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The limit Riemann solutions to nonisentropic Chaplygin Euler equations

  • Maozhou Lin and Lihui Guo EMAIL logo
Published/Copyright: December 31, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

We mainly consider the limit behaviors of the Riemann solutions to Chaplygin Euler equations for nonisentropic fluids. The formation of delta shock wave and the appearance of vacuum state are found as parameter ε tends to a certain value. Different from the isentropic fluids, the weight of delta shock wave is determined by variance density ρ and internal energy H. Meanwhile, involving the entropy inequality, the uniqueness of delta shock wave is obtained.

Keywords: nonisentropic fluids; Riemann solutions; delta shock; vacuum state; entropy inequality
MSC 2010: 35L65; 35L67; 76N15

1 Introduction

One-dimensional compressible Euler equations for nonisentropic fluids can be written as

(1.1) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + P ( ρ , s ) ) x = 0 , ( ρ u 2 / 2 + ρ e ) t + ( ( ρ u 2 / 2 + ρ e + P ( ρ , s ) ) u ) x = 0 ,

where the variables ρ , u, s, P, e stand for the density, velocity, specific entropy, pressure and specific energy, respectively, and P ( ε , ρ ) = ε p satisfies lim ε → 0 P ( ρ , ε ) = 0 . P and e are the functions of ρ and s, and fulfill the thermodynamical constraint

(1.2) d e = T d s − P d 1 ρ ,

where T = T ( ρ , s ) represents the temperature. The equation of state with Chaplygin gas can be expressed as

(1.3) p = − 1 ρ ,

which was introduced by Chaplygin [1] in 1904. In some theories of cosmology, Chaplygin gas explains the acceleration and the dark energy of the universe, and the formation of delta shock wave may be used to illustrate the different periods of evolution of the universe. As for the related results, one can see [2,3,4,5,6].

In 2005, Brenier [7] considered the Riemann problem of the isentropic Chaplygin gas Euler equations

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

and obtained the concentration solutions when the initial value belongs to a certain region in the phase plane. In 2010, Guo et al. [8] put away this restriction to system (1.4) and received the global solutions including the delta shock. Wang and Zhang [9] investigated the Riemann problem with delta initial data and obtained four kinds of the global generalized solutions. In 2014, Nedeljkov [10] studied higher order shadow waves and delta shock blow up in the Chaplygin gas and found that a double shadow wave interacted with an outgoing wave and formed a singled weighted shadow wave, which is in general called delta shock wave. Meanwhile, Nedeljkov proved that this delta shock has a variable strength and variable speed. For more detailed knowledge of delta shock, interested readers can refer to [11,12,13,14,15,16,17].

As the pressure vanishes, equations (1.4) converge to the transport equations

(1.5) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 ) x = 0 ,

which are also called the pressureless Euler equations and can be used to describe the motion of free particles sticking under collision in [18,19,20]. Equations (1.5) have been extensively studied since 1994 such as in [21,22,23]. In 2016, Shen [24] considered the Riemann problem for the Chaplygin gas equations with a source term. Furthermore, Guo et al. [25] studied the vanishing pressure limits of Riemann solutions and analyzed the phenomena of concentration and cavitation to the Chaplygin gas equations with a source term. As for the pressure vanishing limits of the isentropic Euler equations, let us refer to [26,27,28,29,30,31] for more details.

Kraiko [32] studied system (1.1) with P ( ρ , s ) = 0 in 1979. In order to construct a solution for any initial data, they needed the discontinuities which are different from classical waves that carry mass, impulse and energy. In 2012, Cheng [33] solved the Riemann problem for (1.1) with P ( ρ , s ) = 0 and found two kinds of solutions containing vacuum state and delta shock with Dirac delta function in both the density and the internal energy. We replace internal energy ρ e by H, therefore, system (1.1) can be transformed into the following equations:

(1.6) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + P ) x = 0 , ( ρ u 2 / 2 + H ) t + ( ( ρ u 2 / 2 + H + P ) u ) x = 0 ,

where H denoted the internal energy and H ≥ 0 . Pang [34] considered the system of (1.6) for Chaplygin gas equations with the following initial data

(1.7) ( ρ , u , H ) ( x , 0 ) = ( ρ − , u − , H − ) , x < 0 , ( ρ + , u + , H + ) , x > 0 ,

where ρ ± > 0 , u ± > 0 and H ± > 0 are different constants. For more detailed information on the nonisentropic Euler equations, interested readers can refer to [35,36,37,38].

In this article, we mainly focus our attention to the vanishing pressure limits of Riemann solutions for system (1.6)–(1.7), when the pressure vanishes, equation (1.6) can be translated into (1.5), and an additional conservation law

(1.8) ( ρ u 2 / 2 + H ) t + ( ρ u 3 / 2 + H u ) x = 0 .

As pressure vanishes, we identify and analyze the formation of delta shock waves and vacuum states in the Riemann solutions. Furthermore, in the sense of distributions, entropy inequality corresponding to equation (1.8) will be verified

(1.9) ( ρ u 2 ) t + ( ρ u 3 ) x ≥ 0 .

The remainder of this article can be organized as follows: in Sections 2 and 3, we review the Riemann solutions to (1.5) and (1.6), respectively. In Section 4, we consider the vanishing pressure limits of Riemann solutions to (1.6) and (1.7). In Section 5, we give some discussions.

2 Riemann problem for (1.5)

In this section, we review some results on Riemann solution to system (1.5) with initial data

(2.1) ( ρ , u ) ( x , 0 ) = ( ρ − , u − ) , x < 0 , ( ρ + , u + ) , x > 0 ,

where ρ ± > 0 , the details can be referred to in [23].

For the case u − < u + , we know that the Riemann solutions of (1.5) contain two-contact discontinuities J 1 , J 2 and a vacuum state between two-contact discontinuities, and J 1 , J 2 satisfy

(2.2) J 1 : u = u − , J 2 : u = u + .

For the case u − = u + , the Riemann solution include a contact discontinuity J that connects ( ρ − , u − ) to ( ρ + , u + ) , and J satisfies

(2.3) J : u = u − = u + .

