NON-RELATIVISTIC GLOBAL LIMITS OF THE ENTROPY SOLUTIONS TO THE RELATIVISTIC EULER EQUATIONS WITH γ -LAW

. We analyze the limit as the speed of light c → ∞ of the global entropy solutions of the 2 × 2 relativistic Euler equations for the state p = κ 2 ρ γ (1 < γ < 2), and ﬁnd that the limit is the entropy solution of the corresponding non-relativistic Euler equations.


1.
Introduction. The relativistic Euler equations for a perfect fluid in two-dimensional Minkowski space-time have the form (cf. [3,4,14,21,22,23,24,25]): where v = v(x, t) is the classical velocity of the fluid, ρ = ρ(x, t) is the mass-energy density of the fluid, p = p(ρ) is the pressure, and c is the speed of light. The equation of state is p = p(ρ), where p(ρ) is a smooth function of ρ. For a perfect gas, where κ > 0 is constant, γ is the adiabatic exponent, γ = 1 models a barotropic (or isothermal) gas, and γ > 1 a polytropic gas. The corresponding physical region is where ρ max = sup{ρ : p ′ (ρ) ≤ c 2 }, which means that the sound speed p ′ (ρ) is less than the light speed c. For the Cauchy problem of system (1.1) with initial data t = 0 : ρ = ρ 0 (x), v = v 0 (x), (1.4) Smoller-Temple [21] established the existence of global BV solutions when γ = 1 for any large initial data with bounded variation. Chen [4] established the existence of global BV solutions when γ > 1 for the initial data satisfying for some C > 0 independent of γ. For other related problems about system (1.1), we refer the reader to [1,2,7,8,9,10,11,12,13,16,17,18] and the references cited therein.
On the other hand, the classical non-relativistic isentropic Euler system is One of the motivations of this paper is that the classical mechanics is regarded as the limit of the relativistic mechanics when c → ∞, and in particular, it is easy to check that the relativistic Euler system (1.1) reduces formally to system (1.5) when c → ∞. However, rigorous mathematical proof of the singular limit for global weak solutions has been an open challenging mathematical problem. Min-Ukai [15] discussed the global convergence of weak solutions of (1.1) when γ = 1 as c → ∞, and proved that the limit is a weak solution of (1.5). The purpose of this paper is to establish the same results for entropy solutions when γ > 1.

