EXTENDED DIVERGENCE-MEASURE FIELDS AND THE EULER EQUATIONS FOR GAS DYNAMICS

GUI-QIANG CHEN & HERMANO FRID

Abstract. A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L^{p} and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields arise naturally in the study of the behavior of entropy solutions to the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.

1. Introduction

We are concerned with a class of extended vector fields, called extended divergencemeasure fields, or DM-fields for short. These fields include vector fields in L^{p}, 1 ≤ p ≤ ∞, and vector-valued Radon measures, whose divergences are Radon measures. The DM-fields arise naturally in the study of the behavior of entropy solutions of nonlinear hyperbolic systems of conservation laws, which take the form

u_{t} + ∇_{x} · f(u) = 0, u ∈ ℝ^{m}, x ∈ ℝ^{n},

where f : ℝ^{m} → (ℝ^{m})^{n} is a nonlinear map. One of its most important prototypes is the Euler equations for gas dynamics in Lagrangian coordinates:

τ_{t} − v_{x} = 0, (1.1)

v_{t} + p_{x} = 0, (1.2)

(e + \frac{v^{2}}{2} )_{t} + (pv)_{x} = 0, (1.3)

where τ = 1/ρ is the specific volume with the density ρ, and v, p, e are the velocity, the pressure, the internal energy, respectively; the other two gas dynamical variables are the temperature θ and the entropy S. For ideal polytropic gases, the system (1.1)–(1.3) is closed by the following constitutive relations:

(1.4)  pτ = Rθ,  e = c_{v}θ,  p(τ, S) = κτ^{−γ}e^{S/c_{v}},

1991 Mathematics Subject Classification. Primary: 26B20,28C05, 35L65, 35B35; Secondary: 26B35, 26B12 35L67.

Key words and phrases. divergence-measure fields, normal traces, Gauss-Green theorem, product rule, Euler equations, gas dynamics, entropy solutions, stability, vacuum.

where c_{v}, R, and κ are positive constants, and γ = 1 + c_{v}/R > 1. For isentropic gases, the Euler equations become

τ_{t} − v_{x} = 0, (1.5)

v_{t} + p(τ)_{x} = 0, (1.6)

where p(τ) = κτ^{−γ}, γ > 1.

The main feature of nonlinear hyperbolic conservation laws, especially (1.1)– (1.3), is that, no matter how smooth the initial data are, solutions may develop singularities and form shock waves in finite time. One may expect solutions in the space of functions of bounded variation. This is indeed the case by the Glimm theorem [15] which establishes that, when the initial data have sufficiently small total variation and stay away from vacuum for (1.1)–(1.3), there exists a global entropy solution in BV satisfying the Clausius inequality:

(1.7)  S_{t} ≥ 0

in the sense of distributions. However, when the initial data are large, still away from vacuum, the solutions may develop vacuum in finite time, even instantaneously as t > 0. In this case, the specific volume τ = 1/ρ may then become a Radon measure or an L^{1} function, rather than a function of bounded variation. This indicates that solutions of nonlinear hyperbolic conservation laws are generally either in M(ℝ_{+} × ℝ^{n}), the space of signed Radon measures, or in L^{p}(ℝ_{+} × ℝ^{n}), 1 ≤ p ≤ ∞. On the other hand, the fact that (1.1)–(1.3), and (1.7) hold in the sense of distributions implies, in particular, that the divergences of the fields (τ, −v), (v, p), (e + v^{2}/2, pv), (S, 0), in the (t, x) variables, are Radon measures, in which the first three are the trivial null measure and the last one is a nonnegative measure as a consequence of the Schwartz Lemma [21]. This motivates our study of the extended divergence-measure fields (see Definition 1.1 below).

In this connection, we recall that Wagner [25] has proved that the well known Lagrangian transformation carries entropy solutions of the Euler equations in Eulerian coordinates to entropy solutions of (1.1)–(1.3), in a one-to-one and onto manner. However, since solutions of the first which contain vacuum are carried into solutions of (1.1)–(1.3) which are vector-valued measures, the concept of entropy solutions for the latter has to be strengthened. We will return to this point in Section 4.

Understanding more properties of DM-fields can advance our understanding of the behavior of entropy solutions (cf. [5, 6, 7]). One of the fundamental questions is whether the normal traces can still be defined and the Gauss-Green formula, i.e., integration by parts, still works for these extended fields, which are very weak.

We begin with the definition of DM-fields. For open sets A, B ⊂ ℝ^{N}, by the relation A ⋐ B we mean that the closure of A, Ā, is a compact subset of B.

Definition 1.1. Let D ⊂ ℝ^{N} be open. For F ∈ L^{p}(D; ℝ^{N}), 1 ≤ p ≤ ∞, or F ∈ M(D; ℝ^{N}), set

|div F |(D) := sup{ ∫_{D} ∇φ · F : φ ∈ C_{0}^{1}(D), |φ(x)| ≤ 1, x ∈ D }.

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

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

We say that F is an extended divergence-measure field over D, i.e., F ∈ DM^{ext}(D), if F ∈ M(D; ℝ^{N}) and

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

If F ∈ DM^{p}(D) for any open set D ⋐ ℝ^{N}, then we say F ∈ DM_{loc}^{p}(ℝ^{N}); and, if F ∈ DM^{ext}(D) for any open set D ⋐ ℝ^{N}, we say F ∈ DM_{loc}^{ext}(ℝ^{N}).

It is easy to check that these spaces under the norms (1.8) and (1.9), respectively, are Banach spaces. These spaces are larger than the BV -space. The establishment of the Gauss-Green theorem, traces, and other properties of BV functions in the middle of last century (see Federer [13]) 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 normal traces and the Gauss-Green formula. At a first glance, it seems unclear.

Example 1.1: 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 [26], for Ω = (0, 1) × (0, 1),

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

if one understands F · ν in the classical sense, which implies that the classical Gauss-Green theorem fails. In this paper, we succeed in using the neighborhood information via the Lipschitz deformation to develop a natural notion of normal traces, under which our generalized Gauss-Green theorem holds, even for F ∈ DM^{ext}(D).

Example 1.2: For any μ ∈ M(ℝ) with finite total variation,

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

for any bounded open interval I ⊂ ℝ. A non-trivial example of such fields is provided by the Riemann solutions of the Euler equations (1.1)–(1.3) for gas dynamics, which develop vacuum. See (4.12) below.

Some efforts have been made in generalizing the Gauss-Green Theorem. Some results for several situations can be found in Anzellotti [1] for an abstract formulation for F ∈ L^{∞}, Rodrigues [20] for F ∈ L^{2}, and Ziemer [28] for a related problem for div F ∈ L^{1} (see also Baiocchi-Capelo [2], Brezzi-Fortin [4], and Ziemer [29]). In Chen-Frid [6], we observed an explicit way to calculate the normal traces for F ∈ DM^{∞} by the neighborhood information, under which a generalized Gauss-Green theorem holds.

In this paper, motivated by various nonlinear problems from conservation laws, we propose a natural notion of normal traces by the neighborhood information via Lipschitz deformation under which a generalized Gauss-Green theorem is established for F ∈ DM^{ext}(D) in Section 3, where our main results concerning extended divergence-measure fields are stated and proved, after establishing some auxiliary results in Section 2. In particular, we show an explicit way to calculate the normal traces over any deformable Lipschitz surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. We also show a product rule for these extended fields. Their proofs require some refined properties of Radon measures and the Whitney extension theory, among others.

In Section 4, we give an important application of the theory of DM-fields to the Euler equations (1.1)–(1.3) for gas dynamics and establish the uniqueness and stability of Riemann solutions of large oscillation that may contain two rarefaction waves and one contact discontinuity or vacuum states (i.e. measure solutions) in the class of entropy solutions, which may not belong to either BV_{loc} or L^{∞}, without specific reference on the method of construction of the solutions. The proof, motivated by [11] and [7]–[9], is heavily based on our explicit approach to calculate normal traces over Lipschitz deformable surfaces, in the generalized Gauss-Green theorem, and the product rule for DM-fields. The same arguments clearly also yield the uniqueness and stability of Riemann solutions in the class of entropy solutions for the Euler equations (1.5) and (1.6) for isentropic gas dynamics.

Before closing this introduction, we recall some correlated results. In DiPerna [11], a uniqueness theorem of Riemann solutions was first established for 2 × 2 systems in the class of entropy solutions in L^{∞} ∩ BV_{loc} with small oscillation. We also refer to Dafermos [10] for the stability of Lipschitz solutions for hyperbolic systems of conservation laws. In [8, 9], the uniqueness and stability of Riemann solutions of large oscillation without vacuum (possibly containing shocks) was proved for the 3 × 3 Euler equations, in the class of entropy solutions in L^{∞} ∩ BV_{loc} which stay away from vacuum. Another related connection is the recent results on the L^{1}-stability of the solutions in L^{∞} ∩BV obtained either by the Glimm scheme [15], the wave front-tracking method, the vanishing viscosity method, or more generally satisfying an additional regularity, with small total variation in x uniformly for all t > 0 (see the recent references cited in Bianchini-Bressan [3] and Dafermos [10]).

2. Radon Measures and the Whitney Extension

In this section, we establish some auxiliary properties about Radon measures and the Whitney extension of Lipschitz continuous functions, which are required for our analysis on the extended divergence-measure vector fields in Section 3. We begin with some properties about Radon measures. Let Ω, D ⊂ ℝ^{N} be open. For μ^{ϵ}, μ ∈ M(Ω), we denote μ^{ϵ} ⇀ μ the weak convergence of μ^{ϵ} to μ in M(Ω). The next three lemmas are standard, but we include their proofs for completeness

Lemma 2.1. Let μ^{ε}, μ ∈ M(Ω) be signed Radon measures over Ω with μ^{ε} ⇀ μ. Then

|μ|(Ω) ≤ lim inf_{ε→0} |μ^{ε}|(Ω).

This can be seen as follows: For any ϕ ∈ C_{0}(Ω), |ϕ| ≤ 1,

⟨μ, ϕ⟩ = lim_{ε→0}⟨μ^{ε}, ϕ⟩ ≤ lim inf_{ε→0} |μ^{ε}|(Ω).

Lemma 2.2. Let μ^{ε}, μ ∈ M(Ω) be such that

μ^{ε} ⇀ μ,  lim_{ε→0} |μ^{ε}|(Ω) = |μ|(Ω).

Then, for every open set A ⊂ Ω,

|μ|(\overline{A} ∩ Ω) ≥ lim sup_{ε→0} |μ^{ε}|(\overline{A} ∩ Ω).

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

|μ|(A) = lim_{ε→0} |μ^{ε}|(A).

Proof. Set B = Ω − \overline{A}, which is open. Then

|μ|(A) ≤ lim inf_{ε→0} |μ^{ε}|(A), |μ|(B) ≤ lim inf_{ε→0} |μ^{ε}|(B).

On the other hand,

|μ|(Ā ∩ Ω) + |μ|(B) = |μ|(Ω) = lim_{ε→0} |μ^{ε}|(Ω)
≥ lim sup |μ^{ε}|(Ā ∩ Ω) + lim inf |μ^{ε}|(B)
≥ lim sup |μ^{ε}|(Ā ∩ Ω) + |μ|(B),

which yields the desired result. □

Lemma 2.3. Let ω^{ε} be a sequence of positive symmetric mollifiers in ℝ^{N} and μ ∈ M(Ω). Set μ^{ε} = μ ∗ ω^{ε}, the mollified measures. Then, for any open set A ⋐ Ω with |μ|(∂A) = 0,

|μ|(A) = lim_{ε→0} |μ^{ε}|(A).

