Limiting behavior of some strictly hyperbolic systems of conservation laws

Abstract

The objective of this paper is two fold. Firstly, a strictly hyperbolic system of conservation laws known as Euler equation of compressible fluid flow is considered and limiting behavior of its solutions as the pressure like term vanishes is studied. Secondly, the limiting behavior of the solutions for another strictly hyperbolic system which is a perturbed model of the system introduced by Korchinski [Solution of a Riemann problem for a system of conservation laws possessing no classical weak solution, 1977, Adelphi University] is also studied as the perturbation parameter vanishes.

Keywords

Euler equation Riemann problem delta waves shadow waves

1. Introduction

This article deals with the limiting behaviour of two strictly hyperbolic systems of different nature. First one is the one dimensional model for Euler equation of compressible fluid flow and second one is a perturbed version of a non-strictly hyperbolic system of conservation laws, called one dimensional model for the large scale structure formation of the universe, which was first studied by Korchinski [13]. Both the characteristic fields of the first system are genuinely nonlinear whereas the second one does not possess the same property. For the second system the first characteristic field is genuinely nonlinear and the second characteristic field is linearly degenerate.

Euler equation of one-dimensional compressible fluid flow reads $\begin{matrix} (1.1) & \{\begin{matrix} u_{t} + {(\frac{u^{2}}{2} + P (ρ))}_{x} = 0, \\ ρ_{t} + {(ρ u)}_{x} = 0 . \end{matrix} \end{matrix}$

We take the initial conditions $\begin{matrix} (1.2) & u (x, 0) = u_{0} (x), ρ (x, 0) = ρ_{0} (x) . \end{matrix}$ The equation (1.1) was first derived by S. Earnshaw [8,26] for isentropic flow. It is a scaling limit system of a Newtonian dynamics with long range interaction for a continuous distribution of mass [20,21]. This equation is also hydrodynamic limit of Vlasov equation [2]. For smooth nonzero solutions the system (1.1) is equivelent to isentropic gas dynamics equation, namely $\begin{matrix} (1.3) & \{\begin{matrix} ρ_{t} + {(ρ u)}_{x} = 0, \\ {(ρ u)}_{t} + {(ρ u^{2} + \tilde{P} (ρ))}_{x} = 0, \tilde{P} (ρ) = \int_{0}^{ρ} s P^{'} (s) d s . \end{matrix} \end{matrix}$ This is evident from the following. $\begin{matrix} {(ρ u)}_{t} + (ρ u^{2} + {(\tilde{P} (ρ))}_{x} = u [ρ_{t} + {(ρ u)}_{x}] + ρ [u_{t} + {(\frac{u^{2}}{2} + P (ρ))}_{x}] . \end{matrix}$ But for shock solutions the above identity is no more valid and the two systems (1.1) and (1.3) are completely different.

The existence of viscosity solution of (1.1) with initial data $ρ_{0} (x) > 0$ , was shown in [15] and existence of global weak solutions for locally finite bounded variation initial data for the equation (1.1) was by DiPerna [7], where he took $p (ρ) = k^{2} ρ^{γ}$ , $γ \in (1, 3)$ .

In our present work we are interested in the limiting behavior of the solutions of (1.1) as the pressure term P approaches zero. For that purpose we take scalar function P is not only a function of density ρ but also a small parameter $ϵ > 0$ , satisfying ${lim}_{ϵ \to 0} P (ρ, ϵ) = 0$ and we take $P (ρ, ϵ) = ϵ p (ρ)$ , where $p (ρ)$ is a twice differentiable function and satisfies $\begin{matrix} (1.4) & p^{'} (ρ) > 0, 3 p^{'} (ρ) + ρ p^{″} (ρ) > 0 . \end{matrix}$ For the calculation of “entropy-entropy flux pair” we use the following particular form of p which also satisfies (1.4). $\begin{matrix} (1.5) & p (ρ) = \int_{0}^{ρ} \frac{q^{'} (ξ)}{ξ} d ξ, where q (ρ) = \int_{0}^{ρ} ξ^{2} exp (ξ) d ξ . \end{matrix}$

At this point the system (1.1) can be expressed as $\begin{matrix} (1.6) & \{\begin{matrix} u_{t} + {(\frac{u^{2}}{2} + ϵ p (ρ))}_{x} = 0, \\ ρ_{t} + {(ρ u)}_{x} = 0 . \end{matrix} \end{matrix}$ One can readily see that as $ϵ \to 0$ , formally the system (1.6) becomes $\begin{matrix} (1.7) & \{\begin{matrix} u_{t} + {(\frac{u^{2}}{2})}_{x} = 0, & x \in R, t > 0, \\ ρ_{t} + {(ρ u)}_{x} = 0, & x \in R . \end{matrix} \end{matrix}$

The above equation (1.7) is a one dimensional model for the large scale structure formation of universe [27]. This is an example of a non-strictly hyperbolic system, which got a widespread attention, started with the work of Korchinski [13]. For some interesting articles regarding this system we cite [6,10,12,19].

Physical significance for introducing the term $P (ρ, ϵ) = ϵ p (ρ)$ are as follows: The system (1.7) is equivalent to pressureless gas dynamics system (1.3) for smooth positive solutions. In the absense of pressure ( $ϵ = 0$ ) there may be concentration of mass and therefore the solution may not lie in the class of functions of bounded variation. To avoid that difficulty we introduced the pressure term $P (ρ, ϵ) = ϵ p (ρ)$ , so that in the presence of pressure ( $ϵ > 0$ ), there will be no concentration of mass and the solution will lie in the class of functions of bounded variation. Therefore one can employ Lax method to construct the solution. Solution for the original system (1.7) is constructed as the distributional limit of the solutions of the system (1.6) as ϵ approaches zero. This kind of approach can be thought of as an alternative of vanishing viscosity approach.

In this paper, we study the existence of solution for the equation (1.6) for Riemann type initial data, namely, $\begin{matrix} (1.8) & (\begin{matrix} u_{0} (x) \\ ρ_{0} (x) \end{matrix}) = \{\begin{matrix} (\begin{matrix} u_{l} \\ ρ_{l} \end{matrix}), & if x < 0, \\ (\begin{matrix} u_{r} \\ ρ_{r} \end{matrix}), & if x > 0 . \end{matrix} \end{matrix}$ Note that for $ϵ > 0$ , the system (1.6) is strictly hyperbolic and both the characteristics fields are genuinely nonlinear. For a strictly hyperbolic system whose characteristics field are either genuinely nonlinear or linearly degenerate, the theory [1,4] demonstrates the existence of solution for close-by Riemann type initial data. But for our system (1.6), large Riemann data is not an obstruction.

In this paper first we find solution for the system (1.6) for any Riemann type initial data and the solution is a combination of shock and rarefaction waves. Then we study the limiting behavior of these solutions as the parameter ϵ approaches to zero. It turns out that this limit is a solution for (1.7) and agrees with vanishing viscosity limit [10]. This kind of method is not very common in the literature and can be used to construct solution for non-strictly hyperbolic systems. In this regard, we refer two interesting articles [16,23] on isentropic gas dynamics.

The second strictly hyperbolic system we study in this paper is $\begin{matrix} (1.9) & \{\begin{matrix} u_{t} + {(\frac{{(u + ϵ)}^{2}}{2})}_{x} = 0, \\ ρ_{t} + {(ρ u)}_{x} = 0 . \end{matrix} \end{matrix}$

Though the system (1.9) is strictly hyperbolic for $ϵ > 0$ , it can be solved only for close by Riemann data. We observe that if $u_{l} - u_{r} > 2 ϵ$ , one can not get Lax type solution consisting of shock and rarefaction waves. This is an example of a system where smallness condition is required on the initial data to get Lax type solution. For large Riemann data, the solution is not a function of bounded variation. There are many methods such as Colombeau generalized functions [3,18], weak asymptotic method [22], Volpert product [5,25] and shadow wave approach [17] to overcome such difficulties. We cite [24] which deals with a highly non-strictly hyperbolic system of conservation laws using some of these methods. Here in this case shadow wave approach [17] will be our method of choice.

Shadow wave is a family of piecewise continuous functions $(u^{η}, ρ^{η})$ , $η > 0$ such that the equations (1.9) holds in the sense of distribution as η approaches zero. It turns out that the distributional limit of $(u^{η}, ρ^{η})$ as η tends to zero satisfies the equation.

The paper is organized as follows. In Section 2, shock and rarefaction curves are described for the system (1.6). In Section 3, shock-wave solution is constructed for (1.6)–(1.8), when $u_{l} > u_{r}$ and the distributional limit is obtained when the parameter ϵ approaches to zero and it is shown that limit satisfies (1.7) in the sense of distribution. In Section 4, entropy-entropy flux pair is found for (1.6) and it satisfies entropy condition for small ϵ. In Section 5, the solution for the case $u_{l} ⩽ u_{r}$ is obtained by using other elementary waves. Finally, in Section 6, we explicitly determine the solution for the system (1.9) for any Riemann type initial data and also the distributional limit of the solutions as ϵ vanishes, are obtained.

2. The Riemann solution

The co-efficient matrix $A (u, ρ)$ of the equation (1.6) is given by $\begin{matrix} A (u, ρ) = (\begin{matrix} u & ϵ p^{'} (ρ) \\ ρ & u \end{matrix}) . \end{matrix}$ Eigenvalues for this co-efficient matrix are the following: $λ_{1} (u, ρ) = u - \sqrt{ϵ p^{'} (ρ) ρ}$ and $λ_{2} (u, ρ) = u + \sqrt{ϵ p^{'} (ρ) ρ}$ and the eigenvectors corresponding to $λ_{1}$ and $λ_{2}$ are $X_{1} = (- \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, 1)$ and $X_{2} = (\sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, 1)$ respectively and $\nabla λ_{i} . X_{i} \neq 0$ for $i = 1, 2$ .

Each characteristic field is genuinely nonlinear for problem (1.6).

Shock curves: The shock curves $s_{1}$ , $s_{2}$ through $(u_{l}, ρ_{l})$ are derived from the Rankine–Hugoniot conditions $\begin{matrix} (2.1) & \begin{matrix} λ (u - u_{l}) = (\frac{u^{2}}{2} + ϵ p (ρ)) - (\frac{u_{l}^{2}}{2} + ϵ p (ρ_{l})), \\ λ (ρ - ρ_{l}) = ρ u - ρ_{l} u_{l} . \end{matrix} \end{matrix}$ Eliminating λ from (2.1), the admissible part of the shock curves passing through $(u_{l}, ρ_{l})$ are computed as $\begin{array}{l} s_{1} = {(u, ρ) : {(u - u_{l})}^{2} \frac{(ρ + ρ_{l})}{2} = ϵ (ρ - ρ_{l}) (p (ρ) - p (ρ_{l})), ρ > ρ_{l}; u < u_{l}}, \\ s_{2} = {(u, ρ) : {(u - u_{l})}^{2} \frac{(ρ + ρ_{l})}{2} = ϵ (ρ - ρ_{l}) (p (ρ) - p (ρ_{l})), ρ < ρ_{l}; u < u_{l}} . \end{array}$

Rarefaction curves: The Rarefaction curves $R_{1}$ , $R_{2}$ passing through $(u_{l}, ρ_{l})$ are the following:

1-Rarefaction curve: The first Rarefaction curve passing through $(u_{l}, ρ_{l})$ is derived by solving $\begin{array}{l} \frac{d u}{d ρ} = - \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, u (ρ_{l}) = u_{l}; \\ R_{1} = {(u, ρ) : u - u_{l} = - \int_{ρ_{l}}^{ρ} \sqrt{\frac{ϵ p^{'} (ξ)}{ξ}} d ξ, ρ < ρ_{l}} . \end{array}$

2-Rarefaction curve: The second Rarefaction curve $R_{2}$ passing through $(u_{l}, ρ_{l})$ is derived by solving $\begin{array}{l} \frac{d u}{d ρ} = \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, u (ρ_{l}) = u_{l}; \\ R_{2} = {(u, ρ) : u - u_{l} = \int_{ρ_{l}}^{ρ} \sqrt{\frac{ϵ p^{'} (ξ)}{ξ}} d ξ, ρ > ρ_{l}} . \end{array}$

To solve the equation (1.6) with (1.8), three cases are required to be considered, that is (I) $u_{l} > u_{r}$ , (II) $u_{l} = u_{r}$ and (III) $u_{l} < u_{r}$ . In case (I) for sufficiently small ϵ, we have solutions as a combination of two shock waves, if the Riemann type initial data are fixed. For case (II) solutions are given as the combination of 1-rarefaction and 2-shock curves or 1-shock and 2-rarefaction curves depending upon $ρ_{l} > ρ_{r}$ or $ρ_{l} < ρ_{r}$ respectively. And finally in case (III) for sufficiently small ϵ and with fixed Riemann type initial data, the solution consists of two rarefaction waves and vacuum state. We obtain the limit for the solutions in each case and it is exactly equal to the vanishing viscosity limit found in [10] which satisfies the equation in the sense of definition (3.4).

