FORMATION OF δ-SHOCKS AND VACUUM STATES IN THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE

EULER EQUATIONS FOR ISENTROPIC FLUIDS

GUI-QIANG CHEN HAILIANG LIU

Abstract. The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic 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 the δ-shock. By contrast, any tworarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of δ-shocks are also presented.

1. Introduction

We are concerned with the phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for compressible fluids as the pressure vanishes. In this paper, we consider the Euler equations of isentropic gas dynamics in Eulerian coordinates:

∂_{t}ρ + ∂_{x}(ρv) = 0, (1.1)

∂_{t}(ρv) + ∂_{x}(ρv^{2} + P) = 0, (1.2)

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.

For concreteness, we focus on the prototypical pressure function for polytropic gases:

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

Date: To appear in SIAM J. Math. Anal. 2002. 1991 Mathematics Subject Classification. Primary: 35L65, 35B30, 76E19, 35Q35, 35L67; Sec-

ondary: 35B25,65M06.

Key words and phrases. Concentration, cavitation, δ-shocks, vacuum states, Euler equations, vanishing pressure limit, transport equations, measure solutions, isentropic fluids, pressureless fluids, numerical simulations.

System (1.1)–(1.3) is an archetype of hyperbolic systems of conservation laws with form

(1.4)  ∂_{t}u + ∂_{x}f(u, ϵ) = 0,

with u = (ρ, ρv) and f(u, ϵ) = (ρv, ρv^{2} + ϵp(ρ)). Observe that system (1.1)–(1.3) with parameter ϵ > 0 is generic in the sense that such a system can also be obtained under the scaling:

(x, t) ⟶ (αx, αt), ρ ⟶ αρ,

with α = ϵ^{−1/(γ−1)} from system (1.1)–(1.2) with p = p(ρ).

In Chang-Chen-Yang [4, 5], a phenomenon of concentration in solutions of the two-dimensional Riemann problem was first observed numerically, 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. One of the main objectives of this paper is to show rigorously that the phenomenon of concentration in solutions, observed numerically in [4, 5], for inviscid compressible fluid flow is fundamental, which occurs not only in the multidimensional situations, but also even naturally in the one-dimensional case.

The limit system as ϵ → 0 formally becomes the following transport equations:

∂_{t}ρ + ∂_{x}(ρv) = 0, (1.5)

∂_{t}(ρv) + ∂_{x}(ρv^{2}) = 0, (1.6)

which is also called the one-dimensional system of pressureless Euler equations, modeling the motion of free particles which stick under collision (see [3, 10, 30]).

The transport equations (1.5)–(1.6) have been analyzed extensively since 1994; for example, see [1, 2, 3, 10, 11, 12, 16, 17, 18, 20, 22, 29] and the references cited therein. In particular, the existence of measure solutions of the Riemann problem was first presented in Bouchut [1], and a connection of (1.5)–(1.6) with adhesion particle dynamics and the behavior of global weak solutions with random initial data were discussed in E-Rykov-Sinai [10]. Also see [13, 14, 15, 21, 26, 27] for related equations and results. It has been shown that, for the transport equations (1.5)–(1.6), δ-shocks and vacuum states do occur in the Riemann solutions. Since the two eigenvalues of the transport equations coincide, the occurrence of δ-shocks and vacuum states as t > 0 can be regarded as a result of resonance between the two characteristic fields.

δ-SHOCKS AND VACUUM STATES IN THE VANISHING PRESSURE LIMIT 3

In this paper, we rigorously analyze the phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for isentropic fluids as the pressure vanishes. The vanishing pressure limit can be regarded as a singular flux-function limit for hyperbolic systems of conservation laws (1.4). We show that such phenomena occur naturally in the one-dimensional case as the pressure vanishes: any two-shock Riemann solution to the Euler equations for isentropic 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; by contrast, any tworarefaction-wave Riemann solution to the Euler equations for isentropic fluids tends to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state even when the initial data stays away from the vacuum. These results show that the δ-shocks for the transport equations result from a phenomenon of concentration, while the vacuum states result from a phenomenon of cavitation in the process of vanishing pressure limit; both are fundamental and physical in fluid dynamics.

