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

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Normal modes analysis 425

subscript l for the state past the shock, and the subscript r for the state behind,

the latter being most often omitted for simplicity.

From now on, we consider a reference subsonic state W = W

,andasuper-

sonic state W

on the corresponding Hugoniot locus such that the mass-transfer

ﬂux j is positive (consistently with the Lax shock inequalities in (13.4.48)). By

a change of frame, we can assume without loss of generality that the normal

propagation speed of the shock σ is zero, and that the tangential velocity of the

ﬂuid

u is zero (recall that the tangential velocity is continuous across the shock

front.) In this way, we merely have

j = ρu = ρ

> 0and0<M =

< 1 . (15.1.1)

15.1.1 The stable subspace for interior equations

The story is the same as for Initial Boundary Value Problems, since we only need

to consider the equations behind the shock. This simpliﬁcation always occurs for

extreme shocks. (It does not occur for the phase transitions studied in [9, 10, 12]

for instance, but the method is analogous.)

We brieﬂy recall the derivation of the stable subspace associated with interior

equations, for readers who would have skipped Section 14.3. There is one addi-

tional simpliﬁcation here, due to the null tangential velocity

u, which amounts

to replacing τ by τ . Linearizing the Euler equations written in the variables

(v, u,s), (14.3.2), about our constant reference state (v, u =(0,u),s), we get











( ∂

+ u∂

)˙v − v ∇·

u =0,

( ∂

+ u∂

)

u + vp



∇˙v + vp



∇˙s =0,

( ∂

+ u∂

)˙s =0,

(15.1.2)

where z denotes the co-ordinate normal to the shock front. By deﬁnition,

for Re τ>0andη ∈ R

d−1

, the sought stable subspace, E

−

(τ,η) is the space

spanned by vectors ( ˙v,

u,s) such that there exists a mode ω of positive real part

for which

exp( τt) exp( i η · y)exp(−ωz)(˙v,

u,s)

solves (15.1.2). Here, we have denoted y := (x

,...,x

d−1

). We are thus led to

the system











( τ − uω)˙v − v (i η ·

u)+vω ˙u =0,

( τ − uω)

u + vp



i ˙v η + vp



i ˙s η =0,

( τ − uω)˙u − vp



ω ˙v − vp



ω ˙s =0,

( τ − uω)˙s =0,

426 Shock stability in gas dynamics

where

u =(

u, ˙u). Because of (15.1.1), the only non-trivial solutions with Re ω>

0correspondtoeitherω = τ/u or the only root of positive real part of the

dispersion equation

( τ − uω)

= c

( ω

−η

) . (15.1.3)

And the corresponding invariant subspace admits the simple characterization

−

(τ,η)=(τ,η)

⊥

(τ,η):=



a, −ivuη

,vτ,ap





(15.1.4)

where

a := uτ + ω ( c

− u

) . (15.1.5)

15.1.2 The linearized jump conditions

With

σ =

∂



1+

∇X

and n =



1+

∇X



−

∇X



the ﬁrst (d + 1) equations in the Rankine–Hugoniot conditions (13.4.30) may be

rewritten as











u −

u ·

∇X

∂

u −

u ·

∇X

− [ p ]

∇X = ∂

u −

u ·

∇X

+[p ]=∂

As to the last equation in (13.4.30), it may equivalently be replaced by the

purely thermodynamical equation (13.4.32).

Again, we emphasize that we need not perturb the state past the shock, which

simpliﬁes the linearized jump conditions below. This is because we ultimately

look for solutions of the linearized jump conditions that belong to the stable

subspace E

−

(τ,η), which does not depend on the state past the shock, (v

, u

This simpliﬁcation is speciﬁc to extreme Lax shocks. However, the method is

similar for other kinds of shocks.

Linearizing all jump conditions about v

= v,

=0, u

= u, s

= s (with

=0,u

, s

kept ﬁxed) and X = 0, and looking for front modes of the form

Normal modes analysis 427

exp( τt) exp( i η · y), we obtain the equations











−

˙v +

˙u =[ρ] τ

u = i [p]

X η ,



− u

)˙v +2

˙u + p



˙s =0,

( p



[v] − [p]) ˙v +(T +



[v]) ˙s =0,

(15.1.6)

where the jumps now stand for jumps between the reference states. The last

equation has been obtained by linearizing (13.4.32), using the fundamental

relation (13.1.1). Note that the Rankine–Hugoniot condition applied to the

reference states implies

[p]=−

[v]. (15.1.7)

(This is (13.4.34) with j = u/v.) So we shall substitute −

[v]for[p]inthe

system (15.1.6).

