ

s

)2dx +

3∑i=1

Πτi = 0, (2.126)

where

Πτ1 :=

1∫0

(τ 3aτ

3 · ψτs + τ 2aτ

6 · ητx

) · (ητ + j τ) · ητ

s dx, (2.127)

Πτ2 :=

1∫0

(aτ

5 · ψτx − aτ

7 · eτx

) · ητs dx, (2.128)

Πτ3 := −

1∫0

Fτ · ητs dx. (2.129)

It is easy to obtain that

∣∣Πτ1

∣∣ ≤ M ·1∫

0

((ψτ

s

)2 + (ητ

x

)2)dx + ε

1∫0

(τητ

s

)2dx, (2.130)

for the arbitrarily positive constant ε. By integration by parts,

Πτ2 = d

ds

1∫0

(aτ

5 · ψτx − aτ

7 · eτx

) · ητ dx +1∫

0

((aτ

5 · ητ)x

· ψτs − aτ

7 · ητ eτxs

)dx

≥ d

ds

1∫0

(aτ

5 · ψτx − aτ

7 · eτx

) · ητ dx − M

1∫0

((ψτ

s

)2 + (eτxs

)2 + (ητ

x

)2)dx − ε

1∫0

(ητ

)2dx.

(2.131)

From (2.109), it follows that

−1∫

0

Fτ1 · ψτ

x ητs dx = −τ 2

1∫0

hτ2

(ρτ

) · (ητ + 2j τ) · ητψτ

x ητs dx +

1∫0

hτ1 · ψτ

x ψτητs dx.

(2.132)


The first term on the right side is bounded by M∫ 1

0 (ψτx )2dx + ε

∫ 10 (τητ

s )2dx. The second term can be computed as

1∫0

hτ1 · ψτψτ

x ητs dx

= d

ds

1∫0

hτ1 · ψτψτ

x ητ dx −1∫

0

(hτ

1 · ψτ)s· ητψτ

x dx +1∫

0

(h1 · ψτητ

)x

· ψτs

≥ d

ds

1∫0

hτ1 · ψτψτ

x ητ dx − M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ητ

x

)2)dx − ε

1∫0

(τητ

s

)2dx,

(2.133)


−1∫

0

Fτ1 · ψτ

x ητs dx

≥ d

ds

1∫0

hτ1 · ψτητψτ

x − M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ητ

x

)2)dx − ε

1∫0

(τητ

s

)2dx. (2.134)

Similarly, we obtain from (2.109) and (2.110)

−1∫

0

Fτ2 · ητ

s dx ≥ d

ds

1∫0

(hτ

3 · φx − hτ1 · ρx

) · ψτητ dx −1∫

0

((ψτ

s

)2 + (ψτ

s

)2)dx

− ε

1∫0

(ητ

)2dx, (2.135)

1∫0

Fτ3 · ητητ

s dx ≤ M · (N(T ) + δ) 1∫

0

(ητ

)2dx + ε

1∫0

(τητ

s

)2dx, (2.136)

−1∫

0

Fτ4 · eτ

xητs dx ≥ − d

ds

1∫0

hτ3 · ψτeτ

xητ dx − M

1∫0

((ψτ

s

)2 + (eτxs

)2 + (ητ

x

)2)dx

− ε

1∫ (τητ

)2dx − M · (N(T ) + δ

) 1∫ (ητ

)2dx. (2.137)

0 0


Thus, (2.129) together with (2.134)–(2.137), (2.112) and (2.114) yields

Π3 ≥1∫

0

(hτ

1 · (ψτx + ρτ

x

) − hτ3 · (φτ

x + eτx

))ψτητ dx − M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

x

)2)dx

− M · (N(T ) + δ) 1∫

0

(ητ

)2dx − ε

1∫0

((ητ

)2 + (τητ

s

)2)dx. (2.138)


d

ds

1∫0

((ητ )2

2+ (

aτ5 · ψτ + aτ

7 · eτx

) · ητ

)dx

+1∫

0

(hτ

1 · (ψτx + ρτ

x

) − hτ3 · (φτ

x + eτx

))ψτητ dx + 1

2

1∫0

((ητ

)2 + aτ4 · (τητ

s

)2)dx

≤ M

1∫0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx + τ 2Λ1∗

1∫0

(τητ

s

)2dx, (2.139)

for suitably small N(T ) + δ and ε. Due to (2.100)–(2.101), there exists a positive constant m4independent of τ , such that aτ

4 ≥ m4. Finally, we take 0 < τ 2 ≤ m44Λ1∗

and obtain (2.116). �We state the estimates for the higher order derivatives as follows.

Lemma 2.8. There exist positive constants δ1∗ , β1∗ , τ 1∗ and M independent of s, δ and τ , such that for any fixed T > 0, if

N(T ) + δ ≤ δ1∗ and 0 < τ ≤ τ 1∗ ,

then the following estimates hold for any s ∈ [0, T ]∥∥(

τ 2ψτss, τψτ

xs,ψτs ,ψτ

x ,ψτ , ητ)(s, ·)∥∥2 ≤ M · (∥∥(ψ,η)

∥∥21 + ∥∥(ψxx, τηxx)

∥∥2)e−β1∗ s , (2.140)∥∥(

ητx, τ 2ητ

s

)(s, ·)∥∥2 ≤ M · (∥∥(ψ,η)

∥∥21 + ∥∥(ψxx, τηxx)

∥∥2)e−β1∗ s , (2.141)∥∥eτ

x(s, ·)∥∥21 + ∥∥eτ

xs(s, ·)∥∥2

1 ≤ M · (∥∥(ψ,η)∥∥2

1 + ∥∥(ψxx, τηxx)∥∥2)

e−β1∗ s , (2.142)∥∥(ψτ

xx, τ2ητ

xs, τ4ητ

ss, τητxx

)(s, ·)∥∥2 ≤ M · (∥∥(ψ,η)

∥∥21 + ∥∥(ψxx, τηxx)

∥∥2)e−β1∗ s , (2.143)

s∫0

∥∥(τητ

s , τ 3ηss,ψτxs

)(t, ·)∥∥2

dt ≤ M · (1 + s). (2.144)


Proof. Differentiating (2.106) with respect to x, it holds that(τ 3aτ

3 · (ητ + j τ) · ψτ

s

)x

+ τ 2aτ4 · ητ

xs + ητx + (

aτ5 · ψτ

x

)x

− aτ7 · eτ

xx

+ τ 2(aτ6 · (ητ + j τ

) · ητx

)x

+ τ 2(aτ4

)x

· ητs − (

aτ7

)x

· eτx − Fτ

x = 0. (2.145)

It is hard to deal with the terms of ητxx and ητ

xs in (2.145). Thus, we technically transform this equation into the second order quasi-linear wave equation (2.147) of ψτ . Differentiating (2.121)with respect to s, we have

ητxs = −Aτ

(Bτ

)−1 · ψτss + aτ

2 aτ5

(Bτ

)−1 · (ητ + j τ) · ψτ

xs − τaτ2 aτ

7

(Bτ

)−1 · (ητ + j τ) · eτ

xs

+ τaτ2

(Bτ

)−1 · (ητ + j τ) · ητ

s − τaτ2

(Bτ

)−1 · (ητ + j τ) · Fτ

s − (τAτ

(Bτ

)−1)s· ψτ

s

+ (τaτ

2 aτ5

(Bτ

)−1 · (ητ + j τ))

s· ψτ

x + (τaτ

2

(Bτ

)−1 · (ητ + j τ))

s· ητ

− (τaτ

2 aτ7

(Bτ

)−1 · (ητ + j τ))

s· eτ

x − (τaτ

2

(Bτ

)−1 · (ητ + j τ))

s· Fτ . (2.146)