While for the case u − > u + , the superposition of S and J leads to the singularity for ρ on the line x = x ( t ) t as a weighted Dirac delta function, which was named as the so-called delta shock wave. Thus, the delta shock wave solution to the Riemann problem (1.6) and (1.7) should be constructed when u − > u + . Then, let us recollect the definition of delta shock wave in [13,22].

Definition 2.1

For arbitrary ψ ( x , t ) ∈ C 0 ∞ ( R + R + ) , the two-dimensional weighted Dirac delta function β ( s ) δ Γ with the support on a parameterized smooth curve Γ = { ( x ( s ) , t ( s ) ) : a < s < b } is defined by

〈 β ( s ) δ Γ , ψ ( x , t ) 〉 = ∫ β ( s ) ψ ( x ( s ) , t ( s ) ) d s .

By virtue of the above definition, the Riemann solution of (1.6) and (1.7) contains a delta shock wave. It can be briefly expressed by

(2.4) ( ρ − , u − ) + δ S + ( ρ + , u + ) ,

namely,

(2.5) ( ρ , u ) ( x , t ) = ( ρ − , u − ) , x < x ( t ) , ( ω ( t ) δ ( x − x ( t ) ) , u δ ) , x = x ( t ) , ( ρ + , u + ) , x ( t ) < x ,

where w ( t ) and σ ( t ) denote the weight and velocity of delta shock wave, respectively.

While for the case u − > u + , singularity must happen. We use a delta shock wave to construct the Riemann solution. The details can be found in [22]. The location, weight and velocity of the delta shock are given by computing generalized Rankine-Hugoniot relations, which are

(2.6) x ( t ) = ρ + u + + ρ − u − ρ + + ρ − t , ω ( t ) = t 1 + σ 2 ( σ [ ρ ] − [ ρ u ] ) , σ = ρ + u + + ρ − u − ρ + + ρ − .

In addition, the delta shock wave satisfies the generalized entropy condition

(2.7) u + < σ < u − ,

which means that all characteristics on both sides of the δ-shock wave curve are incoming. Furthermore, the uniqueness of delta shock wave can be obtained.

3 Riemann problem for (1.6)–(1.7) for the Chaplygin gas

From thermodynamical constraint (1.2), we derive

T d s = d e − ε 2 ρ 2 ,

thus, there is a function f ( s ) satisfying

T = f ′ ( s ) , e = ε 2 ρ 2 + f ( s ) .

Due to the value of e is positive, which means that the function g ( x ) = 1 2 x 2 + f ( s ) ≥ 0 when x ∈ ( 0 , + ∞ ) , so f ( s ) > 0 , namely, e − ε 2 ρ 2 ≥ 0 . Then, the physically relevant region can be expressed as

ℵ = ( ρ , u , H ) | ρ > 0 , H ≥ ε 2 ρ , u ∈ R .

In this section, we review results on the Riemann problem of (1.6) for the Chaplygin gas, see [34] for the details. Equations (1.6) have three eigenvalues

(3.1) λ 1 = u − ε ρ , λ 2 = u , λ 3 = u + ε ρ ,

with corresponding right eigenvectors

r → 1 = − ρ 2 ε , 1 , ε − 1 ε H ρ T , r → 2 = ( 0 , 0 , 1 ) T , r → 3 = ρ 2 ε , 1 , − ε + 1 ε H ρ T .

Direct calculation yields ∇ λ i ⋅ r → i = 0 , for i = 1, 2, 3, which indicates that all the characteristic fields are contact discontinuous.

For any given constant state ( ρ − , u − , H − ) in the phase plane, we can derive three families of contact discontinuities

(3.2) J 1 ε ( ρ − , u − , H − ) : σ 1 ε = u − ε ρ = u − − ε ρ − , ( 2 H ρ − ε ) ρ − 2 = ( 2 H − ρ − − ε ) ρ 2 ,

(3.3) J 2 ε ( ρ − , u − , H − ) : σ 2 ε = u = u − ,    ρ = ρ − , H ≠ H − ,

(3.4) J 3 ε ( ρ − , u − , H − ) : σ 3 ε = u + ε ρ = u − + ε ρ − , ( 2 H ρ − ε ) ρ − 2 = ( 2 H − ρ − − ε ) ρ 2 .

On the physical correlation region, that is ( ρ − , u − , H − ) ∈ ℵ , from given state ( ρ − , u − , H − ) , we can draw the one-contact discontinuity curve J 1 ε that satisfies (3.2) and the three-contact discontinuity curve J 3 ε that satisfies (3.4). And from the point ρ − , u − − 2 ε ρ − , H − draw three-contact discontinuity curve S δ ε that satisfies (3.4). In fact, this curve S δ ε consists of some states that can be connected to the states ( ρ − , u − , H − ) on the right by a δ S .

We project these curves onto the ( ρ , u ) -plane. J 1 ε has two asymptotes u = u − − ε ρ − and ρ = 0 , J 3 ε has two asymptotes u = u − + ε ρ − and ρ = 0 , and S δ ε satisfies

(3.5) u + ε ρ = u − − ε ρ − ,

which has two asymptotic lines u = u − − ε ρ − and ρ = 0 . Thus, the phase plane can be divided into five regions.

When the projection of ( ρ + , u + , H + ) belongs to I ( ρ − , u − ) ∪ I I ( ρ − , u − ) ∪ I I I ( ρ − , u − ) ∪ I V ( ρ − , u − ) in the ( ρ , u ) -plane, the Riemann solution can be briefly expressed by

(3.6) ( ρ − , u − , H − ) + J 1 ε + ( ρ ⁎ 1 ε , u ⁎ 1 ε , H ⁎ 1 ε ) + J 2 ε + ( ρ ⁎ 2 ε , u ⁎ 2 ε , H ⁎ 2 ε ) + J 3 ε + ( ρ + , u + , H + ) ,

where ( ρ ⁎ 1 ε , u ⁎ 1 ε , H ⁎ 1 ε ) and ( ρ ⁎ 2 ε , u ⁎ 2 ε , H ⁎ 2 ε ) are the intermediate states.

For the projection of ( ρ + , u + , H + ) belongs to V ( ρ − , u − ) in the ( ρ , u ) -plane, the Riemann solution can be given by

(3.7) ( ρ − , u − , H − ) + δ S + ( ρ + , u + , H + ) .

