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

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From FBP to IBVP 335

in mind an iterative scheme, we are going to linearize about a special solution

that is not necessarily the (planar) reference one. Substituting u + ε ˙u for u

and χ + ε ˙χ for χ in (12.1.3) and (12.1.4), diﬀerentiating these equations with

respect to ε and evaluating at ε = 0, we get the linearized problem

d−1



j=0

) ∂

˙u

± A

, dχ) ∂

˙u

d−1



j=0

(dA

) · ˙u

) ∂

± (d

, dχ) · ˙u

) ∂

∓

d−1



j=0

(∂

˙χ) A

) ∂

= 0 for z>0 ,

(12.1.5)

with the boundary conditions

d−1



j=0

( f

) − f

−

))∂

˙χ = A

, dχ) · ˙u

− A

−

, dχ) · ˙u

−

(12.1.6)

at z =0.

Of course, if u

= u

are constant and χ = χ, with χ(y,t):=σt, (12.1.5)

and (12.1.6) simplify and become

d−1



j=0

) ∂

˙u

± A

,σ,0) ∂

˙u

=0, (12.1.7)

d−1



j=0

( f

) − f

−

))∂

˙χ = A

,σ,0) · ˙u

− A

−

,σ,0) · ˙u

−

. (12.1.8)

Observe that the derivatives of ˙χ appear here only in the boundary conditions

(12.1.8), whereas for the general problem (12.1.5) and (12.1.6) the derivatives of

˙χ do appear in the interior equations (unless we use Alinhac’s trick, as described

below).

The messy appearance of (12.1.5) and (12.1.6) can be lessened by using

additional shorter notations. Let us ﬁrst deﬁne the diﬀerential operators L

−

and L

(v, dχ)=

d−1



j=0

(v) ∂

± A

(v, dχ) ∂

Note: with these notations, the non-linear equations in (12.1.3) equivalently read

, dχ) u

=0.

336 Persistence of multidimensional shocks

We also introduce a shortcut for the zeroth-order terms in (12.1.5):

(v, dχ) · ˙u =

d−1



j=0

(dA

(v) · ˙u) ∂

v ± (d

(v, dχ) · ˙u) ∂

Then the interior equations in (12.1.5) read

, dχ)˙u

+ D

, dχ) · ˙u

∓ A(u

,∂

˙χ, ∂

˙χ,...,∂

d−1

˙χ, 0) ∂

=0.

(12.1.9)

As pointed out by Alinhac [5], there is a way to replace the derivatives of ˙χ

by zeroth-order contributions of ˙χ in these interior equations. This is done by

changing the unknown ˙u

to the so-called good unknown

˙v

:= ˙u

∓ ˙χ∂

Proposition 12.1 With all the notations introduced above,

, dχ)˙u

+ D

, dχ) · ˙u

∓ A(u

,∂

˙χ, ∂

˙χ,...,∂

d−1

˙χ, 0) ∂

= L

, dχ)˙v

+ D

, dχ) · ˙v

+˙χ∂



, dχ) u



for all u

, χ, ˙u

, ˙χ for which both sides make sense.

Proof The computations for either one of the signs + or − are similar. We

do it for the + sign, and to facilitate the reading we omit the subscript +. By

deﬁnition of ˙v =˙u − ˙χ∂

L(u, dχ)˙v = L(u, dχ)˙u − ˙χ∂



L(u, dχ) u,



+˙χ [∂

,L(u, dχ)] u −(L(u, dχ)˙χ)∂

and

D(u, dχ) · ˙v = D(u, dχ) · ˙u − ˙χD(u, dχ) · (∂

u).

Summing these two equalities, and reordering terms we get

L(u, dχ)˙v + D(u, dχ) · ˙v +˙χ∂



L(u, dχ) u



= L(u, dχ)˙u + D(u, dχ) · ˙u − ( L(u, dχ)˙χ ) ∂

+˙χ [ ∂

,L(u, dχ)]u − ˙χD(u, dχ) · (∂

u).