Substituting (2.94), (2.107), (2.121) and (2.146) into (2.145),

Dτ1 · τ 2ψτ

ss − (aτ

5 · ψτx

)x

+ aτ7 aτ

8 · ψτ + aτ1 · ψτ

s − (Dτ

2 · τ 2ψτs

)x

− (Dτ

3 · ψτx

)x

+ Dτ4 · eτ

xx + Dτ5 · τ 3eτ

xs + Dτ6 · eτ

x + Dτ7 · τ 2ψτ

s + Dτ8 · ψτ

x + Dτ9 · ψτ

+ Dτ10 · τ 2ητ

s + Dτ11 · ητ

x + Dτ12 · ητ = 0, (2.147)

where Dτ1 = Gτ

1(ψτ + ρτ , τ 2(ητ + j τ )2) is given by (2.43) and

Dτ2 := (

τ(aτ

3 + aτ2 aτ

4 aτ5

(Bτ

)−1) − aτ6 · Aτ

(Bτ

)−1) · (ητ + j τ) + τΓ τ

1 · Fτ1 , (2.148)

Dτ3 := τ 3aτ

2 aτ5 aτ

6

(Bτ

)−1 · (ητ + j τ)2 + Γ τ

2 · Fτ1 ,

Dτ4 := τ 3aτ

2 aτ6 aτ

7

(Bτ

)−1 · (ητ + j τ)2 + Γ τ

2 · Fτ4 , (2.149)

Dτ5 := aτ

2 aτ4 aτ

7

(Bτ

)−1 · (ητ + j τ) + Γ τ

1 · Fτ4 , (2.150)

Dτ6 := τ 3(aτ

2 aτ6 aτ

7

(Bτ

)−1 · (ητ + j τ)2)

x+ τ 3aτ

4 · (aτ2 aτ

7

(Bτ

)−1 · (ητ + j τ))

s

+ τ 3Γ τ1 · ∂sF

τ4 + Γ τ

2 · ∂xFτ4 + Γ τ

3 · Fτ4 + ∂xa

τ7 , (2.151)

Dτ7 := (

τaτ2 aτ

4 aτ5

(Bτ

)−1 · (ητ + j τ))

x+ aτ

4 · (Aτ(Bτ

)−1)s+ τΓ τ

1 · Fτ2

− (τΓ τ

1 · Fτ1

)x, (2.152)

Dτ8 := −τ 3aτ

4 · (aτ2 aτ

5

(Bτ

)−1 · (ητ + j τ))

s+ τ 3Γ τ

1 · ∂sFτ1

+ ∂xΓτ

2 · Fτ1 + Γ τ

2 · Fτ2 + Γ τ

3 · Fτ1 , (2.153)

Dτ9 := τ 3aτ

5 · Fτ5 + τ 3Γ τ

1 · ∂sFτ2 + Γ τ

2 · ∂xFτ2 + τ 3Γ τ

3 · Fτ2 , (2.154)

Dτ10 := −τaτ

2 aτ4

(Bτ

)−1 · (ητ + j τ) − aτ

2 · (ητ + j τ) − ∂xa

τ4 + τΓ τ

1 · Fτ3 , (2.155)

Dτ := −τ 3aτ aτ(Bτ

)−1 · (ητ + j τ)2 + Γ τ · Fτ , (2.156)
11 2 6 2 3


Dτ12 := −τ 3Γ τ

3 · (1 − Fτ3

) + τ 3aτ7 · Fτ

6 + Γ τ1 · ∂sF

τ3 + Γ τ

2 · ∂xFτ3 + τ 3Γ τ

3 · Fτ3 , (2.157)

Γ τ1 := aτ

2 aτ4

(Bτ

)−1 · (ητ + j τ), Γ τ

2 := τ 3aτ2 aτ

6

(Bτ

)−1 · (ητ + j τ)2 + 1,

Γ τ3 := aτ

4 · (aτ2

(Bτ

)−1(ητ + j τ

))s+ (

aτ2 aτ

6

(Bτ

)−1 · (ητ + j τ)2)

x.

Multiplying (2.147) by ψτ , integrating it over (0, 1) and integrating by parts, we have

d

dt

1∫0

(Dτ

1 · τ 2ψτs ψτ + aτ

1 · (ψτ )2

2

)dx

+1∫

0

(aτ

7 aτ8 · (ψτ

)2 + aτ5 · (ψτ

x

)2 − Dτ1 · (ψτ

s

)2)dx +

3∑i=0

χτi = 0, (2.158)

where

χτ1 :=

1∫0

(Dτ

2 · τ 2ψτs ψτ

x + Dτ3 · (ψτ

x

)2 − ∂sDτ1 · τ 2ψτ

s ψτ − ∂saτ1 · (ψτ )2

2

)dx, (2.159)

χτ2 :=

1∫0

(Dτ

4 · eτxx + Dτ

5 · eτxs + Dτ

6 · eτx

) · ψτdx, (2.160)

χτ3 :=

1∫0

(Dτ

7 · τ 2ψs + Dτ6 · ψτ

x + Dτ9 · ψτ + Dτ

10 · τ 2ητs + Dτ

11 · ητx

) · ψτdx

+1∫

0

Dτ12 · ητψτdx. (2.161)

Due to (2.148), (2.149) and the chain rule,∣∣∂sDτ1

∣∣ ≤ M · (∣∣ψτs

∣∣ + τ 2∣∣ητ

s

∣∣), (2.162)∣∣∂saτ1

∣∣ ≤ M · (τ ∣∣ψτs

∣∣ + τ 2∣∣ητ

s

∣∣), (2.163)∣∣(Dτ2 ,Dτ

3

)∣∣ ≤ M · (∣∣ητ∣∣ + ∣∣j τ

∣∣ + ∣∣ψτ∣∣), (2.164)

Thus, (2.159) together with (2.162)–(2.164) shows

∣∣χτ1

∣∣ ≤ M · (N(T ) + δ) 1∫

0

((ψτ

)2 + (ψτ

x

)2 + (ψτ

s

)2)dx, (2.165)

with the help of (2.100)–(2.101). By (2.149)–(2.157), we similarly obtain from (2.160) and (2.161)


∣∣χτ2

∣∣ ≤ M · (N(T ) + δ) 1∫

0

((eτxx

)2 + (eτxs

)2 + (eτx

)2)dx, (2.166)

and

∣∣χτ3

∣∣ ≤ M · (N(T ) + δ) 1∫

0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + τ 4(ητs

)2 + (ητ

x

)2 + (ητ

)2)dx.

(2.167)


d

dt

1∫0

(Dτ

1 · τ 2ψτs ψτ + aτ

1 · (ψτ )2

2

)dx +

1∫0

(aτ

5 · (ψτx

)2 + aτ7 aτ

8 · (ψτ)2 − Dτ

1 · (τψτs

)2)dx

≤ M · (N(T ) + δ) 1∫

0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

)2)dx, (2.168)

with the help of (2.112)–(2.115).Multiplying (2.147) by ψτ

s , integrating the resulting equation over (0, 1), similar to (2.168), we obtain

d

ds

1∫0

(Dτ

1 · (τψs)2

2+ (

aτ5 + Dτ

3

) · (ψτx )2

2+ aτ

7 aτ8 · (ψτ )2

2

)dx +

1∫0

aτ1 · (ψτ

s

)2dx

≤ M · (N(T ) + δ) 1∫

0

((ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

)2)dx. (2.169)

Multiplying this estimate by a large positive constant K1 and combining the resulting equation with (2.168) gives

d

ds

1∫0

(K1D

τ1 · (τψs)

2

2+ Dτ

1 · τ 2ψτs ψτ + K1 · (aτ

5 + Dτ3

) · (ψτx )2

2

+ (aτ

1 + K1aτ7 aτ

8

) · (ψτ )2

2

)dx

+1∫

0

(aτ

5 · (ψτx

)2 + aτ7 aτ

8 · (ψτ)2 + (

K1aτ1 − τ 2Dτ

1

) · (ψτs

)2)dx

≤ M · (N(T ) + δ) 1∫ ((

ψτs

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

)2)dx. (2.170)