The details can be referred to in [34]. The delta shock wave holds the generalized Rankine-Hugoniot conditions

(3.8) d x ( t , ε ) d t = u δ ( t , ε ) , d ω ( t , ε ) d t = u δ ( t , ε ) [ ρ ] − [ ρ u ] , d ( ω ( t , ε ) u δ ( t ) ) d t = u δ ( t , ε ) [ ρ u ] − [ ρ u 2 + P ] , d ( ω ( t ) u δ 2 ( t ) / 2 + h ( t ) ) d t = u δ ( t , ε ) ρ u 2 2 + H − ρ u 2 2 + H + P u , ( x , ω , u δ , h ) ( 0 ) = ( 0 , 0 , 0 , 0 ) ,

where ω ( t , ε ) and u δ ( t , ε ) are weight and velocity of delta shock wave, respectively.

It can be derived from (3.8) that

(3.9) x ( t , ε ) = 1 [ ρ ] ( [ ρ u ] t + ω ( t , ε ) ) , u δ ( t , ε ) = 1 [ ρ ] ( [ ρ u ] + ω ′ ( t , ε ) ) , ω ( t , ε ) = ρ + ρ − u − + ε ρ − − u + + ε ρ + u − − ε ρ − − u + − ε ρ + t , h ( t , ε ) = − ω ( t , ε ) u δ ( t , ε ) 2 / 2 + x ( t , ε ) [ ρ u 2 / 2 + H ] − [ ( ρ u 2 / 2 + H + P ) u ] t ,

for [ ρ ] = ρ + − ρ − ≠ 0 , and

(3.10) x ( t , ε ) = ( u − + u + ) t 2 , u δ ( t , ε ) = u − + u + 2 , ω ( t , ε ) = ρ − ( u − − u + ) t , h ( t , ε ) = − ω ( t , ε ) u δ 2 ( t ) / 2 + x ( t , ε ) [ ρ u 2 / 2 + H ] − [ ( ρ u 2 / 2 + H + P ) u ] t ,

for [ ρ ] = ρ + − ρ − = 0 .

In addition, it is easy to see that the delta shock wave satisfies the generalized entropy condition

(3.11) u + + ε ρ + ≤ u δ ( t , ε ) ≤ u − − ε ρ − ,

which ensures the uniqueness of Riemann solutions.

4 Limits of Riemann solutions to (1.6)–(1.7)

In this section, we concentrate on the limit behavior of Riemann solutions to equations (1.6)–(1.7), and the formation of delta shock and the vacuum phenomenon are considered in the case u − > u + and the case u − < u + .

4.1 Limits of Riemann solutions in the case u − > u +

Lemma 4.1

Assume u − > u + , and then there exist two constant values ε 1 , ε 2 , ε 1 > ε 2 > 0 , such that the projection of ( ρ + , u + , H + ) belongs to I V ( ρ − , u − ) when ε 2 < ε < ε 1 , and belongs to V ( ρ − , u − ) when 0 < ε < ε 2 .

Proof

Suppose u − > u + , the states ( ρ + , u + , H + ) connect with ( ρ − , u − , H − ) by contact discontinuities that satisfy

(4.1) u + = u − − ε ρ − + ε ρ + , ρ + > ρ − ,

(4.2) u + = u − + ε ρ − − ε ρ + , ρ + < ρ − .

If ρ − ≠ ρ + , the projection pertains to IV ( ρ − , u − ) or V ( ρ − , u − ) , we have

(4.3) ε 1 = ( u − − u + ) ρ + ρ − ρ − − ρ + 2 ,

that is, the projection belongs to IV ( ρ − , u − ) or V ( ρ − , u − ) when 0 < ε < ε 1 .

If the projection pertains to V ( ρ − , u − ) , we have

(4.4) ε 2 = ( u − − u + ) ρ + ρ − ρ − + ρ + 2 ,

that is, projection is located in IV ( ρ − , u − ) when ε 2 < ε < ε 1 , and projection belongs to V ( ρ − , u − ) when 0 < ε < ε 2 .

If ρ − = ρ + , the conclusion is clearly valid.□

From Lemma 4.1, we know there is no delta shock wave when ε > ε 2 . We find that the curves of two-contact discontinuities become steeper when ε decreases, that is, when ε decreases, the projection of ( ρ + , u + , H + ) must belong to IV ( ρ − , u − ) or V ( ρ − , u − ) .

First, we consider the situation ε 2 < ε < ε 1 , namely, the projection of ( ρ + , u + , H + ) pertains to IV ( ρ − , u − ) . In this situation, the Riemann solution to (1.6)–(1.7) is

(4.5) ( ρ − , u − , H − ) + J 1 ε + ( ρ ⁎ 1 ε , u ⁎ 1 ε , H ⁎ 1 ε ) + J 2 ε + ( ρ ⁎ 2 ε , u ⁎ 2 ε , H ⁎ 2 ε ) + J 3 ε + ( ρ + , u + , H + ) ,

the intermediate states satisfy the following formulae

(4.6) J 1 ε : u ⁎ 1 ε − u − = ε 1 ρ ⁎ 1 ε − 1 ρ − , ρ ⁎ 1 ε > ρ − , 2 H ⁎ 1 ε ρ ⁎ 1 ε − ε ρ − 2 = ( 2 H − ρ − − ε ) ρ ⁎ 1 ε 2 ,

(4.7) J 2 ε : ρ ⁎ 1 ε = ρ ⁎ 2 ε , u ⁎ 1 ε = u ⁎ 2 ε , H ⁎ 1 ε ≠ H ⁎ 2 ε ,

(4.8) J 3 ε : u + − u ⁎ 2 ε = ε 1 ρ ⁎ 2 ε − 1 ρ + , ρ ⁎ 2 ε > ρ + , ( 2 H + ρ + − ε ) ρ ⁎ 2 ε 2 = 2 H ⁎ 2 ε ρ ⁎ 2 ε − ε ρ + 2 ,

where ρ ⁎ 1 ε , u ⁎ 1 ε , H ⁎ 1 ε and ρ ⁎ 2 ε , u ⁎ 2 ε , H ⁎ 2 ε are the intermediate states.