Proof. Since μ^{ε} ⇀ μ, Lemma 2.1 implies

|μ|(A) ≤ lim inf_{ε→0} |μ^{ε}|(A).

Notice that, for any g ∈ C_{0}(A), |g| ≤ 1,

⟨μ^{ε}, g⟩ = ⟨μ, g^{ε}⟩,

where g^{ε} = g ∗ ω^{ε}. Since |g^{ε}| ≤ 1 and spt (g^{ε}) ⊂ A_{ε} := {x : dist (x, A) ≤ ε} ⊂ Ω for ε ≪ 1, then

|⟨μ^{ε}, g⟩| ≤ |μ|(A_{ε}),

which implies

|μ^{ε}|(A) ≤ |μ|(A_{ε}).

Hence,

lim sup_{ε→0} |μ^{ε}|(A) ≤ lim_{ε→0} |μ|(A_{ε}) = |μ|(Ā),

which yields

lim sup_{ε→0} |μ^{ε}|(A) ≤ |μ|(A),

since |μ|(∂A) = 0. □

In particular, Lemma 2.3 indicates that, if μ ≥ 0 and μ(∂A) = 0, then

μ(A) = lim_{ε→0} μ^{ε}(A).

Proposition 2.1. Let D ⊂ ℝ^{N} be open, μ ∈ M(D), and μ^{ε} = μ ∗ ω^{ε}. Let Ω ⋐ D be open and |μ|(∂Ω) = 0. Then, for any ϕ ∈ C(D),

lim_{ε→0}⟨μ^{ε}, ϕχ_{Ω}⟩ = ⟨μ, ϕχ_{Ω}⟩.

Proof. Write

μ^{ε} = μ^{ε+} − μ^{ε−},

where μ^{ε±} = μ^{±} ∗ ω^{ε} are nonnegative measures. The condition |μ|(∂Ω) = 0 implies

μ^{±}(∂Ω) = 0.

From Lemma 2.3, we have

lim_{ε→0} μ^{ε±}(Ω) = μ^{±}(Ω).

Hence

lim_{ε→0} μ^{ε}(Ω) = lim_{ε→0} μ^{ε+}(Ω) − lim_{ε→0} μ^{ε−}(Ω) = μ^{+}(Ω) − μ^{−}(Ω) = μ(Ω).

Let A be open with Ω ⋐ A ⋐ D. Then, for ϕ ∈ C(D) and δ > 0, we may construct a partition of ℝ^{N} by means of parallelepipeds:

Q_{α} = [x_{1}^{α_{1}}, x_{1}^{α_{1}+1}] × ⋯ × [x_{N}^{α_{N}} , x_{N}^{α_{N} +1}], α = (α_{1}, ⋯, α_{N}) ∈ ℤ^{N},

such that

∪_{Q_{α}∩Ω≠∅} Q_{α} ⊂ A,
|μ|(∂Q_{α} ∩ D) = 0, α ∈ ℤ^{N},

and

|ϕ(x) − ϕ(y)| < δ, x, y ∈ Q_{α}.

Let a_{α} = ϕ(x_{α}^{∗}) for some x_{α}^{∗} ∈ Q_{α} if Q_{α} ∩ Ω ≠ ∅, and a_{α} = 0, otherwise. Then

⟨μ^{ε}, ϕχ_{Ω}⟩ = ⟨μ^{ε}, ∑ a_{α}χ_{Q_{α}∩Ω}⟩ + ⟨μ^{ε}, (ϕ − ∑ a_{α}χ_{Q_{α}})χ_{Ω}⟩,

and

lim sup_{ε→0} ⟨μ^{ε}, ϕχ_{Ω}⟩ ≤ ⟨μ, ∑ a_{α}χ_{Q_{α}∩Ω}⟩ + δ|μ|(A)
≤ ⟨μ, ϕχ_{Ω}⟩ + 2δ|μ|(A).

Analogously,

lim inf_{ε→0} ⟨μ^{ε}, ϕχ_{Ω}⟩ ≥ ⟨μ, ϕχ_{Ω}⟩ − 2δ|μ|(A).

Since δ > 0 is arbitrary, we complete the proof. □

We now discuss some concepts and facts about the extension of Lipschitz continuous functions defined on a closed set C ⊂ ℝ^{N} to ℝ^{N}, following the theory set forth by Whitney [27] (see also [23]), which play an important role in Section 3.

Let k be a nonnegative integer and γ ∈ (k, k + 1]. We say that a function f, defined on C, belongs to Lip (γ, C) if there exist functions f^{(j)}, 0 ≤ |j| ≤ k, defined on C, with f^{(0)} = f, such that, if

f^{(j)}(x) = ∑_{|j+l|≤k} \frac{f^{(j+l)}(y)}{l!}(x − y)^{l} + R_{j}(x, y),

then

(2.1)  \{ \begin{matrix}|f^{(j)}(x)| ≤ M, \\ |R_{j}(x, y)| ≤ M|x − y|^{γ−|j|}, for any x, y ∈ C, |j| ≤ k. \end{matrix}

Here j and l denote multi-indices j = (j_{1}, ⋯, j_{N}) and l = (l_{1}, ⋯, l_{N}) with j! = j_{1}! ⋯ j_{N}!, |j| = j_{1}+j_{2}+⋯+j_{N}, and x^{l} = x_{1}^{l_{1}}x_{2}^{l_{2}} ⋯ x_{N}^{l_{N}}. By an element of Lip (γ, C) we mean the collection {f^{(j)}(x)}_{|j|≤k}. The norm of an element in Lip (γ, C) is defined as the smallest M for which the inequality (2.1) holds. We notice that Lip (γ, C) with this norm is a Banach space. For the case C = ℝ^{N}, since the functions f^{(j)} are determined by f^{(0)}, this collection is then identified with f^{(0)}.

The Whitney extension of order k is defined as follows. Let {f^{(j)}}_{|j|≤k} be an element of Lip (γ, C). The linear mapping E_{k} : Lip (γ, C) → Lip (γ, ℝ^{N}) assigns to any such collection a function E_{k}(f^{(j)}) defined on ℝ^{N} which is an extension of f^{(0)} = f to ℝ^{N}. The definition of E_{k} is the following:

\begin{cases}E_{0}(f)(x) = f(x),  x ∈ C, \\ E_{0}(f)(x) = ∑_{i} f(p_{i})φ_{i}(x),  x ∈ ℝ^{N} − C, \end{cases}

and, for k > 0,

\begin{cases}E_{k}(f^{(j)})(x) = f^{(0)}(x),  x ∈ C, \\ E_{k}(f^{(j)})(x) = ∑′_{i} P (x, p_{i})φ_{i}(x),  x ∈ ℝ^{N} − C. \end{cases}

Here P (x, y) denotes the polynomial in x, which is the Taylor expansion of f about the point y ∈ C:

P (x, y) = ∑_{|l|≤k} \frac{f^{(l)}(y)(x − y)^{l}}{l!} , x ∈ ℝ^{N}, y ∈ C.

The functions φ_{i} form a partition of unity of ℝ^{N} − C with the following properties:

(i) spt (φ_{i}) ⊂ Q_{i} where Q_{i} is a cube with edges parallel to the coordinate axes

and

c_{1} diam (Q_{i}) ≤ dist (Q_{i}, C) ≤ c_{2} diam (Q_{i}),

for certain positive constants c_{1} and c_{2} independent of C; (ii) each point of ℝ^{N} − C is contained in at most N_{0} cubes Q_{i}, for certain

number N_{0} depending only on the dimension N; (iii) the derivatives of φ_{i} satisfy

(2.2)  |∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{N}}^{α_{N}}φ_{i}(x)| ≤ A_{α}(diam Q_{i})^{−|α|}.

Here p_{i} ∈ C is such that dist (Q_{i}, C) = dist (p_{i}, Q_{i}), |α| = α_{1} + ⋯ + α_{N}, and the symbol ∑′ indicates that the summation is taken only over those cubes whose distances to C are not greater than one.

The following theorem, whose proof can also be found in [23], is due to Whitney [27].

Theorem 2.1. Suppose that k is a nonnegative integer, γ ∈ (k, k + 1], and C is a closed set. Then the mapping E_{k} is a continuous linear mapping from Lip (γ, C) to Lip (γ, ℝ^{N}) which defines an extension of f^{(0)} to ℝ^{N}, and the norm of this mapping has a bound independent of C.

We will need the following proposition, which seems to be of interest in itself and is useful in establishing the generalized Gauss-Green theorem in Section 3.

Proposition 2.2. Let C be a closed set in ℝ^{N} and

C_{δ} := { x ∈ ℝ^{N} : dist (x, C) ≤ δ }, for δ > 0.

Let E_{k} : Lip (γ, C) → Lip (γ, ℝ^{N}) with γ ∈ (k, k + 1] be the Whitney extension of order k. Then, for any ϕ ∈ Lip (γ, ℝ^{N}) and any γ′ ∈ (k, γ),

(2.3)  ∥E_{k}(ϕ|C) − ϕ∥_{Lip (γ′,C_{δ})} → 0, as δ → 0.

Proof. We will prove the proposition in detail only for the cases k = 0 and k = 1 since the case k > 1 can then be obtained by induction.

For k = 0, E_{0} is given by E_{0}(f)(x) = f(x) if x ∈ C, and

(2.4)  E_{0}(f)(x) = ∑_{i=1}^{∞} f(p_{i})φ_{i}(x),  if x ∈ ℝ^{N} − C,

where p_{i} and φ_{i} are as above. Now, for any ϕ ∈ Lip (γ, ℝ^{N}), we have

E_{0}(ϕ|C)(x) − ϕ(x) = ∑_{i=1}^{∞}(ϕ(p_{i}) − ϕ(x))φ_{i}(x).

Clearly,

(2.5)  sup_{x∈C_{δ}} |E_{0}(ϕ|C)(x) − ϕ(x)| ≤ c sup_{\begin{matrix}y∈C,x∈C_{δ} \\ |x−y|≤c_{0}δ \end{matrix}} |ϕ(y) − ϕ(x)| → 0, as δ → 0.

Here and in what follows in this proof c_{0} and c are positive constants, which do not depend on either δ > 0 or C, whose values may change at each appearance.

Set g = E_{0}(ϕ|C) − ϕ. We now show that

(2.6)  |g(x) − g(y)| ≤ M(δ)|x − y|^{γ′},  x, y ∈ C_{δ},

where M(δ) → 0 as δ → 0. Indeed,

g(x) − g(y) = ∑_{x∈Q_{i}}(ϕ(p_{i}) − ϕ(x))φ_{i}(x) − ∑_{y∈Q_{i}}(ϕ(p_{i}) − ϕ(y))φ_{i}(y).

We split each of these sums into two, respectively:

∑_{x∈Q_{i}} = ∑_{\begin{matrix}x∈Q_{i} \\ y∉Q_{i} \end{matrix}} + ∑_{\begin{matrix}x∈Q_{i} \\ y∈Q_{i} \end{matrix}} ,  ∑_{y∈Q_{i}} = ∑_{\begin{matrix}y∈Q_{i} \\ x∉Q_{i} \end{matrix}} + ∑_{\begin{matrix}x∈Q_{i} \\ y∈Q_{i} \end{matrix}} ,

and denote

∑^{0} = ∑_{\begin{matrix}x∈Q_{i} \\ y∈Q_{i} \end{matrix}} ,  ∑^{1} = ∑_{\begin{matrix}x∈Q_{i} \\ y∉Q_{i} \end{matrix}} ,  ∑^{2} = ∑_{\begin{matrix}y∈Q_{i} \\ x∉Q_{i} \end{matrix}} .