Our main result is
Main Theorem. Suppose ρ 0 (x) and v 0 (x) are independent of c and satisfy Then there exists a constant M 0 > 0 such that, when there exists c 0 > 0 such that, for any c ≥ c 0 , there is an L ∞ entropy solution (ρ c , v c ) of (1.1) and (1.4) satisfying for all t ≥ 0, where M is a constant depending only on the initial data (ρ 0 , v 0 ) but independent of c > c 0 . Moreover, there exists a subsequence {c k }, c k → ∞ as k → ∞, such that as k → ∞, and the limit (ρ, v) is an entropy solution of the Cauchy problem (1.5) and (1.4).
The key point in the proof of the main theorem is based on the total variation estimates, independent of large c, on the approximate solutions constructed by the Glimm scheme. To achieve the desired estimates, we need to improve the estimates in [4] for the wave length in the approximate solutions, since the bounds in [4] for the total variations of the approximate solutions depend on the light speed c.
We organize this paper as follows. In §2, we review some basic and important properties of the system. In §3, we study the Riemann problem, analyze the global geometric behavior of nonlinear waves, and obtain some estimates for wave lengths independent of large c. In §4 and §5, we define the approximate solutions of the Cauchy problem based on the Glimm scheme and establish some essential estimates independent of c ≥ c 0 on the approximate solutions. Finally, we use the BVcompactness to get the convergence, thus to prove the main theorem.
2. The Relativistic Euler System. In this section we review some basic properties and important features of system (1.1), and introduce the notion of entropy solutions.
We rewrite system (1.1) in the general form of conservation laws by setting where we already plug p = κ 2 ρ γ into the equations in (1.1). For the mapping (ρ, v) → (U 1 , U 2 ), it is easy to have is 1-1, and the Jacobian of the mapping is continuous and non-zero in the region V {ρ > 0}. Moreover, the convergence  6) and the corresponding right eigenvectors are so system (1.1) is strictly hyperbolic and genuinely nonlinear. The Riemann invariants of system (1.1) are defined as Here we remark that the Riemann invariants r and s defined by (2.7) and (2.8) are different from those in [4], so that the limit of our pair makes sense as c → ∞ and converges to the pair of Riemann invariants for system (1.5): It is also easy to see that Lemma 2.2. The mapping (ρ, v) → (s, r) is 1−1, and the Jacobian of this mapping is nonzero in the region V {ρ > 0}.
So we can choose (ρ, v), or (s, r), according to our convenience, as a coordinate system.
One of the important features of system (1.1) is its Lorentz invariance. If a barred coordinates (t,x) moves with velocity τ as measured in the unbarred coordinates (t, x), and if v denotes the velocity of a particle as measured in the unbarred frame, andv denotes the velocity of a particle as measured in the barred frame, then under the Lorentz transformation (see [3]): . We recall that an entropy-entropy flux pair for (2.1) is a pair of C 1 functions (η(U ), q(U )) satisfying ∇η(U )∇F (U ) = ∇q(U ). In particular, it is easy to check that the following pair (η * (U ), q * (U )) is the physical entropy-entropy flux pair of system (1.1) and c → ∞ which is the mechanical energy-energy flux pair of the non-relativistic system (1.5).
The Riemann problem can be solved in the class of functions consisting of constant states, separated by discontinuities determined by both Rankine-Hugoniot jump conditions 3) and the Lax entropy conditions denotes the jump of the function f (U ) between the left and right hand states along the curve of discontinuity in the x − t plane, while σ 1 and σ 2 represent the shock speed of 1-shock and 2-shock, respectively. Rarefaction waves are continuous solutions of the form U (x/t).
Due to the Lorentz invariance, we can always, without loss of generality, assume that the velocity state on the left-hand side of the shock wave is v l = 0. We denote by (ρ, v) the state on the right-hand side (ρ r , v r ). The Lax entropy inequality and the Rankine-Hugoniot conditions imply that (see e.g. [3,4,12]) Lemma 3.1. The shock curves S 1 and S 2 are given by where p = κ 2 ρ γ . The rarefaction wave curves R 1 and R 2 are The wave curves are sketched in Figure 1.
We further describe the nonlinear waves in the r − s plane. By definition (2.7)-(2.8) of the Riemann invariants, we know that, s is constant along the 1-rarefaction wave curve and r is constant along the 2-rarefaction wave curve. The geometric behavior of shock curves can be expressed for general pressure p with p ′ (ρ) > 0 and p ′′ (ρ) > 0.