From the point of view of hyperbolic conservation laws, since the limit system loses hyperbolicity, the phenomena of concentration and cavitation in the process of vanishing pressure limit can be regarded as a phenomenon of resonance between the two characteristic fields. These phenomena show 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^{∞}; and the space of Radon measures, for which the divergences of certain entropy and entropy flux fields are also Radon measures, is a natural space in order to deal with such a limit in general. In this regard, a theory of divergence-measure fields has been established in Chen-Frid [6, 7, 8].

The organization of this paper is the following. In Section 2, we discuss the δ-shocks and vacuum states for the transport equations (1.5)–(1.6) and examine the dependence of the Riemann solutions on the parameter ϵ > 0 for the Euler equations (1.1)–(1.3). In Section 3, we analyze the formation of δ-shocks in the Riemann solutions to the Euler equations (1.1)–(1.3) as the pressure vanishes. In Section 4, we analyze the formation of vacuum states in the Riemann solutions to (1.1)–(1.3), even when the initial data stays away from the vacuum, as the pressure decreases. In Section 5, we present some representative numerical results, produced by using the higher order ENO scheme in [23, 24], to examine the phenomenon of concentration and the formation process of δ-shocks in the level of the Euler dynamics (1.1)–(1.3) as the pressure decreases.

The main observations and results in this paper were reported at the International Conference on Nonlinear Evolutionary Partial Differential Equations, Academia Sinica (China), June 10–15, 2001, and at the First Joint Meeting of the American Mathematical Society and the Societe Mathematique de France at ENS de Lyon, France, July 17–20, 2001.

2. δ-Shocks, Vacuum States, and Riemann Solutions

In this section, we first discuss δ-shocks and vacuum states in the Riemann solutions to the transport equations (1.5)–(1.6), and then we examine the dependence of the Riemann solutions on the parameter ϵ > 0 to the Euler equations (1.1)–(1.3).

2.1. δ-Shocks and Vacuum States for the Transport Equations. Consider the Riemann problem of the transport equations (1.5)–(1.6) with Riemann initial data

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

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

(x, t) → (βx, βt), β constant,

we consider the self-similar solutions of (1.5), (1.6), and (2.1):

(ρ, v)(x, t) = (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_{±}).

As shown in [22], in the case v_{−} < v_{+}, we can obtain a solution (ρ, v)(ξ) that consists of two contact discontinuities and a vacuum state which are uniquely determined by the Riemann data (ρ_{±}, v_{±}). That is,

(ρ, v)(ξ) = \{ \begin{matrix}(ρ_{−}, v_{−}), & −∞ < ξ ≤ v_{−}, \\ (0, ξ), & v_{−} ≤ ξ ≤ v_{+}, \\ (ρ_{+}, v_{+}), & v_{+} ≤ ξ < ∞. \end{matrix}

In the case v_{−} > v_{+}, a key observation in [22] 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 from the mathematical point of view (also see [26, 27]).

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

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

for any ψ ∈ C_{0}^{∞}((−∞, ∞) × [0, ∞)).

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

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

where S = {(σt, t) : 0 ≤ t < ∞},

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

in which [h] := h_{+} −h_{−} denote 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.

It was shown in [22] that the δ-measure solutions (ρ, v) constructed above satisfy

⟨ρ, ϕ_{t}⟩ + ⟨ρv, ϕ_{x}⟩ = 0, (2.2)

⟨ρv, ϕ_{t}⟩ + ⟨ρv^{2}, ϕ_{x}⟩ = 0, (2.3)

for any ϕ ∈ C_{0}^{∞}((−∞, ∞) × (0, ∞)), where

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

and

⟨ρv, ϕ⟩ = ∫_{0}^{∞} ∫_{−∞}^{∞} ρ_{0}v_{0}ϕ dxdt + ⟨σwδ_{S}, ϕ⟩.