0


Differentiating (2.147) with respect to s, we have

Dτ1 · τ 2ψτ

sss − (aτ

5 · ψτxs

)x

+ aτ7 aτ

8 · ψτs + aτ

1 · ψτss − (

Dτ2 · τ 2ψτ

ss

)x

− (Dτ

3 · ψτxs

)x

+ Dτ4 · eτ

xxs + Dτ5 · τ 3eτ

xss + Dτ6 · eτ

xs +3∑

i=1

I τi = 0, (2.171)

where

I τ1 := ∂sD

τ1 · τ 2ψτ

ss − (∂sD

τ2 · τ 2ψτ

s

)x

− (∂sD

τ3 · ψτ

x

)x

+ ∂saτ1 · ψτ

s + ∂sDτ4 · eτ

xx

+ ∂sDτ5 · eτ

xs + ∂sDτ6 · eτ

x , (2.172)

I τ2 := Dτ

7 · τ 2ψτss + Dτ

8 · ψτxs + Dτ

9 · ψτs + Dτ

10 · τ 2ητss + Dτ

11 · ητxs + Dτ

12 · ητs , (2.173)

I τ3 := ∂sD

τ7 · τ 2ψτ

s + ∂sDτ8 · ψτ

x + ∂sDτ9 · ψτ + ∂sD

τ10 · τ 2ητ

s + ∂sDτ11 · ητ

x

+ ∂sDτ12 · ητ . (2.174)

Multiplying (2.171) by ψτs , integrating it over (0, s) × (0, 1) and integrating by parts, we obtain

d

ds

1∫0

(Dτ

1 · τ 2ψτssψ

τs + aτ

1 · (ψτs )2

2

)dx +

1∫0

(aτ

5 · (ψτxs

)2 + aτ7 aτ

8 · ψ2s − Dτ

1 · τ 2(ψτss

)2)dx

+ I τ +3∑

i=1

1∫0

I τi · ψτ

s dx = 0, (2.175)

where

I τ :=1∫

0

(Dτ

2 · τ 2ψτssψ

τxs + Dτ

3 · (ψτxs

)2 − ∂sDτ1 · τ 2ψτ

ssψτs − ∂sa

τ1 · (ψτ

s )2

2+ Dτ

4 · eτxxsψ

τs

)dx

+1∫

0

(Dτ

5 · eτxssψ

τs + Dτ

6 · eτxsψ

τs

)dx. (2.176)

Due to (2.148)–(2.151),

∣∣I τ∣∣ ≤ M · (N(T ) + δ

) 1∫0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2)dx

+1∫ ((

eτxxs

)2 + τ 6(eτxss

)2 + (eτxs

)2)dx. (2.177)

0


Differentiating (2.118) with respect to s, similar to (2.119),

1∫0

(eτxss

)2dx ≤ M

1∫0

(ψτ

ss

)2dx + M · (N(T ) + δ

) 1∫0

((ψτ

s

)2 + (τητ

s

)2 + τ 6(ητss

)2)dx

(2.178)

which in combination with (2.177), (2.115) and (2.112) implies

∣∣I τ∣∣ ≤ M · (N(T ) + δ

) 1∫0

((τψτ

ss

)2 + (ψτ

x

)2 + (ψτ

s

)2 + (ψτ

)2 + τ 8(ητss

)2 + (ητ

)2)dx.

.(2.179)

By (2.148)–(2.157) and the chain rule, the straightforward calculations show∣∣(∂sDτ1 , ∂sD

τ4

)∣∣ ≤ M · (∣∣ψτs

∣∣ + τ 3∣∣ητ

s

∣∣), (2.180)∣∣(∂sDτ2 , ∂sD

τ5

)∣∣ ≤ M · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣),∣∣∂sDτ3

∣∣ ≤ M · (∣∣ψτs

∣∣ + τ 2∣∣ητ

s

∣∣), (2.181)∣∣∂sDτ6

∣∣ ≤ M · ((∣∣ψτs

∣∣ + τ 3∣∣ητ

s

∣∣) · (∣∣ψτx

∣∣ + ∣∣ρτx

∣∣ + τ 3∣∣ητ

x

∣∣) + ∣∣ψτxs

∣∣)+ τ 3M · (∣∣ψτ

ss

∣∣ + ∣∣ητss

∣∣ + ∣∣ητxs

∣∣ + τ 3∣∣ητ

s

∣∣2)+ τ 3M · (∣∣ψτ

s

∣∣ + ∣∣ητxs

∣∣) · (∣∣ψτs

∣∣ + ∣∣ητx

∣∣), (2.182)∣∣∂sDτ7

∣∣ ≤ τM · (∣∣ψτx

∣∣ + ∣∣φτx

∣∣ + τ 2∣∣ητ

x

∣∣ + ∣∣ρτx

∣∣) · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣)+ M · (∣∣ψτ

ss

∣∣ + τ 3∣∣ητ

ss

∣∣ + τ∣∣ψτ

xs

∣∣ + τ∣∣ητ

xs

∣∣)+ M · (∣∣ψτ

s

∣∣ + τ 3∣∣ητ

s

∣∣) · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣), (2.183)∣∣∂sDτ8

∣∣ ≤ τ 3M · (∣∣ψτs

∣∣ + ∣∣ψτx

∣∣ + ∣∣ητx

∣∣ + ∣∣ρτx

∣∣) · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣)+ τ 3M · (∣∣ψτ

ss

∣∣ + ∣∣ητss

∣∣ + τ∣∣ψτ

xs

∣∣ + ∣∣ητxs

∣∣ + τ 2∣∣ητ

s

∣∣2)+ M · (∣∣ψτ

s

∣∣ + τ 3∣∣ητ

s

∣∣) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣), (2.184)∣∣∂sDτ9

∣∣ ≤ τ 3M · (∣∣ψτxs

∣∣ + τ 3∣∣ψτ

ss

∣∣ + τ 3∣∣ητ

ss

∣∣ + τ 3∣∣ητ

xs

∣∣ + τ 3∣∣ητ

s

∣∣2) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣)+ τ 3M · (∣∣ψτ

x

∣∣ + ∣∣ρτx

∣∣ + ∣∣ητx

∣∣) · (∣∣ψτs

∣∣ + τ 3∣∣ητ

s

∣∣) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣)+ M · (∣∣φτ

xx

∣∣ + ∣∣ρτxx

∣∣) · (∣∣ψτs

∣∣ + τ 3∣∣ητ

s

∣∣) + τ 3M · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣), (2.185)∣∣∂sDτ10

∣∣ ≤ τM · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣)+ τ 3M · ((∣∣ψτ

s

∣∣ + ∣∣ητs

∣∣) · ∣∣ητx

∣∣ + ∣∣ητxs

∣∣)+ τM · (∣∣ψτ

s

∣∣ + ∣∣ητs

∣∣ + ∣∣ψτxs

∣∣), (2.186)∣∣∂sDτ

∣∣ ≤ τ 3M · (∣∣ψτ∣∣ + ∣∣ητ

∣∣) · (∣∣φτ∣∣ + ∣∣ρτ

∣∣) + τ 3M · (∣∣ψτ∣∣ + ∣∣ητ

∣∣), (2.187)
11 s s x x s s


∣∣∂sDτ12

∣∣ ≤ τ 3M · (∣∣ψτx

∣∣ + ∣∣ψτs

∣∣ + ∣∣ρτx

∣∣ + ∣∣ητx

∣∣) · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣ + 1)

+ τ 3M · (∣∣ψτss

∣∣ + ∣∣ητxs

∣∣ + τ 3∣∣ητ

x

∣∣2) · (∣∣φτx

∣∣ + ∣∣ρτx

∣∣ + 1)

