Singularity for the macroscopic production model with Chaplygin gas

Xiaoli Liu; Lihui Guo; Yanbo Hu

doi:10.1515/anona-2025-0098

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Singularity for the macroscopic production model with Chaplygin gas

Xiaoli Liu , Lihui Guo and Yanbo Hu

Published/Copyright: August 13, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we investigate the formation of singularities in solutions of homogeneous and inhomogeneous systems for a macroscopic production model with Chaplygin gas using the method of characteristic decomposition. It is a model originating from production planning in supply chains. Since the model with the Chaplygin gas equation of state is a strictly hyperbolic system with both linearly degenerate characteristic field and genuinely nonlinear characteristic field, we can simultaneously study the blow-up of the solution itself and the blow-up of the derivative of the solution, in which the blow-up of the solution itself corresponds to the bottlenecks in the supply chains. Besides, we obtain the C 1 estimates of the solution before the solution itself blows up. In addition, we give some representative examples of numerical simulations concerning the generation of singularities of solutions.

Keywords: macroscopic production model; Chaplygin gas; singularity formation; characteristic decomposition; numerical simulation

MSC 2010: 35L65; 35L67

1 Introduction

Production planning in supply chains is a significant focus in industrial engineering and operations research. It aims to optimize the production process of a given good by planning the supply network (production of goods and their distributions to final users) to ensure a specific behavior of the supply chains, such as avoiding bottlenecks [1,2,8,10]. In previous phases, a fluid-like continuous model [1] was used to analyze high-volume product flows for forecasting production behavior and developing production planning. To compensate for the inability of the fluid-like continuous model to account for the diffusion of data, Forestier-Coste et al. [9] proposed a macroscopic production model consisting of two conservation laws:

(1.1) ρ t + ( ρ u ) x = 0 , ( ρ u ( 1 + P ( ρ ) ) ) t + ( ρ u 2 ( 1 + P ( ρ ) ) ) x = 0 ,

in which the variables ρ and u correspond to the product density and velocity, respectively. The variable x denotes the degree of completion or stage of production, and the pressure term P ( ρ ) is an anticipated factor in the production line. Since the product cannot be passed backward through the production line, it is rational to assume that it propagates at a positive velocity. The first conservation law equation in system (1.1) represents the conservation of mass. This is because, in a long production line with multiple steps, we consider the factory a pipe and the parts moving through the pipe to be fluids. It is assumed that defective parts are sorted out at the end of the production process in the factory and that there are no sources or sinks, which ensures that the law of conservation of mass is satisfied during the production process [3]. The second equation is a conservation equation obtained under the assumption that the information Z = u ( 1 + P ( ρ ) ) is transported at the rate of the product, where Z describes the change of the production capacity and ρ Z = ρ u ( 1 + P ( ρ ) ) is the conservative variable [9]. More studies on the macroscopic production model can be found in [6,25,27–30,35].

In this article, we consider the singularity generation problem of the solution with the following Chaplygin gas equation of state:

(1.2) P ( ρ ) = 1 ρ 0 − 1 ρ ,

which is introduced by Chaplygin [5], Tsien [31], and von Karman [32]. The ρ 0 in (1.2) is a positive constant that can be chosen arbitrarily small according to our needs [26]. Using (1.2), system (1.1) can be simplified to the following form:

(1.3) ρ t + ( ρ u ) x = 0 , 1 + 1 ρ 0 ρ u − u t + 1 + 1 ρ 0 ρ u 2 − u 2 x = 0 .

In [33,34], the authors studied the perturbed Riemann problem of system (1.1) with P = − 1 ρ . This system is a strictly hyperbolic system of temple type, which has both linearly degenerate and genuinely nonlinear eigenvalues. We know that the study of singularity formation of solution with this feature differs from previous theoretical results. In [12,13,15,23,24], the authors delved into the generation of the singularity of solution for the hyperbolic equations with linearly degenerate eigenvalues and obtained solutions with mass concentration phenomenon using appropriate initial data. Previous studies [4,7,14,22] imply that the solutions of the Euler system with different equations of state with genuinely nonlinear eigenvalues lead to the creation of shock wave in finite time, indicating the blow-up of the derivative of the solution.

In [33], we find that ρ tends to positive infinity when the positive state converges to the asymptote. Besides, when the right state is greater than the left state, Wang and Shen derived a Riemann solution consisting of a shock wave and a contact discontinuity. Therefore, while studying the macroscopic production model (1.3) with both linear degenerate and genuinely nonlinear eigenvalues, we obtain the C 1 estimates of the solution before the blow-up of the solution itself and demonstrate that when the solution itself blows up, the derivative of the solution also blows up by giving suitable initial values, which means that there is a δ shock wave. This phenomenon corresponds to the bottleneck in the supply chains, implying that when the inflow exceeds the total capacity, the density will blow up [1,2,8,10]. Furthermore, we show that the derivative of the solution blows up in finite time by giving different initial data. This is a consequence of a sudden change of the first-order derivative of the solution, while the solution itself is bounded in the region under consideration.

In addition, we consider the following relaxation model [9]:

(1.4) ρ t + ( ρ u ) x = 0 , ( ρ u ( 1 + P ( ρ ) ) ) t + ( ρ u 2 ( 1 + P ( ρ ) ) ) x = 1 ε ( μ ρ − ρ u ( 1 + P ( ρ ) ) ) .

It is an extension of system (1.1) to the situation where the current capacity of the production equipment is adjusted to the desired value μ at a given time ε , where ε > 0 and μ ≥ 0 are independent of ( x , t ) . For simplicity, we take μ = 0 ; then using (1.2), this system is rewritten as

(1.5) ρ t + ( ρ u ) x = 0 , 1 + 1 ρ 0 ρ u − u t + 1 + 1 ρ 0 ρ u 2 − u 2 x = − 1 ε 1 + 1 ρ 0 ρ − 1 u .

Similarly, we prove that the solution itself and the derivative of the solution of system (1.5) blow up in finite time under the given initial value conditions.