By deﬁnition of L(u, dχ)andD(u, dχ), the last two terms cancel out, and

L(u, dχ)˙χ = A(u, ∂

˙χ, ∂

˙χ,...,∂

d−1

˙χ, 0),

hence the claimed equality. 

Therefore, (12.1.9) equivalently reads

, dχ)˙v

+ D

, dχ) · ˙v

+˙χ∂



, dχ)u



=0. (12.1.10)

Normal modes analysis 337

This is a ﬁrst-order linear PDE in ( ˙v

, ˙χ), of principal part L

, dχ)˙v

.The

actual PDE on ˙χ comes from the boundary conditions in (12.1.6). We introduce

the notation

B(u

−

) · (ξ

,...,ξ



j=0

( f

) − f

−

)),

in such a way that the non-linear boundary conditions (12.1.4) merely read

B(u

−

) · (dχ, −1) = 0,

where dχ has been identiﬁed with the row vector (∂

χ, ∂

χ,...,∂

d−1

χ). The

linearized boundary conditions (12.1.6) thus read

B(u

−

) · (d ˙χ, 0) = A

, dχ) · ˙u

− A

−

, dχ) · ˙u

−

which we shorten even more into

b(u, d˙χ)+M(u, dχ) · (˙u

−

, ˙u

)=0, (12.1.11)

with the obvious deﬁnitions

b(u, d˙χ):=B(u

−

) · (d ˙χ, 0),

M(u, dχ) · (˙u

−

, ˙u

):=A

−

, dχ) · ˙u

−

− A

, dχ) · ˙u

12.2 Normal modes analysis

In the 1980s, Majda showed how to extend Kreiss’ method to non-standard BVPs

associated with the shock-persistence problem (here (12.1.3) and (12.1.4)), or

more precisely their linearized versions (here (12.1.10) and (12.1.11)). As for

standard BVP, a crucial, preliminary step is the so-called normal analysis of the

constant coeﬃcients problems (here (12.1.7) and (12.1.8)).

The main purpose of this whole section is to describe the normal modes

analysis for (12.1.7) and (12.1.8). To simplify the writing, in this section we

omit underlining the states of the reference planar shock: u

−

and u

are to be

understood as u

−

and u

in what follows.

12.2.1 Comparison with standard IBVP

In order to derive a generalized version of the Kreiss–Lopatinski˘ı condition

we look for special solutions, or ‘normal modes’ of (12.1.7) and (12.1.8). This

amounts to applying to (12.1.7) and (12.1.8) a Laplace transform in t and Fourier

transform in y. The resulting equations are, if

and

X denote the Fourier–

Laplace transforms of ˙u

and ˙χ, respectively,

A(u

,τ,iη,0)

± A

,σ,0) ∂

= 0 for z>0 , (12.2.12)

Xb(u, τ, iη) − M(u, σ, 0) · (

−

)=0 at z =0, (12.2.13)

338 Persistence of multidimensional shocks

where we have used the shortcuts deﬁned in the previous section, τ is the complex

variable dual to t and η ∈ R

d−1

is the wave vector associated to y. Furthermore,

we have assumed that ˙u

and ˙χ equal zero at t = 0: otherwise, there would

have been a right-hand side in both (12.2.12) and (12.2.13); we shall introduce

right-hand sides later.