+ τ 3M · ((τ ∣∣φτxx

∣∣ + ∣∣ρτxx

∣∣) · ∣∣ητs

∣∣ + τ∣∣ψτ

xs

∣∣∣∣φτx

∣∣ + τ∣∣φτ

xx

∣∣∣∣ψτs

∣∣)+ τ 3M · ((∣∣φτ

x

∣∣ + ∣∣ρτx

∣∣)τ 2∣∣ητ

ss

∣∣ + ∣∣ητxs

∣∣) + τ 3M · (∣∣ψτs

∣∣ + ∣∣ητs

∣∣). (2.188)

Thus,

1∫0

I τ1 · ψτ

s dx ≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2 + (eτxx

)2 + (eτxs

)2)dx

+ M · (N(T ) + δ) 1∫

0

(τ 4(ητ

xs

)2 + τ 6(ητss

)2 + τ 4(ητs

)2)dx, (2.189)

1∫0

I τ2 · ψτ

s dx ≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2)dx

+ M · (N(T ) + δ) 1∫

0

(τ 4(ητ

xs

)2 + τ 6(ητss

)2 + τ 4(ητs

)2)dx, (2.190)

and

1∫0

I τ3 · ψτ

s dx ≤ M · (N(T ) + δ) 1∫

0

(τ 4(ητ

xs

)2 + τ 6(ητss

)2 + τ 4(ητs

)2 + (τητ

s

)2 + (ητ

x

)2)dx

+ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2)dx. (2.191)

Substituting (2.189)–(2.191) and (2.179) into (2.175), we have

d

ds

1∫0

(Dτ

1 · τ 2ψτssψ

τs + aτ

1 · (ψτs )2

2

)dx +

1∫0

(aτ

5 · (ψτxs

)2 + aτ7 aτ

8 · ψ2s − Dτ

1 · (τψτss

)2)dx

≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2 + (ψτ

)2)dx

+ M · (N(T ) + δ) 1∫

0

(τ 4(ητ

xs

)2 + τ 6(ητss

)2 + τ 4(ητs

)2)dx, (2.192)

with the help of (2.112), (2.114) and (2.115).


On the other hand, multiplying (2.171) by τ 2ψτss , integrating it over (0, 1) and integrating by

parts yield

d

ds

1∫0

(Dτ

1 · τ 4 (ψτss)

2

2+ (

aτ5 + Dτ

3

) · (τψτxs)

2

2+ aτ

7 aτ8 · (τψτ

s )2

2

)dx +

1∫0

aτ1 · (τψτ

ss

)2dx

+ Λτ +3∑

i=1

1∫0

Ii · τ 2ψτssdx = 0, (2.193)

where

Λτ :=1∫

0

(Dτ

4 · eτxxs + Dτ

5 · τ 3eτxss + Dτ

6 · eτxs

) · τ 2ψτssdx

−1∫

0

(∂sD

τ1 · τ 4 (ψτ

ss)2

2+ ∂xD

τ2 · τ 4 (ψτ

ss)2

2+ ∂sD

τ3 · (τψτ

xs)2

2

)dx. (2.194)


d

ds

1∫0

(Dτ

1 · τ 4 (ψτss)

2

2+ (

aτ5 + Dτ

3

) · (τψτxs)

2

2+ aτ

7 aτ8 · (τψτ

s )2

2

)dx +

1∫0

aτ1 · (τψτ

ss

)2dx

≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xx

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx

+ M · (N(T ) + δ) 1∫

0

(τ 8(ητ

ss

)2 + τ 4(ητxs

)2 + (τητ

s

)2 + (ητ

)2)dx. (2.195)

Multiplying (2.146) by τ 4ητxs and integrating it over (0, 1) and using (2.112) and (2.115), the

Cauchy–Schwarz inequality shows

1∫0

τ 4(ητxs

)2dx ≤ M

1∫0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

)2)dx,

(2.196)

for the sufficiently small N(T ) + δ. Similarly, differentiating (2.106) with respect to s and mul-tiplying the resulting equation by τ 4ητ

ss ,


1∫0

τ 6(ητss

)2dx ≤ M

1∫0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx

+1∫

0

((τητ

s

)2 + (ητ

)2)dx. (2.197)

Taking L2-norm over (2.147) with respect to x, we have

1∫0

(ψτ

xx

)2dx ≤ M

1∫0

((τψτ

ss

)2 + (ψτ

xs

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2 + (ητ

)2)dx (2.198)

with the help of (2.112) and (2.114). Substituting (2.196)–(2.198) into (2.195) yields

d

ds

1∫0

(Dτ

1 · τ 4 (ψτss)

2

2+ (

aτ5 + Dτ

3

) · (τψτxs)

2

2+ aτ

7 aτ8 · (τψτ

s )2

2

)dx +

1∫0

aτ1 · (τψτ

ss

)2dx

≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xx

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx

+ M · (N(T ) + δ) 1∫

0

((τητ

s

)2 + (ητ

)2)dx. (2.199)

Multiplying (2.199) by a large positive constant K2 and combining the resulting equation with (2.192), it follows that

d

ds

1∫0

(K2D

τ1 · τ 4 (ψτ

ss)2

2+ Dτ

1 · τ 2ψτssψ

τs + K2 · (aτ

5 + Dτ3

) (τψτxs)

2

2

)dx

+ d

ds

1∫0

(τ 2K2a

τ7 aτ

8 + aτ1

) · (ψτs )2

2dx

+1∫

0

(aτ

5 · (ψτxs

)2 + aτ7 aτ

8 · ψ2s + (

K2aτ1 − τ 2Dτ

1

) · (ψτss

)2)dx

≤ M · (N(T ) + δ) 1∫

0

((τψτ

ss

)2 + (ψτ

xx

)2 + (ψτ

s

)2 + (ψτ

x

)2 + (ψτ

)2)dx

+ M · (N(T ) + δ) 1∫ ((

τητs

)2 + (ητ

)2)dx, (2.200)

0


with the help of (2.196)–(2.197), which in combination with (2.116) yields

d

ds

∥∥(τ 2ψτ

ss, τψτxs,ψ

τs ,ψτ

x ,ψτx ,ψτ , ητ

)(s, ·)∥∥2

+ β1∗∥∥(

τ 2ψτss, τψτ

xs,ψτs ,ψτ

x ,ψτx ,ψτ , ητ

)(s, ·)∥∥2 ≤ 0, (2.201)

for some positive constant β1∗ , provided that the positive constants K1 and K2 are so large and N(T ) + δ is suitably small.

We easily derive (2.140) from (2.201). Then, combining (2.140) with (2.114), (2.115)gives (2.141). The estimate in (2.142) follows from (2.140) and (2.112). Due to (2.121), (2.140)–(2.142) and (2.196)–(2.198), we easily obtain (2.143). Therefore, integrating (2.126)over (0, s) with respect to s gives

s∫0

1∫0

(τητ

s

)2dxds ≤ M · (1 + s), (2.202)


s∫0

1∫0

τ 6(ητss

)2dxds ≤ M · (1 + s). (2.203)

Similarly, we obtain from (2.192) that

s∫0

1∫0

(ψτ

xs

)2dxds ≤ M · (1 + s). (2.204)

Summing up (2.202)–(2.204), (2.144) follows. This is the end of proof. �2.2. Convergence rate estimates

In this subsection, we prove that the global smooth solution to system (2.2) given by Theo-rem 2.2 converges to the solution of system (2.9) with the rate in (2.24)–(2.27) as τ → 0.

Set

Rτ := ρτ − ρ0, J τ := jτ − j0, Φτ := φτ − φ0.

Subtracting (2.9) from (2.2), we get by Taylor’s expansion⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Rτ

s + J τx + τ 3aτ

2 · jτ j τs + o

(τ 3)ρτ

s = 0,

σ 2Rτx + J τ + o(τ)ρτ

x + τ 3aτ3 · jτ ρτ

s + τ 2aτ6 · jτ j τ

x

+ Rτφτx + ρ0Φτ

x + o(τ 3)φτ

x + aτ4 · τ 2jτ

s = 0,

Φτ = Rτ + o(τ 3).

