Vanishing Viscosity Limit for Initial-Boundary Value Problems for Conservation Laws

Gui-Qiang Chen and Hermano Frid

Abstract. The convergence of the vanishing viscosity method for initialboundary value problems is analyzed for nonlinear hyperbolic conservation laws through several representative systems. Some techniques are developed to construct the global viscous solutions and establish the H^{−1} compactness of entropy dissipation measures for the convergence of the viscous solutions with general initial-boundary conditions. The representative examples considered include the systems of isentropic gas dynamics, nonlinear elasticity, and chromatography.

1. Introduction

We are concerned with the convergence of the vanishing viscosity method for initial-boundary value problems for nonlinear hyperbolic systems of conservation laws. Physical motivation is the vanishing viscosity limit from viscous compressible fluids to the inviscid ones with initial-boundary conditions, which is a natural way to determine entropy solutions for the inviscid equations in the interior of fluid domains under consideration. Our analysis focuses on several representative examples of nonlinear systems including the models for isentropic gas dynamics, nonlinear elasticity, and chromatography. The main objective in addressing the particular systems is to expose a general procedure of analysis for such a problem with general boundary conditions, especially nonhomogeneous ones. In order to make the main steps clearer and to avoid superfluous technicalities, we restrict our analysis here to the standard domains of form Q_{T} = (0, T) × (0, 1) or Q = (0, ∞) × (0, 1). Many other examples may be treated by following the same procedure.

We start with a general system of conservation laws in one space variable:

∂_{t}u + ∂_{x}f(u) = 0, (t, x) ∈ Q, (1.1)

with u ∈ U ⊂ R^{m}, f ∈ C^{1}(U; R^{m}), for some domain U ⊂ R^{m}. We consider the following initial-boundary conditions for (1.1):

u|_{t=0} = u_{0}(x), (1.2)

u|_{x=0} = a_{0}(t), u|_{x=1} = a_{1}(t). (1.3)

1991 Mathematics Subject Classification. Primary: 35L65, 35L50; Secondary: 35B25, 35L67.

Key words and phrases. Initial-boundary value problems, vanishing viscosity limit, conservation laws, convergence, compactness, entropy solutions, estimates.

We assume that the initial-boundary data satisfy

u_{0} ∈ L^{∞}((0, 1); R^{m}), a_{0}, a_{1} ∈ L^{∞}((0, ∞); R^{m}). (1.4)

Since the boundary layers generally exist for arbitrarily given a_{i}, i = 1, 2, our focus here is to expose a procedure to construct weak entropy solutions in the interior of Q such that the solutions obtained are natural for the case that there is no boundary layer (see [5]). With this in mind, a definition of entropy solutions for the initial-boundary value problem was given in [5] for general multidimensional systems of conservation laws in more general (not necessarily cylindrical) domains, motivated from [1] for the scalar case. Some discussions about the initial-boundary problem in different contexts, related to [1], have been made for hyperbolic systems of conservation laws (e.g. [2, 13, 19, 20, 30, 34, 36]).

We say that η ∈ C^{1}(R^{m}) is an entropy for (1.1), with associated entropy flux q ∈ C^{1}(R^{m}), if

∇q(u) = ∇η(u)∇f(u). (1.5)

We call F (u) = (η(u), q(u)) an entropy pair. If η(u) is convex, we say F (u) is a convex entropy pair. An entropy pair F(u, v) = (α(u, v), β(u, v)) is called a boundary entropy pair if, for each fixed v ∈ R^{m}, α(u, v) is convex with respect to u, and

α(v, v) = β(v, v) = ∂_{u}α(v, v) = 0. (1.6)

We say that F(u, v) = (α(u, v), β(u, v)) is a generalized boundary entropy pair if it is the uniform limit of a sequence of boundary entropy pairs over compact sets.

Definition 1.1. We say that u ∈ L^{∞}(Q_{T}; R^{m}) is a weak entropy solution of (1.1)-(1.3) in Q_{T} if it satisfies

• Conservation Laws (1.1): For all ϕ ∈ C_{0}^{∞}(Q_{T}), ϕ ≥ 0, and any convex entropy pair (η, q),

∫ ∫_{Q_{T}} (η(u)∂_{t}ϕ + q(u)∂_{x}ϕ) dx dt ≥ 0; (1.7)

• Initial Condition (1.2):

ess lim_{t→0+} ∫_{0}^{1} |u(t, x) − u_{0}(x)| dx = 0; (1.8)

• Boundary Condition (1.3): For any γ(t) ∈ L^{1}(0, T), γ(t) ≥ 0, a.e., and any boundary entropy pair F = (α, β),

ess lim_{x→0+} ∫_{0}^{T} β(u(t, x), a_{0}(t))γ(t) dt ≤ 0, ess lim_{x→1−} ∫_{0}^{T} β(u(t, x), a_{1}(t))γ(t) dt ≥ 0. (1.9)

To illustrate some features of the above definition, we consider a simple example. Let (1.1) be strictly hyperbolic, which means that the Jacobian ∇f is diagonalizable and all eigenvalues are real and distinct. In (1.2)-(1.3), let a_{0}, u_{0}, and a_{1} be constant states u_{l}, u_{m}, and u_{r}, respectively, such that u_{l} and u_{m} are connected by a shock with a negative speed, while u_{m} and u_{r} are connected by a shock with a positive speed. To simplify, we assume that both shocks belong to genuinely nonlinear families, that is, the corresponding eigenvalues of ∇f are monotone along the integral curves of the associated right-eigenvectors (cf. [21]). Then it is easy to prove that u(t, x) ≡ u_{m}, (t, x) ∈ Q, is an entropy solution for the corresponding initial-boundary value problem, according to the above definition, provided that u_{l}, u_{m}, and u_{r} are sufficiently close to each other.

Indeed, (1.7) and (1.8) are trivially satisfied. For (1.9), it suffices to check the inequality for the left boundary, x = 0, since the other is similar. Then this reduces to showing that β(u_{m}, u_{l}) ≤ 0. Now, from the Lax shock condition, it follows that β(u_{m}, u_{l}) ≤ sα(u_{m}, u_{l}), for |u_{m} − u_{l}| sufficiently small (see [22]). Now, from the properties of boundary entropies, one has α(u, v) ≥ 0, and hence the desired inequality follows since s < 0.