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

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

that satisfies the δ−entropy condition:

(2.5)  v_{+} < σ < v_{−}.

The entropy condition (2.5) means that, in the (x, t)-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.

2.2. Riemann Solutions to the Euler Equations for Isentropic Fluids. The eigenvalues of system (1.1)–(1.3) are

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

with

c(ρ, ϵ) = \sqrt{ϵ p′(ρ)} = \sqrt{ϵ} ρ^{θ}, θ = \frac{γ − 1}{2}.

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, (2.6)

− ξ(ρv)_{ξ} + (ρv^{2} + ϵ p(ρ))_{ξ} = 0, (2.7)

(ρ, v)(±∞) = (ρ_{±}, v_{±}). (2.8)

Shock Curves. The Rankine-Hugoniot conditions for discontinuous solutions of (1.1)–(1.3) are

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

The Lax entropy inequalities imply

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.

We now turn to analyzing the Riemann solutions that consist of rarefaction waves and constant states. There are also two families of rarefaction waves, corresponding to characteristic fields λ_{1} and λ_{2}, respectively.

Rarefaction Wave Curves. A rarefaction wave is a continuous solution of (2.6)– (2.8) of form (ρ, ρv)(ξ), ξ = x/t, satisfying

ξ = v ∓ \sqrt{ϵ p′(ρ)}, −ξρ_{ξ} + (ρv)_{ξ} = 0.

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

1-rarefaction wave curve R_{1}(u_{−}):

v − v_{−} = − ∫_{ρ_{−}}^{ρ} \frac{\sqrt{ϵ p′(s)}}{s}ds, ρ < ρ_{−};

2-rarefaction wave curve R_{2}(u_{−}):

v − v_{−} = ∫_{ρ_{−}}^{ρ} \frac{\sqrt{ϵ p′(s)}}{s}ds, ρ > ρ_{−}.

The rarefaction wave curves are concave or convex, respectively, in the ρ − m plane.

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; and, when u_{+} ∈ R_{2}R_{1}(u_{−}), R(x/t) contains a 1-rarefaction wave, a 2-rarefaction wave, and an intermediate constant state that may be a vacuum state. Since the other two regions S_{2}R_{1}(u_{−}) and R_{2}S_{1}(u_{−}) have empty interior when ϵ → 0, it suffices to analyze the limit process for the two cases u_{+} ∈ S_{2}S_{1}(u_{−}) (in Section 3) and u_{+} ∈ R_{2}R_{1}(u_{−}) (in Section 4). For more details about Riemann solutions, see [9, 25].

3. Formation of δ-Shocks

In this section, we study the formation of δ-shocks in the Riemann solutions to the Euler equations for isentropic fluids in the case u_{+} ∈ S_{2}S_{1}(u_{−}) with v_{−} > v_{+} and ρ_{±} > 0 as the pressure vanishes.

3.1. Limiting Behavior of the Riemann Solutions as ϵ → 0. 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}) are determined by

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

and

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

Define g(s, τ) = \sqrt{(\frac{1}{s} − \frac{1}{τ})(p(τ) − p(s))} for s, τ > 0. Thus, a combination of the jump conditions (3.1) and (3.2) gives

(3.3)  v_{−} − v_{+} = \sqrt{ϵ} (g(ρ_{∗}^{ϵ}, ρ_{−}) + g(ρ_{∗}^{ϵ}, ρ_{+})) > 0.

Then one must have lim_{ϵ→0} g(ρ_{∗}^{ϵ}, ρ_{±}) = ∞, which yields lim_{ϵ→0} ρ_{∗}^{ϵ} = ∞. Letting ϵ → 0 in (3.3) yields

lim_{ϵ→0} ϵ(ρ_{∗}^{ϵ})^{γ} = \frac{\sqrt{ρ_{−}ρ_{+}}}{\sqrt{ρ_{−}} + \sqrt{ρ_{+}}} (v_{−} − v_{+}).