(2.205)

xx


This system is supplemented with the following initial and boundary conditions

Rτ (0, x) = 0, J τ (0, x) = J (x) := j0(x) − j0(x), x ∈ (0,1), (2.206)

Rτ (s,0) = Rτ (s,1) = 0, s ≥ 0, (2.207)

Φτ (s,0) = Φτ (s,1) = 0, s ≥ 0. (2.208)

Before stating the main result of this subsection, we recall the existence of global smooth solution to the IBVP (2.9), (2.10), (2.3), (1.8) and (1.9) which was given by Theorem 2.4 in [23].

Lemma 2.9. Let (ρ0, j0, φ0) be the steady state solution to the BVP (2.16), (1.8) and (1.9). Suppose that the initial data ρ ∈ H 2(0, 1) and the boundary conditions (1.8) and (1.9) hold. Then there exists a positive constant δ0 depending on infx∈[0,1] ρ and supx∈[0,1] ρsuch that if δ ≤ δ0, the IBVP (2.9), (2.10), (2.3), (1.8) and (1.9) has a unique solution (ρ0, j0, φ0) ∈ C([0, +∞); H 2(0, 1)) × C([0, +∞); H 1(0, 1)) × C([0, +∞); H 2(0, 1)). More-over, the solution (ρ0, j0, φ0) satisfies additional regularities ρ0

s ∈ C((0, +∞); H 2(0, 1)) ∩L2((0, ∞); H 1(0, 1)) ∩L2

loc((0, ∞); H 2(0, 1)) and φ0 −φ0 ∈ C([0, +∞); H 4(0, 1)). Moreover, it verifies the estimates

min{ρr, ρl, c, inf

x∈[0,1]ρ}

≤ ρ0(s, x) ≤ max{ρr, ρl, c, sup

x∈[0,1]ρ(x)

}, (2.209)

∥∥(ρ0 − ρ0)(s, ·)∥∥2

2 + ∥∥(j0 − j0)(s, ·)∥∥2

1 + ∥∥(φ0 − φ0)(s, ·)∥∥2

4 ≤ Ce−βs, (2.210)

s∫0

t∥∥(

ρ0xxs, ρ

0ss

)(t, ·)∥∥2

dt ≤ C(1 + s), (2.211)

for x ∈ [0, 1] and s ≥ 0, where C and β are positive constants independent of s and δ.

We first obtain the convergence rates for s ∈ (0, ∞) as

Lemma 2.10. Let (ρτ , jτ , φτ ) be a solution given by Theorem 2.2 to the IBVP (2.2), (2.3)–(2.4),(1.8)–(1.9) and (2.6)–(2.7). Then there exist positive constants m1, m2, δ, τ and M independent of s, δ and τ , such that if δ ≤ δ and 0 < τ ≤ τ , then the following estimates hold for any s ∈(0, ∞)

∥∥Rτ (s, ·)∥∥21 ≤ τ 2Me−βs,

∥∥Rτs (s, ·)∥∥2 ≤ τM ·

(1 + 1

s

)+ τMeβs, (2.212)

∥∥J τ (s, ·)∥∥2 ≤ M‖J‖2e− s

τ2 + τ 2Meβs, (2.213)∥∥J τx (s, ·)∥∥2 ≤ τM ·

(1 + 1

s

)+ τMeβs, (2.214)

∥∥τ 2jτs (s, ·)∥∥2 ≤ M · (τ + τ 2eβs + ‖J‖2e

− s

τ2). (2.215)


Proof. Multiplying (2.205)2 by J τ and integrating it over (0, s) × (0, 1), we get

s∫0

1∫0

σ 2RτxJ τ dxdt +

s∫0

1∫0

(J τ

)2dxdt + Wτ

1 = 0, (2.216)

where

Wτ1 : =

s∫0

1∫0

(τ 3aτ

3 · jτ ρτs + τ 2aτ

6 · jτ j2x + Rτφτ

x + ρ0Φτx

)Rτdxdt

+s∫

0

1∫0

(o(τ)ρτ

x + o(τ 3)φτ

x

)dxdt. (2.217)

Due to (2.205)1,

s∫0

1∫0

σ 2RτxJ τ dxdt ≥

1∫0

σ 2 (Rτ )2

2dx − M

s∫0

1∫0

(Rτ

)2dxdt − τ 2Ms. (2.218)

From now on, we denote M as generic positive constants independent of s, δ and τ . By (2.20)–(2.23) and the Cauchy–Schwarz inequality, it holds that

∣∣Wτ1

∣∣ ≤ τ 2M · (1 + s) + M

s∫0

1∫0

((Rτ

)2 + (Φτ

x

)2)dxdt + ε

s∫0

1∫0

(J τ

)2dxdt, (2.219)

for the arbitrarily positive constant ε. In virtue of (2.205)3,

1∫0

(Φτ

x

)2dx ≤ M

1∫0

(Rτ

)2dx + τ 3M (2.220)


∣∣Wτ1

∣∣ ≤ τ 2M · (1 + s) + M

s∫0

1∫0

(Rτ

)2dxdt + ε

s∫0

1∫0

(J τ

)2dxdt. (2.221)

Thus, (2.216) together with (2.218) and (2.221) gives

1∫σ 2 (Rτ )2

2dx + 1

2

s∫ 1∫ (J τ

)2dxdt ≤ τ 2M · (1 + s) + M

s∫ 1∫ (Rτ

)2dxdt, (2.222)

0 0 0 0 0


provided that ε is sufficiently small. The Gronwall inequality implies

∥∥Rτ (s, ·)∥∥2 +s∫

0

1∫0

(J τ

)2dxdt ≤ τ 2Meβs, (2.223)

for some positive constant β independent of s, δ and τ .Differentiating (2.9)2 with respect to x, we obtain

ρ0s − a2ρ0

xx + (ρ0)2 + ρ0

xφ0x − ρ0C(x) = 0. (2.224)

Subtracting (2.224) from (2.42), we have by Taylor’s expansion

Rτs − (

a2Rτx

)x

+ (ρτ + ρ0) · Rτ + Gτ

1 · τ 2ρτss − τ 2(Gτ

2 · J τρτs

)x

+ Rτxφτ

x + ρ0xΦτ

x

− (o(τ)ρτ

x

)x

+ o(τ 3)(ρτ + ρτ

s − C(x)) + o

(τ 2)φτ

x − RτC(x) + Ωτ = 0, (2.225)

where

Ωτ := τ 3Gτ5 · jτφτ

xs + Gτ7 · τ 2ρτ

s + Gτ8 · τ 3ρτ

x + Gτ9 · τ 2jτ

s + Gτ10 · τ 3jτ + Gτ

11 · τ 3jτx .

(2.226)

Multiplying (2.225) by Rτs , integrating it over (0, s) × (0, 1) and integrating by parts, we have

1∫0

σ 2 (Rτx )2

2dx +

s∫0

1∫0

(Rτ

s

)2dxdt +

3∑i=2

Wτi +

s∫0

1∫0

Ωτ · Rτs dxdt = 0, (2.227)

where

Wτ2 :=

s∫0

1∫0

[(ρτ + ρ0) · Rτ + Gτ

1 · τ 2ρτss − τ 2(Gτ

2 · jτ ρτs

)x

]Rτ

s dxdt

+s∫

0

1∫0

[Rτ

xφτx + ρ0

xΦτx − RτC(x)

]Rτ

s dxdt, (2.228)

Wτ3 :=

s∫0

1∫0

[(o(τ)ρτ

x

)x

+ o(τ 3)(ρτ

s + ρτ − C(x)) + o(τ)φτ

x

]Rτ

s dxdt. (2.229)


1∫0

σ 2 (Rτx )

2dx + 1

2

s∫0

1∫0

(Rτ

s

)2dxdt ≤ τ 2M · (1 + s) + M

s∫0

1∫0

(Rτ

)2dxdt, (2.230)

which in combination with (2.223) yields the first estimate in (2.212).