Lemma 4.2

The intermediate density ρ ⁎ ε becomes unbounded as ε → ε 2 , that is,

(4.9) lim ε → ε 2 ρ ⁎ ε = ∞ ,

where ρ ⁎ ε ≔ ρ ⁎ 1 ε = ρ ⁎ 2 ε , the intermediate internal energy H ⁎ 1 ε , H ⁎ 2 ε become unbounded, i.e.,

(4.10) lim ε → ε 2 H ⁎ 1 ε = ∞ , lim ε → ε 2 H ⁎ 2 ε = ∞ .

Proof

From (4.6) to (4.8), it is easy to calculate that

(4.11) ρ ⁎ ε ≔ ρ ⁎ 1 ε = ρ ⁎ 2 ε , u ⁎ ε ≔ u ⁎ 1 ε = u ⁎ 2 ε ,

(4.12) u ⁎ ε = 1 2 u + + ε ρ + + 1 2 u − − ε ρ − ,

(4.13) ε ρ ⁎ ε = 1 2 u + + ε ρ + − 1 2 u − − ε ρ − .

Therefore,

(4.14) lim ε → ε 2 ε ρ ⁎ ε = lim ε → ε 2 1 2 u + + ε ρ + − 1 2 u − − ε ρ − = 0 ,

which implies that lim ε → ε 2 ρ ⁎ ε = ∞ .

Using (4.6)2, we derive

(4.15) 2 H ⁎ 1 ε ρ ⁎ ε − ε ( ρ ⁎ ε ) 2 = 2 H − ρ − − ε ρ − 2 ,

thus

(4.16) lim ε → ε 2 2 H ⁎ 1 ε ρ ⁎ ε = 2 H − ρ − − ε 2 ρ − 2 ,

namely,

(4.17) lim ε → ε 2 H ⁎ 1 ε = ∞ .

By using (4.8)2, we obtain the same conclusion

(4.18) lim ε → ε 2 H ⁎ 2 ε = ∞ .

□

Lemma 4.3

Let

(4.19) u δ = ρ + u + + ρ − u − ρ + + ρ − ,

then

(4.20) lim ε → ε 2 u ⁎ ε = lim ε → ε 2 σ 1 ε = lim ε → ε 2 σ 2 ε = lim ε → ε 2 σ 3 ε = u δ .

Proof

Involving the first term of (3.2)–(3.4), the following equations are obtained

(4.21) lim ε → ε 2 σ 1 ε = lim ε → ε 2 u − − ε ρ − = u − − ρ + ( u − − u + ) ρ + + ρ − = ρ + u + + ρ − u − ρ + + ρ − = u δ ,

(4.22) lim ε → ε 2 σ 3 ε = lim ε → ε 2 u + + ε ρ + = u + + ρ − ( u − − u + ) ρ + + ρ − = ρ + u + + ρ − u − ρ + + ρ − = u δ ,

(4.23) lim ε → ε 2 σ 2 ε = lim ε → ε 2 u ⁎ ε = u δ .

□

Lemma 4.4

(4.24) lim ε → ε 2 ρ ⁎ ε ( σ 2 ε − σ 1 ε ) = ( u − − u + ) ρ + ρ − ρ + + ρ − ,

(4.25) lim ε → ε 2 ρ ⁎ ε ( σ 3 ε − σ 2 ε ) = ( u − − u + ) ρ + ρ − ρ + + ρ − .

Proof

The expressions of σ i (i = 1, 2, 3) are employed again, and the following discussions will be presented

(4.26) lim ε → ε 2 ρ ⁎ ε ( σ 2 ε − σ 1 ε ) = lim ε → ε 2 ρ ⁎ ε u ⁎ ε − u ⁎ ε + ε ρ ⁎ ε = lim ε → ε 2 ε = ( u − − u + ) ρ + ρ − ρ + + ρ − ,

(4.27) lim ε → ε 2 ρ ⁎ ε ( σ 3 ε − σ 2 ε ) = lim ε → ε 2 ρ ⁎ ε u ⁎ ε + ε ρ ⁎ ε − u ⁎ ε = lim ε → ε 2 ε = ( u − − u + ) ρ + ρ − ρ + + ρ − .

□

Lemma 4.5

(4.28) lim ε → ε 2 ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ( σ 3 ε − σ 1 ε ) = u δ ρ u 2 2 + H − ρ u 2 2 + H − ε 2 ρ u ,

where

H ⁎ ε = H ⁎ 1 ε , σ 1 ε < ξ < σ 2 ε , H ⁎ 2 ε , σ 2 ε < ξ < σ 3 ε .

Proof

Using the Rankine-Hugoniot conditions of (1.6), we obtain the following forms:

(4.29) σ 1 ε ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 1 ε − ρ − u − 2 2 − H − = ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 1 ε − ε ρ ⁎ ε u ⁎ ε − ρ − u − 2 2 + H − − ε ρ − u − , σ 2 ε ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 2 ε − ρ ⁎ ε ( u ⁎ ε ) 2 2 − H ⁎ 1 ε = ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 2 ε − ε ρ ⁎ ε u ⁎ ε − ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 1 ε − ε ρ ⁎ ε u ⁎ ε , σ 3 ε ρ + u + 2 2 + H + − ρ ⁎ ε ( u ⁎ ε ) 2 2 − H ⁎ 2 ε = − ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 2 ε − ε ρ ⁎ ε u ⁎ ε + ρ + u + 2 2 + H + − ε ρ + u + .

When ε → ε 2 , utilizing (4.29) and taking limits, we have

(4.30) lim ε → ε 2 ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 1 ε ( σ 2 ε − σ 1 ε ) + ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ 2 ε ( σ 3 ε − σ 2 ε ) = u δ ρ u 2 2 + H − ρ u 2 2 + H − ε 2 ρ u ,

which implies that

lim ε → ε 2 ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ( σ 3 ε − σ 1 ε ) = u δ ρ u 2 2 + H − ρ u 2 2 + H − ε 2 ρ u .