3. Formation of shock waves for $u_{l} > u_{r}$

In this section the limiting behavior for the solution of the equations (1.6)–(1.8) for $u_{l} > u_{r}$ as ϵ tends to zero has been studied. We assume $p (ρ)$ is a twice differentiable function and satisfies (1.4). First, we find solution for the system (1.6) satisfying Lax- entropy condition for the case $u_{l} > u_{r}$ . $ρ_{l}$ and $ρ_{r}$ are taken positive through out this section. The key result of this section is the following:

Theorem 3.1.
If $u_{l} > u_{r}$ , there exists an $η > 0$ such that for any $ϵ < η$ , we have a unique intermediate state $(u_{ϵ}^{}, ρ_{ϵ}^{})$ which connects $(u_{l}, ρ_{l})$ to $(u_{ϵ}^{}, ρ_{ϵ}^{})$ by 1-shock and $(u_{ϵ}^{}, ρ_{ϵ}^{})$ to $(u_{r}, ρ_{r})$ by 2-shock and satisfies Lax-entropy condition.
Proof.
The admissible 1-shock curve passing through $(\bar{u}, \bar{ρ})$ satisfies the following: $\begin{matrix} (3.1) & \begin{matrix} (u - \bar{u}) s_{1} = (\frac{u^{2}}{2} + ϵ p (ρ)) - (\frac{{\bar{u}}^{2}}{2} + ϵ p (\bar{ρ})), \\ (ρ - \bar{ρ}) s_{1} = ρ u - \bar{ρ} \bar{u}, \end{matrix} \end{matrix}$ and satisfies the inequality $\begin{matrix} (3.2) & s_{1} < λ_{1} (\bar{u}, \bar{ρ}), λ_{1} (u, ρ) < s_{1} < λ_{2} (u, ρ) . \end{matrix}$

Eliminating $s_{1}$ from (3.1) and simplifying yields $\begin{matrix} (3.3) & {(u - \bar{u})}^{2} = 2 ϵ \frac{ρ - \bar{ρ}}{ρ + \bar{ρ}} (p (ρ) - p (\bar{ρ})) . \end{matrix}$ We show that for a given $u < \bar{u}$ , there exists a unique $ρ > \bar{ρ}$ such that equation (3.3) holds. For that let us define a function $\begin{matrix} (3.4) & F (ρ) : = 2 ϵ \frac{ρ - \bar{ρ}}{ρ + \bar{ρ}} (p (ρ) - p (\bar{ρ})) . \end{matrix}$ We see that $F (\bar{ρ}) = 0$ and $F (ρ) \to \infty$ as $ρ \to \infty$ . So by intermediate value theorem we have $F ([\bar{ρ}, \infty)) = [0, \infty)$ . Hence for a given u there exist a $ρ > \bar{ρ}$ such that $\begin{matrix} F (ρ) = {(u - \bar{u})}^{2} . \end{matrix}$ This proves the existence of ρ. To prove the uniqueness, now we differentiate the equation (3.4) with respect to ρ to get $\begin{matrix} F^{'} (ρ) = 2 ϵ \frac{2 \bar{ρ}}{{(ρ + \bar{ρ})}^{2}} (p (ρ) - p (\bar{ρ})) + 2 ϵ \frac{ρ - \bar{ρ}}{ρ + \bar{ρ}} p^{'} (ρ) . \end{matrix}$ As $ρ > \bar{ρ}$ and $p^{'} (ρ) > 0$ , $F^{'} (ρ)$ is positive. So ${(u - \bar{u})}^{2}$ will be achieved only once in the interval $[\bar{ρ}, \infty)$ , which proves the uniqueness. The conditions (3.1) and (3.2) hold if and only if $u < \bar{u}$ and $ρ > \bar{ρ}$ . In fact, $s_{1}$ satisfies (3.2) if $\begin{matrix} (3.5) & \begin{matrix} \frac{ρ u - \bar{ρ} \bar{u}}{ρ - \bar{ρ}} < \bar{u} - \sqrt{ϵ p^{^{'}} (\bar{ρ})} \bar{ρ}, \\ u - \sqrt{ϵ p^{^{'}} (ρ)} ρ < \frac{ρ u - \bar{ρ} \bar{u}}{ρ - \bar{ρ}} < u + \sqrt{ϵ p^{'} (ρ) ρ} . \end{matrix} \end{matrix}$ Now from the first inequality of (3.5) one can get, $\begin{matrix} \frac{ρ (u - \bar{u})}{(ρ - \bar{ρ})} < - \sqrt{ϵ p^{'} (\bar{ρ}) \bar{ρ}} . \end{matrix}$ Since $u < \bar{u}$ , the above inequality implies $\begin{matrix} (3.6) & \frac{ρ^{2} {(u - \bar{u})}^{2}}{{(ρ - \bar{ρ})}^{2}} > ϵ p^{'} (\bar{ρ}) \bar{ρ} . \end{matrix}$ Then the equation (3.3) yields $\begin{matrix} (3.7) & \frac{2 ϵ ρ^{2} (p (ρ) - p (\bar{ρ})}{(ρ + \bar{ρ}) (ρ - \bar{ρ})} > ϵ p^{'} (\bar{ρ}) \bar{ρ} . \end{matrix}$ To show that the inequality (3.7) holds let us define $\begin{matrix} G (ρ) : = 2 ρ^{2} (p (ρ) - p (\bar{ρ})) - p^{'} (\bar{ρ}) \bar{ρ} (ρ^{2} - {\bar{ρ}}^{2}) . \end{matrix}$ Differentiating above equation one can get $\begin{matrix} G^{'} (ρ) : = 2 p^{'} (ρ) ρ (ρ - \bar{ρ}) + 4 ρ (p (ρ) - p (\bar{ρ})) > 0 . \end{matrix}$ The above inequality is evident since p is an increasing function and $ρ > \bar{ρ}$ . So, G is an increasing function and $G (ρ) > G (\bar{ρ}) = 0$ . Thus we are done.

Again from the second inequality of (3.5) one can get $\begin{matrix} \frac{{\bar{ρ}}^{2} {(u - \bar{u})}^{2}}{{(ρ - \bar{ρ})}^{2}} < ϵ p^{'} (ρ) ρ . \end{matrix}$ To prove this inequality, let us define the following: $\begin{matrix} H (ρ) : = 2 {\bar{ρ}}^{2} (p (ρ) - p (\bar{ρ})) - p^{'} (ρ) ρ (ρ^{2} - {\bar{ρ}}^{2}) . \end{matrix}$ Differentiating the above equation, $\begin{matrix} H^{'} (ρ) = - (ρ^{2} - {\bar{ρ}}^{2}) (3 p^{'} (ρ) + p^{″} (ρ) ρ) < 0, \end{matrix}$ since $ρ > \bar{ρ}$ and $3 p^{'} (ρ) + p^{″} (ρ) ρ > 0$ . So, H is a decreasing function and $H (ρ) < H (\bar{ρ}) = 0$ .

Therefore, the branch of the curve satisfying (3.1) and (3.2) can be parameterized by a $C^{1}$ function $ρ_{1} : (- \infty, \bar{u}] \to [\bar{ρ}, \infty)$ with the parameter u.

From the equation (3.3), $ρ_{1} (u)$ satisfies $\begin{matrix} (3.8) & \frac{{(u - \bar{u})}^{2}}{ϵ} \frac{\bar{ρ} + ρ_{1} (u)}{2 (ρ_{1} (u) - \bar{ρ}))} + p (\bar{ρ}) = p (ρ_{1} (u)) . \end{matrix}$

Differentiating the above equation with respect to u, we get $\begin{matrix} \frac{(u - \bar{u}) (\bar{ρ} + ρ_{1} (u))}{ϵ (ρ_{1} (u) - \bar{ρ})} + \frac{- \bar{ρ} ρ_{1}^{'} (u) {(u - \bar{u})}^{2}}{ϵ {(ρ_{1} (u) - \bar{ρ})}^{2}} = p^{'} (ρ_{1} (u)) ρ_{1}^{'} (u) . \end{matrix}$ Since $ρ_{1} (u) > \bar{ρ}$ , $\bar{ρ} + ρ_{1} (u)$ and $- \bar{ρ} + ρ_{1} (u)$ are positive, left-hand side of the above equation is negative. This implies $ρ_{1}^{'} (u)$ is negative, because $p^{'}$ is positive.

Similarly the branch of the curve satisfying $\begin{matrix} s_{1} > λ_{2} (u, ρ), λ_{1} (\bar{u}, \bar{ρ}) < s_{1} < λ_{2} (\bar{u}, \bar{ρ}), \end{matrix}$ is the admissible 2-shock curve which can be parameterized by a $C^{1}$ function $ρ_{2} : (- \infty, \bar{u}] \to (- \infty, \bar{ρ}]$ with the parameter u.

Also $ρ_{2}$ satisfies the following equation: $\begin{matrix} (3.9) & \frac{{(u - \bar{u})}^{2}}{ϵ} \frac{\bar{ρ} + ρ_{2} (u)}{2 (ρ_{2} (u) - \bar{ρ}))} + p (\bar{ρ}) = p (ρ_{2} (u)) . \end{matrix}$

Differentiating the above equation (3.9) with respect to u, we get $\begin{matrix} \frac{(u - \bar{u}) (\bar{ρ} + ρ_{2} (u))}{ϵ (ρ_{2} (u) - \bar{ρ})} + \frac{- \bar{ρ} ρ_{2}^{'} (u) {(u - \bar{u})}^{2}}{ϵ {(ρ_{2} (u) - \bar{ρ})}^{2}} = p^{'} (ρ_{2} (u)) ρ_{2}' (u) . \end{matrix}$ That is, $\begin{matrix} (3.10) & \frac{(u - \bar{u}) (\bar{ρ} + ρ_{2} (u))}{ϵ (ρ_{2} (u) - \bar{ρ})} = ρ_{2}^{'} (u) [p^{'} (ρ_{2} (u)) + \frac{\bar{ρ} {(u - \bar{u})}^{2}}{ϵ {(ρ_{2} (u) - \bar{ρ})}^{2}}] . \end{matrix}$

From (3.9), we get $\begin{matrix} (3.11) & \frac{(u - \bar{u})}{ϵ} \frac{\bar{ρ} + ρ_{2} (u)}{2 (ρ_{2} (u) - \bar{ρ}))} = \frac{p (ρ_{2} (u)) - p (\bar{ρ})}{u - \bar{u}} . \end{matrix}$ Since p is increasing and $u < \bar{u}$ , from (3.11) and (3.10) $\begin{matrix} ρ_{2}^{'} (u) [p^{'} (ρ_{2} (u)) + \frac{\bar{ρ} {(u - \bar{u})}^{2}}{ϵ {(ρ_{2} (u) - \bar{ρ})}^{2}}] > 0 . \end{matrix}$ This implies $ρ_{2}^{'} (u) > 0$ on $(- \infty, \bar{u})$ .