Multiplying (2.205)2 by es/τ 2J τ and integrating it over (0, s) × (0, 1), we get

s∫0

1∫0

et/τ 2(J τ

)2dxdt +

s∫0

1∫0

aτ4 · τ 2jτ

s et/τ 2J τ dxdt +

5∑i=4

Wτi = 0, (2.231)

where

Wτ4 :=

s∫0

1∫0

et/τ 2(σ 2Rτ

x + o(τ)ρτx + Rτφτ

x + ρ0Φτx

)J τ dxdt, (2.232)

Wτ5 :=

s∫0

1∫0

et/τ 2(τ 3aτ

3 · jτ ρτs + τ 2aτ

6 · jτ j τx − o

(τ 3)φτ

x

)J τ dxdt. (2.233)

We first estimate Wτi (i = 4, 5) as follows

∣∣Wτi

∣∣ ≤ τ 4M · ses/τ 2eβs + ε

s∫0

1∫0

es/τ 2(J τ

)2dxdt, (2.234)

where we have used the fact ∫ s

0 et/τ 2dt ≤ τ 2es/τ 2

. A straightforward calculation shows

s∫0

1∫0

aτ4 · et/τ 2

τ 2jτs J τ dxdt

=1∫

0

aτ4 · es/τ 2

τ 2 (J τ )2

2dx −

1∫0

aτ4 · τ 2 (J τ )2

2dx

−s∫

0

1∫0

aτ4 · et/τ 2 (J τ )2

2dxdt −

s∫0

1∫0

∂saτ4 · et/τ 2 (J τ )2

2dxdt

−s∫

0

1∫0

aτ4 · et/τ 2

τ 2j0s J τ dxdt. (2.235)

Due to (1.17) and Taylor’s expansion, there exists a positive constant Λ1 independent of s, δand τ , such that ∣∣aτ

4

∣∣ ≤ 1 + Λ1τ, (2.236)



s∫0

1∫0

aτ4 · et/τ 2

τ 2jτs J τ dxdt ≥

1∫0

aτ4 · es/τ 2

τ 2 (J τ )2

2dx −

1∫0

aτ4 · τ 2 (J τ )2

2dx

− 1

2(1 + Λ1τ)

s∫0

1∫0

et/τ 2 (J τ )2

2dxdt − τ 2M · (1 + s)es/τ 2

− ε

s∫0

1∫0

et/τ 2(J τ

)2dxdt. (2.237)

Thus, (2.231) together with (2.235) and (2.236) gives the first estimate in (2.214) provided that 0 < τ ≤ 1

8Λ1and ε is sufficiently small.

Multiplying (2.225) by −sRτxss and integrating the resulting equation over (0, s) × (0, 1), we

get

σ 2s

1∫0

(Rτxx)

2

2dx +

s∫0

1∫0

t(Rτ

xs

)2dxdt −

s∫0

1∫0

σ 2 (Rτxx)

2

2dxdt

+s∫

0

1∫0

τ 2(Gτ2 · jτ ρτ

s

)x

· tRτxxsdxdt +

s∫0

1∫0

τ 2Gτ1 · ρτ

ss tRτxxsdxdt

+8∑

i=6

Wτi −

s∫0

1∫0

Ωτ · tRτxxsdxdt = 0, (2.238)

where

Wτ6 := −

s∫0

1∫0

((ρτ + ρ0)Rτ − RτC(x)

)tRτ

xxsdxdt, (2.239)

Wτ7 := −

s∫0

1∫0

(Rτ

xφτx + ρ0

xΦτx

)tRτ

xxsdxdt, (2.240)

Wτ8 :=

s∫0

1∫0

[(o(τ)ρτ

x

)x

− o(τ 3)(ρτ + ρτ

s − C(x)) + o

(τ 2)φτ

x

]tRτ

xxsdxdt. (2.241)

First, we estimate the third term of (2.238). Multiplying (2.225) by Rτxx and integrating the

resulting equation over (0, s) × (0, 1), the Cauchy–Schwarz inequality shows

s∫ 1∫ (Rτ

xx

)2dxdt ≤ τ 2M(1 + s)eβs, (2.242)

0 0


with the help of (2.48)–(2.50) and (2.20)–(2.23). Now, we estimate the fourth term of (2.238). A straightforward calculation shows

s∫0

1∫0

τ 2(Gτ2 · jτ ρτ

s

)x

· tRτxxsdxdt

= τ 2

s∫0

1∫0

tGτ2 · jτ ρτ

xsρτxxsdxdt − τ 2

s∫0

1∫0

tGτ2 · jτ ρτ

xsρ0xxsdxdt

+ τ 2

s∫0

1∫0

(Gτ

2 · jτ)x

· tρτs Rτ

xxsdxdt. (2.243)

Due to (2.20)–(2.23), (2.211) and the integration by parts,

−τ 2

s∫0

1∫0

Gτ2jτ ρτ

xs · tρ0xxsdxdt ≤ τM · s(1 + s), (2.244)

τ 2

s∫0

1∫0

(Gτ

2 · jτ)x

· tρτs Rτ

xxsdxdt ≤ τM · s(1 + s). (2.245)

We turn to the estimate for the first term on the right side of (2.243). It is hard to estimate this

term because we can’t apply the integration by parts to it. Multiply (2.64) by tτ 2 Gτ2

Gτ3

· jτ ρτxs and

integrate it over (0, s) × (0, 1) to get

τ 2

s∫0

1∫0

tGτ2 · jτ ρτ

xsρτxxsdxdt

= τ 4

s∫0

1∫0

tGτ

1Gτ2

Gτ3

· jτ ρτsssρ

τxsdxdt − τ 4

s∫0

1∫0

tGτ

2

Gτ3

· (Gτ2 · jτ ρτ

ss

)xj τ ρτ

xsdxdt

− τ 2

s∫0

1∫0

tGτ

2

Gτ3

· (∂xGτ3 · ρτ

xs − aτ1 ρτ

ss − Gτ4 · ρτ

xs

)dxdt

+ τ 24∑

i=1

s∫0

1∫0

tGτ

2

Gτ3

· Rτi j τ ρτ

xsdxdt. (2.246)

Now, we handle the right side of (2.246). The simple calculations show


τ 4

s∫0

1∫0

tGτ

1Gτ2

Gτ3

jτ ρτsssρ

τxsdxdt

= τ 4s

1∫0

tGτ

1Gτ2

Gτ3

· jτ ρτssρ

τxsdx − τ 4

s∫0

1∫0

Gτ1Gτ

2

Gτ3

jτ ρτssρ

τxsdxdt

− τ 4

s∫0

1∫0

tGτ

1Gτ2

Gτ3

· jτs ρτ

ssρτxsdxdt − τ 4

s∫0

1∫0

t

(Gτ

1Gτ2

Gτ3

)s

· jτ ρτssρ

τxsdxdt

+ τ 4

s∫0

1∫0

t

(Gτ

1Gτ2

Gτ3

· jτ

)x

(ρτss)

2

2dxdt

≤ τM · s(1 + s), (2.247)

−τ 4

s∫0

1∫0

tGτ

2

Gτ3

· (Gτ2 · jτ ρτ

xx

)xj τ ρτ

xsdxdt

= −τ 4s

1∫0

t(Gτ

2)2

Gτ3

· (jτ)2 (ρτ

xs)2

2dx − τ 4

s∫0

1∫0

tGτ

2

Gτ3

· (Gτ2 · jτ

)xj τ ρτ

xsρτssdxdt

− τ 4

s∫0

1∫0

(Gτ2)2

Gτ3

· jτ (ρτxs)

2

2dxdt − τ 4

s∫0

1∫0

t

((Gτ

2)2

Gτ3

· (jτ)2

)s

(ρτxs)

2

2dxdt

≤ τM · s(1 + s). (2.248)