15.1.3 The Lopatinski˘ı determinant

The existence of non-trivial vectors belonging to the space E

−

(τ,η) deﬁned in

(15.1.4) and satisfying the system (15.1.6) is clearly equivalent to the existence

of non-trivial solutions of a linear, purely algebraic, system. To simplify the

computation, we can ﬁrst eliminate

u by means of the second equation in (15.1.6),

which allows us to substitute u

[v] η

X for iuvη ·

u in the equation (τ,η) ·

(˙v,

u, ˙u, ˙s ) = 0 (deﬁning E

−

(τ,η)). Using again the relation c

= −v



,we

readily obtain the linear system







0 −[ρ] τ

+ u



( c

− u

)[v]0T +



[v]0

−av

vτ −ap



−u

[v] η













−˙v/v

˙u

˙s



















(15.1.8)

428 Shock stability in gas dynamics

The existence of non-trivial solutions of (15.1.8) is then equivalent to the

vanishing of its determinant, which we denote by ∆(τ,η).

‘One-dimensional’ case The expansion of ∆(τ,0) is straightforward. Using

the fact that a = τcfor η =0,wehave

∆(τ,0) =

v [v][ρ] τ



+ u



− u

0 p



2 T

[v]

−cv 1 −p



v/c



= τ

[v]

c (c + u)(1+M + M



[v]).

(Recall that M = u/c denotes the Mach number.) Rewriting p



/T =Γ/v,the

one-dimensional stability condition thus reads

1+M +ΓM

[v]

=0. (15.1.9)

This condition holds true at least for weak shocks, as expected from the

general theory, or can be checked directly. For, the Mach number M admits the

ﬁnite limit 1 and thus

1+M +ΓM

[v]

> 0 (15.1.10)

for |[v]| small enough. The condition (15.1.9) might break down for large shocks

though. It depends of course on the equation of state.

An easy consequence of the previous argument is the following.

Proposition 15.1 Lax shocks of arbitrary strength are stable in one space

dimension, provided that γ ≥ Γ ≥ 0 (at least at the endstates).

Proof Surprisingly enough, the inequality (15.1.10) trivially holds true for an

expansive shock, that is, if [v] > 0. But more ‘standard’ shocks are compressive.

And if [v] < 0, we need an upper bound for Γ M

/v to prove that (15.1.10) holds

true. This bound can be found by noting that

= γpv ≥ Γ pv

by assumption. Therefore,

Γ M

[v]

Γ v

[v] ≥

[v]=−

[p]

because of the Rankine–Hugoniot condition in (13.4.34). And consequently,

1+M +ΓM

[v]

≥ M +

> 0 .



Normal modes analysis 429

Observe that ideal gases trivially satisfy γ ≥ Γ > 0 (provided that c

> 0).

For van der Waals gases, a short computation shows that γ ≥ Γ if and only if

≥ 2

v − b

−

a(v − b)

RT .

This is obviously true for large enough temperatures, a rough bound being the

critical temperature T

27bR

, but false for ‘small’ temperatures.

In view of (13.2.20), an equivalent statement of the strict inequalities

γ>Γ > 0is

∂p

∂v



< 0and

∂p

∂e



> 0 .

It is notable that this is precisely the assumptions that enabled Liu [118] to show

the unique solvability of the Riemann problem (using his admissibility criterion

for shocks).

Multidimensional case Assuming that η = 0, we may introduce

V :=

i η

and A :=

i η

to simplify the writing. In fact, it will be clearer to handle only quantities

homogeneous to velocities. This is why we set

A = cW .

Recalling the deﬁnition of a, (14.3.10), where ω is the root of positive real part

of (13.2.23), we ﬁnd that W has the simple representation

= V

− (c

− u

) , Im W<0whenImV<0 , (15.1.11)

which extends analytically to {V ∈ R ; V

− u

}. By a standard argu-

ment, using Cauchy–Riemann equations, we also ﬁnd that W has a continuous

extension to {V ∈ R ; V

≥ c

− u

}, given by

W = sign (V )



− (c

− u

) .

Using these notations, the genuinely multidimensional Lopatinski˘ı determi-

nant reads

∆(τ,η):=−η



0 −[ρ] V

+ u



( c

− u

)[v]0T +



[v]0

−cW v

vV −Wp



[v]



430 Shock stability in gas dynamics

that is,

∆(τ,η):=−η

[v]



+ u



− u

0 p



[v]

−cvW V −W





The expansion of ∆(τ,η) results in

∆(τ,η)= −η

[v]

(2 + ΓM

[v]

)(V/u + W/c ) V/u

− (1 − M

)((V/u)

+ v

/v )

(15.1.12)

where we have substituted T Γ/v for p



, as in the 1D-case.

