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

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Normal modes analysis 415

where z stands for the co-ordinate x

, normal to the boundary, and

∇ is

the gradient operator along the boundary. The tangential co-ordinates will be

denoted by y ∈ R

d−1

. 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

amodeω of positive real part for which

exp( τt) exp( i η · y)exp(−ωz)(˙v,

u,s)

solves (14.3.4). We are thus led to the system











( τ + i η ·

u − uω)˙v − v (i η ·

u)+vω ˙u =0,

( τ + i η ·

u − uω)

u + vp



i ˙v η + vp



i ˙s η =0,

( τ + i η ·

u − uω)˙u − vp



ω ˙v − vp



ω ˙s =0,

( τ + i η ·

u − uω)˙s =0,

(14.3.5)

where we have used the obvious (although ugly) notation

u =(

u, ˙u). To simplify

the writing we introduce

τ := τ + i η ·

u .

(Observe that Re τ =Reτ.) The only non-trivial modes are thus obtained

for ω = τ/u and

u (i η ·

u) − τ ˙u =0 and p



˙v + p



˙s =0, (14.3.6)

for ω solution of the dispersion relation

( τ − uω)

+ v



( ω

−η

)=0,

and











( τ − uω)

u + vp



i ˙v η =0,

( τ − uω)˙u − vp



ω ˙v =0

˙s =0.

(14.3.7)

We see that the dispersion equation, which also reads

( τ − uω)

= c

( ω

−η

) (14.3.8)

has no purely imaginary root when Re τ>0. Looking at the easier case η =0

and using our usual continuity argument, we ﬁnd that because of the subsonicity

condition (14.3.3), (14.3.8) has exactly one root of positive real part, which we

denote by ω

, and one root of negative real part, ω

−

. By deﬁnition, the stable

subspace E

−

(τ,η) involved in the Kreiss–Lopatinski˘ı condition is made of normal

modes with Re ω>0forReτ>0. (Recall that – with our notation – decaying

416 Boundary conditions for Euler equations

modes at z =+∞ are obtained for Re ω>0.) So the root ω

−

does not con-

tribute to E

−

(τ,η), and we only need to consider ω

and, if u>0, ω

:= τ/u.

To simplify again the writing, we simply denote ω

by ω when no confusion is

possible. (We shall come back to the notations ω

in Section 14.4.)

If u<0 (outﬂow case), E

−

(τ,η) is a line, spanned by the solution e(τ,η)=

(˙v,

u, ˙u, ˙s)

of (14.3.7) deﬁned by

e(τ,η)=







v ( τ − uω)

−c







. (14.3.9)

If u>0 (inﬂow case), E

−

(τ,η) is a hyperplane. For convenience, we intro-

duce the additional notation

a := u τ + ω ( c

− u

) . (14.3.10)

An elementary manipulation of (14.3.8) then shows that

a ( τ − uω)=c

( τω − u η

) .

Using this relation and combining (14.3.6) and (14.3.7) together, we get the very

simple description

−

(τ,η)=(τ,η)

⊥

(τ,η):=



a, −ivuη

,vτ,ap





(Observe that  is homogeneous degree 1 in (τ,η), like a.) This description has

the advantage of unifying the treatment of regular points and Jordan points

τ = u η –whereω coincides with ω

. In the particular ‘one-dimensional’ case,

i.e. with η = 0, one easily checks that

ω =

u + c

,a= τc and (τ, 0) := τ ( c, 0 ,v,cp



) .

Omitting the null coeﬃcient, we recover in the latter a left eigenvector associated

with the incoming eigenvalue λ

= u − c of the one-dimensional Euler equations











∂

v + u∂

v − v∂

u =0,

∂

u + u∂

u + vp



∂

v + vp



∂

s =0,

∂

s + u∂

s =0.

14.3.2 Derivation of the Lopatinski˘ı determinant

Once we have the description of E

−

(τ,η) we easily arrive at the Lopatinski˘ı

condition. We consider the two cases separately.

Normal modes analysis 417

Outﬂow case If u<0, one boundary condition b(v,u,s) is required. The

existence of non-trivial normal modes in the line E

−

(τ,η) is thus equivalent to

∆(τ,η):=db · e(τ,η) =0.

We see in particular that this condition does not depend on ∂b/∂s. By deﬁnition

of e,

∆(τ,η)=v ( τ − uω)

∂b

∂v

+ ic

b · η − c

∂b

∂u

We thus recover (as pointed out in Section 14.2) that prescribing the pressure

ensures the uniform Lopatinski˘ı condition, since for b(v, u,s)=p(v,s)wehave

∆(τ,η)=v ( τ − uω) p



= 0 for Re τ ≥ 0 , (τ,η) =(0, 0) .

