Acta Mathematicae Applicatae Sinica, English Series

Vol. 19, No. 1 (2003) 1–12

Entropy Production and Admissibility of ShocksTai-Ping Liu1, Tommaso Ruggeri2

1Institute of Mathematics, Academia Sinica, Taipei, Taiwan & Department of Mathematics, University of Stan-

ford, USA (E-mail: tpliu@math.sinica.edu.tw; liu@math.stanford.edu)2Department of Mathematics and Research Center of Applied Mathematics, C.I.R.A.M., University of Bologna,

Italy (E-mail: ruggeri@ciram.unibo.it)

Abstract In shock wave theory there are two considerations in selecting the physically relevant shock waves.

There is the admissibility criterion for the well-posedness of hyperbolic conservation laws. Another consideration

concerns the entropy production across the shocks. The latter is natural from the physical point of view, but

is not sufficient in its straightforward formulation, if the system is not genuinely nonlinear. In this paper we

propose the principles of increasing entropy production and that of the superposition of shocks. These principles

are shown to be equivalent to the admissibility criterion.

Keywords Entropy production, admissibility condition, conservation laws, shock waves

2000 MR Subject Classification 35L67, 76L05; 58J45, 74J30

1 Introduction

The theory of hyperbolic balance laws is complicated by the fact that shock waves in generalarise from smooth data and that shock wave theory is well-posed only with the addition of cer-tain physically motivated admissibility criterion. One physically natural criterion is to requirethe entropy to increase across a shock. For physical systems which are genuinely nonlinear,such as the compressible Euler equations for polytropic gases, this is equivalent to the stabilitycriterion of Lax[9].

For physical systems which are not genuinely nonlinear the requirement that entropy in-creases across shocks is not sufficient to yield a well-posed theory for such a system. An admis-sibility criterion has been introduced, Liu[14], which yields a well-posed theory (see [1,12,15]).The increase of entropy across a shock is necessary, but not sufficient, for the shock to be admis-sible. On the other hand, from the physical point of view, the consideration of entropy increaseis natural and should in some form guarantee the admissibility of shocks. The purpose of thispaper is to formulate physically natural principles on entropy increase which are equivalent tothe admissibility criterion. The principles demand that stronger shocks have greater entropyincrease and that shocks of same speed can be superposed.

That the consideration of entropy is of primary physical importance is seen clearly throughthe theory of extended thermodynamics (see [16]). The theory shows that the existence ofentropy function greatly limits the form of the physical systems, and, equally important, auto-

Received September 25, 2002.1 Partially supported by the NSF (Grant DMS-9803323).2 Supported by fondi MIUR Progetto di interesse Nazionale Problemi Matematici Non Lineari di Propagazionee Stabilita nei Modelli del Continuo Coordinatore T. Ruggeri, by the GNFM-INDAM, and by the IstitutoNazionale di Fisica Nucleare (INFN).

2 T.P. Liu, T. Ruggeri

matically implies the existence of a privileged field (main field) such that the original systembecomes symmetric hyperbolic, (see [2,20]). In the next two section we briefly recall this ques-tion with some historical remarks. Taking into account the particular symmetric hyperbolicform, the behavior of entropy across a shock can be studied easily (see [2,20]) and is briefed inSection 4. In Section 5 we recall the admissibility criterion of [14] with some remarks relevantto the present contents. Finally, in Section 6, we formulate the principles of entropy increaseand shock superposition, and prove their equivalence to the admissibility criterion. A differentadmissibility criterion, expressed also in term of the entropy production, is the entropy ratecriterion of [5], which applies to some physically interesting models.

2 Entropy and Balance Laws

The objective of the thermomechanics is the determination of a N -vector field uuu(xxx, t) whichdefines the state of a body at the time t in the point xxx.

The field equations necessary for the determination of the N -vector field uuu(xxx, t) are basedupon N equations of balance of the form

∂uuu

∂t+

∂fff i

∂xi= ppp. (1)

The components of uuu are called densities and the components fff i are called fluxes, while ppprepresents the productions.

In order to provide field equations for the fields uuu, the balance equations must be supple-mented by constitutive equations.