We now start to make some assumptions. First, we suppose that the planar

shock of speed σ between u

−

and u

is not characteristic. This precisely means

that both matrices

,σ,0) = A

) − σA

)

are non-singular, so that the equations in (12.2.12) form a system of 2n indepen-

dent diﬀerential equations, which can be rewritten as

= A(u, η, τ )

with

U :=



−



and

A(u, η, τ):=



−

,σ,0)

−1

A(u

−

,τ,iη,0) 0

0 −A

,σ,0)

−1

A(u

,τ,iη,0)



The second assumption, which will turn out to be a consequence of a stronger

one, is that

X can be eliminated from the boundary condition (12.2.13). This

amounts to requiring the ellipticity of the symbol (τ,η) → b(u, τ, iη), that is,

b(u, τ, iη) = 0 for all (τ,η) =(0, 0),

or equivalently that the jump vectors [f

(u)], [f

(u)], ...,[f

d−1

(u)] be indepen-

dent in R

Remark 12.2 A byproduct of the ellipticity assumption on the symbol b is

the necessary condition d ≤ n (i.e. the space dimension smaller than the size of

the system). This precludes in particular multidimensional scalar conservation

laws! Another observation is that ellipticity is obviously not uniform when the

shock strength goes to zero: this problem was pointed out by M´etivier [133], and

overcome in [56, 133].

Introducing

Π(u, τ, η)=I

−

b(u, τ, iη) b(u, τ, iη)

∗

b(u, τ, iη)

Normal modes analysis 339

the orthogonal projection onto b(u, τ, iη)

⊥

, we see that the boundary condition

in (12.2.13) is equivalent to

Π(u, τ, η) M(u, σ, 0)

U =0 and

X =

b(u, τ, iη)

∗

M(u, σ, 0)

|b(u, τ, iη)|

(where M(u, σ, 0)

U stands for M(u, σ, 0) · (

−

), with a slight abuse of

notation).

Therefore, solving the problem (12.2.12) and (12.2.13) is equivalent to solving

the pure boundary value problem:











= A(u, η, τ)

U for z>0 ,

Π(u, τ, η) M (u, σ, 0)

U =0 at z =0.

(12.2.14)

It is worth pausing for a while, and compare (12.2.14) to











= −A

(v)

−1

A(v, τ, iη, 0)

V for z>0 ,

C(v) ·

V =0 at z =0,

derived by Fourier–Laplace transform from a standard IBVP











d−1



j=0

(v) ∂

˙v + A

(v) ∂

˙v = 0 for z>0 ,

C(v) · ˙v =0 at z =0 and ˙v =0 at t =0.

Except for the size of the interior system (which is doubled in (12.2.14)), the main

diﬀerence is that the boundary condition in (12.2.14) contains the ‘frequencies’ τ

and η. This reﬂects the fact that the corresponding IBVP (obtained by an inverse

Fourier–Laplace transform) has pseudo-diﬀerential boundary conditions. Despite

this non-standard feature, one may derive a (generalized) Kreiss–Lopatinski˘ı

condition by looking for solutions

U ∈ L

) of (12.2.14). This is part of the

normal modes analysis. In fact, the derivation of the uniform Kreiss–Lopatinski˘ı

condition also requires the neutral modes analysis: neutral modes correspond to

solutions of (12.2.14) for Re τ = 0 that are not necessarily square-integrable and

may oscillate in the z-direction, but not all of them are to be considered (this

should be clear from Chapter 4, see also the discussion below).

As far as the normal modes analysis is concerned, we may avoid the cumber-

some projection Π(u, τ, η) and work with the full system (12.2.12) and (12.2.13)

on (

X).

A ﬁrst question is, which values of (η, τ) ensure the hyperbolicity of the matrix

A(u, η, τ)? A preliminary answer is, if the operators L

,σ,0) are hyperbolic

in the t-direction (or equivalently, the operator A

(u) ∂



j=1

(u)∂

340 Persistence of multidimensional shocks

hyperbolic in the t-direction, which also means the system of conservation laws

(12.1.1) is hyperbolic) and if the n × n matrices A

,σ,0) are non-singular

(which we have already assumed), A(u, η, τ) is hyperbolic at least for Re τ>0.