Consider the branch of the curve passing through $(u_{r}, ρ_{r})$ satisfying the condition $u > u_{r}$ , $ρ > ρ_{r}$ . In a similar way as above it can be parameterized by a $C^{1}$ -curve $ρ_{2}^{} : [u_{r}, \infty) \to [ρ_{r}, \infty)$ . The part of the curve $ρ_{2}^{}$ from $(w, z)$ to $(u_{r}, ρ_{r})$ will be the admissible 2-shock curve connecting $(w, z)$ to $(u_{r}, ρ_{r})$ . So it is clear that ${ρ_{2}^{}}^{'} (u)$ is positive.

Let us denote the admissible 1-shock curve passing through $(u_{l}, ρ_{l})$ as $ρ_{1}^{}$ . From the previous analysis, this is parameterized by a $C^{1}$ -curve $ρ_{1}^{} : (- \infty, u_{l}] \to [ρ_{l}, \infty)$ and satisfies ${ρ_{1}^{}}^{'} (u) < 0$ .

$ρ_{1}^{} (u_{r})$ satisfies (3.8) with $ρ_{1} (u)$ and u replaced by $ρ_{1}^{} (u_{r})$ and $u_{r}$ respectively, and $\bar{u}$ , $\bar{ρ}$ replaced by $u_{l}$ and $ρ_{l}$ respectively, i.e., $\begin{matrix} (3.12) & \frac{{(u_{r} - u_{l})}^{2}}{ϵ} \frac{ρ_{l} + ρ_{1}^{} (u_{r})}{2 (ρ_{1}^{} (u_{r}) - ρ_{l})} + p (ρ_{l}) = p (ρ_{1}^{} (u_{r})) . \end{matrix}$ Again $ρ_{2}^{} (u_{l})$ satisfies (3.9) with $ρ_{2} (u)$ and u replaced by $ρ_{2}^{} (u_{l})$ and $u_{l}$ respectively, and $\bar{u}, \bar{ρ}$ replaced by $u_{r}$ and $ρ_{r}$ respectively, i.e., $\begin{matrix} (3.13) & \frac{{(u_{l} - u_{r})}^{2}}{ϵ} \frac{ρ_{l} + ρ_{2}^{} (u_{l})}{2 (ρ_{2}^{} (u_{l}) - ρ_{r})} + p (ρ_{r}) = p (ρ_{2}^{} (u_{l})) . \end{matrix}$ It is evident from (3.12) and (3.13) that $ρ_{1}^{} (u_{r})$ and $ρ_{2}^{} (u_{l})$ tend to ∞ as ϵ tends to zero. Therefore there exists an $η > 0$ such that $ϵ < η$ , we have $ρ_{2}^{} (u_{l}) > ρ_{l}$ and $ρ_{1}^{} (u_{r}) > ρ_{r}$ . Now let us consider the function $ρ_{1}^{} - ρ_{2}^{}$ . Since $ρ_{1}^{} (u_{l}) - ρ_{2}^{} (u_{l}) = ρ_{l} - ρ_{2}^{} (u_{l}) < 0$ and $ρ_{1}^{} (u_{r}) - ρ_{2}^{} (u_{r}) = ρ_{1}^{} (u_{r}) - ρ_{r} > 0$ , by intermediate value theorem there exists a point $u_{ϵ}^{}$ such that $ρ_{1}^{} (u_{ϵ}^{}) = ρ_{2}^{} (u_{ϵ}^{}) = ρ_{ϵ}^{}$ (say). $ρ_{ϵ}^{}$ is unique because $ρ_{1}^{}$ is strictly decreasing and $ρ_{2}^{}$ is strictly increasing. Since we are considering only admissible curves, Lax entropy condition holds. This completes the proof. □

Now we determine the limit of the problem (1.6) for the shock case. For this, first we will define δ-distribution followed by a simple technical lemma which will be useful later.
Definition.
A weighted δ-distribution “ $d (t) δ_{x = c (t)}$ ” concentrated on a smooth curve $x = c (t)$ can be defined by $\begin{matrix} ⟨ d (t) δ_{x = c (t)}, φ (x, t) ⟩ = \int_{0}^{\infty} d (t) φ (c (t), t) d t \end{matrix}$ for all $φ \in C_{c}^{\infty} (R \times (0, \infty))$ .
Lemma 3.2.
Suppose* $a_{ϵ} (t)$ ( $> 0$ ) and $b_{ϵ} (t)$ ( $> 0$ ) converge uniformly to 0 on compact subsets of $(0, \infty)$ as ϵ tends to zero. Also assume that $d_{ϵ} (t)$ converges to $d (t)$ uniformly on compact subsets of $(0, \infty)$ as ϵ tends to zero. Then $\begin{matrix} \frac{1}{b_{ϵ} (t) + a_{ϵ} (t)} d_{ϵ} (t) χ_{(c (t) - a_{ϵ} (t), c (t) + b_{ϵ} (t))} (x) \end{matrix}$ converges to $d (t) δ_{x = c (t)}$ in the sense of distribution.
Proof.
Let us denote $\begin{matrix} Ψ (x, t) = \frac{1}{b_{ϵ} (t) + a_{ϵ} (t)} d_{ϵ} (t) χ_{(c - a_{ϵ} (t), c + b_{ϵ} (t))} (x) . \end{matrix}$ Let us now consider the integral $\begin{array}{l} | \int_{0}^{\infty} \int_{- \infty}^{\infty} (Ψ (x, t) φ (x, t) d x d t - \int_{0}^{\infty} d (t) φ (c (t), t) d t) | \\ ⩽ \int_{0}^{\infty} | \frac{1}{b_{ϵ} (t) + a_{ϵ} (t)} \int_{c (t) - a_{ϵ} (t)}^{c (t) + b_{ϵ} (t)} d_{ϵ} (t) φ (x, t) - d (t) φ (c (t), t) | d x d t . \end{array}$ Now, since $φ (x, t)$ has compact support and $d_{ϵ} (t)$ converges to $d (t)$ uniformly on compact sets as $ϵ \to 0$ , the integral in the right-hand side of the inequality above converges to 0. Since this is true for all test function φ, the proof of this lemma is completed. □
Theorem 3.3.
The pointwise limit $u^{ϵ}$ is u which is also distributional limit and is given by $\begin{matrix} u (x, t) = \{\begin{matrix} u_{l}, & if x < \frac{u_{l} + u_{r}}{2} t, \\ \frac{u_{l} + u_{r}}{2}, & if x = \frac{u_{l} + u_{r}}{2} t, \\ u_{r}, & if x > \frac{u_{l} + u_{r}}{2} t . \end{matrix} \end{matrix}$ The distributional limit of $ρ^{ϵ}$ is ρ and is given by $\begin{matrix} ρ (x, t) = \{\begin{matrix} ρ_{l}, & if x < \frac{u_{l} + u_{r}}{2} t, \\ (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} t δ_{x = \frac{u_{l} + u_{r}}{2} t}, & if x = \frac{u_{l} + u_{r}}{2} t, \\ ρ_{r}, & if x > \frac{u_{l} + u_{r}}{2} t . \end{matrix} \end{matrix}$
Proof.
From the above theorem, $(u_{ϵ}^{}, ρ_{ϵ}^{})$ satisfies the following conditions: $\begin{matrix} (3.14) & \begin{matrix} (u_{ϵ}^{} - u_{l}) \frac{ρ_{ϵ}^{} u_{ϵ}^{} - ρ_{l} u_{l}}{ρ_{ϵ}^{} - ρ_{l}} = (\frac{{u_{ϵ}^{}}^{2}}{2} + ϵ p (ρ_{ϵ}^{})) - (\frac{u_{l}^{2}}{2} + ϵ p (ρ_{l})), \\ (u_{ϵ}^{} - u_{r}) \frac{ρ_{ϵ}^{} u_{ϵ}^{} - ρ_{r} u_{r}}{ρ_{ϵ}^{} - ρ_{r}} = (\frac{{u_{ϵ}^{}}^{2}}{2} + ϵ p (ρ_{ϵ}^{})) - (\frac{u_{r}^{2}}{2} + ϵ p (ρ_{r})) . \end{matrix} \end{matrix}$

We know $u_{ϵ}^{} \in (u_{r}, u_{l})$ . So the sequence $u_{ϵ}^{}$ is bounded. We claim that $ρ_{ϵ}^{}$ is unbounded as ϵ tends to zero. Infact, if $ρ_{ϵ}^{}$ is bounded, then it has a convergent subsequence still denoted by $ρ_{ϵ}^{}$ and it converges to $ρ^{}$ as ϵ tends to zero. Then from the equation (3.3), we get that $ρ_{ϵ}^{}$ satisfies: $\begin{matrix} {(u_{ϵ}^{} - u_{l})}^{2} \frac{(ρ_{ϵ}^{} + ρ_{l})}{2} = ϵ (ρ_{ϵ}^{} - ρ_{l}) (p (ρ_{ϵ}^{}) - p (ρ_{l})) . \end{matrix}$ Now as $ϵ \to 0$ , the above equation becomes $\begin{matrix} {(u^{} - u_{l})}^{2} \frac{(ρ^{} + ρ_{l})}{2} = 0, \end{matrix}$ as right-hand side of the equation is bounded. Now since $ρ^{} + ρ_{l} > 0$ , we get $u^{} = u_{l}$ . Again since $ρ_{ϵ}^{}$ satisfies: $\begin{matrix} {(u_{ϵ}^{} - u_{r})}^{2} \frac{(ρ_{ϵ}^{} + ρ_{r})}{2} = ϵ (ρ_{ϵ}^{} - ρ_{r}) (p (ρ_{ϵ}^{}) - p (ρ_{r})) . \end{matrix}$ By the similar argument as above, we get, $u^{} = u_{r}$ . This implies $u_{l} = u_{r}$ , leads to a contradiction.

So for subsequence of $u_{ϵ}^{}$ and $ρ_{ϵ}^{}$ still labeled as $u_{ϵ}^{}$ and $ρ_{ϵ}^{}$ respectively and that $u_{ϵ}^{}$ converges to $u^{}$ and $ρ_{ϵ}^{}$ tend to $+ \infty$ . Passing to the limit for this subsequence in (3.14), we get $\begin{array}{l} u^{} (u^{} - u_{l}) = \frac{{u^{}}^{2}}{2} - \frac{{u_{l}}^{2}}{2} + l, \\ u^{} (u^{} - u_{r}) = \frac{{u^{}}^{2}}{2} - \frac{{u_{r}}^{2}}{2} + l, \end{array}$ where ${lim}_{ϵ \to 0} ϵ p (ρ_{ϵ}^{}) = l$ . Solving the above two equations we get $\begin{matrix} (3.15) & u^{} = \frac{u_{l} + u_{r}}{2} and l = \frac{1}{8} {(u_{l} - u_{r})}^{2} . \end{matrix}$

The solution for $(u^{ϵ}, ρ^{ϵ})$ is given by $\begin{array}{l} (3.16) & u^{ϵ} (x, t) = \{\begin{matrix} u_{l}, & if x < (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}) t, \\ u_{ϵ}^{}, & if (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}) t < x < (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{⋆}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) t, \\ u_{r}, & if x > (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) t, \end{matrix} \\ (3.17) & ρ^{ϵ} (x, t) = \{\begin{matrix} ρ_{l}, & if x < (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}) t, \\ ρ_{ϵ}^{}, & if (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}) t < x < (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{⋆}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) t, \\ ρ_{r}, & if x > (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) t . \end{matrix} \end{array}$ As $u_{ϵ}^{}$ converges to $u^{} = \frac{u_{l} + u_{r}}{2}$ as $ϵ \to 0$ , we have the limit for $u (x, t)$ as stated in the theorem.