This solution is actually consistent with the natural physical solution in the interior of Q, with boundary layers on the boundaries, via the characteristic analysis. It would be interesting to analyze systematically the uniqueness of entropy solutions in the sense of (1.7)-(1.9) for such problems.

We next recall an important fact about the solutions of (1.1)-(1.3) according to (1.7)-(1.9), established in [5], which holds even for the general multidimensional case in general (not necessarily cylindrical) domains.

Theorem 1.1. Assume that (1.1) is endowed with a strictly convex entropy. A function u(t, x) ∈ L^{∞}(Q_{T}; R^{m}) satisfies (1.7)-(1.9) if and only if the following conditions hold:

(a). u(t, x) satisfies (1.1) in the sense of distributions in Q_{T}; (b). Given any boundary entropy pair (α(u, v), β(u, v)), there exists a constant

M > 0 such that, for any nonnegative ϕ(t, x) ∈ C_{0}^{∞}((−∞, T) × R) and any

v ∈ R^{m},

\begin{matrix}∫_{0}^{T} ∫_{0}^{1}(α(u(t, x), v)∂_{t}ϕ + β(u(t, x), v)∂_{x}ϕ) dx dt \\ + ∫_{0}^{1} α(u_{0}(x), v)ϕ(0, x) dx + M ∫_{Γ} α(u^{b}, v)ϕ dt ≥ 0, \end{matrix} (1.10)

where Γ = ∪_{j=0}^{1}{x = j, t > 0}, and u^{b}(t) = a_{i}(t), i = 0, 1.

In the subsequent sections, we solve problem (1.1)–(1.3), in the sense of Definition 1.1, for the representative systems of nonlinear elasticity, chromatography, and isentropic gas dynamics, according to the following scheme. In Section 2, we establish some general results for parabolic systems obtained from (1.1) with an additional viscosity term in its right-hand side. In particular, we obtain a useful uniform estimate (2.22) for the derivative of the viscous solutions, which is essential in order to establish the H^{−1}-compactness of entropy dissipation measures for nonhomogeneous boundary conditions addressed here (see §3.1). In Section 3, we apply the results of Sections 1-2 and the compensated compactness methods to obtain the existence of entropy solutions when (1.1) is either the 2 × 2 system of nonlinear elasticity, or the m × m system of chromatography in Langmuir coordinates, or some other systems mentioned therein. Finally, in Section 4, we analyze the convergence of the viscous approximate solutions of the initial-boundary value problem for the system of isentropic Euler equations for gas dynamics with the aid of the results in Sections 1-2, especially estimate (2.22).

2. Parabolic Systems

In this section we consider the initial-boundary value problem for the parabolic system obtained from (1.1) with an additional viscosity term. Namely, we are concerned with the following initial-boundary value problem:

∂_{t}u + ∂_{x}f(u) = ε∂_{xx}u, (t, x) ∈ Q, (2.1)

u|_{x=0} = a_{0,ε}(t), u|_{x=1} = a_{1,ε}(t), t > 0, (2.2)

u|_{t=0} = u_{0,ε}(x), x ∈ (0, 1). (2.3)

To simplify the statements of the results, we assume that a_{0,ε}, a_{1,ε} are smooth and

sup_{ε>0} ∥(a_{0,ε}, εa′_{0,ε}, a_{1,ε}, εa′_{1,ε})∥_{L^{∞}(0,∞)} < ∞, (2.4)

and u_{0,ε} ∈ C_{0}^{∞}(0, 1) with

sup_{ϵ>0} ∥u_{0,ε}∥_{L^{∞}(0,1)} < ∞, (2.5)

and the compatibility conditions:

a_{0,ε}^{(k)}(0) = a_{1,ε}^{(k)}(0) = 0, for all k ∈ N. (2.6)

For the purposes of the applications given in this paper, it suffices to consider the situation in which f satisfies:

(2.7) f ∈ C^{3}(R^{m}; R^{m}) is globally Lipschitz, and f(−u) = f(u).

Denote by K^{ε}(t, x) the fundamental solution of the heat equation ∂_{t}v = ε∂_{xx}v, that is,

K^{ε}(t, x) = \frac{1}{\sqrt{4πεt}}e^{− \frac{x^{2}}{4εt}} .

We will make use of the fact that

∥K^{ε}(t)∥_{1} = 1, ∥K_{x}^{ε}(t)∥_{1} = \frac{1}{\sqrt{επt}}, (2.8)

where ∥ ∥_{1} denotes the norm of L^{1}(R).

For a function ζ(x) defined in (0, 1), denote by ζ̃ the function defined in R such that

\begin{cases}ζ̃(x) = ζ(x), & 0 < x < 1, \\ ζ̃(−x) = −ζ̃(x), & x ∈ R, \\ ζ̃(x + 2n) = ζ̃(x), & x ∈ R, n ∈ Z. \end{cases} (2.9)

Set h_{ε}(t, x) = (1 − x)a_{0,ε}(t) + xa_{1,ε}(t), x ∈ (0, 1), t ≥ 0. If u(t, x) is a smooth solution of (2.1)–(2.3) in [0, T] × (0, 1), then w(t, x) = u(t, x) − h_{ε}(t, x) is a smooth solution of the initial-boundary value problem:

∂_{t}w − ε∂_{xx}w = −∂_{x}f(u) − ∂_{t}h_{ε}, (t, x) ∈ Q, (2.10)

w|_{x=0} = 0,  w|_{x=1} = 0,  t > 0, (2.11)

w|_{t=0} = u_{0,ε}(x), x ∈ (0, 1). (2.12)

Hence, we expect that u(t, x) satisfies

u(t) = K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} K^{ε}(t − s) ∗ ({ \atop ∂_{x}f(u)}(s) + ˜∂_{t}h_{ε}(s)) ds + h_{ε}(t), (2.13)

for (t, x) ∈ Q.

