Contemporary Mathematics

Concentration in Solutions to Hyperbolic Conservation Laws

Gui-Qiang Chen

Abstract. The phenomenon of concentration in solutions to hyperbolic systems of conservation laws and related recent analytical techniques for studying such solutions are discussed. In particular, for the Euler equations for both isentropic and nonisentropic fluids, the phenomenon of concentration in solutions is fundamental as the pressure vanishes, which occurs not only in the multidimensional situations, but also even naturally in the one-dimensional case. ¿From the point of view of hyperbolic conservation laws, since the limit system loses hyperbolicity, the phenomenon of concentration in the process of vanishing pressure limit can be regarded as a phenomenon of resonance among the characteristic fields. This phenomenon indicates that the flux-function limit can be very singular: the limit functions of solutions are no longer in the spaces of functions, BV or L^{∞}, yet the space of Radon measures, for which the divergences of certain entropy and entropy flux fields are also Radon measures, is a natural space to deal with such a limit in general. In this regard, a theory of divergence-measure fields developed recently is also presented. This theory especially includes normal traces, a generalized Gauss-Green theorem, and product rules, among others. Some applications of this theory to solving various nonlinear problems in conservation laws and related areas are also discussed.

1. Introduction

We are concerned with the phenomenon of concentration in solutions to hyperbolic systems of conservation laws:

(1.1)  ∂_{t}u + ∇ · f(u) = 0,  x ∈ R^{d}, u ∈ R^{m},

and related recent analytical techniques for studying such solutions for (1.1).

One of the archetypes of hyperbolic systems of conservation laws is the Euler equations of isentropic gas dynamics:

∂_{t}ρ + ∂_{x}(ρv) = 0, (1.2)

∂_{t}(ρv) + ∂_{x}(ρv^{2} + P) = 0, (1.3)

1991 Mathematics Subject Classification. Primary:00-02,35L65,35L80,46E27; Secondary: 35B34,35B25, 35L67,46E40.

Key words and phrases. Concentration, hyperbolic conservation laws, flux limits, δ-shocks, measure solutions, entropy inequality, divergence-measure fields, normal traces, Gauss-Green formula.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

where ρ, P, and m = ρv represent the density, the scalar pressure, and the momentum, respectively; and ρ and m are in the physical region {(ρ, m) : ρ ≥ 0, |m| ≤ V_{0}ρ} for some V_{0} > 0. For ρ > 0, v = m/ρ is the velocity with |v| ≤ V_{0}. The scalar pressure P is a function of the density ρ and a small parameter ϵ > 0 satisfying

lim_{ϵ→0} P (ρ, ϵ) = 0;

its prototypical example is the pressure function for polytropic gases:

(1.4)  P (ρ, ϵ) = ϵp(ρ),  p(ρ) = ρ^{γ}/γ, γ > 1.

Another archetype of hyperbolic systems of conservation laws is the Euler equations of nonisentropic gas dynamics:

∂_{t}ρ + ∂_{x}(ρv) = 0, (1.5)

∂_{t}(ρv) + ∂_{x} (ρv^{2} + P ) = 0, (1.6)

∂_{t}(ρE) + ∂_{x} ((ρE + P) v) = 0, (1.7)

where ρE = ρe + \frac{1}{2}ρv^{2} is the total energy with e the internal energy, ρ and m are in a physical region {(ρ, m) : ρ ≥ 0, |m| ≤ V_{0}ρ} for some V_{0} > 0 and, for ρ > 0, v = m/ρ is the velocity with |v| ≤ V_{0}. Now the scalar pressure P is a function of the density ρ, the internal energy e, and a small parameter ϵ > 0 satisfying

lim_{ϵ→0} P (ρ, e, ϵ) = 0;

its prototypical example is now the pressure function for polytropic gases:

(1.8)  P (ρ, e, ϵ) = ϵ p(ρ, e),  p(ρ, e) = (γ − 1)ρe.

The phenomenon of concentration in solutions of the two-dimensional Riemann problem was first observed numerically in Chang-Chen-Yang [10, 11], which leads to the occurrence of so called smoothed δ-shocks, for the Euler equations of gas dynamics, when the Riemann data produces four initial contact discontinuities with different signs and the initial pressure data is close to zero.

In this paper, we first discuss the results in Chen-Liu [17, 18] to show that the phenomenon of concentration in solutions, observed numerically in [10, 11], for inviscid compressible fluid flow is fundamental, which occurs not only in the multidimensional situations, but also even naturally in the one-dimensional case.

As ϵ → 0, the limit system formally becomes the following transport equations:

∂_{t}ρ + ∂_{x}(ρv) = 0, (1.9)

∂_{t}(ρv) + ∂_{x}(ρv^{2}) = 0, (1.10)

which are also called the one-dimensional system of pressureless Euler equations, modeling the motion of free particles which stick under collision (see [7, 22, 49, 64]). System (1.9)–(1.10) has been analyzed extensively since 1994; for example, see [4, 5, 7, 22, 28, 31, 37, 38, 39, 44, 56, 61] and the references cited therein. In particular, the existence of measure solutions of the Riemann problem was first presented in Bouchut [4], and a connection of (1.9)–(1.10) with adhesion particle dynamics and the behavior of global weak solutions with random initial data were discussed in E-Rykov-Sinai [22]. Also see [33, 34, 35, 48, 54, 55] for related equations and results. It has been shown that, for the transport equations (1.9)– (1.10), δ-shocks do occur in the Riemann solutions.

In Sections 2–4, we discuss the phenomenon of concentration and the formation of δ-shocks in solutions to the Euler equations for both isentropic and nonisentropic fluids as the pressure vanishes. We show that the phenomenon of concentration occurs naturally in the one-dimensional case as the pressure vanishes: any Riemann solution containing two shocks to the Euler equations for isentropic or nonisentropic fluids tends to a δ-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted δ-measure that forms a δ-shock. These results show that the δ-shocks for the transport equations result from the phenomenon of concentration in the process of vanishing pressure limit.

From the point of view of hyperbolic conservation laws, since the limit system loses hyperbolicity, the phenomenon of concentration in the process of vanishing pressure limit can be regarded as a phenomenon of resonance between the characteristic fields. This phenomenon shows that the flux-function limit can be very singular: the limit functions of solutions are no longer in the spaces of functions, BV or L^{∞}, yet the space of Radon measures, for which the divergences of certain entropy and entropy flux fields are also Radon measures, is a natural space to deal with such a limit in general. In this regard, in Section 5, we discuss a theory of divergence-measure fields, developed recently in Chen-Frid [13, 14, 15], for dealing with entropy solutions with concentration, that is, measure solutions satisfying the Lax entropy inequality. Some applications of this theory to solving several nonlinear problems in conservation laws and related areas are also discussed.

2. δ-Shocks for the Euler Equations for Pressureless Fluids

Consider the Riemann problem of the transport equations (1.9)–(1.10) with Riemann initial data

(2.1)  (ρ, v)(0, x) = (ρ_{±}, v_{±}),  ± x > 0,

with ρ_{±} > 0. Since the equations and the Riemann data are invariant under uniform stretching of coordinates:

(t, x) → (βt, βx), β constant,

we consider the self-similar solutions of (1.9), (1.10), and (2.1):

(ρ, v)(t, x) = (ρ, v)(ξ), ξ = x/t.

Then the Riemann problem is reduced to a boundary value problem for ordinary differential equations:

− ξρ_{ξ} + (ρv)_{ξ} = 0,
− ξ(ρv)_{ξ} + (ρv^{2})_{ξ} = 0,
(ρ, v)(±∞) = (ρ_{±}, v_{±}).

