Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity

Nangao Zhang

doi:10.1515/anona-2025-0097

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Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity

Nangao Zhang

Published/Copyright: August 13, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

This article shows time-asymptotic nonlinear stability of rarefaction wave to the Cauchy problem for the one-dimensional relaxed compressible Navier-Stokes equations with density-dependent viscosity. We prove that the solution to this typical system tends time-asymptotically to the rarefaction wave. For this, we technically construct the correction function S ˆ ( x , t ) , which means that S ± can be non-zero. The proof is accomplished by virtue of energy estimates.

Keywords: compressible Navier-Stokes equations; rarefaction wave; correction function; energy estimates

MSC 2010: 35Q35; 35Q60; 76N30; 76W05

1 Introduction

In this article, we study the system of one-dimensional isentropic compressible Navier-Stokes equations with Maxwell’s constitutive relations. The system can be described as follows (cf. [5,6]):

(1.1) v t − u x = 0 , u t + p ( v ) x = S x , τ S t + S = μ ( v ) u x v ,

where ( x , t ) ∈ R × ( 0 , ∞ ) . The unknown functions v ( x , t ) > 0 , u ( x , t ) , and S ( x , t ) represent, respectively, the specific volume, fluid velocity and stress tensor. The pressure function p ( v ) is given by the well-known γ -law: p ( v ) = A v − γ with A > 0 , γ > 1 . The positive parameter τ is the relaxation time describing the time lag in the response of the stress tensor to velocity gradient, μ = μ ( v ) is the viscosity coefficient. Maxwell [25] proposed the constitutive relation (1.1)₃, which is used to describe the relation of stress tensor and velocity gradient for a non-simple fluid. Indeed, it can be viewed as one-dimensional version of the Oldroyd-B model describing viscoelastic fluid (cf. [1,14,26]):

(1.2) τ [ S t + u ⋅ ∇ S + S W ( u ) − W ( u ) S ] + S = μ 1 ∇ u + ( ∇ u ) t − 2 n div u I n + μ 2 div u I n ,

where W ( u ) = 1 2 [ ∇ u − ( ∇ u ) t ] , I n denotes the n th-order identity matrix, and μ 1 and μ 2 are shear and bulk viscosity, respectively. It should be noted that if n = 1 , W ( u ) = 0 , the constitutive (1.2) reduces to (1.1)₃ by change of Euler coordinates to Lagrange coordinates. Actually, even for simple fluid, the time lag does exist but is very short from 1 ps to 1 ns (cf. [24,29]). However, Pelton et al. [28] pointed out that such a time lag cannot be neglected, even for simple fluids, in the experiments of high-frequency (20 GHZ) vibration of nano-scale mechanical devices immersed in water–glycerol mixtures. It was shown that (1.1)₃ provides a general formalism to characterize the fluid–structure interaction of nano-scale mechanical devices vibrating in simple fluids.

Subjected to (1.1), the initial value is proposed as

(1.3) ( v , u , S ) ( x , 0 ) = ( v 0 , u 0 , S 0 ) ( x ) , x ∈ R .

We assume that the initial value in the far field x = ± ∞ is constant, namely

(1.4) lim x → ± ∞ ( v 0 , u 0 , S 0 ) ( x ) = ( v ± , u ± , S ± ) ,

where v ± > 0 .

It is worth noting that if τ = 0 , system (1.1) reduces to the classical isentropic compressible Navier-Stokes equations:

(1.5) v t − u x = 0 , u t + p ( v ) x = μ ( v ) u x v x .

The global existence and asymptotic behavior of the solutions to (1.5) and its non-isentropic form have been extensively studied (cf. [7–10,16,17,20]). In particular, for the stability of viscous shock wave, Matsumura-Nishihara [21] first showed the time-asymptotic stability for small initial perturbations with integral zero. Later on, the assumption on integral zero was removed by Mascia and Zumbrun [19] and Liu and Zeng [18]. Recently, Wang and Wang [30] studied a planar shock wave for the three-dimensional Navier-Stokes equations by utilizing a new method called “ a -contraction with shifts.” For the stability of rarefaction wave, Matsumura and Nishihara [22] considered the asymptotic stability of rarefaction wave with small initial perturbations. Subsequently, the results in [22] were extended to the large data cases (cf. [23,27]). Note that the standard antiderivative methods, used to study the stability of viscous shocks, are not compatible with the energy method used for the stability of rarefactions. Recently, Kang et al. [12] overcame the difficulty of incompatibility between the standard antiderivative method used to study the stability of viscous shock and the energy method used for the stability of rarefaction. They affirmed the stability of composite waves comprising both viscous shock and rarefaction by employing the method of relative entropy and the a -contraction with shifts theory. For other related results, see [2,11,13] and references therein.

When τ is a positive constant, the study in this case is very limited as we know. Hu and Wang [4] showed that the solution to (1.1) exists globally and converges to the equilibrium state for small initial data, while solutions blow up in finite time with some large initial data. Hu and Wang [5] considered the linear stability of the viscous shock wave. Recently, Hu and Wang [6] established the nonlinear stability of rarefaction wave under the stiff condition S + = S − = 0 . This is due to the fact that there exist some gaps between the original solution and the rarefaction wave at x = ± ∞ , and the main difficulty is hard to construct appropriate correction function to eliminate it. However, by a deep observation, we succeed in this article in constructing such a correction function so that we can obtain the stability of rarefaction wave without the restrictive condition S + = S − = 0 (see (2.10)–(2.12) for more details).

Before concluding this section, we point out the main difference between the study in this article and the related results in the previous works. First, we consider the case that S + and S − can be arbitrarily given, which is different from [6]. For this, we first analyze the behavior of solutions to (1.1) and (1.3) and (1.4) at the far fields x = ± ∞ and then guess that the correction function should satisfy system (2.11). By artfully introducing special function S ˆ ( x , 0 ) , we can obtain S ˆ ( x , t ) . Thus, we can prove the stability of the rarefaction wave for (1.1) and (1.3)–(1.4) without the restrictive condition S + = S − = 0 . Second, because we use the idea of constructing a smooth approximation rarefaction wave in Nishihara et al. [27] and skillfully introduce a small positive constant ε 0 , the rarefaction wave strength ∣ v + − v − ∣ + ∣ u + − u − ∣ + ∣ S + − S − ∣ here need not be sufficiently small, which really relaxes the conditions in [6]. Finally, because of the introduction of the correction function S ˆ ( x , t ) and μ = μ ( v ) , the reformulated problem of (3.1)–(3.2) becomes more complicated, which leads to some extra trouble terms in establishing the energy estimates (see (3.12), (3.14), (3.28), and so on). For this, we require a technical condition μ ( v ) ∈ C 3 ( R + ) . To the best of our knowledge, this is the first result on the global existence and large-time behavior of solutions to the Cauchy problem of relaxed compressible Navier-Stokes equations with density-dependent viscosity.

