Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications

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The Euler equations 395

In this case, (p, u,s) are independent variables and we may rewrite the quasilinear

system (13.2.22) in those new variables. We ﬁnd











∂

p + u ·∇p + ρc

∇·u =0,

∂

u +(u ·∇)u + ρ

−1

∇p =0,

∂

s + u ·∇s =0,

(13.2.26)

where c denotes as usual the sound speed (see (13.1.10)). The characteristic

matrix of this system reads





u · n ρc

−1

n (u · n) I

00u · n





which is ‘almost’ symmetric, up to the diagonal symmetrizer diag((ρc

)

−1

ρ, ···,ρ,1).

Special symmetrization for polytropic gases A special symmetrization

procedure was found by Makino et al. [128] for polytropic gases. Their motivation

was to deal with compactly supported solutions, and their symmetrizer supported

vacuum regions (unlike the one hereabove). To explain this in more detail, we

consider again the Euler equations in the variables (p, u,s), as in (13.2.26). For

polytropic gases, we may rewrite the pressure law as

p =(γ − 1) ρ

exp(s/c

which implies p = 0 for ρ = 0 as soon as γ>0. For γ>1, the sound speed c =



γp/ρ is also well-deﬁned up to ρ = 0. Furthermore, the apparently singular

term in the velocity equation can be rewritten as

−1

∇p = γ exp(s/(γc

)) ∇





γ − 1



(γ−1)/γ



and the quantity

π :=



γ − 1



(γ−1)/γ

is well-deﬁned up to p = 0. In view of the equation on p in (13.2.26), we have

∂

π + u ·∇π +(γ − 1) π ∇·u =0.

396 The Euler equations for real ﬂuids

The characteristic matrix of the system made of the equations on π, u and s







u · n (γ − 1) π n

γ e

s/(γc

)

n (u · n) I

00u · n







which is again ‘almost’ symmetric. But as in the previous general situation, the

obvious diagonal symmetrizer blows up at vacuum, since it involves 1/π.The

trick of Makino et al. consists in replacing π by a suitable power of π. For any

α ∈ (0, 1), π := π

is well-deﬁned up to 0, and

∇π =

1−α

∇(π

π

(1−α)/α

∇π

also makes sense. Furthermore, π satisﬁes the equation

∂

π + u ·∇π + α (γ − 1) π ∇·u =0.

We see that the power of π in front of ∇·u is 1, which is the same as in front of

∇π in the velocity equation only if α =1/2. With this choice, the characteristic

matrix for the equations on π, u and s is







u · n

(γ − 1) π n

2 γ e

s/(γc

)

π n (u ·n) I

00u · n







which admits the diagonal symmetrizer

diag



γ − 1

2γ

−s/(γc

)

2γ

−s/(γc

)

, 1



independently of π.

Entropy symmetrization in conservative variables A more sophisticated

way makes use of a mathematical entropy, as suggested by the general theory of

hyperbolic systems of conservation laws (see [184], p. 83–84). The mathematical

entropy to be used is S := ρs, the physical entropy per unit volume. As regards

its convexity properties, we have the following characterization.

Proposition 13.2 If e is a convex function of (v, s) satisfying (13.1.1) with p

and T positive, then S := ρs is a concave mathematical entropy of (13.2.18).

Proof The fact that S is a mathematical entropy follows from the last equation

in (13.2.26) and the mass continuity equation: indeed, S is easily found to be a

conserved quantity for smooth solutions of (13.2.18) (endowed with a complete

The Euler equations 397

equation of state), that is, (13.1.1) implies for smooth solutions of (13.2.18),

∂

S + ∇·( S u )=0. (13.2.27)

The concavity of S is more delicate to obtain, since it has to hold true in the

(complicated) conservative variables

W =(ρ, ρu ,

ρ u

+ ρe).

A very simplifying remark is based on the following fact. Any concave function

can be viewed as the inﬁmum of a family of aﬃne functions. This shows that S

is a concave function of W for ρ ∈ (0, +∞) if and only if s = vS is a concave

function of

( v, u ,

u

+ e ).