In the case v_{−} > v_{+}, a key observation in [56] is that the singularity cannot be a jump with finite amplitude, that is, there is no solution which is piecewise smooth and bounded; hence a solution containing a weighted δ-measure (i.e. δ-shock) supported on a line was constructed in order to establish the existence in a space of measures (also see [54, 55]).

To define the measure solutions, the weighted δ-measure w(t)δ_{S} supported on a smooth curve S = {(t(s), x(s)) : a < s < b} can be defined by

⟨w(·)δ_{S}, ψ(·, ·)⟩ = ∫_{a}^{b} w(t(s))ψ(x(s), t(s))\sqrt{t′(s)^{2} + x′(s)^{2}}ds

for any ψ ∈ C_{0}^{∞}(R_{+} × R).

With this definition, a family of δ-measure solutions with parameter σ in the case v_{−} > v_{+} can be obtained as

ρ(t, x) = ρ_{0}(t, x) + w_{1}(t)δ_{S}, v(t, x) = v_{0}(t, x),

where S = {(t, σt) : 0 ≤ t < ∞},

ρ_{0}(t, x) = ρ_{−}+[ρ] χ(x−σt), v_{0}(t, x) = v_{−}+[v] χ(x−σt), w_{1}(t) = \frac{t}{1 + σ^{2}} (σ[ρ]−[ρv]),

in which [h] := h_{+} − h_{−} denotes the jump of function h across the discontinuity, and χ(x) is the characteristic (or indicator) function that is 0 when x < 0 and 1 when x > 0.

As shown in [56], the δ-measure solutions (ρ, v) constructed above satisfy

⟨ρ, ϕ_{t}⟩ + ⟨ρv, ϕ_{x}⟩ = 0,
⟨ρv, ϕ_{t}⟩ + ⟨ρv^{2}, ϕ_{x}⟩ = 0,

for any ϕ ∈ C_{0}^{∞}(R_{+} × R), where

⟨ρ, ϕ⟩ = ∫_{0}^{∞} ∫_{−∞}^{∞} ρ_{0}ϕ dxdt + ⟨w_{1}δ_{S}, ϕ⟩,

and

⟨ρv, ϕ⟩ = ∫_{0}^{∞} ∫_{−∞}^{∞} ρ_{0}v_{0}ϕ dxdt + ⟨w_{2}δ_{S}, ϕ⟩,

where w_{2}(t) = \frac{t}{1+σ^{2}} (σ[ρv] − [ρv^{2}]).

A unique solution can be singled out by the so called δ-Rankine-Hugoniot condition:

(2.2)  σ = \frac{\sqrt{ρ_{+}}v_{+} + \sqrt{ρ_{−}}v_{−}}{\sqrt{ρ_{+}} + \sqrt{ρ_{−}}}

that satisfies the δ−entropy condition:

(2.3)  v_{+} < σ < v_{−}.

The entropy condition (2.3) means that, in the (t, x)-plane, all the characteristic lines on either side of a δ-shock run into the line of δ-shock, which implies that a δ-shock is an overcompressive shock. The δ-entropy condition can be derived from the Lax entropy inequality:

(2.4)  ∂_{t}(ρv^{2}) + ∂_{x}(ρv^{3}) ≤ 0

in the sense of distributions, that is, for any ϕ ∈ C_{0}^{∞}(R_{+} × R), ϕ ≥ 0, we have

⟨ρv^{2}, ϕ_{t}⟩ + ⟨ρv^{3}, ϕ_{x}⟩ ≥ 0.

3. Concentration in Solutions to the Euler Equations for Isentropic Fluids

In this section, we discuss the phenomenon of concentration in solutions to system (1.2)–(1.4). We start with Riemann solutions.

3.1. Riemann Solutions. The eigenvalues of system (1.2)–(1.4) are

λ_{1} = v − \sqrt{ϵ p′(ρ)}, λ_{2} = v + \sqrt{ϵ p′(ρ)}, for ρ > 0.

The Riemann invariants are

w = v + ∫_{0}^{ρ} \frac{\sqrt{ϵp′(s)}}{s}ds, z = v − ∫_{0}^{ρ} \frac{\sqrt{ϵp′(s)}}{s}ds.

Then the Riemann solutions, which are functions of ξ = x/t, are solutions of

− ξρ_{ξ} + (ρv)_{ξ} = 0, (3.1)

− ξ(ρv)_{ξ} + (ρv^{2} + ϵ p(ρ))_{ξ} = 0, (3.2)

(ρ, v)(±∞) = (ρ_{±}, v_{±}). (3.3)

Shock Curves. The Rankine-Hugoniot conditions for discontinuous solutions of (1.2)–(1.4) are

−σ[ρ] + [ρv] = 0, −σ[ρv] + [ρv^{2} + ϵ p(ρ)] = 0.

The Lax entropy conditions imply

(3.4)  ρ_{+} > ρ_{−} (1-shock),  ρ_{+} < ρ_{−} (2-shock).

Then, given a state u_{−} = (ρ_{−}, ρ_{−}v_{−}), the shock curves in the phase plane, which are the sets of states that can be connected on the right by a 1-shock or 2-shock, are the following.

1-shock curve S_{1}(u_{−}):

v − v_{−} = −\sqrt{\frac{ϵ (p(ρ) − p(ρ_{−}))}{ρ_{−}ρ(ρ − ρ_{−})}}(ρ − ρ_{−}), ρ > ρ_{−};

2-shock curve S_{2}(u_{−}):

v − v_{−} = −\sqrt{\frac{ϵ (p(ρ) − p(ρ_{−}))}{ρ_{−}ρ(ρ − ρ_{−})}}(ρ − ρ_{−}), ρ < ρ_{−}.

Then the shock curves are concave or convex, respectively, with respect to the point u_{−} = (ρ_{−}, ρ_{−}v_{−}) in the ρ−m plane with m = ρv, that is, the quotient \frac{m−m_{−}}{ρ−ρ_{−}} as a function of ρ is monotone.

Similarly, we can define the 1-rarefaction wave curve R_{1}(u_{−}) and the 2-rarefaction wave curve R_{2}(u_{−}).

Given a left state u_{−} = (ρ_{−}, ρ_{−}v_{−}), the set of states that can be connected on the right by a shock or a rarefaction wave in the phase plane consists of the 1-shock curve S_{1}(u_{−}), the 1-rarefaction curve R_{1}(u_{−}), the 2-shock curve S_{2}(u_{−}), and the 2-rarefaction curve R_{2}(u_{−}). These curves divide the phase plane into four regions S_{2}S_{1}(u_{−}), S_{2}R_{1}(u_{−}), R_{2}S_{1}(u_{−}), and R_{2}R_{1}(u_{−}); any right state of the Riemann data staying in one of them yields a unique global Riemann solution R(x/t), which contains a 1-shock (or 1-rarefaction wave) and/or a 2-shock (or 2-rarefaction wave) satisfying

w(R(x/t)) ≤ w(u_{+}), z(R(x/t)) ≥ z(u_{−}), w(R(x/t)) − z(R(x/t)) ≥ 0.

In particular, when u_{+} ∈ S_{2}S_{1}(u_{−}), R(x/t) contains a 1-shock, a 2-shock, and a nonvacuum intermediate constant state. For more details about Riemann solutions, see [20, 51].