In our study, we use the method of characteristic decomposition [19] to demonstrate the blow-up of the solution. The method of characteristic decomposition has a wide range of applications. In [11,16–18,20], the authors used this method to study the interaction of waves and to construct the classical solution for hyperbolic systems. Lai and Zhu [15], and Lv and Hu [23] used this method to prove that the solution itself blows up for the systems with eigenvalues, which are totally linearly degenerate. Lai and Zhao [14] used this approach to demonstrate that the derivative of the solution blows up for the systems with genuinely nonlinear eigenvalues. Moreover, for the systems with genuinely nonlinear eigenvalues, Buckmaster et al. [4], proved the singularity formation of the solution using self-similar transformations and the fact that the Burgers equation dominates two-dimensional purely azimuthal wave motion. Chen et al. [7] introduced a critical strength for nonlinear compression and demonstrated that the singularity develops in finite time if the compression intensity exceeds this critical value. In our study, the method of characteristic decomposition is applied to the system with both linearly degenerate eigenvalues and genuinely nonlinear eigenvalues to prove that the solution itself and the derivative of the solution blow up in finite time by giving different initial values. It is worth noting that in general singularity problems, we cannot avoid the generation of shock waves while studying the blow-up of the solution itself. However, in this article, we can obtain the C 1 estimates of the solution before the blow-up of the solution itself, which is the primary difficulty of this article. The main results of this article are stated as the following theorems.

Theorem 1.1

Let ( ρ ¯ , u 0 ( x ) ) ∈ C 1 ( R ) satisfy

(1.6) ρ ¯ > ρ 0 1 + ρ 0 , u 0 ( x ) = u ¯ ( a ) , x ∈ ( − ∞ , a ) , u ¯ ( x ) , x ∈ [ a , b ] , u ¯ ( b ) , x ∈ ( b , ∞ ) ,

where ρ ¯ is a constant and u ¯ is a C 1 function satisfying

(1.7) u ¯ ( b ) > 0 , u ¯ ′ ( a ) = u ¯ ′ ( b ) = 0 , u ¯ x ( x ) < 0 , ∀ x ∈ ( a , b ) .

Assume that ( ρ , u ) ∈ C 1 is a solution of (1.3) for ( x , t ) ∈ R × [ 0 , T ) with initial data ( ρ ¯ , u 0 ( x ) ) satisfying

(1.8) u ¯ ( k 1 ) − u ¯ ( k 2 ) − u ¯ ( a ) c ¯ ρ 0 1 + ρ 0 − c ¯ ρ 0 > 0 ,

where c ¯ = 1 ρ ¯ , ∀ [ k 1 , k 2 ] ⊂ [ a , b ] . Then, there holds

(1.9) t * = k 2 − k 1 u ¯ ( k 1 ) − u ¯ ( k 2 ) − u ¯ ( a ) c ¯ ρ 0 1 + ρ 0 − c ¯ ρ 0 > T , and t * < t 0 = b − k 1 u ¯ ( a ) − 1 − 2 c ¯ ρ 0 1 + ρ 0 u ¯ ( b ) .

Furthermore, there exists a point ( x * , t * ) ∈ ( a , b ) × ( 0 , t * ) such that

(1.10) lim ( x , t ) → ( x * , t * ) ρ ( x , t ) = + ∞ .

Theorem 1.2

Let ( ρ , u ) ( x , 0 ) = ( ρ ˆ , u ˆ 0 ) ( x ) satisfy

(1.11) ρ ˆ ( x ) > 2 ρ 0 1 + ρ 0 , u ˆ 0 ( x ) = u ˆ ( h 1 ) , x ∈ ( − ∞ , h 1 ) , u ˆ ( x ) , x ∈ [ h 1 , h 2 ] , u ˆ ( h 2 ) , x ∈ ( h 2 , ∞ ) ,

where u ˆ satisfies

(1.12) u ˆ ′ ( h 1 ) = 0 , u ˆ ′ ( h 2 ) = 0 , u ˆ x ( x ) > 0 , ∀ x ∈ ( h 1 , h 2 ) .

Suppose that ( ρ , u ) ( x , t ) is a solution of (1.3) for ( x , t ) ∈ R × ( 0 , T ′ ) with initial data ( ρ ˆ ( x ) , u ˆ 0 ( x ) ) satisfying

(1.13) u ˆ x − u ˆ c ˆ x ρ 0 1 + ρ 0 − c ˆ ρ 0 < 0 ,

where c ˆ = 1 ρ ˆ . Then, there holds

(1.14) t * * = 1 + ρ 0 2 ρ 0 c ˆ 1 u ˆ x 1 > T ′ ,

where c ˆ 1 u ˆ x 1 = inf x ∈ ( h 1 , h 2 ) { c ˆ u ˆ x } . Moreover, there exists a point ( x * * , t * * ) ∈ ( h 1 , h 2 ) × ( 0 , t * * ) such that

(1.15) lim ( x , t ) → ( x * * , t * * ) c x ( x , t ) = + ∞ .

Theorem 1.3

Let ( ρ ˜ , u ˜ 0 ( x ) ) ∈ C 1 ( R ) satisfy

(1.16) ρ ˜ > ρ 0 1 + ρ 0 , u ˜ 0 ( x ) = u ˜ ( a 1 ) , x ∈ ( − ∞ , a 1 ) , u ˜ ( x ) , x ∈ [ a 1 , b 1 ] , u ˜ ( b 1 ) , x ∈ ( b 1 , ∞ ) ,

where ρ ˜ is a constant and u ˜ is a C 1 function satisfying

(1.17) u ˜ ( b 1 ) > 0 , u ˜ ′ ( a 1 ) = u ˜ ′ ( b 1 ) = 0 , u ˜ x ( x ) < 0 , ∀ x ∈ ( a 1 , b 1 ) .

Assume that ( ρ , u ) ∈ C 1 is a solution of (1.5) for ( x , t ) ∈ R × [ 0 , T ˜ ) with initial data ( ρ ˜ , u ˜ 0 ( x ) ) satisfying

(1.18) u ˜ ( m 1 ) − u ˜ ( m 2 ) − u ˜ ( a 1 ) c ˜ ρ 0 1 + ρ 0 − c ˜ ρ 0 ≥ 4 u ˜ ( a 1 ) ( m 2 − m 1 ) ε ,

where c ˜ = 1 ρ ˜ , ∀ [ m 1 , m 2 ] ⊂ [ a 1 , b 1 ] . Then, there holds

(1.19) t ˜ * > T ˜ , and t ˜ * < t ˜ 0 = b 1 − m 1 u ˜ ( a 1 ) − A − 2 c ˜ A u ˜ 1 ,

where u ˜ 1 = inf x ∈ ( a 1 , b 1 ) { u ˜ ( x ) } . Furthermore, there exists a point ( x ˜ * , t ˜ * ) ∈ ( a 1 , b 1 ) × ( 0 , t ˜ * ) such that

(1.20) lim ( x , t ) → ( x ˜ * , t ˜ * ) ρ ( x , t ) = + ∞ .