The last component of these variables,

ε =

u

+ e,

is obviously a convex function of (v, u,s). We claim that, together with the

monotonicity of e with respect to s (since T>0), this implies that s is a concave

function of (v,u,ε). (This generalizes the result that the reciprocal of an increas-

ing convex function is concave.) As a matter of fact, (v, u,s) → ( v, u ,ε(v, u,s))

is a local diﬀeomorphism and we have

ds = T

−1

( p dv − u du +dε ).

(In this equality and similar ones below, u should be viewed as a row vector.) A

short computation then shows that

s(v, u,ε) · (˙v,

u, ˙ε)

[2]

= −



e(v, s) · (˙v, ˙s)

[2]



with

˙s = T

−1

( p ˙v − u ·

u +˙ε ).

Hence the Hessian of s as a function of (v, u,ε) is clearly negative. 

Once we know that S is a concave function of (ρ, m := ρu , E := ρε ), we

may consider its Legendre transform S



. By (13.1.1), we have

T dS =dE − g dρ,

where E = ρedenotes the internal energy per unit volume and g := e − sT + pv

denotes the chemical potential. Rewriting

E = E−

2 ρ

m

398 The Euler equations for real ﬂuids

we arrive at

T dS =(

u

− g )dρ − u dm +dE.

Therefore, by deﬁnition of the Legendre transform,



(

u

− g ) ρ − u · m + E



− S.

After some simpliﬁcations, it appears that



= −

In addition, we have



= ρ dq + m dn + r dE,

where

( q, n ,r):=



u

− g, −u , 1



denote the dual variables of (ρ, m , E ). Then it is easy to check that (13.2.18)

also reads

∂



∂S



∂q



+ ∇·



∂(S



∂q



=0,

∂



∂S



∂n



+ ∇·



∂(S



∂n



=0,

∂



∂S



∂r



+ ∇·



∂(S



∂r



=0.

(13.2.28)

Viewed in the variables ( q, n ,r), the equations in (13.2.28) form a typical

Friedrichs-symmetric system: it admits a straightforward energy estimate and it

ﬁts with Deﬁnition 2.1 when rewritten, in conservative variables, as

∂











α=1



u)D

S∂









=0, (13.2.29)

where D



u) denotes the Hessian of (S



u) as a function of ( q, n ,r), and

S the Hessian of S as a function of (ρ, m , E ). These matrices are symmetric

by the Schwarz Lemma. Hence −D

S is a symmetrizer of (13.2.29) in the sense

of Deﬁnition 2.1. And the system (13.2.29) is simply the quasilinear form of the

Euler equations (13.2.18) in conservative variables.

Shock waves 399

13.3 The Cauchy problem

The Friedrichs symmetrizers described in Section 13.2 allow us, in principle, to

apply the general theory of Chapter 10 for the local existence of H

solutions,

k>d/2 + 1. One has to be careful with vacuum though, because it is not allowed

in the general symmetrization procedure, and supposedly H

solutions should of

course vanish at inﬁnity.

For polytropic gases, Chemin proved the existence of smooth solutions involv-

ing vacuum outside a compact set by using the procedure of Makino et al. [128];

see [32] for a detailed analysis.

For more general materials, it is possible to prove the existence of smooth

solutions away from vacuum by modifying slightly Theorem 10.1. We may impose

non-zero conditions at inﬁnity, as already mentioned in Chapter 10, and look for

solutions in the aﬃne space W

+ H

. (By a change of frame, the velocity at

inﬁnity, u

, may be taken equal to 0.)