3.2. Concentration in the Vanishing Pressure Limit. We now discuss the phenomenon of concentration and the formation of δ-shocks in the Riemann solutions to (1.2)–(1.4) in the case u_{+} ∈ S_{2}S_{1}(u_{−}) with v_{−} > v_{+} and ρ_{±} > 0 as the pressure vanishes.

For fixed ϵ > 0, let u_{∗}^{ϵ} := (ρ_{∗}^{ϵ}, ρ_{∗}^{ϵ}v_{∗}^{ϵ}) be the intermediate state in the sense that u_{−} and u_{∗}^{ϵ} are connected by 1-shock S_{1} with speed σ_{1}^{ϵ}, and that u_{∗}^{ϵ} and u_{+} are connected by 2-shock S_{2} with speed σ_{2}^{ϵ}. Then (ρ_{∗}^{ϵ}, v_{∗}^{ϵ}) is determined by

(3.5)  v_{∗}^{ϵ} − v_{−} = −\sqrt{\frac{ϵ (p(ρ_{∗}^{ϵ}) − p(ρ_{−}))}{ρ_{−}ρ_{∗}^{ϵ}(ρ_{∗}^{ϵ} − ρ_{−})}}(ρ_{∗}^{ϵ} − ρ_{−}),  ρ_{∗}^{ϵ} > ρ_{−},

and

(3.6)  v_{+} − v_{∗}^{ϵ} = −\sqrt{\frac{ϵ (p(ρ_{+}) − p(ρ_{∗}^{ϵ}))}{ρ_{+}ρ_{∗}^{ϵ}(ρ_{+} − ρ_{∗}^{ϵ})}}(ρ_{+} − ρ_{∗}^{ϵ}),  ρ_{∗}^{ϵ} > ρ_{+}.

A combination of the jump conditions (3.5) and (3.6) with the Lax entropy conditions (3.4) yields

Lemma 3.1. (i) lim_{ϵ→0} ϵ^{1/γ}ρ_{∗}^{ϵ} = (\frac{\sqrt{ρ_{−}ρ_{+}}}{\sqrt{ρ_{−}}+\sqrt{ρ_{+}}} (v_{−} − v_{+}))^{1/γ} .
(ii) Set σ = \frac{\sqrt{ρ_{−}}v_{−}+\sqrt{ρ_{+}}v_{+}}{\sqrt{ρ_{−}}+\sqrt{ρ_{+}}} ∈ (v_{+}, v_{−}). Then

lim_{ϵ→0} v_{∗}^{ϵ} = lim_{ϵ→0} σ_{1}^{ϵ} = lim_{ϵ→0} σ_{2}^{ϵ} = σ, lim_{ϵ→0} ρ_{∗}^{ϵ}(σ_{2}^{ϵ} − σ_{1}^{ϵ}) = σ[ρ] − [ρv].

The quantity σ, which is the limit of v_{∗}^{ε}, σ_{1}^{ε}, and σ_{2}^{ε}, uniquely determines the δshock solution of (1.9)–(1.10) as the limit of the Riemann solutions when ε → 0 and is consistent with both the δ-Rankine-Hugoniot condition (2.2) and the δ-entropy condition (2.3), as proposed for the Riemann solutions of the pressureless Euler equations in Section 2.

The following theorem characterizes the vanishing pressure limit in the case v_{−} > v_{+}, which can be proved from the weak formulations and Lemma 3.1 and by tracking the time-dependence of the weights of δ-measures as ϵ → 0.

Theorem 3.1. Let v_{−} > v_{+}. For each fixed ϵ > 0, assume that (ρ^{ϵ}, m^{ϵ}) = (ρ^{ϵ}, ρ^{ϵ}v^{ϵ}) is a two-shock solution of (1.2)–(1.4) with Riemann data u_{±} = (ρ_{±}, ρ_{±}v_{±}). Then, as ϵ → 0, ρ^{ϵ} and m^{ϵ} converge in the sense of distributions, and the limit functions ρ and m are the sums of a step function and a δ-measure with weights

\frac{t}{\sqrt{1 + σ^{2}}} (σ[ρ] − [ρv]) and \frac{t}{\sqrt{1 + σ^{2}}} (σ[ρv] − [ρv^{2}]),

respectively, which form a δ-shock solution of (1.9)–(1.10) with the same Riemann data u_{±}.

The formation process of δ-shocks in the Riemann solutions can be described as the pressure decreases as follows: As ε decreases, the locations of the two shocks become closer, and the density of the intermediate state increases dramatically, while the velocity is closer to a step function. In the vanishing pressure limit, the two shocks coincide to form, along with the intermediate state, a δ-shock of the transport equations (1.9)–(1.10), while the velocity becomes a step function.

4. Concentration in Solutions to the Euler Equations for Nonisentropic Fluids

In this section, we discuss the phenomenon of concentration in solutions to the Euler equations for nonisentropic fluids.

4.1. Riemann Solutions. The Euler equations (1.5)–(1.8) for smooth solutions with ρ > 0 can be written as

∂_{t}u + A(u)∂_{x}u = 0, u = (ρ, ρv, ρE)^{⊤},

where

A(u) = ( \begin{matrix}0 & 1 & 0 \\ \frac{(γ−1)ϵ−2}{2}v^{2} & (2 − ϵ(γ − 1))v & ϵ(γ − 1) \\ −\frac{1}{ϵ(γ−1)}(c^{ε})^{2}v + \frac{ϵ(γ−1)−1}{2}v^{3} & \frac{1}{ϵ(γ−1)}(c^{ε})^{2} + \frac{1−2ϵ(γ−1)}{2}v^{2} & (1 + ϵ(γ − 1))v \end{matrix} ) .

Then the eigenvalues of system (1.5)–(1.8) are

λ_{1} = v − c^{ε}, λ_{2} = v, λ_{3} = v + c^{ε}, for ρ > 0,

with associated right eigenvectors

r_{j} = ( \begin{matrix}1 \\ v + sgn(j − 2)c^{ε} \\ \frac{1}{2}v^{2} + sgn(j − 2)c^{ε}v + \frac{sgn|j−2|}{ϵ(γ−1)}(c^{ε})^{2} \end{matrix} ) ,

and left eigenvectors

l_{j} = (\frac{1}{2}v^{2} − \frac{sgn(j − 2)}{ϵ(γ − 1)}c^{ε}v + \frac{(sgn|j − 2| − 1)}{ϵ(γ − 1)}(c^{ε})^{2}, −v + \frac{sgn(j − 2)}{ϵ(γ − 1)}c^{ε}, 1) ,

where c^{ε} = c(e, ϵ) := \sqrt{εγ_{ϵ}(γ − 1)e} and γ_{ϵ} = 1 + ϵ(γ − 1).

The Riemann invariants along the characteristic fields are

(4.1)  \{S^{ε}, v + \frac{2}{ϵ(γ − 1)}c^{ϵ}\} , \{v, p\}, \{S^{ϵ}, v − \frac{2}{ϵ(γ − 1)}c^{ϵ}\} ,

respectively, where S^{ϵ} is determined by

(4.2)  p = κe^{S^{ϵ}}ρ^{γ_{ε}}, κ > 0.

The Riemann solutions, which are the functions of ξ = x/t, are solutions of

− ξρ_{ξ} + (ρv)_{ξ} = 0,
− ξ(ρv)_{ξ} + (ρv^{2} + ϵp)_{ξ} = 0,
− ξ(ρE)_{ξ} + ((ρE + ϵp)v)_{ξ} = 0,
(ρ, v, E)(±∞) = (ρ_{±}, v_{±}, E_{±}).