We have

∑^{0}((ϕ(p_{i}) − ϕ(x))φ_{i}(x) − (ϕ(p_{i}) − ϕ(y))φ_{i}(y))
= ∑^{0}(ϕ(y) − ϕ(x))φ_{i}(x) + ∑^{0}(ϕ(p_{i}) − ϕ(y))(φ_{i}(x) − φ_{i}(y)).

Since, if x, y ∈ Q_{i} ∩ C_{δ}, |x − y| ≤ c_{0}δ, for a given constant c_{0} > 0, we obtain, for the first sum,

|∑^{0}(ϕ(x) − ϕ(y))φ_{i}(x)| ≤ cδ^{γ−γ′}|x − y|^{γ′},

and, for the second sum,

|∑^{0}(ϕ(p_{i}) − ϕ(y))(φ_{i}(x) − φ_{i}(y))| ≤ c∑^{0} |p_{i} − y|^{γ}(diam Q_{i})^{−1}|x − y|
≤ c |x − y|^{γ} ≤ c δ^{γ−γ′}|x − y|^{γ′}.

Now, for ∑^{1}, we have

∑^{1}(ϕ(p_{i}) − ϕ(x))φ_{i}(x) = ∑^{1}(ϕ(p_{i}) − ϕ(x))(φ_{i}(x) − φ_{i}(q_{i})),

with q_{i} = ∂Q_{i} ∩ [x, y], where [x, y] is the straight line segment connecting x to y. Therefore,

∑^{1}|ϕ(p_{i}) − ϕ(x)||φ_{i}(x) − φ_{i}(q_{i})| ≤ c ∑^{1}|p_{i} − x|^{γ}(diam Q_{i})^{−1}|x − q_{i}|
≤ c ∑^{1}|x − q_{i}|^{γ} ≤ c δ^{γ−γ′}|x − y|^{γ′}.

The sum ∑^{2} is treated similarly. We have then proved (2.6) which, together with (2.5), gives (2.3) for k = 0.

For k = 1, we have E_{1}(f)(x) = f(x) for x ∈ C, and

E_{1}(f)(x) = ∑_{i=1}^{∞} P_{f}(x, p_{i})φ_{i}(x), for x ∈ ℝ^{N} − C,

where

P_{f}(x, y) = f(y) + ∑_{j=1}^{N} ∂_{x_{j}}f(y)(x_{j} − y_{j}),

and the functions φ_{i}, conveniently renumbered together with the Q_{i} containing spt (φ_{i}), satisfy dist (Q_{i} , C) ≤ 1. Since ϕ ∈ Lip (γ, ℝ^{N}), γ > 1, we clearly have

(ϕ|C)^{j}(y) = ∂_{x_{j}}ϕ(y), for y ∈ C.

Hence

E_{1}(ϕ|C)(x) − ϕ(x) = ∑_{i=1}^{∞}(P_{ϕ}(x, p_{i}) − ϕ(x))φ_{i}(x).

Setting g_{1} = E_{1}(ϕ|C) − ϕ, we have

|g_{1}(x)| ≤ ∑_{i=1}^{∞} |P_{ϕ}(x, p_{i}) − ϕ(x)||φ_{i}(x)|

≤ c ∑_{x∈Q_{i}} |x − p_{i}|^{γ}|φ_{i}(x)| ≤ c δ^{γ} → 0, as δ → 0.

Also,

∂_{x_{k}}g_{1}(x) = ∑_{x∈Q_{i}} ∂_{x_{k}}(P_{ϕ}(x, p_{i}) − ϕ(x))φ_{i}(x) + ∑_{x∈Q_{i}}(P_{ϕ}(x, p_{i}) − ϕ(x))∂_{x_{k}}φ_{i}(x)
= h_{1}(x) + h_{2}(x),

where

h_{1}(x) := ∑_{x∈Q_{i}}(∂_{x_{k}}ϕ(p_{i})−∂_{x_{k}}ϕ(x))φ_{i}(x), h_{2}(x) := ∑_{x∈Q_{i}}(P_{ϕ}(x, p_{i})−ϕ(x))∂_{x_{k}}φ_{i}(x).

Hence,

|∂_{x_{k}}g_{1}(x)| ≤ c ∑_{x∈Q_{i}} |p_{i} − x|^{γ−1} + c ∑_{x∈Q_{i}} |x − p_{i}|^{γ}diam (Q_{i})^{−1}
≤ c δ^{γ−1} → 0, as δ → 0.

We now show

|∂_{x_{k}}g_{1}(x) − ∂_{x_{k}}g_{1}(y)| ≤ M(δ)|x − y|^{γ′−1},

with M(δ) → 0 as δ → 0. First we obtain

|h_{1}(x) − h_{1}(y)| ≤ M(δ)|x − y|^{γ′−1}, with M(δ) → 0, as δ → 0,

exactly as in the case k = 0. Now,

h_{2}(x) − h_{2}(y) = ∑_{x∈Q_{i}}(P_{ϕ}(x, p_{i}) − ϕ(x))∂_{x_{k}}φ_{i}(x) − ∑_{x∈Q_{i}}(P_{ϕ}(x, p_{i}) − ϕ(y))∂_{x_{k}}φ_{i}(y)

= ∑_{x∈Q_{i}} R(x, p_{i})∂_{x_{k}}φ_{i}(x) − ∑_{y∈Q_{i}} R(y, p_{i})∂_{x_{k}}φ_{i}(y),

where we put R(x, y) = P_{ϕ}(x, y) − ϕ(x). Again, we split each of the last two sums above into two, respectively:

∑_{x∈Q_{i}} = ∑^{0} + ∑^{1},  ∑_{y∈Q_{i}} = ∑^{0} + ∑^{2},

as in the first part of the proof, and compute

|∑^{0}R(x, p_{i})∂_{x_{k}}φ_{i}(x) − ∑^{0}R(y, p_{i})∂_{x_{k}}φ_{i}(y)|

≤ |∑^{0}(R(x, p_{i}) − R(y, p_{i}))∂_{x_{k}}φ_{i}(x)| + |∑^{0}R(y, p_{i})(∂_{x_{k}}φ_{i}(y) − ∂_{x_{k}}φ_{i}(x))|

≤ c∑^{0} |x − y|^{γ}(diam Q_{i})^{−1} + c∑^{0} |y − p_{i}|^{γ}(diam Q_{i})^{−2}|x − y|

≤ c δ^{γ−γ′}|x − y|^{γ′−1}.

For the remaining sums, we have

|∑^{1}R(x, p_{i})∂_{x_{k}}φ_{i}(x)| = |∑^{1}R(x, p_{i})(∂_{x_{k}}φ_{i}(x) − ∂_{x_{k}}φ_{i}(q_{i}))|
≤ c∑^{1} |x − p_{i}|^{γ}(diam Q_{i})^{−2}|x − q_{i}|
≤ c δ^{γ−γ′}|x − y|^{γ′−1},

where q_{i} is as that in the first part of the proof; and the sum ∑^{2}R(y, p_{i})∂_{x_{k}}φ_{i}(y) is treated similarly. This concludes the proof in the case k = 1. As indicated above, the case k > 1 follows similarly by induction. □

3. Normal Traces and Generalized Gauss-Green Theorem

In this section, we prove our main results concerning extended DM-fields, including a generalized Gauss-Green theorem, a new notion of normal traces, and a product rule for DM-fields. We begin with the definition of deformable Lipschitz boundaries.

Definition 3.1. 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.

The following lemma is a direct corollary of the boundedness of F and div F over Ω as Radon measures. Since the theory of DM^{∞}-fields has been addressed in [5], henceforth we focus on DM^{∗}-fields, where ∗ stands for either p ∈ [1, ∞) or ext.

Lemma 3.1. Let F ∈ DM^{∗}(Ω) with Ω an open set whose boundary ∂Ω has a Lipschitz deformation Ψ with ∂Ω_{s} = Ψ_{s}(∂Ω), s ∈ [0, 1]. Then there exists a countable set T ⊂ (0, 1) such that

|F |(∂Ω_{s}) = |div F |(∂Ω_{s}) = 0, for any s ∈ (0, 1) − T .

We now establish the generalized Gauss-Green theorem for DM^{∗}-fields, by introducing a suitable definition of normal traces over the boundary ∂Ω of a bounded open set Ω with Lipschitz deformable boundary.

Theorem 3.1 (Generalized Gauss-Green Theorem for DM^{∗}-fields). Assume F ∈ DM^{∗}(Ω). Let Ω ⊂ ℝ^{N} is a bounded open set with Lipschitz deformable boundary. Then there exists a continuous linear functional F · ν|_{∂Ω} over Lip (γ, ∂Ω), γ > 1, such that, for any ϕ ∈ Lip (γ, ℝ^{N}),

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

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, if F ∈ DM^{p}(Ω), 1 ≤ p < ∞,

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

for any ψ ∈ Lip (∂Ω), where E(ψ) is any Lipschitz extension of ψ to all ℝ^{N} and ν : Ψ(∂Ω×[0, 1]) → ℝ^{N} is 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]). Formula (3.2) also holds if F ∈ DM^{ext}(Ω), for any ψ ∈ Lip (γ, ∂Ω), γ > 1, and E(ψ) ∈ Lip (γ, ∂ℝ^{N}) any extension of ψ to ℝ^{N}, provided that the set of non-Lebesgue points of ∇h(x) on Ψ(∂Ω × (0, 1)) has |F |-measure zero. 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) ∩ C(∂Ω).

Proof. We divide the proof into four steps.

Step 1. We first treat the more general case F ∈ DM^{ext}(Ω). For ψ ∈ Lip (γ, ∂Ω), let E(ψ) be the Whitney extension of ψ of order k = 1. Then, by the classical Gauss-Green formula, we have

(3.4)  ∫_{∂Ω} F^{ε} · ν ψ dH^{N−1} = ∫_{Ω} E(ψ) div F^{ε} + ∫_{Ω} ∇E(ψ) · F^{ε}.

where F^{ε} = F ∗ ω_{ε} with the standard mollified sequence ω_{ε}. To begin with, we first focus on F ∈ DM^{ext}(D), for Ω ⋐ D ⋐ ℝ^{N}, satisfying

(3.5)  |F |(∂Ω) = |div F |(∂Ω) = 0.

The right-hand side of (3.4) defines a uniformly bounded family of continuous linear operators l^{ε} over Lip (γ, ∂Ω). Moreover, for each ψ ∈ Lip (γ, ∂Ω), the limit lim_{ε→0} l^{ε}(ψ) exists and equals

∫_{Ω} E(ψ) div F + ∫_{Ω} ∇E(ψ) · F,

as a consequence of (3.5) and Proposition 2.1. Hence, we may define

⟨F · ν|_{∂Ω}, ψ⟩ = lim_{ε→0} l^{ε}(ψ), ∀ψ ∈ Lip (γ, ∂Ω).

Now, for any ϕ ∈ Lip (γ, ℝ^{N}), we let ε → 0 in the Gauss-Green formula:

∫_{∂Ω} F^{ε} · ν ϕ dH^{N−1} = ∫_{Ω} ϕ div F^{ε} + ∫_{Ω} ∇ϕ · F^{ε},

and obtain the identity (3.1).

For the general case that F ∈ DM^{ext}(Ω) without the assumption (3.5), we consider a Lipschitz deformation of ∂Ω, Ψ : ∂Ω × [0, 1] → \overline{Ω}. Let s ∈ (0, 1) be such that

(3.6)  |F |(∂Ω_{s}) = |div F |(∂Ω_{s}) = 0.