This is less obvious with the alternative boundary condition b(v, u,s)=u,

because in this case

∆(τ,η)=c

ω,

and it demands a little eﬀort to check that ω does not vanish. For clarity, we

state this point in the following.

Proposition 14.1 For 0 >u>−c, the root ω

of (14.3.8) that is of positive

real part for Re τ>0 has a continuous extension to Re τ =0that does not

vanish for (τ,η) =(0, 0).

Proof The only points where it could happen that ω

vanishes are such that

τ

= −c

η

. In particular, ω

= 0 implies τ ∈ i R, and also −τ

≥ (c

−

) η

– otherwise, ω

is of positive real part – in which case both roots of

(14.3.8) are purely imaginary. To determine which one is ω

, we use the Cauchy–

Riemann equations, which imply that

∂(Im ω

)

∂(Im τ )

> 0 .

Using this selection criterion, it is not diﬃcult to see that

−u τ ± icsign(Im τ )



−τ

− (c

− u

) η

− u

. (14.3.11)

(The same formulae hold for u>0.) For τ

= −c

η

, this gives (using the

fact that u is negative)

−

=0 and ω

−2 u τ

− u

the latter being non-zero unless (τ,η)=(0, 0). 

More generally, we can ﬁnd alternative boundary conditions that satisfy

the uniform Lopatinski˘ı condition without being dissipative. For instance,

418 Boundary conditions for Euler equations

take α ∈ (0, 1) and

b(v, u,s)=

+ h(p(v, s),s) ,

where h = e + pv is the speciﬁc enthalpy. We ﬁnd that

∆(τ,η)=−c

( τ − (1 − α) uω) = 0 for Re τ ≥ 0 , (τ, η) =(0, 0) .

This means that the uniform Lopatinski˘ı condition is satisﬁed. Nevertheless, the

quadratic form deﬁned in (14.2.1) may be non-deﬁnite on the tangent hyperplane

{αu ˙u + v ˙p + T ˙s =0}. More speciﬁcally, this happens for

α ∈



√

1 − M

, 1



Inﬂow case If u<0, (d + 1) boundary conditions b

(v, u,s),...,b

d+1

(v, u,s)

are needed. The existence of non-trivial normal modes in the hyperplane E

−

(τ,η)

is thus equivalent to

∆(τ,η):=det







d+1

(τ,η)







=0.

We may consider, for example, as the ﬁrst d conditions

,... , b

d−1

= s.

Then, up to a minus sign,

∆(τ,η)=−v τ

∂b

d+1

∂v

+ a

∂b

d+1

∂u

In particular, for

d+1

(v, u, s)=

+ h(p(v, s),s) ,

we have

∆(τ,η)=(c

+ αu

) τ + αuω(c

− u

) =0 for Reτ ≥ 0, (τ,η) =(0, 0),

provided that α>0. For α large enough, this gives again an example of

boundary conditions satisfying the uniform Lopatinski˘ı condition without being

dissipative.

Construction of a Kreiss symmetrizer 419

14.4 Construction of a Kreiss symmetrizer

The system (14.3.5) can be put into the abstract form

( A(τ,η)+ωI

d+2

)

U =0,

U =







˙v

˙u

˙s







We do not really need the explicit form of the (d +2)×(d + 2) matrix A(τ,η).

We already know the eigenmodes of A(τ,η) from the calculation in Section

14.3.1. Its eigenvalues are −ω

, of geometric multiplicity d, −ω

−

and −ω

, with

associated eigenvectors deﬁned by (14.3.9), i.e.

(τ,η)=







v ( τ − uω

)

−c







The construction of a Kreiss’ symmetrizer in the neighbourhood of points (τ,η)

such that Re τ>0 follows the general line described in Chapter 5, basically

using the Lyapunov matrix theorem. It is more interesting to look at the con-

struction about points with Re τ = 0, or equivalently Re τ = 0. It will appear

to be much more elementary than for general abstract systems.