Then the Rankine-Hugoniot conditions for discontinuous solutions to (1.5)–(1.8) on a discontinuity are

σ[ρ] = [ρv], σ[ρv] = [ρv^{2} + ϵp], σ[ρE] = [(ρE + ϵp)v],

which can be rewritten in terms of w := v − σ and J := ρw = ρv − σρ as

[J] = 0, [Jw + ϵp] = 0, J [\frac{2}{ϵ(γ − 1)}(c^{ϵ})^{2} + w^{2}] = 0.

Contact Discontinuities. If σ = v_{l} = v_{r}, the discontinuity is a contact discontinuity S_{c}, which corresponds to the second family of characteristic fields. Then

[v] = 0, [p] = 0.

Shock Curves. If σ ≠ v_{l}, v_{r}, then the discontinuity is a shock wave, which corresponds to either the first or the third family of characteristic fields. Then the Lax entropy conditions and the Rankine-Hugoniot conditions imply that, on a 1-shock S_{1},

(4.3)  [p] > 0, [ρ] > 0, [v] < 0,

and, on a 3-shock S_{3},

(4.4)  [p] < 0, [ρ] < 0, [v] < 0.

We now examine the dependence of the two one-parameter families of shock curves on ϵ.

Set

α_{ϵ} = \frac{γ − 1}{2γ_{ϵ}}, β_{ϵ} = \frac{2}{ϵ(γ − 1)} + 1.

Then we obtain the following formulas for the j-shock curves, j = 1, 3, in terms of the parametrization s ≤ 0, respectively.

1-shock curve: For s ≤ 0,

\frac{p_{r}}{p_{l}} = e^{−s}, \frac{ρ_{r}}{ρ_{l}} = f_{1}^{ε}(s) := \frac{β_{ϵ} + e^{s}}{1 + β_{ϵ}e^{s}} , \frac{v_{r} − v_{l}}{c_{l}^{ε}} = h_{1}^{ε}(s) := \frac{2\sqrt{α_{ϵ}/ε}}{γ − 1} \frac{1 − e^{−s}}{\sqrt{1 + β_{ϵ}e^{−s}}} .

3-shock curve: For s ≤ 0,

\frac{p_{r}}{p_{l}} = e^{s}, \frac{ρ_{r}}{ρ_{l}} = f_{3}^{ε}(s) := \frac{β_{ϵ} + e^{−s}}{1 + β_{ϵ}e^{−s}} , \frac{v_{r} − v_{l}}{c_{l}^{ε}} = h_{3}^{ε}(s) := \frac{2\sqrt{α_{ϵ}/ε}}{γ − 1} \frac{e^{s} − 1}{\sqrt{1 + β_{ϵ}e^{s}}} .

Riemann Solutions. Consider the Riemann problem (1.5)–(1.8) with Riemann data:

(4.5)  u(0, x) = u_{±}, ±x > 0,

and u_{+} ∈ S_{2}S_{c}S_{1}(u_{−}). Then the Riemann solution R(x/t) may contain a 1-shock, a 2-contact discontinuity, a 3-shock, and two nonvacuum intermediate constant states between the shocks and the contact discontinuity. Hence, there exist s_{1}^{ε}, s_{2}^{ε}, and s_{3}^{ε} determined by

ρ_{+} = f_{1}^{ε}(s_{1}^{ε})e^{s_{2}^{ε}}f_{3}^{ε}(s_{3}^{ε})ρ_{−}, (4.6)

p_{+} = e^{s_{3}^{ε}−s_{1}^{ε}}p_{−}, (4.7)

v_{+} = v_{−} + c_{−}^{ε} (h_{1}^{ε}(s_{1}^{ε}) + \frac{e^{−(s_{1}^{ε}+s_{2}^{ε})/2}}{\sqrt{f_{1}^{ε}(s_{1}^{ε})}} h_{3}^{ε}(s_{3}^{ε})) . (4.8)

Define

A = ρ_{+}/ρ_{−}, B = p_{+}/p_{−}, C^{ϵ} = (v_{+} − v_{−})/c_{−}^{ϵ}.

Then one can find s_{i}^{ε}, i = 1, 2, 3, successively as follows:

h_{1}^{ε}(s_{1}^{ε}) + \sqrt{\frac{B}{A}} h_{1}^{ε}(s_{1}^{ε} + logB) = C^{ϵ}, (4.9)

s_{3}^{ε} = s_{1}^{ε} + logB, (4.10)

s_{2}^{ε} = log (\frac{A}{f_{1}^{ε}(s_{1}^{ε})f_{3}^{ε}(s_{3}^{ε})}) , (4.11)

where we have used the fact that f_{1}^{ε}(s)f_{3}^{ε}(s) = 1 and h_{3}^{ε}(s) = e^{s/2}h_{1}^{ε}(s)\sqrt{f_{1}^{ε}(s)}. In general, we have

Theorem 4.1. For any fixed ε > 0, there exists a unique Riemann solution R(x/t), staying away from vacuum, of the Riemann problem (1.5)–(1.8) with Riemann data (4.5) if and only if v_{+}−v_{−} < \frac{2(c_{−}^{ϵ}+c_{+}^{ϵ})}{ε(γ−1)}. Otherwise, the vacuum is present in the Riemann solution.

For more details about Riemann solutions, see [51, 52].

4.2. Concentration in the Vanishing Pressure Limit. We now discuss the phenomenon of concentration and the formation of δ-shocks in the Riemann solutions to the Euler equations for nonisentropic fluids in the case u_{+} ∈ S_{2}S_{c}S_{1}(u_{−}) with v_{−} > v_{+} and ρ_{±} > 0 as the pressure vanishes.

Let u_{i}^{ε} := (ρ_{i}^{ε}, ρ_{i}^{ε}v_{∗}^{ε}, ρ_{i}^{ε}E_{i}^{ε}), i = 1, 2, with p_{1}^{ε} = p_{2}^{ε} = p_{∗}^{ε}, be the intermediate states in the sense that (p_{−}, ρ_{−}, v_{−}) and (p_{∗}^{ε}, ρ_{1}^{ε}, v_{∗}^{ε}) determine a 1-shock S_{1} with speed σ_{1}^{ϵ} and that (p_{∗}^{ε}, ρ_{2}^{ε}, v_{∗}^{ε}) and (p_{+}, ρ_{+}, v_{+}) determine a 3-shock S_{3} with speed σ_{3}^{ϵ}. The difference ρ_{2}^{ε} − ρ_{1}^{ε} is the jump of density which determines a 2-contact discontinuity S_{c}. Based on these facts, we can obtain the limits of s_{i}^{ε}, i = 1, 2, 3, as ϵ → 0, respectively.