The proof is complete.□

Theorem 4.6

When u − > u + , the Riemann solution tends to a delta shock wave as ε → ε 2 . The limit functions ρ , ρ u and H are the sums of a step function and a δ -measure with weights

t 1 + u δ 2 u δ [ ρ ] − [ ρ u ] , t 1 + u δ 2 u δ [ ρ u ] − ρ u 2 − ε 2 ρ , t 1 + u δ 2 u δ ρ u 2 2 + H − ρ u 2 2 + H − ε 2 ρ u ,

where u δ = ρ + u + + ρ − u − ρ + + ρ − .

Proof

  1. For ξ = x t , the Riemann solutions are denoted by

( ρ ε ( ξ ) , u ε ( ξ ) ) = ( ρ − , u − ) , ξ < σ 1 ε , ( ρ ⁎ ε , u ⁎ ε ) , σ 1 ε < ξ < σ 3 ε , ( ρ + , u + ) , ξ > σ 3 ε , and H ε ( ξ ) = H − , ξ < σ 1 ε , H ⁎ 1 ε , σ 1 ε < ξ < σ 2 ε , H ⁎ 2 ε , σ 2 ε < ξ < σ 3 ε , H + , ξ > σ 3 ε ,

which satisfies the following weak formulae:

(4.31) 〈 − ξ ρ ξ ε + ( ρ ε u ε ) ξ , ψ 〉 = ∫ − ∞ + ∞ ( − ξ ρ ξ ε + ( ρ ε u ε ) ξ ) ψ d ξ = 0 ,

(4.32) − ξ ( ρ ε u ε ) ξ + ρ ε ( u ε ) 2 − ε ρ ε ξ , ψ = ∫ − ∞ + ∞ − ξ ( ρ ε u ε ) ξ + ρ ε ( u ε ) 2 − ε ρ ε ξ ψ d ξ = 0 ,

(4.33) − ξ ρ ε ( u ε ) 2 2 + H ε ξ + ρ ε ( u ε ) 2 2 + H ε − ε ρ ε u ε ξ , ψ = ∫ − ∞ + ∞ − ξ ( ρ ε ( u ε ) 2 2 + H ε ) ξ + ρ ε ( u ε ) 3 2 + H ε u ε − ε u ε ρ ε ξ ψ d ξ = 0,

for arbitrary ψ ∈ C 0 ∞ ( ℝ ) .

  1. From (4.31), one can obtain

    (4.34) ∫ − ∞ + ∞ ( − ξ ρ ξ ε + ( ρ ε u ε ) ξ ) ψ d ξ = I 1 + I 2 ,

    where

    I 1 = ∫ − ∞ + ∞ ρ ε ψ d ξ , I 2 = ∫ − ∞ + ∞ ρ ε ( ξ − u ε ) ψ ξ d ξ .

    We decompose I 2 as

    I 2 = ∫ − ∞ σ 1 ε + ∫ σ 1 ε σ 3 ε + ∫ σ 3 ε + ∞ ρ ε ( ξ − u ε ) ψ ξ d ξ .

    The total of the first and last terms is

    ∫ − ∞ σ 1 ε ρ − ( ξ − u − ) ψ ξ d ξ + ∫ σ 3 ε + ∞ ρ + ( ξ − u + ) ψ ξ d ξ = ρ + u + ψ ( σ 3 ε ) − ρ − u − ψ ( σ 1 ε ) + ρ − σ 1 ε ψ ( σ 1 ε ) − ρ + σ 3 ε ψ ( σ 3 ε ) − ∫ − ∞ σ 1 ε ρ − ψ d ξ − ∫ σ 3 ε + ∞ ρ + ψ d ξ ,

    when ε → ε 2 , it converges to the following equality:

    (4.35) [ ρ u ] ψ − u δ [ ρ ] ψ − ∫ − ∞ u δ ρ − ψ d ξ − ∫ u δ + ∞ ρ + ψ d ξ = ( [ ρ u ] − u δ [ ρ ] ) ψ ( u δ ) − ∫ − ∞ + ∞ ρ 0 ( ξ − u δ ) ψ ( ξ ) d ξ ,

    where ρ 0 ( ξ − u δ ) = ρ − + [ ρ ] ℱ ( ξ ) and ℱ ( ξ ) is a Heaviside function.

    In addition,

    (4.36) ∫ σ 1 ε σ 3 ε ρ ⁎ ε ( ξ − u ⁎ ε ) ψ ξ d ξ = ρ ⁎ ε ( σ 3 ε − σ 1 ε ) σ 3 ε ψ ( σ 3 ε ) − σ 1 ε ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − u ⁎ ε ψ ( σ 3 ε ) − ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − 1 σ 3 ε − σ 1 ε ∫ σ 1 ε σ 3 ε ψ d ξ ,

    when ε → ε 2 , it leads to

    (4.37) lim ε → ε 2 ∫ σ 1 ε σ 3 ε ρ ⁎ ε ( ξ − u ⁎ ε ) ψ ξ d ξ = ρ ⁎ ε ( σ 3 ε − σ 1 ε ) ( ( u δ ψ ) ′ − u δ ψ ′ − ψ ) = 0 .

    Above all, from (4.34), it yields

    (4.38) lim ε → ε 2 ∫ − ∞ ∞ ρ ε ψ − ρ 0 ( ξ − u δ ) ψ ( ξ ) d ξ = ( u δ [ ρ ] − [ ρ u ] ) ψ ( u δ ) .

  2. We deduce the limit of the momentum m ε = ρ ε u ε from momentum equation (4.32), that is to say,

    (4.39) ∫ − ∞ ∞ − ξ ( ρ ε u ε ) ξ + ρ ε ( u ε ) 2 − ε ρ ε ξ ψ d ξ = ∫ − ∞ ∞ ρ ε u ε ( ξ − u ε ) + ε ρ ε ψ ξ d ξ + ∫ − ∞ ∞ ρ ε u ε ψ d ξ = 0 .

    The first term on the right of equation (4.39) can be rewritten as

    (4.40) ∫ − ∞ ∞ ρ ε u ε ( ξ − u ε ) + ε ρ ε ψ ξ d ξ = ∫ − ∞ σ 1 ε + ∫ σ 1 ε σ 3 ε + ∫ σ 3 ε ∞ ρ ε u ε ( ξ − u ε ) + ε ρ ε ψ ξ d ξ .