In many circumstances and in particular in the modern approach of non-equilibrium ther-modynamics (extended thermodynamics, see [16]), such constitutive equations relate the fluxesfff i and the productions ppp to the fields uuu in a materially dependent manner, i.e.:

fff i ≡ fff i(uuu) and ppp ≡ ppp(uuu), (2)

so that the fluxes fff i and productions ppp at one event (xxx, t) depend only on the values of the fieldsuuu at the same event. We may thus say that the constitutive equations are local in space-time.

If the constitutive functions fff i(uuu) and ppp(uuu) were known explicitly, we could eliminate fff i

and ppp between (1) and (2) and obtain explicit field equations for uuu. They form a quasi-linearsystem of partial differential equation of first order. Every solution of this system is called athermodynamic process.

In reality the constitutive functions are not known and it is the task of the constitutivetheory to determine these functions or, at least, to reduce their generality. The tools of theconstitutive theory are certain universal physical principles which represent expectations whosevalidity we accept from long experience. The main principles are the entropy principle and theprinciple of relativity.

The entropy inequality is a scalar balance law — additional to the equations (1). We writeit as

∂h

∂t+

∂hi

∂xi= Σ ≤ 0 (3)

and require this inequality to hold for all thermodynamic processes. h and hi represents theentropy density and the entropy flux respectively and Σ is the density of entropy production.We assume that all constitutive quantities are also of local type:

h ≡ h(uuu), hi ≡ hi(uuu), Σ = Σ(uuu). (4)

Entropy Production and Admissibility of Shocks 3

Following [6] we require that the entropy density h(uuu) is a convex function of uuu, i.e.1

∂2h

∂uuu∂uuupositive definite. (5)

The meaning of this requirement corresponds, physically, to a thermodynamical stability con-dition.

The principle of relativity states that the field equations (1), (2), and the entropy inequality(3), (4) have the same forms in all Galilean frames or in all Lorentz frames depending on whetherthe theory is non-relativistic or relativistic. Interested readers are referred to the paper [18].

3 Symmetric Hyperbolic System

Taking into account the local constitutive assumptions (2) and (4), the compatibility conditionsbetween (1) and (3) yield:

∂hi

∂uuu= uuu′ · ∂fff i

∂uuuand uuu′ · ppp = Σ ≤ 0, (6)

where we have introduced the field (main field [20])

uuu′ =∂h

∂uuu, (7)

which is globally univalent with respect to the field uuu thanks to the convexity assumptions forh(uuu).

If we rewrite (7) and (6)1in this form

dh = uuu′ · duuu, dhi = uuu′ · dfff i (8)

and introduce, through Legendre transform, the four potentials

h′ = uuu′ · uuu − h, h′i = uuu′ · fff i − hi, (9)

then (8) becomesdh′ = uuu · duuu′, dh′i = fff i · duuu′. (10)

Therefore from (10)1, if we choose the main field uuu′ as field variable, we have

uuu =∂h

∂uuu′ (11)

and from (10)2 and (9)

fff i =∂h

′i

∂uuu′ , h = −h′ + uuu′ · ∂h′

∂uuu′ , hi = −h′i + uuu′ · ∂h

′i

∂uuu′ . (12)

In this way, as functions of the field uuu′, the constitutive functions fff i(uuu′), h(uuu′) and hi(uuu′) maybe derived from four potentials h′(uuu′) and h

′i(uuu′). There is the residual inequality (6)2, namely

Σ = uuu′ · ppp(uuu′) ≤ 0 (13)

1 In the mathematical literature h, hi and Σ are called improperly entropy density, entropy flux and entropyproduction. In the physical case instead the entropy density, the flux and the production coincides with −h, −hi,and −Σ respectively and therefore the proper entropy density is concave function with respect to the densitiesand the entropy production is a non negative function.