The rest of the article is organized as follows. We calculate the rarefaction wave curves in Section 2.1 and state our main result in Section 2.2. In Section 3, we prove the main results (Theorem 2.1).

Notations: Throughout this article, C and c denote generic positive constants independent of x and t . For simplicity, we use H m to denote the usual Sobolev space H m ( R ) for any integer m ≥ 0 , the norm of H m is denoted by ‖ ⋅ ‖ m with ‖ ⋅ ‖ 0 = ‖ ⋅ ‖ . In the same way, we use ‖ ⋅ ‖ L p to denote ‖ ⋅ ‖ L p ( R ) for 1 ≤ p ≤ ∞ .

2 Preliminaries and main results

2.1 Rarefaction wave and smooth approximate profile

Now, we construct the asymptotic profile of the Cauchy problem (1.1) and (1.3)–(1.4). By employing asymptotic analysis arguments, the asymptotic behavior of solutions to this system is expected to be determined by the following Riemann problem:

(2.1) v t − u x = 0 , u t + p ( v ) x = 0 , S = 0 ,

with initial data

(2.2) ( v , u , S ) ( x , 0 ) = ( v + , u + , 0 ) , x > 0 , ( v − , u − , 0 ) , x < 0 .

Let us first consider the Riemann problem of ( v , u ) in (2.1)–(2.2). For any given constant ( v − , u − ) with v − > 0 , u − ∈ R , there exists a suitable neighborhood Ω ⊂ R v , u 2 ≔ R + × R as

R 1 ( v − , u − ) ≔ ( v , u ) ∈ Ω ∣ u = u − − ∫ v − v λ 1 ( z ) d z , u ≥ u − , R 2 ( v − , u − ) ≔ ( v , u ) ∈ Ω ∣ u = u − − ∫ v − v λ 2 ( z ) d z , u ≥ u − ,

and

R R ( v − , u − ) ≔ ( v , u ) ∈ Ω ∣ u ≥ u − − ∫ v − v λ 1 ( z ) d z , u ≥ u − − ∫ v − v λ 2 ( z ) d z ,

where λ 1 ( v ) = − − p ′ ( v ) and λ 2 ( v ) = − p ′ ( v ) (see [20,22,23]).

If ( v + , u + ) ∈ R R ( v − , u − ) , then there is a unique constant state ( v ¯ , u ¯ ) ∈ R 1 ( v − , u − ) and ( v + , u + ) ∈ R 2 ( v ¯ , u ¯ ) , the continuous weak solution (so-called rarefaction wave) to the Riemann problem of ( v , u ) in (2.1)–(2.2) is exactly given by (see [13,22,23])

(2.3) ( v r ( x , t ) , u r ( x , t ) ) = v 1 r x t + v 2 r x t − v ¯ , u 1 r x t + u 2 r x t − u ¯ ,

where ( v i r , u i r ) x t i = 1 , 2 are defined as

v 1 r x t , u 1 r x t = ( v − , u − ) , − ∞ ≤ x t ≤ λ 1 ( v − ) , λ 1 − 1 x t , u − − ∫ v − λ 1 − 1 x t λ 1 ( s ) d s , λ 1 ( v − ) ≤ x t ≤ λ 1 ( v ¯ ) , ( v ¯ , u ¯ ) , λ 1 ( v ¯ ) ≤ x t ≤ ∞

and

v 2 r x t , u 2 r x t = ( v ¯ , u ¯ ) , − ∞ ≤ x t ≤ λ 2 ( v ¯ ) , λ 2 − 1 x t , u − − ∫ v ¯ λ 2 − 1 x t λ 2 ( s ) d s , λ 2 ( v ¯ ) ≤ x t ≤ λ 2 ( v + ) , ( v + , u + ) , λ 2 ( v + ) ≤ x t ≤ ∞ .

Since the rarefaction wave ( v r , u r ) ( x , t ) is not smooth enough, in order to study its stability, we need to construct its smooth approximation. As in [22,23,27], let w i ( x , t ) ( i = 1 , 2) be the solution to the following Cauchy problem:

(2.4) w i t + w i w i x = 0 , w i ( x , 0 ) = w i 0 ( x ) = w i + + w i − 2 + w i + − w i − 2 tanh ( ε x ) ,

where ε > 0 is a small constant to be determined later, and w 1 − = λ 1 ( v − ) , w 1 + = λ 1 ( v ¯ ) , w 2 − = λ 2 ( v ¯ ) , w 2 + = λ 2 ( v + ) . The smooth approximate wave ( V , U ) ( x , t ) of ( v r , u r ) ( x , t ) is defined as

(2.5) ( V ( x , t ) , U ( x , t ) ) = ( V 1 ( x , t + t 0 ) + V 2 ( x , t + t 0 ) − v ¯ , U 1 ( x , t + t 0 ) + U 2 ( x , t + t 0 ) − u ¯ ) .

Here, t 0 = 1 ε 2 and ε given in (2.4), ( V i , U i ) ( x , t ) ( i = 1 , 2 ) satisfies

(2.6) λ i ( V i ( x , t ) ) = w i ( x , t ) , λ i ( v ) = ( − 1 ) i − p ′ ( v ) , i = 1 , 2 , U 1 ( x , t ) = u − + ∫ v − V 1 ( x , t ) − p ′ ( z ) d z , U 2 ( x , t ) = u ¯ − ∫ v ¯ V 2 ( x , t ) − p ′ ( z ) d z .

Then, it is easy to see that ( V , U ) ( x , t ) satisfies the following equation:

(2.7) V t − U x = 0 , U t + p ( V ) x = g ( V ) x ,

where g ( V ) = p ( V ) − p ( V 1 ) − p ( V 2 ) + p ( v ¯ ) . Finally, let us define S ¯ = μ ( V ) U x V , we also call it rarefaction wave for simplicity.

Next, we want to make a disturbance near the smooth rarefaction wave ( V , U , S ¯ ) . However, there are some gaps between the original solution and the rarefaction wave at far fields x = ± ∞ . In order to fill such gaps, as in [3], we need to construct the correction function, which will play a crucial role in the proof. First, let us look into the behavior of the solution to (1.1) and (1.3)–(1.4) at the far fields. By taking the limits of (1.1) with respect to x , and noting that u x , p ( v ) x , S x , and μ u x v will vanish at x = ± ∞ , we obtain the following ODEs for ( v , u , S ) ( ± ∞ , t ) :

(2.8) d d t v ( ± ∞ , t ) = 0 , d d t u ( ± ∞ , t ) = 0 , τ d d t S ( ± ∞ , t ) + S = 0 , ( v , u , S ) ( ± ∞ , 0 ) = ( v 0 , u 0 , S 0 ) ( ± ∞ ) = ( v ± , u ± , S ± ) .

