UNIQUENESS AND ASYMPTOTIC STABILITY OF RIEMANN SOLUTIONS FOR THE COMPRESSIBLE

EULER EQUATIONS

GUI-QIANG CHEN AND HERMANO FRID

Abstract. We prove the uniqueness of Riemann solutions in the class of entropy solutions in L^{∞} ∩ BV_{loc} for the 3 × 3 system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global L^{2}-stability of the Riemann solutions even in the class of entropy solutions in L^{∞} with arbitrarily large oscillation for the 3 × 3 system. We apply the framework established in [4, 5] to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under L^{1} perturbation of the Riemann initial data, as long as the corresponding solutions are in L^{∞} and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x, t), piecewise Lipschitz in x, for any t > 0.

Descriptive Title: Entropy Solutions for the Compressible Euler Equations

1. Introduction

We are concerned with the large-time behavior of entropy solutions in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}), ℝ_{+}^{2} = ℝ×[0, ∞), for the 3×3 system of Euler equations for compressible fluids with a general pressure law, whose initial data are an L^{1} perturbation of Riemann initial data. More specifically, for any entropy solution U(x, t) ∈ L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) which represents the evolution through the Euler equations of perturbed Riemann initial data corresponding to the Riemann solution R(x/t), the problem is whether U(ξt, t) → R(ξ) in L_{loc}^{1}(ℝ) as t → ∞. As shown in [4, 5], this problem can be reduced to proving the uniqueness of the Riemann solution in the class of entropy solutions in L^{∞} ∩BV_{loc}(ℝ_{+}^{2}), provided that U(x, t) satisfies certain natural growth condition for its local total variation (see (1.23)), which guarantees the L_{loc}^{1} compactness of the associated scaling sequence, U^{T} (x, t) = U(T x, T t).

1991 Mathematics Subject Classification. Primary:35B40,35L65; Secondary:35B35,76N15. Key words and phrases. compressible Euler equations, discontinuous entropy solutions, Rie-

mann solutions, uniqueness, asymptotic stability, scaling sequence, compactness, hyperbolic conservation laws.

.

The Euler system for compressible fluids in Lagrangian coordinates reads

\begin{cases}∂_{t}u + ∂_{x}p = 0, \\ ∂_{t}v − ∂_{x}u = 0, \\ ∂_{t}(e + \frac{u^{2}}{2} ) + ∂_{x}(pu) = 0, \end{cases} (1.1)

where u, p, v, and e represent the velocity, the pressure, the specific volume (v = 1/ρ, ρ the density), and the internal energy of the fluids, respectively. Other important physical variables are the temperature θ and the entropy S. To close system (1.1), one needs the basic law of thermodynamics which translates into the differential equation:

de = θ dS − p dv. (1.2)

It follows from relation (1.2) that any two of the variables v, p, e, θ, and S determine the other three, through the constitutive relations. We then choose v and S as the independent variables to obtain the equations of state p = p(v, S), θ = θ(v, S), and e = e(v, S). These functions must be compatible with (1.2):

\begin{cases}∂_{v}e(v, S) = −p(v, S), \\ ∂_{S}e(v, S) = θ(v, S), \\ ∂_{v}θ(v, S) = −∂_{S}p(v, S). \end{cases} (1.3)

For ideal polytropic gases, pv = Rθ, e = c_{v}θ, where R and c_{v} are positive constants, and the choice of v and S as the independent variables implies

p(v, S) = κe^{S/c_{v}}v^{−γ}, γ = 1 + \frac{R}{c_{v}} > 1. (1.4)

Using (1.2), any smooth solution of (1.1) is also a smooth solution of the system

\begin{cases}∂_{t}u + ∂_{x}p = 0, \\ ∂_{t}v − ∂_{x}u = 0, \\ ∂_{t}S = 0. \end{cases} (1.5)

However, for discontinuous solutions of (1.1), the last equation of (1.5) no longer holds, even in the weak sense, and it must be replaced by the so called Clausius inequality

∂_{t}a(S) ≥ 0 (1.6)

in the sense of distributions for any C^{1} function a(S) with a′(S) ≥ 0.

System (1.1) can be written in the general form:

∂_{t}U + ∂_{x}F (U) = 0, (1.7)

by setting U = (u, v, E), with E = \frac{1}{2}u^{2} + e, and F (U) = (p, −u, pu), with p as a function of (u, v, E).

Recall that the Riemann problem for a general system (1.7) is a special Cauchy problem with initial data:

U|_{t=0} ≡ R_{0}(x) = \begin{cases}U_{L}, & x < 0, \\ U_{R}, & x > 0, \end{cases} (1.8)

where U_{L} and U_{R} are constant states. We are interested in the large-time behavior of solutions in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) of the Cauchy problem for (1.1) with initial data:

(u, v, E)|_{t=0} = (u_{0}, v_{0}, E_{0}), (1.9)

which are an L^{1} perturbation of Riemann initial data. That is,

U|_{t=0} = U_{0}(x) ≡ R_{0}(x) + P_{0}(x), with P_{0}(x) ∈ L^{1}(ℝ). (1.10)

To avoid ambiguities, we henceforth denote W = (u, v, S) to distinguish from U = (u, v, E).