4 T.P. Liu, T. Ruggeri

relating the entropy production Σ with the productions ppp.If we substitute (11) and (12)1 into (1) we obtain for the field equations

∂2h′

∂uuu′∂uuu′∂uuu′

∂t+

∂2h′i

∂uuu′∂uuu′∂uuu′

∂xi= ppp. (14)

Inspection shows that the original system of balance laws becomes a special symmetric hyper-bolic system in the main field uuu′. In fact all the matrices in (14) are Hessian matrices and thefirst matrix is positive definite, taking into account that h′ is the Legendre transform of h anduuu and uuu′ are dual fields

(see (9)1, (7) and (11)

):

∂2h′

∂uuu′∂uuu′ δuuu′ · δuuu′ =∂2h

∂uuu∂uuuδuuu · δuuu > 0. (15)

That a system of balance-type field equations, compatible with the entropy inequality witha convex entropy density, becomes symmetric hyperbolic in the chosen privileged field variableswas discovered by Godunov[8] for the special cases of Euler fluids and systems derived froma variational principle. In general this was proved by Boillat[2] and Ruggeri, Strumia[20], thelatter authors were motivated by the paper of [6].

s

χ

u+

u1 u2

u3 u4

u -

uo u*

d

Figure 1. The shock velocity as function of the shock strength and the range of

the (E)-admissibility conditions

4 Entropy Production

For smooth solutions there is the entropy production due to the function Σ. Because of thenonlinearity the weak solutions generate an additional entropy production across the shocks.The Rankine-Hugoniot conditions across a shock are

−s[uuu] + [fff ] = 0, (16)

where [uuu] = uuu− − uuu+ and fff = fff ini and ni are the components of the unity normal to the shockfront.

Entropy Production and Admissibility of Shocks 5

The entropy production across the shock is

η = −s[h] + [q], q = hini, (17)

which is generally non-vanishing, that is, the shock “produces” entropy.For plane waves (ni=const.), the Rankine-Hugoniot conditions are a system of N equations

for the N + 1 unknowns uuu− (perturbed field) and s in terms of the assigned unperturbed fielduuu+. Excluding the characteristic shocks the usual shocks are one-parameter families of the type:

uuu− ≡ uuu−(uuu+, µ), s ≡ s(uuu+, µ). (18)

Here µ is the parameter with µ = µ0 giving the null shock:

uuu−(uuu+, µ0) = uuu+, s(uuu+, µ0) = λ+,

where λ+ = λ(uuu+) and λ(uuu) denotes the generic eigenvalue of the Jacobian matrix

AAA(uuu) =∂fff(uuu)∂uuu

, (AAA − λIII)rrr = 0. (19)

Inserting (18) into (17) we haveη ≡ η(uuu+, µ).

Differentiating η with respect to µ with uuu+ fixed, we obtain (the dot indicates the partialderivative with respect to µ):

η = sw, (20)

w ≡ h(uuu+) − h(uuu−) +∂h

∂uuu(uuu−) · (uuu− − uuu+) > 0, ∀uuu− = uuu+. (21)

The consequences of (20) and (21) are represented in the Figures 1 and 2.The identity (20) follows immediately, proceeding as in [19]: First we differentiate (16) and

(17) with respect to µ

− s [uuu] +(AAA(uuu−) − sIII

)uuu− = 0, (22)

η = −s [h] +(∂q

∂uuu(uuu−) − s

∂h

∂uuu(uuu−)

)uuu−, (23)

then multiply (22) by ∂h∂uuu (uuu−). Taking into account (6) and (7), we obtain

−s∂h

∂uuu(uuu−) · [uuu] +

(∂q

∂uuu(uuu−) − s

∂h

∂uuu(uuu−)

)uuu− = 0

and (23) becomes (20).From (20) we concludes that the function η(µ) increases or decreases along with the function

s(µ). In particular, the stationary points of s(µ) are also stationary points for η(µ). In theranges in which s(µ) is invertible it is possible to choose as the shock parameter µ = s and ηas function of s is always a strictly increasing function:

∂η

∂s= w > 0, ∀uuu− = uuu+. (24)

If we restrict to weak shocks that passing through the null shock, we have s → λ+, η →0, s = λ++λ−

2 and then from (24) we have

η > 0 ⇐⇒ λ+ < s < λ−. (25)

6 T.P. Liu, T. Ruggeri

This shows the equivalence in the case of weak shocks between the entropy growth criterionη > 0 and the Lax conditions λ+ < s < λ−. This criterion is not valid in general in the casewhen the genuine non linearity is lost, i.e., when