By solving (2.8), we have

(2.9) lim x → ± ∞ ( v , u , S ) ( x , t ) = ( v , u , S ) ( ± ∞ , t ) = ( v ± , u ± , S ± e − t τ ) .

Combining (1.1) and (2.5) and (2.6), we can derive

∣ v ( ± ∞ , t ) − V ( ± ∞ , t ) ∣ = 0 , ∣ u ( ± ∞ , t ) − U ( ± ∞ , t ) ∣ = 0 , ∣ S ( ± ∞ , t ) − S ¯ ( ± ∞ , t ) ∣ = ∣ S ± e − t τ ∣ ≠ 0 .

It is obvious that if S ± ≠ 0 , then

S ( x , t ) − S ¯ ( x , t ) ∉ L 2 ( R ) .

To eliminate these gaps, we need to introduce a correction function S ˆ ( x , t ) . By observing the structure of the original solutions at x = ± ∞ , we technically construct the correction function S ˆ ( x , t ) satisfying the following equations for ( x , t ) ∈ R × R + :

(2.10) S ˆ t + 1 τ S ˆ = 0 , S ˆ ( x , t ) → S ± e − t τ , as x → ± ∞ .

Inspired by [3], we consider the following system:

(2.11) S ˆ t + 1 τ S ˆ = 0 , S ˆ ( x , 0 ) ≔ S ˆ 0 ( x ) = S − + ( S + − S − ) ∫ − ∞ ε 0 x m 0 ( y ) d y ,

where ε 0 > 0 is a small constant to be determined later, and m 0 ( x ) satisfies

m 0 ( x ) ∈ C 0 ∞ ( R ) , m 0 ( x ) ⩾ 0 , ∫ R m 0 ( x ) d x = 1 .

By solving (2.11), we obtain m ˆ ( x , t ) as

(2.12) S ˆ ( x , t ) = e − t τ S − + ( S + − S − ) ∫ − ∞ ε 0 x m 0 ( y ) d y .

2.2 Main results

With the above preparation in hand, we can state the main results of this article as follows.

Theorem 2.1

Suppose μ ( v ) ∈ C 3 ( R + ) . For a given constant state ( v − , u − ) ∈ R + × R , there exist constant δ 1 > 0 and C > 0 such that if ( v + , u + ) ∈ R R ( v − , u − ) and

‖ ( v 0 ( x ) − V ( x , 0 ) , u 0 ( x ) − U ( x , 0 ) , S 0 ( x ) − S ¯ ( x , 0 ) − S ˆ 0 ( x ) ) ‖ H 2 ≤ δ 1 ,

then the Cauchy problem (1.1) and (1.3)–(1.4) has a unique global solution ( v , u , S ) satisfying

sup t ≥ 0 ‖ ( v − V , u − U , S − S ¯ − S ˆ ) ( t ) ‖ H 2 ≤ C .

Moreover, the solution ( v , u , S ) ( x , t ) tends time-asymptotically to the rarefaction wave in the sense that

(2.13) lim t → + ∞ sup x ∈ R ∣ ( v , u , S ) ( x , t ) − ( v r , u r , 0 ) ( x , t ) ∣ = 0 .

2.3 Properties of the smooth rarefaction wave and correction function

Before studying the nonlinear stability of the rarefaction wave ( v r , u r , 0 ) ( x , t ) , we need to deduce the properties of the approximate rarefaction wave ( V , U , S ¯ ) ( x , t ) . We start from the following Riemann problem for the inviscid Burgers equation:

(2.14) w i t r + w r w i x r = 0 , w i r ( x , 0 ) = w i 0 r ( x ) = w i − , x < 0 , w i + , x > 0 ,

for i = 1 , 2 . Let w i − < w i + , then (2.14) has a continuous weak solution w i r ( x , t ) with

(2.15) w i r ( x , t ) = w i r x t = w − , x t ≤ w − , x t , w − ≤ x t ≤ w + , w + , w + ≤ x t .

As in [22,27], we approximate w r ( x , t ) by w i ( x , t ) of (2.4). It is easy to see that w i ( x , t ) has the following properties:

Lemma 2.1

Let w i − < w i + and w ˜ i = w i + − w i − ( i = 1 , 2 ) , then the Cauchy problem (2.4) has a unique global smooth solution w i ( x , t ) , which satisfies

w i − < w i ( x , t ) < w i + , w i x ( x , t ) > 0 , for x ∈ R and t ≥ 0 .
For 1 ≤ p ≤ ∞ and t ≥ 0 , there exists a constant C p > 0 depending only on p such that
‖ w i x ( t ) ‖ L p ≤ C p min w ˜ i ε 1 − 1 p , w ˜ i 1 p t − 1 + 1 p , ∥ ∂ x j w i ( t ) ∥ L p ≤ C p min w ˜ i ε j − 1 p , ε j − 1 − 1 p t − 1 , j = 2 , 3 .
If w i − > 0 , then for x ≤ 0 and t ≥ 0 , we have
∣ w i ( x , t ) − w i − ∣ ≤ w ˜ i e − 2 ε ( ∣ x ∣ + ∣ w i − ∣ t ) , ∣ w i x ( x , t ) ∣ ≤ 2 w ˜ i ε e − 2 ε ( ∣ x ∣ + ∣ w i − ∣ t ) , ∣ w i x x ( x , t ) ∣ ≤ 4 w ˜ i ε 2 e − 2 ε ( ∣ x ∣ + ∣ w i − ∣ t ) .
If w i + < 0 , then for x ≥ 0 and t ≥ 0 ,
∣ w i ( x , t ) − w i + ∣ ≤ w ˜ i e − 2 ε ( ∣ x ∣ + ∣ w i + ∣ t ) , ∣ w i x ( x , t ) ∣ ≤ 2 w ˜ i ε e − 2 ε ( ∣ x ∣ + ∣ w i + ∣ t ) , ∣ w i x x ( x , t ) ∣ ≤ 4 w ˜ i ε 2 e − 2 ε ( ∣ x ∣ + ∣ w i + ∣ t ) .
lim t → + ∞ sup x ∈ R ∣ w i ( x , t ) − w i r ( x , t ) ∣ = 0 .

