Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics

Yanbo Hu; Ying Zeng

doi:10.1515/anona-2025-0126

Article Open Access

Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics

Yanbo Hu and Ying Zeng

Published/Copyright: November 15, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

This article is concerned with the singularity formation of smooth solutions for a nonhomogeneous hyperbolic system arising in magnetohydrodynamics. The system owns four linearly degenerate characteristic fields that influence each other in the relations of second derivatives, making the problem difficult to handle. We develop the characteristic decomposition technique to apply the nonlinear hyperbolic equations with more than two wave characteristics. It is verified that the smooth solution can form a singularity in finite time and the density itself tends to infinity at the blowup point for a special kind of initial data.

Keywords: magnetohydrodynamics; Chaplygin gas; singularity; characteristic decomposition

MSC 2020: 35L60; 35Q35; 35B44

1 Introduction

We are interested in the singularity formation of smooth solutions for the one-dimensional hyperbolic system arising in magnetohydrodynamics

(1.1) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + p ( ρ ) − ρ G 2 ) x = κ ρ v , ( ρ v ) t + ( ρ u v − ρ G F ) x = − κ ρ u , ( ρ G ) t = 0 , ( ρ F ) t + ( ρ F u − ρ G v ) x = 0 ,

supplemented with a divergence constraint

(1.2) ( ρ G ) x = 0 .

In system (1.1), ρ ≥ 0 is the density of the fluid, p ( ρ ) is the pressure, which is a given smooth function of ρ , u and v are, respectively, the velocities of the fluid in x and y directions, G and F are the respective magnetic field components along x and y axes, the term ( κ ρ v , − κ ρ u ) T on the right-hand side of the second and third equations corresponds to the Coriolis force generated by the Earth’s rotation, and the constant κ > 0 is the Coriolis parameter. Equation (1.2) originates from the magnetic field divergence-free equation in the initial magnetohydrodynamics equations.

System (1.1) with the equation of state p ( ρ ) = A ρ 2 is known as the one-dimensional rotating shallow water magnetohydrodynamic (SWMHD) equations, which were first introduced by Gilman [13] for describing the phenomena in the solar tachocline, also see the derivation by Rossmanith [43]. Subsequently, this model was widely used in many important incompressible physical contexts, for example, see [17,47] for the neutron-star atmosphere dynamics and [1,53] for the optimization of aluminium production process. In [26], Klimachkova and Petrosyan generalized the shallow water magnetohydrodynamic theory describing incompressible flows of plasma to the case of compressible flows. They [27] further investigated the compressibility effects in the magnetohydrodynamics of a rotating astrophysical plasma in the shallow water approximation. Fedotova and Petrosyan [11] derived the system of magnetohydrodynamic equations for a compressible rotating plasma with a stable stratification in a gravity field in the anelastic approximation, also see [10]. For more magnetohydrodynamic theories in rotating astrophysical plasmas that consider compressibility, we refer the reader to the review paper [9] and the references therein. Mathematically, the hyperbolic theory and the Riemann problem of (1.1) with κ = 0 were investigated among others in [24,48]. In [15], Gu et al. showed the existence of global weak entropy solutions for its Cauchy problem by using the compensated compactness theory. The linear instability of horizontal shear flows for SWMHD equations was discussed in [40], and the structural stability of shock waves and current vortex sheets was explored in [49].

When the magnetic field is ignored, system (1.1) reduces to the one-dimensional rotating Euler equations

(1.3) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + p ( ρ ) ) x = κ ρ v , ( ρ v ) t + ( ρ u v ) x = − κ ρ u ,

which have been extensively researched since they were introduced [12,42] as a model for studying the behavior of large-scale geophysical motions in a thin layer of fluid. In particular, the finite-time singularity formation of classical solutions of (1.3) with polytropic gas equation of state p ( ρ ) = A ρ γ was verified by Cheng et al. [5]. They also established the global existence of classical solutions under the smallness assumptions on the initial data. The axisymmetric case of the 2D rotating shallow water system was analyzed in [23]. It is important and interesting that the effects of rotation may delay or even prevent the formation of singularities of smooth solutions [6,7,38]. Moreover, Hu and Qian [21] established the well-posedness of strong solutions for an initial-boundary value problem of the 2D axisymmetric rotating Euler equations.

In this article, we focus on the singularity formation for system (1.1) with the Chaplygin gas equation of state

(1.4) p ( ρ ) = − 1 ρ ,

which was introduced by Chaplygin [3], Tsien [51], and vonKarman [25] as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. This equation of state also finds its applications in the study of the dark energy in cosmology [8,14]. Furthermore, the Chaplygin gas equation of state makes system (1.1) totally linearly degenerate, which is quite different from the polytropic gas equation of state p ( ρ ) = A ρ γ with γ ≥ 1 . The classical Euler equations with Chaplygin gases (1.4) have been well investigated in the past two decades, see, for example, [2,16,30,44,45]. In particular, the formation of singularities for the system of one-dimensional Chaplygin gas was presented in [28]. In [32], Lai and Zhu applied the characteristic decomposition technique to examine the formation of singularities of smooth solutions for the 1D and 2D axisymmetric Chaplygin Euler equations. With the aid of the idea of dealing with the variational wave equations, Chen et al. [4] probed the formation of singularities for a spatially inhomogeneous Chaplygin Euler system and demonstrated that the density becomes zero at the occurrence of blowup. In [39], we utilized the characteristic decomposition technique to overcome the influence of the rotation terms to verify that the density itself of smooth solutions of (1.3) with (1.4) blows up in finite time.

System (1.1) with the Chaplygin gas equation of state (1.4) is a nonhomogeneous hyperbolic system, and its local-in-time existence of smooth solutions to the Cauchy problem can be obtained by the general theory of quasilinear hyperbolic equations (e.g., [37]). In this article, we consider the singularity formation of its smooth solutions in finite time. Compared to the earlier works on system (1.3) with (1.4), we have to handle the interaction of multiple wave characteristics in the characteristic decompositions. The characteristic decomposition method is a powerful tool created by Li et al. [33], which was employed to successfully solve many hyperbolic problems, see, e.g., [18–20,22,29,31,34–36,46,52]. We point out that all the applications of the characteristic decomposition idea in previous references are solved the hyperbolic systems with two wave characteristics or two wave characteristics accompanied by one flow characteristic. It is easy to check that system (1.1) with (1.2) and (1.4) owns four wave characteristics, which is essentially different from system (1.3) with (1.4). We develop the characteristic decomposition theory to show that the smooth solution can form a singularity in finite time and the density ρ tends to infinity at the blowup point for a special kind of initial data. This is the first time that the characteristic decomposition method has been excitingly applied to the hyperbolic systems with more than two wave characteristics, which is the main innovation and contribution of the article. We comment that the technique developed here may be used for studying other hyperbolic problems to system (1.1) and more general equations.