Indeed, the set of eigenvalues of the 2n × 2n matrix A(u, η, τ ) is obviously the

union of the eigenvalues of the n × n matrices

,η,τ):=∓A

,σ,0)

−1

A(u

,τ,iη) ,

and the hyperbolicity of L

,σ,0) prevents A

,η,τ) from having purely

imaginary eigenvalues when τ is not purely imaginary itself; as already mentioned

in Chapter 9, this observation dates back to Hersh [83].

From now on, we assume that the system of conservation laws (12.1.1) is

hyperbolic and that the matrices A

,σ,0) are non-singular.

The next question concerns the dimension of the stable subspace of A(u, η, τ).

The answer lies in the decomposition

(A(u, η, τ))  E

−

,η,τ)) × E

−

,η,τ))

= E



−

,σ,0)

−1

A(u

−

,τ,iη)



× E



,σ,0)

−1

A(u

,τ,iη)



The dimension of these spaces is constant over the connected set {(τ,η); Re τ>

0 ,η∈ R

d−1

}. So it can be computed at (τ,0). We easily see that a complex

number ω is an eigenvalue of

(v, σ, 0)

−1

A(v, τ, 0) = τ ( A

(v) − σA

(v))

−1

(v)

if and only if ω = τ/(λ − σ), where λ is a root of

det( A

(v) − λA

(v)) = 0,

that is, λ is a characteristic speed of the operator

(v) ∂

+ A

(v)∂

More precisely, we have the following.

Proposition 12.2 Assume that the operator

L = A

∂

+ A

∂

is hyperbolic in the t-direction, and that both A

and A

− σA

are non-singular.

Then the dimension of the stable subspace E



( A

− σA

)

−1



is equal to

the number, counted with multiplicity, of characteristic speeds of L less than σ.

Proof The hyperbolicity of L means that there exist a real diagonal matrix Λ

and a real non-singular matrix P such that A

P = A

P Λ. Since A

− σA

non-singular, we may assume without loss of generality that Λ splits into a ﬁrst

block, say of size k, with coeﬃcients less than σ, and another one with coeﬃcients

greater than σ. By deﬁnition,



( A

− σA

)

−1



= {h ∈ C

; lim

x→+∞

x( A

− σA

)

−1

h =0}.

Normal modes analysis 341

Denoting by φ(x, h):=e

x( A

− σA

)

−1

h the ﬂow of the diﬀerential equation



=(A

− σA

)

−1

we see that ϕ := P

−1

φ is the ﬂow of



=(Λ− σI

)

−1

Indeed, since A

is non-singular,

− σA

) ∂

φ = A

φ ⇔ (Λ − σI

) ∂

ϕ = ϕ.

Therefore, h belongs to E



( A

− σA

)

−1



if and only if P

−1

h belongs to

(Λ − σI

)

−1

, which is clearly of dimension k. 

Corollary 12.1 Assume that (12.1.1) is constantly hyperbolic. Then for all

(τ,η) ∈ C × R

d−1

with Re τ>0, the dimension of E

(A(u, η, τ)) is equal to the

number of characteristics exiting the shock, that is, the number of characteristic

speeds of A

−

) ∂

+ A

−

)∂

less than σ plus the number of characteristic

speeds of A

) ∂

+ A

)∂

greater than σ, all counted with multiplicity.

12.2.2 Nature of shocks

The result of Corollary 12.1 is illustrated in Fig. 12.1 for a Lax shock, in which

the dimension of E

(A(u, η, τ)) is (p − 1) + n − p = n − 1. Indeed, we recall

that Lax shocks [109] are deﬁned as follows.

Deﬁnition 12.1 Assume that (12.1.1) is constantly hyperbolic, and denote by

(u, ν) ≤ ··· ≤ λ

(u, ν)

its characteristic speeds, that is, the roots (repeated according to their multi-

plicities) of det( A(u, ν) − λA

(u)). A planar shock wave between u

−

and u

propagating with speed σ in some direction ν ∈ R

, is called a Lax shock if there

is some integer p ∈{1,...,n} such that the Lax shock inequalities

,ν) <σ<λ

−

,ν) and λ

p−1

−

,ν) <σ<λ

p+1

,ν) (12.2.15)

are satisﬁed. (By convention, λ

= −∞ and λ

n+1

=+∞ when p =1or p = n.)