Due to (2.5)–(2.6) and Lemma 2.1, ( V , U ) ( x , t ) has the following properties (cf. [22,27]):

Lemma 2.2

Let δ = ∣ v + − v − ∣ + ∣ u + − u − ∣ , the smooth approximate profile ( V , U ) ( x , t ) defined in (2.6) satisfies

V t = U x > 0 , ∣ ( V t , U t ) ∣ ≤ C ∣ ( V x , U x ) ∣ , for x ∈ R and t ≥ 0 .
For 1 ≤ p ≤ ∞ and t ≥ 0 , there exists a constant C p > 0 depending only on p such that
‖ ( V x , U x ) ( t ) ‖ L p ≤ C p min δ ε 1 − 1 p , δ 1 p ( t + t 0 ) − 1 + 1 p , ∥ ∂ x j ( V , U ) ( t ) ∥ L p ≤ C p min δ ε j − 1 p , ε j − 1 − 1 p ( t + t 0 ) − 1 , j = 2 , 3 .
There exists a positive constant α depending only on w i ± ( i = 1 , 2 ) such that
∣ g ( V ) x ( x , t ) ∣ ≤ C ε e − α ε ( ∣ x ∣ + t + t 0 ) , ∣ g ( V ) x x ( x , t ) ∣ ≤ C ε 2 e − α ε ( ∣ x ∣ + t + t 0 ) , ∥ g ( V ) x ( t ) ∥ L p ≤ C ε 1 − 1 p e − α ε ( t + t 0 ) ≤ C n α − n ε n + 1 − 1 p e − α ε t , ∥ g ( V ) x x ( t ) ∥ L p ≤ C ε 2 − 1 p e − α ε ( t + t 0 ) ≤ C n α − n ε n + 2 − 1 p e − α ε t , ∫ R + ∥ g ( V ) x ( t ) ∥ L p d t ≤ C ε − 1 p e − α ε t 0 ≤ C n α − n ε n − 1 p , n ∈ Z + , ∫ R + ∥ g ( V ) x x ( t ) ∥ L p d t ≤ C ε 1 − 1 p e − α ε t 0 ≤ C n α − n ε n + 1 − 1 p , n ∈ Z + .
lim t → + ∞ sup x ∈ R ( V , U ) ( x , t ) − ( v r , u r ) x t = 0 .

Proof

We only prove property (iii), the others have been proved in [22,27]. By direct computations, we can deduce

(2.16) g ( V ) x = ( p ′ ( V ) − p ′ ( V 1 ) ) V 1 x + ( p ′ ( V ) − p ′ ( V 2 ) ) V 2 x ≤ C ( ∣ V 2 − v ¯ ∥ V 1 x ∣ + ∣ V 1 − v ¯ ∥ V 2 x ∣ ) ,

(2.17) g ( V ) x x = ( p ″ ( V ) − p ″ ( V 1 ) ) V 1 x 2 + ( p ″ ( V ) − p ″ ( V 2 ) ) V 2 x 2 + 2 p ″ ( V ) V 1 x V 2 x + ( p ′ ( V ) − p ′ ( V 1 ) ) V 1 x x + ( p ′ ( V ) − p ′ ( V 2 ) ) V 2 x x ≤ C ( ∣ V 2 − v ¯ ∣ ∣ V 1 x ∣ 2 + ∣ V 1 − v ¯ ∣ ∣ V 2 x ∣ 2 + ∣ V 1 x V 2 x ∣ + ∣ V 2 − v ¯ ∣ ∣ V 1 x x ∣ + ∣ V 1 − v ¯ ∣ ∣ V 2 x x ∣ ) .

Combining Lemma 2.1 and (i)–(ii) in Lemma 2.2, we can prove (iii).□

Finally, by the definitions of S ˆ ( x , t ) , we immediately obtain the following lemma.

Lemma 2.3

For any 1 ≤ p ≤ + ∞ , let k, j be nonnegative integers. Then, we have

(2.18) ‖ S ˆ ( t ) ‖ L ∞ ≤ max { ∣ S + ∣ , ∣ S − ∣ } e − t τ , ‖ ∂ x k ∂ t j S ˆ ( t ) ‖ L p ≤ C ε 0 k − 1 p e − t τ , k ≥ 1 , j ≥ 0 .

Remark 2.1

From the definition of S ˆ ( x , t ) , we know that they do not belong to any L p space for 1 ≤ p < + ∞ .

3 Global stability of rarefaction wave

In this section, we study the stability of the approximate rarefaction wave ( V , U , S ¯ ) for the Cauchy problem (1.1) and (1.3)–(1.4), and then prove our main result Theorem 2.1. For this purpose, let us define the perturbation as

( ϕ , ψ , ζ ) ( x , t ) = ( v − V , u − U , S − S ¯ − S ˆ ) ( x , t ) .

It is easy to derive that ( ϕ , ψ , ζ ) ( x , t ) satisfies

(3.1) ϕ t − ψ x = 0 , ψ t + [ p ( ϕ + V ) − p ( V ) ] x − ζ x = μ ( V ) U x V x − g ( V ) x + S ˆ x , τ ζ t + ζ − μ ( v ) v ψ x = − τ μ ( V ) U x V t + U x μ ( v ) v − μ ( V ) V ,

with initial data for x ∈ R ,

(3.2) ( ϕ , ψ , ζ ) ∣ t = 0 ≔ ( ϕ 0 , ψ 0 , ζ 0 ) ( x ) = ( v 0 ( x ) − V ( x , 0 ) , u 0 ( x ) − U ( x , 0 ) , S 0 ( x ) − S ¯ ( x , 0 ) − S ˆ ( x , 0 ) ) .

The local existence of solutions to the reformulated Cauchy problem (3.1)–(3.2) can be obtained by the standard iteration argument. To prove Theorem 2.1 for brevity, we only devote ourselves to establishing the global-in-time a priori estimates in the following.

Proposition 3.1

Assume that all the conditions in Theorem 2.1 hold. There are constants ε 1 > 0 and C 0 > 0 such that if the smooth solution ( ϕ , ψ , ζ ) ( x , t ) to the Cauchy problems (3.1)–(3.2) on 0 ≤ t ≤ T for T ≥ 0 satisfies

(3.3) sup 0 ≤ t ≤ T ‖ ( ϕ , ψ , ζ ) ( t ) ‖ 2 2 + ε 0 + ε ≤ ε 1 2 ,

then it holds that

(3.4) sup 0 ≤ t ≤ T ‖ ( ϕ , ψ , ζ ) ( t ) ‖ 2 2 + ∫ 0 T ( ‖ ( V t ϕ ) ( t ) ‖ 2 + ‖ ( ϕ x , ψ x ) ( t ) ‖ 1 2 + ‖ ζ ( t ) ‖ 2 2 ) d t ≤ C 0 ‖ ( ϕ 0 , ψ 0 , ζ 0 ) ‖ 2 2 + ε 0 1 2 + ε 1 6 .

By using the Sobolev inequality

(3.5) ‖ f ‖ L ∞ ≤ 2 ‖ f ‖ 1 2 ‖ f x ‖ 1 2 , f ( x ) ∈ H 1 ( R ) ,

we have

(3.6) ‖ ( ϕ , ψ , ζ ) ( t ) ‖ L ∞ ≤ 2 ε 1 , ‖ ( ϕ x , ψ x , ζ x ) ( t ) ‖ L ∞ ≤ 2 ε 1 .