We recall that an entropy-entropy flux pair for (1.7) is a pair (η(U), q(U)) of C^{1} functions satisfying

∇η(U)∇F (U) = ∇q(U). (1.11)

Clearly, (a(S), 0), a(S) ∈ C^{1}, is an entropy-entropy flux pair for (1.1). Throughout this paper, the following entropy-entropy flux pair for (1.1) plays an important role:

η_{∗}(W) = \frac{u^{2}}{2} − ∫_{v_{0}}^{v} p(σ, S) dσ + K(S), (1.12)

q_{∗}(W) = up(v, S), (1.13)

where K(S), a C^{2} function, and v_{0} > 0 are to be suitably chosen. As usual, we assume that the function p = p(v, S) satisfies

p_{v}(v, S) < 0,  p_{vv}(v, S) > 0,  ∇_{v,S}^{2}p(v, S) ≥ 0, (1.14)

in the domain v > 0. The polytropic gases clearly satisfy (1.14). These assumptions imply that the system is strictly hyperbolic and that the first and third characteristic families are genuinely nonlinear in the sense of Lax [14]. They also imply the uniqueness of the Riemann solution in the class of the self-similar piecewise smooth solutions consisting of shocks, rarefaction waves, and contact discontinuities (cf. [14]).

At the onset, we require that K(S) and v_{0} be such that η_{∗}(W) is a strictly convex function in the bounded domain V ⊂ {v > 0} where the solutions to be considered take their values. This amounts to assume

K′′(S) > ∫_{v_{0}}^{v} p_{SS}(σ, S) dσ − \frac{p_{S}^{2}(v, S)}{p_{v}(v, S)}, (u, v, S) ∈ V, (1.15)

where (1.14) has been used.

Definition 1.1. A bounded measurable function U(x, t) = (u(x, t), v(x, t), E(x, t)) is an entropy solution of (1.1) and (1.9) in Π_{T} = ℝ×[0, T) if U(x, t) takes its values in the physical region {(v, u, E) | v > 0} and satisfies the following. (i). Equations (1.1) hold in the weak sense in Π_{T}, i.e., for all ϕ ∈ C_{0}^{1}(Π_{T}),

∫_{Π_{T}} {U∂_{t}ϕ + F (U)∂_{x}ϕ} dx dt + ∫_{−∞}^{∞} U_{0}(x)ϕ(x, 0)dx = 0, (1.16)

with U = (u, v, E) and F (U) = (p, −u, pu).

(ii). The Clausius inequality holds in the sense of distributions in Π_{T}, i.e., for all nonnegative ϕ ∈ C_{0}^{1}(Π_{T}),

∫_{Π_{T}} a(S)∂_{t}ϕ dx dt + ∫_{−∞}^{∞} a(S_{0}(x))ϕ(x, 0)dx ≤ 0, (1.17)

for any a(S) ∈ C^{1} and a′(S) ≥ 0.

Observe that (1.16) and (1.17) imply that any entropy solution W (x, t) satisfies the entropy inequality

∫_{Π_{T}} {η_{∗}(W)∂_{t}ψ + q_{∗}(W)∂_{x}ψ} dx dt + ∫_{−∞}^{∞} η_{∗}(W_{0}(x))ψ(x, 0) dx ≥ 0, (1.18)

for any nonnegative ψ ∈ C_{0}^{1}(Π_{T}), where (η_{∗}, q_{∗}) is defined in (1.12)-(1.13), provided that K(S) and v_{0} > 0 are chosen such that

K′(S) < θ(v_{0}, S), i.e. ∂_{S}η_{∗}(W) < θ(v, S). (1.19)

Indeed, by (1.3) and (1.12), we have

η_{∗}(W) = E + a_{∗}(S), a_{∗}(S) = K(S) − e(v_{0}, S).

Now, using (1.3) again and (1.19), we have a′_{∗}(S) ≤ 0, which allows to conclude (1.18) from (1.16) and (1.17).

Throughout this paper, we choose K(S) and v_{0} > 0 such that, besides (1.15), both (1.19) and

∂_{S}η_{∗}(W) ≥ 0, i.e. K′(S) ≥ θ(v_{0}, S) − θ(v, S) (1.20)

hold. That is, we assume (1.15) and

θ(v_{0}, S) − θ(v, S) ≤ K′(S) < θ(v_{0}, S), (1.21)

in the bounded physical region V ⊂ {(u, v, S) | v > 0} ⊂ ℝ^{3} under consideration.

It is easy to check that, for the ideal polytropic gases (i.e. the γ-law gases), for any compact subset V ⊂ {v > 0}, (1.15) and (1.21) hold, by choosing, for instance, v_{0} > 0 arbitrary and K(S) = \frac{κc_{v}}{R} v_{0}^{−γ+1}e^{S/c_{v}} − δS with 0 < δ ≤ min θ(v, S). More generally, if the equations of state are such that θ(v, S) and p(v, S) satisfy θ_{S}(v, S)+ \frac{p_{S}^{2}(v,S)}{p_{v}(v,S)} > 0 in \{v > 0\}, then (1.15) and (1.21) hold in any compact subset of {v > 0} with v_{0} > 0 arbitrary and K(S) given by K′(S) = θ(v_{0}, S) − δS.

Our main goal in this paper is to prove the uniqueness of the classical Riemann solution R(ξ), corresponding to the Riemann data (1.8), which in turn implies that, for any entropy solution U(x, t) ∈ L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) of (1.1) and (1.10), whose local total variation satisfies a certain natural growth condition (cf. (1.23) below), one has

(1.22)  ess lim_{t→∞} ∫_{−L}^{L} |U(ξt, t) − R(ξ)|dξ = 0,  for any L > 0.

By the framework established in Chen-Frid [4, 5], for any entropy solution of (1.1) and (1.10), U(x, t) ∈ L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}), the asymptotic stability problem can be reduced to the problem of the uniqueness of the Riemann solution in the class of entropy solutions in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}), provided the local total variation of U(x, t) satisfies a natural growth condition:

\begin{cases}There exists c_{0} > 0 such that, for all c ≥ c_{0}, there is C > 0, \\ depending only on c, such that T V (U|K_{c,T}) ≤ CT, for any T > 0, \end{cases}

(1.23)

where

K_{c,T} = {(x, t) ∈ ℝ_{+}^{2} | |x| < ct, t ∈ (0, T)}. (1.24)

Actually, in [4], the growth condition is imposed on the total variation over the cylinders {|x| < cT } × (0, T) instead of the cones K_{c,T}. But, in this way, condition (1.23) is more general and does not require any modification in the arguments.