The main result of this article is stated as follows:

Theorem 1

Let ( ρ 0 , u 0 , v 0 , G 0 , F 0 ) ( x ) ∈ C 1 ( R ) satisfy

(1.5) ρ 0 ( x ) ≡ 1 , G 0 ( x ) ≡ ν 0 , F 0 ( x ) ≡ 1 , v 0 ( x ) = v ¯ ( x ) , x ∈ ( − ∞ , − 1 ) , 1 2 u ˜ ( 1 − x ) , x ∈ [ − 1 , 1 ] , v ̲ ( x ) , x ∈ ( 1 , + ∞ ) u 0 ( x ) = u ¯ ( x ) , x ∈ ( − ∞ , − 1 ) , u ˆ ( x ) , x ∈ [ − 1 , 1 ] , u ̲ ( x ) , x ∈ ( 1 , + ∞ ) ,

where ν 0 ≥ 4 κ is a constant, v ¯ ( x ) , v ̲ ( x ) , u ¯ ( x ) , and u ̲ ( x ) are some C 1 functions, u ˜ = u ˆ ( − 1 ) = 24 ( 1 + ν 0 ) , and u ˆ ( x ) is a C 1 function satisfying

(1.6) u ˆ ( 1 ) = 1 , u ˆ ′ ( x ) = − 1 4 u ˜ , x ∈ − 1 , − 1 8 , α ( x ) , x ∈ − 1 8 , − 1 16 , − 3 u ˜ , x ∈ − 1 16 , 1 16 , β ( x ) , x ∈ 1 16 , 1 8 , − 1 4 u ˜ , x ∈ 1 8 , 1 .

Figure 1 shows the diagram of u ˆ ( x ) . In (1.6), α ( x ) and β ( x ) are two continuous functions such that u ˆ ′ ( x ) is continuous on the closed interval [ − 1 , 1] and satisfy

(1.7) − 3 u ˜ ≤ α ( x ) , β ( x ) ≤ − 1 4 u ˜ ,

and

(1.8) ∫ − 1 8 − 1 16 − α ( x ) d x = ∫ 1 16 1 8 − β ( x ) d x = 7 + 9 ν 0 4 .

Assume that ( ρ , u , v , G , F ) ( x , t ) ∈ C 1 is a solution of (1.1)–(1.2), (1.4) for ( x , t ) ∈ R × [ 0 , T ) with initial data ( ρ 0 , u 0 , v 0 , G 0 , F 0 ) ( x ) . Then, there holds

(1.9) T < T * = 1 48 ( 1 + ν 0 ) − 16 1 + ν 0 2 .

Moreover, there exists a point ( x * , t * ) ∈ ( − 1 , 1 ) × ( 0 , T * ) such that

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

$Figure 1 The diagram of u ˆ ( x ) \hat{u}\left(x) in the interval [ − 1 -1 , 1].$

Figure 1

The diagram of u ˆ ( x ) in the interval [ − 1 , 1].

Remark 1

The conditions in (1.8) can be satisfied by (1.7) and the continuity of functions α ( x ) and β ( x ) , which are used to ensure u ˆ ( − 1 ) = 24 ( 1 + ν 0 ) by (1.6).

Remark 2

The special initial data taken in Theorem 1 are only for the convenience of considering the problem. The technique employed in the article can handle more general initial data. Specifically, according to the idea of the article, the key is to construct a special region of the solution (Lemma 3), whose boundaries have properties similar to the invariant region except for the boundary 1 ⁄ ρ = 0 . For the general initial data ( ρ 0 , u 0 , v 0 , G 0 , F 0 ) ( x ) ∈ C 1 ( R ) satisfying the following conditions on an interval [ a , b ]

G 0 ( x ) = ν 0 , 0 < c ̲ ≤ 1 ρ 0 ( x ) ≤ c ¯ , ∣ c 0 ′ ( x ) ∣ ≤ c ¯ 1 , ∣ F 0 ( x ) ∣ ≤ F ¯ , ∣ F 0 ′ ( x ) ∣ ≤ F ¯ 1 , u 0 ( b ) ≥ 0 , v 0 ( b ) ≥ 0 , u 0 ′ ( x ) < − κ v 0 ( a ) 1 + ν 0 2 c ̲ − 1 + ν 0 2 c ¯ 1 , v 0 ′ ( x ) < − − κ u 0 ( a ) ν 0 c ̲ − F ¯ 1 ,

where c ̲ , c ¯ , c ¯ 1 , F ¯ , F ¯ 1 are positive constants, one can utilize the characteristic decomposition technique to establish a region of the solution similar to Ω ˜ in Lemma 3 and then show that the smooth solution forms a singularity in finite time. Moreover, the time and location of blowup of the solution depend on the constants κ , ν 0 , c ̲ , c ¯ , c ¯ 1 , F ¯ , F ¯ 1 and u 0 ( a ) , u 0 ( b ) , v 0 ( a ) , v 0 ( b ) .

Remark 3

We shall see that, for the chosen initial data in Theorem 1, the presence of the magnetic field in system (1.1) plays a role in accelerating the formation of singularity.

Remark 4

The formation of singularity of smooth solutions for system (1.1) with a polytropic gas is also a very interesting problem. Compared with system (1.3) handled by Cheng et al. [5], the new difficulty lies in the fact that two different types of singularities can occur simultaneously for system (1.1), as two of its four wave characteristic fields are genuinely nonlinear, while the other two are linearly degenerate. Moreover, the potential vorticity is no longer conserved due to the presence of the magnetic field. We will study this problem in future work.

2 Proof of the main theorem

We show Theorem 1 by using the idea of characteristic decompositions in this section. In Section 2.1, we rewrite the system into the diagonal form and then develop the characteristic decomposition technique to apply it to the hyperbolic system with four wave characteristics. In Section 2.2, we analyze the initial values for the derivatives of the solution by the conditions given in (1.5)–(1.8) and then establish their estimates. The proof of Theorem 1 is completed in Section 2.3. Moreover, some numerical simulation results are presented in Section 2.4.

2.1 The characteristic decompositions

It immediately follows from the fourth equation of (1.1) and (1.2) that

(2.1) ρ G ≡ Const. ,

which, together with the initial data in (1.5), gives

(2.2) G ( x , t ) = ν 0 ρ ( x , t ) .

Putting (2.2) and (1.4) into (1.1), we obtain

(2.3) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ρ u 2 − 1 + ν 0 2 ρ x = κ ρ v , ( ρ v ) t + ( ρ u v − ν 0 F ) x = − κ ρ u , ( ρ F ) t + ( ρ F u − ν 0 v ) x = 0 .