By noticing the smallness of ε 1 , we can also obtain

(3.7) 0 ≤ 1 2 min { v + , v − } ≤ v = ϕ + V ≤ 3 2 max { v + , v − } , ∣ u ∣ ≤ ∣ ψ ∣ + ∣ U ∣ ≤ 3 2 max { ∣ u + ∣ , ∣ u − ∣ } , ∣ S ∣ ≤ ∣ ζ ∣ + ∣ S ¯ ∣ + ∣ S ˆ ∣ ≤ 3 2 max { ∣ S + ∣ , ∣ S − ∣ } ,

which will be frequently used in the sequel.

Proposition 3.1 is a conclusion of several lemmas below. We first give the zero-order energy estimate.

Lemma 3.1

Under the assumptions of Proposition 3.1, it holds for 0 ≤ t ≤ T that

(3.8) ‖ ( ϕ , ψ , ζ ) ( t ) ‖ 2 + ∫ 0 t ( ‖ ( V t ϕ ) ( s ) ‖ 2 + ‖ ζ ( s ) ‖ 2 ) d s ≤ C ε 1 2 ∫ 0 t ‖ ψ x ( s ) ‖ 2 d s + C ( ‖ ( ϕ 0 , ψ 0 , ζ 0 ) ‖ 2 + ε 0 1 2 + ε 1 6 ) .

Proof

Multiplying (3.1)₁, (3.1)₂, (3.1)₃ by p ( V ) − p ( ϕ + V ) , ψ and v μ ζ , respectively, and integrating the resulting equations with respect to x over R , one obtains

(3.9) d d t ∫ R Φ ( v , V ) + 1 2 ψ 2 + τ v 2 μ ( v ) ζ 2 d x + ∫ R v μ ( v ) ζ 2 + ( p ( V + ϕ ) − p ( V ) − p ′ ( V ) ϕ ) V t d x = ∫ R ψ μ ( V ) U x V x − ψ g ( V ) x d x + ∫ R ψ S ˆ x d x + ∫ R τ v 2 μ ( v ) t ζ 2 d x + ∫ R − τ v ζ μ ( v ) μ ( V ) U x V t + v ζ U x μ ( v ) μ ( v ) v − μ ( V ) V d x ≔ ∑ j = 1 4 ,

where

(3.10) Φ ( v , V ) = p ( V ) ϕ − ∫ V v p ( ξ ) d ξ .

It follows from Lemmas 2.2–2.3, (3.6)–(3.7), and Young’s inequality that

(3.11) ∣ I 1 ∣ ≤ C ∫ R ∣ ψ ∣ ( ∣ U x x ∣ + ∣ U x V x ∣ + ∣ g ( V ) x ∣ ) d x ≤ C ‖ ψ ‖ 1 2 ‖ ψ x ‖ 1 2 ( ‖ U x x ‖ L 1 + ‖ U x ‖ ‖ V x ‖ + ‖ g ( V ) x ‖ L 1 ) ≤ C ε 1 ‖ ψ ‖ 2 ‖ ψ x ‖ 2 + ‖ U x x ‖ L 1 4 3 + ‖ U x ‖ 4 3 ‖ V x ‖ 4 3 + ‖ g ( V ) x ‖ L 1 4 3 ≤ C ε 1 2 ‖ ψ x ‖ 2 + C ε 1 6 ( t + t 0 ) − 7 6 + e − c ε ( t + t 0 ) ,

(3.12) ∣ I 2 ∣ ≤ ‖ ψ ‖ ‖ S ˆ x ‖ ≤ C ε 0 1 2 e − t τ ,

(3.13) ∣ I 3 ∣ ≤ C ∫ R ∣ v t ∣ ζ 2 d x ≤ C ∫ R ∣ u x ∣ ζ 2 d x ≤ C ∫ R ( ∣ ψ x ∣ + ∣ U x ∣ ) ζ 2 d x ≤ C ( ‖ ψ x ‖ L ∞ + ‖ U x ‖ L ∞ ) ‖ ζ ‖ 2 ≤ C ( ε 1 + ε ) ‖ ζ ‖ 2 ,

and

(3.14) ∣ I 4 ∣ ≤ C ∫ R ∣ ζ ∣ ( ∣ U x t ∣ + ∣ U x V t ∣ + ∣ U x ϕ ∣ ) d x ≤ C ∫ R ∣ ζ ∣ ( ∣ p ( V ) x x ∣ + ∣ g ( V ) x x ∣ + U x 2 + ∣ U x ϕ ∣ ) d x + C ‖ U x ‖ L ∞ 1 2 ∫ R ∣ ζ ∣ ∣ U x 1 2 ϕ ∣ d x ≤ ( η + ε 1 2 ) ‖ ζ ‖ 2 + C η ( ‖ V x x ‖ 2 + ‖ U x ‖ L ∞ 2 ‖ U x ‖ 2 + ‖ g ( V ) x x ‖ 2 ) + C ε 1 2 ‖ V t 1 2 ϕ ‖ 2 ≤ ( η + ε 1 2 ) ‖ ζ ‖ 2 + C ε 1 2 ‖ V t ϕ ‖ 2 + C η ( ε ( t + t 0 ) − 2 + ε 3 e − c ε ( t + t 0 ) ) .

Substituting (3.11)–(3.14) into (3.9), we obtain

(3.15) d d t ∫ R Φ ( v , V ) + 1 2 ψ 2 + τ v 2 μ ( v ) ζ 2 d x + ∫ R v μ ( v ) ζ 2 + ( p ( V + ϕ ) − p ( V ) − p ′ ( V ) ϕ ) V t d x ≤ C ε 1 2 ‖ ψ x ‖ 2 + ( η + C ( ε 1 2 + ε 1 ) ) ‖ ζ ‖ 2 + C ε 1 2 ‖ V t ϕ ‖ 2 + C η ε 1 6 ( t + t 0 ) − 7 6 + e − c ε ( t + t 0 ) + ε 0 1 2 e − t τ .

Note that

(3.16) Φ ( v , V ) = p ( V ) ϕ − ∫ V v p ( ξ ) d ξ = − p ′ ( V + θ 1 ϕ ) 2 ϕ 2 ≥ C − 1 ϕ 2 ,

(3.17) p ( V + ϕ ) − p ( V ) − p ′ ( V ) ϕ = p ′ ′ ( V + θ 2 ϕ ) 2 ϕ 2 ≥ C − 1 ϕ 2 .

Integrating inequality (3.15) with respect to t and employing (3.6)–(3.7), one can arrive at (3.8). The proof of this lemma is completed.□

Next, we consider high-order energy estimation.