Theorem 1.4

Let ( ρ , u ) ( x , 0 ) = ( ρ ˇ , u ˇ 0 ) ( x ) satisfy

(1.21) ρ ˇ ( x ) > 2 ρ 0 1 + ρ 0 , u ˇ 0 ( x ) = u ˇ ( n 1 ) , x ∈ ( − ∞ , n 1 ) , u ˇ ( x ) , x ∈ [ n 1 , n 2 ] , u ˇ ( n 2 ) , x ∈ ( n 2 , ∞ ) ,

where u ˇ satisfies

(1.22) u ˇ ( n 1 ) > 0 , u ˇ ′ ( n 1 ) = 0 , u ˇ ′ ( n 2 ) = 0 , u ˇ x ( x ) > 0 , ∀ x ∈ ( n 1 , n 2 ) .

Suppose that ( ρ , u ) ( x , t ) is a solution of (1.5) for ( x , t ) ∈ R × ( 0 , T ˜ ′ ) with initial data ( ρ ˇ ( x ) , u ˇ 0 ( x ) ) satisfying

(1.23) u ˇ x − u ˇ c ˇ x ρ 0 1 + ρ 0 − c ˇ ρ 0 < 0 , c ˇ u ˇ x ρ 0 1 + ρ 0 − c ˇ ρ 0 − 1 ε > 0 ,

where c ˇ = 1 ρ ˇ . Then, there holds

(1.24) t ˜ * * = 1 + ρ 0 ρ 0 c ˇ 1 u ˇ x 1 > T ˜ ′ ,

where c ˇ 1 u ˇ x 1 = inf x ∈ ( n 1 , n 2 ) c ˇ u ˇ x . Moreover, there exists a point ( x ˜ * * , t ˜ * * ) ∈ ( n 1 , n 2 ) × ( 0 , t ˜ * * ) such that

(1.25) lim ( x , t ) → ( x ˜ * * , t ˜ * * ) c x ( x , t ) = + ∞ .

Remark 1.5

Some of the special initial data used in these theorems are only for the convenience of considering the problem. The techniques used in this article can handle more general initial data.

This article is structured as follows. In Section 2, we first derive the characteristic decomposition of system (1.3). Then, we obtain the C 1 estimates of the solution and demonstrate that the solution itself of the Cauchy problems (1.3) and (1.6) blows up in finite time if specific initial values are satisfied. Finally, for particular initial data that are different from that in demonstrating the blow-up of the solution itself, we prove that the derivative of the solution tends to be infinite in finite time. In Section 3, using the same steps as in Section 2, we prove that the solution itself and the derivative of the solution blow up in finite time. In Section 4, we present some representative numerical results concerning the theoretical results in Sections 2 and 3. In Section 5, we give the conclusion of the work mentioned in this article.

2 Singularity formation for the homogeneous system

2.1 Characteristic decomposition of the macroscopic production model

Setting A = ( 1 + 1 ρ 0 ) , we rewrite system (1.3) as

(2.1) ρ t + ( ρ u ) x = 0 , ( A ρ − 1 ) u t + ( A ρ − 2 ) u u x = 0 .

The eigenvalues of (2.1) are

(2.2) λ + = u , λ − = A − 2 c A − c u ,

where the function c is defined by c = p ′ ( ρ ) = 1 ρ . By calculation, we obtain the characteristic equations

(2.3) ∂ + u = u A − c ∂ + c , ∂ − u = 0 ,

where ∂ ± = ∂ t + λ ± ∂ x . We also deduce the following operator relations:

(2.4) ∂ t = A − c c ∂ − − A − 2 c c ∂ + , ∂ x = A − c c u ( ∂ + − ∂ − ) .

In light of (2.2) and (2.3), we obtain the commutator relation [18]

(2.5) ∂ + ∂ − − ∂ − ∂ + = ( ∂ t + λ + ∂ x ) ( ∂ t + λ − ∂ x ) − ( ∂ t + λ − ∂ x ) ( ∂ t + λ + ∂ x ) = ( ∂ + λ − − ∂ − λ + ) ∂ x = − A u ∂ + c ( A − c ) 2 ∂ x + ( A − 2 c ) ∂ + u A − c ∂ x .

Based on (2.3)–(2.5), we have the following lemma.

Lemma 2.1

For the variables ( c , u ) , there hold

(2.6) ∂ − ∂ + c = 2 A − c ( ∂ + c ) 2 − 1 A − c ∂ + c ∂ − c , ∂ + ∂ − c = 1 A − c ∂ + c ∂ − c , ∂ − ∂ + u = 2 u ( ∂ + u ) 2 , ∂ + ∂ − u = 0 .

2.2 Blow-up of the density itself of the solution

Let L 1 and L 2 represent two characteristic curves defined as

(2.7) L 1 : d x + ( t ) d t = u , x + ( 0 ) = a , L 2 : d x − ( t ) d t = A − 2 c A − c u , x − ( 0 ) = b .

Let l 1 : x = u ¯ ( a ) t + a , l 2 : x = A − 2 c ¯ A u ¯ ( b ) t + b . Set Ω 1 ≔ { ( x , t ) ∣ l 1 < x < l 2 , 0 < t < T } , for some T > 0 (Figure 1).

Figure 1

Characteristic curves.

Lemma 2.2

Assume that system (1.3) with initial data ( ρ ¯ , u 0 ( x ) ) admits a classical solution in Ω 1 . Then, the solution satisfies

(2.8) 0 < c ( x , t ) < c ¯ < A , u ¯ ( b ) < u ( x , t ) < u ¯ ( a ) , ∂ + c < 0 , ∂ − c < 0 , ∂ + u < 0 , ∂ − u = 0 , ∣ ∂ + c ∣ < A u ¯ ( b ) R 1 , ∣ ∂ − c ∣ < − c ¯ u ¯ x , ∣ ∂ + u ∣ < R 1 ,

for ( x , t ) ∈ Ω 1 , where R 1 = − A − c ¯ c ¯ u ¯ u ¯ x + 2 t u ¯ ( a ) 1 + ( A − 2 c ¯ A u ¯ ( b ) ) 2 − 1 .