    The sum of the first and last terms of equation (4.40) is

    (4.41) ∫ − ∞ σ 1 ε ρ − u − ( ξ − u − ) + ε ρ − ψ ξ + ∫ σ 3 ε ∞ ρ + u + ( ξ − u + ) + ε ρ + ψ ξ d ξ = ρ − u − σ 1 ε ψ ( σ 1 ε ) − ρ + u + σ 3 ε ψ ( σ 3 ε ) + ρ + u + 2 − ε ρ + ψ ( σ 3 ε ) − ρ − u − 2 − ε ρ − ψ ( σ 1 ε ) − ∫ − ∞ σ 1 ε ρ − u − ψ d ξ − ∫ σ 3 ε ∞ ρ + u + ψ d ξ .

    Letting ε → ε 2 , we derive

    (4.42) lim ε → ε 2 ∫ − ∞ σ 1 ε ρ − u − ( ξ − u − ) + ε ρ − ψ ξ + ∫ σ 3 ε ∞ ρ + u + ( ξ − u + ) + ε ρ + ψ ξ d ξ = − u δ [ ρ u ] ψ + ρ u 2 − ε 2 ρ ψ − ∫ − ∞ ∞ m 0 ( ξ − u δ ) ψ ( ξ ) d ξ ,

    where m 0 ( ξ ) = ρ − u − + [ ρ u ] ℱ ( ξ ) and ℱ ( ξ ) is a Heaviside function.

    (4.43) ∫ σ 1 ε σ 3 ε ρ ⁎ ε u ⁎ ε ( ξ − u ⁎ ε ) + ε ρ ⁎ ε ψ ξ d ξ = ρ ⁎ ε u ⁎ ε ( ξ − u ⁎ ε ) + ε ρ ⁎ ε ψ | σ 1 ε σ 3 ε − ∫ σ 1 ε σ 3 ε ρ ⁎ ε u ⁎ ε ψ d ξ = ρ ⁎ ε u ⁎ ε ( σ 3 ε − σ 1 ε ) σ 3 ε ψ ( σ 3 ε ) − σ 1 ε ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − u ⁎ ε ψ ( σ 3 ε ) − ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − ∫ σ 1 ε σ 3 ε ρ ⁎ u ⁎ ψ d ξ σ 3 ε − σ 1 ε + ε ρ ⁎ ε ( ψ ( σ 3 ε ) − ψ ( σ 1 ε ) ) ,

    it converges to

    (4.44) ρ ⁎ ε u ⁎ ε ( σ 3 ε − σ 1 ε ) ( u δ ψ ( u δ ) ) ′ − u δ ψ ′ − ψ = 0 ,

    as ε → ε 2 .

    Above all, from (4.39), we obtain

    (4.45) lim ε → ε 2 ∫ − ∞ ∞ ρ ε u ε ψ ( ξ ) − m 0 ( ξ − u δ ) ψ ( ξ ) d ξ = u δ [ ρ u ] − ρ u 2 − ε 2 ρ ψ ( σ ) .

  3. Next, let us consider the conservation of energy (4.33), we have

    (4.46) ∫ − ∞ ∞ − ξ ρ ε ( u ε ) 2 2 + H ε ξ + ρ ε ( u ε ) 3 2 + H ε u ε − ε u ε ρ ε ξ ψ d ξ = ∫ − ∞ ∞ ρ ε ( u ε ) 2 2 + H ε ( ξ − u ε ) ψ ξ + ε u ε ρ ε ψ ξ d ξ + ∫ − ∞ ∞ ρ ε ( u ε ) 2 2 + H ε ψ d ξ .

    The first integral in (4.46) can be decomposed into

    (4.47) ∫ − ∞ σ 1 ε + ∫ σ 1 ε σ 3 ε + ∫ σ 3 ε ∞ ρ ε ( u ε ) 2 2 + H ε ( ξ − u ε ) + ε u ε ρ ψ ξ d ξ .

    The sum of the first and the last terms in (4.47) is

    (4.48) ∫ − ∞ σ 1 ε + ∫ σ 3 ε ∞ ρ ε ( u ε ) 2 2 + H ε ( ξ − u ε ) + ε u ε ρ ε ψ ξ d ξ = ρ − u − 2 2 + H − σ 1 ε ψ ( σ 1 ε ) − ρ + u + 2 2 + H + σ 3 ε ψ ( σ 3 ε ) − ρ − u − 3 2 + H − u − − ε u − ρ − ψ ( σ 1 ε ) + ρ + u + 3 2 + H + u + − ε u + ρ + ψ ( σ 3 ε ) − ∫ − ∞ σ 1 ε ρ − u − 2 2 + H − ψ d ξ − ∫ σ 3 ε ∞ ρ + u + 2 2 + H + ψ d ξ ,

    which converges to

    (4.49) − u δ ρ u 2 2 + H + ρ u 2 2 + H − ε 2 ρ u ψ ( u δ ) − ∫ − ∞ ∞ ℰ 0 ( ξ − u δ ) ψ ( ξ ) d ξ

    as ε → ε 2 , with ε 0 ( ξ ) = ρ − u − 2 2 + H − + [ ρ u 2 2 + H ] ℱ ( ξ ) and ℱ ( ξ ) is a Heaviside function.

    For the second term in (4.47), we get

    (4.50) lim ε → ε 2 ∫ σ 1 ε σ 3 ε ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ( ξ − u ⁎ ε ) + ε u ⁎ ε ρ ⁎ ε ψ ξ d ξ = lim ε → ε 2 ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ( σ 3 ε − σ 1 ε ) σ 3 ε ψ ( σ 3 ε ) − σ 1 ε ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − u ⁎ ε ψ ( σ 3 ε ) − ψ ( σ 1 ε ) σ 3 ε − σ 1 ε − ψ ( σ 3 ε − σ 1 ε ) σ 3 ε − σ 1 ε + ε u ⁎ ε ρ ⁎ ε ( ψ ( σ 3 ε ) − ψ ( σ 1 ε ) ) = ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ( σ 3 ε − σ 1 ε ) ( ( u δ ψ ) ' − u δ ψ ' − ψ ) + 0 = 0 ,

    where

    (4.51) H ⁎ ε = H ⁎ 1 ε , σ 1 ε < ξ < σ 2 ε , H ⁎ 2 ε , σ 2 ε < ξ < σ 3 ε .