(∂λ

∂uuu· rrr

)(uuu) = 0

for some states. For strong shocks the range where the entropy growth is satisfied is, in general,larger than the set in which the Lax condition are verified. Ruggeri, Muracchini and Secciahave noted that the two range are the same if we modify the entropy growth criterion requiringalso the most natural, stronger condition

η > 0,∂η

∂χ> 0, (26)

where χ = |µ − µ0| is the shock strength. The meaning of (26) is that in the physical shocksthe entropy production created by the shock must be not only positive but also monotonicallyincreasing when the shock strength grows. We will come back to this basic point later in Section6 when it is formulated as our first principle of admissibility of shocks.

χ

η

u+

u1

u2

u3

u4

Figure 2. The production of entropy as function of the shock strength and

the range of the (E)-admissibility conditions

5 Admissibility Criterion

The above considerations arising from physical assumptions are now related to the analyticalconsideration of the shock wave theory for hyperbolic partial differential equations. Since theproduction function ppp plays no role in the Rankine-Hugoniot condition and entropy productionΣΣΣ along the shocks, and since shocks are locally plane waves, we consider the one-dimensionalhyperbolic conservation laws

∂uuu

∂t+

∂fff(uuu)∂x

= 0.

It is well known that the weak solution of the initial value problem to the hyperbolic conservationlaws is not unique. A fundamental question is to find an admissibility criterion for the shock

Entropy Production and Admissibility of Shocks 7

waves, the entropy condition, to select the unique, physical solution. This is done so as to have awell-posed theory for the hyperbolic conservation laws. We now explain the entropy condition:let us denote by H(uuu0) the Hugoniot curve:

H(uuu0) ≡{uuu : −σ(uuu − uuu0) + fff(uuu) − fff(uuu0) = 0 for some scalar σ ≡ σ(uuu0, uuu)

}.

Thus the Rankine-Hugoniot condition becomes

uuu+ ∈ H(uuu−), uuu− ∈ H(uuu+), s = σ(uuu−, uuu+) = σ(uuu+, uuu−).

Definition 1. A shock (uuu−, uuu+) is admissible if it satisfies the following equivalent entropyconditions (E):

• σ(uuu+, uuu) ≤ σ(uuu+, uuu−) for any uuu on the Hugoniot curve H(uuu+) between uuu+ and uuu−;

• σ(uuu−, uuu) ≥ σ(uuu−, uuu+) for any uuu on the Hugoniot curve H(uuu−) between uuu− and uuu+.

u1

u2

d

s

χFigure 3. See proof of Theorem 1

Here we are concerned with shocks which are formed by a characteristic value λ(uuu), an eigen-value of AAA(uuu), which is a distance away from other characteristic values. This is so, for instance,if the system is strictly hyperbolic, with all characteristic values distinct. For physical systemssuch as gas dynamics equations, the characteristic values which are not simple correspond tothe linear fields and do not give rise to shocks. Other systems, such as magneto-hydrodynamics,characteristics for nonlinear fields and linear fields may be equal and the issue of admissibilityis more complex and is not considered here, cf. [11] and references therein. In our situation,the entropy condition (E) implies the following generalized Lax condition[9],

λ(uuu−) ≥ s ≥ λ(uuu+).

This is so because the shock speed is close to the characteristic speed when it is weak[9]:σ(uuu0, uuu0) = λ(uuu0). That the above two definitions are equivalent is due to the fact that σ(uuu0, uuu)

8 T.P. Liu, T. Ruggeri

increases as uuu moves away from uuu0 along the Hugoniot curve if and only if σ(uuu0, uuu) < λ(uuu) asseen by differentiating the jump condition along the Hugoniot curve:

σ(uuu − uuu0) =(AAA(uuu) − σIII

)uuu.

At least for shock of moderate strength, uuu − uuu0 and uuu are close to the characteristic directionrrr(uuu), the eigenvector of AAA(uuu). This implies that the shock speed increases, σ > 0, if and only ifσ − λ(uuu) < 0 as claimed.