Therefore, we have

Lemma 3.1. lim_{ϵ→0} ϵ^{1/γ}ρ_{∗}^{ϵ} = (\frac{\sqrt{ρ_{−}ρ_{+}}}{\sqrt{ρ_{−}}+\sqrt{ρ_{+}}} (v_{−} − v_{+}))^{1/γ} .

Lemma 3.2. 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,

and

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

Proof. First, Lemma 3.1 and (3.1)–(3.2) immediately imply that

lim_{ϵ→0} v_{∗}^{ϵ} = σ.

On the other hand, using the Lax entropy inequalities for the shocks, we have

(3.4)  v_{∗}^{ϵ} − \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ} < σ_{1}^{ϵ} < min(v_{∗}^{ϵ} + \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ}, v_{−} − \sqrt{ϵ} (ρ_{−})^{θ}),

and

(3.5)  max(v_{∗}^{ϵ} − \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ}, v_{+} + \sqrt{ϵ} (ρ_{+})^{θ}) < σ_{2}^{ϵ} < v_{∗}^{ϵ} + \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ}.

Noting that \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ} = ϵ^{\frac{1}{2γ}} (ϵ^{1/γ}ρ_{∗}^{ϵ})^{θ}, we can see from Lemma 3.1 that, for γ > 1,

(3.6)  lim_{ϵ→0} \sqrt{ϵ} (ρ_{∗}^{ϵ})^{θ} = 0.

Then (3.4)–(3.6) yield

(3.7)  lim_{ϵ→0} v_{∗}^{ϵ} = lim_{ϵ→0} σ_{1}^{ϵ} = lim_{ϵ→0} σ_{2}^{ϵ} = σ.

The Rankine-Hugoniot conditions for (1.1) on the shocks and the results (3.7) yield

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

This completes the proof of Lemma 3.2.

Remark 3.1. The quantity σ that is the limit of v_{∗}^{ϵ}, σ_{1}^{ϵ}, and σ_{2}^{ϵ} uniquely determines the δ-shock solution of (1.5)–(1.6) as the limit of the Riemann solutions when ϵ → 0 and is consistent with the δ-Rankine-Hugoniot condition (2.4) and the δ-entropy condition (2.5), as proposed for the Riemann solutions for pressureless Euler equations.

3.2. Weighted δ-Shocks. We now show the following theorem characterizing the vanishing pressure limit in the case v_{−} > v_{+}.

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 (1.1)–(1.3) with Riemann data u_{±} = (ρ_{±}, ρ_{±}v_{±}), constructed in §2.2. Then, when 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 (1.5)–(1.6) with the same Riemann data u_{±}.

Proof. 1. Set ξ = x/t. Then, for each fixed ϵ > 0, the Riemann solution can be written as

ρ^{ϵ}(ξ) = \{ \begin{matrix}ρ_{−},  for ξ < σ_{1}^{ϵ}, \\ ρ_{∗}^{ϵ}(ξ), for σ_{1}^{ϵ} < ξ < σ_{2}^{ϵ}, \\ ρ_{+},  for ξ > s_{2}^{ϵ}, \end{matrix}

and

v^{ϵ}(ξ) = \{ \begin{matrix}v_{−},  for ξ < σ_{1}^{ϵ}, \\ v_{∗}^{ϵ}(ξ), for σ_{1}^{ϵ} < ξ < σ_{2}^{ϵ}, \\ v_{+},  for ξ > σ_{2}^{ϵ}, \end{matrix}

satisfying the following weak formulations: For any ψ ∈ C_{0}^{1}(−∞, ∞),

(3.8)  − ∫_{−∞}^{∞}(v^{ϵ}(ξ) − ξ)ρ^{ϵ}(ξ) ψ′(ξ)dξ + ∫_{−∞}^{∞} ρ^{ϵ}(ξ) ψ(ξ) dξ = 0,

and

− ∫_{−∞}^{∞}(v^{ϵ}(ξ) − ξ)ρ^{ϵ}(ξ)v^{ϵ}(ξ) ψ′(ξ)dξ + ∫_{−∞}^{∞} ρ^{ϵ}(ξ)v^{ϵ}(ξ) ψ(ξ)dξ (3.9)

= ϵ ∫_{−∞}^{∞} p(ρ^{ϵ}(ξ)) ψ′(ξ)dξ.