Lemma 3.2. [4]
For shock curves, it holds that We can depict the shock curves and the rarefaction wave curves in Fig. 2 in the r − s plane.
A standard analysis leads to the following existence theorem for the Riemann problem of system (1.1).
p(s)+s ds and s(ρ l , v l ) ≥ r(ρ r , v r ), then there exists a solution of the Riemann problem for system (1.1). The solution is unique in the class of constant states separated by rarefaction waves and shock waves.
Remark 3.1. In Theorem 3.2, we have a constant C independent of large c, which is of vital importance for this paper. To prove this, we need to check the constants in the proof of the corresponding theorem in [4].
Remark 3.2. In the Glimm scheme, the initial data are constants in small segments, and two neighboring segments give rise to a Riemann problem. Thus, if we let s max and r min be the maximum value of s and minimum value of r in that Riemann problem in the Glimm scheme, and let then we have s ≤ s max ≤ s sup , r ≥ r min ≥ r inf . Now we sketch the proof of Theorem 3.2.
Proof. When the point (r 0 , s 0 ) is given, the shape of the shock curve depends on ǫ, s is determined by r and certainly is a function of s 1 . We can express this as Then domain D(f ) of the function f is From the definition of Riemann invariants, we know that D(f ) is independent of large c. Next we define Since the only possible singular points of g(ǫ, s 1 , r) are ǫ = 0, s 1 − s 0 = 0 and r − r 0 = 0, we only need to show that the following limits: are bounded continuous functions of (ǫ, s 1 , r). We first consider lim ǫ→0 f (ǫ,s1,r) ǫ . We notice that ρ 2 and ρ are functions of ǫ, and, at the case ǫ = 0, ρ/ρ l is uniquely determined by ∆r, i.e., (3.14) So ln ρ 2 − ln ρ 1 = ln ρ − ln ρ 0 .
where ρ(0) is ρ and ρ 2 (0) is ρ 2 in (3.14)-(3.15). The first term is zero due to (3.15). From (3.15) we also know that the coefficient of the second term is Thus the coefficient of the second term is bounded and independent of c. Now we consider the coefficient of the third term: It is easy to see that | κ 1+( κ c ) 2 | < κ. Next we consider dρ dǫ /ρ at ǫ = 0. We know from Figure 3 that △r is constant. Thus We can calculate as in [4] that where C is a positive constant depending only on ρ but independent of c when c > c 0 for some c 0 > 0 from Lemma 3.3 and the fact that the numerator and denominator have the same power of parameter c. Meanwhile, where C is a positive constant depending only on ρ but independent of c when c > c 0 for some c 0 > 0. Therefore, lim ǫ→0 f (ǫ,s1,r) ǫ is a bounded continuous function of s 1 and r, and can be bounded by a constant which is independent of large c.
Next we consider lim s→s0 f (ǫ,s1,r) Noticing that and r is constant, we have

Similar calculation gives
Thus that is, where we regard v as a function of ρ and ρ 0 . From this, we have Thus, (3.20) Since v ρ < 0, v ρ0 > 0 on S 1 , we have , whose second term has the same power of parameter c in the numerator and the denominator, then is a bounded continuous function, and can be bounded by a constant that is independent of large c.
Finally, from and since the composite shock and rarefaction curve is C 2 smooth, and s = constant on rarefaction wave curves, we obtain that, on shock curves, Figure 3, it is obvious that and so Thus, f (ǫ,s1,r) r0−r is bounded independent of c.