To see this fact, we denote the right-hand side of (2.10) by r(t, x) and approximate it in L_{loc}^{1}(Q) by a sequence r^{n}(t, x) in C_{0}^{∞}(Q). If w^{n}(t, x) is the corresponding solution for the modified (2.10) with conditions (2.11)-(2.12), then we clearly have

w^{n}(t) = K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} K^{ε}(t − s) ∗ r̃^{n}(s) ds.

Now, letting n → ∞, we see that w^{n} converges in L_{loc}^{1}(Q) to a certain w satisfying

w(t) = K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} K^{ε}(t − s) ∗ r̃(s) ds.

From the properties of the heat kernel and the function r̃(t, x), we deduce that w satisfies (2.10) in the sense of distributions in Q. Then w − (u − h_{ε}) satisfies the homogeneous heat equation. By the standard regularity theory, w is then smooth. Since (2.11) and (2.12) are easily verified from (2.13), the uniqueness of the solution of (2.10)–(2.12) implies then that w = u − h_{ε}. On the other hand, if u satisfies (2.13) and has continuous derivatives of first order in t and up to second order in x, throughout Q, then applying the heat operator ∂_{t} − ε∂_{xx}^{2} to both sides of (2.13), for (t, x) ∈ Q, we deduce that u satisfies (2.1) in Q in the sense of distributions and, hence, in the classical sense. Conditions (2.2)-(2.3) are also immediately deduced from (2.13).

Then, for smooth solutions in [0, T] × (0, 1), (2.13) is equivalent to

(2.14) u(t) = K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ f(ũ)(s) ds

− ∫_{0}^{t} K^{ε}(t − s) ∗ ˜∂_{t}h_{ε}(s) ds + h_{ε}(t),

where we have used the definition of the extension { \atop ∂_{x}f(u)}(s) of ∂_{x}f(u)(s), from (0, 1) to R according to (2.9). Indeed, to pass the derivative from { \atop ∂_{x}f(u)}(s) to the heat kernel to obtain (2.14) from (2.13), we write the convolution as a sum of integrals in (0, 1), using (2.9), then apply integration by parts, and observe that the sums like

∑_{n∈Z} ∫_{0}^{t}{K^{ε}(t − s, x − 2n − 1) − K^{ε}(t − s, x − 2n + 1)}f(a_{1,ε}(s)) ds,

resulting also from this process, vanish identically. Hence a possible strategy for solving (2.1)–(2.3) is to obtain first a solution of (2.14), and then to prove that it possesses the required regularity.

Let G_{T} = L^{∞}((0, T); L^{∞}(0, 1)) and define the operator L : G_{T} → G_{T} by

(2.15) L(v)(t) = K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ f(ṽ)(s) ds

− ∫_{0}^{t} K^{ε}(t − s) ∗ ˜∂_{t}h_{ε}(s) ds + h_{ε}(t).

Let ∥∇f(u)∥_{L^{∞}} = C_{0}. We have

∥L(v_{1})(t) − L(v_{2})(t)∥_{L^{∞}} ≤ C_{0}\sqrt{\frac{T}{επ}} ∥v_{1} − v_{2}∥_{L^{∞}},

and so L is a contraction in G_{T} as long as

C_{0}\sqrt{\frac{T}{επ}} < 1. (2.16)

By the Banach Fixed Point Theorem, there exists a unique u ∈ G_{T}, which satisfies L(u) = u when 0 ≤ t ≤ T.

Lemma 2.1. Let u be the unique fixed point of L in G_{T} with T = α_{0}ε for α_{0} << 1 independent of ε, such that (2.16) holds. Then there exists C_{1} > 0 such that,

∥∂_{x}u(t)∥_{L^{∞}} ≤ \frac{C_{1}}{\sqrt{εt}}, 0 < t ≤ T, (2.17)

with C_{1} independent of ε. Furthermore, there exists a constant C_{2}, depending on ε, such that

∥∂_{x}u(t)∥_{L^{∞}} ≤ C_{2}, 0 < t ≤ T. (2.18)

Proof. The proof of (2.17) reduces to proving the following assertion: there exists a constant C_{1} > 0 such that, if v ∈ G_{T} ∩ C((0, T] × (0, 1)), v|_{x=0} = a_{0,ε}, v|_{x=1} = a_{1,ε}, and

∥∂_{x}v(t)∥_{L^{∞}} ≤ \frac{C_{1}}{\sqrt{εt}}, (2.19)

then L(v) also satisfies these properties.

Indeed, since u is the unique fixed point of L in G_{T}, we have u = lim_{k→∞} u^{k} in G_{T}, where u^{1} = h, u^{k+1} = L(u^{k}). Hence, from the assertion, we have that (2.19) is satisfied for v = u^{k}, u^{k}|_{x=0} = a_{0,ε}(t), and u^{k}|_{x=1} = a_{1,ε}(t), for all k ∈ N. Therefore, the standard arguments yield that u must also satisfy these properties.

We now pass to the proof of the assertion. From the hypothesis on v, one has

∂_{x}L(v)(t) = ∂_{x}K^{ε}(t) ∗ ũ_{0,ε} − ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ { \atop ∂_{x}f(v)}(s) ds

− ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ ∂_{t}h̃_{ε}(s) ds + ∂_{x}h_{ε}(t),

and so

∥∂_{x}L(v)(t)∥_{L^{∞}} ≤\frac{∥ũ_{0,ε}∥_{L^{∞}}}{\sqrt{πεt}} + \frac{CC_{1}}{ε} + 2\sqrt{\frac{T}{πε}}∥∂_{t}h_{ε}∥_{L^{∞}} + ∥∂_{x}h_{ε}∥_{L^{∞}}

≤\frac{1}{\sqrt{πεt}}(∥u_{0,ε}∥_{L^{∞}} + CC_{1}\sqrt{\frac{πT}{ε}} + 2T ∥∂_{t}h_{ε}∥_{L^{∞}} + \sqrt{πεT} ∥∂_{x}h_{ε}∥_{L^{∞}})
≤\frac{C_{1}}{\sqrt{εt}},

provided that

C_{1} ≥ \frac{∥u_{0,ε}∥_{L^{∞}} + 2T ∥∂_{t}h_{ε}∥_{L^{∞}} + \sqrt{επT} ∥∂_{x}h_{ε}∥_{L^{∞}}}{\sqrt{π}(1 − C\sqrt{\frac{T}{ε}})} , (2.20)

where C > 0 depends only on C_{0}, independent of ε and v. Since T = α_{0}ε for α_{0} << 1 independent of ε, the fact that C_{1} can be taken independent of ε is clearly seen from (2.20), because of (2.4) and (2.5).