Since we have H^{N−1}(∂Ω_{s}) < +∞, Federer’s extension of the Gauss-Green formula (see [13]) holds for ϕF^{ε} over Ω_{s}. Thus, we know from the previous analysis that F · ν|_{∂Ω_{s}} is defined as a continuous linear functional over Lip (γ, ∂Ω_{s}), whose norm is bounded, independent of s ∈ (0, 1). Now, for ψ ∈ Lip (γ, ∂Ω), we have

(3.7)  ⟨F · ν|_{∂Ω_{s}}, (E(ψ)|∂Ω_{s})⟩ = ∫_{Ω_{s}} E(ψ) div F + ∫_{Ω_{s}} ∇E(ψ) · F.

Again, the right-hand side of (3.7) defines a uniformly bounded family of continuous linear functionals l^{s} over Lip (γ, ∂Ω), for s ∈ (0, 1)−T , where T is defined in Lemma 3.1. Furthermore, lim_{s→0} l^{s}(ψ) exists for any ψ ∈ Lip (γ, ∂Ω), as a consequence of the Dominated Convergence Theorem applied to both integrals on the right-hand side of (3.7). Hence, we may define

⟨F · ν|_{∂Ω}, ψ⟩ = lim_{s→0} l^{s}(ψ),

which is then a continuous linear functional over Lip (γ, ∂Ω). Finally, for any ϕ ∈ Lip (γ, ℝ^{N}), we obtain (3.1) by taking the limit as s → 0 in the formula:

⟨F · ν|_{∂Ω_{s}}, (ϕ|∂Ω_{s})⟩ = ∫_{Ω_{s}} ϕ div F + ∫_{Ω_{s}} ∇ϕ · F,

observing that

|⟨F · ν|_{∂Ω_{s}}, (ϕ|∂Ω_{s}) − (E(ϕ|∂Ω)|∂Ω_{s})⟩|
≤ c ∥(ϕ|∂Ω_{s}) − (E(ϕ|∂Ω)|∂Ω_{s})∥_{Lip (γ′,∂Ω_{s})}
≤ c ∥ϕ − E(ϕ|∂Ω)∥_{Lip (γ′,Ψ(∂Ω×[0,s]))} → 0, as s → 0,

for 1 < γ′ < γ, as a consequence of Proposition 2.2.

Step 2. We now consider the more regular case that F ∈ DM^{p}(Ω), 1 ≤ p < ∞. Let F^{ε} be as above. Again, for any ϕ ∈ Lip (ℝ^{N}), we have

(3.8)  ∫_{∂Ω_{s}} ϕF^{ε} · ν dH^{N−1} = ∫_{Ω_{s}} ϕ div F^{ε}dx + ∫_{Ω_{s}} ∇ϕ · F^{ε} dx.

Now we integrate (3.8) in s ∈ (0, δ), 0 < δ < 1, and use the coarea formula (see, e.g. [12, 13]) in the left-hand side to obtain (3.9)

− ∫_{Ψ(Ω×(0,δ))} ϕF^{ε} · ∇h dx = ∫_{0}^{δ} {∫_{Ω_{s}} ϕ div F^{ε} dx} ds + ∫_{0}^{δ} {∫_{Ω_{s}} ∇ϕ · F^{ε} dx} ds.

Let ε → 0. Observing that, by Proposition 2.1, the integrand of the first integral converges for a.e. s ∈ (0, δ) to the corresponding integral for F, we obtain

(3.10) − ∫_{Ψ(Ω×(0,δ))} ϕF ·∇h dx = ∫_{0}^{δ} {∫_{Ω_{s}} ϕ div F } ds+∫_{0}^{δ} {∫_{Ω_{s}} ∇ϕ · F dx} ds.

We then divide (3.10) by δ, let δ → 0, and observe that both terms in the righthand side converge to the corresponding integrals inside the brackets over Ω, by the dominated convergence theorem. Hence, the left-hand side also converges, which yields

(3.11)  − lim_{δ→0} \frac{1}{δ} ∫_{Ψ(Ω×(0,δ))} ϕ F · ∇h dx = ∫_{Ω} ϕ div F + ∫_{Ω} ∇ϕ · F dx.

Now, for ψ ∈ Lip (∂Ω), let E(ψ) ∈ Lip (ℝ^{N}) be a Lipschitz extension of ψ preserving the norm ∥ · ∥_{Lip} := ∥ · ∥_{∞} + Lip (·) (see, e.g., [12, 13]). We then define

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

Because the right-hand side of (3.11) does not depend on the particular deformation Ψ for ∂Ω, we see that the normal trace defined by (3.12) is also independent of the deformation.

We still have to prove that the normal trace as defined by (3.12) also does not depend on the specific Lipschitz extension E(ψ) of ψ. This will be accomplished if we prove that the right-hand side of (3.11) vanishes for ϕ|_{∂Ω} ≡ 0. Denote it by [F, ϕ]_{∂Ω}, that is,

[F, ϕ]_{∂Ω} := ⟨div F, ϕ⟩_{Ω} + ⟨F, ∇ϕ⟩_{Ω}.

We claim that [F, ϕ]_{∂Ω} = 0 if ϕ|_{∂Ω} ≡ 0. In fact, we may approximate such ϕ by a sequence ϕ^{j} ∈ C_{0}^{∞}(Ω), with ∥ϕ^{j}∥_{∞} ≤ ∥ϕ∥_{∞}, such that ϕ^{j} → ϕ locally uniformly in Ω and ∇ϕ^{j} → ∇ϕ in L^{q}(Ω)^{N}, with \frac{1}{p} + \frac{1}{q} = 1. Hence, [F, ϕ]_{∂Ω} = lim_{j→∞}[F, ϕ^{j}] = 0, as asserted. In particular, for 1 ≤ p ≤ ∞, the values of the normal trace, ⟨F · ν|_{∂Ω}, ϕ|_{∂Ω}⟩, depend only on the values of ϕ over ∂Ω.

Step 3. The fact that formula (3.2) also holds if F ∈ DM^{ext}(Ω), for any ψ ∈ Lip (γ, ∂Ω), γ > 1, provided that the set of non-Lebesgue points of ∇h(x) on Ψ(∂Ω × (0, 1)) has |F |-measure zero, is clear from the above proof for DM^{p}-fields, 1 ≤ p ≤ ∞.

Step 4. As for the last assertion, we recall a well-known result of Gagliardo [14] which indicates, in particular, that, if ∂Ω is Lipschitz (that is, satisfies (i) of Definition 3.1) and ψ ∈ W^{1−1/p, p}(∂Ω), then it can be extended into Ω to a function E(ψ) ∈ W^{1,p}(Ω), and

(3.13)  ∥E(ψ)∥_{W^{1,p}(Ω)} ≤ c∥ψ∥_{W^{1−1/p, p}(∂Ω)},

for some positive constant c independent of ψ. ¿From the definition of E(ψ) given in [14], it is easy to verify that, when ψ ∈ C(∂Ω), E(ψ) ∈ C(Ω) and ∥E(ψ)∥_{L^{∞}(Ω)} ≤ ∥ψ∥_{L^{∞}(∂Ω)}. Hence, using these facts and (3.11), we easily deduce the last assertion.

□

Remark 3.1. When F ∈ DM^{∞}(D), the normal trace F ·ν|_{∂Ω} is in fact in L^{∞}(∂Ω, H^{N−1}) as the weak-star limit of (F ·ν_{s})∘Ψ_{s} in L^{∞}(∂Ω, H^{N−1}) for any Lipschitz deformation Ψ_{s}:

F · ν|_{∂Ω} = w^{∗} − lim (F · ν_{s}) ∘ Ψ_{s}, L^{∞}(∂Ω, H^{N−1}),

which is independent of Ψ_{s}; and the weak-star topology for this limit is optimal to define F · ν|_{∂Ω} in general (see Chen-Frid [5]). However, for F ∈ DM^{∗}(D), the normal traces F · ν|_{∂Ω} may no longer be functions in general. This can be seen in Example 1.1 for F ∈ DM_{loc}^{1}(ℝ^{2}) with Ω = (0, 1) × (0, 1), for which

F · ν|_{∂Ω} = \frac{π}{2} δ_{(0,0)} − dH^{1}|_{∂Ω},

where H^{1} is the one-dimensional Hausdorff measure on ∂Ω.

Remark 3.2. As mentioned in the proof of Theorem 3.1, if F ∈ DM^{p}(Ω), for 1 ≤ p ≤ ∞, the values of the normal trace, ⟨F · ν|_{∂Ω}, ϕ|∂Ω⟩, depend only on the values of ϕ over ∂Ω. In contrast, for F ∈ DM^{ext}(Ω), the values of ⟨F · ν|_{∂Ω}, ϕ|∂Ω⟩ also depend, in principle, on the values of the first derivatives of ϕ over ∂Ω, since ϕ|∂Ω must be viewed as elements of Lip (γ, ∂Ω), for some γ > 1.

Finally, we establish the following useful product rule.

Theorem 3.2 (Product Rule). Let F = (F_{1}, ⋯, F_{N}) ∈ DM^{∗}(D). Let g ∈ C(D) 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. Then gF ∈ DM^{∗}(D) and

(3.14)  div (gF) = g div F + ∇g · F.

Proof. For any ϕ ∈ C_{0}^{1}(D), we have

(3.15)  ⟨div (gF), ϕ⟩ = −⟨F, g∇ϕ⟩ = −⟨F, ∇(gϕ)⟩ + ⟨F, ϕ∇g⟩.

Therefore, it suffices to show that

(3.16)  ⟨F, ∇(gϕ)⟩ = −⟨div F, gϕ⟩.

Set g^{ε} = g ∗ ω^{ε}. We have

(3.17)  −⟨div F, g^{ε}ϕ⟩ = ⟨F, ∇(g^{ε}ϕ)⟩ = ∑_{j=1}^{N}⟨F_{j}, ∂_{x_{j}}(g^{ε}ϕ)⟩

= ∑_{j=1}^{N}{⟨F_{j}, ϕ∂_{x_{j}}g^{ε}⟩ + ⟨F_{j}, g^{ε}∂_{x_{j}}ϕ⟩}.

Let ε → 0. Then the right-hand side converges to

∑_{j=1}^{N}{⟨F_{j}, ϕ∂_{x_{j}}g⟩ + ⟨F_{j}, g∂_{x_{j}}ϕ⟩} = ⟨F, ∇(gϕ)⟩,

by the assumption on the set of non-Lebesgue points of ∂_{x_{j}}g, while the left-hand side of (3.17) converges to −⟨div F, gϕ⟩ by the Dominated Convergence Theorem, which implies (3.16). Then (3.14) follows. □

Remark 3.3. The continuity assumption of g(x) in Theorem 3.3 can be relaxed. In particular, when F ∈ DM^{∞}(D), it requires only that g ∈ BV (D; ℝ) to have gF ∈ DM^{∞}(D).

Remark 3.4. The results in this section for the extended vector fields over ℝ^{N} extend to a general context over Riemannian manifolds. In particular, if a_{ij}(x) are smooth functions on the open set D ⊂ ℝ^{N} and the N × N matrix (a_{ij})(x) is symmetric, positive definite, the results can easily be generalized to the extended vector fields F = (F_{1}, ⋯, F_{N}) in L^{p}(D; ℝ^{N}), 1 ≤ p ≤ ∞, or M(D; ℝ^{N}), satisfying the condition:

∑_{i,j} a_{ij}∂_{i}F_{j} ∈ M(D).

This is clear from the fact that no specific property of the Euclidean metric has been used in our analysis.