For smooth solutions, system (2.3) can be rewritten as

(2.4) ρ u v F t + u ρ 0 0 ( 1 + ν 0 2 ) c 3 u 0 0 0 0 u − ν 0 c 0 0 − ν 0 c u ρ u v F x = 0 κ v − κ u 0 ,

where c = p ′ ( ρ ) = 1 ⁄ ρ is the speed of sound. We acquire the four wave eigenvalues of (2.4)

(2.5) λ ± = u ± 1 + ν 0 2 c , λ ˜ ± = u ± ν 0 c .

Clearly, there holds for c > 0

(2.6) λ − < λ ˜ − < λ ˜ + < λ + .

By direct calculation, the corresponding right and left eigenvectors of λ ± and λ ˜ ± are

(2.7) r ± = ( ± ρ , 1 + ν 0 2 c , 0 , 0 ) T , r ˜ ± = ( 0 , 0 , 1 , ∓ 1 ) T ,

and

(2.8) ℓ ± = ( 1 + ν 0 2 c , ± ρ , 0 , 0 ) , ℓ ˜ ± = ( 0 , 0 , 1 , ∓ 1 ) .

One can easily check that there has ∇ λ ± ⋅ r ± ≡ 0 and ∇ λ ˜ ± ⋅ r ˜ ± ≡ 0 , which means that all four eigenvalues of (2.3) are linearly degenerate in the sense of Lax. We left-multiply (2.3) by ( ℓ + , ℓ − , ℓ ˜ + , ℓ ˜ − ) T and do simplifications to achieve the characteristic form

(2.9) ∂ + ( u − 1 + ν 0 2 c ) = κ v , ∂ − ( u + 1 + ν 0 2 c ) = κ v , ∂ ˜ + ( v − F ) = − κ u , ∂ ˜ − ( v + F ) = − κ u , or ∂ + u = 1 + ν 0 2 ∂ + c + κ v , ∂ − u = − 1 + ν 0 2 ∂ − c + κ v , ∂ ˜ + v = ∂ ˜ + F − κ u , ∂ ˜ − v = − ∂ ˜ − F − κ u ,

where

(2.10) ∂ ± = ∂ t + λ ± ∂ x , ∂ ˜ ± = ∂ t + λ ˜ ± ∂ x .

Furthermore, the following operator relations are valid:

(2.11) ∂ t = ( u + 1 + ν 0 2 c ) ∂ − − ( u − 1 + ν 0 2 c ) ∂ + 2 1 + ν 0 2 c , ∂ x = ∂ + − ∂ − 2 1 + ν 0 2 c , ∂ t = ( u + ν 0 c ) ∂ ˜ − − ( u − ν 0 c ) ∂ ˜ + 2 ν 0 c , ∂ x = ∂ ˜ + − ∂ ˜ − 2 ν 0 c ,

and

(2.12) ∂ ˜ + = ( 1 + ν 0 2 + ν 0 ) ∂ + + ( 1 + ν 0 2 − ν 0 ) ∂ − 2 1 + ν 0 2 , ∂ ˜ − = ( 1 + ν 0 2 + ν 0 ) ∂ − + ( 1 + ν 0 2 − ν 0 ) ∂ + 2 1 + ν 0 2 ,

and

(2.13) ∂ + = ( 1 + ν 0 2 + ν 0 ) ∂ ˜ + − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ − 2 ν 0 , ∂ − = ( 1 + ν 0 2 + ν 0 ) ∂ ˜ − − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ + 2 ν 0 .

In addition, there has

Lemma 1

The following commutator relations of second-derivative operators hold

(2.14) ∂ + ∂ − = ∂ − ∂ + , ∂ ˜ + ∂ ˜ − = ∂ ˜ − ∂ ˜ + .

Proof

It suggests by (2.5) and (2.10) that

∂ + ∂ − − ∂ − ∂ + = ( ∂ t + λ + ∂ x ) ( ∂ t + λ − ∂ x ) − ( ∂ t + λ − ∂ x ) ( ∂ t + λ + ∂ x ) = ( ∂ + λ − − ∂ − λ + ) ∂ x = ( κ v − κ v ) ∂ x = 0 ,

which leads to the first conclusion in (2.14).

Moreover, we calculate by utilizing (2.12) and (2.9)

(2.15) ∂ ˜ + λ ˜ − = ( 1 + ν 0 2 + ν 0 ) ∂ + + ( 1 + ν 0 2 − ν 0 ) ∂ − 2 1 + ν 0 2 ( u − ν 0 c ) = ( 1 + ν 0 2 + ν 0 ) 2 1 + ν 0 2 [ ( 1 + ν 0 2 − ν 0 ) ∂ + c + κ v ] + ( 1 + ν 0 2 − ν 0 ) 2 1 + ν 0 2 [ − ( 1 + ν 0 2 + ν 0 ) ∂ − c + κ v ] = ∂ + c − ∂ − c 2 1 + ν 0 2 + κ v ,

and

(2.16) ∂ ˜ − λ ˜ + = ( 1 + ν 0 2 + ν 0 ) ∂ − + ( 1 + ν 0 2 − ν 0 ) ∂ + 2 1 + ν 0 2 ( u + ν 0 c ) = ( 1 + ν 0 2 + ν 0 ) 2 1 + ν 0 2 [ − ( 1 + ν 0 2 − ν 0 ) ∂ − c + κ v ] + ( 1 + ν 0 2 − ν 0 ) 2 1 + ν 0 2 [ ( 1 + ν 0 2 + ν 0 ) ∂ + c + κ v ] = ∂ + c − ∂ − c 2 1 + ν 0 2 + κ v .

Combining (2.15) and (2.16) gives

(2.17) ∂ ˜ + λ ˜ − = ∂ ˜ − λ ˜ + .

Thanks to (2.5), (2.10), and (2.17), we see that

∂ ˜ + ∂ ˜ − − ∂ ˜ − ∂ ˜ + = ( ∂ t + λ ˜ + ∂ x ) ( ∂ t + λ ˜ − ∂ x ) − ( ∂ t + λ ˜ − ∂ x ) ( ∂ t + λ ˜ + ∂ x ) = ( ∂ ˜ + λ ˜ − − ∂ ˜ − λ ˜ + ) ∂ x = 0 ,

which arrives at the second conclusion in (2.14). The proof of the lemma is finished.□

We then obtain the characteristic decompositions for the variables ( c , F ) and ( u , v )

Lemma 2

For the variables c and u, there hold

(2.18) ∂ + ∂ − c = ∂ − ∂ + c = κ 2 ν 0 ( ∂ ˜ + F + ∂ ˜ − F ) , ∂ ˜ + ∂ ˜ − F = ∂ ˜ − ∂ ˜ + F = − κ ν 0 2 ( ∂ + c + ∂ − c ) ,

and

(2.19) ∂ + ∂ − u = ∂ − ∂ + u = κ 2 ( ∂ ˜ + v + ∂ ˜ − v ) , ∂ ˜ + ∂ ˜ − v = ∂ ˜ − ∂ ˜ + v = − κ 2 ( ∂ + u + ∂ − u ) .