The second part of the statement follows similarly. We only need to prove that there exists a constant C_{2} such that, if ∥∂_{x}v∥_{L^{∞}} ≤ C_{2}, v|_{x=0} = a_{0,ε} and v|_{x=1} = a_{1,ε}, then L(v) has also these properties. To this end, we observe that we may write

∂_{x}L(v)(t) = K^{ε}(t) ∗ { \atop ∂_{x}u_{0,ε}} − ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ { \atop ∂_{x}f(v)}(s) ds

− ∫_{0}^{t} ∂_{x}K^{ε}(t − s) ∗ ∂_{t}h̃_{ε}(s) ds + ∂_{x}h_{ε}(t).

Hence

∥∂_{x}L(v)∥_{L^{∞}} ≤ ∥{ \atop ∂_{x}u_{0,ε}}∥_{L^{∞}} + CC_{2}\sqrt{\frac{T}{ε}} + 2\sqrt{\frac{T}{πε}}∥∂_{t}h_{ε}∥_{L^{∞}} + ∥∂_{x}h_{ε}∥_{L^{∞}} ≤ C_{2}, (2.21)

provided that

C_{2} ≥ \frac{∥∂_{x}u_{0,ε}∥_{L^{∞}} + 2\sqrt{T/(πε)}∥∂_{t}h_{ε}∥_{L^{∞}} + ∥∂_{x}h_{ε}∥_{∞}}{1 − C\sqrt{T/ε}} ,

where C depends only on C_{0}. This concludes the proof.

Let u be the unique fixed point of L in G_{T}. By Lemma 2.1, ∂_{x}u is bounded in [0, T] × (0, 1). Clearly, u is a weak solution of (2.1)–(2.3) in the sense that u belongs to the space

W (T) = {v ∈ L^{2}([0, T]; W^{1,2}(0, 1)) | ∂_{t}v ∈ L^{2}([0, T]; W^{−1,2}(0, 1))},

satisfying

⟨∂_{t}u(t), ϕ⟩ + ε ∫_{0}^{1} ∂_{x}u∂_{x}ϕ dx = ∫_{0}^{1} f(u)(t)∂_{x}ϕ dx,

for almost all t ∈ [0, T] and all ϕ ∈ W_{0}^{1,2}(Ω), u(0) = u_{0,ε}, and u(t) − h_{ε}(t) ∈ W_{0}^{1,2}(0, 1), for almost all t ∈ [0, T]. The fact that ∂_{t}u ∈ L^{2}([0, T]; W^{−1,2}(0, 1)) follows the observation that u is the limit in L^{∞} of a sequence u^{n} satisfying

∂_{t}u^{n+1} − ε∂_{xx}u^{n+1} = −∂_{x}f(u^{n}),

with ∥∂_{x}u^{n}∥_{L^{∞}} uniformly bounded in n. Therefore, ∂_{t}u^{n}(t, x) is uniformly bounded in L^{2}([0, T]; W^{−1,2}(0, 1)) and must converge weakly to ∂_{t}u(t, x).

Applying the regularity theory for parabolic equations (see [17, 23, 28]), one deduces that u is a smooth solution of (2.1)–(2.3) in [0, T] × (0, 1). It is easy to see that the hypothesis that f is globally Lipschitz allows one to repeat the above procedures step by step in time, indefinitely, to obtain a global smooth solution to (2.1)–(2.3) satisfying (2.17)-(2.18).

If system (2.1) is endowed with a bounded invariant region (see [8]), then the global smooth solution u is uniformly bounded. We now prove that this allows us to obtain a useful estimate for ∂_{x}u.

Theorem 2.1. Let u be the unique smooth solution of (2.1)-(2.3). Assume that u is uniformly bounded in [0, ∞) × (0, 1), independently of ε, and that (2.4)-(2.6) hold. Then, for all δ > 0 sufficiently small, there exists a positive constant M, independent of ε, such that

\begin{cases}∥ε∂_{x}u(t)∥_{L^{∞}} ≤ M, & for t > εδ, \\ ∥ε∂_{x}u(t)∥_{L^{∞}} ≤ M\sqrt{\frac{ε}{t}}, & for 0 < t ≤ εδ. \end{cases} (2.22)

The same result holds for the smooth solution of the Cauchy problem.

Proof. For any t_{0} > 0, consider the operator L in G_{T} = L^{∞}([t_{0}, t_{0}+T]; L^{∞}(Ω)), given by

L(v)(t) =K^{ε}(t − t_{0}) ∗ ũ(t_{0}) − ∫_{t_{0}}^{t} ∂_{x}K^{ε}(t − s) ∗ f(ṽ)(s) ds

− ∫_{t_{0}}^{t} K^{ε}(t − s) ∗ ˜∂_{t}h_{ε}(s) ds + h_{ε}(t). (2.23)

Exactly as above, we easily see that L is a contraction mapping in G_{T} if (2.16) is satisfied. Also, identically as in the proof of Lemma 2.1, we prove the assertion that, for C_{1} satisfying (2.20) with ∥u_{0,ε}∥_{L^{∞}} replaced by ∥u(t_{0})∥_{L^{∞}}, if v ∈ G_{T} ∩ C((t_{0}, t_{0} + T] × Ω), v|_{x=0} = a_{0,ε}, v|_{x=1} = a_{1,ε}, and

∥∂_{x}v(t)∥_{L^{∞}} ≤ \frac{C_{1}}{\sqrt{ε(t − t_{0})}}, (2.24)

then L(v) also satisfies these properties. Thus, using the fact that u is the unique fixed point of L, we deduce from the standard arguments that u must satisfy (2.24). Take T = 2εδ, for any δ > 0 such that (2.16) holds. Then, for any t > εδ, we take some t_{0} = t − T/2 in (2.24) to obtain

∥ε∂_{x}u(t)∥_{L^{∞}} ≤ \frac{C_{1}}{\sqrt{δ}} . (2.25)

On the other hand, for 0 < t ≤ εδ, we have that (2.17) holds. From Lemma 2.1, C_{1} can be taken independent of ε. Therefore, taking M = C_{1}/\sqrt{δ}, we conclude the proof of (2.22).