Proof

Recall from (2.6) that

∂ + ∂ − c ∂ − c = 1 A − c ∂ + c , ∂ − ∂ + c ∂ + c = 2 A − c ∂ + c − 1 A − c ∂ − c .

Integrating and combining with the initial data (1.6) and (1.7), we find

∂ − c = ∂ − c ∣ t = 0 exp ∫ 0 t 1 A − c ∂ + c d x + ( t ) = c ¯ u ¯ x exp ∫ 0 t 1 A − c ∂ + c d x + ( t ) < 0 , ∂ + c = ∂ + c ∣ t = 0 exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c d x − ( t ) = c ¯ u ¯ x exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c d x − ( t ) < 0 ,

and we therefore deduce 0 < c < c ¯ < A . From u ¯ x < 0 , we know that u ¯ ( x ) is a decreasing function satisfying u ¯ ( x ) > u ¯ ( b ) > 0 . Then, utilizing the equations in (2.3), we find ∂ + u < 0 , ∂ − u = 0 , and we thus have u ¯ ( b ) < u < u ¯ ( a ) . Observing from our analysis mentioned earlier, we have

(2.9) ∂ + c < 0 , ∂ − c < 0 , ∂ + u < 0 , ∂ − u = 0 , 0 < c < c ¯ < A , u ¯ ( b ) < u < u ¯ ( a ) , ∀ ( x , t ) ∈ Ω 1 .

By integrating the third equation in (2.6), we conclude

− 1 ∂ + u + 1 ∂ + u t = 0 = ∫ 0 t 2 u 1 + A − 2 c A − c u 2 d τ .

Hence, we derive

∂ + u > − R 1 ,

where R 1 = − A − c ¯ c ¯ u ¯ ( a ) u ¯ x + 2 t u ¯ ( a ) 1 + ( A − 2 c ¯ A u ¯ ( b ) ) 2 − 1 . Remembering ∂ + u < 0 , we deduce ∣ ∂ + u ∣ < R 1 . This estimate, combining with (2.3), gives

∣ ∂ + c ∣ ≤ A u ¯ ( b ) ∣ ∂ + u ∣ < A u ¯ ( b ) R 1 .

From (2.6) and (2.9), we obtain

(2.10) ∂ + ∂ − c = 1 A − c ∂ + c ∂ − c > 0 .

Integrating (2.10), we deduce

∂ − c > ∂ − c ∣ t = 0 = c ¯ u ¯ x .

Combining with ∂ − c < 0 , we have

□ ∣ ∂ − c ∣ < − c ¯ u ¯ x .

By means of ∣ ∂ + u ∣ < R 1 , ∂ − u = 0 , ∣ ∂ + c ∣ < A u ¯ ( b ) R 1 , ∣ ∂ − c ∣ < − c ¯ u ¯ x , and (2.4), we conclude

∣ ∂ x u ∣ < A c u ¯ ( b ) R 1 , ∣ ∂ x c ∣ < A c u ¯ ( b ) A u ¯ ( b ) R 1 − c ¯ u ¯ x .

The remainder of this section will be devoted to proving that ρ tends to infinity for some point ( x * , t * ) in ( a , b ) × ( 0 , t * ) . Let K 1 : x = x 1 ( t ) and K 2 : x = x 2 ( t ) be two characteristic curves defined as follows:

(2.11) K 1 : d x 1 ( t ) d t = u ( x 1 ( t ) , t ) , x 1 ( t ) ∣ t = 0 = k 1 , K 2 : d x 2 ( t ) d t = u ( x 2 ( t ) , t ) , x 2 ( t ) ∣ t = 0 = k 2 ,

where [ k 1 , k 2 ] ⊂ ( a , b ) (Figure 2). Since we want the moment of streamline intersection to be in the region of existence of the solution, it is easy to find that the time of intersection of K 1 with l 2 must be earlier than the time of intersection of K 2 with l 2 ; thus, we only need to estimate the intersection time of K 1 with l 2 . Set K 1 ′ : x = u ¯ ( a ) t + k 1 , then, the insert time of l 2 and K 1 ′ is t 0 = b − k 1 u ¯ ( a ) − A − 2 c ¯ A u ¯ ( b ) . From (1.9), we obtain t * < t 0 , implying that the intersection time of K 1 and l 2 is greater than t * . Therefore, the region defined by K 1 , K 2 , t = 0 and t = t * is in Ω 1 .

Figure 2

Flow characteristic curves.

Recalling the first equation of (2.1), we see that

d d t ∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x = 0 ,

which indicates

∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x = ∫ x 1 ( 0 ) x 2 ( 0 ) ρ ( x , 0 ) d x = ρ ¯ ( k 2 − k 1 ) .

Thus,

(2.12) max x 1 ( t ) < x ( t ) < x 2 ( t ) ρ ( x , t ) ≥ ρ ¯ ( k 2 − k 1 ) x 2 ( t ) − x 1 ( t ) .

We next estimate x 2 ( t ) − x 1 ( t ) for t < t * . From (2.3) and Lemma (2.2), we have

(2.13) ∂ + u = u A − c ∂ + c > u ¯ ( a ) A − c ¯ ∂ + c , ∂ + u < 0 .

Integrate (2.13) along K 1 and K 2 , respectively,

(2.14) u ( x 1 ( t ) , t ) > u ¯ ( k 1 ) − u ¯ ( a ) c ¯ A − c ¯ , u ( x 2 ( t ) , t ) < u ¯ ( k 2 ) .

Now, integrate (2.14) in time between 0 and t to derive

x 1 ( t ) > k 1 + ( u ¯ ( k 1 ) − u ¯ ( a ) c ¯ A − c ¯ ) t , x 2 ( t ) < k 2 + u ¯ ( k 2 ) t .

We therefore obtain

(2.15) x 2 ( t ) − x 1 ( t ) < k 2 − k 1 − u ¯ ( k 1 ) − u ¯ ( k 2 ) − u ¯ ( a ) c ¯ A − c ¯ t ,

with

k 2 − k 1 − u ¯ ( k 1 ) − u ¯ ( k 2 ) − u ¯ ( a ) c ¯ A − c ¯ t < k 2 − k 1 − u ¯ ( k 1 ) − u ¯ ( k 2 ) − u ¯ ( a ) c ¯ A − c ¯ t * = 0 .