From (3.14) and (3.15), one can show that $\begin{matrix} lim_{ϵ \to 0} [\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}] = \frac{u_{l} + u_{r}}{2}, \end{matrix}$ and $\begin{matrix} lim_{ϵ \to 0} [\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}] = \frac{u_{l} + u_{r}}{2} . \end{matrix}$

Let us denote $\begin{array}{l} c (t) = \frac{u_{l} + u_{r}}{2} t, \\ a_{ϵ} (t) = c (t) - (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}) t, \\ b_{ϵ} (t) = (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) t - c (t), \\ d_{ϵ} (t) = [\frac{u_{r} - u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}} - \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}] ρ_{ϵ}^{} t . \end{array}$

With the above notations, the formula for $ρ^{ϵ}$ in equation (3.17) can be written in the following form as in Lemma 3.2: $\begin{array}{rcl} ρ^{ϵ} & = & ρ_{l} χ_{(- \infty, c (t) - a_{ϵ} (t))} (x) + \frac{d_{ϵ} (t)}{b_{ϵ} (t) + a_{ϵ} (t)} χ_{(c (t) - a_{ϵ} (t), c (t) + b_{ϵ} (t))} (x) \\ (3.18) & + ρ_{r} χ_{(c (t) + b_{ϵ} (t), \infty)} (x) . \end{array}$ Observe that $a_{ϵ} (t)$ and $b_{ϵ} (t)$ satisfies the condition of the lemma, i.e, $a_{ϵ} (t) > 0$ and $b_{ϵ} (t) > 0$ for small ϵ.

Now we will determine the limit of $d_{ϵ} (t)$ as $ϵ \to 0$ .

The equation (3.14) can also be written in the following form: $\begin{matrix} (3.19) & \begin{matrix} (ρ_{ϵ}^{} - ρ_{l}) {\frac{u_{ϵ}^{} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}} = ρ_{ϵ}^{} u_{ϵ}^{} - ρ_{l} u_{l}, \\ (ρ_{ϵ}^{} - ρ_{r}) {\frac{u_{ϵ}^{} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}} = ρ_{ϵ}^{} u_{ϵ}^{} - ρ_{r} u_{r} . \end{matrix} \end{matrix}$ Subtracting second equation from the first in (3.19), we get $\begin{array}{l} [\frac{u_{r} - u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}} - \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}] ρ_{ϵ}^{} \\ = ρ_{l} u_{l} - ρ_{r} u_{r} + ρ_{r} (\frac{u_{ϵ}^{} + u_{r}}{2}) - ρ_{l} (\frac{u_{ϵ}^{} + u_{l}}{2}) \\ + ρ_{r} \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}} - ρ_{l} \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}} . \end{array}$ Passing to the limit as $ϵ \to 0$ , we get $\begin{matrix} (3.20) & lim_{ϵ \to 0} [\frac{u_{r} - u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}} - \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}] ρ_{ϵ}^{} = \frac{1}{2} (u_{l} - u_{r}) (ρ_{l} + ρ_{r}) . \end{matrix}$ This implies $\begin{matrix} (3.21) & lim_{ϵ \to 0} d_{ϵ} (t) = \frac{1}{2} (u_{l} - u_{r}) (ρ_{l} + ρ_{r}) t . \end{matrix}$ Here in the calculation of (3.21), we have used the fact that ${lim}_{ϵ \to 0} ϵ p (ρ_{ϵ}^{}) = \frac{1}{8} {(u_{l} - u_{r})}^{2}$ and ${lim}_{ϵ \to 0} u_{ϵ}^{} = \frac{u_{l} + u_{r}}{2}$ from the equation (3.15).

The first and the third terms of (3.18) converge to $ρ_{l} χ_{(- \infty, \frac{u_{l} + u_{r}}{2} t)} (x)$ and $ρ_{r} χ_{(\frac{u_{l} + u_{r}}{2} t, \infty)} (x)$ respectively. Hence, employing the above lemma to the second term of (3.18), we get the distribution limit $ρ (x, t)$ as given in the theorem. Note that all the analysis has been done for a subsequence. Since the limit is same for any subsequence, this implies the sequence itself converges to the same limit. Thus the proof of Theorem 3.3 is completed. □

The limit $(u, ρ)$ satisfies the equation in the sense of Volpert is available in [11,14]. There it was shown that $R_{t} + \overline{u} R_{x} = 0$ , where $ρ = R_{x}$ and $\overline{u} R_{x}$ is known as Volpert product [25]. Then $ρ = R_{x}$ satisfies the equation (1.7) in the sense of distribution. For completeness, we intend here to show that the limit $(u, ρ)$ satisfies the equation by formulating as follows. Definition 3.4.
Let u is a Borel measurable function and $ρ = d ν$ is a Radon measure on $R \times [0, \infty)$ . Then $(u, ρ = d ν)$ is said to be a solution for the system (1.7) with initial data (1.2) if the following conditions hold. $\begin{matrix} (3.22) & \begin{matrix} \int_{R \times [0, \infty)} (u ϕ_{t} + u ϕ_{x}) d x d t + \int_{R} u_{0} (x) ϕ (x, 0) d x = 0, \\ \int_{R \times [0, \infty)} (ϕ_{t} + u ϕ_{x}) d ν + \int_{R} ρ_{0} (x) ϕ (x, 0) d x = 0, \end{matrix} \end{matrix}$ for any test function ϕ supported in $R \times [0, \infty)$ .
Theorem 3.5.
For* $u_{l} > u_{r}$ , the pointwise limit u of $u^{ϵ}$ and distributional limit of ρ of $ρ^{ϵ}$ satisfies the equation ( 3.22 ).
Proof.
From Theorem 3.3, the pointwise limit u of $u^{ϵ}$ and the distributional limit ρ of $ρ^{ϵ}$ are given by $\begin{matrix} u (x, t) = \{\begin{matrix} u_{l}, & if x < s t, \\ s, & if x = s t, \\ u_{r}, & if x > s t, \end{matrix} ρ (x, t) = \{\begin{matrix} ρ_{l}, & if x < s t, \\ (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} t δ_{x = s t}, & if x = s t, \\ ρ_{r}, & if x > s t, \end{matrix} \end{matrix}$ where $s = \frac{u_{l} + u_{r}}{2}$ .

Let ϕ be any test function supported in $R \times [0, \infty)$ . It is well known that u satisfies the first equation of (3.22). Now we show that $(u, ρ)$ satisfies the second equation of (3.22). Observe that the limit ρ is a Radon measure and can be written in the following way. $\begin{matrix} ρ = d ν = {ρ_{l} + (ρ_{r} - ρ_{l}) H (x - s t)} d x d t + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} t δ_{x = s t} d t, \end{matrix}$ where H is the Heaviside function and δ is the Dirac delta distribution. Note that u is a Borel measurable function and defined everywhere on the domain. We calculate $\begin{array}{l} \int_{R \times [0, \infty)} ϕ_{t} d ν \\ = \int_{x < s t} ρ_{l} ϕ_{t} d x d t + \int_{x > s t} ρ_{r} ϕ_{t} d x d t + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} \int_{R} t ϕ_{t} (x, t) δ_{x = s t} d t \\ (3.23) & = \int_{x < s t} ρ_{l} ϕ_{t} d x d t + \int_{x > s t} ρ_{r} ϕ_{t} d x d t + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{t} (s t, t) d t . \end{array}$ Using integration by parts for the first two integrals of the equation (3.23), we get $\begin{array}{l} \int_{R \times [0, \infty)} ϕ_{t} d ν \\ = - s ρ_{l} \int_{0}^{\infty} ϕ (s t, t) d t + s ρ_{r} \int_{0}^{\infty} ϕ (s t, t) d t - \int_{R} ρ_{0} (x) ϕ (x, 0) d x \\ + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{t} (s t, t) d t \\ = s (ρ_{r} - ρ_{l}) \int_{0}^{\infty} ϕ (s t, t) d t - \int_{R} ρ_{0} (x) ϕ (x, 0) d x \\ (3.24) & + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{t} (s t, t) d t, \end{array}$ where $ρ_{0} (x) = ρ_{l} + (ρ_{r} - ρ_{l}) H (x)$ . Similarly we calculate $\begin{array}{l} \int_{R \times (0, \infty)} u ϕ_{x} d ν \\ = \int_{x < s t} ρ_{l} u_{l} ϕ_{x} d x d t + \int_{x > s t} ρ_{r} u_{r} ϕ_{x} d x d t + (u_{l} - u_{r}) s \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{x} (s t, t) d t \\ = ρ_{l} u_{l} \int_{0}^{\infty} ϕ (s t, t) d t - ρ_{r} u_{r} \int_{0}^{\infty} ϕ (s t, t) d t + s (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{x} (s t, t) d t \\ (3.25) & = (ρ_{l} u_{l} - ρ_{r} u_{r}) \int_{0}^{\infty} ϕ (s t, t) d t + s (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t ϕ_{x} (s t, t) d t, \end{array}$ where in the third step we used integration by parts. From equations (3.24) and (3.25), we get $\begin{array}{l} \int_{R \times [0, \infty)} (ϕ_{t} + u ϕ_{x}) d ν + \int_{R} ρ_{0} (x) ϕ (x, 0) d x \\ = [s (ρ_{r} - ρ_{l}) + ρ_{l} u_{l} - ρ_{r} u_{r}] \int_{0}^{\infty} ϕ (s t, t) d t \\ + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t (s ϕ_{x} (s t, t) + ϕ_{t} (s t, t)) d t \\ = [s (ρ_{r} - ρ_{l}) + ρ_{l} u_{l} - ρ_{r} u_{r}] \int_{0}^{\infty} ϕ (s t, t) d t \\ + (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2} \int_{0}^{\infty} t \frac{d}{d t} (ϕ (s t, t)) d t \\ = [s (ρ_{r} - ρ_{l}) + ρ_{l} u_{l} - ρ_{r} u_{r} - (u_{l} - u_{r}) \frac{ρ_{l} + ρ_{r}}{2}] \int_{0}^{\infty} ϕ (s t, t) d t = 0 . \end{array}$ This completes the proof. □

4. Entropy and entropy flux pairs

In this section we show that the solution constructed for the system (1.6) for Riemann type data is entropy admissible. For the sake of completeness we start with the following definitions [1] restricted to $2 \times 2$ system, namely $\begin{matrix} (4.1) & \begin{matrix} u_{t} + {(f_{1} (u, ρ))}_{x} = 0, \\ ρ_{t} + {(f_{2} (u, ρ))}_{x} = 0 . \end{matrix} \end{matrix}$

Definition 4.1.
A continuously differentiable function $η : R^{2} \mapsto R$ is called an entropy for the system (4.1) with entropy flux $q : R^{2} \mapsto R$ if $\begin{matrix} D η (u, ρ) . D f (u, ρ) = D q (u, ρ), \end{matrix}$ where $f (u, ρ) = (f_{1} (u, ρ), f_{2} (u, ρ))$ . We say $(η, q)$ as entropy-entropy flux pair of the system (4.1).
Definition 4.2.
A weak solution $(u, ρ)$ of the system (4.1) is called entropy admissible if $\begin{matrix} \iint_{R \times (0, \infty)} η (u, ρ) φ_{t} + q (u, ρ) φ_{x} d x d t ⩾ 0, \end{matrix}$ for every non-negative test function $φ : R \times (0, \infty) \to R$ with compact support in $R \times (0, \infty)$ , where $(η, q)$ is the entropy-entropy flux pair as in Definition 4.1.