Thus, (2.246) together with (2.247) and (2.248) yields

τ 2

s∫0

1∫0

tGτ2 · jτ ρτ

xsρτxxsdxdt ≤ τM · s(1 + s). (2.249)

Substituting (2.244)–(2.245) and (2.249) into (2.243),

s∫0

1∫0

(τ 2Gτ

2 · jτ ρτs

)x

· tRτxxsdxdt ≤ τM · s(1 + s). (2.250)

We turn to the estimate for the fifth term on the right side of (2.238). By integration by parts,

−s∫ 1∫

τ 2Gτ1 · ρτ

ss tRτxxsdxdt ≥ τ 2s

1∫Gτ

1 · (ρτxs)

2

2dx − τM · (1 + s + s2). (2.251)

0 0 0


Next, we deal with Wτi (i = 6, 7, 8) in (2.238). By integration by parts, it holds that

Wτ6 =

s∫0

1∫0

((ρτ + ρ0)Rτ − RτC(x)

)x

· tRτxsdxdt

≤ τ 2M · s2eβs + ε

s∫0

1∫0

t(Rτ

xs

)2dxdt. (2.252)

Due to (2.60), (2.220), the first estimate in (2.214), (2.242) and the Sobolev inequality,

Wτ7 = s

1∫0

(Rτ

xφτx − ρ0

xΦτx

)Rτ

xxdx −s∫

0

1∫0

(Rτ

xφτx + ρ0

xΦτx

)Rτ

xxdxdt

−s∫

0

1∫0

(Rτ

xφτx + ρ0

xΦτx

)sRτ

xxdxdt

≤ τ 2M · (s + (1 + s)2eβs) + ε

s∫0

1∫0

t(Rτ

xs

)2dxdt

+ ε

s∫0

1∫0

t(Rτ

xx

)2dxdt. (2.253)

Similarly,

Wτ8 =

s∫0

1∫0

[(o(τ)ρτ

x

)x

− o(τ 3) · (ρτ + ρτ

s − C(x)) + o

(τ 2)φτ

x

] · t(ρτxxs − ρ0

xxs

)dxdt

≤ o(τ)

s∫0

1∫0

∣∣ρτs

∣∣(ρτxx

)2dxdt + τM · s(1 + s). (2.254)

Multiplying (2.42) by o(τ)|ρτ

s |Gτ

3· ρτ

xx and integrating over (0, s) × (0, 1), we have

s∫0

1∫0

o(τ)∣∣ρτ

s

∣∣(ρτxx

)2dxdt ≤ τM · s(1 + s). (2.255)

Thus, it follows from (2.254) that ∣∣Wτ∣∣ ≤ τM · s(1 + s). (2.256)
8


Finally, we estimate the last term in (2.238). Due to (2.60), (2.72) and integration by parts,

−s∫

0

1∫0

τ 3Gτ5 · jτφτ

xs · tRτxxsdxdt

= −τ 3s

s∫0

Gτ5 · jτφτ

xsρτxxdx + τ 3

s∫0

1∫0

(Gτ

5 · jτφτxs

)s· tRτ

xxdxdt

+ τ 3

s∫0

1∫0

Gτ5 · jτφτ

xsρτxxdxdt + τ 3

s∫0

1∫0

Gτ5 · jτφτ

xs · tρ0xxsdxdt

≤ τM · s(1 + s), (2.257)

−s∫

0

1∫0

Gτ7 · τ 2ρτ

s · tRτxxsdxdt

= −s∫

0

1∫0

(Gτ

7 · τ 2ρτs

)x

· tρτxsdxdt +

s∫0

1∫0

Gτ7 · τ 2ρτ

s · tρ0xxsdxdt

≤ τM · s(1 + s). (2.258)

Multiplying (2.64) by τ 3 Gτ8

Gτ3

· sρτx and τ 3 Gτ

9Gτ

3· sjτ

s , similar to (2.249),

s∫0

1∫0

τ 3(Gτ8 · τ 3ρτ

x + Gτ9 · τ 2jτ

s

) · tRτxxsdxdt ≤ τM · s(1 + s). (2.259)

Similarly,

s∫0

1∫0

τ 3(Gτ10 · jτ + Gτ

11 · jτx

) · tRτxxsdxdt ≤ τM · s(1 + s). (2.260)

Thus, (2.226) together with (2.257)–(2.260) yields

s∫0

1∫0

Ωτ · tRτxxsdxdt ≤ τM · s(1 + s). (2.261)

Substituting (2.242), (2.250)–(2.253), (2.256) and (2.261) into (2.238) concludes that

∥∥Rτxx(s, ·)

∥∥2 ≤ τM ·(

1 + 1)

+ τ 2M(1 + s)2

eβs . (2.262)
s s


Multiplying (2.225) by ses/τ 2Rτ

s and integrating it over (0, s) × (0, 1), we have

τ 2ses/τ 2

1∫0

Gτ1 · Rτ

s

2dx +

(1

4− Λ2

2τ

) s∫0

1∫0

tet/τ 2 (Rτs )2

2dxdt

≤ τ 3M · (1 + s)es/τ 2 + τ 3M · ses/τ 2eβs (2.263)

for some positive constant Λ2 independent of s, δ and τ . We take 0 < τ ≤ 14Λ2

and obtain the

second estimate in (2.212). Thus, taking L2-norm over (2.205)1, we have the second estimate in (2.214). Taking L2-norm over (2.205)2, we have (2.215). This completes the proof of the lemma. �Lemma 2.11. There exist positive constants α and M independent of τ such that the estimates in (2.24)–(2.27) hold.

Proof. We take β so large as s ≤ eβs for s ≥ 0. Then, we obtain from (2.212)–(2.215) that∥∥Rτ (s, ·)∥∥21 ≤ τ 2Meβs,

∥∥Rτs (s, ·)∥∥2 ≤ τs−1Meβs,∥∥J τ (s, ·)∥∥2 ≤ ‖J‖2e−s/τ 2 + τMeβs,

∥∥J τx (s, ·)∥∥2 ≤ τs−1Meβs,∥∥τ 2jτ

s (s, ·)∥∥2 ≤ M‖J‖2e−s/τ 2 + τMeβs.

We chose s0 = log 1/τα

βwith 0 < α < 1 and obtain that if s ≤ s0 then

∥∥Rτ (s, ·)∥∥21 ≤ Mτ 2−α,

∥∥Rτs (s, ·)∥∥2 ≤ s−1Mτ 1−α,∥∥J τ (s, ·)∥∥2 ≤ ‖J‖2e−s/τ 2 + τMτ 1−α,

∥∥J τx (s, ·)∥∥2 ≤ s−1Mτ 1−α,∥∥τ 2jτ

s (s, ·)∥∥2 ≤ M‖J‖2e−s/τ 2 + Mτ 1−α,

if s0 ≤ s, then

∥∥Rτs (s, ·)∥∥2 ≤ ∥∥(

ρτ − ρτ , ρτ − ρ0, ρ0 − ρ0)(s, ·)∥∥2 ≤ M · (τ ββα + τ

)where we have used (2.17), (2.19) and (2.210). We take α = min{1 − α, β

βα} with 0 < α <

1 and obtain the first estimate in (2.24). Similarly, we obtain the remaining estimates in (2.24)–(2.27). �3. Non-relativistic limit

We study the non-relativistic limit c → ∞ for system (1.14). We obtain that the global smooth solution to the IBVP (1.14), (1.13), (1.8)–(1.9) and (1.22)–(1.23) with properties (1.21) converges to the subsonic smooth solution to system (1.6) with (1.8), (1.9) and (1.13). To describe this limit, we denote the solution of system (1.14) as (ρc, jc, φc).


For convenience, we take the relaxation time τ to be the constant one. For the combined zero-relaxation and non-relativistic limit, the limit system (2.9) is a coupled system of a uni-formly parabolic equation and the Poisson equation. However, the limit system (1.6) for the non-relativistic limit is a coupled system of the compressible Euler equations and the Poisson equation. Thus, all of results in this section can be proven by the arguments similar to that in Section 2.