2. The first integral in (3.8) can be decomposed into

(3.10)  − {∫_{−∞}^{σ_{1}^{ϵ}} + ∫_{σ_{1}^{ϵ}}^{σ_{2}^{ϵ}} + ∫_{σ_{2}^{ϵ}}^{∞}} (v^{ϵ}(ξ) − ξ)ρ^{ϵ}(ξ)ψ′(ξ)dξ.

The sum of the first and last term of (3.10) is

− ∫_{−∞}^{σ_{1}^{ϵ}}(v_{−} − ξ)ρ_{−}ψ′(ξ)dξ − ∫_{σ_{2}^{ϵ}}^{∞}(v_{+} − ξ)ρ_{+}ψ′(ξ)dξ
= −ρ_{−}v_{−}ψ(σ_{1}^{ϵ}) + ρ_{+}v_{+}ψ(σ_{2}^{ϵ}) + ρ_{−}σ_{1}^{ϵ}ψ(σ_{1}^{ϵ}) − ρ_{+}σ_{2}^{ϵ}ψ(σ_{2}^{ϵ})
− ∫_{−∞}^{σ_{1}^{ϵ}} ρ_{−}ψ(ξ)dξ − ∫_{σ_{2}^{ϵ}}^{∞} ρ_{+}ψ(ξ)dξ,

which converges as ϵ → 0 to

([ρv] − σ[ρ]) ψ(σ) − ∫_{−∞}^{∞} ρ_{0}(ξ − σ) ψ(ξ)dξ

with

ρ_{0}(ξ) = ρ_{−} + [ρ] χ(ξ),

where χ(ξ) is the characteristic function.

For the second term of (3.10),

lim_{ϵ→0} ∫_{σ_{1}^{ϵ}}^{σ_{2}^{ϵ}}(v^{ϵ}(ξ) − ξ)ρ^{ϵ}(ξ)ψ′(ξ)dξ

= ρ_{∗}^{ϵ}(σ_{2}^{ϵ} − σ_{1}^{ϵ}) \{v_{∗}^{ϵ} \frac{ψ(σ_{2}^{ϵ}) − ψ(σ_{1}^{ϵ})}{σ_{2}^{ϵ} − σ_{1}^{ϵ}} − \frac{σ_{2}^{ϵ}ψ(σ_{2}^{ϵ}) − σ_{1}^{ϵ}ψ(σ_{1}^{ϵ})}{σ_{2}^{ϵ} − σ_{1}^{ϵ}} + \frac{1}{σ_{2}^{ϵ} − σ_{1}^{ϵ}} ∫_{σ_{1}^{ϵ}}^{σ_{2}^{ϵ}} ψ(ξ)dξ\} ,

which converges as ϵ → 0 to

([ρv] − σ[ρ]){−σψ′(σ) + σψ′(σ) + ψ(σ) − ψ(σ)} = 0,

since ψ ∈ C_{0}^{1}(−∞, ∞), lim_{ϵ→0} v_{∗}^{ϵ} = σ, and lim_{ϵ→0} σ_{j}^{ϵ} = σ for j = 1, 2.

Then, the integral identity (3.8) yields

lim_{ϵ→0} ∫_{−∞}^{∞}(ρ^{ϵ}(ξ) − ρ_{0}(ξ − σ)) ψ(ξ)dξ = (σ[ρ] − [ρv]) ψ(σ),

for any function ψ ∈ C_{0}^{∞}(−∞, ∞).

3. We now turn to justifying the limit of momentum m^{ϵ} = ρ^{ϵ}v^{ϵ} using the weak formulation (3.9). As done previously, we can obtain the limit for the first term on the left of (3.9) as

− lim_{ϵ→0} ∫_{−∞}^{∞}(v^{ϵ}(ξ) − ξ) ρ^{ϵ}(ξ)v^{ϵ}(ξ)ψ′(ξ)dξ

= ψ(σ) ([ρv^{2}] − σ[ρv]) − ∫_{−∞}^{σ} ρ_{−}v_{−} ψ(ξ)dξ − ∫_{σ}^{∞} ρ_{+}v_{+} ψ(ξ)dξ.