For our system (1.6), $f (u, ρ) = (\frac{u^{2}}{2} + ϵ p (ρ), u ρ)$ , where p is of the form (1.5). Therefore $(η, q)$ will be an entropy-entropy flux pair of (1.6) if $\begin{matrix} (\frac{\partial η}{\partial u} u + \frac{\partial η}{\partial ρ} ρ, ϵ ρ e^{ρ} \frac{\partial η}{\partial u} + u \frac{\partial η}{\partial ρ}) = (\frac{\partial q}{\partial u}, \frac{\partial q}{\partial ρ}) . \end{matrix}$ That is, $\begin{matrix} (4.2) & \begin{matrix} \frac{\partial q}{\partial u} = \frac{\partial η}{\partial u} u + \frac{\partial η}{\partial ρ} ρ, \\ \frac{\partial q}{\partial ρ} = ϵ ρ e^{ρ} \frac{\partial η}{\partial u} + u \frac{\partial η}{\partial ρ} . \end{matrix} \end{matrix}$ Eliminating q from (4.2), we have $\begin{matrix} \frac{\partial^{2} η}{\partial ρ^{2}} = ϵ e^{ρ} \frac{\partial^{2} η}{\partial u^{2}} . \end{matrix}$

One can see that $\begin{matrix} η (u, ρ) = \frac{1}{2} u^{2} + ϵ e^{ρ} \end{matrix}$ is a solution of above the equation which is a strictly convex (since $D^{2} η > 0$ ) and the corresponding entropy flux is $\begin{matrix} q (u, ρ) = \frac{1}{3} u^{3} + ϵ ρ u e^{ρ} . \end{matrix}$

We show here that our solution constructed in the previous section for Riemann type initial data ( $u_{l} > u_{r}$ ) is entropy admissible as in Definition 4.1.

We calculate $\begin{array}{l} η_{t} + q_{x} & = - s_{1} (\frac{1}{2} u_{ϵ}^{* 2} + ϵ e^{ρ_{ϵ}^{}} - \frac{1}{2} u_{l}^{2} - ϵ e^{ρ_{l}}) δ_{x = s_{1} t} \\ - s_{2} (\frac{1}{2} u_{r}^{2} + ϵ e^{ρ_{r}} - \frac{1}{2} u_{ϵ}^{ 2} - ϵ e^{ρ_{ϵ}^{}}) δ_{x = s_{2} t} \\ + (\frac{1}{3} u_{ϵ}^{ 3} + ϵ ρ_{ϵ}^{} u_{ϵ}^{} e^{ρ_{ϵ}^{}} - \frac{1}{3} u_{l}^{3} - ϵ ρ_{l} u_{l} e^{ρ_{l}}) δ_{x = s_{1} t} \\ (4.3) & + (\frac{1}{3} u_{r}^{3} - ϵ ρ_{r} u_{r} e^{ρ_{r}} - \frac{1}{3} u_{ϵ}^{ 3} - ϵ ρ_{ϵ}^{} u_{ϵ}^{} e^{ρ_{ϵ}^{}}) δ_{x = s_{2} t}, \end{array}$ where $\begin{array}{l} s_{1} = (\frac{{u^{}}_{ϵ} + u_{l}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{l}))}{u_{ϵ}^{} - u_{l}}), \\ s_{2} = (\frac{{u^{}}_{ϵ} + u_{r}}{2} + \frac{ϵ (p (ρ_{ϵ}^{}) - p (ρ_{r}))}{u_{ϵ}^{} - u_{r}}) . \end{array}$ One should note that to show $η (u, ρ)$ and $q (u, ρ)$ satisfies the entropy inequality for small ϵ, we separately show that the coefficients $δ_{x = s_{1} t}$ and $δ_{x = s_{2} t}$ are negative. Therefore it is enough if the limiting values of the coefficients are negative as ϵ tends to zero. $\begin{array}{l} Coefficient of δ_{x = s_{1} t} \\ (4.4) & = \underset{I}{\underset{︸}{- s_{1} (\frac{1}{2} u_{ϵ}^{ 2} + ϵ e^{ρ_{ϵ}^{}} - \frac{1}{2} u_{l}^{2} - ϵ e^{ρ_{l}})}} + \underset{II}{\underset{︸}{(\frac{1}{3} u_{ϵ}^{ 3} + ϵ ρ_{ϵ}^{} u_{ϵ}^{} e^{ρ_{ϵ}^{}} - \frac{1}{3} u_{l}^{3} - ϵ ρ_{l} u_{l} e^{ρ_{l}})}} . \end{array}$ From (3.15), $\begin{matrix} (4.5) & ϵ e^{ρ_{ϵ}^{}} (ρ_{ϵ}^{} - 1) \to \frac{{(u_{l} - u_{r})}^{2}}{8} as ϵ \to 0 . \end{matrix}$

Since $ρ_{ϵ}^{}$ is unbounded, we get $\begin{matrix} (4.6) & ϵ e^{ρ_{ϵ}^{}} = \frac{ϵ e^{ρ_{ϵ}^{}} (ρ_{ϵ}^{} - 1)}{(ρ_{ϵ}^{} - 1)} \to 0 as ϵ \to 0 . \end{matrix}$

Now using (4.5) and observing that $s_{1} \to \frac{(u_{l} + u_{r})}{2}$ , one can see $\begin{matrix} I \to \frac{- (u_{l} + u_{r}) (3 u_{l} + u_{r}) (u_{r} - u_{l})}{16} as ϵ \to 0 . \end{matrix}$

Again using (4.5) and (4.6), a simple calculation yields $\begin{matrix} II \to \frac{(u_{r} - u_{l}) (7 u_{l}^{2} + 4 u_{l} u_{r} + u_{r}^{2})}{24} + \frac{(u_{r} - u_{l}) (u_{r}^{2} - u_{l}^{2})}{16} as ϵ \to 0 . \end{matrix}$

Therefore from the equation (4.4), $\begin{matrix} Coefficient of δ_{x = s_{1} t} = I + II \to \frac{2 (u_{r} - u_{l}) {(u_{l} - u_{r})}^{2}}{48} as ϵ \to 0 . \end{matrix}$ Since $u_{l} > u_{r}$ , $Coefficient of δ_{x = s_{1} t} = I + II < 0$ for small ϵ. In a similar way the coefficients of $δ_{x = s_{2} t}$ can be handled. Remark 4.3.
It is well known that if η be a smooth entropy of the system (1.6) with the entropy flux q and if one assumes that the Hessian $D^{2} η > 0$ , then for genuinely non-linear characteristic fields the entropy inequality $η {(u)}_{t} + q {(u)}_{x} ⩽ 0$ is satisfied for sufficiently close initial data. For details one can see [1]. But in our case, initial data need not to be sufficiently close. Our proof relies only on the smallness of ϵ.

5. Solution for the case $u_{l} ⩽ u_{r}$

This section is devoted to discuss other two cases, i.e, $u_{l} = u_{r}$ and $u_{l} < u_{r}$ . We assume the same conditions on $p (ρ)$ as in section 3, i.e., $p (ρ)$ is a twice differentiable function and satisfies (1.4). In this section our proof goes in the spirit of [16].

Case I $(u_{l} = u_{r})$ : For $u_{l} = u_{r}$ , initial data is $\begin{matrix} (\begin{matrix} u_{0} (x) \\ ρ_{0} (x) \end{matrix}) = \{\begin{matrix} (\begin{matrix} u_{l} \\ ρ_{l} \end{matrix}), & if x < 0, \\ (\begin{matrix} u_{l} \\ ρ_{r} \end{matrix}), & if x > 0 . \end{matrix} \end{matrix}$ Now if $ρ_{l} = ρ_{r}$ , we have the trivial solution $u (x, t) = u_{l}$ and $ρ (x, t) = ρ_{l}$ . Another two possibilities are $ρ_{r} < ρ_{l}$ or $ρ_{r} > ρ_{l}$ .

Subcase I ( $ρ_{r} < ρ_{l}$ ): For this case, we start traveling from the state $(u_{l}, ρ_{l})$ in the curve $R_{1}$ to reach at $(u_{ϵ}^{*}, ρ_{ϵ}^{*})$ , then from $(u_{ϵ}^{*}, ρ_{ϵ}^{*})$ we travel by $S_{2}$ to reach at $(u_{l}, ρ_{r})$ . 1-rarefaction curve $R_{1}$ through $(u_{l}, ρ_{l})$ is obtained solving the differential equation $\begin{matrix} (5.1) & \frac{d u}{d ρ} = - \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, u (ρ_{l}) = u_{l} . \end{matrix}$ So, the branch of the curve satisfying (5.1) can be parameterized by a continuous function $u_{1} : [ρ_{r}, ρ_{l}] \to [u_{l}, \infty)$ with parameter ρ. Since $p^{'} (ρ) > 0$ , we see that $u_{1}$ is decreasing. Therefore, $u_{1} (ρ_{r}) > u_{l}$ .

Any state $(u, ρ)$ connected to the end state $(u_{l}, ρ_{r})$ by admissible 2-shock curve $S_{2}$ satisfies the following equation: $\begin{matrix} (5.2) & (u - u_{l}) \frac{ρ u - ρ_{r} u_{l}}{ρ - ρ_{r}} = (\frac{u^{2}}{2} + ϵ p (ρ)) - (\frac{u_{l}^{2}}{2} + ϵ p (ρ_{r})), u > u_{l}, and ρ > ρ_{r}, \end{matrix}$ and $\begin{matrix} (5.3) & s > λ_{2} (u, ρ), λ_{1} (u_{l}, ρ_{r}) < s < λ_{2} (u_{l}, ρ_{r}), where s = \frac{ρ u - ρ_{r} u_{l}}{ρ - ρ_{r}} . \end{matrix}$ Equation (5.2) implies $\begin{matrix} (5.4) & {(u - u_{l})}^{2} = 2 ϵ \frac{(ρ - ρ_{r})}{(ρ + ρ_{r})} (p (ρ) - p (ρ_{r})) . \end{matrix}$ Our claim is that for every fixed $ρ > ρ_{r}$ there exists a unique $u > u_{l}$ such that the equation (5.2) holds. Let us define $\begin{matrix} F (u) : = {(u - u_{l})}^{2} . \end{matrix}$ Since $F (u_{l}) = 0$ and $F (u) \to \infty$ as $u \to \infty$ , we have $F ([u_{l}, \infty)) = [0, \infty)$ . Since p is increasing and $ρ > ρ_{r}$ , right-hand side of (5.4) is positive. Therefore for the given $ρ > ρ_{r}$ , there exists a $u > u_{l}$ such that $\begin{matrix} F (u) = 2 ϵ \frac{(ρ - ρ_{r})}{(ρ + ρ_{r})} (p (ρ) - p (ρ_{r})) . \end{matrix}$ Also since $F (u)$ is an increasing function for $u > u_{l}$ , u is unique for the given ρ.