Proof

According to (2.14), we first acquire

(2.20) ∂ + ∂ − u = ∂ − ∂ + u , ∂ + ∂ − c = ∂ − ∂ + c .

Differentiating the first two equations of (2.9) yields

(2.21) ∂ − ∂ + u = 1 + ν 0 2 ∂ − ∂ + c + κ ∂ − v , ∂ + ∂ − u = − 1 + ν 0 2 ∂ + ∂ − c + κ ∂ + v .

One combines (2.20) and (2.21) and applies (2.13) and (2.9) to find that

(2.22) ∂ − ∂ + c = ∂ + ∂ − c = κ ( ∂ + v − ∂ − v ) 2 1 + ν 0 2 = κ 2 1 + ν 0 2 ( 1 + ν 0 2 + ν 0 ) ∂ ˜ + v − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ − v 2 ν 0 − ( 1 + ν 0 2 + ν 0 ) ∂ ˜ − v − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ + v 2 ν 0 = κ 2 ν 0 ( ∂ ˜ + v − ∂ ˜ − v ) = κ 2 ν 0 ( ∂ ˜ + F + ∂ ˜ − F ) .

Analogically, we gain by (2.14)

(2.23) ∂ ˜ + ∂ ˜ − v = ∂ ˜ − ∂ ˜ + v , ∂ ˜ + ∂ ˜ − F = ∂ ˜ − ∂ ˜ + F .

One differentiates the last two equations of (2.9) to find that

(2.24) ∂ ˜ − ∂ ˜ + v = ∂ ˜ − ∂ ˜ + F − κ ∂ ˜ − u , ∂ ˜ + ∂ ˜ − v = − ∂ ˜ + ∂ ˜ − F − κ ∂ ˜ + u ,

which, along with (2.23) and (2.12), (2.9) obtains

(2.25) ∂ ˜ − ∂ ˜ + F = ∂ ˜ + ∂ ˜ − F = − κ 2 ( ∂ ˜ + u − ∂ ˜ − u ) = − κ 2 ( 1 + ν 0 2 + ν 0 ) ∂ + u + ( 1 + ν 0 2 − ν 0 ) ∂ − u 2 1 + ν 0 2 − ( 1 + ν 0 2 + ν 0 ) ∂ − u + ( 1 + ν 0 2 − ν 0 ) ∂ + u 2 1 + ν 0 2 = − κ ν 0 2 1 + ν 0 2 ( ∂ + u − ∂ − u ) = − κ ν 0 2 ( ∂ + c + ∂ − c ) .

Then, we have derived the two equations in (2.18).

We now insert (2.22) into (2.21) and employ (2.12) and (2.13) to attain

(2.26) ∂ − ∂ + u = ∂ + ∂ − u = 1 + ν 0 2 ⋅ κ ( ∂ + v − ∂ − v ) 2 1 + ν 0 2 + κ ∂ − v = κ 2 ( ∂ + v + ∂ − v ) = κ 2 ( 1 + ν 0 2 + ν 0 ) ∂ ˜ + v − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ − v 2 ν 0 + ( 1 + ν 0 2 + ν 0 ) ∂ ˜ − v − ( 1 + ν 0 2 − ν 0 ) ∂ ˜ + v 2 ν 0 = κ 2 ( ∂ ˜ + v + ∂ ˜ − v ) .

The second equation of (2.19) can be derived in a similar way. The proof of the lemma is complete.□

2.2 The estimates

We first analyze the initial data of ( ∂ ± c , ∂ ˜ ± F ) and ( ∂ ± u , ∂ ˜ ± v ) on the closed interval [ − 1 , 1]. Thanks to (1.5), one has on [ − 1 , 1]

(2.27) c 0 ( x ) = 1 ρ 0 ( x ) = 1 , F 0 ( x ) = 1 , v 0 ( x ) = 1 2 u ˜ ( 1 − x ) , u 0 ( x ) = u ˆ ( x ) ,

with u ˆ ( 1 ) = 1 and u ˆ ( − 1 ) = u ˜ . Furthermore, it follows by (1.6) and (1.8) that

(2.28) u ˜ = u ˆ ( − 1 ) = u ˆ ( 1 ) − ∫ − 1 1 u ˆ ′ ( x ) d x = 1 − ∫ − 1 − 1 8 − 1 4 u ˜ d x − ∫ − 1 8 − 1 16 α ( x ) d x − ∫ − 1 16 1 16 − 3 u ˜ d x − ∫ 1 16 1 8 β ( x ) d x − ∫ 1 8 1 − 1 4 u ˜ d x = 1 + 2 ⋅ 7 8 ⋅ 1 4 u ˜ + 2 ⋅ 7 + 9 ν 0 4 + 1 8 ⋅ 3 u ˜ = 13 16 u ˜ + 9 ( 1 + ν 0 ) 2 ,

which indicates that u ˜ = 24 ( 1 + ν 0 ) . With the help of (2.27) and (1.6), we observe that

(2.29) c x ∣ t = 0 = 0 , F x ∣ t = 0 = 0 , v x ∣ t = 0 = − 1 2 u ˜ , u x ∣ t = 0 = u ˆ ′ ( x ) ∈ − 3 u ˜ , − 1 4 u ˜ .

Hence, v 0 ( x ) and u ˆ ( x ) are two strictly decreasing functions satisfying 0 ≤ v 0 ( x ) ≤ u ˜ and 1 ≤ u ˆ ( x ) ≤ u ˜ on [ − 1 , 1].

Now, we recall the first and fourth equations of (2.4) to acquire

(2.30) c t + u c x = c u x , F t + u F x = ν 0 c v x ,

which, together with (2.27) and (2.29), yields

(2.31) c t ∣ t = 0 = u ˆ ′ ( x ) , F t ∣ t = 0 = − ν 0 2 u ˜ , ∀ x ∈ [ − 1 , 1 ] .

Thus, we achieve the data ( ∂ ± c , ∂ ˜ ± F ) ∣ t = 0 and ( ∂ ± u , ∂ ˜ ± v ) ∣ t = 0 on [ − 1 , 1]

(2.32) ∂ ± c ∣ t = 0 = u ˆ ′ ( x ) , ∂ ± u ∣ t = 0 = ± 1 + ν 0 2 u ˆ ′ ( x ) + κ v 0 ( x ) ,

and

(2.33) ∂ ˜ ± F ∣ t = 0 = − ν 0 2 u ˜ , ∂ ˜ ± v ∣ t = 0 = ∓ ν 0 2 u ˜ − κ u ˆ ( x ) .