4.
The difference approximation. We now use Glimm's scheme to construct an approximate solution U △x (x, t) for the problem (1.1) and (1.4), and derive some estimates on U △x (x, t), which will be used in the next section. Let △x = l denote a mesh length in x and △t = h a mesh length in t, and let x j = j△x, t n = n△t, denote the mesh points for the approximate solution. We start from where U 0 j = U 0 (x j +). For t n−1 < t < t n , let U △x (x, t) be the solution of the Riemann Problem posed at time t = t n−1 . Then define where U n j = U △x (x j + a n △x, t n −) for some random sequence a n ∈ (0, 1), and use this as the initial data for the Riemann problem posed at t = t n , Thus, U △x (x, t) can be defined for all x ∈ R and t > 0 by induction, if the waves do not interact within one time step. It suffices to require that However, this does not make sense when we consider the limit c → ∞. Actually it suffices to choose △x/(2△t) to be larger than the eigenvalues of the system (1.1). Next we study the interaction of the two families of waves. Thus if R 1 and R 2 are rarefaction waves corresponding to the first and second characteristic families, respectively, then we must study the following six nontrivial interactions (here the "first" wave is considered to be on the left of the "second" wave): (i) S 2 interacts with S 1 ; (ii)S 2 interacts with R 1 (or R 2 interacts with S 1 ); (iii)S 2 interacts with S 2 (or S 1 interacts with S 1 ); (iv)S 2 interacts with R 2 (or R 1 interacts with S 1 ); (v)R 2 interacts with S 2 (or S 1 interacts with R 1 ); (vi)R 2 interacts with R 1 ; Also see Figure 4. The interactions obtained by interchanging the indices 1 and 2 can be treated similarly.
Let J 2 and J 1 be mesh curves, with J 2 an immediate successor to J 1 . Let β (resp.γ) be an S 1 (resp.S 2 ) shock on J 1 , and let β ′ (resp.γ ′ ) denote an S 1 (resp.S 2 ) shock on J 2 . Let the absolute value in terms of the Riemann invariant r (resp.s), denote the strengths of the S 1 (resp.S 2 ) shock. See Figure 5.  We denote by γ + β → β ′ + γ ′ the interaction of an S 2 with an S 1 which produces an S 1 and an S 2 ; the other cases can be written in a similar way, while rarefaction waves are denoted by 0.
Now we have the following interaction estimates.
Remark 4.1. We remark that Lemma 3.2 and Theorem 3.2 play the key role in the process of finding the constants C and C 0 . The proof is similar as the proof in [19].
Remark 4.2. It suffices to consider only shock waves in the above functionals since only shocks contribute to the decreasing variation of the solution across J, and the total variation is controlled by twice the decreasing variation, plus the difference in the value of the functions at ±∞.
We now give the following decreasing estimate without proof. The proof is similar as the proof in [19].
Lemma 4.2. If ǫF (0) is sufficiently small, then F (J 2 ) < F (J 1 ), where J 1 and J 2 are mesh curves and J 2 is an immediate successor to J 1 .
Next we will estimate the total variation of v △x and ρ △x .
Then there exists a constant M 0 > 0 such that, when Suppose |v △x | ≤ M and c ≥ c 0 . Then the eigenvalues λ i (ρ, v) of the system (1.1) given in (2.6) satisfy Now we choose △x △t Thus, our choice of △x/△t is independent of c for c > c 0 , and we see that Theorem 4.1 holds with this choice. Moreover, it allows us to show that the approximate solutions (ρ △x , v △x ) are L 1 − Lipschitz continuous in t through a standard procedure.
where c 0 is the constant in Theorem 4.1.

5.
Convergence. In this section we can complete the proof of the Main Theorem. First, for any initial data (ρ 0 , v 0 ) which satisfies the assumption of the Main Theorem, there exist positive constant c 0 and M such that, for any c > c 0 , the approximate solutions (ρ c △x , v c △x ) of (1.1) generated by the Glimm's method satisfy the following (Theorem 4.1 and Lemma 3.2): where M depends only on the initial data (ρ 0 , v 0 ) and is independent of c > c 0 . For each c > c 0 , we now apply Glimm's theorem [6] to obtain the entropy solutions (ρ c , v c ) of system (1.1). Let a ≡ {a k } ∈ A denote a (fixed) random sequence, 0 < a k < 1, 1 < k < ∞, where A denotes the infinite product of intervals [0, 1] endowed with Lebesgue measure.
Theorem 5.1. Assume that the approximate solution (ρ c △x , v c △x ) of (1.1) satisfies (5.1)-(5.4). Then there exists a subsequence of mesh lengths △x i → 0, (5.5) such that (ρ c △xi , v c △xi ) → (ρ c , v c ), (5.6) where (ρ c , v c ) also satisfies (5.1)-(5.4). The convergence is pointwise a.e., and in L 1 loc (R) at each time t, uniformly on bounded x and t sets. Moreover, there exists a set N ⊂ A of Lebesgue measure zero such that, if a ∈ A − N , then (ρ c , v c ) is an entropy solution of the initial value problem (1.1)-(1.4).
Based on Theorem 5.1, we can consider the limit as c → ∞. Then there exists a subsequence {c k } such that {(ρ c k , v c k )} converges strongly to a pair of function (ρ, v) a.e. in L 1 loc (R) at each time t and in L 1 loc (R×R + ). Moreover, (ρ, v) is an entropy solution satisfying (5.1)-(5.4) with the same constant.
The Main Theorem follows immediately from Theorem 5.1 and Theorem 5.2.
Remark 5.1. With the initial data depending on c, it is easy to see that the same conclusion holds if (ρ c 0 (x), v c 0 (x)) → (ρ 0 (x), v 0 (x)) (5.8) strongly in L 1 loc as c → ∞.