We recall that a local construction amounts to ﬁnding local coordinates, i.e.

locally invertible matrices T (τ,η), and Hermitian matrices



R(τ,η), such that in

those co-ordinates the new matrix



A = T

−1

AT enjoys a local estimate

Re (



A)  γI, γ =Reτ, (14.4.12)

and the boundary matrix



B = BT satisﬁes



R + C



∗



B ≥ βI (14.4.13)

for β and C>0. For this construction, we of course assume that the Lopatinski˘ı

condition holds at the point (τ

, η

) considered, which means that we have an

algebraic estimate

P

U

 (I − P

) U

+ 



BU

, ∀U ∈ C

d+2

where P

denotes a projector onto E

−

(τ

, η

). If P is a smoothly deﬁned projector

in the neighbourhood of (τ

, η

) such that P

= P (τ

, η

), this also implies the

locally uniform estimate

PU

 (I − P ) U

+ 



BU

, ∀U ∈ C

d+2

. (14.4.14)

This will be our working assumption, with a projector P to be speciﬁed.

We also assume (14.3.3). For Re τ = 0, we always have Re ω

=0,andwe

have locally uniform bounds on Re ω

depending on the location of (τ,η) with

420 Boundary conditions for Euler equations

respect to glancing points,where

−τ

=(c

− u

) η

,ω

−

= ω

If −τ

< (c

− u

) η

Re ω

−

 −1andReω

 1 ,

whereas, if −τ

> (c

− u

) η

,weonlyhave

Re ω

−

 −γ and Re ω

 γ. (14.4.15)

Both ω

−

and ω

remain bounded away from ω

though.

Of course the weaker estimates (14.4.15) are satisﬁed in the ﬁrst case. So we

can address the two open cases simultaneously. The glancing points will be dealt

with separately. Away from glancing points, ω

, ω

and ω

−

are separated, so we

can choose a smooth basis of and consider the diagonal matrix similar to A,



A =





−ω

−





The construction of a matrix



R satisfying (14.4.12) and (14.4.13) thus depends

on the sign of Re ω

. We shall use the spectral projectors Π

according to ω

Outﬂow (u<0 hence Re ω

 −γ) E

−

is one-dimensional and spanned by

the ﬁrst vector in the new basis, and the basic estimate deriving from the

Lopatinski˘ı condition is (14.4.14) with P =Π

, i.e.

U





d+2



j=2

U



+ 



BU

Then, taking



R =





−1

µI





with µ large enough ensures (14.4.12) and (14.4.13).

Inﬂow (u>0 hence Re ω

 γ) E

−

is the hyperplane spanned by the ﬁrst

(d + 1) vectors of the new basis, and the estimate deriving from the Lopatinski˘ı

condition is (14.4.14) with P = I − Π

−

, i.e.

d+1



j=1

U



 U

d+2



+ 



BU

Construction of a Kreiss symmetrizer 421

Then, taking



R =





−1

−I





with µ large enough ensures (14.4.12) and (14.4.13).

The construction of a symmetrizer about glancing points is, in general, a

hard piece of algebra/algebraic geometry. Here, it will be greatly simpliﬁed by

the special structure of our system, and more speciﬁcally the nice form of the

eigenvectors e

, which depend linearly on ω

. There is a very simple way to get a

Jordan basis at glancing points and extend it smoothly in their neighbourhoods.

It merely consists in considering

:= i

+ e

−

and e

− e

−

− ω

−

instead of e

and e

−

.Bothe

and e

are well-deﬁned and independent of each

other in the neighbourhood of glancing points, e

is even constant







−vu

−c







and e

does belong to E

−

at glancing points. (The interest of the factor i in e

will appear afterwards.) It is not diﬃcult to ﬁnd the reduced 2 × 2 matrix a of

A on the invariant plane spanned by e

and e

. To do the calculation even more

easily, we may use the standard notations

ω =

+ ω

−

, e =

+ e

−

, etc.,

[ω]=ω

− ω

−

, [e]=e

− e

−

, etc.

On the one hand, we have

A[e]=−[ω e]

and thus, by standard manipulation,

A[e]=−[ω] e−[ω] ω

[e]

[ω]

On the other hand, we have

Ae = −ω e = −ω e +(ω 

−ω

)

[e]

[ω]

422 Boundary conditions for Euler equations

Therefore,

a =



−ω  i

i (ω 

−ω

) −ω 



Recalling from (13.2.23) that

+ ω

−

= −

2 u τ

− u

and ω

−

= −

τ

+ c

η

− u

, (14.4.16)

the matrix a is explicitly given by

a =







u τ

− u

−ic

τ

+(c

− u

) η

− u

)

u τ

− u







(Note that, at glancing points, a/i is exactly a 2 ×2 Jordan block: we have

performed here an explicit calculation of the reduction pointed out by Ralston

[160] in an abstract framework.) Completing {e

, e

} into a whole basis of C

d+2

by means of independent eigenvectors of A associated with −ω

, we get the

reduced – block-diagonal – matrix in that basis



A =



−ω



Then we look for a local symmetrizer



R that has the same structure as



A.We

shall use here the projectors Π

1,2

onto the co-ordinate axes spanned by e

1,2

Outﬂow (u<0henceReω

 −γ) From the Lopatinski˘ı condition we have

(14.4.14) with P =Π

, i.e.