In view of the choice of the initial data and ν 0 ≥ 4 κ , one observes that there hold on [ − 1 , 1]

(2.34) − 3 u ˜ ≤ ∂ ± c ∣ t = 0 ≤ − 1 4 u ˜ , ∂ ˜ ± F ∣ t = 0 = − ν 0 2 u ˜ < − κ u ˜ < 0 , ∂ + u ∣ t = 0 ≤ 1 + ν 0 2 ⋅ − 1 4 u ˜ + κ v 0 ( − 1 ) = − 1 + ν 0 2 4 u ˜ + κ u ˜ < 0 , ∂ − u ∣ t = 0 ≥ 1 + ν 0 2 − 1 4 u ˜ + κ v 0 ( 0 ) > 0 , ∂ ˜ + v ∣ t = 0 ≤ − ν 0 2 u ˜ − κ u ˆ ( 1 ) < 0 , ∂ ˜ − v ∣ t = 0 ≥ ν 0 2 u ˜ − κ u ˆ ( − 1 ) = κ u ˜ > 0 .

The main strategy of the article is to show that c reaches zero in the region where ∂ ˜ ± F is less than − κ u ˜ .

Let l ± be two lines defined as follows:

(2.35) l + : x = [ u ˜ + 1 + ν 0 2 ] t − 1 , l − : x = [ 1 − 1 + ν 0 2 ] t + 1 .

Denote the intersection time of l ± by T ˜ , where

(2.36) T ˜ = 2 u ˜ − 1 + 2 1 + ν 0 2 .

Recalling u ˜ = 24 ( 1 + ν 0 ) , it is not difficult to verify that

(2.37) T ˜ < 1 4 ν 0 ≕ T ^ .

We use Ω ˜ to represent the region bounded by l ± and t = 0 . Then

Lemma 3

Suppose that system (1.1), (1.2), and (1.4) with initial data ( ρ 0 , u 0 , v 0 , G 0 , F 0 ) ( x ) admits a smooth solution in Ω ˜ . Then, the solution in Ω ˜ satisfies

(2.38) ∂ ± c < − κ u ˜ 1 + ν 0 2 , ∂ ˜ ± F < − κ u ˜ , ∂ + u < 0 , ∂ − u > 0 , ∂ ˜ + v < 0 , ∂ ˜ − v > 0 , 0 < c < 1 , 1 < u < u ˜ , 0 < v < u ˜ .

Moreover, the region Ω ˜ is a strong determinate domain, that is, for any point ( ζ , τ ) ∈ Ω ˜ , the four characteristic curves x ± ( t ; ζ , τ ) and x ˜ ± ( t ; ζ , τ ) defined by

(2.39) d x ± ( t ; ζ , τ ) d t = λ ± ( x ± ( t ; ζ , τ ) , t ) , x ± ( τ ; ζ , τ ) = ζ , d x ˜ ± ( t ; ζ , τ ) d t = λ ˜ ± ( x ˜ ± ( t ; ζ , τ ) , t ) , x ˜ ± ( τ ; ζ , τ ) = ζ ,

for t ∈ [ 0 , τ ] only intersect the line t = 0 and not the lines l ± .

Proof

The proof of the lemma mainly lies in the inequalities of ∂ ± c and ∂ ˜ ± F . Set Ω ˜ ε = Ω ˜ ∩ { t < ε } for any ε > 0 . By means of (2.34) and the continuity, we know that

(2.40) ∂ ± c < − κ u ˜ 1 + ν 0 2 , ∂ ˜ ± F < − κ u ˜ , ∀ ( x , t ) ∈ Ω ˜ ε ′ ,

for sufficiently small ε ′ . Now, we move the line t = ε from ε ′ to T ˜ . Assume that the point P in Ω ˜ is the first time so that ∂ + c ∣ P = − κ u ˜ 1 + ν 0 2 and

(2.41) ∂ + c < − κ u ˜ 1 + ν 0 2 , ∂ − c ≤ − κ u ˜ 1 + ν 0 2 , ∂ ˜ ± F ≤ − κ u ˜ , ∀ ( x , t ) ∈ Ω ˜ t P .

Then, it suggests by (2.9) and (2.41) that

(2.42) ( c , u , v ) ∈ ( 0,1 ) × ( 1 , u ˜ ) × ( 0 , u ˜ ) , ∀ ( x , t ) ∈ Ω ˜ t P .

In fact, if there exists a first point Q ( t Q < t P ) such that one of c , u , or v touches the boundary of ( 0,1 ) × ( 1 , u ˜ ) × ( 0 , u ˜ ) . Without the loss of generality, suppose that u ∣ Q = u ˜ and

( c , u , v ) ∈ ( 0,1 ) × ( 1 , u ˜ ) × ( 0 , u ˜ ) , ∀ ( x , t ) ∈ Ω ˜ t Q .

Thus, from (2.6), the four characteristic curves x ± ( t ; Q ) and x ˜ ± ( t ; Q ) for t ∈ [ 0 , t Q ] must intersect the line t = 0 and not intersect the lines l ± by the fact that

(2.43) 1 − 1 + ν 0 2 < λ − ( x − ( t ) , t ) < λ ˜ − ( x ˜ − ( t ) , t ) < λ ˜ + ( x ˜ + ( t ) , t ) < λ + ( x + ( t ) , t ) < 1 + 1 + ν 0 2 .

Denote the point ( x + ( 0 ; Q ) , 0 ) by Q + . Then, we see that the point Q + is on the interval ( − 1 , 1 ) , which means that u ∣ Q + < u ˜ . On the other hand, by (2.9) and (2.41), one obtains on the characteristic curve x = x + ( t ; Q )

∂ + u = 1 + ν 0 2 ∂ + c + κ v < 1 + ν 0 2 ⋅ − κ u ˜ 1 + ν 0 2 + κ u ˜ = 0 ,

which implies that the function u is strictly decreasing along x + ( t ; Q ) . Thus,

u ˜ = u ∣ Q < u ∣ Q + < u ˜ ,

a contradiction. Therefore, (2.42) is valid, which indicates that the four characteristic curves x ± ( t ; P ) and x ˜ ± ( t ; P ) for t ∈ [ 0 , t P ] must intersect the line t = 0 and not intersect the lines l ± . Along the characteristic curve x − ( t ; P ) , we clearly gain by (2.41) that

(2.44) ∂ − ∂ + c ∣ P > 0 .

On the other hand, one recalls (2.18) and applies (2.41) again to acquire

∂ − ∂ + c ∣ P = κ 2 ν 0 ( ∂ ˜ + F + ∂ ˜ − F ) ∣ P ≤ κ 2 ν 0 ⋅ 2 ( − κ u ˜ ) = − κ 2 u ˜ ν 0 < 0 ,

which contradicts to (2.44).