We recall that the growth condition (1.23), which is necessary only for the study of the asymptotic behavior, is natural since any solution obtained by the Glimm method or related methods satisfies (1.23). In particular, (1.23) is verified for BV solutions satisfying the Glimm-Lax condition [13], namely,

T V_{x}(U(·, t)|(−L, L)) ≤ C_{0} \frac{L}{t},

for any t > 0 and L > c_{0}t, for some fixed C_{0}, c_{0} > 0, where T V_{x} denotes the total variation in x, for fixed t > 0.

Hence, in the main part of this paper, we focus on the uniqueness problem for solutions in L^{∞} ∩ BV_{loc} of (1.1) and (1.8) requiring only (1.16)-(1.17) (see Sections 3-4). To achieve this goal we extend the method developed by DiPerna [9] for the 2 × 2 case to our 3 × 3 system. We remark that our uniqueness result for Riemann solutions can easily be extended to the uniqueness of entropy solutions of (1.1) which are piecewise Lipschitz in x, for any t > 0, in the same spirit as DiPerna’s theorem in [9] for the 2 × 2 case. We also remark that all the results in this paper have a straightforward equivalent version for system (1.1) written in Eulerian coordinates, namely,

\begin{cases}∂_{τ} ρ + ∂_{y}(ρu) = 0, \\ ∂_{τ} (ρu) + ∂_{y}(p + ρu^{2}) = 0, \\ ∂_{τ} (ρ(\frac{1}{2}u^{2} + e)) + ∂_{y}(ρu(\frac{1}{2}u^{2} + e) + pu) = 0, \end{cases} (1.25)

where ρ = 1/v is the density and

τ = t, y = ∫_{0}^{x} v(s, t) ds + ∫_{0}^{t} u(0, σ) dσ.

In order to avoid repetitions, we will not state the corresponding results for (1.25) which can are obtained using the well-known equivalence between (1.1) and (1.25) (see, e.g., [25]).

Finally, we recall recent fundamental results for m × m systems in Bressan-Crasta-Piccoli [1] (also see [2]), Bressan-Liu-Yang [3], and Liu-Yang [19] on the L^{1}-stability of entropy solutions in L^{∞} ∩ BV obtained by the Glimm scheme and the wave front-tracking method, or satisfying an additional regularity, with small total variation in x for all t > 0. We remark that the uniqueness results in this paper cannot be directly obtained from those stability results since the results presented here neither impose smallness restrictions on the total variation and additional regularity of the solutions, nor need specific reference to any particular method for constructing the entropy solutions. In this connection we also recall that, for special cases of system (1.1), including the one for polytropic gases, there are many existence results of solutions in L^{∞} ∩BV_{loc} with non-small total variation (see, e.g., [13, 17, 22, 20, 23]). We also refer the reader to Dafermos [8] for the stability of classical solutions of hyperbolic systems of conservation laws.

This paper is organized as follows. In Section 2, we discuss the relation between the uniqueness of Riemann solutions in the class of entropy solutions in L^{∞} ∩BV_{loc} and the asymptotic stability of Riemann solutions with respect to L^{1} initial perturbation. In Section 3, in order to make our analysis clearer, we address separately the case when the Riemann solution is shock-free, i.e., it may contain only rarefaction waves of the first and third characteristic families and, possibly, a contact discontinuity of the second family. Then, in Section 4, we consider the general case in which the Riemann solution may contain shock waves.

2. Asymptotic Stability and Uniqueness of Riemann Solutions

One of the main features of hyperbolic systems of conservation laws is the invariance under self-similar scaling. That is, if U(x, t) solves (1.7), then

U^{T} (x, t) = U(T x, T t) (2.1)

also solves (1.7). In particular, since the Riemann initial data are also self-similar scaling invariant, a natural question is whether any entropy solution of the Riemann problem is self-similar invariant. This question is answered in the present situation by our uniqueness results proved in Sections 3-4.

The following proposition establishes the relationship among uniqueness of entropy solutions of the Riemann problem, compactness of the scaling sequence U^{T} (x, t), and asymptotic stability in the sense of (1.22).

Proposition 2.1. Assume that the Cauchy problems (1.7)-(1.10) satisfy the following. (i). System (1.7) is endowed with a strictly convex entropy; (ii). The Riemann solution R(ξ), ξ = x/t, is piecewise Lipschitz in the variable

ξ and is unique in the class of entropy solutions of (1.7)-(1.8) in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2});

(iii). Given an entropy solution U(x, t) ∈ L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) of (1.7)-(1.8), the sequence U^{T} (x, t) is compact in L_{loc}^{1}(ℝ_{+}^{2}), and any limit function of its subse-

quences is still in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}).

Then the Riemann solution R(x/t) is asymptotically stable in the sense of (1.22) in the class of entropy solutions in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) corresponding to initial perturbations P_{0}(x) ∈ L^{1}.

Proof. This fact can be seen as follows. Take any subsequence {U^{T_{k}}(x, t)}_{k=1}^{∞} ⊂ {U^{T} (x, t)}_{T >0}. Condition (iii) implies that there exists a further subsequence converging in L_{loc}^{1} to Ũ(x, t) ∈ L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) satisfying the same initial data as R(x/t). Condition (ii) then ensures that Ũ(x, t) = R(x/t) a.e., which is unique. This indicates that the whole sequence {U^{T} (x, t)}_{T >0} converges to the Riemann solution R(x/t) in L_{loc}^{1}(ℝ_{+}^{2}). For any 0 < r < ∞, we have

\frac{1}{T^{2}} ∫_{0}^{T} ∫_{|ξ|≤r} |U(ξt, t) − R(ξ)|tdξdt = \frac{1}{T^{2}} ∫_{0}^{T} ∫_{|x|≤rt} |U(x, t) − R(x/t)|dxdt

= ∫_{0}^{1} ∫_{|x|≤rt} |U^{T} (x, t) − R(x/t)|dxdt → 0, when T → ∞,

which implies

\frac{1}{T} ∫_{0}^{T} ∫_{|ξ|≤r} |U(ξt, t) − R(ξ)| dξdt → 0, when T → ∞.