The term on the right of (3.9) equals to

ϵ ∫_{−∞}^{∞} p(ρ^{ϵ}) ψ′(ξ)dξ = ϵ {∫_{−∞}^{σ_{1}^{ϵ}} + ∫_{σ_{1}^{ϵ}}^{σ_{2}^{ϵ}} + ∫_{σ_{2}^{ϵ}}^{∞}} p(ρ^{ϵ}) ψ′(ξ)dξ,

which converges to

ϵ {p(ρ_{−})ψ(σ_{1}^{ϵ}) + p(ρ_{∗}^{ϵ})(ψ(σ_{2}^{ϵ}) − ψ(σ_{1}^{ϵ})) − p(ρ_{+})ψ(σ_{2}^{ϵ})}
= o(ϵ) + ϵ p(ρ_{∗}^{ϵ})(ψ(σ_{2}^{ϵ}) − ψ(σ_{1}^{ϵ})) → 0, as ϵ → 0,

where we used the fact that ϵp(ρ_{∗}^{ϵ}) is bounded and lim_{ϵ→0} σ_{j}^{ϵ} = σ for j = 1, 2. Returning to the weak formulation (3.9), one has

lim_{ϵ→0} ∫_{−∞}^{∞} ((ρ^{ϵ}v^{ϵ})(ξ) − (ρ_{0}v_{0})(ξ − σ)) ψ(ξ)dξ = ψ(σ)(σ[ρv] − [ρv^{2}]).

4. Finally, we are in the position to study the limits of ρ^{ϵ} and m^{ϵ} = ρ^{ϵ}v^{ϵ} by tracking the time-dependence of the weights of the δ-measures as ϵ → 0.

Let ϕ(x, t) ∈ C_{0}^{∞}((−∞, ∞) × [0, ∞)) be a smooth test function and ϕ̃(ξ, t) := ϕ(ξt, t). Then we have

lim_{ϵ→0} ∫_{0}^{∞} ∫_{−∞}^{∞} ρ^{ϵ}(x/t)ϕ(x, t)dxdt = lim_{ϵ→0} ∫_{0}^{∞} t (∫_{−∞}^{∞} ρ^{ϵ}(ξ)ϕ̃(ξ, t)dξ) dt.

On the other hand, we have

lim_{ϵ→0} ∫_{−∞}^{∞} ρ^{ϵ}(ξ)ϕ̃(ξ, t)dξ = ∫_{−∞}^{∞} ρ_{0}(ξ − σ)ϕ̃(ξ, t)dξ + (σ[ρ] − [ρv])ϕ̃(σ, t)

= t^{−1} ∫_{−∞}^{∞} ρ_{0}(x − σt)ϕ(x, t)dx + (σ[ρ] − [ρv])ϕ(σt, t).

Combining the two relations above yields

lim_{ϵ→0} ∫_{0}^{∞} ∫_{−∞}^{∞} ρ^{ϵ}(x/t)ϕ(x, t)dxdt

= ∫_{0}^{∞} ∫_{−∞}^{∞} ρ_{0}(x − σt)ϕ(x, t)dxdt + ∫_{0}^{∞} t([ρv] − σ[ρ])ϕ(σt, t)dt.

The last term, by the definition, equals to

< w_{1}(·)δ_{S}, ϕ(·, ·) >

with

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

Similarly, we can show that

lim_{ϵ→0} ∫_{0}^{∞} ∫_{−∞}^{∞} m^{ϵ}(x/t)ϕ(x, t)dxdt

= ∫_{0}^{∞} ∫_{−∞}^{∞}(ρ_{0}v_{0})(x − σt)ϕ(x, t)dxdt + < w_{2}(·)δ_{S}, ϕ(·, ·) >

with

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

This completes the proof of Theorem 3.1.

