Solutions of the Ultra-Relativistic Euler Equations in Riemann Invariants

we studied the Riemann invariants for the ultra-relativistic Euler system. In fact we show that the Riemann invariants have interesting relations with the representations of nonlinear elementary waves (shocks and rarefaction waves). Namely, we points out the relation between Riemann invariants and nonlinear elementary waves. This turns out to be the basic ingredient of our paper [9]. One of the main applications of the Riemann invariants is to derive the diagonal form of system (1). We hope that these formula will be useful in various studies of the ultra-relativistic Euler system, for example, in developing numerical methods, [10]. We show that the Riemann invariants, play a pivotal role in the solution of the ultra-relativistic Euler system (1).

Consider x=x(t) is a shock-discontinuity of the weak solution of (1) with speed = ( ) s x t  , (p_,u_)the state lower to the shock and (p + , u + ) the state upper to the shock with p ± >0, respectively. Then the Rankine-Hugoniot jump (RHj) conditions are The entropy inequality at singular points is ( , ) = 1 , ( , ) = , h p u p u p u p u ϕ + which is equivalent to u_ >u + , [2].
We can rewrite the 2×2 system for p and u in (1) in the quasilinear form The characteristic velocities λ 1 and λ 3 are corresponding to the 1 and 3 family of waves, respectively. The decoupled equation for the particle density n > 0 gives rise for contact discontinuities with Lemma 1.1 Suppose that (p_,u_)=(p_,0) and (p+, u+)= (p, u(p)) satisfy condition (2). Then the shock curves satisfy [2] 3( The +ve sign in (7) with p < p_ gives a 3-shock. These 3-shocks satisfy both the RHj conditions (2) and the entropy condition (3), or in a similar way u_> u + .
The -ve sign in (7) with p_ < p gives a 1-shock. These 1-shocks satisfy both the RHj conditions (2) and the entropy condition (3), or in a similar way u_> u + . Furthermore < 0 du dp on shock curves S 1 and > 0 du dp on shock waves S 3 , where we studied the Riemann invariants for the ultra-relativistic Euler system. In fact we show that the Riemann invariants have interesting relations with the representations of nonlinear elementary waves (shocks and rarefaction waves). Namely, we points out the relation between Riemann invariants and nonlinear elementary waves. This turns out to be the basic ingredient of our paper [9]. One of the main applications of the Riemann invariants is to derive the diagonal form of system (1). We hope that these formula will be useful in various studies of the ultra-relativistic Euler system, for example, in developing numerical methods, [10]. We show that the Riemann invariants, play a pivotal role in the solution of the ultra-relativistic Euler system (1).
The two equations are dependent since Thus we obtain the following differential equation, This is the 1-rarefaction curve. Similarly, we can determine the 3-rarefaction as follows Since Riemann invariants are functions, which are constant along rarefaction waves, we can define and are the 1 and 3-Riemann invariant for system (1), respectively.

Remark 2.1
The function w=w (p,u) is constant across 1-rarefaction waves and z=z (p,u) is constant across 3-rarefaction waves.

Applications of Riemann Invariants
In this section we will show how Riemann invariants can be used to solve various problems related to the system (1).

Geometry of the shock waves
Here we study the geometry of the shock waves of the ultrarelativistic Euler system (1) in the Riemann invariants coordinates (w,z). We first derive the new parametrization of the ultra-relativistic Euler system (1) in Lemma 3.1. In fact this representation turns out to be very valuable in order to study the geometric properties of the solution in a unified way. In other words, in this paper we pose the following questions: 1. What are the Riemann invariants.
2. For what purpose they are useful.
The rest of this paper is given as follows : In Section 2, we derive the Riemann invariants for the ultra-relativistic Euler equations (1). In fact this topic plays a useful role in studying the ultra-relativistic Euler system (1) in a completely unified way. In Section 3 we give some applications of the Riemann invariants for the system (1). The first one, we study the geometric properties of the solution in the Riemann invariants coordinates. we give a new parametrization of the system (1), namely Lemma 3.1. This parametrization plays an important role in order to study these properties in useful way. The second one, is to give the diagonal form of the ultra-relativistic Euler system (1). Finally, in Section 4 we give the conclusions.

Riemann Invariants
We derive the Riemann invariants for the system (1), which plays a main role in this paper.
We consider our rarefaction waves. If we assume = x t ξ , then = ( ) x W W t satisfies the ordinary differential equation is an eigenvector of JW -1 JF for the eigenvalue ξ. Since JW -1 JF has two distinct real eigenvalues, λ 1 <λ 3 , there are two families of rarefaction waves, 1-rarefaction waves and 3-rarefaction waves.
We first consider 1-rarefaction waves. The eigenvector p u gives a 1-shock curve S 1 given in (8). The +ve sign in (16) and p < p_ gives a 3-shock curve S 3 given in (8).
Proof. Using the RHj conditions (2) After a straight but tedious computation, we get the result explained in Figure 1 The following lemma shows that the differences Z -Z_ and w -w_ through a shock curve depend only on the parameters α, and thus the geometric aspect of the shock wave in the zw-plane is independent of the base point. To give this lemma in a useful way we define the functions, K S :  +   + by [9] is given as follows: While the 3-shock curves S 3 based at (z_w_) has the parametrization = ln ( ) ln ( ), = ln ( ) ln ( ).
The 3-shock wave based at an arbitrary point (w_, z_) is the reflection in the wz-plane of the 1-shock wave based at the same point, where the axis of reflection is the line passing through (w_, z_), parallel to the line w=z.
Proof. Using (16), then the result follows immediately from the following: The following lemma presents further important features of the shock wave.

Lemma 3.4
For shock curves of system (1) we have [9]  Therefore we can use either the pu-plane or the zw-plane to study our model, see Figure 1. Thus we conclude that the shock waves are independent of the base point (z_w_).

Diagonalization of the ultra-relativistic Euler equations
Here we present the ultra-relativistic Euler system (1) in diagonalized form. This form enables us to develop numerical methods to study the ultra-relativistic Euler system (1), [10]. This will be presented in a forthcoming paper. (4) is said to be diagonalizable, if there exists a smooth transformationR=(w,z) T with non-vanishing Jacobian such that (4) can be rewritten as follows 2 =1 ( ) = 0, = 1, 2,

Definition 3.1 System
Where λ i (R i ) are smooth function of Riemann invariants R.
The diagonal system (21) is so important possessing so interesting properties. For example, it is easier to find exact solutions and study uniqueness of solutions, [11,12]. In fact, not all quasilinear systems can represent in diagonal form. Hence, it is so interesting to study this problem.