4. Applications to the Euler Equations for Gas Dynamics

In this section, as a direct application of the theory developed in Section 3, we establish the uniqueness and stability of Riemann solutions that may contain vacuum for the Euler equations for gas dynamics in Lagrangian coordinates. We will address more applications of the theory in forthcoming papers.

Denote ℝ_{+}^{2} = (0, ∞) × ℝ and \overline{ℝ_{+}^{2}} = [0, ∞) × ℝ. We consider τ ∈ M^{+}(ℝ_{+}^{2}) satisfying τ ≥ c L^{2} for some c > 0, where L^{k} is the k-dimensional Lebesgue measure. Let v ∈ L^{∞}(ℝ_{+}^{2}) and τ_{0} ∈ M^{+}(ℝ) with τ_{0} ≥ c L^{1}. We assume that τ, v, and τ_{0} satisfy

(4.1)  ∫∫_{ℝ_{+}^{2}}(ϕ_{t}τ − v ϕ_{x} dt dx) + ∫_{ℝ} ϕ(0, x) τ_{0}(x) = 0,

for any ϕ ∈ C_{0}^{1}(ℝ^{2}).

Definition 4.1. Let τ and τ_{0} be as above. We say that a function ϕ(t, x) defined on \overline{ℝ_{+}^{2}} is a τ-test function if it satisfies the following:

(1) spt (ϕ) is a compact subset of \overline{ℝ_{+}^{2}} and ϕ is continuous on ℝ_{+}^{2};

(2) ϕ_{t} and ϕ_{x} are τ-measurable; and ϕ_{t} is τ-integrable over ℝ_{+}^{2}, that is, the

integrals ∫∫_{ℝ_{+}^{2}}(ϕ_{t})_{±}τ exist and at least one of them is finite;
(3) lim_{\begin{matrix}t→0 \\ x→a \end{matrix}} ϕ(t, x) = ϕ(0, a) for τ_{0}–a.e. a ∈ ℝ.

Theorem 4.1. Let τ, v, and τ_{0} be as above. Then

(1) the nonnegative measure τ admits a slicing of the form τ = dt ⊗ μ_{t}(x) with μ_{t} ∈ M^{+}(ℝ) for L^{1}-a.e. t > 0. More precisely, for all ϕ ∈ C_{0}(ℝ_{+}^{2}),

∫∫ ϕ(t, x) τ = ∫ (∫ ϕ(t, x) μ_{t}(x)) dt.

(2) the points (t, x) ∈ ℝ_{+}^{2} such that μ_{t}(x) > 0, with the exception of a set of H^{1}-measure zero, form a countable union of vertical line segments, called vacuum lines. In particular, τ(l) = 0 for any non-vertical straight line segment l.

(3) the identity (4.1) holds for any τ-test function ϕ(t, x).

The proof of Theorem 4.1 is given in Section 5. As a corollary, we have Corollary 4.1. Let τ, v, and τ_{0} be as above. Let p̄(t, x) be a nonnegative function over \overline{ℝ_{+}^{2}}, continuous on ℝ_{+}^{2}, such that ϕ p̄ is a τ-test function for any ϕ ∈ C_{0}^{1}(ℝ^{2}), p̄_{t} ≤ 0, τ-a.e., and p̄_{x} ∈ L_{loc}^{1}(ℝ_{+}^{2}). Then, for any nonnegative function ζ ∈ C_{0}^{1}(ℝ),

(4.2)  lim sup_{t→0+} ∫ ζ(x) p̄(t, x) μ_{t}(x) ≤ ∫ ζ(x) p̄(0, x) τ_{0}(x).

Proof. First we have from Theorem 4.1 that (4.1) holds for ϕ = p̄ψ with ψ ∈ C_{0}^{1}(ℝ^{2}). Then we choose ψ = ψ^{ε}(t, x) := σ^{ε}(t − t_{0})ζ(x) with ζ(x) ≥ 0, t_{0} > 0, and

0 ≤ σ^{ε} ∈ C_{0}^{1}(ℝ),

σ^{ε}(t − t_{0}) → χ_{(−∞,t_{0})}(t),

δ^{ε}(t − t_{0}) = −\frac{d}{dt}σ^{ε}(t − t_{0}) ⇀ δ_{t_{0}}, as ε → 0,

where δ_{t_{0}} is the Dirac measure concentrated at t_{0}, and the convergences are in L^{1}(ℝ) and M(ℝ), respectively. We obtain from (4.1) that

0 = ∫∫_{t>0} ((ψ^{ε}p̄)_{t}μ_{t} dt − (ψ^{ε}p̄)_{x}v dt dx) + ∫ (ψ^{ε}p̄_{0}) τ_{0}

= − ∫ δ^{ε} (∫ ζ p̄μ_{t}) dt + ∫∫ p̄_{t}ψ^{ε} μ_{t} dt − ∫∫ (ψ^{ε} p̄)_{x}v dt dx + ∫ (ψ^{ε} p̄_{0}) τ_{0}.

Now, using that p̄_{t} ≤ 0, τ-a.e., we obtain

− ∫ δ^{ε} (∫ ζ p̄μ_{t}) dt + C_{0} ∫∫_{\begin{matrix}0≤t≤t_{0} \\ |x|≤|spt (ζ)| \end{matrix}} (|p̄_{x}| + |ζ′|) dt dx + ∫ (ψ^{ε} p̄_{0}) τ_{0} ≥ 0.

Assuming that t_{0} is a Lebesgue point of g(t) = ∫ ζ p̄μ_{t} and letting ε → 0 yield

(4.3)  ∫ ζ p̄μ_{t_{0}} ≤ C ∫∫_{\begin{matrix}0≤t≤t_{0} \\ |x|≤|spt (ζ)| \end{matrix}} (|p̄_{x}| + |ζ′|) dt dx + ∫ ζ p̄_{0} τ_{0}.

Now, taking the lim sup as t_{0} → 0 in both sides of (4.3), we finally arrive at (4.2). □

We now consider the solutions of the Euler equations (1.1)–(1.3) for gas dynamics in the sense of distributions such that τ is a nonnegative Radon measure, with τ ≥ cL^{2} for some c > 0, and v(t, x) and S(t, x) are bounded τ-measurable functions, along with our understanding that the constitutive relations (1.4) for (τ, p, e, θ, S)(t, x) hold L^{2}-almost everywhere out of the vacuum lines, in the set where τ is absolutely continuous with respect to L^{2}, and both p(t, x) and e(t, x) are defined as zero on the remaining set with measure zero in ℝ_{+}^{2}, including the vacuum lines.

We consider the Cauchy problem for (1.1)–(1.3):

(4.4)  (τ, v, S)|_{t=0} = (τ_{0}, v_{0}, S_{0})(x),

where τ_{0}(x) is a nonnegative Radon measure over ℝ, τ_{0} ≥ cL^{1} for some c > 0, v_{0}(x) and S_{0}(x) are bounded τ_{0}-measurable functions, and e_{0}(x) = e(τ_{0}(x), S_{0}(x)) a.e. out of the countable points {x_{k}} such that τ_{0}(x_{k}) > 0, the initial vacuum set.

Set Π_{T} = (0, T) × ℝ and Π_{T}^{∗} = (−∞, T) × ℝ for T > 0. Let D and F be functions or measures over Π_{T}. Let D_{0} be a function or a measure over ℝ. By weak formulation on Π_{T} for the Cauchy problem:

D_{t} + F_{x} = 0, (4.5)

D|_{t=0} = D_{0}, (4.6)

we mean that, for a suitable set of test functions ϕ(t, x) defined on Π_{T}^{∗},

(4.7)  ∫∫_{Π_{T}} (ϕ_{t}D + ϕ_{x}F) + ∫_{ℝ} ϕ(0, x)D_{0} = 0.

Analogously, if the identity “ = ” in (4.5) is replaced by “ ≥ ” or “ ≤ ”, the weak formulation of the corresponding problem (4.5) and (4.6) is (4.7) with “ = ” replaced by “ ≤ ” or “ ≥ ”, respectively, for a suitable set of nonnegative test functions defined on Π_{T}^{∗}.

Denote W = (τ, v, S), f(W) = (−v, p(τ, S), 0), η(W) = e(τ, S) + \frac{v^{2}}{2} , q(W) = vp(τ, S), and

α(W, \overline{W}) = η(W) − η(\overline{W}) − ∇η(\overline{W}) · (W − \overline{W}),
β(W, \overline{W}) = q(W) − q(\overline{W}) − ∇η(\overline{W}) · (f(W) − f(\overline{W})).

Observe that ∇η(\overline{W}) = (−p̄, −v̄, θ̄).

Definition 4.2. We say that W (t, x) is a distributional entropy solution of (1.1)– (1.3), and (4.4) in Π_{T} if τ is a Radon measure on Π_{T} with τ ≥ cL^{2} for some c > 0, v and S are bounded τ-measurable functions such that the weak formulation of (1.1)–(1.3), (1.7), and (4.4) is satisfied for all test functions in C_{0}^{1}(Π_{T}^{∗}), and S(t, ·) ⇀ S_{0}( ·), as t → 0, in the weak-star topology of L^{∞}(ℝ).

Observe that the weak formulation implies that μ_{t} ⇀ τ_{0} in M(ℝ), and v(t, ·) ⇀ v_{0}( ·), and E(t, ·) ⇀ E_{0}( ·) in the weak-star topology of L^{∞}(ℝ), as t → 0, where E = e + v^{2}/2. We also remark that these convergences can be strengthened to the convergences in L_{loc}^{1}(ℝ) in the case that τ is a bounded measurable function, as an easy consequence of the DM^{∞} theory (cf. [5]).

As shown by Wagner [25], by means of the transformation from Eulerian to Lagrangian coordinates, bounded measurable entropy solutions of the Euler equations in Eulerian coordinates transform into distributional entropy solutions of (1.1)–(1.3) and (4.4), satisfying the additional restriction that the weak formulation of (1.2), (1.3), and (1.7) holds for test functions with compact support in Π_{T} such that ϕ_{t} = g, ϕ_{x} = h τ, where g, h ∈ L^{∞}(Π_{T} , τ). It is also shown through an example in [25] that distributional entropy solutions without the additional restriction may have no physical meaning.

Now we consider the Riemann solution \overline{W} (t, x) associated to the Riemann problem for (1.1)–(1.3) with initial condition

(4.8)  \overline{W}_{0}(x) = \begin{cases}W_{L}, & x < 0, \\ W_{R}, & x > 0, \end{cases}

where W_{L} and W_{R} are two constant states in the physical domain {W = (τ, v, S) : τ > 0\}. First, we address the case that \overline{W} (t, x) is a bounded self-similar entropy solution of (1.1)–(1.3) which consists of at most two rarefaction waves, one corresponding to the first characteristic family and another corresponding to the third one, and possibly one contact discontinuity on the line x = 0. Then, \overline{W} (t, x) has the following general form:

(4.9)  \overline{W} (x, t) = \begin{cases}W_{L}, & x/t < ξ_{1}, \\ R_{1}(x/t), & ξ_{1} ≤ x/t < ξ_{2}, \\ W_{M}, & ξ_{2} ≤ x/t < 0, \\ W_{N}, & 0 < x/t < ξ_{3}, \\ R_{3}(x/t), & ξ_{3} ≤ x/t < ξ_{4}, \\ W_{R}, & x/t ≥ ξ_{4}. \end{cases}

In what follows we use the notation W^{p} = (τ, S), \overline{W}^{p} = (τ̄, \overline{S}).