Lemma 4.1. If v_{−} > v_{+}, then

(i) lim_{ϵ→0} ϵ e^{−s_{1}^{ε}} = q_{1} := \frac{ρ_{+}ρ_{−}(v_{+}−v_{−})^{2}}{p_{−}(\sqrt{ρ_{+}}+\sqrt{ρ_{−}})^{2}} , lim_{ϵ→0} ϵ e^{−s_{3}^{ε}} = q_{3} := \frac{ρ_{+}ρ_{−}(v_{+}−v_{−})^{2}}{p_{+}(\sqrt{ρ_{+}}+\sqrt{ρ_{−}})^{2}} ,

lim_{ϵ→0} e^{s_{2}^{ε}} = q_{2} := \frac{2+(γ−1)q_{3}}{2+(γ−1)q_{1}} ; (ii) As ϵ → 0, the intermediate densities ρ_{1}^{ε} and ρ_{2}^{ε} become unbounded, i.e.,

lim_{ϵ→0} ϵρ_{1}^{ε} = \frac{2ρ_{−}q_{1}}{2 + (γ − 1)q_{1}} ,  lim_{ϵ→0} ϵρ_{2}^{ε} = \frac{2ρ_{+}q_{3}}{2 + (γ − 1)q_{3}} ;

(iii) The intermediate velocity v_{∗}^{ε} and pressure p_{∗}^{ε} satisfy

lim_{ϵ→0} v_{∗}^{ε} = σ := \frac{\sqrt{ρ_{−}}v_{−} + \sqrt{ρ_{+}}v_{+}}{\sqrt{ρ_{+}} + \sqrt{ρ_{−}}} , lim_{ϵ→0} ϵp_{∗}^{ε} = \frac{ρ_{+}ρ_{−}(v_{+} − v_{−})^{2}}{(\sqrt{ρ_{+}} + \sqrt{ρ_{−}})^{2}} ;

(iv) lim_{ϵ→0} σ_{1}^{ε} = lim_{ϵ→0} σ_{2}^{ε} = lim_{ϵ→0} σ_{3}^{ε} = \frac{\sqrt{ρ_{−}}v_{−}+\sqrt{ρ_{+}}v_{+}}{\sqrt{ρ_{+}}+\sqrt{ρ_{−}}} = σ ∈ (v_{+}, v_{−});
(v) lim_{ϵ→0} ρ_{1}^{ε}(σ_{1}^{ε} − σ_{2}^{ε}) = \frac{ρ_{−}\sqrt{ρ_{+}}}{\sqrt{ρ_{−}}+\sqrt{ρ_{+}}} (v_{+} − v_{−}),

lim_{ϵ→0} ρ_{2}^{ε}(σ_{2}^{ε} − σ_{3}^{ε}) = \frac{ρ_{+}\sqrt{ρ_{−}}}{\sqrt{ρ_{−}}+\sqrt{ρ_{+}}} (v_{+} − v_{−}).

Again, the quantity σ, which is the limit of v_{∗}^{ε} and σ_{j}^{ε}, j = 1, 2, 3, in Lemma 4.1, uniquely determines the δ-shock solution as the limit of the Riemann solutions when ε → 0 and is consistent with both the δ-Rankine-Hugoniot condition (2.2) and the δ-entropy condition (2.3).

The following theorem characterizes the vanishing pressure limit for the case v_{−} > v_{+}.

Theorem 4.2. Let (ρ^{ϵ}, ρ^{ϵ}v^{ϵ}, ρ^{ε}E^{ϵ}) be the Riemann solutions to (1.5)–(1.8) and (4.5), which contain two shocks and possibly one contact discontinuity. Then ρ^{ϵ}, ρ^{ϵ}v^{ϵ}, and ρ^{ε}E^{ϵ} converge in the sense of distributions, respectively, and the limit functions are all sums of a step function and a δ-measure with weights

\frac{t}{\sqrt{1 + σ^{2}}} ([ρv] − σ[v]), \frac{t}{\sqrt{1 + σ^{2}}} ([ρv^{2}] − σ[ρv]), \frac{t}{\sqrt{1 + σ^{2}}} ([ρvE] − σ[ρE]),

respectively, where σ = (\sqrt{ρ_{−}}v_{−} + \sqrt{ρ_{+}}v_{+})/(\sqrt{ρ_{+}} + \sqrt{ρ_{−}}).

Furthermore, we can conclude the entropy consistency by proving that the conservation law of energy (1.7) actually yields the correct entropy inequality (2.4) for the transport equations (1.9) and (1.10).

Theorem 4.3. The limit functions (ρ, v) are a measure solution of the transport equation (1.9)–(1.10) satisfying

∂_{t}(ρv^{2}) + ∂_{x}(ρv^{3}) ≤ 0

in the sense of distributions.

This can be seen as follows. Since ∂_{t}(ρ^{ε}E^{ε}) + ∂_{x}(ρ^{ε}v^{ε}E^{ε}) = 0 in the sense of distributions and ρE = \frac{p}{γ−1} + \frac{1}{2}ρv^{2}, then, for any nonnegative test function ϕ ∈ C_{0}^{∞}(R_{+} × R),

− (⟨ρ^{ε}(v^{ε})^{2}, ϕ_{t}⟩ + ⟨ρ^{ε}(v^{ε})^{3}, ϕ_{x}⟩) = \frac{2}{γ − 1} (⟨p^{ε}, ϕ_{t}⟩ + ⟨p^{ε}v^{ε}, ϕ_{x}⟩) .

Note that

lim_{ϵ→0} ∫_{0}^{∞} ∫_{−∞}^{∞}(p^{ε}ϕ_{t} + p^{ε}v^{ε}ϕ_{x})dxdt

= lim_{ϵ→0} ∫_{0}^{∞} (σ_{1}^{ε}(p_{∗}^{ε} − p_{−}) − (p_{∗}^{ε}v_{∗}^{ε} − p_{−}v_{−})) ϕ(σ_{1}^{ε}t, t)dt

+ lim_{ϵ→0} ∫_{0}^{∞} σ_{2}^{ε} ((p_{+} − p_{∗}^{ε}) − (p_{+}v_{+} − p_{∗}^{ε}v_{∗}^{ε})) ϕ(σ_{2}^{ε}t, t)dt

= ∫_{0}^{∞}(v_{−} − v_{+})(αp_{+} + (1 − α)p_{−})ϕ(σt, t)dt < 0,

since α = \frac{\sqrt{ρ_{−}}}{\sqrt{ρ_{+}}+\sqrt{ρ_{−}}} ∈ (0, 1). This verifies the entropy consistency as claimed.

The process of concentration in the Riemann solutions to the Euler equations (1.5)–(1.8) can be seen when the pressure vanishes as follows: As ε decreases, the locations of the two shocks become closer to the contact discontinuity, and the densities of the intermediate states increase dramatically, while the velocity is closer to a step function; in the vanishing pressure limit, the two shocks and the contact discontinuity coincide to form, along with the intermediate states, a δ-shock of the transport equations (1.9) and (1.10), while the velocity becomes a step function.

5. Divergence-Measure Fields for Conservation Laws

In this section, we discuss a theory of divergence-measure fields (DM-fields, for short), developed recently in Chen-Frid [13, 14, 15], in order to deal with entropy solutions with concentration.

5.1. Conservation Laws and Divergence-Measure Fields. We first discuss some connection between hyperbolic conservation laws and DM-fields.

The DM-fields arise naturally in the study of entropy solutions of nonlinear hyperbolic systems of conservation laws, which take the form (1.1). The main feature of nonlinear hyperbolic conservation laws, especially systems (1.2)–(1.4) and (1.5)– (1.8), is that solutions may develop singularities and form shock waves in finite time, no matter how smooth the initial data is. One may expect solutions in the space of functions of bounded variation. This is indeed the case by the Glimm theorem [26] which indicates that, when the initial data has sufficiently small total variation and the one -dimensional system is strictly hyperbolic, there exists a global entropy solution in BV. However, when the initial data becomes large, and/or strict hyperbolicity of the system fails (especially for the multidimensional case), the solutions are no longer in the space of functions of bounded variation and may become measures in finite time, even instantaneously as t > 0 (see [14, 45]). Furthermore, as discussed in Sections 2–4, the concentration occurs and the δ-shocks form in the vanishing pressure limit of solutions to the Euler systems (1.2)–(1.4) and (1.5)–(1.8). These indicate that solutions of nonlinear hyperbolic conservation laws are generally in M(R_{+} × R^{d}), the space of signed Radon measures.