Remark 12.3 In space dimension d = 1, a shock wave of the form

u(x, t)=u

for x ≷ σt

satisfying (12.2.15) with ν = 1 is called a p-shock. In space dimension d ≥ 2,

there is no ‘natural’ choice for left and right. Indeed, a shock between u

−

and

propagating with speed σ in the direction ν may be equivalently regarded as

a shock between u

and u

−

propagating with speed −σ in the direction −ν.In

gas dynamics, for instance, in which the characteristic speeds are

(u, ν)=u · ν − c ν,λ

(u, ν)=u · ν and λ

(u, ν)=u · ν + c ν

342 Persistence of multidimensional shocks

λp+1

p−1

−

Figure 12.1: Characteristic speeds (λ

) / shock speed (σ) for a p-Lax shock

(where u is the ﬂuid velocity and c is the sound speed), the inequalities in

(12.2.15) with p = 1 are equivalent, through the change of notations

−

,σ,ν) ↔ (u

−

, −σ, −ν),

to the inequalities in (12.2.15) with p = 3. This shows that speaking of a p-

shock in several space dimensions is meaningless. What is most important is to

distinguish between the states on either side of the discontinuity by means of an

intrinsic criterion (in gas dynamics, one may speak about the state behind the

shock with respect to the ﬂow of the gas, see Section 13.4).

Proposition 12.3 Assume that (12.1.1) is constantly hyperbolic, and that the

matrices A

), ( A

) − σA

)) are non-singular. Then the dimension

of E

(A(u, η, τ)) for (τ,η) ∈ C × R

d−1

with Re τ>0 is equal to n − 1 if and

only if u is a Lax shock.

Proof The ‘if’ part has already been pointed out. The ‘only if’ part is also easy.

If the dimension of E

(A(u, η, τ)) is n − 1, there must be an integer p ∈{1,...,d}

such that σ<λ

−

,ν) (otherwise, there would be at least n characteristics

exiting the shock, contradicting Corollary 12.1). Assume that p is the smallest

one. Similarly, there must be an integer q ∈{1,...,d} such that λ

−

,ν) <σ.

Assume that q is the greatest one. Then we have

,ν) <σ<λ

−

,ν)andλ

p−1

−

,ν) <σ<λ

q+1

,ν)

Normal modes analysis 343

(with the convention λ

= −∞ or λ

d+1

=+∞ if either p =1orq = d). These

inequalities imply that the number of characteristics exiting the shock is (p −

1) + n − q, which is of course equal to (n − 1) only if p = q. In this case, the

previous inequalities are precisely the Lax-shock inequalities (12.2.15). 

People aware of Majda’s work and in particular of Proposition 12.3, used

to consider non-Laxian shocks – now called non-classical shocks – as unstable

and thus of no interest. At ﬁrst glance, this is indeed a tempting conclusion. For,

having E

(A(u, η, τ)) of dimension (n − 1) is necessary for the non-homogeonous

boundary value problem











= A(u, η, τ)

U + F for z>0 ,

Xb(u, τ, iη) − M(u, σ, 0)

U = G at z =0

(12.2.16)

to be well-posed in L

) – a condition needed for the well-posedness of

the original, free boundary value problem: if dimE

(A(u, η, τ)) = n − 1, either

(A(u, η, τ)) is too big, and the problem (12.2.16) suﬀers from non-uniqueness,

or E

(A(u, η, τ)) is too small and (12.2.16) has no solution in L

). However,

non-classical shocks are often physically relevant. So what is the trick?