Theorem 4.2. Let \overline{W} (t, x) be the Riemann solution (4.9), and let W (t, x) be any distributional entropy solution of (1.1)–(1.3) and (4.4) with W_{0} ∈ L^{∞}(ℝ; ℝ^{3}). Then there exist positive constants C and K_{0}, and a function ω ∈ L^{∞}(Π_{T}), positive a.e. in Π_{T}, such that, for any X > 0 and a.e. t > 0,

(4.10) ∫_{|x|≤X} (|v(t, x) − v̄(t, x)|^{2} + |W_{a.c.}^{p}(t, x) − W̄^{p}(t, x)|^{2}ω(t, x)) dx

≤ C ∫_{|x|≤X+K_{0}t} (|v_{0}(x) − v̄_{0}(x)|^{2} + |W_{0}^{p}(x) − \overline{W} _{0}^{p}(x)|^{2}ω(0, x)) dx.

Proof. We divide the proof into four steps.

Step 1. Consider the measure

ϖ := α(W, \overline{W})_{t} + β(W, \overline{W})_{x}.

Given any X > 0 and t > 0, let t_{0} ∈ (0, t) and

Ω_{t_{0},t} = {(σ, x) : |x| < X + K_{0}(t − σ), t_{0} < σ < t},

for K_{0} > 0 to be suitably chosen later.

First, by the Gauss-Green formula (Theorem 3.1), we have

ϖ(Ω_{t_{0},t}) = ⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, 1⟩

Now, let ζ_{i}, i = 1, ⋯, 4, be nonnegative functions in C_{0}^{∞}(ℝ^{2}) such that

∑_{i=1}^{4} ζ_{i} = 1, over ∂Ω_{t_{0},t},

ζ_{1} = 1, over {(0, x) : |x| < X + K_{0}(t − t_{0}) },

ζ_{2} = 1, over {(t, x) : |x| < X },

(spt (ζ_{3}) ∪ spt (ζ_{4})) ∩ (({t} × ℝ) ∪ ({t_{0}} × ℝ)) = ∅.

We choose ζ_{3} and ζ_{4} so that spt (ζ_{3}) intersects the left lateral side of Ω_{t_{0},t} but not the right, and spt (ζ_{4}) intersects its right lateral side but not the left. We have

ϖ(Ω_{t_{0},t}) = ⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{1} + ζ_{2} + ζ_{3} + ζ_{4}⟩.

In what follows, we will use the notations:

ĝ = ∫_{0}^{1}(1 − σ)g(στ + (1 − σ)τ̄, σS + (1 − σ)\overline{S}) dσ,

for any function g = g(τ, S), and

g̃ = ∫_{0}^{1} g(στ + (1 − σ)τ̄, σS + (1 − σ)\overline{S}) dσ.

Step 2. We now prove the following five estimates.

1. ⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{3}⟩ ≥ 0 and ⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{4}⟩ ≥ 0.

Indeed, let z_{∗} = (t_{∗}, x_{∗}) be the center of Ω_{t_{0},t}. We consider the following deformation of ∂Ω_{t_{0},t}:

Ψ(z, s) := z + εs(z_{∗} − z), z ∈ ∂Ω_{t_{0},t}, s ∈ [0, 1],

where ε is chosen so small that

(spt (ζ_{3})∪ spt (ζ_{4}))∩({(t − εs(t − t_{∗}), x) : x ∈ ℝ} ∪ {(t_{0} + εs(t_{∗} − t_{0}), x) : x ∈ ℝ}) = ∅,

for s ∈ [0, 1]. Then

⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{3}⟩ = lim_{s→0} \frac{1}{s} ∫∫_{Ψ(∂Ω_{t_{0},t}×(0,s))}(K_{0}α − β)|∇h| ζ_{3}dσdx ≥ 0,

by choosing K_{0} such that K_{0}α ≥ β, which is possible by what follows.

2. Given δ > 0 and bounded sets B_{1} ⊂ V_{δ} := {(τ, v, S) : τ > δ} and B_{2} ⊂ ℝ^{2}, there exists a constant K_{0} = K_{0}(δ, B_{1}, B_{2}) > 0 such that K_{0}α(W, \overline{W}) ≥ β(W, \overline{W}), for any \overline{W} ∈ B_{1} and W ∈ V_{δ} with (v, S) ∈ B_{2}.

In fact, we first have

α(W, \overline{W}) = \frac{1}{2}(v − v̄)^{2} + ê_{ττ} (τ − τ̄)^{2} + 2ê_{τS}(τ − τ̄)(S − \overline{S}) + ê_{SS}(S − \overline{S})^{2},

β(W, \overline{W}) = (v − v̄)(p − p̄).

If W (t, x) and \overline{W} (t, x) belong to a bounded set B in V_{δ}, we can find K_{0} depending only on B such that K_{0}α ≥ β. Now, since (τ̄, v̄, \overline{S}) ∈ B_{1} and (v, S) ∈ B_{2}, it suffices to show that, for τ sufficiently large, we have β(W, \overline{W}) ≤ α(W, \overline{W}). Notice that

β ≤ \frac{1}{2}(v − v̄)^{2} + \frac{1}{2}p̃_{τ}^{2}(τ − τ̄)^{2} + p̃_{τ} p̃_{S}(τ − τ̄)(S − \overline{S}) + \frac{1}{2}p̃_{S}^{2}(S − \overline{S})^{2}

≤ \frac{1}{2}(v − v̄)^{2} + p̃_{τ}^{2}(τ − τ̄)^{2} + p̃^{2}(S − \overline{S})^{2}.

On the other hand,

α ≥ \frac{1}{2}(v − v̄)^{2} + \frac{ê_{ττ}}{2(K + 1)}(τ − τ̄)^{2} + \frac{1}{2}(ê_{SS} − \frac{K + 1}{K}\begin{matrix} \\ ( & \frac{e_{τS}^{2}}{e_{ττ}} & ) \end{matrix})(S − \overline{S})^{2},

for any K > 0. Now, p̃_{τ}^{2} decays faster than ê_{ττ} = −p̂_{τ} as τ → ∞, p̃_{S}^{2} decays faster than ê_{SS} as τ → ∞, and, for K sufficiently large,

ê_{SS} − \frac{K + 1}{K}\begin{matrix} \\ ( & \frac{e_{τS}^{2}}{e_{ττ}} & ) \end{matrix} > c ê_{SS},

for some c > 0 sufficiently small, since γ > 1, as one can easily check from (1.4).

Hence, β(W, \overline{W}) ≤ α(W, \overline{W}) for τ larger than a certain τ_{∗}. Then the assertion follows.

3. Similarly, we have ⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{4}⟩ ≥ 0.

4. As for ζ_{2}, we have

⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{2}⟩ = lim_{s→0} \frac{1}{s} ∫∫_{Ψ(∂Ω_{t_{0},t}×(0,s))}(α, β) · ∇h ζ_{2}dσdx

≥ lim_{s→0} \frac{1}{ε(t − t_{∗})s} ∫_{t−εs(t−t_{∗})}^{t}dσ ∫_{|x|≤X−εK_{1} \frac{T}{2}} \{η(W) − η(\overline{W}) − v̄(v − v̄) + p̄(τ_{a.c.} − τ̄) + θ̄(S − \overline{S})\} dx

= ∫_{\begin{matrix}|x|≤X−εK_{1} \frac{T}{2} \\ σ=t \end{matrix}} \{η(W_{a.c.}) − η(\overline{W}) − v̄(v − v̄) + p̄(τ_{a.c.} − τ̄) + θ̄(S − \overline{S})\} dx

= ∫_{\begin{matrix}|x|≤X−εK_{1} \frac{T}{2} \\ σ=t \end{matrix}} (\frac{1}{2}|v − v̄|^{2} + Q(W_{a.c.}^{p}(t, x) − \overline{W}^{p}(t, x))) dx,

if t is a Lebesgue point of

g(s) = ∫_{|x|≤X−εK_{1} \frac{T}{2}} α(W_{a.c.}, \overline{W})(s, x) dx,

where Q is the quadratic form associated with the symmetric matrice

A = [\begin{matrix}ê_{ττ} & ê_{τS} \\ ê_{τS} & ê_{SS} \end{matrix}] .

5. For ζ_{1}, we have

⟨(α, β) · ν|_{∂Ω_{t_{0},t}}, ζ_{1}⟩ = lim_{s→0} \frac{1}{s} ∫∫_{Ψ(∂Ω_{t_{0},t}×(0,s))}(α, β) · ν ζ_{1}|∇h| dσ dx

≥ − lim_{s→0} \frac{1}{s} ∫_{t_{0}}^{t_{0}+εs(t_{∗}−t_{0})} ∫_{|x|≤X+K_{0}(t−t_{0})} α(W, \overline{W})\frac{1}{ε(t_{∗} − t_{0})} dσ dx

= − ∫_{\begin{matrix}|x|≤X+K_{0}(t−t_{0}) \\ σ=t_{0} \end{matrix}} \{η(W) − η(\overline{W}) − v̄(v − v̄) + p̄(μ_{t_{0}} − τ̄) − θ̄(S − \overline{S})\},

where we have also used that μ_{σ} ⇀ μ_{t_{0}} as σ → t_{0} + 0, for a.e. t_{0} > 0, with μ_{t} as in Theorem 4.1, and that p̄ is continuous on [t_{0}, t] × ℝ.

Step 3. On the other hand,

ϖ(Ω_{t_{0},t}) = ∑_{i=1}^{4} ϖ(l̃_{i} ∩ Ω_{t_{0},t}) + ϖ(l ∩ Ω_{t_{0},t}) + ϖ(Ω_{1} ∩ Ω_{t_{0},t})
+ ϖ(Ω_{3} ∩ Ω_{t_{0},t}) + ϖ(Ω_{t_{0},t} − (∪_{i=1}^{4}l̃_{i} ∪ l ∪ Ω_{1} ∪ Ω_{3})),

where Ω_{1} and Ω_{3} are the left and right rarefaction regions, l̃_{i}, 1 ≤ i ≤ 4, are the lines bounding the rarefaction regions Ω_{1} and Ω_{3}, and l is the line {x = 0} where \overline{W} (t, x) has a contact discontinuity.

We first observe that, on Ω_{t_{0},t} − (∪_{i=1}^{4}l̃_{i} ∪ l ∪ Ω_{1} ∪ Ω_{3}), the measure ϖ reduces to −θ̄∂_{t}S which is nonpositive. Now, we have

ϖ = −div (F_{1} + F_{2} + F_{3}),

where

F_{1} = v̄(v − v̄, p − p̄), F_{2} = −p̄(τ − τ̄, v̄ − v), F_{3} = θ̄(S − \overline{S}, 0),

and div := div_{t,x}. Applying the product rule (Theorem 3.2), we get

div F_{1} = v̄_{t}(v − v̄) + v̄_{x}(p − p̄), div F_{2} = −p̄_{t}(τ − τ̄) + p̄_{x}(v − v̄).

Hence,

div F_{1}(l̃_{j} ∩ Ω_{t_{0},t}) = div F_{1}(l ∩ Ω_{t_{0},t}) = 0, j = 1, ⋯, 4,

since div F_{1} is absolutely continuous with respect to L^{2}. Also,

div F_{3}(l̃_{j} ∩ Ω_{t_{0},t}) ≥ 0, j = 1, ⋯, 4.

On the other hand, since F_{3} ∈ DM^{∞}(Ω_{t_{0},t}) and ν|_{l} = (0, 1), we have

div F_{3}(l ∩ Ω_{t_{0},t}) = [⟨F_{3} · ν|_{l}, 1⟩] = 0,

where the square-bracket denotes the difference between the normal traces from the right and the left, which make sense for F_{3} ∈ DM^{∞} because the normal traces of DM^{∞} fields are functions in L^{∞} over the boundaries.