Similarly in Theorem 3.1, the branch of the curve satisfying (5.2) and (5.3) can be parameterized by a $C^{1}$ -function $u_{2} (ρ) = u_{2} : [ρ_{r}, ρ_{l}] \to [u_{l}, \infty)$ satisfying $\begin{matrix} (5.5) & F (u_{2} (ρ)) = {(u_{2} (ρ) - u_{l})}^{2} = 2 ϵ \frac{(ρ - ρ_{r})}{(ρ + ρ_{r})} (p (ρ) - p (ρ_{r})) . \end{matrix}$ Note that $u_{2} (ρ_{r}) = u_{l}$ and it is clear that the function $u_{2}$ is well defined. The function $u_{2}$ is increasing in the interval $(ρ_{r}, ρ_{l})$ . Infact, differentiating the above equation (5.5) we get, $\begin{matrix} (u_{2} (ρ) - u_{l}) {u_{2}}^{'} (ρ) = ϵ [p^{'} (ρ) \frac{ρ - ρ_{r}}{ρ + ρ_{r}} + (p (ρ) - p (ρ_{r})) \frac{2 ρ_{r}}{{(ρ + ρ_{r})}^{2}}] . \end{matrix}$ Since $ρ > ρ_{r}$ and $p (ρ)$ is an increasing function, i.e, $p^{'} (ρ) > 0$ , RHS of above equation is positive for small $ϵ > 0$ . That is, $(u_{2} (ρ) - u_{l}) {u_{2}}^{'} (ρ) > 0$ . Since $u_{2} (ρ) > u_{l}$ , ${u_{2}}^{'} (ρ) > 0$ .

From the above analysis, there exists an intermediate state $ρ_{ϵ}^{*} \in (ρ_{r}, ρ_{l})$ such that $u_{1} (ρ_{ϵ}^{*}) = u_{2} (ρ_{ϵ}^{*}) = u_{ϵ}^{*}$ . Hence the solution for (1.6) is given by: $\begin{matrix} u^{ϵ} = \{\begin{matrix} u_{l}, & if x < λ_{1} (u_{l}, ρ_{l}) t, \\ R_{1}^{u} (x / t) (u_{l}, ρ_{l}), & if λ_{1} (u_{l}, ρ_{l}) t < x < λ_{1} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t, \\ u_{ϵ}^{*}, & if λ_{1} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t < x < \frac{ρ_{r} u_{l} - ρ_{ϵ}^{*} u_{ϵ}^{*}}{ρ_{r} - ρ_{ϵ}^{*}} t, \\ u_{r}, & if x > \frac{ρ_{r} u_{l} - ρ_{ϵ}^{*} u_{ϵ}^{*}}{ρ_{r} - ρ_{ϵ}^{*}} t \end{matrix} \end{matrix}$ and $\begin{matrix} ρ^{ϵ} = \{\begin{matrix} ρ_{l}, & if x < λ_{1} (u_{l}, ρ_{l}) t, \\ R_{1}^{ρ} (x / t) (u_{l}, ρ_{l}), & if λ_{1} (u_{l}, ρ_{l}) t < x < λ_{1} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t, \\ ρ_{ϵ}^{*}, & if λ_{1} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t < x < \frac{ρ_{r} u_{l} - ρ_{ϵ}^{*} u_{ϵ}^{*}}{ρ_{r} - ρ_{ϵ}^{*}} t, \\ ρ_{r}, & if x > \frac{ρ_{r} u_{l} - ρ_{ϵ}^{*} u_{ϵ}^{*}}{ρ_{r} - ρ_{ϵ}^{*}} t . \end{matrix} \end{matrix}$ Where $R_{1} (ξ) (\bar{u}, \bar{ρ}) = (R_{1}^{u} (ξ) (\bar{u}, \bar{ρ}), R_{1}^{ρ} (ξ) (\bar{u}, \bar{ρ}))$ and $R_{1}^{u} (ξ) (\bar{u}, \bar{ρ})$ is obtained by solving $\begin{matrix} \frac{d u}{d ξ} = \frac{2 p^{'} (ρ)}{3 p^{'} (ρ) + ρ p^{″} (ρ)}, u (λ_{1} (\bar{u}, \bar{ξ})) = \bar{u} \end{matrix}$ and $R_{1}^{ρ} (ξ) (\bar{u}, \bar{ρ})$ is obtained by solving $\begin{matrix} \frac{d ρ}{d ξ} = - \frac{2 \sqrt{ϵ p^{'} (ρ) ρ}}{3 ϵ p^{'} (ρ) + ϵ ρ p^{″} (ρ)}, ρ (λ_{1} (\bar{u}, \bar{ξ})) = \bar{ρ} . \end{matrix}$ Subcase II ( $ρ_{l} < ρ_{r}$ ): This can be handled in a similar way. In fact, here we start from $(u_{l}, ρ_{l})$ and reach at $(u_{ϵ}^{*}, ρ_{ϵ}^{*})$ by $S_{1}$ and from $(u_{ϵ}^{*}, ρ_{ϵ}^{*})$ to $(u_{l}, ρ_{r})$ by $R_{2}$ . So, the solution is given by: $\begin{matrix} u^{ϵ} = \{\begin{matrix} u_{l}, & if x < \frac{ρ_{ϵ}^{*} u_{ϵ}^{*} - ρ_{l} u_{l}}{ρ_{ϵ}^{*} - ρ_{l}} t, \\ u_{ϵ}^{*}, & if \frac{ρ_{ϵ}^{*} u_{ϵ}^{*} - ρ_{l} u_{l}}{ρ_{ϵ}^{*} - ρ_{l}} t < x < λ_{2} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t, \\ R_{2}^{u} (x / t) (u_{ϵ}^{*}, ρ_{ϵ}^{*}), & if λ_{2} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t < x < λ_{2} (u_{l}, ρ_{r}) t, \\ u_{r}, & if x > λ_{2} (u_{l}, ρ_{r}) t \end{matrix} \end{matrix}$ and $\begin{matrix} ρ^{ϵ} = \{\begin{matrix} ρ_{l}, & if x < \frac{ρ_{ϵ}^{*} u_{ϵ}^{*} - ρ_{l} u_{l}}{ρ_{ϵ}^{*} - ρ_{l}} t, \\ ρ_{ϵ}^{*}, & if \frac{ρ_{ϵ}^{*} u_{ϵ}^{*} - ρ_{l} u_{l}}{ρ_{ϵ}^{*} - ρ_{l}} t < x < λ_{2} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t, \\ R_{2}^{ρ} (x / t) (u_{ϵ}^{*}, ρ_{ϵ}^{*}), & if λ_{2} (u_{ϵ}^{*}, ρ_{ϵ}^{*}) t < x < λ_{2} (u_{l}, ρ_{r}) t, \\ ρ_{r}, & if x > λ_{2} (u_{l}, ρ_{r}) t, \end{matrix} \end{matrix}$ where $R_{2} (ξ) (\bar{u}, \bar{ρ}) = (R_{2}^{u} (ξ) (\bar{u}, \bar{ρ}), R_{2}^{ρ} (ξ) (\bar{u}, \bar{ρ}))$ and $R_{2}^{u} (ξ) (\bar{u}, \bar{ρ})$ is obtained by solving $\begin{matrix} \frac{d u}{d ξ} = \frac{2 p^{'} (ρ)}{3 p^{'} (ρ) + ρ p^{″} (ρ)}, u (λ_{2} (\bar{u}, \bar{ξ})) = \bar{u} \end{matrix}$ and $R_{2}^{ρ} (ξ) (\bar{u}, \bar{ρ})$ is obtained by solving $\begin{matrix} \frac{d ρ}{d ξ} = \frac{2 \sqrt{ϵ p^{'} (ρ) ρ}}{3 ϵ p^{'} (ρ) + ϵ ρ p^{″} (ρ)}, ρ (λ_{2} (\bar{u}, \bar{ξ})) = \bar{ρ} . \end{matrix}$ Now our aim is to find the limit of $(u^{ϵ}, ρ^{ϵ})$ as $ϵ \to 0$ in both of the above cases. Since $ρ_{ϵ}^{*} \in (ρ_{l}, ρ_{r})$ or $ρ_{ϵ}^{*} \in (ρ_{r}, ρ_{l})$ this implies $ρ_{ϵ}^{*}$ is bounded. Also $ρ_{ϵ}^{*}$ and $u_{ϵ}^{*}$ satisfies $\begin{matrix} {(u_{ϵ}^{*} - u_{l})}^{2} \frac{(ρ_{ϵ}^{*} + ρ_{r})}{2} = ϵ (ρ_{ϵ}^{*} - ρ_{r}) (p (ρ_{ϵ}^{*}) - p (ρ_{r})) . \end{matrix}$ Since RHS is bounded, as $ϵ \to 0$ we get, $\begin{matrix} lim_{ϵ \to 0} {(u_{ϵ}^{*} - u_{l})}^{2} \frac{(ρ_{ϵ}^{*} + ρ_{r})}{2} = 0, \end{matrix}$ that is, ${lim}_{ϵ \to 0} u_{ϵ}^{*} = u_{l}$ . Therefore the solution $(u^{ϵ}, ρ^{ϵ}) \to (u, ρ)$ as $ϵ \to 0$ where $(u, ρ)$ is given by: $\begin{matrix} ρ = \{\begin{matrix} ρ_{l}, & if x < u_{l} t, \\ ρ_{r}, & if x > u_{l} t \end{matrix} \end{matrix}$ and $\begin{matrix} u = \{\begin{matrix} u_{l}, & if x < u_{l} t, \\ u_{r}, & if x > u_{l} t . \end{matrix} \end{matrix}$ Since here $u_{l} = u_{r}$ we have $u \equiv u_{l}$ .

Case II $(u_{l} < u_{r})$ : The 1st-rarefaction curve passing through $(u_{l}, ρ_{l})$ is given by the solution of the following Cauchy problem: $\begin{matrix} \frac{d u}{d ρ} = - \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, ρ < ρ_{l}, u (ρ_{l}) = u_{l} . \end{matrix}$ Note that for this case it does not matter whether $ρ_{l} < ρ_{r}$ or $ρ_{l} > ρ_{r}$ . So, without loss of generality we assume $ρ_{l} > ρ_{r} > 0$ . Now a branch of $R_{1}$ can be parameterized by a differentiable function $u_{1} : [0, ρ_{l}] \to [u_{l}, \infty)$ with a parameter ρ. Explicitly $u_{1}$ can be written as $\begin{matrix} u_{1} (ρ) - u_{l} = - \int_{ρ_{l}}^{ρ} \sqrt{\frac{ϵ p^{'} (ξ)}{ξ}} d ξ . \end{matrix}$ Since $ρ \in [0, ρ_{l}]$ is bounded and p is increasing, we have $u_{1} (ρ) \to u_{l}$ as $ϵ \to 0$ decreasingly. Similarly, the 2nd-rarefaction curve is given by the solution of then Cauchy problem: $\begin{matrix} (5.6) & \frac{d u}{d ρ} = \sqrt{\frac{ϵ p^{'} (ρ)}{ρ}}, ρ < ρ_{r}, u (ρ_{r}) = u_{r} . \end{matrix}$ Let $u_{2} : [0, ρ_{r}] \to (- \infty, u_{r}]$ is differentiable and parameterized branch of $R_{2}$ satisfying (5.6) and can be written as $\begin{matrix} u_{2} (ρ) - u_{r} = \int_{ρ}^{ρ_{r}} \sqrt{\frac{ϵ p^{'} (ξ)}{ξ}} d ξ . \end{matrix}$ Since $ρ \in [0, ρ_{r}]$ and p is increasing, we have $u_{2} (ρ) \to u_{r}$ as $ϵ \to 0$ increasingly. Since $u_{l} < u_{r}$ , by the above calculation one can see $u_{1} (0) < u_{2} (0)$ for small ϵ. In this case the complete solution is the following: $\begin{matrix} (5.7) & u^{ϵ} = \{\begin{matrix} u_{l}, & if x < λ_{1} (u_{l}, ρ_{l}) t, \\ R_{1}^{u} (x / t) (u_{l}, ρ_{l}), & if λ_{1} (u_{l}, ρ_{l}) t < x < λ_{1} (u_{ϵ}^{* (1)}, 0) t, \\ x / t, & if λ_{1} (u_{ϵ}^{* (1)}, 0) t < x < λ_{2} (u_{ϵ}^{* (2)}, 0) t, \\ R_{2}^{u} (x / t) (u_{ϵ}^{* (2)}, 0), & if λ_{2} (u_{ϵ}^{* (2)}, 0) t < x < λ_{2} (u_{r}, ρ_{r}) t, \\ u_{r}, & if x > λ_{2} (u_{r}, ρ_{r}) t \end{matrix} \end{matrix}$ and $\begin{matrix} ρ^{ϵ} = \{\begin{matrix} ρ_{l}, & if x < λ_{1} (u_{l}, ρ_{l}) t, \\ R_{1}^{ρ} (x / t) (u_{l}, ρ_{l}), & if λ_{1} (u_{l}, ρ_{l}) t < x < λ_{1} (u_{ϵ}^{* (1)}, 0) t, \\ 0, & if λ_{1} (u_{ϵ}^{* (1)}, 0) t < x < λ_{2} (u_{ϵ}^{* (2)}, 0) t, \\ R_{2}^{ρ} (x / t) (u_{ϵ}^{* (2)}, 0), & if λ_{2} (u_{ϵ}^{* (2)}, 0) t < x < λ_{2} (u_{r}, ρ_{r}) t, \\ ρ_{r}, & if x > λ_{2} (u_{r}, ρ_{r}) t, \end{matrix} \end{matrix}$ where $R_{1}^{u} (\cdot)$ , $R_{1}^{ρ} (\cdot)$ , $R_{2}^{u} (\cdot)$ , $R_{2}^{ρ} (\cdot)$ are as previously.