Theorem 13.1 Considering the Euler equations (13.2.18) endowed with a

complete equation of state e = e(v,s) such that, in some open domain

U ⊂{(v,u,s) ∈ (0, +∞) × R

× R },

∂

∂v

= −

∂p

∂v



> 0,

we assume that

g ∈ V

+ H

) , V

=(v

, u

) ∈ U ,

with k>1+d/2 and g is valued in K ⊂⊂ U . Then there exists T>0 and a

unique classical solution V ∈ C

× [0,T]) of the Cauchy problem associated

with (13.2.18) and the initial data u(0) = g. Furthermore, V − V

belongs to

C ([0,T]; H

) ∩ C

([0,T]; H

k−1

There is a wide literature on the continuation of smooth solutions, mostly for

polytropic gases though. In some cases, global smooth solutions arise [74, 183].

On the other hand, there are numerous blow-up results [5, 32, 194].

Still, the understanding of the Cauchy problem for (multidimensional) Euler

equations (with general pressure laws) is a wide open question. This makes the

study of special, piecewise smooth, solutions, like curved shocks as in Chapter

15 interesting. In the next section, we review some basic facts on (planar) shock

waves in gas dynamics.

13.4 Shock waves

13.4.1 The Rankine–Hugoniot condition

A function U =(ρ, u,e) of class C

outside a moving interface Σ is a weak

solution of (13.2.18) if and only if it satisﬁes (13.2.18) outside Σ and if the

Rankine–Hugoniot jump conditions hold across Σ. If n ∈ R

denotes a (unit)

400 The Euler equations for real ﬂuids

normal vector to Σ and σ the normal speed of propagation of Σ, the Rankine–

Hugoniot condition associated with (13.2.18) reads











[ ρ ( u · n − σ )]=0,

[ ρ ( u · n − σ ) u + p n ]=0,

)

ρ ( u · n − σ )(

u

+ e )+ p u · n

=0,

(13.4.30)

where the brackets [·] stand as usual for jumps across Σ. The ﬁrst equation in

(13.4.30) implies

j := ρ

( u

· n − σ )=ρ

( u

· n − σ ) , (13.4.31)

where by convention, the subscripts l and r are such that the vector n points to

the state indexed by r,andj thus corresponds to the mass-transfer ﬂux from

the state indexed by l to the state indexed by r.

We may distinguish between two kinds of discontinuities, depending on the

value of j. The ﬁrst kind corresponds to discontinuities moving accordingly with

the ﬂuid, also called contact discontinuities, for which the mass-transfer ﬂux

j is in fact equal to zero. Their normal speed of propagation is σ = u

r,l

· n.

When their tangential velocity does experience a jump across Σ, which causes a

singularity of the vorticity ∇×u, they are called vortex sheets.

The second kind of discontinuity corresponds to j = 0. We call them dynam-

ical discontinuities. The fact that j is non-zero implies the tangential velocity

must be continuous and only the normal velocity u · n =: u experiences a jump

across Σ. Furthermore, some manipulations of the Rankine–Hugoniot equations

(13.4.30) yield the relation

[e]+p[v]=0, (13.4.32)

where p denotes the arithmetic mean value of pressures on both sides. We recall

the derivation of (13.4.32) for completeness. First, we observe that (13.4.31)

implies

[u]=j [v] . (13.4.33)

Then, substituting in the momentum jump condition j [u]+[p]=0,weget

[p]=−j

[v] . (13.4.34)

So that a necessary condition for such a discontinuity to exist is [p][v] < 0.

Additionally, the energy jump condition can be rewritten as

j u[u]+j [e]+[pu]=0.

Using (13.4.33) and (13.4.34), we see that

j u[u]+[pu]=j p[v].

Shock waves 401

Therefore, we have

j ([e]+p[v]) = 0,

which proves (13.4.32) since j =0.

Two diﬀerent states are thus connected by a dynamical discontinuity if and

only if there exists j ∈ R\{0} so that (13.4.32), (13.4.33) and (13.4.34) hold.

Observe that (13.4.32) and (13.4.34) are purely thermodynamic. We call the set

of all possible states connected to a reference state by a dynamical discontinuity

a Hugoniot locus.

13.4.2 The Hugoniot adiabats

The structure of Hugoniot loci is of great importance in the theory of shock waves,

and relies heavily on the so-called Hugoniot adiabats, characterized by (13.4.32).