Now, if there exists a point P in Ω ˜ , which is the first time so that ∂ ˜ − F = − κ u ˜ and

(2.45) ∂ ± c ≤ − κ u ˜ 1 + ν 0 2 , ∂ ˜ + F ≤ − κ u ˜ , ∂ ˜ − F < − κ u ˜ , ∀ ( x , t ) ∈ Ω ˜ t P .

We can similarly show as before that (2.42) is still true. Thus, the four characteristic curves x ± ( t ; P ) and x ˜ ± ( t ; P ) for t ∈ [ 0 , t P ] must intersect the interval ( − 1 , 1 ) on t = 0 . Denote the point ( x ˜ + ( 0 ; P ) , 0 ) by P ˜ + , which is on the interval ( − 1 , 1 ) . For any point P ′ on the curve P P ˜ + ^ , one can draw the characteristic curves x ± ( t ; P ′ ) up to the interval ( − 1 , 1 ) at P ± ′ . Since ∂ ± c ≤ − κ u ˜ 1 + ν 0 2 in the region Ω ˜ t P , we use (2.18) to achieve

∂ ˜ + ∂ ˜ − F = ∂ ˜ − ∂ ˜ + F = − κ ν 0 2 ( ∂ + c + ∂ − c ) > 0 , ∀ ( x , t ) ∈ Ω ˜ t P ,

which means that ∂ ˜ − F and ∂ ˜ + F are strictly increasing and then by the initial data in (2.34)

(2.46) − ν 0 2 u ˜ ≤ ∂ ˜ + F ≤ − κ u ˜ , − ν 0 2 u ˜ ≤ ∂ ˜ − F ≤ − κ u ˜ , ∀ ( x , t ) ∈ Ω ˜ t P .

Integrating the equations of ∂ − c and ∂ + c in (2.18) along x + ( t ; P ′ ) and x − ( t ; P ′ ) from P + ′ and P − ′ to P ′ , respectively, we have by (2.34) and (2.46)

(2.47) ∂ − c ∣ P ′ = ∂ − c ∣ P + ′ + ∫ 0 t P ′ κ 2 ν 0 ( ∂ ˜ + F + ∂ ˜ − F ) ( x + ( t ; P ′ ) , t ) d t ≥ − 3 u ˜ + ∫ 0 t P ′ κ 2 ν 0 ⋅ 2 ⋅ ( − 2 κ u ˜ ) d t = − 3 u ˜ − 2 κ 2 u ˜ ν 0 t P ′ ,

and

(2.48) ∂ + c ∣ P ′ = ∂ + c ∣ P ′ + ∫ 0 t P ′ κ 2 ν 0 ( ∂ ˜ + F + ∂ ˜ − F ) ( x − ( t ; P ′ ) , t ) d t ≥ − 3 u ˜ + ∫ 0 t P ′ κ 2 ν 0 ⋅ 2 ⋅ ( − 2 κ u ˜ ) d t = − 3 u ˜ − 2 κ 2 u ˜ ν 0 t P ′ .

Therefore, we integrate the equation of ∂ ˜ − F in (2.18) along x ˜ + ( t ; P ) from P ˜ + to P and utilize (2.47) and (2.48) to attain by (2.37)

(2.49) − κ u ˜ = ∂ ˜ − F ∣ P = ∂ ˜ − F ∣ P ˜ + + ∫ 0 t P − κ ν 0 2 ( ∂ + c + ∂ − c ) ( x ˜ + ( t ; P ) , t ) d t ≤ − 2 κ u ˜ + ∫ 0 T ˜ κ ν 0 2 ⋅ 2 ⋅ 3 u ˜ + 2 κ 2 u ˜ ν 0 t d t < − 2 κ u ˜ + ∫ 0 T ˜ 2 κ 2 ⋅ 2 ⋅ ( 3 u ˜ + u ˜ ) d t ≤ − 2 κ u ˜ + κ u ˜ ⋅ 4 ν 0 T ˜ < − 2 κ u ˜ + κ u ˜ ⋅ 4 ν 0 T ^ = − κ u ˜ ,

a contradiction. Hence, the inequalities of ∂ ± c and ∂ ˜ ± F in (2.38) hold in the region Ω ˜ .

Based on the inequalities of ∂ ± c and ∂ ˜ ± F in (2.38), we can employ the contradiction argument again to show that

(2.50) ( c , u , v ) ∈ ( 0,1 ) × ( 1 , u ˜ ) × ( 0 , u ˜ ) , ∀ ( x , t ) ∈ Ω ˜ .

Indeed, if there exists a point P in Ω ˜ such that v ∣ P = 0 and

( c , u , v ) ∈ ( 0,1 ) × ( 1 , u ˜ ) × ( 0 , u ˜ ) , ∀ ( x , t ) ∈ Ω ˜ ∩ { t < t P } ,

then one obtains

(2.51) ∂ ˜ − v ∣ P ≤ 0 .

On the other hand, we apply the equation of v in (2.9) and the inequality of ∂ ˜ − F in (2.38) to conclude that

∂ ˜ − v ∣ P = ( − ∂ ˜ − F − κ u ) ∣ P > κ ( u ˜ − u ∣ P ) ≥ 0 ,

which leads to a contradiction with (2.51) and then (2.50) is true. Furthermore, the inequalities of ∂ ± u and ∂ ˜ ± v in (2.38) can be derived directly by (2.9)

(2.52) ∂ + u = 1 + ν 0 2 ∂ + c + κ v < 1 + ν 0 2 − κ u ˜ 1 + ν 0 2 + κ u ˜ = 0 , ∂ − u = − 1 + ν 0 2 ∂ − c + κ v > 1 + ν 0 2 ⋅ κ u ˜ 1 + ν 0 2 + κ ⋅ 0 = κ u ˜ > 0 , ∂ ˜ + v = ∂ ˜ + F − κ u < − κ u ˜ − κ ⋅ 1 = − κ ( u ˜ + 1 ) < 0 , ∂ ˜ − v = − ∂ ˜ − F − κ u > κ u ˜ − κ u ≥ κ ( u ˜ − u ˜ ) = 0 ,

which completes the proof of (2.38). According to (2.50) and (2.43), we easily see that the region Ω ˜ is a strong determinate domain. The proof of the lemma is ended.□

Lemma 4

Suppose that system (1.1), (1.2), and (1.4) with initial data ( ρ 0 , u 0 , v 0 , G 0 , F 0 ) ( x ) admits a smooth solution in Ω ˜ . Then, the solution in Ω ˜ satisfies

(2.53) ∣ c t ∣ , ∣ c x ∣ , ∣ u t ∣ , ∣ u x ∣ , ∣ v t ∣ , ∣ v x ∣ , ∣ F t ∣ , ∣ F x ∣ ≤ M c ,

for some uniform constant M > 0 .