In view of condition (i) and the piecewise Lipschitz continuity of R(ξ) given by condition (ii), we can use Theorem 3.1 of Chen-Frid [5] (also see [4]) to conclude that the Riemann solution is asymptotically stable in the sense of (1.22) with respect to the initial perturbation P_{0}(x).

For entropy solutions in L^{∞} ∩BV_{loc} satisfying condition (1.23), the compactness of the scaling sequence is guaranteed by the following observation.

Proposition 2.2. Assume that U(x, t) ∈ L^{∞} ∩BV_{loc}(ℝ×(0, ∞)) satisfies that, for some fixed c_{0} > 0, for any c ≥ c_{0}, there exists C > 0, depending only on c, such that

T V (U|K_{c,T_{∗}}) ≤ CT_{∗}, for all T_{∗} > 0. (2.2)

Then, for each T > 0, U^{T} (x, t), defined by (2.1), also satisfies (2.2) with the same constant C, depending on c. Furthermore, the sequence U^{T} is compact in L_{loc}^{1}(ℝ_{+}^{2}).

Proof. The proof of the first part is immediate from a scaling argument. Then, the compactness in L_{loc}^{1} of U^{T} is a consequence of the well-known compactness in L_{loc}^{1} of bounded sets in BV (cf. [10]).

As remarked above, condition (1.23) holds for the entropy solutions constructed by Glimm’s method (see [12, 13, 17, 22, 23]). Therefore, according to Proposition 2.2, condition (iii) in Proposition 2.1 is verified in all these cases.

As we indicated above, for solutions in L^{∞} ∩ BV_{loc}, the asymptotic problem reduces to the uniqueness problem of Riemann solutions in the class of entropy solutions in L^{∞} ∩ BV_{loc} of (1.7)-(1.8). In what follows we focus on the uniqueness problem.

3. Uniqueness of Shock-Free Riemann Solutions

In this section we consider the uniqueness of classical Riemann solutions of (1.1) and (1.8), which are shock-free. We recall that (1.1) has the eigenvalues

λ_{1} = −\sqrt{−p_{v}}, λ_{2} = 0, λ_{3} = \sqrt{−p_{v}},

and the corresponding right eigenvectors r_{1}, r_{2}, r_{3}. The first and third families of (1.1) are genuinely nonlinear, i.e., ∇λ_{i} · r_{i} ≠ 0, i = 1, 3, and the second family is linearly degenerate, i.e., ∇λ_{2} · r_{2} = 0. For the eigenvectors, we have

r_{i} = ∇gr̃_{i}, i = 1, 2, 3,

where U = g(W) is the transformation which takes (1.1) into (1.5) and

r̃_{1} = c(W)(−λ_{1}, 1, 0)^{⊤}, r̃_{2} = (0, p_{S}, −p_{v})^{⊤}, r̃_{3} = c(W)(−λ_{1}, −1, 0)^{⊤}.

Here c(W) = 2\sqrt{−p_{v}}/p_{vv} is a normalizing factor such that ∇_{W} λ_{i} · r̃_{i} = 1, which is equivalent to ∇_{U}λ_{i} · r_{i} = 1, i = 1, 3.

Under conditions (1.14), which guarantee the strict hyperbolicity of (1.1) and the genuine nonlinearity of the first and third characteristic families, it follows (see, e.g., [14, 21]) that a classical shock-free Riemann solution generally has the form:

R(x/t) ≡ \begin{cases}U_{L}, & x/t ≤ λ_{1}(U_{L}), \\ R_{1}(x/t), & λ_{1}(U_{L}) < x/t < λ_{1}(U_{M}), \\ U_{M}, & λ_{1}(U_{M}) ≤ x/t < 0, \\ U_{N}, & 0 < x/t ≤ λ_{3}(U_{N}), \\ R_{3}(x/t), & λ_{3}(U_{N}) < x/t < λ_{3}(U_{R}), \\ U_{R}, & x/t ≥ λ_{3}(U_{R}), \end{cases} (3.1)

where R_{1}(ξ) and R_{3}(ξ) are the solutions of the boundary value problems

\{ \begin{matrix}\frac{d}{dξ} R_{1}(ξ) = r_{1}(R_{1}(ξ)), λ_{1}(U_{L}) < ξ, \\ R_{1}(λ_{1}(U_{L})) = U_{L}, \end{matrix} (3.2)

and

\{ \begin{matrix}\frac{d}{dξ} R_{3}(ξ) = r_{3}(R_{3}(ξ)), ξ < λ_{3}(U_{R}), \\ R_{3}(λ_{3}(U_{R})) = U_{R}. \end{matrix} (3.3)

The states U_{M} and U_{N} are also completely determined by (3.2) and (3.3). The best way to see this fact is first to recall that S is constant over rarefaction curves of the first and third families, since S is a Riemann invariant of the first and third families (see [21]). Similarly, u and p are both constant over the wave curves of the second (linearly degenerate) family. Hence, in the space (u, p, S), we can project the curves R_{1} and R_{3} in the plane (u, p), find the intersection of these projected curves, and immediately obtain the two points in the planes S = S(R_{1}) and S = S(R_{3}).