Lemma 3.2

Under the assumptions of Proposition 3.1, it holds that

(3.18) ‖ ( ϕ x , ψ x , ζ x ) ( t ) ‖ 1 2 + ∫ 0 t ‖ ζ x ( s ) ‖ 1 2 d s ≤ C ( ε 1 + ε 1 2 + ε 0 1 2 ) ∫ 0 t ‖ ( ϕ x , ψ x ) ( s ) ‖ 1 2 d s + C ( ‖ ( ϕ 0 , ψ 0 , ζ 0 ) ‖ 2 2 + ε 1 2 + ε 0 1 2 ) .

Proof

By applying ∂ x k ( k = 1 , 2 ) to (3.1), we can deduce the following equation:

(3.19) ∂ t ∂ x k ϕ − ∂ x k + 1 ψ = 0 , ∂ t ∂ x k ψ + ∂ x k ( p ′ ( ϕ + V ) ϕ x ) + ∂ x k ( ( p ′ ( ϕ + V ) − p ′ ( V ) ) V x ) − ∂ x k + 1 ζ = ∂ x k + 1 μ ( V ) U x V − g ( V ) + S ˆ , τ ∂ t ∂ x k ζ + ∂ x k ζ − ∂ x k μ ( v ) v ψ x = − τ ∂ t ∂ x k μ ( V ) U x V + ∂ x k U x μ ( v ) v − μ ( V ) V .

Multiplying (3.19)₁, (3.19)₂, (3.19)₃ by − p ′ ( ϕ + V ) ∂ x k ϕ , ∂ x k ψ , v μ ∂ x k ζ ( k = 1 , 2 ) , respectively, and integrating it with respect to x over R give

(3.20) d d t ∫ R − 1 2 p ′ ( ϕ + V ) ( ∂ x k ϕ ) 2 + 1 2 ( ∂ x k ψ ) 2 + τ v 2 μ ( v ) ( ∂ x k ζ ) 2 d x + ∫ R v μ ( v ) ( ∂ x k ζ ) 2 d x = − ∫ R 1 2 p ″ ( ϕ + V ) ( ( ϕ t + V t ) ( ∂ x k ϕ ) 2 − ( ϕ x + V x ) ∂ x k ψ ∂ x k ϕ ) d x + ∫ R τ v 2 μ ( v ) t ( ∂ x k ζ ) 2 d x − ∫ R ( ∂ x k ( p ′ ( ϕ + V ) ϕ x ) − p ′ ( ϕ + V ) ∂ x k + 1 ϕ ) ∂ x k ψ d x − ∫ R ∂ x k ( ( p ′ ( ϕ + V ) − p ′ ( V ) ) V x ) ∂ x k ψ d x + ∫ R ∂ x k μ ( v ) v ψ x − μ ( v ) v ∂ x k + 1 ψ v ∂ x k ζ μ ( v ) d x + ∫ R ∂ x k + 1 μ ( V ) U x V − g ( V ) + S ˆ ∂ x k ψ d x − ∫ R τ ∂ t ∂ x k μ ( V ) U x V − ∂ x k U x μ ( v ) v − μ ( V ) V ∂ x k ψ d x ≔ ∑ j = 5 11 I j k .

By using Lemmas 2.2–2.3, (3.6)–(3.7), and Young’s inequality, we have

(3.21) I 5 k + I 6 k ≤ C ∫ R ( ∣ ψ x ∣ + ∣ ϕ x ∣ + ∣ V x ∣ + ∣ U x ∣ ) ( ( ∂ x k ϕ ) 2 + ( ∂ x k ψ ) 2 + ( ∂ x k ζ ) 2 ) d x ≤ ‖ ( ψ x , ϕ x , V x , U x ) ‖ L ∞ ( ‖ ∂ x k ϕ ‖ 2 + ‖ ∂ x k ψ ‖ 2 + ‖ ∂ x k ζ ‖ 2 ) ≤ C ( ε 1 + ε ) ( ‖ ∂ x k ϕ ‖ 2 + ‖ ∂ x k ψ ‖ 2 + ‖ ∂ x k ζ ‖ 2 ) .

Similarly, we can prove that

(3.22) I 7 1 = − ∫ R p ″ ( ϕ + V ) ( ϕ x + V x ) ϕ x ψ x d x ≤ C ∫ R ( ∣ ϕ x ∣ + ∣ V x ∣ ) ∣ ϕ x ψ x ∣ d x ≤ C ‖ ( ϕ x , V x ) ‖ L ∞ ‖ ϕ x ‖ ‖ ψ x ‖ ≤ C ( ε 1 + ε ) ( ‖ ϕ x ‖ 2 + ‖ ψ x ‖ 2 ) ,

(3.23) I 7 2 = − ∫ R ( p ′ ′ ′ ( v ) ( ϕ x + V x ) 2 ϕ x + p ′ ′ ( v ) ( 3 ϕ x x ϕ x + V x x ϕ x + 2 V x ϕ x x ) ) ψ x x d x ≤ C ‖ ( ϕ x , V x , V x x ) ‖ L ∞ ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 ) ≤ C ( ε 1 + ε ) ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 ) ,

(3.24) I 8 1 = − ∫ R ( ( p ′ ′ ( v ) ( ϕ x + V x ) − p ′ ′ ( V ) V x ) V x + ( p ′ ( v ) − p ′ ( V ) ) V x x ) ψ x d x ≤ C ∫ R ( ∣ ϕ ∣ V x 2 + ∣ ϕ x V x ∣ + ∣ ϕ V x x ∣ ) ∣ ψ x ∣ d x ≤ C ‖ V x ‖ L ∞ ‖ ϕ x ‖ ‖ ψ x ‖ + C ‖ ϕ ‖ L ∞ ‖ ψ x ‖ ( ‖ V x x ‖ + ‖ V x ‖ L ∞ ‖ V x ‖ ) ≤ C ( ε 1 + ε ) ( ‖ ϕ x ‖ 2 + ‖ ψ x ‖ 2 ) + C ε ( t + t 0 ) − 2 ,

(3.25) I 8 2 ≤ C ∫ R ( ( ϕ x 2 + V x 2 + ∣ ϕ x x ∣ + ∣ V x x ∣ ) ∣ V x ∣ + ( ∣ ϕ x ∣ + ∣ V x ∣ ) ∣ V x x ∣ + ∣ ϕ V x x x ∣ ) ∣ ψ x x ∣ d x ≤ C ‖ V x ‖ L ∞ ‖ ψ x x ‖ ( ‖ ϕ x x ‖ + ‖ V x x ‖ + ‖ ϕ x ‖ L ∞ ‖ ϕ x ‖ + ‖ V x ‖ L ∞ ‖ V x ‖ ) + C ‖ ϕ ‖ L ∞ ‖ ψ x x ‖ ‖ V x x x ‖ ≤ C ( ε 1 + ε ) ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 ) + C ε ( t + t 0 ) − 2 ,