Proof

Attributed to (2.18) and (2.38), we know that ∂ ˜ − F and ∂ ˜ + F are strictly increasing in the region Ω ˜ . Then,

(2.54) − ν 0 2 u ˜ ≤ ∂ ˜ + F ≤ − κ u ˜ , − ν 0 2 u ˜ ≤ ∂ ˜ − F ≤ − κ u ˜ , ∀ ( x , t ) ∈ Ω ˜ .

Making use of (2.18) and (2.54) gives

(2.55) − 1 2 κ u ˜ ≤ ∂ − ∂ + c = ∂ + ∂ − c ≤ − κ 2 ν 0 u ˜ ,

from which one has

(2.56) − 1 2 κ u ˜ T ˜ ≤ ∂ + c , ∂ − c ≤ − κ 2 ν 0 u ˜ T ˜ ,

where T ˜ is given in (2.36). We combine (2.9), (2.38), (2.54), and (2.56) to obtain

(2.57) 0 > ∂ + u = 1 + ν 0 2 ∂ + c + κ v ≥ 1 + ν 0 2 ∂ + c ≥ − 1 + ν 0 2 κ u ˜ T ˜ , 0 < ∂ − u = − 1 + ν 0 2 ∂ − c + κ v ≤ ( 1 + ν 0 2 T ˜ + 1 ) κ u ˜ , 0 > ∂ ˜ + v = ∂ ˜ + F − κ u ≥ − ν 0 2 u ˜ − κ u ˜ = − ν 0 2 + κ u ˜ , 0 < ∂ ˜ − v = − ∂ ˜ − F − κ u ≤ − ∂ ˜ − F ≤ ν 0 2 u ˜ .

Hence, there exists a uniformly constant M 1 > 0 such that

(2.58) ∣ ∂ ± c ∣ , ∣ ∂ ± u ∣ , ∣ ∂ ˜ ± v ∣ , ∣ ∂ ˜ ± F ∣ ≤ M 1

holds, which, together with (2.11)–(2.13), yields (2.53). The proof of the lemma is completed.□

Remark 5

Lemma 4 implies that the smooth solution is well-posed in Ω ˜ as long as c > 0 , or equivalently, ρ < ∞ .

2.3 The proof of Theorem 1

We now show that ρ tends to infinity at some point ( x * , t * ) in the region Ω ˜ . Set ∂ 0 = ∂ t + u ∂ x . Then, one employs (2.11) to acquire

(2.59) ∂ 0 = ( u + 1 + ν 0 2 c ) ∂ − − ( u − 1 + ν 0 2 c ) ∂ + 2 1 + ν 0 2 c + u ⋅ ∂ + − ∂ − 2 1 + ν 0 2 c = ∂ + + ∂ − 2 ,

which, together with (2.38) and (2.9), arrives at

(2.60) ∂ 0 c = ∂ + c + ∂ − c 2 < 0 ,

and

(2.61) ∂ 0 u = ∂ + u + ∂ − u 2 = 1 + ν 0 2 ( ∂ + c − ∂ − c ) 2 + κ v = 1 + ν 0 2 ( ∂ 0 c − ∂ − c ) + κ v = 1 + ν 0 2 ( ∂ + c − ∂ 0 c ) + κ v .

Combining (2.38) and (2.61) yields

(2.62) ∂ 0 u = 1 + ν 0 2 ( ∂ 0 c − ∂ − c ) + κ v ≥ 1 + ν 0 2 ∂ 0 c + κ v > 1 + ν 0 2 ∂ 0 c ,

and

(2.63) ∂ 0 u = 1 + ν 0 2 ( ∂ + c − ∂ 0 c ) + κ v ≤ 1 + ν 0 2 ( − ∂ 0 c ) + κ v < − 1 + ν 0 2 ∂ 0 c + κ u ˜ .

We next verify the following lemma, which results in Theorem 1 directly.

Lemma 5

There exists a point ( x * , t * ) ∈ Ω ˜ with t * < T * such that

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

Here, T * is given in (1.9).

Proof

Let L 1 : x = x 1 ( t ) and L 2 : x = x 2 ( t ) be two curves defined as follows:

(2.65) d x 1 ( t ) d t = u ( x 1 ( t ) , t ) , x 1 ( 0 ) = − 1 16 , d x 2 ( t ) d t = u ( x 2 ( t ) , t ) , x 2 ( 0 ) = 1 16 , ( x i ( t ) , t ) ∈ Ω ˜ .

In view of (2.38), one can obtain the ranges of curves L 1 and L 2

(2.66) t − 1 16 ≤ x 1 ( t ) ≤ u ˜ t − 1 16 , t + 1 16 ≤ x 2 ( t ) ≤ u ˜ t + 1 16 .

We use T i j ( i = 1 , 2 ; j = ± ) to represent the intersection times of curves L i and the lines l j . Combining (2.35) and (2.66) achieves the estimates

(2.67) T i j ≥ 15 16 u ˜ − 1 + 1 + ν 0 2 ≕ T ¯ ( < T ˜ ) , ( i = 1 , 2 ; j = ± ) ,

where T ˜ is given in (2.36). One recalls the value u ˜ = 24 ( 1 + ν 0 ) and the definition of T * in (1.9) to obtain

(2.68) T ¯ = 15 16 24 ( 1 + ν 0 ) − 1 + 1 + ν 0 2 > 1 48 ( 1 + ν 0 ) − 16 1 + ν 0 2 = T * ,

which implies that the region bounded by L 1 , L 2 , t = 0 , and t = T * is in the region Ω ˜ .

Now, we estimate the upper bound of x 2 ( t ) − x 1 ( t ) for t ≤ T * . Integrating (2.62) and (2.63) along x 1 ( t ) and x 2 ( t ) , respectively, yields

(2.69) u ( x 1 ( t ) , t ) > u ˆ − 1 16 − 1 + ν 0 2 , u ( x 2 ( t ) , t ) < u ˆ 1 16 + 1 + ν 0 2 + κ u ˜ T ˜ ,

from which, together with (1.6), (2.37), we obtain

(2.70) u ( x 1 ( t ) , t ) − u ( x 2 ( t ) , t ) > u ˆ − 1 16 − u ˆ 1 16 − 2 1 + ν 0 2 − κ u ˜ T ˜ > ∫ − 1 16 1 16 3 u ˜ d x − 2 1 + ν 0 2 − 1 16 u ˜ > 1 4 u ˜ − 2 1 + ν 0 2 .