The proof of estimate (2.22) for the smooth solution of the Cauchy problem is completely similar.

In order to prove (1.10) for hyperbolic systems, we need to get a corresponding inequality for the associated parabolic systems. To this end, we will make use of a construction as in [28, 30]. For δ > 0 sufficiently small, define

d(x) = \begin{cases}x, & 0 < x < δ, \\ δ, & δ < x < 1 − δ, \\ 1 − x, & 1 − δ < x < 1, \end{cases}

and, for some M > 0, set

ξ_{ε}(x) ≡ 1 − e^{− \frac{M}{ε} d(x)}.

For any φ ∈ C_{0}(R), φ ≥ 0, the function ξ_{ε}(x) satisfies

M ∫_{0}^{1} |ξ′_{ε}(x)|φ(x) dx ≤ ε ∫_{0}^{1} ξ′_{ε}(x)φ′(x) dx + M(φ(0) + φ(1)). (2.26)

Indeed,

∫_{0}^{1} ξ′_{ε}(x)φ′(x) dx = ∫_{{0<d<δ}} ξ′_{ε}(x)φ′(x) dx
= − ∫_{\{0<d<δ\}} ξ′′_{ε}(x)φ(x) dx − \frac{M}{ε} (φ(1) + φ(0))

+ \frac{M}{ε} e^{− \frac{Mδ}{ε}} (φ(1 − δ) + φ(δ))

≥\frac{M^{2}}{ε^{2}} ∫_{\{0<d<δ\}} e^{− \frac{Md(x)}{ε}} φ(x) dx − \frac{M}{ε} (φ(1) + φ(0))

=\frac{M}{ε} ∫_{O}^{1} |ξ′_{ε}(x)|φ(x) dx − \frac{M}{ε} (φ(1) + φ(0)),

which immediately give (2.26).

Theorem 2.2. Let u be the smooth solution of (2.1)–(2.3), and let (α(u, v), β(u, v)) be a boundary entropy pair for (1.1). Then there exists a constant M > 0 such that, for all ϕ ∈ C_{0}^{∞}((−∞, T) × R), ϕ ≥ 0, and v ∈ R^{m},

\begin{matrix}− ∫_{0}^{T} ∫_{0}^{1} \{α(u, v)∂_{t}ϕ + β(u, v)∂_{x}ϕ + εα(u, v)∂_{xx}ϕ\} ξ_{ε} dx dt \\ ≤ ∫_{0}^{1} α(u_{0,ε}, v)ϕ(x, 0)ξ_{ε} dx + M ∫_{Γ} α(u_{ε}^{b}, v)ϕ dt \\ +2ε ∫_{0}^{T} ∫_{0}^{1} α(u, v)ξ′_{ε}∂_{x}ϕ dx dt, \end{matrix} (2.27)

where Γ = ∪_{j=0}^{1}{x = j, t > 0}, and u_{ε}^{b} = a_{i,ε}, i = 0, 1.

Proof. Denote η(u) = α(u, v), q(u) = β(u, v). By the convexity of α with respect to u and (1.6), we easily see that there must exist a constant M > 0, independent of v, such that |q(u)| ≤ Mη(u). Now, multiplying (2.1) by ∇η(u), one obtains

∂_{t}η(u) + ∂_{x}q(u) ≤ ε∂_{xx}η(u). (2.28)

Then, multiplying (2.28) by ξ_{ε}ϕ, integrating in Q_{T} = (0, T) × (0, 1), and using integration by parts, we obtain

− ∫_{0}^{T} ∫_{0}^{1} {η(u)∂_{t}ϕ + q(u)∂_{x}ϕ + εη(u)∂_{xx}ϕ} ξ_{ε} dx dt
≤ ∫_{0}^{1} η(u_{0,ε})ϕ(x, 0)ξ_{ε} dx + M ∫_{0}^{T} ∫_{0}^{1} η(u)ϕ|ξ′_{ε}| dx dt
−ε ∫_{0}^{T} ∫_{0}^{1} ∂_{x}(η(u)ϕ)ξ′_{ε} dx dt + 2ε ∫_{0}^{T} ∫_{0}^{1} η(u)ξ′_{ε}∂_{x}ϕ dx dt,

where we have used |q(u)| ≤ Mη(u). Applying (2.26) with η(u)ϕ replacing φ in the inequality displayed above, we then obtain (2.27).

3. Nonlinear Elasticity, Chromatography and Other Systems

In this section we apply the results in Sections 1-2 to solving the initialboundary value problem for two specific systems: the one arising in one-dimensional nonlinear elasticity and the other appearing in chromatography with Langmuir coordinates. We also discuss other applications which follow in a similar fashion.

3.1. Nonlinear Elasticity. Consider the one-dimensional nonlinear elasticity system:

\begin{cases}∂_{t}u_{1} − ∂_{x}σ(u_{2}) = 0, \\ ∂_{t}u_{2} − ∂_{x}u_{1} = 0, \end{cases} (3.1)

where σ is a smooth function satisfying σ′(τ) > 0, and τσ′′(τ) > 0 if τ ≠ 0. Then, in this case, f(u) = (−σ(u_{2}), −u_{1})^{⊤}. System (3.1) is endowed with the following strictly convex entropy:

η_{∗}(u) = u_{1}^{2} + ∫_{0}^{u_{2}} σ(τ) dτ,

with entropy-flux:

q_{∗}(u) = u_{1}σ(u_{2}).