Definition 5.1. A function η : R^{m} → R is called an entropy of (1.1) if there exists q : R^{m} → R^{d} such that

(5.1)  ∇q_{k}(u) = ∇η(u)∇f_{k}(u),  k = 1, 2, ⋯, d.

The function q(u) is called the entropy flux associated with the entropy η(u), and the pair (η(u), q(u)) is called an entropy pair. The entropy pair (η(u), q(u)) is called a convex entropy pair on the domain K ⊂ R^{m} if the Hessian matrix ∇^{2}η(u) ≥ 0, for u ∈ K. The entropy pair (η(u), q(u)) is called a strictly convex entropy pair on the domain K if ∇^{2}η(u) > 0 for u ∈ K.

Friedrichs-Lax [25] observed that most of the systems of conservation laws that result from continuum mechanics are endowed with a globally defined, strictly convex entropy.

Available existence theories show that the solutions u(t, x) of (1.1) are generally in the following class of entropy solutions:

i) u(t, x) ∈ M(R_{+} × R^{d}), or L^{p}(R_{+} × R^{d}), 1 ≤ p ≤ ∞;

ii) u(t, x) satisfies the Lax entropy inequality:

(5.2)  ∂_{t}η(u(t, x)) + ∇_{x} · q(u(t, x)) ≤ 0

in the sense of distributions, for any entropy pair (η, q) : R^{m} → R×R^{d} with convex η, ∇^{2}η(u) ≥ 0, so that (η(u(t, x)) and q(u(t, x)) are distributional functions. If we take η = ±u, we see that any entropy solution is a weak solution.

One of the main issues in conservation laws is to study the behavior of solutions in this class to explore all possible information of solutions, including large-time behavior, uniqueness, stability, and traces of solutions, among others. The Schwartz lemma indicates from (5.2) that the distribution

∂_{t}η(u(t, x)) + ∇_{x} · q(u(t, x))

is in fact a Radon measure, that is,

(5.3)  div_{(t,x)}(η(u(t, x)), q(u(t, x))) ∈ M(R_{+} × R^{d}).

Divergence-measure fields also arise in various nonlinear problems involving some extended vector fields whose divergences are Radon measures. For example, see Bouchitté-Buttazzo [3] for such fields in the characterization of optimal shapes and masses through the Monge-Kantorovich equation. It is clear that understanding more properties of DM-fields can advance our understanding of the behavior of entropy solutions for hyperbolic conservation laws and other related nonlinear equations.

5.2. Divergence-Measure Fields. Divergence-measure fields are extended vector fields, including vector fields in L^{p} and vector-valued Radon measures, whose divergences are Radon measures. More precisely, we have

Definition 5.2. Let Ω ⊂ R^{N} be open. For F ∈ L^{p}(Ω; R^{N}), 1 ≤ p ≤ ∞, or F ∈ M(Ω; R^{N}), set

|div F |(Ω) := sup{ ⟨F, ∇φ⟩ : φ ∈ C_{0}^{1}(Ω), |φ(x)| ≤ 1, x ∈ Ω }.

For 1 ≤ p ≤ ∞, we say that F is an L^{p} divergence-measure field over Ω, i.e., F ∈ DM^{p}(Ω), if

(5.4)  ∥F ∥_{DM^{p}(Ω)} := ∥F ∥_{L^{p}(Ω;R^{N})} + |div F |(Ω) < ∞.

We say that F is an extended divergence-measure field over Ω, i.e., F ∈ DM^{ext}(Ω), if

(5.5)  ∥F ∥_{DM^{ext}(Ω)} := |F |(D) + |div F |(D) < ∞.

If F ∈ DM^{p}(Ω) for any open set Ω with Ω ⋐ D ⊂ R^{N}, then we say F ∈ DM_{loc}^{p}(D); and, if F ∈ DM^{ext}(Ω) for any open set Ω with Ω ⋐ D ⊂ R^{N}, we say F ∈ DM_{loc}^{ext}(D). We denote F ∈ DM(Ω) either F ∈ DM^{p}(Ω) or F ∈ DM^{ext}(Ω). Here, for open sets A, B ⊂ R^{N}, the relation A ⋐ B means that the closure of A, Ā, is a compact subset of B.

These spaces under the norms (5.4) and (5.5) are Banach spaces, respectively. These spaces are larger than the space of vector fields of bounded variation. The establishment of the Gauss-Green theorem, traces, and other properties of BV functions in the middle of last century (see Federer [24]) has advanced significantly our understanding of solutions of nonlinear partial differential equations and nonlinear problems in calculus of variations, differential geometry, and other areas. A natural question is whether the DM-fields have similar properties, especially the traces and the Gauss-Green formula as for the BV functions. At a first glance, it seems unclear.

First, observe that one cannot define the traces for each component of a DM field over any Lipschitz boundary in general, as opposed to the case of BV fields. This fact can be easily seen through the following example.

Example 5.1: The field F (x, y) = (cos(\frac{1}{x−y}), cos(\frac{1}{x−y})) belongs to DM^{∞}(R^{2}). It is impossible to define any reasonable notion of traces over the line x = y for the component cos(\frac{1}{x−y}).

The following example indicates that the classical Gauss-Green theorem may fail.

Example 5.2: The field F (x, y) = (\frac{x}{x^{2}+y^{2}} , \frac{y}{x^{2}+y^{2}} ) belongs to DM_{loc}^{1}(R^{2}). As remarked in Whitney [62], for Ω = (0, 1) × (0, 1),

∫_{Ω} div F dxdy = 0 ≠ ∫_{∂Ω} F · ν dH^{1} = \frac{π}{2} ,

if one understands F · ν|_{∂Ω} in the classical sense.

Example 5.3: Let μ ∈ M. Then, for any bounded open interval I ⊂ R,

F (x, y) = (μ(y), 0) ∈ DM^{ext}(I × R).

Some efforts have been made in generalizing the Gauss-Green theorem. Some results for certain situations can be found in Anzellotti [1] for an abstract formulation for F ∈ L^{∞}, Rodrigues [46] for F ∈ L^{2}, and Ziemer [65] for a related problem for div F ∈ L^{1} (see also Baiocchi-Capelo [2], Brezzi-Fortin [8], and Ziemer [66]). Also see Harrison [30], Harrison-Norton [29], Jurkat-Nonnenmacher [32], Nonnenmacher [42], Pfeffer [43], and Shapiro [50] for related problems and references.

5.3. Basic Properties of Divergence-Measure Fields. Now we discuss some basic properties of divergence-measure fields in the spaces DM^{p}(Ω), 1 ≤ p ≤ ∞, and DM^{ext}(Ω).

Proposition 5.1. (i) Let {F_{j}} be a sequence in DM^{p}(Ω) such that

F_{j} ⇀ F L_{loc}^{p}(Ω; R^{N}), for 1 ≤ p < ∞, (5.6)

F_{j} ⇀^{∗} F L_{loc}^{∞}(Ω; R^{N}), for p = ∞. (5.7)

Then

∥F ∥_{L^{p}(Ω)} ≤ lim inf_{j→∞} ∥F_{j}∥_{L^{p}(Ω)}, |div F |(Ω) ≤ lim inf_{j→∞} |div F_{j}|(Ω).