4. Formation of Vacuum States

In this section, we show the formation of vacuum states in the Riemann solutions of (1.1)–(1.3) in the case u_{+} ∈ R_{2}R_{1}(u_{−}) with v_{−} < v_{+} and ρ_{±} > 0 as the pressure decreases.

As stated previously, on the rarefaction waves, the solution satisfies

(4.1)  ξ = x/t = v^{ϵ} ± \sqrt{ϵ p′(ρ^{ϵ})},

for each fixed ϵ > 0. More precisely, we have that, on the 1-rarefaction wave,

ξ = v^{ϵ} − \sqrt{ϵ p′(ρ^{ϵ})},  v_{−} − \sqrt{ϵ p′(ρ_{−})} < ξ < v_{∗}^{ϵ} − \sqrt{ϵ p′(ρ_{∗}^{ϵ})},  ρ_{−} > ρ_{∗}^{ϵ};

and, on the 2-rarefaction wave,

ξ = v^{ϵ} + \sqrt{ϵ p′(ρ^{ϵ})},  v_{∗}^{ϵ} + \sqrt{ϵ p′(ρ_{∗}^{ϵ})} < ξ < v_{+} + \sqrt{ϵ p′(ρ_{+})},  ρ_{∗}^{ϵ} < ρ_{+},

where (ρ_{∗}^{ϵ}, ρ_{∗}^{ϵ}v_{∗}^{ϵ}) is the intermediate state in the Riemann solutions. Since (ρ_{∗}^{ϵ}, ρ_{∗}^{ϵ}v_{∗}^{ϵ}) is on the curve R_{1}(u_{−}), we have

v_{∗}^{ϵ} = v_{−} − ∫_{ρ_{−}}^{ρ_{∗}^{ϵ}} \frac{\sqrt{ϵ p′(s)}}{s}ds ≤ v_{−} + ∫_{0}^{ρ_{−}} \frac{\sqrt{ϵ p′(s)}}{s}ds = v_{−} + \sqrt{ϵ}\frac{ρ_{−}^{θ}}{θ} ≡ A^{ϵ}.

When v_{−} < v_{+} < A^{ϵ}, that is,

(4.2)  ϵ > (\frac{θ(v_{+} − v_{−})}{ρ_{−}^{θ}})^{2} ≡ ϵ_{0}(u_{−}, u_{+}),

there is no vacuum in the solution. This implies that, for a fluid with strong pressure, no vacuum occurs in the solution generically.

However, when ϵ decreases so that ϵ < ϵ_{0}(u_{−}, u_{+}), then A^{ϵ} < v_{+}, and the intermediate state (ρ_{∗}^{ϵ}, ρ_{∗}^{ϵ}v_{∗}^{ϵ}) becomes a vacuum state with

(ρ_{∗}^{ϵ}, v_{∗}^{ϵ})(ξ) = (0, ξ), v_{1}^{ϵ} ≤ ξ ≤ v_{2}^{ϵ},

where

v_{1}^{ϵ} = v_{−} + ∫_{0}^{ρ_{−}} \frac{\sqrt{ϵp′(s)}}{s}ds, v_{2}^{ϵ} = v_{+} − ∫_{0}^{ρ_{+}} \frac{\sqrt{ϵp′(s)}}{s}ds.

The uniform boundedness of ρ^{ϵ}(ξ) with respect to ϵ in this case leads to

lim_{ϵ→0} v_{1}^{ϵ} = v_{−},  lim_{ϵ→0} v_{2}^{ϵ} = v_{+},

and

lim_{ϵ→0} v^{ϵ}(ξ) = ξ, for ξ ∈ (v_{−}, v_{+}).

In summary, the limit function (ρ, v) in this case is

(ρ, v)(ξ) = \{ \begin{matrix}(ρ_{−}, v_{−}), & −∞ < ξ ≤ v_{−}, \\ (0, ξ), & v_{−} ≤ ξ ≤ v_{+}, \\ (ρ_{+}, v_{+}), & v_{+} ≤ ξ < ∞, \end{matrix}

which is a solution to the transport equations (1.5)–(1.6) containing a vacuum state that fills up the region formed by the two contact discontinuities ξ = x/t = v_{±}.