From the expression of the Lopatinski˘ı determinant in (15.1.12), we see

that multidimensional stability conditions must involve the two non-dimensional

quantities

r :=

and k := 2 + Γ M

[v]

It is to be noted that k is related in a simple way to the ratio R deﬁned in

Proposition 13.5,

R =

[p]

[v]

v +

Γ[v]

Γ[p] − γp

Indeed, using (15.1.7), we ﬁnd that

R =1−

(1 − M

) .

15.2 Stability conditions

15.2.1 General result

Thanks to the preliminary work of the previous section, we can derive a complete

hierarchy of stability conditions. As expected from the theory in [186], the one-

dimensional stability threshold in (15.1.9) turns out to be a transition point

from strong multidimensional instability to weak stability of WRtype. Attentive

readers will also note that the transition from neutral to uniform stability occurs

at a glancing point – in that the neutral mode found for k =1+M

(r − 1)

corresponds to ω = τ/u (see Chapter 13 for a more detailed discussion of

Stability conditions 431

glancing points). (This kind of transition was pointed out in an abstract setting

in [13].) To summarize, we have the following.

Theorem 15.1 (Majda) The stability of a Lax shock depends on the Mach

number behind the shock

M =

|u · n − σ|

∈ (0, 1)

and on the volumes behind and past the shock, respectively denoted by v and v

through the quantities

r :=

and k := 2 + Γ M

v − v

. (15.2.13)

One-dimensional stability is equivalent to

k =1− M. (15.2.14)

Weak (non-uniform) multidimensional stability is equivalent to

1 − M<k≤ 1+M

( r − 1). (15.2.15)

Uniform multidimensional stability is equivalent to

1+M

( r − 1) <k. (15.2.16)

Remark 15.1 If the lower inequality in (15.2.15) is violated, and more precisely

k<1 − M,

the shock is violently unstable with respect to multidimensional perturbations,

even though it is stable to one-dimensional perturbations. As mentioned by

Majda in [125], this striking result was known since the work of Erpenbeck [53]

(also see the earlier work of D



yakov [52]).

Proof The one-dimensional stability condition has already been obtained

above, in the equivalent form (15.1.9).

It will turn out that multidimensional stability conditions can be infered

from the properties of a second-order algebraic curve. We strongly advocate this

mostly algebraic, and elementary point of view, which is a convenient alternative

to the analytical approach of Jenssen and Lyng [92], for instance. Of course, all

methods require some care and are a little lengthy.

The expression in (15.1.12) of the Lopatinski˘ı determinant in terms of W ,

deﬁned in (15.1.11), shows that multidimensional stability is encoded by the

zeroes of the function

f(z):=k ( z + g(z))z − (1 − M

)(z

+ r ) , (15.2.17)

where

g(z)

= M

− (1 − M

) , Im g(z) < 0 (15.2.18)

432 Shock stability in gas dynamics

on the lower half-plane {Im z<0}. On the real axis, g (and thus also f)is

extended by continuity, which implies that

zg(z) > 0 for all z ∈ R ; z

> (1 − M

)/M

. (15.2.19)

Of course, g is analytic up to z ∈ R if z

< (1 − M

)/M

. And on that part of

the real axis, g is purely imaginary. So it is clear that f cannot vanish for z ∈ R

and z

< (1 − M

)/M

. Besides (15.2.19), it will also be useful to have in mind

that

zg(z) < 0 for all z ∈ i R

−

. (15.2.20)

We shall obtain stability conditions mainly by algebraic arguments, using

the following observation. If z is a zero of f,thenx = z

must be a zero of the

polynomial p(·,k) deﬁned by

p(x, k)=



( k − (1 − M

))x − (1 − M

) r



− k

x ( M

x − (1 − M

)).

(15.2.21)

For simplicity, we have stressed here only the dependence of p upon the parameter

k, even though it is also obviously a polynomial in r and M

. This simpliﬁes the

writing and is confusionless since r and M

are kept ﬁxed in the discussion below.

Readers gifted in algebra will have checked that p is the resultant with respect

to y of two polynomials F (x, y)andG(x, y),

F (x, y)=k ( x + y ) − (1 − M

)(x + r ) , (15.2.22)

which is obtained by replacing z

by x and zg(z)byy in f(z), and

G(x, y)=y

− x (M

x − (1 − M

)) , (15.2.23)

which vanishes simultaneously with y

− z

g(z)

Conversely, if x is a zero of p(·,k)andz

= x then, either

( k − (1 − M

))z

− (1 − M

) r = −kzg(z) ,

which means that f(z) = 0, or

( k − (1 − M

))z

− (1 − M

) r =+kzg(z)

and z is not a zero of f (unless zg(z) = 0). If x is real, the inequalities (15.2.19)

and (15.2.20) will enable us to decide which is the case. This discussion can be

summarized as follows, where for convenience we denote

(1 − M

) r

k − (1 − M

)

If x is a negative real root of p(·,k)andz

= x, z ∈ i R

−

is a zero of f if

and only if

( k − (1 − M

))(x − x

) > 0 .