Given a reference speciﬁc volume v

, the corresponding density ρ

=1/v

,and

a reference energy e

, we deﬁne p

:= p(ρ

) and consider the set of states

(v =1/ρ, e) such that

e − e

p(ρ, e)+p



−



=0. (13.4.35)

This set is called a Hugoniot adiabat. By the implicit function theorem, it is

locally a smooth curve, parametrized by ρ,inthe(ρ, e)-plane. Its global behaviour

is crucial in the resolution of Riemann problems.

Examples

For a Bethe–Weyl ﬂuid, using the fact that v → p(v, s) is a global diﬀeo-

morphism, Hugoniot adiabats are equivalently deﬁned in the (p, s)-plane

by the zero set of the function

(p, s):=e(v(p, s),s) − e

p + p

(v(p, s) − v

) . (13.4.36)

This will be used below in the discussion of admissibility criteria.

In the case of polytropic gases with γ ≥ 1, the Hugoniot adiabats are merely

hyperboles in the (ρ, p)-plane, given by

ρ = ρ

(γ +1)p

+(γ − 1) p

(γ +1)p +(γ − 1) p

13.4.3 Admissibility criteria

For real compressible ﬂuids, the admissibility of shocks is still a controversial

topic. The purpose of this section is to compare several admissibility criteria

under various assumptions on the equation of state.

We start with a most elementary observation on the variations of thermo-

dynamic quantities across dynamical discontinuities: because of (13.4.32) and

402 The Euler equations for real ﬂuids

(13.4.34) we have indeed

sign [ρ] = sign [p] = sign [e].

This allows us to speak about compressive discontinuities without ambiguity, in

that both the density and the pressure should increase along particle paths. On

the contrary, density and pressure decrease along particle paths for expansive (or

rarefaction) discontinuities.

Entropy criterion

A ﬁrst admissibility criterion comes from the second principle of thermody-

namics, or equivalently from the Lax entropy criterion associated with the

mathematical entropy S = ρs, which is concave (see Proposition 13.2) if e is

a convex function of (v, s), a ‘minimal’ thermodynamical stability assumption

that we make from now on. The entropy criterion requires that

j [s] ≥ 0 . (13.4.37)

(Observe that the inequality (13.4.37) is trivially satisﬁed (and is an equality)

for static discontinuites.) It is shown below that for dynamical discontinuities in

Smith ﬂuids, the entropy criterion (13.4.37) is equivalent to compressivity.

Proposition 13.3 (Henderson–Menikoﬀ) For a Smith ﬂuid, we have

sign [p]=sign [s]

on Hugoniot loci, which means that the function h

deﬁned in (13.4.36) does not

vanish for (p, s) such that (p − p

)(s − s

) < 0.

Proof By (13.1.2) and (13.1.5), we have

v dp = −γpdv +ΓT ds. (13.4.38)

So, substituting dv into

=de +

v − v

dp +

p + p

dv = T ds +

v − v

dp +

− p

(13.4.39)

we obtain



1+Γ

− p

2γp



T ds +



v − v

γp

(p − p

)



dp. (13.4.40)

Therefore, because of Smith’s condition in (13.1.15), h

is increasing with s at

constant p.Wethushave

( h

(p, s) − h

(p, s

))(s − s

) > 0.

Shock waves 403

Now, if we look at h

:= h

(·,s

), preferably in the v variable, we see that it

has an inﬂexion point at v = v

, and its second derivative reduces to

∂

(v − v

)

∂

∂v

By the convexity of p, this shows that

( h

(v) − h

))(v − v

) > 0,

or equivalently

( h

(p, s

) − h

))(p − p

) < 0.

To conclude, assume that (p − p

)(s − s

) < 0. Then

(p, s)=(h

(p, s) − h

(p, s

)) + (h

(p, s

) − h

))

is the sum of two terms of the same sign as (s − s

), which cannot vanish unless

s = s

. 