(3.26) I 9 1 = ∫ R μ ( v ) v ′ v v x ψ x ζ x μ ( v ) d x ≤ C ∫ R ( ∣ ϕ x ∣ + ∣ V x ∣ ) ∣ ψ x ζ x ∣ d x ≤ C ‖ ( ϕ x , V x ) ‖ L ∞ ∫ R ‖ ψ x ‖ ‖ ζ x ‖ d x ≤ C ( ε 1 + ε ) ( ‖ ψ x ‖ 2 + ‖ ζ x ‖ 2 ) ,

(3.27) I 9 2 ≤ C ∫ R ( ∣ v x x ψ x ∣ + v x 2 ∣ ψ x ∣ + ∣ v x ψ x x ∣ ) ∣ ζ x x ∣ d x ≤ C ‖ ( ϕ x , V x ) ‖ L ∞ ∫ R ( ‖ ψ x ‖ + ‖ ψ x x ‖ ) ‖ ζ x x ‖ d x + C ‖ ψ x ‖ L ∞ ∫ R ( ‖ ϕ x x ‖ + ‖ V x x ‖ ) ‖ ζ x x ‖ d x ≤ C ( ε 1 + ε ) ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 + ‖ ζ x ‖ 1 2 ) + C ε ( t + t 0 ) − 2 ,

(3.28) I 10 ≤ C ∫ R ( ∣ V x x ∣ + ∣ U x x ∣ + V x 2 ∣ U x ∣ + ∣ ∂ x 3 V ∣ + ∣ ∂ x 3 U ∣ + ∣ ∂ x 4 U ∣ ) ( ∣ ψ x ∣ + ∣ ψ x x ∣ ) d x + C ∑ k = 1 2 ∫ R ( ∣ ∂ x k + 1 g ( V ) ∣ + ∣ ∂ x k + 1 S ˆ ∣ ) ∣ ∂ x k ψ ∣ d x ≤ C ( ε 1 2 + ε 0 1 2 ) ‖ ψ x ‖ 1 2 + C ( ε 1 2 + ε 0 1 2 ) ( t + t 0 ) − 2 ,

and

(3.29) I 11 + I 12 ≤ C ( ε 0 + ε 1 2 ) ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 ) + C ε 1 2 ( t + t 0 ) − 2 .

Substituting (3.21)–(3.29) into (3.20), we obtain

(3.30) d d t ∫ R − 1 2 p ′ ( ϕ + V ) ( ∂ x k ϕ ) 2 + 1 2 ( ∂ x k ψ ) 2 + τ v 2 μ ( v ) ( ∂ x k ζ ) 2 d x + ∫ R v μ ( v ) ( ∂ x k ζ ) 2 d x ≤ C ( ε 1 + ε 1 2 ) ( ‖ ϕ x ‖ 1 2 + ‖ ψ x ‖ 1 2 + ‖ ζ x ‖ 1 2 ) + C ε 1 2 ( t + t 0 ) − 2 .

Integrating (3.30) with respect to t and using the smallness of ε 1 and ε , we can obtain (3.18).□

The next lemma gives the dissipative estimates of ϕ x and ψ x .

Lemma 3.3

Under the assumptions of Proposition 3.1, it holds that

(3.31) ∫ 0 t ‖ ( ϕ x , ψ x ) ( s ) ‖ 1 2 d s ≤ C ‖ ( ϕ , ψ , ζ ) ( t ) ‖ 2 2 + C ∫ 0 t ‖ ζ ( s ) ‖ 2 2 d s + C ( ‖ ( ϕ 0 , ψ 0 , ζ 0 ) ‖ 2 2 + ε 1 2 + ε 0 1 2 ) .

Proof

Multiplying the equation (3.19)₂ by ∂ x k + 1 ϕ for k = 0 or 1, and integrating over ( 0 , t ) × R show

(3.32) ∫ R − p ′ ( v ) ( ∂ x k + 1 ϕ ) 2 d x = ∫ R ∂ t ∂ x k ψ ∂ x k + 1 ϕ d x + ∫ R ( ∂ x k ( p ′ ( v ) ϕ x ) − p ′ ( v ) ∂ x k + 1 ϕ ) ∂ x k + 1 ϕ d x + ∫ R ∂ x k ( ( p ′ ( v ) − p ′ ( V ) ) V x ) ∂ x k + 1 ϕ d x + ∫ R ∂ x k + 1 ζ ∂ x k + 1 ϕ d x + ∫ R ∂ x k + 1 μ ( V ) U x V − g ( V ) + S ˆ ∂ x k + 1 ϕ d x ≔ ∑ j = 13 17 I j k .

By integrating by parts, we first obtain

(3.33) I 13 = d d t ∫ R ∂ x k ψ ∂ x k + 1 ϕ d x + ∫ R ∂ x k + 1 ψ ∂ t ∂ x k ϕ d x ≤ d d t ∫ R ∂ x k ψ ∂ x k + 1 ϕ d x + ∫ R ( ∂ x k + 1 ψ ) 2 d x .

Second, for k = 0 , I 14 0 vanishes and for k = 1 ,

(3.34) I 14 1 = ∫ 0 t ∫ R p ″ ( ϕ + V ) ( ϕ x + V x ) ϕ x ϕ x x d x ≤ C ‖ ( ϕ x , V x ) ‖ L ∞ ∫ R ∣ ϕ x ϕ x x ∣ d x ≤ C ( ε 1 + ε ) ‖ ϕ x ‖ 1 2 .

Similarly, we have for k = 0 that

(3.35) I 15 0 ≤ C ∫ R ∣ ϕ ϕ x V x ∣ d x ≤ η ‖ ϕ x ‖ 2 + C η ‖ V x ‖ L ∞ 2 ‖ ϕ ‖ 2 ≤ η ‖ ϕ x ‖ 2 + C η ε 1 2 ( t + t 0 ) − 3 2 ,

and k = 1 ,

(3.36) I 15 1 ≤ C ∫ R ( ( ∣ ϕ x ∣ + ∣ V x ∣ ) ∣ V x ∣ + ∣ ϕ V x x ∣ ) ∣ ϕ x x ∣ d x ≤ η ‖ ϕ x x ‖ 2 + C η ( ‖ V x ‖ L ∞ 2 ( ‖ ϕ x ‖ 2 + ‖ V x ‖ 2 ) + ‖ V x x ‖ 2 ) ≤ η ‖ ϕ x x ‖ 2 + C η ε 2 ‖ ϕ x ‖ 2 + C η ε 1 2 ( t + t 0 ) − 2 .