Let W (x, t) = (u(x, t), v(x, t), S(x, t)) be a certain entropy solution of the Riemann problem (1.1) and (1.8). Denote by \overline{W} (x, t) = (\overline{u}(x, t), \overline{v}(x, t), \overline{S}(x, t)) the classical self-similar solution of this problem, that is, \overline{W} (x, t) equals R(x/t) written in the coordinates (u, v, S). We use the quadratic entropy-entropy flux pairs obtained from (η_{∗}, q_{∗}) through a procedure introduced by Dafermos (cf. [8, 9])

α(W, \overline{W}) = η_{∗}(W) − η_{∗}(\overline{W}) − ∇η_{∗}(\overline{W})(W − \overline{W}), (3.4)

β(W, \overline{W}) = q_{∗}(W) − q_{∗}(\overline{W}) − ∇η_{∗}(\overline{W})(f(W) − f(\overline{W})), (3.5)

where f(W) = (p(v, S), −u, 0)^{⊤}.

Motivated by [9], we consider the measures

θ = ∂_{t}η_{∗}(W (x, t)) + ∂_{x}q_{∗}(W (x, t)) − ∂_{S}η_{∗}(\overline{W} (x, t))∂_{t}S, (x, t) ∈ ℝ_{+}^{2} − \{x = 0\},
γ = ∂_{t}α(W (x, t), \overline{W} (x, t)) + ∂_{x}β(W (x, t), \overline{W} (x, t)).

Hypothesis (1.20) also guarantees that θ ≤ 0, as a Radon measure in {x ≠ 0}, and that γ ≤ 0, in any domain in which \overline{W} is constant, due to (1.16) and (1.18).

The first important fact is that γ{ℓ} = 0, where ℓ = {(0, t) | t > 0}, since γ\{ℓ\} = ∫_{ℓ}[β(W, \overline{W})] dH^{1} and [β(W, \overline{W})] = 0, H^{1}-a.e. over ℓ. The latter follows from β(W, \overline{W}) = (u − ū)(p − p̄) and the fact that u, p, ū, p̄ cannot change across the jump discontinuities of W and \overline{W} over ℓ, because of the Rankine-Hugoniot relations. Here and in what follows we use the notation [H] = H_{−} − H_{+}, where H_{−} and H_{+} are the values of the function H on the left-hand side and the right-hand side of the discontinuity curve, respectively.

Set

Ω_{1} = { (x, t) | λ_{1}(U_{L}) < x/t < λ_{1}(U_{M}), t > 0 },
Ω_{3} = { (x, t) | λ_{3}(U_{N}) < x/t < λ_{j}(U_{R}), t > 0 },

the rarefaction wave regions of the classical Riemann solution. Over these regions, γ satisfies

\begin{matrix}γ = & ∂_{t}α(W, \overline{W}) + ∂_{x}β(W, \overline{W}) \\ = & ∂_{t}η_{∗}(W) + ∂_{x}q_{∗}(W) − ∂_{S}η_{∗}(\overline{W})∂_{t}S \\ − ∇^{2}η_{∗}(\overline{W})(∂_{t}\overline{W} , W − \overline{W}) − ∇^{2}η_{∗}(\overline{W})(∂_{x}\overline{W} , f(W) − f(\overline{W})) \\ = & θ − ∇^{2}η_{∗}(\overline{W})(∂_{x}\overline{W} , Qf(W, \overline{W})), \end{matrix} (3.6)

where we used the fact that ∇^{2}η_{∗}∇f is symmetric, and Qf(W, \overline{W}) = f(W) − f(\overline{W}) − ∇f(\overline{W})(W − \overline{W}) is the quadratic part of f at \overline{W}. Recall that

∇^{2}η(W)(r̃_{i}(W), r̃_{j}(W)) = 0, i ≠ j, (3.7)

for any entropy η. We notice that, because of (3.7),

l̃_{j}(\overline{W}) = r̃_{j}(\overline{W})∇^{2}η_{∗}(\overline{W}) (3.8)

is a left eigenvector of ∇f(\overline{W}) corresponding to the eigenvalue λ_{j}(\overline{W}), j = 1, 3. We also easily see that, for (x, t) ∈ Ω_{j}, one has

\frac{∂\overline{W} (x, t)}{∂x} = \frac{1}{t}r̃_{j}(\overline{W} (x, t)), j = 1, 3.

Then, by (3.1) and (3.6), for any Borel set E ⊂ Ω_{j}, j = 1, 3, we have

γ(E) = θ(E) − ∫_{E} \frac{1}{t}l̃_{j}(\overline{W})Qf(W, \overline{W}) dxdt. (3.9)

As a direct consequence of (3.9), we have the following result.

Theorem 3.1. Let R(x/t) be the classical shock-free solution of the Riemann problem (1.1) and (1.8), given by (3.1). Let U(x, t) be any entropy solution in L^{∞} ∩ BV_{loc}(ℝ × (0, T)) of (1.1) and (1.8) in Π_{T} = ℝ × (0, T). Assume (1.14)-(1.15) and (1.21) are satisfied. Then U(x, t) = R(x/t), a.e. in Π_{T}.

Proof. Let Π_{t} denote the strip { (x, s) | x ∈ ℝ, 0 < s < t} and Ω_{j}(t) = Ω_{j} ∩ Π_{t}, j = 1, 3, ℓ_{t} = ℓ ∩ Π_{t}. By the Gauss-Green formula for BV functions (cf. [10, 24]) and the finiteness of propagation speed of the solution (see [9]), we have

γ\{Π_{t}\} = ∫_{−∞}^{∞} α(W (x, t), \overline{W} (x, t)) dx. (3.10)

On the other hand, we have from (3.9)

γ\{Ω_{j}(t)\} = θ\{Ω_{j}(t)\} − ∫_{Ω_{j}(t)} \frac{1}{s}l̃_{j}(\overline{W})Qf(W, \overline{W}) dx ds, j = 1, 3.