    From (4.46), we obtain that

    (4.52) lim ε → ε 2 ∫ − ∞ ∞ ρ ε ( u ε ) 2 2 + H ε ψ ( ξ ) + ℰ 0 ( ξ − u δ ) ψ ( ξ ) d ξ = u δ ρ u 2 2 + H ψ + ρ u 2 2 + H − ε 2 ρ u ψ .

  4. By considering the time dependence of weights of δ -measures, the limits of density, momentum and energy are obtained.

For any ψ ∈ C 0 ∞ ( ℝ × ℝ + ) , letting ψ ˜ ( ξ , t ) = ψ ( ξ t , t ) , we obtain

(4.53) 〈 ρ ε , ψ 〉 = lim ε → ε 2 ∫ 0 + ∞ ∫ − ∞ + ∞ ρ ε x t ψ ( x , t ) d ( ξ t ) d t = lim ε → ε 2 ∫ 0 + ∞ t ∫ − ∞ + ∞ ρ ε ( ξ ) ψ ˜ ( ξ , t ) d ξ d t ,

as ( ρ ε , u ε , H ε ) is a self-similar solution.

∫ − ∞ + ∞ ρ ε ( ξ ) ψ ˜ ( ξ , t ) d ξ = ∫ − ∞ + ∞ ρ 0 ( ξ − u δ ) ψ ˜ ( ξ , t ) d ξ + ( σ [ ρ ] − [ ρ u ] ) ψ ˜ ( u δ , t ) = t − 1 ∫ − ∞ + ∞ ρ 0 ( ξ − u δ ) ψ ( x , t ) d x + ( σ [ ρ ] − [ ρ u ] ) ψ ( u δ t , t ) ,

(4.54) 〈 ρ ε , ψ 〉 = ∫ 0 + ∞ ∫ − ∞ + ∞ ρ 0 ( ξ − u δ t ) ψ ( x , t ) d x d t + ∫ 0 + ∞ t ( u δ [ ρ ] − [ ρ u ] ) ψ ( u δ t , t ) d t ,

(4.55) ρ ( x , t ) = ρ 0 ( x , t ) + ω 0 ( t ) δ s ,

and

(4.56) 〈 ω 0 ( t ) δ s , ψ 〉 = ∫ 0 + ∞ ω 0 ( t ) x ′ ( t ) 2 + 1 ψ ( x , t ) d t ,

utilizing (4.54)–(4.56), we have

(4.57) 〈 ρ , ψ 〉 = ∫ 0 + ∞ ∫ − ∞ + ∞ ρ 0 ( x , t ) ψ ( x , t ) d x d t + ∫ 0 + ∞ ω 0 ( t ) x ′ ( t ) 2 + 1 ψ ( x , t ) d t = ∫ 0 + ∞ ∫ − ∞ + ∞ ρ 0 ( ξ − σ t ) ψ ( x , t ) d x d t + ∫ 0 + ∞ ω 0 ( t ) ψ ( x , t ) u δ 2 + 1 ψ ( x , t ) d t ,

therefore,

(4.58) ω 0 ( t ) u δ 2 + 1 = t ( u δ [ ρ ] − [ ρ u ] ) ,

namely,

(4.59) ω 0 ( t ) = t 1 + u δ 2 ( u δ [ ρ ] − [ ρ u ] ) .

Similarly, it can be shown that

(4.60) lim ε → ε 2 ∫ − ∞ ∞ ∫ 0 ∞ ρ ⁎ ε u ⁎ ε ψ ( x , t ) d x d t = ∫ − ∞ ∞ ∫ 0 ∞ m 0 ( x − u δ t ) ψ ( x , t ) d x d t + ∫ 0 ∞ ω 1 ( t ) ψ ( u δ t , t ) u δ 2 + 1 d t ,

with

(4.61) ω 1 ( t ) = t 1 + u δ 2 u δ [ ρ u ] − ρ u 2 − ε 2 ρ .

(4.62) lim ε → ε 2 ∫ − ∞ ∞ ∫ 0 ∞ ρ ⁎ ε ( u ⁎ ε ) 2 2 + H ⁎ ε ψ ( x , t ) d x d t = ∫ − ∞ ∞ ∫ 0 ∞ ε 0 ( x − σ t ) ψ ( x , t ) d x d t + ∫ 0 ∞ ω 2 ( t ) ψ ( u δ t , t ) u δ 2 + 1 d t ,

with

(4.63) ω 2 ( t ) = t 1 + u δ 2 u δ ρ u 2 2 + H − ρ u 2 2 + H − ε 2 ρ u .

When 0 < ε < ε 2 , the projection of the state ( ρ + , u + , H + ) belongs to V ( ρ − , u − ) onto the ( ρ , u ) -plane. From [34], when ε → 0 , we know that the limit solutions of (1.6) satisfy the following generalized Rankine-Hugoniot condition:

(4.64) d x ( t ) d t = σ ( t ) , d ω ( t ) d t = σ ( t ) [ ρ ] − [ ρ u ] , d ( ω ( t ) σ ( t ) ) d t = σ ( t ) [ ρ u ] − [ ρ u 2 ] , d ( ω ( t ) σ 2 ( t ) / 2 + h ( t ) ) d t = σ ( t ) ρ u 2 2 + H − ρ u 2 2 + H u ,

from (4.64), it is easy to calculate that the solutions can be expressed as

(4.65) x ( t ) = ρ + u + + ρ − u − ρ + + ρ − t , σ ( t ) = ρ + u + + ρ − u − ρ + + ρ − , ω ( t ) = ρ + ρ − ( u − − u + ) t , h ( t ) = − ω ( t ) σ 2 / 2 + x ( t ) [ ρ u 2 / 2 + H ] − [ ( ρ u 2 / 2 + H ) u ] t ,

where σ satisfies the entropy condition.

Finally, we show that the law of conservation of energy (1.6)3 actually produces the entropy inequality (1.9) of the transport equation (1.5) and obtains the entropy consistency.