We can also prove that

(3.37) I 16 ≤ ‖ ∂ x k + 1 ζ ‖ ‖ ∂ x k + 1 ϕ ‖ ≤ η ‖ ∂ x k + 1 ϕ ‖ 2 + C η ‖ ∂ x k + 1 ζ ‖ 2

and

(3.38) I 17 ≤ C ∫ R ( ∣ V x x ∣ + ∣ U x x ∣ + V x 2 + ∣ V x U x ∣ + ∣ ∂ x 3 U ∣ ) ( ∣ ϕ x ∣ + ∣ ϕ x x ∣ ) d x + C ∫ R ( ∣ ∂ x k + 1 g ( V ) ∣ + ∣ ∂ x k + 1 S ˆ ∣ ) ∣ ∂ x k + 1 ϕ ∣ d x ≤ C ( ε 1 2 + ε 0 1 2 ) ‖ ϕ x ‖ 1 2 + C ( ε 1 2 + ε 0 1 2 ) ( t + t 0 ) − 2 .

Combining (3.32)–(3.38) together, we obtain

(3.39) ∫ R − p ′ ( v ) ( ∂ x k + 1 ϕ ) 2 d x ≤ d d t ∫ R ∂ x k ψ ∂ x k + 1 ϕ d x + ( 2 η + C ( ε 1 2 + ε 1 ) ) ‖ ϕ x ‖ 1 2 + C ‖ ∂ x k + 1 ψ ‖ 2 + C η ‖ ∂ x k + 1 ζ ‖ 2 + C η ε 1 2 ( t + t 0 ) − 3 2 .

Now, we turn to estimate ψ x . Multiplying (3.19)₃ by ∂ x k + 1 ψ , we obtain

(3.40) ∫ R μ ( v ) v ( ∂ x k + 1 ψ ) 2 d x = ∫ R τ ∂ t ∂ x k ζ ∂ x k + 1 ψ d x + ∫ R ∂ x k ζ ∂ x k + 1 ψ d x + ∫ R μ ( v ) v ( ∂ x k + 1 ψ ) 2 − ∂ x k μ ( v ) v ψ x ∂ x k + 1 ψ d x + ∫ R τ ∂ t ∂ x k μ ( V ) U x V − ∂ x k U x μ ( v ) v − μ ( V ) V ∂ x k + 1 ψ d x ≔ ∑ j = 18 21 I j k .

Some tedious calculations give

I 18 k = d d t ∫ R τ ∂ x k ζ ∂ x k + 1 ψ d x + ∫ R τ ∂ x k + 1 ζ ∂ t ∂ x k ψ d x = ∫ R τ ∂ x k + 1 ζ − ∂ x k ( p ′ ( v ) ϕ x ) − ∂ x k ( ( p ′ ( v ) − p ′ ( V ) ) V x ) + ∂ x k + 1 ζ + ∂ x k + 1 μ ( V ) U x V − g ( V ) + S ˆ d x ≤ C ∫ R ∣ ∂ x k + 1 ζ ∣ ( ∣ ϕ x ∣ + ∣ ϕ x x ∣ + ∣ ϕ V x ∣ + ∣ ∂ x k + 1 ζ ∣ + V x 2 + ∣ V x U x ∣ + ∣ V x x ∣ + ∣ U x x ∣ + ∣ ∂ x 3 U ∣ ) d x + C ∫ R ∣ ∂ x k + 1 ζ ∣ ( ∂ x k + 1 g ( V ) + ∂ x k + 1 S ˆ ) d x ≤ η ‖ ϕ x ‖ 1 2 + C η ‖ ζ ‖ 1 2 + ‖ V x ‖ L ∞ 2 ( ‖ ϕ ‖ 2 + ‖ V x ‖ 2 + ‖ U x ‖ 2 ) + ‖ V x x ‖ 2 + ‖ U x x ‖ 1 2 + ‖ g ( V ) x ‖ 1 2 + ‖ S ˆ x ‖ 1 2 ≤ η ‖ ϕ x ‖ 1 2 + C η ‖ ζ x ‖ 1 2 + C ( ε 1 2 + ε 0 1 2 ) ( t + t 0 ) − 3 2 ,

I 19 ≤ ‖ ∂ x k ζ ‖ ‖ ∂ x k + 1 ψ ‖ ≤ η ‖ ∂ x k + 1 ψ ‖ 2 + C η ‖ ζ ‖ 1 2 ,

I 19 ≤ C ∫ R ∣ v x ψ x ψ x x ∣ d x ≤ C ( ‖ V x ‖ L ∞ + ‖ ϕ x ‖ L ∞ ) ‖ ψ x ‖ 1 2 ≤ C ( ε 1 + ε ) ‖ ψ x ‖ 1 2 ,

and

I 20 ≤ C ε 1 2 ‖ ψ x ‖ 1 2 + C ε 1 2 ( t + t 0 ) − 2 .

Combining the above formula, we can obtain

(3.41) ∫ R μ ( v ) v ( ∂ x k + 1 ψ ) 2 d x ≤ d d t ∫ R τ ∂ x k ζ ∂ x k + 1 ψ d x + η ‖ ϕ x ‖ 1 2 + ( η + C ( ε 1 2 + ε 1 ) ) ‖ ψ x ‖ 1 2 + C η ‖ ζ ‖ 2 2 + C η ( ε 1 2 + ε 0 1 2 ) ( t + t 0 ) − 3 2 .

Adding up (3.39) and (3.41), and integrating the resulting formula with respect to t , we obtain (3.31).□

Proof of Proposition 3.1

We combine Lemmas 3.1–3.3, and choose ε , ε 1 , and ε 0 small enough to establish the a priori estimates (3.4). Thus, the proof of Proposition 3.1 is completed.□

Proof of Theorem 2.1

The existence of the solution follows from the standard continuity ar gument based on the local existence and the a priori estimate in Proposition 3.1. Therefore, it suffices to show the large time behavior (2.13). For this, we begin with the following estimates:

(3.42) lim t → + ∞ ∥ ( ϕ x , ψ x , ζ x ) ( t ) ∥ 2 = 0 .

Indeed, from (3.4) and (3.30), one can show that

∫ 0 ∞ ∥ ( ϕ x , ψ x , ζ x ) ( t ) ∥ 2 + d d t ∥ ( ϕ x , ψ x , ζ x ) ( t ) ∥ 2 d t < ∞ ,

which implies (3.42). Then, by using (3.5), (3.42), Lemma 2.2 (iv) and (2.18), we can obtain (2.13). This ends the proof of Theorem 2.1.□

Funding information: The research was supported by the National Natural Science Foundation of China #12401283 and the Natural Science Foundation of Hubei Province #2024AFB208.
Author contribution: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data were used for the research described in the article.

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Received: 2025-01-07

Accepted: 2025-06-24

Published Online: 2025-08-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0097

Keywords for this article

compressible Navier-Stokes equations; rarefaction wave; correction function; energy estimates

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