Concerning F_{2}, we have

div F_{2}(l̃_{j} ∩ Ω_{t_{0},t}) = 0, j = 1, ⋯, 4,

since p̄_{t} is τ-integrable and τ(l̃_{j}) = 0, j = 1, ⋯, 4. On the other hand, p̄_{t} vanishes on l so that

div F_{2}(l ∩ Ω_{t,t_{0}}) = 0.

Finally, using the product rule (Theorem 3.2), the fact that W (t, x) and \overline{W} (t, x) are distributional solutions of (1.1)–(1.3) and (4.4), and \overline{S}(t, x)_{t} = 0, we obtain, for j = 1, 3,

ϖ(Ω_{j} ∩ Ω_{t_{0},t}) = − ∫∫_{Ω_{j}∩Ω_{t_{0},t}}\{(v̄(v − v̄))_{t} + (v̄(p − p̄))_{x} − (p̄(τ − τ̄))_{t} + (p̄(v − v̄))_{x} + (θ̄(S − \overline{S}))_{t}\} (4.11)

= − ∫∫_{Ω_{j}∩Ω_{t_{0},t}}\{v̄_{t}(v − v̄) + v̄_{x}(p − p̄) − p̄_{t}(τ − τ̄) + p̄_{x}(v − v̄) + θ̄_{t}(S − \overline{S}) + θ̄S_{t}\}

≤ − ∫∫_{Ω_{j}∩Ω_{t_{0},t}}\{v̄_{x}(p − p̄) − p̄_{τ} v̄_{x}(τ − τ̄)dxds + θ̄_{τ} v̄_{x}(S − \overline{S})\}

≤ − ∫∫_{Ω_{j}∩Ω_{t_{0},t}} v̄_{x} (p − p̄ − p̄_{τ} (τ_{a.c.} − τ̄) − p̄_{S}(S − \overline{S})) dxds + ∫∫_{Ω_{j}∩Ω_{t_{0},t}} v̄_{x}p̄_{τ} τ_{sing}

≤ 0,

since p̄_{τ} < 0 and v̄_{x} is bounded and v̄_{x}(t, x) ≥ 0 everywhere over Ω_{1} and Ω_{3}, and ∇_{W^{p}}^{2} p > 0.

Step 4. Putting all these estimates together, we have

∫_{|x|≤X} (|v(t, x) − v̄(t, x)|^{2} + |W_{a.c.}(t, x) − \overline{W} (t, x)|^{2}ω(t, x)) dx

≤ 2 ∫_{\begin{matrix}|x|≤X+K_{0}(t−t_{0}) \\ σ=t_{0} \end{matrix}} \{η(W) − η(\overline{W}) − v̄(v − v̄) + p̄(μ_{t_{0}} − τ̄) − θ̄(S − \overline{S})\}.

Now, applying Corollary 4.1, we finally get (4.10). □

Corollary 4.2. Let W (t, x) and \overline{W} (t, x) satisfy the conditions of Theorem 4.2 and W_{0}(x) = \overline{W}_{0}(x). Then τ(t, x) is absolutely continuous with respect to L^{2} in Π_{T} and W (t, x) = \overline{W} (t, x) a.e. in Π_{T}.

Proof. From Theorem 4.2, we have

W_{a.c.}(t, x) = \overline{W} (t, x), a.e. in Π_{T} .

Hence, τ_{sing} must satisfy the weak formulation of

τ_{t} = 0, τ|_{t=0} = 0,

for the test functions in C_{0}^{1}(Π_{T}^{∗}). In particular, there exists y ∈ BV_{loc}(Π_{T}) such that

∂_{x}y = τ_{sing}, ∂_{t}y = 0.

Therefore,

τ_{sing} = dt ⊗ ν_{t}(x),

where ν_{t}(·) = \frac{dy}{dx}(t, ·), a.e. t ∈ (0, T). Since y does not depend on t, we have that ν_{t} also does not depend on t, say, ν_{t}(x) = ν_{0}(x) and

τ_{sing} = dt ⊗ ν_{0}.

Furthermore, since ν_{t} ⇀ 0 as t → 0, we conclude

ν_{0} ≡ 0.
□

We now consider the case that the Riemann solution, with the initial condition (4.8), has a vacuum line at x = 0. In this case, the Riemann solution W̄ (t, x) has the following form: (4.12)

\overline{W} (x, t) = \begin{cases}W_{L}, & x/t < ξ_{L}, \\ R_{1}(x/t), & ξ_{L} ≤ x/t < 0, \\ ((v̄_{1} + v̄_{2})/2, (v̄_{2} − v̄_{1})tdt ⊗ δ_{0}(x), (\overline{S}_{L} + \overline{S}_{R})/2), & x = 0, \\ R_{3}(x/t), & 0 < x/t ≤ ξ_{R}, \\ W_{R}, & x/t > ξ_{R}. \end{cases}

Here R_{1}(x/t) and R_{3}(x/t) are as above the rarefaction waves of the first and third characteristic families, respectively, v̄_{1} = lim_{ξ→0_{−}} v̄(ξ), v̄_{2} = lim_{ξ→0_{+}} v̄(ξ), and δ_{0}(x) is the Dirac measure over ℝ concentrated at 0. It is easy to check that \overline{W} (t, x) is a distributional solution of (1.1)–(1.3) and (4.4). The values of v̄ and \overline{S} on the line x = 0 could be taken as any other constants instead of \frac{v̄_{1}+v̄_{2}}{2} and \frac{\overline{S}_{L}+\overline{S}_{R}}{2} , respectively, while the formula of τ̄ at x = 0, (v̄_{2} − v̄_{1})tdt ⊗ δ_{0}(x), is dictated by the fact that (1.1) must hold in the sense of distributions.

Theorem 4.3. Let \overline{W} (t, x) be a Riemann solution containing vacuum as described in (4.12). Let W (t, x) be a distributional entropy solution of (1.1)–(1.3) and (4.4) in Π_{T} with W_{0} ∈ L^{∞}(ℝ; ℝ^{3}). Then there exist positive constants C and K_{0}, and a function ω ∈ L^{∞}(Π_{T}), positive a.e. in Π_{T}, such that, for all X > 0 and a.e. t > 0,

(4.13) ∫_{|x|≤X} (|v(t, x) − v̄(t, x)|^{2} + |W_{a.c.}^{p}(t, x) − W̄^{p}(t, x)|^{2}ω(t, x)) dx

≤ C ∫_{|x|≤X+K_{0}t} (|v_{0}(x) − v̄_{0}(x)|^{2} + |W_{0}^{p}(x) − \overline{W} _{0}^{p}(x)|^{2}ω(0, x)) dx.

Proof. Let F_{1}, F_{2}, and F_{3} be as in the proof of Theorem 4.2. We observe that, since p̄ = 0 for x = 0, we have

F_{2} = −p̄(τ − τ̄_{a.c.}, v̄ − v),

so that the analysis for F_{2} remains the same. Also, nothing needs to be changed concerning F_{3}. As for F_{1} = v̄(v − v̄, p − p̄), we have a new aspect which is the fact that v̄(t, x) is discontinuous at l. Then we have

div F_{1}(l_{t_{0},t}) = [⟨F_{1} · ν|_{l_{t_{0},t}}, 1⟩]

where l_{t_{0},t} = l ∩ Ω_{t_{0},t} and again ν|_{l} = (0, 1). Let p_{−}(t, x) and p_{+}(t, x) denote the functions in L^{∞}(l), given by the theory of DM^{∞} fields developed in [5], such that

⟨p_{−}, ζ⟩ = ⟨(v, p) · ν|_{l}, ζ⟩_{l},
⟨p_{+}, ζ⟩ = ⟨(v, p) · (−ν)|_{l}, ζ⟩_{l}, for any ζ ∈ C_{0}(l).

Hence, we have

div F_{1}(l_{t_{0},t}) = v_{2} ∫_{t_{0}}^{t} p_{+}(s) ds − v_{1} ∫_{t_{0}}^{t} p_{−}(s) ds.

On the other hand,

p_{+}(s) = p_{−}(s), a.e. s > 0,  and  v_{2} > v_{1},

where the first follows from (1.2) and the second is a consequence of the construction of the Riemann solution containing vacuum. Therefore, we conclude

div F_{1}(l_{t_{0},t}) ≥ 0.

The remaining of the proof follows exactly as in Theorem 4.2. □

Again, we have the following corollary.

Corollary 4.3. Let \overline{W} (t, x) and W (t, x) satisfy the hypotheses of Theorem 4.3 and W_{0}(x) = \overline{W}_{0}(x). Then (v, S)(t, x) = (v̄, \overline{S})(t, x), L^{2}-a.e. in Π_{T}, and τ = τ̄ in M(Π_{T}).

Proof. From Theorem 4.3, we deduce that W_{a.c}(t, x) = \overline{W}_{a.c}(t, x), L^{2}-a.e. in Π_{T}. Thus, as in the proof of Corollary 4.2, we deduce that τ_{sing} must be concentrated at {x = 0}. Then τ_{sing} must be equal to (v̄_{2} − v̄_{1})tdt ⊗ δ_{0}(x) as a consequence of (1.1) in the sense of distributions. □

5. Proof of Theorem 4.1

In this section, we give a detailed proof of Theorem 4.1. The arguments are strongly motivated by those in Wagner [25].

Proof. We divide the proof into ten steps.

Step 1. There exists y ∈ BV_{loc}([0, ∞) × ℝ) such that

(5.1)  ∂_{x}y = τ,  ∂_{t}y = v,

in the sense of distributions for t > 0.

Indeed, let ω^{ε} be a positive symmetric mollifier in ℝ^{2}, and set τ^{ε} = τ ∗ ω^{ε} and v^{ε} = v ∗ ω^{ε}, where we have extended τ and v as zero for t < 0. Define

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

Then y^{ε} ∈ BV ∩ C^{1}(ℝ_{+}^{2}). We easily check that y^{ε}(t, x) satisfies

∂_{x}y^{ε} = τ^{ε},  ∂_{t}y^{ε} = v^{ε},  t > ε.

Now, ∥y^{ε}∥_{BV (Ω)} ≤ M_{Ω}, for any open set Ω ⋐ ℝ_{+}^{2}, where M_{Ω} is a positive constant independent of ε. Hence, by the compact embedding of BV (Ω) into L^{1}(Ω), there exists y ∈ BV_{loc}(ℝ_{+}^{2}) such that

y^{ε}(t, x) → y(t, x), in L_{loc}^{1}(ℝ_{+}^{2}),

by passing to a subsequence if necessary. Clearly, y(t, x) satisfies (5.1) in the sense of distributions in ℝ_{+}^{2}.

Step 2. The measure τ(t, x) admits a slicing of the form τ = dt ⊗ μ_{t}, where, for L^{1}-a.e. t > 0, μ_{t} ∈ M^{+}(ℝ).

Let y(t, x) be a solution of (5.1). Since y ∈ BV_{loc}(ℝ_{+}^{2}), then, for a.e. t > 0, y(t, ·) ∈ BV_{loc}(ℝ). Hence,

τ = dt ⊗ \frac{dy}{dx}(t, ·).

Step 3. The points (t, x) ∈ ℝ_{+}^{2} such that μ_{t}(x) > 0, with the possible exception of a set of H^{1}-measure zero, form a countable union of vertical line segments.

Again, since y ∈ BV_{loc}(ℝ_{+}^{2}), then, for a.e. x ∈ ℝ, y(·, x) ∈ BV_{loc}(ℝ_{+}). Hence, ∂_{t}y admits a slicing as

∂_{t}y = \frac{dy}{dt} (·, x) ⊗ dx = v(·, x)dt ⊗ dx.

