The limit Riemann solutions to nonisentropic Chaplygin Euler equations

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 \varepsilon tends to a certain value. Different from the isentropic fluids, the weight of delta shock wave is determined by variance density ρ \rho and internal energy H. Meanwhile, involving the entropy inequality, the uniqueness of delta shock wave is obtained.


Introduction
One-dimensional compressible Euler equations for nonisentropic fluids can be written as 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]. 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 ρe by H, therefore, system (1.1) can be transformed into the following equations: 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 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 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 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 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 , .
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].

Γ
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

4)
namely, 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 In addition, the delta shock wave satisfies the generalized entropy condition

7)
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 Due to the value of e is positive, which means that the function ( ) = . Then, the physically relevant region can be expressed as 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) with corresponding right eigenvectors , which indicates that all the characteristic fields are contact discontinuous.
For any given constant state ( ) in the phase plane, we can derive three families of contact discontinuities On the physical correlation region, that is ( , we can draw the one-contact discontinuity curve J ε 1 that satisfies (3.2) and the three-contact discontinuity curve J ε 3 that satisfies (3.4). And from the point ( )  When the projection of ( ) ,plane, the Riemann solution can be briefly expressed by 2 are the intermediate states. For the projection of ( ) , -plane, the Riemann solution can be given by The details can be referred to in [34]. The delta shock wave holds the generalized Rankine-Hugoniot conditions δ are weight and velocity of delta shock wave, respectively. It can be derived from (3.8) that  for [ ] = − = + − ρ ρ ρ 0. In addition, it is easy to see that the delta shock wave satisfies the generalized entropy condition that is, the projection belongs to ( ) when < < ε ε ε 2 1 , and projection belongs to ( ) 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 twocontact discontinuities become steeper when ε decreases, that is, when ε decreases, the projection of ( ) First, we consider the situation < < ε ε ε 2 1 , namely, the projection of ( ) 2 are the intermediate states.
Proof. From (4.6) to (4.8), it is easy to calculate that Therefore, Using Proof. Involving the first term of (3.2)-(3.4), the following equations are obtained Proof. The expressions of σ i (i = 1, 2, 3) are employed again, and the following discussions will be presented Proof. Using the Rankine-Hugoniot conditions of (1.6), we obtain the following forms: When → ε ε 2 , utilizing (4.29) and taking limits, we have which implies that 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, ρu and H are the sums of a step function and a δ-measure with weights Proof.
We decompose I 2 as The total of the first and last terms is when → ε ε 2 , it leads to Above all, from (4.34), it yields 3. We deduce the limit of the momentum = m ρ u ε ε ε from momentum equation (4.32), that is to say, The first term on the right of equation (4.39) can be rewritten as The sum of the first and last terms of equation (   (4.78)

□
The limit Riemann solutions to nonisentropic Chaplygin Euler equations  1785 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).