Given a convex entropy η(u), a boundary entropy pair (α(u, v), β(u, v)) can be defined by taking the quadratic part of η and its associated flux. That is,

\begin{matrix}α(u, v) = η(u) − η(v) − ∇η(v)(u − v), \\ β(u, v) = q(u) − q(v) − ∇η(v)(f(u) − f(v)). \end{matrix}

Also, system (3.1) is endowed with a pair of independent Riemann invariants (i.e. the functions whose gradient are left-eigenvectors of ∇f) given by

w_{1} = u_{1} + ∫_{0}^{u_{2}} \sqrt{σ′(τ)} dτ, w_{2} = u_{1} − ∫_{0}^{u_{2}} \sqrt{σ′(τ)} dτ.

The regions given by

R = {u ∈ R^{2} | |w_{1}| < M, |w_{2}| < M},

for any M > 0, are invariant under the flow of the parabolic system (2.1) corresponding to (3.1) (cf. [8, 11, 14]). Given uniformly bounded initial-boundary data, we take a region R like the above with M large enough so that the initial-boundary data assume values in R. In order to have the flux function f of (3.1) satisfying condition (2.7), we first change from the coordinates u to

ū = u − u_{∗}, (3.2)

where u_{∗} = (0, −A), A > 0, is any point of the axis u_{1} = 0 which does not belong to R. Then we replace f by

f̄(ū) = φ(ū)f(ū + u_{∗}), if ū > 0;  f̄(−ū) = f̄(ū), (3.3)

with φ ∈ C_{0}^{∞}(R^{2}) and φ(ū) = 1, if ū ∈ R − u_{∗}, and such that (0, −A) does not belong to the support of φ. The function f̄ satisfies (2.7) and coincides with f in the invariant region R. Now, by the invariant region arguments (cf. [8]), any smooth solution of system (2.1) associated with (3.1), with f̄ replacing f, takes its values in R, as long as the initial-boundary data take values in R. Hence, replacing f by f̄ has no real effect, and the solution of the modified system is also a solution of the original one.

For given initial-boundary data (1.2)-(1.4), we can find smooth approximate functions a_{0,ε}, a_{1,ε}, and u_{0,ε}, which converge to a_{0}, a_{1}, and u_{0}, respectively, in L_{loc}^{1}(0, ∞) and L^{1}(0, 1) and satisfy (2.4)-(2.6), using the standard techniques of cutoff and mollification. We now consider the compactness of the smooth solution sequence u^{ε} of the viscous systems (2.1) corresponding to (3.1). First, this sequence is uniformly bounded in L^{∞}(Q; R^{2}), because all of the functions u^{ε} assume values in R, which is a bounded region of R^{2}. That is,

∥u^{ε}∥_{L^{∞}(Q)} ≤ B_{1}, (3.4)

for some B_{1} > 0 independent of ε. To apply DiPerna’s compactness result in [11], it suffices to verify the following:

(3.5) ∂_{t}η(u^{ε}) + ∂_{x}q(u^{ε}) lies in a compact subset of H_{loc}^{−1}(Q),

for any smooth entropy-entropy flux pair (η, q).

With the aid of our estimate (2.22), property (3.5) can be seen as follows. We first multiply (2.1) by ∇η(u) to obtain

∂_{t}η(u^{ε}) + ∂_{x}q(u^{ε}) = ε∂_{xx}η(u^{ε}) − ε(∂_{x}u^{ε})^{⊤}∇^{2}η(u^{ε})∂_{x}u^{ε}. (3.6)

If η is strictly convex (e.g. η = η_{∗} given above), integrating (3.6) in Q_{T} with any T > 0, we have

c_{0} ∫ ∫_{Q_{T}} ε(∂_{x}u^{ε})^{2}dxdt ≤ε ∫_{0}^{T} (η′(u^{ε})∂_{x}u^{ε})|_{0}^{1}dt − ∫_{0}^{1} η(u^{ε})|_{0}^{T} dx − ∫_{0}^{T} q(u^{ε})|_{0}^{1}dt

≤A_{1} ∫_{0}^{ε} \sqrt{\frac{ε}{t}}dt + A_{2} ∫_{ε}^{T} Mdt + A_{3} ≤ B_{2},

using estimate (2.22), where A_{i}, i = 1, 2, 3, and B_{2} are independent of ε. Thus, we have

\sqrt{ε}∥∂_{x}u^{ε}∥_{L^{2}(Q_{T})} ≤ B_{3}, (3.7)

for some constant B_{3} > 0, depending on T, but independent of ε.

Now, from (3.7), we obtain as usual that, for any smooth entropy η,

ε(∂_{x}u^{ε})^{⊤}∇^{2}η(u^{ε})∂_{x}u^{ε}

is uniformly bounded in M(Q_{T}), the space of signed Radon measures in Q_{T}. Therefore, by Sobolev’s embeddings, it is compact in W^{−1,p}(Q_{T}), for 1 < p < 2. Also, from (3.7), we obtain that, for any smooth entropy η(u),

ε∂_{xx}η(u^{ε})

is compact in W^{−1,2}(Q_{T}) (in fact it converges to 0). Thus the right-hand side of (3.6) is compact in W^{−1,p}(Q_{T}), using again Sobolev’s embeddings. Now, because of (3.4), the left-hand side of (3.6) is uniformly bounded in W^{−1,∞}(Q_{T}). Then, an interpolation argument gives (3.5) (see [10, 29]).

Once we have proved (3.5), we can use DiPerna’s compactness result in [11] to conclude the compactness of the sequence u^{ε} in L_{loc}^{1}(Q). Let u be the limit of a subsequence u^{ε_{k}} in L_{loc}^{1}(Q) with ε_{k} → 0 as k → ∞. Hence, from (2.27) in Theorem 2.2, we obtain (1.10), using the fact that ξ_{ε} and εξ′_{ε} are uniformly bounded and converge pointwise to 1 and 0, respectively.

Thus, Theorem 1.1 can be applied to conclude that u is an entropy solution of the initial-boundary value problem for (3.1).