We can clearly see from the analysis above that, when ϵ decreases, the left boundary of the 1-rarefaction wave and the right boundary of the 2-rarefaction wave are fixed, the right boundary of the 1-rarefaction wave becomes closer to the left boundary of the 1-rarefaction wave, and the left boundary of the 2-rarefaction wave becomes closer to the right boundary of the 2-rarefaction wave; while the state between the right boundary of the 1-rarefaction wave and the left boundary of the 2-rarefaction wave in the Riemann solution is a vacuum state; and, in the limit, the left boundary of the 1-rarefaction wave and the right boundary of the 2-rarefaction wave become two contact discontinuities of the transport equation (1.5)–(1.6) and the vacuum state fills up the region between the two contact discontinuities.

5. Formation Process of δ-Shocks: Numerical Simulations

After the cavitation process in the Riemann solutions of (1.1)–(1.3) has been described clearly as the pressure decreases in Section 4, understanding the formation process of δ-shocks in the Riemann solutions as the pressure decreases becomes more constructive for comparison. For this purpose, in this section we present a selected group of representative numerical results in the level of Euler dynamics (1.1)–(1.3) starting with Riemann initial data. We have performed many more numerical tests to make sure what we present are not numerical artifacts.

We solve the Riemann problem for (1.1)–(1.3) with p(ρ) = ρ^{γ}/γ and γ = 1.4 for an ideal gas. The Riemann initial data is

(ρ, v)(x, 0) = \{ \begin{matrix}(1.0, 1.5), & for x < 0, \\ (0.2, 0.0), & for x > 0. \end{matrix}

To discretize the system, we use the higher order ENO scheme to obtain a method-of-line ODE in time and then discretize the ODE by the classical higher order explicit Runge-Kutta method (see [23, 24]). We calculate by the third-order ENO scheme, [24], up to t = 0.2 with mesh 100. The numerical simulations with different choices of ϵ are presented in Figs. 5.1–5.3. These figures show the formation process of a δ-shock in the two-shock Riemann solutions for the Euler equations (1.1)–(1.3) for isentropic fluids as the pressure decreases. We start with ϵ/γ = 1.0 and choose ϵ/γ = 0.05 then and ϵ/γ = 0.001 finally. Figs. 5.1a–5.3a show the concentration process of the density yielding a weighted δ-measure in the limit, in which the horizontal axis stands for the space variable x and the vertical axis stands for the density. Figs. 5.1b–5.3b show the change of the velocity as ϵ decreases yielding a step function in the limit, in which the horizontal axis stands for the space variable x and the vertical axis stands for the velocity.

We can see clearly from these numerical results that, when 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 (1.5)–(1.6), while the velocity is a step function.

We remark that it is delicate to calculate solutions of hyperbolic systems of conservation laws that strict hyperbolicity fails, for which the system of pressureless Euler equations for (1.5)–(1.6) is an example. In this section, we have proposed an efficient numerical approach to calculate the solutions containing δ-shocks for (1.5)–(1.6) via the vanishing pressure limit. It would be interesting to apply the approach and ideas developed here to develop efficient numerical algorithms to calculate solutions for more complex physical models.

Acknowledgments. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0204225, INT-9987378, and DMS-0204455. Hailiang Liu’s research was supported in part by the National Science Foundation under Grant DMS-0107917.

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Gui-Qiang Chen Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730, USA gqchen@math.northwestern.edu Homepage: http://www.math.northwestern.edu/∼gqchen/preprints

Hailiang Liu Department of Mathematics, University of California at Los Angeles Los Angeles, CA 90095-1555, USA hliu@math.ucla.edu Homepage: http://www.math.ucla.edu/∼hliu

(a) (b) Figure 5.1. Density and velocity for ϵ = 1.4

(a) (b) Figure 5.2. Density and velocity for ϵ = 0.07

(a) (b) Figure 5.3. Density and velocity for ϵ = 0.0014