Since \overline{W} (x, t) is constant in each component of Π_{t} − \{Ω_{1}(t) ∪ ℓ_{t} ∪ Ω_{3}(t)\}, one has

γ\{Π_{t}\} = γ\{ℓ_{t}\} + θ\{Π_{t} − \{ℓ_{t}\}\} − ∑_{j=1,3} ∫_{Ω_{j}(t)} \frac{1}{s}l̃_{j}(\overline{W})Qf(W, \overline{W}) dxds. (3.11)

Then, since θ ≤ 0 and γ\{ℓ_{t}\} = 0, it suffices to prove that l̃_{j}(\overline{W})Qf(W, \overline{W}) ≥ 0. Now, one can easily check that l̃_{j}(\overline{W}) is a positive multiple of

(1, ±\sqrt{−p_{v}(\overline{v}, \overline{S})}, ∓p_{S}(\overline{v}, \overline{S})/\sqrt{−p_{v}(\overline{v}, \overline{S})}) ,

and

Qf(W, \overline{W}) = (p(v, S) − p(\overline{v}, \overline{S}) − p_{v}(\overline{v}, \overline{S})(v − \overline{v}) − p_{S}(\overline{v}, \overline{S})(S − \overline{S}), 0, 0)^{⊤}.

Hence, by (1.14), we actually have l̃_{j}(\overline{W})Qf(W, \overline{W}) ≥ 0, j = 1, 3. This completes the proof.

Remark 3.1. From the proof of Theorem 3.1 above, it is easy to see that the same arguments lead to an even stronger result. Actually, it is immediate to deduce from the above proof that the classical shock-free Riemann solution is in fact globally stable in L^{2} with respect to its initial perturbations. More specifically, if U(x, t) is an entropy solution in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) of (1.1) and (1.10), then there exists a constant C > 0 such that

∫_{|x|≤L} |U(x, t) − R(x/t)|^{2} dx ≤ C ∫_{|x|≤L+Mt} |U_{0}(x) − R_{0}(x)|^{2} dx, (3.12)

where C, L, M > 0 are independent of t. Moreover, if the classical Riemann solution consists of at most rarefaction waves and contact discontinuities, similar arguments to those in the above proof yield (3.12) for any L^{∞} entropy solution of (1.1) and (1.10), with the aid of the theory of the divergence-measure fields developed in [6].

Combining Theorem 3.1 with Proposition 2.1, we conclude

Theorem 3.2. Let R(x/t) be the classical shock-free solution of the Riemann problem (1.1) and (1.8), given by (3.1). Let U(x, t) be any entropy solution in L^{∞} ∩ BV_{loc}(ℝ_{+}^{2}) of (1.1) and (1.10) satisfying (1.23). Assume (1.14)-(1.15) and (1.21) are satisfied. Then U(x, t) asymptotically tends to R(x/t) in the sense of (1.22).

4. Uniqueness of General Riemann Solutions

In this section we prove the uniqueness of entropy solutions of the Riemann problem (1.1) and (1.8) for the general case in which the corresponding self-similar classical Riemann solution may also contain shock waves. For concreteness, we assume that the classical Riemann solution has the following generic form

R(x/t) = \begin{cases}U_{L}, & x/t < σ_{1}, \\ U_{M}, & σ_{1} < x/t < 0, \\ U_{N}, & 0 < x/t ≤ λ_{3}(U_{N}), \\ R_{3}(x/t), & λ_{3}(U_{N}) < x/t < λ_{3}(U_{R}), \\ U_{R}, & x/t ≥ λ_{3}(U_{R}), \end{cases} (4.1)

where σ_{1} = σ_{1}(U_{L}, U_{M}) is the shock speed, determined by the Rankine-Hugoniot relation,

σ_{1}(U_{L} − U_{M}) − (F (U_{L}) − F (U_{M})) = 0, (4.2)

and R_{3}(ξ) is the solution of the boundary value problem (3.3). The 1-shock wave connecting U_{L} and U_{M} satisfies the Lax entropy condition:

λ_{1}(U_{M}) < σ < λ_{1}(U_{L}) < 0. (4.3)

To deal with shock waves, we use the concept of generalized characteristics introduced by Dafermos (cf. [7]) as in [9]. A generalized j-characteristic associated with a solution U(x, t) ∈ L^{∞} ∩ BV_{loc} of (1.1) is defined as a trajectory of the equation

ẋ(t) = λ_{j}(U(x(t), t)), (4.4)

where (4.4) is interpreted in the sense of Filippov [11]. Thus, a j-characteristic is a Lipschitz continuous curve (x(t), t) whose speed of propagation ẋ(t) satisfies

ẋ(t) ∈ [m_{x}{λ_{j}(U(x(t), t))}, M_{x}{λ_{j}(U(x(t), t))}], (4.5)

where m_{x}{λ_{j}(U(x(t), t))} and M_{x}{λ_{j}(U(x(t), t))} denote the essential minimum and the essential maximum of λ_{j}(U(·, t)) at the point x(t). As it was proved by Filippov [11], among all solutions of (4.5) passing through a point (x_{0}, t_{0}), there is an upper solution x̄(t) and a lower solution \underline{x}(t), that is, the solutions of (4.5) such that any other solution x(t) of (4.5) satisfies the inequality \underline{x}(t) ≤ x(t) ≤ x̄(t). The lower and upper solutions, for t > t_{0}, are called the minimal and maximal forward j-characteristics, respectively. An important feature about solutions in L^{∞} ∩ BV_{loc} is that, given any generalized i-characteristic y(t), it must propagate either with shock speed or with characteristic speed (cf. [7]). This allows one to treat (y(t), t) simply as a shock curve of U(x, t) in the (x, t)-plane.