Stability conditions 433

If x is a positive real root of p(·,k)andz

= x>x

∗

:= (1 − M

)/M

z ∈ R is a zero of f if and only if

( k − (1 − M

))(x − x

) < 0 .

Therefore, the conclusion will depend on

 the sign of k − (1 − M

)

and on the position of x

(k) with respect to the real roots of p(·,k). From

the deﬁnition (15.2.21) of p(·,k), we see that

p(x

(k),k)= −k

(k)(M

(k) − (1 − M

))

= −k

(1 − M

)

( k − (1 − M

))

( M

r − k +1− M

and we easily compute that the dominant coeﬃcient of p(·,k)is

=(1− M

)(k − 1 − M )(k − 1+M ) .

Consequently, the position of x

(k) with respect to the real roots of p(·,k)is

determined by

 the sign of ( k − 1 − M )(k − 1+M )(M

r − k +1− M

Since M ∈ (0, 1) we obviously have

1 − M<1 − M

< 1+M.

So there will be a priori only two cases to consider, depending on the position of

1+M with respect to M

r +1− M

Case 1 rM < 1+M, hence

1 − M<1 − M

r +1− M

< 1+M,

Case 2 rM > 1+M, hence

1 − M<1 − M

< 1+M<M

r +1− M

Regarding the other coeﬃcients of p(·,k), at the zeroth order we have

=(1− M

)

> 0 ,

and

(k)=(1− M

)(k

− 2 r ( k − (1 − M

)).

Thus it is only

sign a

(k) = sign {( k − 1 − M )(k − 1+M ) }

that determines the sign of the product of the roots of p(·,k). The sign of a

important at singular points of a

, in order to localize the inﬁnite branches of

434 Shock stability in gas dynamics

the algebraic curve P := {(x, k); p(x, k)=0}. In fact, we have

(1 − M )=(1− M

)(1 − M )(1 − M +2rM) > 0 ,

but the sign of

(1 + M )=(1− M

)(1 + M )(1 + M − 2 rM)

depends on r. This leads us to split Case 1 into

Case 1a rM < (1 + M)/2, hence a

(1 + M ) > 0 ,

Case 1b (1 + M)/2 <rM<1+M, hence a

(1 + M ) < 0 .

We now have all the ingredients to infer the needed qualitative features of P .

There is always a parabolic branch with vertical asymptotes x = −1andx =0.

Another remark is that x

(k) is a root of p(·,k)onlyatk =0,wherex

= −r,

and at k = M

r +1− M

,wherex

= x

∗

=(1− M

)/M

. (The latter will

appear to be a transition point from neutral stability to uniform stability.)

We shall consider successively Cases 1a, 1b, and 2, and arrive at the conclusion

that

• a necessary and suﬃcient condition for f to have only real roots is (15.2.15);

• a necessary and suﬃcient condition for f to have no root at all is (15.2.16).

The reader may refer to the corresponding ﬁgures (Figs. 15.1–15.3) in order to

visualize the tedious but elementary arguments involved in the discussion below.

Case 1a

for k<1 − M,wehavea

(k) > 0anda

(k) p(x

(k),k) < 0. Therefore,

p(·,k) has two real roots of the same sign and x

(k) < 0 is in between.

Consequently, the smallest root, x

of p satisﬁes

( k − (1 − M

))(x − x

) > 0

and thus yields a root z ∈ i R

−

of f . This root corresponds to a strongly

unstable mode. When k goes to 1 − M, x

goes to −sign (a

) ∞ = −∞.

for 1 − M<k<1 − M

,wehavea

(k) < 0anda

(k) p(x

(k),k) > 0.

Therefore, p(·,k) has two real roots of opposite signs (one going to +∞

when k goes to 1 − M)andx

(k) is smaller than both of them. The

negative root of p, still denoted x

for simplicity, now satisﬁes

( k − (1 − M

))(x − x

) < 0 ,

which shows it does not give rise to a root of f. The positive one satisﬁes

the same inequality, and thus its square root (and the opposite) is a real

zero of f. This zero corresponds to a neutral mode.

for 1 − M

<k<M

r +1− M

, we still have a

(k) < 0and

(k) p(x

(k),k) > 0 and thus two real roots of opposite signs for p(·,k).