That is, for a.e. x ∈ ℝ, y(·, x) is a Lipschitz function on [0, ∞) whose derivative is v(·, x). On the other hand, the jump set of y(t, x), with the possible exception of a set of H^{1}-measure zero, is a countable union of C^{1} curves {l_{k}}_{k∈ℕ}, by the structure theory of BV functions (see, e.g., [12, 13]). We then conclude that the lines l_{k} must be vertical, because, otherwise, we would have a subset A ⊂ ℝ of positive measure such that, for x ∈ A, \frac{dy}{dt} (·, x) would be a singular measure, rather than an L^{∞} function v(·, x), which proves the assertion.

Observe that μ_{t} ⇀ τ_{0} as t → 0+, which follows from standard arguments by choosing suitable test functions in (4.1).

Step 4. Let x_{0} ∈ ℝ be such that τ_{0}(x_{0}) = μ_{t}(x_{0}) = 0, a.e. t > 0, and x_{0} is a Lebesgue point of ∫_{0}^{t} v(σ, ·) dσ, for all rational t > 0 and hence all fixed t > 0.

Then there exists a solution of (5.1) satisfying

(5.3)  lim_{t→0+} y(t, x) = ∫_{x_{0}}^{x} τ_{0},  for a.e. x ∈ ℝ.

Consider a mollifier ω^{ε}, with ε = (ε_{1}, ε_{2}), of the form ω^{ε}(t, x) = δ^{ε_{1}}(t)δ^{ε_{2}}(x), where δ^{ε_{j}}, j = 1, 2, are standard positive symmetric mollifiers in ℝ. If y^{ε}(t, x) is defined by (5.2), with 0 replaced by x_{0}, then sending ε_{1} to 0 first and next letting ε_{2} → 0 yield

lim_{ε_{2}→0} lim_{ε_{1}→0} y^{ε}(t, x) = ∫_{x_{0}}^{x} μ_{t} + ∫_{0}^{t} v(σ, x_{0}) dσ, a.e. (t, x) ∈ ℝ_{+}^{2}.

Hence,

y(t, x) = ∫_{x_{0}}^{x} μ_{t} + ∫_{0}^{t} v(σ, x_{0}) dσ,

provides the desired solution because of Step 4.

Now, since, for each t ≥ 0, y(t, x) is a strictly increasing function of x, there exists a well-defined monotone increasing continuous function x(t, y) such that

(5.4)  x(t, y(t, x)) = x,  (t, x) ∈ ℝ_{+}^{2},

where we have defined y(t, x) on the vacuum lines by, say, (y(t, x+0)+y(t, x−0))/2.

Let Q and T be the transformations of ℝ_{+}^{2}, Q(t, x) = (t, y(t, x)) and T (t, y) = (t, x(t, y)) so that T (Q(t, x)) = (t, x). Observe that, given an open rectangle R = (t_{1}, t_{2}) × (a, b) ⊂ ℝ_{+}^{2}, if a and b are such that y(t, a) and y(t, b) are continuous functions of t, then Q(R) is an open set in ℝ_{+}^{2}, which implies that T is continuous in ℝ_{+}^{2}. Let

T_{0}(x) = lim_{t→0+} y(t, x), Q_{0}(y) = lim_{t→0_{+}} x(t, y).

Step 5. T_{#}L^{2} = τ and T_{0#}L^{1} = τ_{0}, with notation from [13].

In fact, for a.e. t > 0, y(t, x) is a strictly increasing BV_{loc} function of the variable x, and τ = dt ⊗ μ_{t}(x) with μ_{t}(x) = \frac{dy}{dx}(t, ·). Now, T (t, y) = (t, x(t, y)) and, for any open interval (a, b) and for each fixed t > 0,

x(t, ·)^{−1}(a, b) = (y(t, a + 0), y(t, b − 0)) = τ(a, b).

Hence, the measure T_{#}L^{2} equals dt⊗μ̃_{t}, where μ̃_{t} is the Stieltjes measure associated with y(t, x), that is, μ̃_{t} = μ_{t}, which implies T_{#}L^{2} = τ. Similarly, we obtain T_{0#}L^{1} = τ_{0}.

Step 6. The map T is proper and onto.

The fact that T is onto follows from the fact that it is the inverse of Q which is defined everywhere in ℝ_{+}^{2}. Now, since Q(t, x) is a continuous function of t, for a.e. x ∈ ℝ, given any compact K ⊂ ℝ_{+}^{2}, we may find a rectangle [t_{1}, t_{2}] × [a, b] containing K such that Q(t, a) and Q(t, b) are continuous functions of t. Hence, the set

{(t, y) | y(t, a) ≤ y ≤ y(t, b), t_{1} ≤ t ≤ t_{2} }

is compact and contains T^{−1}(K), which is then also compact by the continuity of T.

Let ρ = Q_{#}L^{2}. Let ρ̃ be the density of L^{2} with respect to τ in ℝ_{+}^{2} and let τ̃ be the density of τ_{a.c.} with respect to L^{2} so that ρ̃τ̃ = 1, L^{2}-a.e. in ℝ_{+}^{2}. The following result was proved by Wagner ([25], p.132).

Step 7. ρ = ∂_{y}x = ρ̃ ∘ T.

Indeed, ρ = Q_{#}L^{2} admits a slicing of the form ρ = dt ⊗ μ̃_{t}, where μ̃_{t} is the Stieltjes measure associated with the inverse of y(t, ·), which is x(t, y), and hence μ̃_{t} = ∂_{y}x. Furthermore, The Lipschitz continuity of x(t, y) yields

ρ = ∂_{y}x,

where as usual we have identified absolutely continuous measures with their densities, with respect to the corresponding Lebesgue measure. Also, we have

T_{#}ρ = T_{#}Q_{#}L^{2} = L^{2} = ρ̃τ̃ = ρ̃τ = ρ̃T_{#}L^{2}.

Now, for all ϕ ∈ C_{0}((0, ∞) × ℝ),

⟨ρ, ϕ⟩ = ⟨T_{#}ρ, ϕ ∘ Q⟩,

while

⟨ρ̃ ∘ T, ϕ⟩ = ⟨ρ̃T_{#}L^{2}, ϕ ∘ Q⟩,

which implies

ρ = ρ̃ ∘ T,

where we have used that ρ = 0 a.e. over T^{−1}(V) and V is the union of the vacuum lines, because of the first part of the assertion.

Now, let ρ_{0} = Q_{0#}L^{1} and \underline{v}(t, y) = v(T (t, y)).

Step 8. For all ϕ ∈ C_{0}^{1}(ℝ^{2}), we have

(5.5)  ∫∫_{t>0}(ρϕ_{t} + ρ\underline{v}ϕ_{y}) dt dy + ∫_{t=0} ϕ(0, y)ρ_{0}(y) dy = 0.

Observe that ρ and ρ_{0} are uniformly bounded since ρ = ρ̃∘ T, ρ̃τ̃ = 1, and τ̃ ≥ c for some c > 0. The same holds for ρ_{0}. Now, for any ϕ ∈ C_{0}^{1}(ℝ^{2}), we have

∫∫_{t>0}(ϕ_{t} + ϕ_{y}\underline{v})ρ dt dy + ∫_{t=0} ϕ(0, y)ρ_{0}(y) dy
= ∫∫_{t>0}(ϕ_{t} ∘ Q + ϕ_{y} ∘ Q\underline{v} ∘ Q) dt dx + ∫_{t=0} ϕ(0, Q_{0}(x)) dx
= ∫∫_{t>0}(ϕ_{t} ∘ Q + vϕ_{y} ∘ Q) dt dx + ∫_{t=0} ϕ(0, Q_{0}(x)) dx
= ∫∫_{t>0} ∂_{t}(ϕ ∘ Q) dt dx + ∫_{t=0} ϕ(0, Q_{0}(x)) dx = 0,

where we have used that \underline{v} ∘ Q = v. In particular, we have

(5.6)  ρ_{t} + (ρ\underline{v})_{y} = 0,

in the sense of distributions. Now, let x_{0} ∈ ℝ be such that y(t, x_{0}) is a Lipschitz continuous function on [0, ∞) and, for a.e. t, y(t, x_{0}) is a Lebesgue point for both ρ(t, y) and \underline{v}(t, y).

Step 9. There exists a solution x̄(t, y) of

(5.7)  ∂_{y}x = ρ,  ∂_{t}x = −ρ\underline{v},

such that x̄(t, y(t, x_{0})) = x_{0}. In particular, x̄(t, y) = x(t, y).

Indeed, let ρ^{ε} = ρ ∗ ω^{ε} and (ρ\underline{v})^{ε} = (ρ\underline{v}) ∗ ω^{∗}, where ω^{ε} is a standard mollifier. Set

x^{ε}(t, y) = ∫_{y(t,x_{0})}^{y} ρ^{ε}(t, s) ds + x_{0}.

Then x^{ε}(t, y) is a Lipschitz function satisfying

∂_{y}x^{ε} = ρ^{ε}(t, y),
∂_{t}x^{ε} = −(ρ\underline{v})^{ε}(t, y) − ρ^{ε}(t, y(t, x_{0}))\underline{v}(t, y(t, x_{0})) + (ρ\underline{v})^{ε}(t, y(t, x_{0})).

Also,

|x^{ε}(t, y) − x_{0}| ≤ M|y − y(t, x_{0})|.

Hence, x^{ε}(t, y) converges in L_{loc}^{1}(ℝ_{+}^{2}) to a Lipschitz function x̄(t, y) which satisfies

|x̄(t, y) − x_{0}| ≤ M|y − y(t, x_{0})|.

Thus, we have x̄(t, y(t, x_{0})) = x_{0}. Since

|ρ^{ε}(t, y(t, x_{0}))\underline{v}(t, y(t, x_{0})) − (ρ\underline{v})^{ε}(t, y(t, x_{0}))| → 0, as ε → 0,

for a.e. t > 0, we conclude that x̄(t, y) satisfies (5.7). Now, since ∂_{y}x(t, y) = ρ(t, y) and x(t, y(t, x_{0})) = x_{0}, we must have x(t, y) = x̄(t, y).

Step 10. Equation (4.1) holds for any τ-test function ϕ(t, x).

Indeed, for any τ-test function ϕ(t, x), we have

∫∫_{t>0}(ϕ_{t}τ − vϕ_{x} dt dx) + ∫_{t=0} ϕ(0, x)τ_{0}

= ∫∫_{t>0}(ϕ_{t} − v ρ̃ϕ_{x})τ + ∫_{t=0} ϕ(0, x)τ_{0}

= ∫∫_{t>0}(ϕ_{t} ∘ T − ϕ_{x} ∘ T ρ \underline{v}) dt dy + ∫_{t=0} ϕ ∘ T_{0} dy

= ∫∫_{t>0} \frac{∂}{∂t}(ϕ ∘ T) dt dy + ∫_{t=0} ϕ ∘ T_{0} dy

= − ∫_{ℝ} lim_{δ→0} ϕ ∘ T (y, δ) dy + ∫_{t=0} ϕ ∘ T_{0} dy = 0.

□

Acknowledgments. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0204225, INT-9987378, and INT-9726215. Hermano Frid’s research was supported in part by CNPq through the grants 352871/96-2, 465714/00-5, 479416/01-0 and FAPERJ through the grant E-26/151.890/2000.

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(G.-Q. Chen) Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730, USA

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

(H. Frid) Instituto de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil

E-mail address: hermano@impa.br