We know from (2.12), (2.15), and (1.8) that there exists a point ( x * , t * ) with t * < t * and x * ∈ [ x 1 ( t ) , x 2 ( t ) ] such that (1.10) holds. Then, Theorem 1.1 is proved.

2.3 Blow-up of the derivative of solution

We consider now system (1.3) with initial data (1.11). Let H 1 and H 2 be two characteristic curves determined by

(2.16) H 1 : d x ( t ) d t = u , x ( t ) ∣ t = 0 = h 1 , and H 2 : d x ( t ) d t = A − 2 c A − c u , x ( t ) ∣ t = 0 = h 2 ,

respectively. Let Ω 2 represent the region bounded by H 1 , H 2 and t = 0 .

Lemma 2.3

Assume that problems (1.3) and (1.11) admit a solution. Then, the solution satisfies

(2.17) ∂ + c > 0 , ∂ − c < 0 , ∀ ( x , t ) ∈ Ω 2 .

Proof

Remembering (2.6), we derive that

∂ + ∂ − c ∂ − c = 1 A − c ∂ + c , ∂ − ∂ + c ∂ + c = 2 A − c ∂ + c − 1 A − c ∂ − c .

Integrating and combining with the initial values (1.12) and (1.13), we obtain

∂ − c = ∂ − c ∣ t = 0 exp ∫ 0 t 1 A − c ∂ + c d x + ( t ) = ( c ˆ u ˆ x − c ˆ u ˆ A − c ˆ c ˆ x ) exp ∫ 0 t 1 A − c ∂ + c d x + ( t ) < 0 ,

∂ + c = ∂ + c ∣ t = 0 exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c d x − ( t ) = c ˆ u ˆ x exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c d x − ( t ) > 0 .

Therefore, we have ∂ + c > 0 , ∂ − c < 0 and this lemma is concluded.□

Now, in view of Lemma 2.3 and (2.6), we conclude

(2.18) ∂ − ∂ + c > 2 A − c ( ∂ + c ) 2 .

In addition, (1.11), (1.12), Lemma 2.3, and (2.3) combine to yield

(2.19) 0 < c ( x , t ) < A 2 , u ˆ ( h 1 ) < u ( x , t ) < u ˆ ( h 2 ) , ∀ ( x , t ) ∈ Ω 2 .

Integrating (2.18), we produce the inequality

(2.20) 0 < 1 ∂ + c < 1 c ˆ u ˆ x − 2 A t < 1 c ˆ 1 u ˆ x 1 − 2 A t ,

where c ˆ 1 u ˆ x 1 = inf x ∈ ( h 1 , h 2 ) { c ˆ u ˆ x } . It turns out that there exists a t * * < t * * such that

lim ( x , t ) → ( x * * , t * * ) ∂ + c = + ∞ ,

where t * * is determined by

1 c ˆ 1 u ˆ x 1 − 2 A t = 0 .

Combining (2.4) and Lemma 2.3, we have

c t < 0 .

In light of the foregoing analysis, we deduce

lim ( x , t ) → ( x * * , t * * ) c x ( x , t ) = + ∞ .

This completes the proof of Theorem 1.2.

3 Singularity formation for the inhomogeneous system

3.1 Characteristic decomposition of the inhomogeneous macroscopic production model

Similarly, setting A = ( 1 + 1 ρ 0 ) , we can rewrite (1.5) to read

(3.1) ρ t + ( ρ u ) x = 0 , ( A ρ − 1 ) u t + ( A ρ − 2 ) u u x = − 1 ε ( A ρ − 1 ) u .

The eigenvalues of system (3.1) are

(3.2) λ + = u , λ − = A − 2 c A − c u ,

where the notion c is the same as defined in Section 2 by c = 1 ρ . The characteristic equations of system (3.1) are

(3.3) ∂ + u = u A − c ∂ + c − 1 ε u , ∂ − u = − 1 ε u ,

where ∂ ± = ∂ t + λ ± ∂ x . From the definition of ∂ ± and (3.2), we have

(3.4) ∂ t = A − c c ∂ − − A − 2 c c ∂ + , ∂ x = A − c c u ( ∂ + − ∂ − ) .

Owing to (3.2) and (3.3), we derive the commutator relation [18]

(3.5) ∂ + ∂ − − ∂ − ∂ + = ( ∂ t + λ + ∂ x ) ( ∂ t + λ − ∂ x ) − ( ∂ t + λ − ∂ x ) ( ∂ t + λ + ∂ x ) = ( ∂ + λ − − ∂ − λ + ) ∂ x = − A u ∂ + c ( A − c ) 2 ∂ x + ( A − 2 c ) ∂ + u A − c ∂ x − ∂ − u ∂ x .

Now, in view of (3.3)–(3.5), we obtain the following lemma.

Lemma 3.1

For the variables ( c , u ) , there hold

(3.6) ∂ − ∂ + c = 2 A − c ( ∂ + c ) 2 − 1 A − c ∂ + c ∂ − c − 1 ε ∂ + c , ∂ + ∂ − c = 1 A − c ∂ + c ∂ − c − 1 ε ∂ − c , ∂ − ∂ + u = 2 u ( ∂ + u ) 2 + A − 3 c c u ∂ + u ∂ − u − A − c c u ( ∂ − u ) 2 + A − c ε c ∂ + u − A ε c ∂ − u , ∂ + ∂ − u = − 1 ε ∂ + u .

3.2 Blow-up of the density itself of the solution

Let L 1 ′ and L 2 ′ represent two characteristic curves defined as

(3.7) L 1 ′ : d x + ( t ) d t = u , x + ( 0 ) = a 1 , L 2 ′ : d x − ( t ) d t = A − 2 c A − c u , x − ( 0 ) = b 1 .

Let l 1 ′ : x = u ˜ ( a ) t + a 1 , l 2 ′ : x = A − 2 c ˜ A u ˜ 1 t + b 1 , where u ˜ 1 = inf x ∈ ( a 1 , b 1 ) { u ˜ ( x ) } . Set Ω ˜ 1 ≔ { ( x , t ) ∣ l 1 ′ < x < l 2 ′ , 0 < t < T ˜ } , for some T ˜ > 0 .