The condition (E) was first formulated in connection with the Riemann problem for thehyperbolic conservation laws with initial values

uuu(x, 0) ={

uuul, x < 0,uuur, x > 0.

It is shown for general system that the Riemann problem has a unique solution when uuul and uuur

are close[14], and for gas dynamics equations of any data[13]. The general initial value problemis also studied using the Glimm scheme[7], and the wave tracing method[12]. It is shown thatthe solution is continuous outside countable Lipschitz continuous shock curves, and that theshocks satisfy condition (E). There is now a definitive well-posed theory by [1]. This is relatedto the viscosity criterion described below.

u1

u2

uu~^

d

χ

s

Figure 4. See proof of Theorem 1

There is the justification of the above entropy condition (E) through the viscosity criterionof relating the solution of the hyperbolic conservation laws to the viscous conservation laws

∂uuu

∂t+

∂fff(uuu)∂x

=∂

∂x

(BBB(uuu, ε)

∂uuu

∂x

).

The viscosity matrix BBB(uuu, ε) vanishes when the dissipation parameters ε are set zero. There isno need to supplement the viscous conservation laws with admissibility criterion, because thesystem does not give rise to discontinuous shocks. The effect of the viscosity is to smooth outthe shocks into smooth travelling waves. It is shown, cf. [3,13,17], that a shock satisfies theentropy condition (E) if and only if it is the zero dissipation limits of the travelling waves ofviscous conservation laws. The aforementioned work[1] shows that, for the artificial viscositymatrix BBB(uuu, ε) = εIII, the solutions to the general initial value problem of the viscous conservationlaws converge, in the zero dissipation limit as ε goes zero, to the solutions of the hyperbolicconservation laws. As we mentioned before, such inviscid solutions automatically satisfy thecondition (E). Moreover, it is shown in [1] that the solutions to the hyperbolic conservationlaws depend continuously on their initial values.

Entropy Production and Admissibility of Shocks 9

The condition (E) implies that the entropy production η is positive across the shock[10].This is seen easily as follows: We differentiate (21) along the Hugoniot curve H(uuu+) to obtain

dw

dµ=

∂2h

∂uuu∂uuu(uuu−)(uuu− − uuu+)

duuu−dµ

.

At least for shocks of moderate strength the vector uuu− − uuu+ and the tangent duuu−/dµ are closeto the eigen-direction rrri(u) and therefore the RHS of the above identity is positive due to theconvexity of the entropy function h(uuu). This implies that, for the same variation in shockspeed, the entropy production η is greater for stronger shocks. For instance η(uuu2) > η(uuu1),η(uuu4) > η(uuu3), see Figures 1 and 2. This observation and (21) yield the conclusion that condition(E) for a shock implies the positivity of entropy production η across the shock. It is also clearthat the positivity of η is not sufficient to guarantee the admissibility of the shock in the senseof condition (E), except for the case that η is strictly increasing from uuu+ to uuu−. This is so, forinstance, for uuu− between uuu+ and uuu1 in Figures 1 and 2. Even in the case of two conservationlaws, for which there are infinitely many entropy functions, the positivity of entropy productionis not equivalent to the admissibility condition (E), see [4].

6 Entropy Principles

In this section we want to relate the entropy production rate η to the condition (E). Thisis done under the following two principles of admissibility:Principle of Entropy Production. A shock is admissible if it can be homotopied fromanother admissible shock by continuously increasing the entropy production rate η as one of theend state uuu0 is fixed and another moving away from uuu0 along the Hugoniot curve H(uuu0).Principle of Superposition. Suppose that (uuu1, uuu2), (uuu2, uuu3) and (uuu1, uuu3) are shocks with thesame speed. Then any one of these shocks is admissible provided that the other two are admis-sible.Remark. The first principle is physically plausible, the entropy production for an admissibleshock should be greater as the strength of the shock increases. The second principle makesthe natural requirement that a shock is admissible if it is the superposition of two admissibleshocks with the same speed.

Our main result is the following:

Theorem 1. A shock is admissible according to the above two Principles if and only if itsatisfies the entropy condition (E).