U

d+1





j=1

U



+ U

d+2



+ 



BU

So we look for



R =



µI



with

Re (ra)  γI

and r 



−10

0 µ



. (14.4.17)

Inﬂow (u>0 hence Re ω

 γ.) From the Lopatinski˘ı condition we have

(14.4.14) with P = I − Π

, i.e.

d+1



j=1

U



 U

d+2



+ 



BU

Construction of a Kreiss symmetrizer 423

and we look for



R =



−I



with again (14.4.17).

We now perform the construction of r in the two cases simultaneously. We ﬁrst

expand a, writing as before τ = γ + iδ. Denoting for simplicity

a :=

− u

=0,b:=

2 c

− u

)

> 0, and ε := c

− (c

− u

)η

− u

)

we have

a = a

+ a

= i



aδ 1

εaδ



, a

= γ



a 0

bδ a



, a

= −



b 0



The matrix a

only contributes to higher-order terms in γ. Noting that the

lower-left corner of a

/γ,

bδ ∼ b



− u

η

is non-zero, we consider a matrix r of the form

r =







εg

bδ

+ iγag

bδ

− iγag h







with g, h some real numbers to be speciﬁed. Then

Re (ra)=γ





1 bδh +

bδ

bδh +

bδ

a (g + h)





+ O(γ

) .

So by the Cauchy–Schwarz inequality, the ﬁrst requirement in (14.4.17) will

hold for small enough γ, provided that g is chosen of the same sign as a, with

|g|$h

>h. Observing that, at the glancing point,

r =



01/(bδ)

1/(bδ) h



the second requirement in (14.4.17) will hold with µ of the same order as h

(large), in a small enough neighbourhood of the glancing point.

SHOCK STABILITY IN GAS DYNAMICS

The general theory (Chapter 12) has shown that the stability of shocks requires

that the number of outgoing characteristics (counting with multiplicity) be equal

to the number of jump conditions minus one: recall indeed that there is one degree

of freedom for the unknown front. With Rankine–Hugoniot jump conditions, a

necessary condition to have this equality is the Lax shock criterion.

In this chapter, we deal with the stability of Lax shocks for the full Euler

equations, as was done by Majda [126] (and independently by Blokhin [18]). For

the stability analysis of undercompressive shocks (submitted to generalized jump

conditions), and in particular of subsonic liquid–vapour phase boundaries, the

reader may refer to the series of papers [10,12,39–41] (also see [215]). Otherwise,

we recommend the very nice review paper by Barmin and Egorushkin [8] (in

which the reader will ﬁnd, if not afraid of the cyrillic alphabet, numerous

interesting references to the Soviet literature), addressing also the stability of

viscous (Lax) shocks.

15.1 Normal modes analysis

Regarding the stability of shocks in gas dynamics, normal modes analysis

dates back to the 1940s (and the atomic bomb research): stability conditions

were derived in particular by Bethe [16], Erpenbeck [53], and independently

by D



yakov [52], Iordanski˘ı [89], Kontorovich [101], etc. It was completed by

Majda and coworkers, who paid attention to neutral modes. However, they did

not publish the complete analysis for the full Euler equations (Majda referring

in [126] to unpublished computations by Oliger and Sundstr¨om), but only for

the isentropic Euler equations. We provide below this analysis, from a mostly

algebraic point of view, in which the isentropic case shows up as a special, easier

case – corresponding to a section of the algebraic manifold considered. (For a

more analytical point of view, see [92].)

With the full Euler equations in space dimension d, the number of outgoing

characteristics for Lax shocks is d + 1. Furthermore, it is easy to check that

outgoing characteristics correspond to the state behind (according to the termi-

nology introduced in Chapter 13) the shock only: depending on the choice of the

normal vector, pointing to the region where states are indexed by r, outgoing

characteristics would be denoted λ

and λ

,orλ

and λ

. Or equivalently, the

stable subspace to be considered in the normal modes analysis only involves

modes associated with the state behind the shock. We shall (arbitrarily) use the