Theorem 3.1. Let a_{0}, a_{1}, and u_{0} satisfy (1.4). Then there exists a global entropy solution of the initial-boundary problem (3.1) and (1.2)-(1.3) in the sense of (1.7)-(1.9).

3.2. Chromatography: The m × m chromatography system for Langmuir isotherms (cf. [31]) is given by

∂_{t}u_{i} + ∂_{x} (\frac{k_{i}u_{i}}{1 + ∑_{j=1}^{m} u_{j}} ) = 0, x ∈ R, t ≥ 0, 1 ≤ i ≤ m, (3.8)

where 0 < k_{1} < k_{2} < ⋯ < k_{m} are given numbers. It is well known (cf. [18]) that (3.8) is endowed with m linearly independent Riemann invariants w_{1}, …, w_{m}, which have the property that the level surfaces w_{i} = const. are affine hyperplanes in R^{m} (also see Temple [38]). For these systems, using the maximum principle (see [33]), it is easy to show that the regions

R = {u ∈ R^{m} | |w_{i}(u) − w̄_{i}| ≤ M_{i}, i = 1, ⋯, m}

are invariant under the flow of the associated parabolic system (2.1), where w̄ = (w̄_{1}, ⋯, w̄_{n}) is a constant state in R^{m} and M_{i} > 0 are arbitrary constants, as long as they are contained in the domain {u ∈ R^{m} | u_{i} ≥ 0, i = 1, ⋯, m}. Then, the same procedures as the one for the system of nonlinear elasticity can yield the existence of entropy solutions of the initial-boundary value problem for (3.8), where we apply the compactness theorem of James-Peng-Perthame [18], with the aid of Theorem 2.1 (i.e. (2.22)).

Theorem 3.2. Let a_{0}, a_{1}, and u_{0} satisfy (1.4). Then there exists a global entropy solution of the initial-boundary problem (3.8) and (1.2)-(1.3) in the sense of (1.7)-(1.9).

3.3. Other Systems: The same techniques can be used to prove the corresponding results for other systems such as the quadratic systems with umbilic degeneracy studied in [6], the class of conjugate type systems considered in [15], and the systems addressed in [32]. All of these systems have bounded invariant regions over which the flux functions are smooth, say, at least C^{3} in the interior of the invariant regions and C^{2} up to the boundaries.

4. System of Isentropic Euler Equations

The system of isentropic Euler equations reads

\begin{cases}∂_{t}ρ + ∂_{x}m = 0, \\ ∂_{t}m + ∂_{x}(\frac{m^{2}}{ρ} + p(ρ)) = 0, \end{cases} (4.1)

where ρ represents the density, m is the momentum, and p(ρ) is the pressure. The behavior of the pressure function p(ρ) depends on the fluids under consideration. We assume at the onset that p(ρ) satisfies

(4.2) p′(ρ) > 0 (hyperbolicity), ρp′′(ρ) + 2p′(ρ) > 0 (genuine nonlinearity)

away from the vacuum ρ = 0 and, when ρ → 0+,

p(ρ) ≈ κρ^{γ}(1 + P (ρ)), |P^{(n)}(ρ)| ≤ Cρ^{1−n}, 0 ≤ n ≤ 4, (4.3)

for some γ > 1. This means that, when ρ → 0, the pressure law p(ρ) has the same principal singularity as the γ-law, but allows additional singularities in the derivatives.

System (4.1) is endowed with a pair of independent Riemann invariants given by

w = \frac{m}{ρ} + ∫_{0}^{ρ} \frac{1}{s}\sqrt{p′(s)} ds, z = \frac{m}{ρ} − ∫_{0}^{ρ} \frac{1}{s}\sqrt{p′(s)} ds. (4.4)

Given positive constants M_{i}, i = 1, 2, consider the region R of the plane ρ–m given by −z ≤ M_{1}, w ≤ M_{2}, that is,

R = \{(ρ, m) | − M_{1}ρ + ρ ∫_{0}^{ρ} \frac{1}{s}\sqrt{p′(s)} ds ≤ m ≤ M_{2}ρ − ρ ∫_{0}^{ρ} \frac{1}{s}\sqrt{p′(s)} ds\}.

Then the region is invariant under smooth flows of the parabolic system (2.1) associated with (4.1) (cf. [8]), provided that we can show

ρ^{ε}(t, x) ≥ δ^{ε}(t), 0 < t < ∞, (4.5)

where δ^{ε}(t) > 0 depends on ε and t. Thus, we assume that the initial-boundary data (2.2)-(2.3), for the viscous systems (2.1) associated with (4.1), take values in R, for large M_{i}, i = 1, 2.

We notice that, in the region R, the flux function of (4.1) is only Lipschitz continuous because of the singularity in ρ = 0. In order to have the flux function f(ρ, m) = (m, m^{2}/ρ + p(ρ))^{⊤} partially satisfying (2.7), we artificially extend it to the half-plane {ρ < 0} as an even function. The resultant function is smooth only away from the vacuum line {ρ = 0}. Hence local (in time) smooth solutions of the problem (2.1)–(2.3), corresponding to (4.1), can be extended only while they stay in the region ρ > 0. This is the main difference between the analyses for system (4.1) and for the systems in Section 3.

For given initial-boundary data a_{0}, a_{1}, and u_{0} satisfying (1.4) and

\{ \begin{matrix}ρ_{0}(x) ≥ 0, |m_{0}(x)| ≤ C_{0}ρ_{0}(x), C_{0} > 0, \\ ρ(t, i) ≥ 0, |m(t, i)| ≤ C_{0}ρ(t, i), i = 0, 1, \end{matrix} (4.6)

there exists sequences a_{0,ε}, a_{1,ε}, and u_{0,ε} that converge to a_{0}, a_{1}, and u_{0}, respectively, in L_{loc}^{1}(0, ∞) and L^{1}(0, 1) and that satisfy (2.4)-(2.6), and

ρ_{0,ε}(x), ρ^{ε}(t, 0), ρ^{ε}(t, 1) ≥ α^{ε} > 0, (4.7)

where α^{ε} → 0 as ε → 0. Under (4.7), there exists a unique global smooth solution of the Cauchy problem for the viscous system satisfying (4.5) (see [12, 4]). Actually, the key point to construct such a solution is to show that any local smooth solution, assuming values in {ρ > 0}, defined up to a certain time T > 0 must satisfy,

ρ^{ε}(t, x) ≥ δ^{ε}(T) > 0, for 0 ≤ t < T, (4.8)

for all x ∈ (0, 1) and some δ^{ε}(T) > 0 depending on both ε and T. The proof of (4.8) in [12, 4] can be easily adapted for the initial-boundary value problem with the help of an obvious version of Theorem 2.1 for local smooth solutions. Nevertheless, we will give an alternate proof here for (4.8) for our initial-boundary problem.