Proof. Let A be the set of shocks satisfying (E), and B the set of shocks constructed fromthe null shocks by the Principles. We first show that any element in A is also in B. Take, forexample, the shock (uuu−, uuu+) in A, uuu− between uuu2 and uuu3 in Figure 1. Since we can homotopyfrom (uuu2, uuu+) to obtain (uuu−, uuu+) by the first Principle, it suffices to show that (uuu2, uuu+) is in B.

Note also that the shock (uuu1, uuu+) can be constructed from the null shock (uuu+, uuu+) throughthe first Principle. Since the three shocks (uuu1, uuu+), (uuu2, uuu1) and (uuu2, uuu+) have the same speed,to show that (uuu2, uuu1) is in the set B using the second Principle, we need to verify that (uuu2, uuu1)is in B.

The construction of (uuu2, uuu1) using the Principles follows the same line of reasoning as above:Note first that (uuu2, uuu1) satisfies condition (E) and that its speed equals that of the null shock(uuu1, uuu1), Figure 3. Instead of starting the Hugoniot curve from the right state uuu1 as in Figure3, we draw the Hugoniot curve starting from the left state uuu2, Figure 4. Similar to the abovereasoning, the construction of the shock (uuu2, uuu1) from the Principle is reduced to the construction

10 T.P. Liu, T. Ruggeri

u+

u1

u2 u

3

u-

χ

s

d

Figure 5. See proof of Theorem 1

u1

u2

u3

u-

u+

f(u)

uFigure 6. See proof of Theorem 1

of the shock (uuu, uuu), Figure 4. The same procedure works for the shock (uuu, uuu) with the alternatingway by starting with the right state uuu. We have thus established a procedure whereby theconstruction of the shock (uuu−, uuu+) by the Principles is reduced inductively to the constructionof shocks with decreasing strength. This procedure of reducing to shocks with decreasingstrength need to be slightly modified in the exceptional case as indicates in Figure 5. Theconstruction of the shock (uuu3, uuu2) is the same as that (uuu2, uuu1) above. The shock (uuu2, uuu1) inFigure 5 is exceptional in that the shock speed reaches local maximum at both states (uuu1

and uuu2). The above procedure of construction does apply straightforwardly. To explain amodification of the procedure we consider, for simplicity, the scalar conservation law. Figure 5is then redraw in terms of the scalar flux f(u) in Figure 6. Figure 7 is a magnification of Figure

Entropy Production and Admissibility of Shocks 11

6 around (u2, u1). Choose states uk, k odd, tending to u1, and uk, k even, tending to u2, ask → ∞. (u2n, u2n−1) can be homotopied from (u2n, u2n+1) according the first Principle withshock speed reaching minimum at u2n+1. (u2n, u2n+1) can be homotopied from (u2n, u2n+1)with shock speed reaching maximum at u2n+2. This way, in the limit, the shock (u2, u1) isconstructed by the first Principle from the null shock. This completes the proof that A is asubset of B.

Next we show that any shock which is not in A cannot be constructed according to thetwo Principles. Take, for simplicity, a state uuu∗ between uuu1 and uuu2 and close to uuu1. The shock(uuu∗, uuu+) does not satisfy condition (E). Suppose, otherwise, that it can be constructed basedon the Principles. Let uuu0 be the state on H(uuu+) between uuu+ and uuu1 with the property thatthe speeds of (uuu0, uuu+) and (uuu∗, uuu+) are the same. Since (uuu0, uuu+) can be homotopied from thenull shock according to the first Principle, it follows from the second Principle of superpositionthat (uuu∗, uuu1) is admissible according to the Principles. This is, however, not possible for thefollowing reason: If we fix uuu∗ and move uuu0 along H(uuu∗), then the entropy production η decreasesas the shock strength increases, thus violating the first Principle. This completes the proof ofthe theorem.

u1

u2

u2n+2

u2n+1

u2n

u2n-1

f(u)

uFigure 7. See proof of Theorem 1

Acknowledgements. The paper was written while the first author was visiting C.I.R.A.M.as visiting professor of G.N.F.M. The first author would like to thank the G.N.F.M. and theC.I.R.A.M. for supports and the hospitality he has received there.

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