Internal structure of shocks

Another, physically relevant, admissibility criterion is based on dissipative eﬀects

due to viscosity and/or heat conductivity. Indeed, dynamical ‘discontinuities’ in

ﬂuids are not exactly sharp: because of dissipative phenomena they have an

internal structure, also called a shock layer. Mathematically, a shock layer is a

smooth solution of the full equations of motion – the (compressible) Navier–

Stokes–Fourier equations, consisting of the Euler equations supplemented with

viscous terms in the stress tensor and with a heat-ﬂux term in the energy

equation – which is ‘almost’ a solution of the Euler equations outside a thin

region of high gradients. A more precise deﬁnition is delicate to formulate, and

it is a tough task to actually show the existence of arbitrarily curved shock

layers (see the recent series of papers by Gu`es et al. [77–79]). But for planar

discontinuities, the search for a layer is a much simpler, ODE problem, which

was addressed a long time ago by Gilbarg [69]. Planar shock layers are indeed to

be sought as heteroclinic travelling wave solutions of the Navier–Stokes–Fourier

equations











∂

ρ + ∇·(ρ u)=0,

∂

(ρu)+∇·(ρ u ⊗ u)+∇p = ∇(λ ∇·u)+∇·(µ (∇u +

(∇u))) ,

∂



u

+ e



+ ∇·





u

+ e



+ p



= ∇·( κ ∇T )

+ ∇·((λ ∇·u + µ (∇u +(∇u)

))u ) .

(13.4.41)

Here, λ and µ are viscosity coeﬃcients and κ is heat conductivity, and these

three coeﬃcients may depend (in a smooth way) on ρ.

404 The Euler equations for real ﬂuids

Theorem 13.2 (Gilbarg) For Bethe–Weyl ﬂuids, all planar compressive dis-

continuities admit shock layers for any λ, µ, κ such that λ +2µ>0 and κ>0.

We do not reproduce the proof here, which is rather long but very clear in [69].

The interested reader will easily check that our deﬁnition of Bethe–Weyl ﬂuids

ﬁts with the assumptions I,Ia., II, III, IV ( [69] p. 271) of Gilbarg. To recast our

problem in Gilbarg’s framework, which is purely one-dimensional, we ﬁrst write

the ODE system governing travelling waves of speed σ in a given direction n.

Decomposing the velocity ﬁeld u into its normal component u := u · n and its

tangential component

u, we get the equations











( ρ (u − σ))



=0,

( ρ (u − σ)



=(µ



)



( ρ (u − σ) u)



+ p



=((λ +2µ) u



)





ρ (u − σ)(e +

u

)





+(pu)



=(κT



+(λ +2µ) uu



)



(13.4.42)

where



denotes the derivative with respect to ξ := x · n − σt. For solutions of

(13.4.42), we have

ρ (u − σ) ≡ const. =: j, (13.4.43)

consistently with the notation previously introduced (see (13.4.31)). Regarding

the tangential velocity, we have

µ (



− j

u ≡ const.

Thus for non-zero mass-tranfer ﬂux j, we must have

u(−∞)=

u(+∞)=:

∞

which is again consistent with the Rankine–Hugoniot condition (13.4.30). Fur-

thermore, the only homoclinic orbit of the ODE

µ (



= j (

u −

∞

)

is its ﬁxed point istself, that is, we must have

u ≡

∞

. Taking this into account,

as well as (13.4.43), the last equation in (13.4.42) equivalently reads

(ρu − σ)(e +

)+(pu) − κT



− (λ +2µ) uu



≡ const.

Additionally, by change of frame, we can assume without loss of generality that

σ = 0. In this way, we have exactly recovered the equations solved by Gilbarg

(namely, (1), (2) and (3) p. 257 in [69]).

In view of Proposition 13.3 and Theorem 13.2, for Smith ﬂuids the entropy

criterion (13.4.37) is equivalent to compressivity and ensures the existence of a

planar shock layer.