Lemma 4.1 (DiPerna [9]). Let (1.7) be an m × m strictly hyperbolic system endowed with a strictly convex entropy pair, whose characteristic fields are either genuinely nonlinear or linearly degenerate. Suppose U(x, t) ∈ L^{∞} ∩ BV_{loc}(Π_{T}) is an entropy solution of (1.7)-(1.8) in Π_{T}. Let x_{max}^{m}(t) denote the maximal forward m-characteristic through (0, 0). Let x_{min}^{1}(t) denote the minimal forward 1-characteristic passing through (0, 0). Then U(x, t) = U_{L}, for a.e. (x, t) with x < x_{min}^{1}(t), 0 ≤ t < T, and U(x, t) = U_{R}, for a.e. (x, t) with x > x_{max}^{m}(t), 0 ≤ t < T.

We now state and prove the uniqueness result for the general case.

Theorem 4.1. Let U(x, t) = (u(x, t), v(x, t), E(x, t)) ∈ BV_{loc}(Π_{T}; V), V ⊂ {(u, v, S) | v > 0} ⊂ ℝ^{3}, be an entropy solution of (1.1) and (1.8) in Π_{T}. Assume (1.14)-(1.15) and (1.21) are satisfied. Then U(x, t) = R(x/t), a.e. in Π_{T}, provided V is small.

Proof. Following an idea in [9], we consider the auxiliary function in Π_{T}:

Ũ(x, t) = \begin{cases}U_{L}, & x < x(t), \\ U_{M}, & x(t) < x < max\{x(t), σ_{1}t\}, \\ R(x/t), & x > max\{x(t), σ_{1}t\}, \end{cases}

where x(t) is the minimal 1-characteristic of U(x, t), and x = σt is the line of 1-shock discontinuity in R(x/t). We then consider the measures

θ̃ = ∂_{t}η_{∗}(W (x, t)) + ∂_{x}q_{∗}(W (x, t)) − ∂_{S}η_{∗}(W̃(x, t))∂_{t}S, (x, t) ∈ Π_{T} − {ℓ_{T} ∪ L_{T} },
γ̃ = ∂_{t}α(W (x, t), W̃(x, t)) + ∂_{x}β(W (x, t), W̃(x, t)), (x, t) ∈ Π_{T} ,

where W̃ equals Ũ written in the coordinates (u, v, S), L_{t} = {(x(s), s) | 0 < s < t}, and α(W, W̃) and β(W, W̃) are defined by (3.4)-(3.5). Our problem essentially reduces to analyzing the measure γ̃ over the region where the Riemann solution is a rarefaction wave and over the curve (x(t), t), which for simplicity may be taken as the jump set of W̃(x, t).

Again, using the Gauss-Green formula for BV functions and the finiteness of propagation speeds of the solutions, we have

γ̃{Π_{t}} = ∫_{−∞}^{∞} α(W (x, t), W̃(x, t)) dx. (4.6)

On the other hand, since γ̃ reduces to the measure θ̃ on the open sets where W̃ is a constant, and W̃(x, t) = \overline{W} (x, t) over Ω̄_{3},

γ̃{Π_{t}} = γ̃{L_{t}} + γ̃{Ω̄_{3}(t)} + θ̃{Π_{t} − (L_{t} ∪ ℓ_{t} ∪ Ω̄_{3}(t))}, (4.7)

where we have used the fact that γ̃{ℓ_{t}} = 0. Hence, if one shows that

γ̃{L_{t}} ≤ 0, (4.8)

the problem will reduce to the same verification as in the shock-free case. Thus, we consider the functional

D(σ, W_{−}, W_{+}, W̃_{−}, W̃_{+}) = σ[α(W, W̃)] − [β(W, W̃)].

We will prove that

D(σ, W_{−}, W_{+}, W̃_{−}, W̃_{+}) ≤ 0, (4.9)

if W_{−}, W_{+} are connected by a 1-shock of speed σ = x′(t), and W̃_{−}, W̃_{+} are connected by a 1-shock of speed σ̄, and also W_{−} = W̃_{−}. Using Lemma 4.1, it is then clear that (4.9) immediately implies (4.8). Thus, when W_{−} = W̃_{−}, an easy calculation shows that

D(σ, W_{−}, W_{+}, W̃_{−}, W̃_{+}) =d(σ, W_{−}, W_{+}) − d(σ̄, W_{−}, W̃_{+}) − (σ − σ̄)α(W_{−}, W̃_{+})

− ∂_{S}η(W̃_{+}) (σ(S_{−} − S_{+}) − σ̄(S_{−} − S̃_{+})) , (4.10)

where d(σ, W_{−}, W_{+}) = σ[η(W)] − [q(W)], and (η, q) = (η_{∗}, q_{∗}) is the entropy pair in (1.12)-(1.13). From the Rankine-Hugoniot relations for (1.1), we may view the state W_{+} = (u_{+}, v_{+}, S_{+}) connected on the right by a 1-shock to a state W_{−} = (u_{−}, v_{−}, S_{−}) as parameterized by the shock speed σ, with σ ≤ λ_{1}(W_{−}) < 0. We recall that, through this parameterization, S(σ) satisfies (see [21])

S(σ) = S(λ_{1}(W_{−})) − \frac{\begin{matrix}... \\ S \end{matrix}(λ_{1}(W_{−}))}{6}(λ_{1}(W_{−}) − σ)^{3} + O((λ_{1}(W_{−}) − σ)^{4}), (4.11)

and

\begin{matrix}... \\ S \end{matrix}(λ_{1}(W_{−})) = −p_{vv}(v̇)^{3}/2θ < 0. (4.12)

According to the parameterization, we set W_{+} = W_{+}(σ) and W̃_{+} = W_{+}(σ̄) in (4.10). For concreteness, we assume σ̄ > σ. Now, we have

− ∂_{S}η(W̃_{+}) (σ(S_{−} − S_{+}(σ)) − σ̄(S_{−} − S_{+}(σ̄)))

= −∂_{S}η(W̃_{+})(σ − σ̄) (S_{−} − S_{+}(σ̄) − σ \frac{S_{+}(σ̄) − S_{+}(σ)}{σ̄ − σ})

= −∂_{S}η(W̃_{+})(σ − σ̄) (S_{−} − S_{+}(σ̄) − σṠ(σ̂)) ,

where σ̂ satisfies σ ≤ σ̂ ≤ σ̄ ≤ λ_{1}(W̄_{−}). Define

h(σ) ≡ d(σ, W_{−}, W_{+}(σ)) = σ[η(W)] − [q(W)].