Theorem 4.7

There are limit functions ( ρ , u ) which are a measure solution of the transport equation (1.5), and these meet

(4.66) ( ρ u 2 ) t + ( ρ u 3 ) x ≥ 0 ,

in the sense of distributions.

Proof

Due to ( ρ ε ( u ε ) 2 2 ) t + ( ρ ε ( u ε ) 3 2 + H ε u ε ) x = 0 in the sense of distributions,

(4.67) ( ρ ε ( u ε ) 2 ) t + ( ρ ε ( u ε ) 3 ) x = − 2 ( H t ε + ( H ε u ε ) x ) .

Furthermore, for each given positive test function ψ ∈ C 0 ∞ ( ℝ × ℝ + )

(4.68) 〈 ( ρ ε ( u ε ) 2 ) t + ( ρ ε ( u ε ) 3 ) x , ψ 〉 = 2 〈 H ε , ψ t 〉 + 2 〈 H ε u ε , ψ x 〉 .

We note that

(4.69) lim ε → ε 2 ∫ 0 ∞ ∫ − ∞ ∞ ( H ε ψ t + H ε u ε ψ x ) d x d t = lim ε → ε 2 ∫ 0 ∞ ( σ 1 ε ( H ⁎1 ε − H − ) − ( H ⁎1 ε u ⁎ ε − H − u − ) ) ψ ( σ 1 ε t , t ) d t + lim ε → ε 2 ∫ 0 ∞ ( σ 2 ε ( H ⁎2 ε − H ⁎1 ε ) − ( H ⁎2 ε u ⁎ ε − H ⁎1 ε u ⁎ ε ) ) ψ ( σ 2 ε t , t ) d t + lim ε → ε 2 ∫ 0 ∞ ( σ 3 ε ( H + − H ⁎2 ε ) − ( H + u + − H ⁎2 ε u ⁎ ε ) ) ψ ( σ 3 ε t , t ) d t = ∫ 0 ∞ ( u δ [ H ] − [ H u ] ) ψ ( u δ t , t ) d t = ∫ 0 ∞ ( u − − u + ) ( α H + + (1 − α ) H − ) ψ ( u δ t , t ) d t ≥ 0,

for α = ρ − ρ − + ρ + ∈ ( 0 , 1 )

u δ = ρ + u + + ρ − u − ρ + + ρ − .

Through the aforementioned discussions, we verify the consistency of entropy.□

4.2 Limits of Riemann solutions in the case u − < u +

In this section, we show the phenomenon of cavitation of Riemann solutions for (1.6)–(1.7), in the case u − < u + as the pressure decreases.

Lemma 4.8

Suppose that u − < u + , then there exists a ε 3 , when 0 < ε < ε 3 , the projection of ( ρ , u , H ) onto the ( ρ , u ) -plane belongs to I ( ρ − , u − ) .

Proof

If ρ + = ρ − , then the conclusion is obviously true. Next, we discuss the situation ρ + ≠ ρ − . Assume that u − < u + , the states ( ρ + , u + , H + ) connect with ( ρ − , u − , H − ) by contact discontinuities that satisfy

(4.70) u + = u − − ε ρ − + ε ρ + , ρ + < ρ − ,

(4.71) u + = u − + ε ρ − − ε ρ + , ρ + > ρ − .

The projection of ( ρ + , u + , H + ) belongs to I ( ρ − , u − ) , we have

(4.72) ε 3 = ( u + − u − ) ρ + ρ − ρ − − ρ + 2 ,

it is easy to see that the projection of ( ρ + , u + , H + ) onto the ( ρ , u ) -plane belongs to I ( ρ − , u − ) as ε < ε 3 .□

Lemma 4.9

When u − < u + , the cavitation occurs as ε → 0 . Namely,

(4.73) lim ε → 0 ρ ⁎ 1 ε = lim ε → 0 ρ ⁎ 2 ε = 0 .

Proof

From the first equation of (4.6)–(4.8), respectively, we derive

(4.74) u ⁎ 1 − ε ρ ⁎ 1 = u − − ε ρ − ,

(4.75) ρ ⁎ 1 ε = ρ ⁎ 2 ε , u ⁎ 1 ε = u ⁎ 2 ,

(4.76) u + + ε ρ + = u ⁎ 2 + ε ρ ⁎ 2 .

Hence, one can easily see

(4.77) lim ε → 0 ε ρ ⁎ 1 = lim ε → 0 ε ρ ⁎ 2 = lim ε → 0 u + + ε ρ + − u − − ε ρ − 2 = u − − u + 2 ,

thus, we have

(4.78) lim ε → 0 ρ ⁎ 1 ε = lim ε → 0 ρ ⁎ 2 ε = 0 .

□

5 Discussion

We have considered the limit behavior of Riemann solutions to Chaplygin Euler equations for nonisentropic fluids, when u − > u + and u − < u + , and we studied the formation of delta shock wave and the appearance of vacuum state for equations (1.6)–(1.7), respectively. When u − > u + and the parameter ε tends to ε 2 , the weight of delta shock wave for (1.6)–(1.7) is analyzed. When u − < u + and ε → 0 , we show the phenomenon of cavitation for (1.6)–(1.7).

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (11761068) and the Natural Science Foundation of Xinjiang (2017D01C053 and 2019Q015). The authors would like to thank the anonymous reviewers for their constructive comments.

  1. Competing interest: The authors declare that they have no conflicts of interest.

  2. Availability of supporting data: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

  3. Funding: This work was supported by the National Natural Science Foundation of China (11761068, 11401508 and 11461066), China Scholarship Council, the Natural Science Foundation of Xinjiang (2017D01C053).

  4. Author contributions: Lihui Guo conceived of the study. Maozhou Lin and Lihui Guo carried out the main results, participated in the sequence alignment and drafted the manuscript. Moreover, all the authors approved the final manuscript.

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Received: 2019-03-22
Revised: 2020-10-02
Accepted: 2020-10-26
Published Online: 2020-12-31

© 2020 Maozhou Lin and Lihui Guo, published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0113
Keywords for this article
nonisentropic fluids; Riemann solutions; delta shock; vacuum state; entropy inequality
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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