Now it remains to find the limit of $(u^{ϵ}, ρ^{ϵ})$ as $ϵ \to 0$ . Since $u_{ϵ}^{* (1)} = u_{1} (0)$ , we have $u_{ϵ}^{* (1)} \to u_{l}$ and in the same way $u_{ϵ}^{* (2)} \to u_{r}$ as $ϵ \to 0$ . Passing to the limit as ϵ tends to zero, we get $\begin{matrix} u (x, t) = \{\begin{matrix} u_{l}, & if x < u_{l} t, \\ x / t, & if u_{l} t < x < u_{r} t, \\ u_{r}, & if x > u_{r} t \end{matrix} \end{matrix}$ and $\begin{matrix} ρ (x, t) = \{\begin{matrix} ρ_{l}, & if x < u_{l} t, \\ 0, & if u_{l} t < x < u_{r} t, \\ ρ_{r}, & if x > u_{r} t . \end{matrix} \end{matrix}$

Remark 5.1.
In equation (5.7), one has to take $u^{ϵ} (x, t) = \frac{x}{t}$ in the region $λ_{1} (u_{ϵ}^{* (1)}, 0) t < x < λ_{2} (u_{ϵ}^{* (2)}, 0) t$ . This kind of selection gives unique entropy solution. In fact, since $ρ = 0$ in this region, the first equation of (1.6) turns out to be the Burgers equation and $u (x, t) = \frac{x}{t}$ is the unique entropy solution for the rarefaction case [9].

6. Limiting behavior of another strictly hyperbolic model: Shadow waves

Aim of this section is to study the limiting behavior of the solution for the following strictly hyperbolic system. $\begin{matrix} (6.1) & \begin{matrix} u_{t} + {(\frac{{(u + ϵ)}^{2}}{2})}_{x} = 0, \\ ρ_{t} + {(ρ u)}_{x} = 0, \end{matrix} \end{matrix}$ with Riemann type initial data: $\begin{matrix} (6.2) & (\begin{matrix} u_{0} (x) \\ ρ_{0} (x) \end{matrix}) = \{\begin{matrix} (\begin{matrix} u_{l} \\ ρ_{l} \end{matrix}), & if x < 0, \\ (\begin{matrix} u_{r} \\ ρ_{r} \end{matrix}), & if x > 0 . \end{matrix} \end{matrix}$ In compare to the previous system dealt in the first part of this article, this system is very much different in nature and only retains the strict hyperbolicity property.

The eigenvalues and the eigenvectors for the system (6.1) are the following:

The first eigenvalue $λ_{1} (u) = u$ and the corresponding eigenvector is $r_{1} (u) = (0, 1)$ and the second eigenvalue $λ_{2} (u) = u + ϵ$ and the corresponding eigenvector is $r_{1} (u) = (1, ρ / ϵ)$ . Again, $\nabla λ_{1} (u) . r_{1} (u) = 0$ and $\nabla λ_{2} (u) . r_{2} (u) = 1$ . So, the first characteristic field is linearly degenerate and the second characteristic field is genuinely nonlinear, where as the previous system is genuinely nonlinear in both of the characteristic fields. The main difficulty is that for certain cases, Lax type solutions don’t exist. In those cases we use a recent technique introduced in [17] called Shadow Wave solution. Now we describe explicitly shock and rarefaction curves for the system (6.1).

1-rarefaction curve: 1-rarefaction curve is the solution of the ODE; $\begin{matrix} \dot{w} (ξ) = r_{1} (w (ξ)), w (λ_{1} (u_{l}, ρ_{l})) = (u_{l}, ρ_{l}), \end{matrix}$ where $w (ξ) = (w_{1} (ξ), w_{2} (ξ))$ . So, solving the following pair of ODEs, $\begin{array}{l} \dot{w_{1}} (ξ) = 0, w_{1} (u_{l}) = u_{l}, \\ \dot{w_{2}} (ξ) = 1, w_{2} (u_{l}) = ρ_{l}, \end{array}$ we get the 1-rarefaction curve $R_{1}$ passing through $(u_{l}, ρ_{l})$ , $\begin{matrix} R_{1} (ξ) = (u_{l}, ξ + ρ_{l} - u_{l}) . \end{matrix}$ 2-rarefaction curve: 2-rarefaction curve is the solution of the ODE; $\begin{matrix} \dot{w} (ξ) = r_{2} (w (ξ)), w (λ_{2} (u_{l}, ρ_{l})) = (u_{l}, ρ_{l}), \end{matrix}$ where $w (ξ) = (w_{1} (ξ), w_{2} (ξ))$ .

This gives the following system of ODEs with initial conditions. $\begin{array}{l} \dot{w_{1}} (ξ) = 1, w_{1} (u_{l} + ϵ) = u_{l}, \\ \dot{w_{2}} (ξ) = \frac{w_{2} (ξ)}{ϵ}, w_{2} (u_{l} + ϵ) = ρ_{l} . \end{array}$ Solving the above pair of ODEs, we get the 2-rarefaction curve $R_{2}$ passing through $(u_{l}, ρ_{l})$ . $\begin{matrix} (6.3) & R_{2} (ξ) = (ξ - ϵ, ρ_{l} . exp (\frac{ξ - (u_{l} + ϵ)}{ϵ})) . \end{matrix}$ The admissibe part of the curve is $R_{2}^{+}$ . $\begin{matrix} (6.4) & R_{2}^{+} (ξ) = (ξ - ϵ, ρ_{l} . exp (\frac{ξ - (u_{l} + ϵ)}{ϵ})), ξ > u_{l} + ϵ . \end{matrix}$

Since the first characteristics field is linearly degenerate, the 1st-Shock curve and the 1st-Rarefaction curve will coincide, i.e., $R_{1} (ξ) = S_{1} (ξ)$ .

Admissible 2-shock curve: Admissible 2-shock curve passing through $(u_{l}, ρ_{l})$ is given by: $\begin{matrix} ρ (u) = \frac{ρ_{l} (\frac{u - u_{l}}{2} + ϵ)}{ϵ - \frac{u - u_{l}}{2}}, u < u_{l}, u_{l} - u ⩽ 2 ϵ . \end{matrix}$ Now, a brief description of the concept of Shadow waves is given bellow.

Definition 6.1.

Let $f \in C (Ω : R^{n})$ and $U : R_{+}^{2} \to Ω \subset R^{n}$ be a piecewise constant function given by $\begin{matrix} U^{η} (x, t) = (u^{η}, ρ^{η}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < (c - a_{η}) t, \\ (u_{1}^{η}, ρ_{1}^{η}), & if (c - a_{η}) t < x < c t, \\ (u_{2}^{η}, ρ_{2}^{η}), & if c t < x < (c + b_{η}) t, \\ (u_{r}, ρ_{r}), & if x > (c + b_{η}) t . \end{matrix} \end{matrix}$ If ${(U^{η})}_{t} + (f {(U^{η})}_{x} \to 0$ in the sense of distribution, then $U^{η}$ is called a Shadow Wave solution to the conservation laws $\begin{matrix} U_{t} + f {(U)}_{x} = 0 . \end{matrix}$

Main result of this section is the following.

Theorem 6.2.

The solutions $(u^{ϵ}, ρ^{ϵ})$ of the system ( 6.1 ) with Riemann type initial data ( 6.2 ) are given below case by case:

For $u_{l} < u_{r}$ , $\begin{matrix} (6.5) & (u^{ϵ}, ρ^{ϵ}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < u_{l} t, \\ (u_{l}, ρ_{r} . exp (\frac{(u_{l} - u_{r})}{ϵ})), & if u_{l} t < x < (u_{l} + ϵ) t, \\ (x / t - ϵ, ρ_{r} . exp (\frac{x / t - (u_{r} + ϵ)}{ϵ})), & if (u_{l} + ϵ) t < x < (u_{r} + ϵ) t, \\ (u_{r}, ρ_{r}), & if x > (u_{r} + ϵ) t . \end{matrix} \end{matrix}$ For $u_{l} = u_{r}$ , the solution is given by $\begin{matrix} (6.6) & (u^{ϵ}, ρ^{ϵ}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < u_{l} t, \\ (u_{l}, ρ_{r}), & if u_{l} t < x < (u_{l} + ϵ) t, \\ (u_{l}, ρ_{r}), & if x > (u_{l} + ϵ) t . \end{matrix} \end{matrix}$ If $u_{l} > u_{r}$ and $u_{l} - u_{r} ⩽ 2 ϵ$ , then the solution is given by: $\begin{matrix} (6.7) & (u^{ϵ}, ρ^{ϵ}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < u_{l} t, \\ (u_{l}, ρ^{⋆}), & if u_{l} t < x < (\frac{u_{l} + u_{r}}{2} + ϵ) t, \\ (u_{r}, ρ_{r}), & if x > (\frac{u_{l} + u_{r}}{2} + ϵ) t . \end{matrix} \end{matrix}$ where $\begin{matrix} ρ^{⋆} = ρ_{r} \frac{\frac{u_{l} - u_{r}}{2} + ϵ}{\frac{u_{r} - u_{l}}{2} + ϵ} . \end{matrix}$

If $u_{l} > u_{r}$ and $u_{l} - u_{r} > 2 ϵ$ , then the system admits shadow wave solution ${(u^{η, ϵ}, ρ^{η, ϵ})}_{η > 0}$ . The pointwise limit of $u^{η, ϵ}$ is $u^{ϵ}$ and the distributional limit of $ρ^{η, ϵ}$ is $ρ^{ϵ}$ and are given as follows: $\begin{array}{l} (6.8) & u^{ϵ} (x, t) = \{\begin{matrix} u_{l}, & if x < (\frac{u_{l} + u_{r}}{2} + ϵ) t, \\ \frac{u_{l} + u_{r}}{2} + ϵ, & if x = (\frac{u_{l} + u_{r}}{2} + ϵ) t, \\ u_{r}, & if x > (\frac{u_{l} + u_{r}}{2} + ϵ) t, \end{matrix} \\ (6.9) & ρ^{ϵ} = \{\begin{matrix} ρ_{l}, & if x < (\frac{u_{l} + u_{r}}{2} + ϵ) t, \\ (\frac{(u_{l} - u_{r}) (ρ_{l} + ρ_{r})}{2} + ϵ (ρ_{r} - ρ_{l})) t δ_{x = (\frac{u_{l} + u_{r}}{2} + ϵ) t}, & if x = (\frac{u_{l} + u_{r}}{2} + ϵ) t, \\ ρ_{r}, & if x > (\frac{u_{l} + u_{r}}{2} + ϵ) t . \end{matrix} \end{array}$ Further $(u^{ϵ}, ρ^{ϵ})$ satisfies the equation ( 6.1 ) with initial data ( 6.2 ) in the sense of Definition 3.4 .