(ii) Let {F_{j}} be a sequence in DM^{ext}(Ω) such that

F_{j} ⇀ F M_{loc}(Ω; R^{N}).

Then

|F |(Ω) ≤ lim inf_{j→∞} |F_{j}|(Ω), |div F |(Ω) ≤ lim inf_{j→∞} |div F_{j}|(Ω).

This proposition immediately implies that the spaces DM^{p}, 1 ≤ p ≤ ∞, and DM^{ext}(Ω) are Banach spaces under the norms (5.4) and (5.5), respectively.

Proposition 5.2. Let {F_{j}} be a sequence in DM(Ω) satisfying

lim_{j→∞} |div F_{j}|(Ω) = |div F |(Ω)

and one of the following three conditions:

F_{j} ⇀ F L_{loc}^{p}(Ω; R^{N}), for 1 ≤ p < ∞,

F_{j} ⇀^{∗} F L_{loc}^{∞}(Ω; R^{N}), for p = ∞,

F_{j} ⇀ F M_{loc}(Ω; R^{N}).

Then, for every open set A ⊂ Ω,

(5.8)  |div F |(Ā ∩ Ω) ≥ lim sup_{j→∞} |div F_{j}|(Ā ∩ Ω).

In particular, if |div F |(∂A ∩ Ω) = 0, then

(5.9)  |div F |(A) = lim_{j→∞} |div F_{j}|(A).

We use the so-called positive symmetric mollifiers ω : R^{N} → R satisfying ω(x) ∈ C_{0}^{∞}(R^{N}), ω(x) ≥ 0, ω(x) = ω(|x|), ∫_{R^{N}} ω(x) dx = 1, supp ω(x) ⊂ B_{1} ≡ \{x ∈ R^{N} : |x| < 1\}. We denote ω_{ε}(x) = ε^{−N}ω(\frac{x}{ε}) and F_{ε} = F ∗ ω_{ε}, that is,

(5.10)  F^{ε}(x) = ε^{−N} ∫_{R^{N}} F (y)ω(\frac{x − y}{ε} ) dy = ∫_{R^{N}} F (x + εy)ω(y) dy.

Then F^{ε} ∈ C^{∞}(A; R^{N}) for any A ⋐ Ω when ε is sufficiently small. We recall that, for any f, g ∈ L^{1}(R^{N}),

(5.11)  ∫_{R^{N}} f_{ε}g dx = ∫_{R^{N}} fg_{ε} dx.

The following fact for DM fields is analogous to a well-known property of BV functions.

Proposition 5.3. Let F ∈ DM(Ω). Let A ⋐ Ω be open and |div F |(∂A) = 0. Then, for any φ ∈ C(Ω; R),

lim_{ε→0} ⟨div F^{ε}, φχ_{A}⟩ =< div F, φχ_{A} > .

Furthermore, if F ∈ DM^{ext}(Ω) and |F |(∂A) = 0, then, for any φ ∈ C(Ω; R^{N}),

lim_{ε→0} < F^{ε}, φχ_{A} > = < F, φχ_{A} > .

Now we discuss some product rules for divergence-measure fields.

Proposition 5.4. Let F = (F_{1}, ⋯, F_{N}) ∈ DM(Ω). Let g ∈ BV ∩ L^{∞}(Ω) be such that ∂_{x_{j}}g(x) is |F_{j}|-integrable, for each j = 1, ⋯, N, and the set of non-Lebesgue points of ∂_{x_{j}}g(x) has |F_{j}|-measure zero; and g(x) is (|F | + |div F |)- integrable and the set of non-Lebesgue points of g(x) has (|F | + |div F |)-measure zero. Then gF ∈ DM(Ω) and

(5.12)  div (gF) = g div F + ∇g · F.

In particular, if F ∈ DM^{∞}(Ω), gF ∈ DM^{∞}(Ω) for any g ∈ BV ∩ L^{∞}(Ω); moreover, if g is also Lipschitz over any compact set in Ω,

(5.13)  div (gF) = g div F + F · ∇g.

In fact, for F ∈ DM^{∞}(Ω), one may refine the above result to yield that (5.13) holds a.e. in a more general case, not only for local Lipschitz functions. In this case, we must take the absolutely continuous part of ∇g. For g ∈ BV, let (∇g)_{ac} and (∇g)_{sing} denote the absolutely continuous part and the singular part of the Radon measure ∇g, respectively. Then

Proposition 5.5. Given F ∈ DM^{∞}(Ω) and g ∈ BV (Ω)∩L^{∞}(Ω), the identity

div (gF) = ḡdiv F + \overline{F · ∇g}

holds in the sense of Radon measures in Ω, where ḡ is the limit of a mollified sequence for g through a positive symmetric mollifier, and \overline{F · ∇g} is a Radon measure absolutely continuous with respect to |∇g|, whose absolutely continuous part with respect to the Lebesgue measure in Ω coincides with F · (∇g)_{ac} almost everywhere in Ω.

5.4. Normal Traces and the Gauss-Green Formula. We now discuss the generalized Gauss-Green theorem for DM-fields over Ω ⊂ R^{N} by introducing a suitable definition of normal traces over the boundary ∂Ω of a bounded open set with Lipschitz deformable boundary, established in Chen-Frid [13, 14, 15].

Definition 5.3. Let Ω ⊂ R^{N} be an open bounded subset. We say that ∂Ω is a deformable Lipschitz boundary, provided that

(i) ∀ x ∈ ∂Ω, ∃ r > 0 and a Lipschitz map γ : R^{N−1} → R such that, after rotating and relabeling coordinates if necessary,

Ω ∩ Q(x, r) = {y ∈ R^{N} : γ(y_{1}, ⋯, y_{N−1}) < y_{N} } ∩ Q(x, r),

where Q(x, r) = {y ∈ R^{N} : |x_{i} − y_{i}| ≤ r, i = 1, ⋯, N };

(ii) ∃ Ψ : ∂Ω × [0, 1] → \overline{Ω} such that Ψ is a homeomorphism bi-Lipschitz over its image and Ψ(ω, 0) = ω for all ω ∈ ∂Ω. The map Ψ is called a Lipschitz deformation of the boundary ∂Ω.

Denote ∂Ω_{s} ≡ Ψ(∂Ω × {s}), s ∈ [0, 1], and Ω_{s} the open subset of Ω whose boundary is ∂Ω_{s}. We call Ψ a Lipschitz deformation of ∂Ω.

Definition 5.4. We say that the Lipschitz deformation is regular if

(5.14)  lim_{s→0+} DΨ_{s} ∘ γ̃ = Dγ̃,  in L_{loc}^{1}(B),

where γ̃ is a map as in Condition (i) of Definition 5.3, and Ψ_{s} denotes the map of ∂Ω into Ω, given by Ψ_{s}(x) = Ψ(x, s). Here B denotes the greatest open set such that γ̃(B) ⊂ ∂Ω.

Remark 5.1. The domains with regular deformable Lipschitz boundaries clearly include bounded domains with Lipschitz boundaries, the star-shaped domains, and the domains whose boundaries satisfy the cone property. It is also clear that, if Ω is the image through a bi-Lipschitz map of a domain Ω̄ with a (regular) Lipschitz deformable boundary, then Ω itself possesses a (regular) Lipschitz deformable boundary.

We first discuss the normal traces and the Gauss-Green formula for DM^{p} fields with 1 < p ≤ ∞.