One easily verifies that

ḣ(σ̄) = α(W_{−}, W_{+}(σ̄)) + ∂_{S}η(W̃_{+}) (S_{−} − S_{+}(σ̄) − σ̄Ṡ_{+}(σ̄)) .

Now, from 0 > λ_{1}(W_{−}) ≥ σ̄ ≥ σ and 0 ≥ Ṡ(σ̄) ≥ Ṡ(σ̂) due to (4.11) and (4.12), it follows that

∂_{S}η(W̃_{+})σ̄Ṡ_{+}(σ̄) ≤ ∂_{S}η(W̃_{+})σṠ(σ̂),

and hence

ḣ(σ̄) ≥ α(W_{−}, W̃_{+}) + ∂_{S}η(W̃_{+}) (S_{−} − S_{+} − σṠ(σ̂)) .

Therefore, we have

D(σ, W_{−}, W_{+}, W̃_{−}, W̃_{+}) ≤ h(σ) − h(σ̄) − ḣ(σ̄)(σ − σ̄).

Observe that the above inequality is also true in the case where σ > σ̄. Now, we will show that, for |λ − λ_{1}(W_{−})| sufficiently small,

ḧ(λ) ≤ 0. (4.13)

Indeed, it is easy to verify

\frac{d}{dλ}α(W_{−}, W_{+}(λ)) = −∇^{2}η(W_{+}(λ)) (Ẇ_{+}(λ), W_{−} − W_{+}(λ)) .

Then

ḧ(λ) = \frac{d}{dλ}α(W_{−}, W_{+}(λ)) + ∂_{S}∇η(W_{+}(λ))Ẇ_{+}(λ)(S_{−} − S_{+}(λ) − λṠ_{+}(λ))

− ∂_{S}η(W_{+}(λ))(2Ṡ_{+}(λ) + λS̈_{+}(λ))

= −∇^{2}η(W_{+}(λ)) (Ẇ_{+}(λ), W_{−} − W_{+}(λ)) − ∂_{S}η(W_{+}(λ))λS̈_{+}(λ)

+ O((λ_{1}(W_{−}) − λ)^{2})

= \{−∇^{2}η(W_{−})(Ẇ_{+}(λ_{1}(W_{−})), Ẇ_{+}(λ_{1}(W_{−}))) + ∂_{S}η(W_{−})\begin{matrix}... \\ S \end{matrix}(λ_{1}(W_{−}))λ_{1}(W_{−})\}

× (λ_{1}(W_{−}) − λ) + O((λ_{1}(W_{−}) − λ)^{2}).

Thus we only need to show that

∇^{2}η(W_{−})(Ẇ_{+}(λ_{1}(W_{−})), Ẇ_{+}(λ_{1}(W_{−}))) − ∂_{S}η(W_{−})\begin{matrix}... \\ S \end{matrix}(λ_{1}(W_{−}))λ_{1}(W_{−}) > 0.

Hence, the left-hand side of the inequality above equals to

(u̇(λ_{1}(W_{−})))^{2} − p_{v}(v̇(λ_{1}(W_{−})))^{2} − \sqrt{−p_{v}} \frac{p_{vv}}{2θ} (v̇(λ_{1}(W_{−})))^{3}∂_{S}η. (4.14)

Now, from the normalization ∇λ_{1}(W) · r̃_{1}(W) = 1, one obtains

u̇(λ_{1}(W_{−})) = −\frac{4p_{v}}{p_{vv}} ,  v̇(λ_{1}(W_{−})) = \frac{4\sqrt{−p_{v}}}{p_{vv}} ,

and so (4.14) equals to

(u̇(λ_{1}(W_{−})))^{2} − p_{v}(v_{−}, S_{−})(v̇(λ_{1}(W_{−})))^{2}(1 − \frac{2∂_{S}η(W_{−})}{θ(v_{−}, S_{−})}).

Therefore, hypothesis (1.19) guarantees the positiveness of (4.14), which implies (4.13) and in turn gives (4.9). Now, by (4.6), we get that W (x, t) = W̃(x, t), a.e. in Π_{T}. In particular, W̃ is an entropy solution of (1.1) and (1.8), and then Rankine-Hugoniot relations imply that W̃ must coincide with the classical Riemann solution \overline{W}. This concludes the proof.

Combining (4.1) with Proposition 2.1, we arrive at the following conclusion.

Theorem 4.2. Let U(x, t) = (u(x, t), v(x, t), E(x, t)) ∈ BV_{loc}(ℝ_{+}^{2}; V) be an entropy solution of (1.1) and (1.10), satisfying (1.23). Assume (1.14)-(1.15) and (1.21) are satisfied. Then U(x, t) asymptotically tends to the corresponding Riemann solution R(x/t) in the sense of (1.22), provided V is small.

Acknowledgments. Gui-Qiang Chen’s research was supported in part by the National Science Foundation grants DMS-9971793, INT-9987378, and INT-9726215. Hermano Frid’s research was supported in part by CNPq-Brazil, proc. 352871/96-2.

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(G.-Q. Chen) Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

  E-mail address: gqchen@math.nwu.edu

(H. Frid) Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. Postal 68530, Rio de Janeiro, RJ 21945-970, Brazil

  E-mail address: hermano@im.ufrj.br