Proof.

Case 1: $u_{l} < u_{r}$ : The state $(u_{l}, ρ_{l})$ can be joined to $(u_{l}, ρ_{ϵ}^{*})$ by 1-shock curve and $(u_{l}, ρ_{ϵ}^{*})$ can be joined to $(u_{r}, u_{r})$ by 2-rarefaction curve. Then by (6.4), $(u_{l}, ρ_{ϵ}^{*})$ will satisfy the following equations. $\begin{matrix} u_{r} = ξ - ϵ, ρ_{r} = ρ_{ϵ}^{*} . exp (\frac{ξ - (u_{l} + ϵ)}{ϵ}) . \end{matrix}$ Which yields $\begin{matrix} ξ = u_{r} + ϵ, ρ_{ϵ}^{*} = ρ_{r} . exp (\frac{(u_{l} - u_{r})}{ϵ}) . \end{matrix}$

So the solution for the perturbed problem is given by: $\begin{matrix} (6.10) & (u^{ϵ}, ρ^{ϵ}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < λ_{1} (u_{l}, ρ_{l}) t, \\ (u_{l}, ρ_{ϵ}^{*}), & if λ_{1} (u_{l}, ρ_{l}) t < x < λ_{2} (u_{l}, ρ_{ϵ}^{*}) t, \\ R_{2} (ξ) (u_{l}, ρ_{ϵ}^{*}), & if λ_{2} (u_{l}, ρ_{ϵ}^{*}) < x < λ_{2} (u_{r}, ρ_{r}) t, \\ (u_{r}, ρ_{r}), & if x > λ_{2} (u_{r}, ρ_{r}) t, \end{matrix} \end{matrix}$ where $λ_{2} (R_{2} (ξ) (u_{l}, ρ_{l}) = x / t$ , i.e, $ξ = x / t$ . Using equation (6.4) and putting the values of $λ_{1}$ and $λ_{2}$ yields (6.5).

Case 2: $u_{l} = u_{r}$ : The state $(u_{l}, ρ_{l})$ is connected to $(u_{l}, ρ_{r})$ by 1-shock and $(u_{l}, ρ_{r})$ to $(u_{l}, ρ_{r})$ by 2-shock, $\begin{matrix} (6.11) & (u^{ϵ}, ρ^{ϵ}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < u_{l} t, \\ (u_{l}, ρ_{r}), & if u_{l} t < x < (u_{l} + ϵ) t, \\ (u_{l}, ρ_{r}), & if x > (u_{l} + ϵ) t . \end{matrix} \end{matrix}$

Case 3: $u_{l} > u_{r}$ , $u_{l} - u_{r} ⩽ 2 ϵ$ : In this case $(u_{l}, ρ_{l})$ can be connected $(u_{l}, ρ^{*})$ by 1-shock and $(u_{l}, ρ^{*})$ can be connected to $(u_{r}, ρ_{r})$ by 2-shock and a simple calculation yields $\begin{matrix} ρ^{⋆} = ρ_{r} \frac{\frac{u_{l} - u_{r}}{2} + ϵ}{\frac{u_{r} - u_{l}}{2} + ϵ} . \end{matrix}$

Case 4: $u_{l} > u_{r}$ , $u_{l} - u_{r} > 2 ϵ$ : In this case Lax method cannot be used. This situation is handled by using shadow wave approach. Use ansatz, $\begin{matrix} (u^{η}, ρ^{η}) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < (c - η) t, \\ (u, \frac{ρ}{η}), & if (c - η) t < x < (c + η) t, \\ (u_{r}, ρ_{r}), & if x > (c + η) t . \end{matrix} \end{matrix}$ We want to determine c, u and ρ such that the following limits hold in the sense of distribution. As $η \to 0$ , $\begin{matrix} \{\begin{matrix} {(u^{η})}_{t} + {(\frac{{(u^{η} + ϵ)}^{2}}{2})}_{x} ⇀ 0, \\ {(ρ^{η})}_{t} + {(u^{η} ρ^{η})}_{x} ⇀ 0 . \end{matrix} \end{matrix}$ That is, as $η \to 0$ , $\begin{matrix} (6.12) & \{\begin{matrix} ⟨ {(u^{η})}_{t} + {(\frac{{(u^{η} + ϵ)}^{2}}{2})}_{x}, φ ⟩ \to 0, \\ ⟨ {(ρ^{η})}_{t} + {(u^{η} ρ^{η})}_{x}, φ ⟩ \to 0, \end{matrix} \end{matrix}$ for all test function $φ \in C_{c}^{\infty} (R \times (0, \infty))$ . Now we calculate $\begin{array}{rcl} ⟨ {(u^{η})}_{t} + {(\frac{{(u^{η} + ϵ)}^{2}}{2})}_{x}, φ ⟩ & = & - (c - η) (u - u_{l}) \int_{0}^{\infty} φ ((c - η) t, t) d t \\ - (c + η) (u_{r} - u) \int_{0}^{\infty} φ ((c + η) t, t) d t \\ + (\frac{{(u + ϵ)}^{2}}{2} - \frac{{(u_{l} + ϵ)}^{2}}{2}) \int_{0}^{\infty} φ ((c - η) t, t) d t \\ (6.13) & + (\frac{{(u_{r} + ϵ)}^{2}}{2} - \frac{{(u + ϵ)}^{2}}{2}) \int_{0}^{\infty} φ ((c + η) t, t) d t . \end{array}$ Since φ has compact support, passing to the limit as η tends to 0, and using the first equation of (6.12) and equation (6.13), we get $\begin{matrix} (6.14) & [- c (u_{r} - u_{l}) + \frac{{(u_{r} + ϵ)}^{2}}{2} - \frac{{(u_{l} + ϵ)}^{2}}{2}] \int_{0}^{\infty} φ (c t, t) d t = 0 . \end{matrix}$ But equation (6.14) is true for all φ having compact support in $R \times (0, \infty)$ . Therefore $\begin{matrix} c = \frac{u_{l} + u_{r}}{2} + ϵ . \end{matrix}$ Next, $\begin{array}{rcl} ⟨ {(ρ^{η})}_{t} + {(u^{η} ρ^{η})}_{x}, φ ⟩ & = & - (c - η) (\frac{ρ}{η} - ρ_{l}) \int_{0}^{\infty} φ ((c - η) t, t) d t \\ - (c + η) (ρ_{r} - \frac{ρ}{η}) \int_{0}^{\infty} φ ((c + η) t, t) d t \\ + (\frac{u ρ}{η} - u_{l} ρ_{l}) \int_{0}^{\infty} φ ((c - η) t, t) d t \\ (6.15) & + (u_{r} ρ_{r} - \frac{u ρ}{η}) \int_{0}^{\infty} φ ((c + η) t, t) d t . \end{array}$ Taylor expansions of the functions $φ ((c - η) t, t)$ and $φ ((c + η) t, t)$ about $(c t, t)$ are $\begin{array}{l} φ ((c - η) t, t) = φ (c t, t) - η t φ_{x} ((c t, t) + η^{2} / 2 \int_{0}^{t} φ_{x x} (c t - η r, t) {(t - r)}^{2} d r, \\ φ ((c + η) t, t) = φ (c t, t) + η t φ_{x} ((c t, t) + η^{2} / 2 \int_{0}^{t} φ_{x x} (c t + η r, t) {(t - r)}^{2} d r . \end{array}$ Using the expansions in (6.15), we get $\begin{array}{l} ⟨ {(ρ^{η})}_{t} + {(u^{η} ρ^{η})}_{x}, φ ⟩ \\ = [- (c - η) (ρ / η - ρ_{l}) - (c + η) (ρ_{r} - ρ / η) - u_{l} ρ_{l} + u_{r} ρ_{r}] \int_{0}^{\infty} φ (c t, t) d t \\ + t [(c - η) (ρ - ρ_{l} η) - (u ρ - η u_{l} ρ_{l}) - (c + η) (η ρ_{r} - ρ) + (η u_{r} ρ_{r} - u ρ)] \\ (6.16) & \times \int_{0}^{\infty} φ_{x} (c t, t) d t + O (η) . \end{array}$ Passing to the limit as $η \to 0$ in the equation (6.16) and comparing the coefficients of the integrals as above, we get $\begin{matrix} u = c and ρ = \frac{u_{l} ρ_{l} - u_{r} ρ_{r} - c (ρ_{l} - ρ_{r})}{2} . \end{matrix}$ Now (6.8) and (6.9) follows easily. $(u, ρ)$ given in (6.8) and (6.9) satisfies the integral formulation (3.22) in Definition 3.4 is exactly similar to the proof of Theorem 3.5 and is omitted. This completes the proof. □

Remark 6.3.

From the formula for $(u^{ϵ}, ρ^{ϵ})$ given in the Theorem 6.2, one can easily verify that for the case $u_{l} ⩽ u_{r}$ , the distributional limit of $(u^{ϵ}, ρ^{ϵ})$ as ϵ tends to zero, converges to $\begin{matrix} (u (x, t), ρ (x, t)) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < u_{l} t, \\ (x / t, 0), & if u_{l} t < x < u_{r} t, \\ (u_{r}, ρ_{r}), & if x > u_{r} t \end{matrix} \end{matrix}$ and for the case $u_{l} > u_{r}$ , the distributional limit of $(u^{ϵ}, ρ^{ϵ})$ as ϵ tends to zero, converges to $\begin{matrix} (u, ρ) (x, t) = \{\begin{matrix} (u_{l}, ρ_{l}), & if x < (\frac{u_{l} + u_{r}}{2}) t, \\ (\frac{u_{l} + u_{r}}{2}, \frac{(u_{l} - u_{r}) (ρ_{l} + ρ_{r})}{2} δ_{x = (\frac{u_{l} + u_{r}}{2}) t}), & if x = (\frac{u_{l} + u_{r}}{2}) t, \\ (u_{r}, ρ_{r}), & if x > (\frac{u_{l} + u_{r}}{2}) t \end{matrix} \end{matrix}$ in the sense of distribution. This is the vanishing viscosity limit for the large scale structure formation of the universe, see [10,12]. This limit satisfies the equation (1.7) is already proved in Theorem 3.5.

Footnotes

Acknowledgements

The authors wish to express their sincere gratitude to anonymous referees for their valuable suggestions and comments.

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Limiting behavior of some strictly hyperbolic systems of conservation laws

Abstract

Keywords

1. Introduction

2. The Riemann solution

3. Formation of shock waves for u l > u r

Footnotes

Acknowledgements

References

3. Formation of shock waves for $u_{l} > u_{r}$