Theorem 5.1. Let F ∈ DM^{p}(Ω), 1 < p ≤ ∞. Let Ω ⊂ R^{N} be a bounded open set with Lipschitz deformable boundary. Then there exists a continuous linear functional F · ν|_{∂Ω} over Lip(∂Ω) such that, for any ϕ ∈ Lip(R^{N}),

(5.15)  ⟨F · ν|_{∂Ω}, ϕ⟩ = ⟨div F, ϕ⟩ + ∫_{Ω} ∇ϕ · F dx.

Moreover, let ν : Ψ(∂Ω × [0, 1]) → R^{N} be such that ν(x) is the unit outer normal to ∂Ω_{s} at x ∈ ∂Ω_{s}, defined for a.e. x ∈ Ψ(∂Ω × [0, 1]). Let h : R^{N} → R be the level set function of ∂Ω_{s}, that is,

h(x) := \{ \begin{matrix}0,  for x ∈ R^{N} − \overline{Ω}, \\ 1,  for x ∈ Ω − Ψ(∂Ω × [0, 1]), \\ s,  for x ∈ ∂Ω_{s}, 0 ≤ s ≤ 1. \end{matrix}

Then, for any ψ ∈ Lip(∂Ω),

(5.16)  ⟨F · ν|_{∂Ω}, ψ⟩ = − lim_{s→0} \frac{1}{s} ∫_{Ψ(∂Ω×(0,s))} E(ψ) ∇h · F dx,

where E(ψ) is any Lipschitz extension of ψ to all R^{N}.

In the case p = ∞, the normal trace F ·ν|_{∂Ω} is a function in L^{∞}(∂Ω) satisfying ∥F · ν∥_{L^{∞}(∂Ω)} ≤ C∥F ∥_{L^{∞}(Ω)}, for some constant C independent of F; if ∂Ω admits a regular Lipschitz deformation, then C = 1. Furthermore, for any field F ∈ DM^{∞}(Ω),

(5.17) ⟨F · ν|_{∂Ω}, ψ⟩ = esslim_{s→0} ∫_{∂Ω_{s}} ψ ∘ Ψ_{s}^{−1} F · ν dH^{N−1},  for any ψ ∈ L^{1}(Ω).

Finally, for F ∈ DM^{p}(Ω) with 1 < p < ∞, F · ν|_{∂Ω} can be extended to a continuous linear functional over W^{1−1/p,p}(∂Ω) ∩ L^{∞}(∂Ω).

Remark 5.2. As an example, consider the normal trace of F (x, y) in Example 5.1 over the line x = y where there is no reasonable notion of traces for the component cos(\frac{1}{x−y}). Nevertheless, the unit normal ν_{s} to the line x − y = s is the vector (−1/\sqrt{2}, 1/\sqrt{2}) so that the scalar product F (x, x − s) · ν_{s} is identically zero over this line. Hence, we find that F · ν ≡ 0 over the line x = y and the Gauss-Green formula in this case implies that, for any ϕ ∈ C_{0}^{1}(R^{2}),

0 = ⟨div F |_{x>y}, ϕ⟩ = − ∫_{x>y} F · ∇ϕ dxdy.

This identity could be also directly obtained by applying the dominated convergence theorem to the analogous identity, which can be derived from the classical Gauss-Green formula, for the domain { (x, y) | x > y + s } when s → 0.

As indicated by Example 5.2, it is more delicate for fields in DM^{1} and DM^{ext}. Then we have to define the normal traces as functionals over the spaces Lip(γ, ∂Ω) with γ > 1 (see Stein [53]). For 1 < γ ≤ 2, the elements of Lip(γ, ∂Ω) are (N + 1)-components vectors, where the first component is the function itself, and the other N components are its “first-order partial derivatives”. In particular, as a functional over Lip(γ, ∂Ω), the values of the normal trace of a field in DM^{1} or DM^{ext} on ∂Ω depend not only on the values of the respective functions over ∂Ω, but also on the values of their first-order derivatives over ∂Ω. To define the normal traces for F ∈ DM^{1} or DM^{ext}, we resort to the properties of the Whitney extensions of functions in Lip(γ, ∂Ω) to Lip(γ, R^{N}). We have the following analogue of Theorem 5.1 which covers fields in DM^{1} and DM^{ext} (see [14] for the proof).

Theorem 5.2. Let F ∈ DM^{1}(Ω) or F ∈ DM^{ext}(Ω). Let Ω ⊂ R^{N} be a bounded open set with Lipschitz deformable boundary. Then there exists a continuous linear functional F ·ν|_{∂Ω} over Lip(γ, ∂Ω) for any γ > 1 such that, for any ϕ ∈ Lip(γ, R^{N}),

(5.18)  ⟨F · ν|_{∂Ω}, ϕ⟩ = ⟨div F, ϕ⟩_{Ω} + ⟨F, ∇ϕ⟩_{Ω}.

Moreover, let h : R^{N} → R be the level set function as in Theorem 5.1; and, in the case that F ∈ DM^{ext}(Ω), we also assume that ∂_{x_{i}}h is |F_{i}|-measurable and its set of non-Lebesgue points has |F_{i}|-measure zero, i = 1, ⋯, N. Then, for any ψ ∈ Lip(γ, ∂Ω), γ > 1,

(5.19)  ⟨F · ν|_{∂Ω}, ψ⟩ = − lim_{s→0} \frac{1}{s}⟨F, E(ψ) ∇h⟩_{Ψ(∂Ω×(0,s))},

where E(ψ) ∈ Lip(γ, R^{N}) is the Whitney extension of ψ to all R^{N}.

Remark 5.3. In general, for F ∈ DM(D), the normal traces F · ν|_{∂Ω} may be no longer functions. This can be seen in Example 5.2 for F ∈ DM_{loc}^{1}(R^{2}) with Ω = (0, 1) × (0, 1), for which F · ν|_{∂Ω} is a measure.

Finally, as a corollary of the normal traces and the generalized Gauss-Green formula in DM^{∞}, we have

Proposition 5.6. Let Ω ⊂ R^{N} be a bounded open set with Lipschitz boundary and F_{1} ∈ DM^{∞}(Ω), F_{2} ∈ DM^{∞}(R^{N} − Ω̄). Then

(5.20)  F (y) = \{ \begin{matrix}F_{1}(y),  y ∈ Ω, \\ F_{2}(y),  y ∈ R^{N} − Ω̄ \end{matrix}

belongs to DM^{∞}(R^{N}), and

∥F ∥_{DM^{∞}(R^{N})} ≤ ∥F_{1}∥_{DM^{∞}(Ω)}+∥F_{2}∥_{DM^{∞}(R^{N} −Ω̄)}+∥F_{1}·ν−F_{2}·ν∥_{L^{∞}(∂Ω)}H^{N−1}(∂Ω).

The theory of divergence-measure fields presented above has been applied to solving various nonlinear problems for conservation laws and related nonlinear equations. These especially include (1) the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics in [14, 15]; (2) the decay of periodic entropy solutions for hyperbolic conservation laws in [13];

(3) Initial and boundary layer problems for hyperbolic conservation laws in [19] and [57]; (4) initial-boundary value problems for hyperbolic conservation laws in [13]; and (5) nonlinear degenerate parabolic-hyperbolic equations in [9] and [41]. It would be interesting to apply the theory to solving further various problems in partial differential equations and related areas whose solutions are only measures or L^{p} functions.

Acknowledgments. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0204225, DMS-0204455, and INT-9987378.

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Gui-Qiang Chen

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA Email: gqchen@math.northwestern.edu Homepage: http://www.math.northwestern.edu/∼gqchen/preprints