Making use of (2.65) and (2.70), we acquire for t ∈ ( 0 , T * )

(2.71) x 2 ( t ) − x 1 ( t ) = 1 16 + ∫ 0 t u ( x 2 ( t ) , t ) d t − − 1 16 + ∫ 0 t u ( x 1 ( t ) , t ) d t = 1 8 + ∫ 0 t [ u ( x 2 ( t ) , t ) − u ( x 1 ( t ) , t ) ] d t < 1 8 − 1 4 u ˜ − 2 1 + ν 0 2 t = 1 8 − 1 8 ( 2 u ˜ − 16 1 + ν 0 2 ) t ,

with

1 8 − 1 8 ( 2 u ˜ − 16 1 + ν 0 2 ) t > 1 8 − 1 8 ( 2 u ˜ − 16 1 + ν 0 2 ) T * = 0 .

Finally, we verify that ρ goes to infinity at some time t * < T * . It concludes by the mass-conservation equation that

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

from which one obtains

(2.72) ∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x = ∫ x 1 ( 0 ) x 2 ( 0 ) ρ ( x , 0 ) d x = x 2 ( 0 ) − x 1 ( 0 ) = 1 8 ,

which, together with (2.71), arrives at

(2.73) max x 1 ( t ) ≤ x ≤ x 2 ( t ) ρ ( x , t ) ≥ ∫ x 1 ( t ) x 2 ( t ) ρ ( x , t ) d x x 2 ( t ) − x 1 ( t ) = 1 8 x 2 ( t ) − x 1 ( t ) > 1 1 − ( 2 u ˜ − 16 1 + ν 0 2 ) t = T * T * − t .

From (2.73), we know that there exists a point ( x * , t * ) with t * < T * and x * ∈ [ x 1 ( t ) , x 2 ( t ) ] ⊂ ( − 1 , 1 ) such that (2.64) holds, which finishes the proof of the lemma.□

With the help of Lemma 5, we complete the proof of Theorem 1.

2.4 Numerical simulations

In this subsection, we provide some typical numerical results to verify the theoretical analysis of the article.

We consider the Lax-Friedrichs scheme (see, e.g., [41,50]) and discretize system (2.3) into the following explicit form:

ρ i j + 1 = ρ i + 1 j + ρ i − 1 j 2 − Δ t 2 Δ x ⋅ { ( ρ u ) i + 1 j − ( ρ u ) i − 1 j } , ( ρ u ) i j + 1 = ( ρ u ) i + 1 j + ( ρ u ) i − 1 j 2 + κ ρ i j v i j ⋅ Δ t − Δ t 2 Δ x ⋅ ( ρ u 2 ) i + 1 j − 1 + ν 0 2 ρ i + 1 j − ( ρ u 2 ) i − 1 j + 1 + ν 0 2 ρ i − 1 j , ( ρ v ) i j + 1 = ( ρ v ) i + 1 j + ( ρ v ) i − 1 j 2 − κ ρ i j u i j ⋅ Δ t − Δ t 2 Δ x ⋅ { ( ρ u v ) i + 1 j − ν 0 F i + 1 j − ( ρ u v ) i − 1 j + ν 0 F i − 1 j } , ( ρ F ) i j + 1 = ( ρ F ) i + 1 j + ( ρ F ) i − 1 j 2 − Δ t 2 Δ x ⋅ { ( ρ F u ) i + 1 j − ν 0 v i + 1 j − ( ρ F u ) i − 1 j + ν 0 v i − 1 j } ,

where the notation ( f ) i j represents the value of the function f at the grid point ( x i , t j ) and the space step Δ x and time step Δ t satisfy the well-known CFL condition

∣ ( u ± 1 + ν 0 2 c ) i j ∣ ⋅ Δ t Δ x ≤ 1 .

We set the initial data on [ − 1 , 1] as follows:

ρ 0 ( x ) = 1 , F 0 ( x ) = 1 , v 0 ( x ) = 12 ( 1 + ν 0 ) ( 1 − x ) ,

and

u 0 ( x ) = ( 1 + ν 0 ) ( 18 − 6 x ) , x ∈ − 1 , − 1 8 , − ( 5632 + 1536 ν 0 ) x 3 − ( 2112 + 960 ν 0 ) x 2 − ( 270 + 174 ν 0 ) x + 7 + 9 ν 0 , x ∈ − 1 8 , − 1 16 , ( 1 + ν 0 ) ( 12 − 72 x ) + 1 2 , x ∈ − 1 16 , 1 16 , − ( 5632 + 1536 ν 0 ) x 3 + ( 2112 + 960 ν 0 ) x 2 − ( 270 + 174 ν 0 ) x + 8 + 15 ν 0 , x ∈ 1 16 , 1 8 , 6 ( 1 + ν 0 ) ( 1 − x ) + 1 , x ∈ 1 8 , 1 .

We conduct two tests: Test 1, with κ = 0.1 , ν 0 = 0.9 , and Test 2, with κ = 0.1 , ν 0 = 1.8 , and the results are presented in Figures 2 and 3, respectively.

$Figure 2 The numerical results for Test 1 ( κ = 0.1 \kappa =0.1 , ν 0 = 0.9 {\nu }_{0}=0.9 ). The left and right figures are at t = 0.007 t=0.007 and t = 0.028 t=0.028 , respectively.$

Figure 2

The numerical results for Test 1 ( κ = 0.1 , ν 0 = 0.9 ). The left and right figures are at t = 0.007 and t = 0.028 , respectively.

$Figure 3 The numerical results for Test 2 ( κ = 0.1 \kappa =0.1 , ν 0 = 1.8 {\nu }_{0}=1.8 ). The left and right figures are at t = 0.007 t=0.007 and t = 0.028 t=0.028 , respectively.$

Figure 3

The numerical results for Test 2 ( κ = 0.1 , ν 0 = 1.8 ). The left and right figures are at t = 0.007 and t = 0.028 , respectively.

From Figures 2 and 3, we can see that the rough process of blow-up of the density ρ . Moreover, by comparing Figures 2 and 3, one can also see that, at the same times, the larger the value of ν 0 , the larger the value of ρ . This means that, for the chosen initial data in Theorem 1, the presence of the magnetic field plays a role in accelerating the formation of singularity.

Acknowledgments

The authors would like to thank the editor and the reviewers for their valuable comments and suggestions to improve the quality of the article.

Funding information: This work was partially supported by the Natural Science Foundation of Zhejiang Province of China (LMS25A010014) and the National Natural Science Foundation of China (12171130).
Author contributions: Y.H.: writing, editing, and supervision. Y.Z.: writing, editing, and reviewing.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: The authors declare that the data supporting the findings of this study are available within the article.

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Received: 2025-01-14

Revised: 2025-08-25

Accepted: 2025-10-13

Published Online: 2025-11-15

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

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0126

Keywords for this article

magnetohydrodynamics; Chaplygin gas; singularity; characteristic decomposition

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