Lemma 3.2

Assume that system (1.5) with initial data ( ρ ˜ , u ˜ 0 ( x ) ) admits a classical solution in Ω ˜ 1 . Then, the solution satisfies

(3.8) 0 < c ( x , t ) < c ˜ < A , 0 < u ( x , t ) < u ˜ ( a 1 ) , ∂ + c < 0 , ∂ − c < 0 , ∂ + u < 0 , ∂ − u < 0 , ∣ ∂ + c ∣ < R 2 , ∣ ∂ − c ∣ < − c ˜ u ˜ x , ∣ ∂ + u ∣ < u ˜ ( a 1 ) A − c ˜ R 2 + 1 ε u ˜ ( a 1 ) , ∣ ∂ − u ∣ < 1 ε u ˜ ( a 1 ) ,

for ( x , t ) ∈ Ω ˜ 1 , where R 2 = − c ˜ u ˜ x exp − c ˜ u ˜ x t A − c ˜ 1 + A A − c ˜ u ˜ ( a 1 ) 2 .

Proof

We deduce from (3.6) that

∂ + ∂ − c ∂ − c = 1 A − c ∂ + c − 1 ε , ∂ − ∂ + c ∂ + c = 2 A − c ∂ + c − 1 A − c ∂ − c − 1 ε .

Integrating and combining with the initial data (1.16) and (1.17), we conclude that

∂ − c = ∂ − c ∣ t = 0 exp ∫ 0 t 1 A − c ∂ + c − 1 ε d x + ( t ) = c ˜ u ˜ x exp ∫ 0 t 1 A − c ∂ + c − 1 ε d x + ( t ) < 0 , ∂ + c = ∂ + c ∣ t = 0 exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c − 1 ε d x − ( t ) = c ˜ u ˜ x exp ∫ 0 t 2 A − c ∂ + c − 1 A − c ∂ − c − 1 ε d x − ( t ) < 0 ,

and therefore, we have 0 < c < c ˜ < A . Besides, we know from (3.3) that

u = u ˜ ( x ) exp − 1 ε ∫ 0 t d x − ( t ) ,

which combines with u ˜ x < 0 and u ˜ ( b 1 ) > 0 leads to u > 0 . In light of (3.3) and the fact that ∂ + c < 0 , we obtain

∂ + u < 0 , ∂ − u < 0 .

Thus, we have 0 < u < u ˜ ( a 1 ) .

Observing from our analysis mentioned earlier, we see that

(3.9) ∂ + c < 0 , ∂ − c < 0 , ∂ + u < 0 , ∂ − u < 0 , 0 < c < c ˜ < A , 0 < u ( x , t ) < u ˜ ( a 1 ) , ∀ ( x , t ) ∈ Ω ˜ 1 .

We then deduce from (3.3) and (3.9) that

∣ ∂ − u ∣ < 1 ε u ˜ ( a 1 ) .

From the second equation in (3.6) and (3.9), we have ∂ + ∂ − c > 0 , which means ∂ − c > ∂ − c ∣ t = 0 = c ˜ u ˜ x ; thus we obtain

(3.10) ∣ ∂ − c ∣ < − c ˜ u ˜ x .

In view of (3.6), (3.9), and (3.10), we conclude that

∂ − ∂ + c = 2 A − c ( ∂ + c ) 2 − 1 A − c ∂ + c ∂ − c − 1 ε ∂ + c > − 1 A − c ∂ + c ∂ − c > − c ˜ u ˜ x A − c ˜ ∂ + c .

We therefore derive

∣ ∂ + c ∣ < R 2 ,

where R 2 = − c ˜ u ˜ x exp − c ˜ u ˜ x t A − c ˜ 1 + ( A A − c ˜ u ˜ ( a 1 ) ) 2 . This estimate, combined with (3.3), gives

□ ∣ ∂ + u ∣ ≤ u A − c ∣ ∂ + c ∣ + 1 ε u < u ˜ ( a 1 ) A − c ˜ R 2 + 1 ε u ˜ ( a 1 ) .

By means of ∣ ∂ − u ∣ < 1 ε u ˜ ( a 1 ) , ∣ ∂ + u ∣ < u ˜ ( a 1 ) A − c ˜ R 2 + 1 ε u ˜ ( a 1 ) , ∣ ∂ − c ∣ < − c ˜ u ˜ x , ∣ ∂ + c ∣ < R 2 and (3.4), we have

∣ ∂ x u ∣ < A u ˜ ( a 1 ) c u ˜ 1 R 2 A − c ˜ + 2 ε , ∣ ∂ x c ∣ < A c u ˜ 1 ( R 2 − c ˜ u ˜ x ) ,

where u ˜ 1 = inf x ∈ ( a 1 , b 1 ) { u ˜ ( x ) } .

We next turn our attention to the case that ρ tends to infinity for some point ( x ˜ * , t ˜ * ) in ( a 1 , b 1 ) × ( 0 , t ˜ * ) . Let M 1 : x = x 1 ( t ) and M 2 : x = x 2 ( t ) be two characteristic curves defined as follows :

(3.11) M 1 : d x 1 ( t ) d t = u ( x 1 ( t ) , t ) , x 1 ( t ) ∣ t = 0 = m 1 , M 2 : d x 2 ( t ) d t = u ( x 2 ( t ) , t ) , x 2 ( t ) ∣ t = 0 = m 2 ,

where [ m 1 , m 2 ] ⊂ ( a 1 , b 1 ) . Similar to Section 2.2, we find that the streamlines are in the region of existence of the solution when the time of intersection of M 1 with l 2 ′ is earlier than the time of intersection of M 2 with l 2 ′ . Consequently, we only estimate the intersection moment of M 1 with l 2 ′ . Set M 1 ′ : x = u ˜ ( a 1 ) t + m 1 ; then, the insert time of l 2 and M 1 ′ is t ˜ 0 = b 1 − m 1 u ˜ ( a 1 ) − A − 2 c ˜ A u ˜ 1 . From (1.19), we obtain t ˜ * < t ˜ 0 , which means that the intersection time of M 1 and l 2 ′ is greater than t ˜ * . Therefore, the region defined by M 1 , M 2 , t = 0 , and t = t ˜ * is in Ω ˜ 1 .

We conclude from (3.1) that

d d t ∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x = 0 ,

which means

∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x = ∫ x 1 ( 0 ) x 2 ( 0 ) ρ ( x , 0 ) d x = ρ ˜ ( m 2 − m 1 ) .