Before stating the main results of this section, we recall the non-relativistic limit for the steady state system (2.11) as follows, which was given by Theorem 2.1 in [13].

Lemma 3.1. Assume that (1.8), (1.9) and (1.10) hold. Then, there exist positive constants c0, J (c0), δ(c0) and K(c0) independent of c, such that for any c ≥ c0 and 0 ≤ δ < δ(c0), the sys-tem (2.11) with (1.8) and (1.9) has a unique solution (ρc, jc, φc) ∈ C1[0, 1] ×[−J (c0), J (c0)] ×C2[0, 1], which satisfies

ρ∗ = min{ρr, ρl,C} ≤ ρc ≤ max{ρr, ρl,C},∣∣J (c0)∣∣ ≤

√σ 2ρ2∗/2,∥∥(

ρcx, ρ

cxx, j

c, φcx, φ

cxx

)∥∥C0[0,1] ≤ K(c0)δ.

Furthermore, it holds that as c → ∞∥∥ρc − ρ

∥∥1 + ∣∣j c − j

∣∣ + ∥∥φc − φ∥∥

2 ≤ K(c0)c−2,

where (ρ, j , φ) is a unique solution to the system (1.28) with (1.8) and (1.9).

Remark 3.2. Li, Markowich and Mei [9] proved that there exist positive constants δ0, J0 and K0 such that for any 0 ≤ δ < δ0, the system (1.28) with (1.8) and (1.9) admits a unique solution (ρ, j, φ) ∈ C1[0, 1] × [−J0, J0] × C2[0, 1] satisfying

ρ∗ = min{ρr, ρl,C} ≤ ρ ≤ max{ρr, ρl,C},|J0| ≤

√σ 2ρ2∗/2,∥∥(ρx, ρxx, j , φx, φxx)

∥∥C0[0,1] ≤ K0δ.

The main results of this section are as follows. The uniform asymptotic stability of the steady state solution with respect to c is stated in Theorem 3.3 and the non-relativistic limit for the global smooth solution is described in Theorem 3.4.

Theorem 3.3. Let (ρc, j c, φc) be the solution of the BVP (2.11), (1.8) and (1.9) obtained by Lemma 3.1. Assume that (ρ, j) ∈ H 2(0, 1) satisfies (1.13) and the boundary conditions (1.8)and (1.9) hold. In addition, the compatibility conditions (1.22) and (1.23) hold. Then, there exist positive constants c0, δ(c0), M(c0) and β(c0) independent of c, such that if

∥∥(ρ − ρc, j − j c

)∥∥2 + δ < δ(c0) and c0 ≤ c, (3.1)
2


then the IBVP (1.14), (1.13), (1.8)–(1.9) and (1.22)–(1.23) admits a unique global smooth solu-tion (ρc, jc, φc) ∈ X2[0, ∞) satisfying

∥∥(ρc − ρc, j c − j c, φc − φc

)(t, ·)∥∥2

2 ≤ M(c0)∥∥(

ρ − ρc, j − j c)∥∥2

2e−β(c0)t

for any t ∈ (0, ∞) and c0 ≤ c.

Proof. This theorem can be proved by the procedure similar to that of Theorem 2.2. �Theorem 3.4. Let (ρc, jc, φc) be the solution to the IBVP (1.14), (1.13), (1.8)–(1.9) and(1.22)–(1.23) given by Theorem 3.3. Then, there exist positive constants c0, m1(c0), m2(c0), M∗(c0) and α(c0) independent of t , δ and c, such that if δ(c0) in (3.1) is suitably small, c0 in (3.1) is so large and ‖j‖L∞(0,1) � 1, then the following estimates hold for t ∈ (0, ∞) and c0 ≤ c

m1(c0) ≤ infx∈[0,1]ρ

c, supx∈[0,1]

ρc ≤ m2(c0), (3.2)

∣∣jc∣∣ ≤

√σ 2m2(c0)/2, (3.3)∥∥(

ρc, jc)(t, ·)∥∥2

2 + ∥∥(ρc

t , jct

)(t, ·)∥∥2

1 + ∥∥(ρc

tt , jctt

)(t, ·)∥∥2 ≤ M∗(c0). (3.4)

Moreover,

∥∥((ρc − ρ

)t,(ρc − ρ

)x,(ρc − ρ

))(t, ·)∥∥2 ≤ M∗(c0)c

−α(c0), (3.5)∥∥((jc − j

)t,(jc − j

)x,(jc − j

))(t, ·)∥∥2 ≤ M∗(c0)c

−α(c0), (3.6)∥∥(φc − φ

)(t, ·)∥∥2

2 ≤ M∗(c0)c−α(c0), (3.7)

where (ρ, j, φ) with j = ρυ is a unique subsonic smooth solution of system (1.6) with (1.13), (1.8) and (1.9).

Remark 3.5. Li, Markowich and Mei [9] and Nishibata and Suzuki [22] proved that if (ρ, j , φ)

is a solution to the BVP (1.28), (1.8) and (1.9) given by Remark 3.2, then there exist positive constants δ0, β0, M0 such that if

∥∥(ρ − ρ, j − j)∥∥2

2 + δ < δ0,

then, the IBVP (1.6), (1.13), (1.8) and (1.9) admits a unique subsonic global smooth solution (ρ, j, φ) ∈ X2[0, ∞) satisfying

∥∥(ρ − ρ, j − j , φ − φ)(t, ·)∥∥22 ≤ M0

∥∥(ρ − ρ, j − j)∥∥2

2e−β0t (3.8)

for any t ∈ (0, ∞).


Proof of Theorem 3.4. Due to Lemma 3.1 and Theorem 3.3, the estimates in (3.2)–(3.4) are easily obtained. We only construct the estimates in (3.5)–(3.7).

By (3.2)–(3.4) and Taylor’s expansion, we have, similar to (2.225), that

Rctt − (

a5 · Rcx + Fc

1 · ρcx

)x

− (a6 · jRc

t + Fc2 · ρt

)x

+ Rct + ρRc

+ o(c−1) · ρtt + Rc

xφx + ρx · Φcx + o

(c−2) = 0, (3.9)

where

Rc := ρc − ρ, J c := jc − j, Φc := φc − φ,

F c1 := hc

1 · jcφcx + Gc

2 + o(c−2),

F c2 := hc

2 · Rc + hc3 · J c + o

(c−2),

hc1 := −2

ρ + ρc

(ρcρ)2, hc

2 = ρ + ρc

(ρcρ)2j,

hc3 := −j + jc

(ρc)2, a5 := σ 2 − j2

ρ2, a6 := 2

ρ.

Compared to (2.225), Eq. (3.9) is the semi-linear wave equation of Rc, so we only choose test functions Rc and Rc

t and obtain, similar to (2.212)–(2.215),

∥∥(Rc

t ,Rcx,R

c)(t, ·)∥∥2 +

t∫0

∥∥(Rc

t ,Rcx,R

c)(s, ·)∥∥2

dxds ≤ M(c0)c−4eβ(c0)t ,

∥∥J cx (t, ·)∥∥2 ≤ M(c0)c

−4eβ(c0)t ,∥∥(J c

t , J c)(t, ·)∥∥2 ≤ M(c0)c

−4(1 + t)eβ(c0)t (3.10)

where we have used ‖jx‖L∞(0,1) ≤ 1/2, which can be obtained by the estimate in (3.8) and assumption ‖j‖L∞(0,1) � 1, in order to obtain the estimate for J c. The remaining proof of this theorem is as similar as that of Theorem 2.3. This completes the proof. �Acknowledgments

Zhang’s work is partially supported by the National Natural Science Foundation of China(No. 11371082). Li’s work is supported by the National Natural Science Foundation of China (Nos. 11171228, 11231006 and 11225102), and the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (No. CIT&TCD20140323).

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