Consider the equation

∂_{t}ρ + ∂_{x}(ρu) = ε∂_{xx}ρ. (4.9)

Multiplying (4.9) by v′(ρ) with v(ρ) = 1/ρ, we obtain

∂_{t}v − ε∂_{xx}v = ∂_{x}(uv) + v′′(ρu∂_{x}ρ − ε(∂_{x}ρ)^{2}) ≤ ∂_{x}(uv) + \frac{2vu^{2}}{2ε} . (4.10)

Consider the equation

∂_{t}g − ε∂_{xx}g = ∂_{x}(ug) + \frac{2gu^{2}}{ε} , x ∈ (0, 1), (4.11)

together with the conditions:

g|_{x=0} = v^{ε}(t, 0), g|_{x=1} = v^{ε}(t, 1), (4.12)

g|_{t=0} = v^{ε}(0, x), x ∈ (0, 1). (4.13)

If g is a smooth solution of (4.11)–(4.13) defined for 0 ≤ t < T, the maximum principle, applied to the difference v − g, gives v(t, x) ≤ g(t, x), for (t, x) ∈ Q_{T}. Thus, to get (4.8), all we have to do is to prove that

g(t, x) ≤ N^{ε}(T), for (t, x) ∈ Q_{T} , (4.14)

for some positive number N^{ε}(T), depending on both ε and T.

Now take any t_{0} ∈ (0, T) and consider the operator L in G_{τ} = L^{∞}((t_{0}, t_{0} + τ) × (0, 1)), with t_{0} + τ ≤ T, given by

(4.15)  L(h) =K^{ε}(t − t_{0}) ∗ g̃(t_{0}) + ∫_{t_{0}}^{t} K^{ε}(t − s) ∗ (2\frac{h̃u^{2}(s)}{ε} − ∂_{t}ζ̃(s)) ds
− ∫_{t_{0}}^{t} ∂_{x}K^{ε}(t − s) ∗ (ũh̃) ds + ζ(t),

where ζ(t, x) = (1 − x)g(t, 1) + xg(t, 0), for 0 < x < 1, t > 0, and the ˜ has the same meaning as in Section 2. This operator is a contraction mapping in G_{τ} if

2 max\{2M^{2}, MC_{0}, 1\}\sqrt{\frac{τ}{ε}} < 1, (4.16)

as one can easily verify, where M is a constant such that |u| ≤ M in R. In this case g is its unique fixed point. Now take t_{0} = T − τ_{0} with

τ_{0} = ε/[8(max{2M^{2}, MC_{0}, 1})^{2}]

and τ = τ_{0}. Define

N(t_{0}) = sup_{0≤t≤t_{0}} ∥g(s)∥_{L^{∞}(0,1)} + ∥ζ∥_{L^{∞}(Q)}.

We now show that there exists a constant N(T) > N(t_{0}) such that, if h ∈ G_{τ} satisfies

∥h(t)∥_{L^{∞}} ≤ N(T), 0 < t < T, (4.17)

then L(h) also satisfies this inequality. We can seen this as follows.

∥L(h)(t)∥_{L^{∞}} ≤N(t_{0}) + 2M^{2} \frac{τ_{0}}{ε} N(T) + τ_{0}∥∂_{t}ζ∥_{L^{∞}(Q)} + MN(T)\sqrt{\frac{τ_{0}}{ε}}
≤N(t_{0}) + τ_{0}∥∂_{t}ζ∥_{L^{∞}(Q)} + N(T)M̄\sqrt{\frac{τ_{0}}{ε}} ,

where M̄ = 2 max{2M^{2}, MC_{0}, 1}. Therefore, one deduces that the assertion is true, provided

N(T) ≥ \frac{N(t_{0}) + τ_{0}∥∂_{t}ζ∥_{L^{∞}(Q)}}{1 − M̄\sqrt{\frac{τ_{0}}{ε}}} . (4.18)

Since L is a contraction mapping in G_{τ}, bound (4.17) must also hold for g, which then proves (4.8).

Once we have proven (4.8), we can easily show the existence of a unique global smooth solution of problem (2.1)–(2.3), corresponding to (4.1). The remaining of the proof of the existence of a solution to problem (1.1)–(1.3) follows the same procedure as the one for the systems in Section 3. In the polytropic case p = κρ^{γ}, after proving (3.5) with the help of Theorem 2.1, we may use the results in [12] (γ = 1 + 2/(2k + 1), k > 1), [3] (1 < γ ≤ 5/3), [24] (γ ≥ 3), and [25] (5/3 < γ < 3) for the reduction of the Young measures to Dirac measures. The same can be done for more general pressure law p(ρ) satisfying (4.2)-(4.3), by using the reduction procedure in the recent paper [7].

Theorem 4.1. Let a_{0}, a_{1}, and u_{0} satisfy (1.4) and (4.6). Then there exists a global entropy solution of the initial-boundary problem (4.1)-(4.3) and (1.2)-(1.3) in the sense of (1.7)-(1.9) and

ρ(t, x) ≥ 0, |m(t, x)| ≤ Cρ(t, x), for some C > 0 independent of t.

Acknowledgments

Gui-Qiang Chen’s research was supported in part by the National Science Foundation grants DMS-9623203 and DMS-9708261, and by an Alfred P. Sloan Foundation Fellowship. Hermano Frid’s research was supported in part by CNPq-Brazil, proc. 352871/96-2.

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(Gui-Qiang Chen) Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

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

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

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