From this, it follows that

(3.12) max x 1 ( t ) < x ( t ) < x 2 ( t ) ρ ( x , t ) ≥ ρ ˜ ( m 2 − m 1 ) x 2 ( t ) − x 1 ( t ) .

We next estimate x 2 ( t ) − x 1 ( t ) for t < t ˜ * . From (3.3) and Lemma (3.2), we have

∂ + u = u A − c ∂ + c − 1 ε u > u ˜ ( a 1 ) A − c ˜ ∂ + c − 1 ε u ˜ ( a 1 ) , ∂ + u < 0 .

Integrate along M 1 and M 2 , respectively, to obtain

(3.13) u ( x 1 ( t ) , t ) > u ˜ ( m 1 ) − u ˜ ( a 1 ) c ˜ A − c ˜ − 1 ε u ˜ ( a 1 ) t , u ( x 2 ( t ) , t ) < u ˜ ( m 2 ) .

Integrate (3.13) in time between 0 and t to derive

x 1 ( t ) > m 1 + u ˜ ( m 1 ) − u ˜ ( a 1 ) c ˜ A − c ˜ − 1 ε u ˜ ( a 1 ) t t , x 2 ( t ) < m 2 + u ˜ ( m 2 ) t .

Thus,

(3.14) x 2 ( t ) − x 1 ( t ) < m 2 − m 1 − u ˜ ( m 1 ) − u ˜ ( m 2 ) − u ˜ ( a 1 ) c ˜ A − c ˜ t + u ˜ ( a 1 ) ε t 2 .

Remembering (1.18), we have

t 1 = ε u ˜ ( m 1 ) − u ˜ ( m 2 ) − u ˜ ( a 1 ) c ˜ A − c ˜ − Δ 2 u ˜ ( a 1 ) , t 2 = ε u ˜ ( m 1 ) − u ˜ ( m 2 ) − u ˜ ( a 1 ) c ˜ A − c ˜ − Δ 2 u ˜ ( a 1 ) ,

where Δ = u ˜ ( m 2 ) − u ˜ ( m 1 ) + u ˜ ( a 1 ) c ˜ ρ 0 1 + ρ 0 − c ˜ ρ 0 2 − 4 ε u ˜ ( a 1 ) ( m 2 − m 1 ) . Denote t ˜ * = min { t 1 , t 2 } . We deduce from (3.12) and (3.14) that there exists a point ( x ˜ * , t ˜ * ) with t ˜ * < t ˜ * and x ˜ * ∈ [ x 1 ( t ) , x 2 ( t ) ] such that (1.20) holds. This completes the proof of Theorem 1.3.

3.3 Blow-up of the derivative of solution

In this section, we investigate system (1.5) with the initial data (1.21). Let N 1 and N 2 be two characteristic curves determined by

(3.15) N 1 : d x ( t ) d t = u , x ( t ) ∣ t = 0 = n 1 , and N 2 : d x ( t ) d t = A − 2 c A − c u , x ( t ) ∣ t = 0 = n 2 ,

respectively. Let Ω ˜ 2 represent the region bounded by N 1 , N 2 , and t = 0 .

Lemma 3.3

Assume that problems (1.5) and (1.21) admit a solution. Then, the solution satisfies

(3.16) ∂ + c > 0 , ∂ − c < 0 , ∀ ( x , t ) ∈ Ω ˜ 2 .

Proof

The results are proved in much the same way as Lemma 2.3.□

From (1.22), we see that 0 < u ˇ ( n 1 ) < u ˇ ( x ) . Besides, we have

u = u ˇ ( x ) exp − 1 ε ∫ 0 t 1 + A − 2 c A − c u 2 d τ ,

according to (3.3); therefore, we obtain u > 0 . Then, utilizing the identity in (3.3), we find ∂ − u < 0 . From (1.23), we have ∂ + u ∣ t = 0 = u ˇ c ˇ A − c ˇ u ˇ x − 1 ε u ˇ > 0 . Suppose that point F is the first point where ∂ + u > 0 fails, then ∂ + u ∣ F = 0 . But (3.6) and (3.3) imply

∂ − ∂ + u ∣ F = 2 u ( ∂ + u ) 2 + 2 ε ∂ + u + u ε 2 F = u ε 2 F > 0 ,

which yields a contradiction. In light of the foregoing analysis, we deduce ∂ − u < 0 , ∂ + u > 0 , and thus,

(3.17) u ˇ ( n 1 ) < u < u ˇ ( n 2 ) in Ω ˜ 2 .

By remembering (3.16), we have 0 < c < c ˇ < A 2 . From Lemma 3.3, ∂ + u > 0 , and u > 0 , we conclude

∂ − ∂ + c > ∂ + c 2 A − c ∂ + c − 1 ε = ∂ + c 1 A − c ∂ + c + 1 A − c ∂ + c − 1 ε > 1 A − c ( ∂ + c ) 2 > 1 A ( ∂ + c ) 2 .

By integrating, we obtain

(3.18) 0 < 1 ∂ + c < 1 c ˇ u ˇ x − 1 A t < 1 c ˇ 1 u ˇ x 1 − 1 A t ,

where c ˇ 1 u ˇ x 1 = inf x ∈ ( n 1 , n 2 ) c ˇ u ˇ x . It turns out that there exists a t * * < t * * such that

(3.19) lim ( x , t ) → ( x * * , t * * ) ∂ + c = + ∞ ,

where t * * is determined by

1 c ˇ 1 u ˇ x 1 − 1 A t = 0 .

Combining (3.4) and Lemma (3.3), we deduce

(3.20) c t < 0 .

In view of (3.18)–(3.20), we conclude

lim ( x , t ) → ( x * * , t * * ) c x ( x , t ) = + ∞ .

Therefore, Theorem 1.4 is proved.

4 Numerical simulation

In this section, we show some numerical results of the formation of singularities in Sections 2 and 3. We use the first-order upwind scheme based on the split-coefficient matrix method (SCMM) [21] to discretize systems (1.3) and (1.5).

4.1 Numerical simulation for the homogeneous system

We first transform system (1.3) into the following quasi-linear form:

U t + B U x = 0 ,

in which

(4.1) U = ρ u , B = u ρ 0 ( 1 + 1 ρ 0 ) ρ − 2 ( 1 + 1 ρ 0 ) ρ − 1 u .

Moreover, we have

B = R Λ L ,

where

R = 1 − ρ 1 + 1 ρ 0 ρ − 1 u 0 1 , Λ = u 0 0 ( 1 + 1 ρ 0 ) ρ − 2 ( 1 + 1 ρ 0 ) ρ − 1 u , L = 1 ρ 1 + 1 ρ 0 ρ − 1 u 0 1 .

According to SCMM, the first-order upwind scheme can be written as

U j n + 1 = U j n − △ t △ x { B j − ( U j + 1 n − U j n ) + B j + ( U j n − U j − 1 n ) } ,

where

(4.2) B j + = B j n + ∣ B j n ∣ 2 , B j − = B j n − ∣ B j n ∣ 2 , ∣ B j n ∣ = R j n ∣ Λ j n ∣ L j n .

For the blow-up of the solution itself, we take the following initial data:

u 0 ( x ) = 15 , x < 0 , 5 cos x + 10 , 0 ≤ x ≤ π , 5 , x > π , ρ ¯ = 2 , and ρ 0 = 1 .

For the blow-up of the derivative of the solution, we take the following initial data:

(4.3) u ˆ ( x ) = 116 , x < 1 , x 2 − x + 116 , 1 ≤ x ≤ 5 , 136 , x > 5 , ρ ˆ ( x ) = 10 , x < 1 , 10 x , 1 ≤ x ≤ 5 , 2 , x > 5 , and ρ 0 = 1 .

The numerical results are displayed in Figures 3 and 4. As depicted in Figure 3, the curves of the density function are smooth initially but undergo a sharp increase over time, corresponding to the blow-up of the density itself of the solution. Besides, we see from Figure 4 that the curves of the density function are smooth initially but appear discontinuity when time increases, which coincides with the blow-up of the derivative of the solution.

$Figure 3 Values of ρ \rho and u u in problems (1.3) and (1.6) when t = 0.1 t=0.1 , t = 0.2 t=0.2 , t = 0.3 t=0.3 , and t = 0.4 t=0.4 .$

Figure 3

Values of ρ and u in problems (1.3) and (1.6) when t = 0.1 , t = 0.2 , t = 0.3 , and t = 0.4 .

$Figure 4 Values of ρ \rho and u u in problems (1.3) and (1.11) when t = 0.2 t=0.2 , t = 0.4 t=0.4 , t = 0.6 t=0.6 , and t = 0.8 t=0.8 .$

Figure 4

Values of ρ and u in problems (1.3) and (1.11) when t = 0.2 , t = 0.4 , t = 0.6 , and t = 0.8 .

4.2 Numerical simulation for the inhomogeneous system

For system (1.5), we first transform it into the following quasi-linear form:

U t + B U x = E , with E = 0 − u ε ,

where U and B are defined by (4.1). According to SCMM, the first-order upwind scheme can be written as

U j n + 1 = U j n − △ t △ x { B j − ( U j + 1 n − U j n ) + B j + ( U j n − U j − 1 n ) } + △ t E j n ,

in which B j + , B j − , and ∣ B j n ∣ are defined by (4.2). For the blow-up of the solution itself, we take the following initial data:

u ˜ 0 ( x ) = 43 , x < 0 , x 2 − 13 x + 43 , 0 ≤ x ≤ 6 , 1 , x > 6 , ρ ˜ = 3 2 , ε = 1 , and ρ 0 = 1 .

For the blow-up of the derivative of the solution, we take the following initial data:

u ˇ 0 ( x ) = 546 , x < 2 , x 2 + 12 x + 518 , 2 ≤ x ≤ 8 , 678 , x > 8 , ρ ˇ ( x ) = 8 , x < 2 , 16 x , 2 ≤ x ≤ 8 , 2 , x > 8 , ε = 1 , and ρ 0 = 1 .

The numerical results are shown in Figures 5 and 6. From Figure 5, we discover that the curves of the density function are smooth initially but undergo a sharp increase over time, corresponding to the blow-up of the density itself of the solution. In addition, by observing Figure 6, we find that the curves of the density function are smooth initially but appear discontinuity when time increases, which coincides with the blow-up of the derivative of the solution.

$Figure 5 Values of ρ \rho and u u in problems (1.5) and (1.16) when t = 0.1 t=0.1 , t = 0.12 t=0.12 , t = 0.13 t=0.13 , and t = 0.14 t=0.14 .$

Figure 5

Values of ρ and u in problems (1.5) and (1.16) when t = 0.1 , t = 0.12 , t = 0.13 , and t = 0.14 .

$Figure 6 Values of ρ \rho and u u in problems (1.5) and (1.21) when t = 0.01 t=0.01 , t = 0.08 t=0.08 , t = 0.15 t=0.15 , and t = 0.20 t=0.20 .$

Figure 6

Values of ρ and u in problems (1.5) and (1.21) when t = 0.01 , t = 0.08 , t = 0.15 , and t = 0.20 .

5 Conclusion

In this work, we have studied the problem of singularity generation for the solution of the macroscopic production model with Chaplygin gas using the method of characteristic decomposition and have theoretically proven that physical phenomena like bottlenecks can occur. Initially, we consider the homogeneous system (1.3) and, giving different initial values, show that the solution itself and the derivative of the solution blow up in finite time, respectively. Subsequently, we analyze the inhomogeneous system (1.5), utilizing the same steps used in the study of the singularity generation problem for the homogeneous system, to demonstrate that the solution itself and the derivative of the solution blow up in finite time. Finally, we present several representative examples of numerical simulations to enhance the understanding of our theoretical findings.

Acknowledgments

The authors are very grateful to the anonymous referees to their careful reading of the manuscript and valuable comments.

Funding information: This work was partially supported by the National Natural Science Foundation of China (12161084, 12071106, 12171130), the Natural Science Foundation of Xinjiang, PR China (2022D01E42), and the Natural Science Foundation of Zhejiang Province of China (LMS25A010014).
Author contributions: The authors contributed equally to this manuscript.
Conflict of interest: The authors of this article warrant that there is no conflict of interest in this article.
Originality statement: The authors confirm that all the images used in the manuscript are original.

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Received: 2024-08-19

Revised: 2025-01-14

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-2025-0098

Keywords for this article

macroscopic production model; Chaplygin gas; singularity formation; characteristic decomposition; numerical simulation

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BY 4.0