When the dimension of E

(A(u, η, τ)) is greater than (n − 1), the shock is

called undercompressive – a term inspired from gas dynamics, meaning that there

are more characteristics exiting the shock than for the usual, compressive shocks

of Lax type. Then the homogeneous problem (12.2.16) with F =0andG =0

admits non-trivial solutions (

X) ∈ L

) × C. Those solutions are given by

U(z)=e

z A(u,η,τ )

;

Xb(u, τ, iη)=M(u, σ, 0)

with

∈ E

(A(u, η, τ)). There are non-trivial ones just because the linear

algebraic system of n equations

M(u, σ, 0)

U −

Xb(u, τ, iη) = 0 (12.2.17)

is underdetermined in E

(A(u, η, τ)) × C. In fact, this just means that the

Rankine–Hugoniot conditions are not suﬃcient as jump conditions for under-

compressive shocks: they should be supplemented with extra jump conditions

(also called kinetic relations, see [113]), based on further modelling arguments;

this is the role played by the so-called viscosity-capillarity criterion introduced in

the 1980s by Slemrod [196], and independently by Truskinosky [214], for phase

boundaries). It was pointed out by Freist¨uhler [58–60] that well-chosen extra

jump conditions could indeed restore stability of undercompressive shocks (for an

application to subsonic liquid-vapour interfaces, see [9,10,12]). The detailed proof

of the persistence of undercompressive shocks has been done by Coulombel [40],

by adapting M´etivier’s method [136].

When E

(A(u, η, τ)) has dimension less than (n − 1), the algebraic system

(12.2.17) is overdetermined on E

(A(u, η, τ)) × C and the shock is termed

344 Persistence of multidimensional shocks

overcompressive: it turns out – see [61, 120] – that the stability analysis of

overcompressive shocks must include viscous eﬀects, but this is far beyond the

scope of this book.

In what follows, we concentrate on Lax shocks for simplicity. As regards

undercompressive shocks, techniques are similar, but there are more jump con-

ditions to deal with: the reader is referred in particular to [40, 58–60] for more

details.

12.2.3 The generalized Kreiss–Lopatinski˘ı condition

A natural extension to (12.2.14) of the uniform Kreiss–Lopatinski˘ı condition is

(UKL

) there exists C>0 so that for all

(η, τ) ∈ P

:= {(η,τ) ∈ R

× C with Re τ ≥ 0and η

+ |τ|

=1},



U≤C Π(u, τ, iη) M(u, σ, 0)

U  (12.2.18)

for all

U ∈ E

(A(u, η, τ)), the stable subspace of A(u, η, τ )ifReτ>0,

extended by continuity to imaginary values of τ.

As for standard IBVP, it is not easy to prove directly the estimate in (12.2.18).

A preliminary step is of course to construct the stable subspace E

(A(u, η, τ)),

at ﬁrst for Re τ>0. A second step would be to formulate the existence of non-

trivial solutions

U ∈ E

(A(u, η, τ)) of the algebraic system

Π(u, τ, iη) M(u, σ, 0)

U =0

as being equivalent to the vanishing of a (Lopatinski˘ı) determinant ∆

(τ,η),

depending analytically on (τ,η) (and being C

∞

with respect to the parame-

ters (u, σ)). Then, one would check (UKL

) by showing that the function ∆

extended carefully by continuity up to the boundary of P

, does not have any

zero in P

. Indeed, this would mean that for all (η, τ) ∈ P

the linear mapping

Π(u, τ, iη) M(u, σ, 0) is invertible when restricted to E

(A(u, η, τ)), hence the

inequality (12.2.18) with a uniform C on the compact set P

. (Observe this also

implies (12.2.18) with the same constant C for all (τ, η) =(0, 0) with Re τ ≥ 0,

since both E

(A(u, η, τ)) and Π(u, τ, iη) are by deﬁnition homogeneous degree 0

in (τ,η).)

As we said before, handling the projection operator Π(u, τ, iη)isnotvery

convenient. Furthermore, the condition (UKL

) above is in fact not suﬃcient for

the well-posedness of the complete problem (12.2.16). A more complete condition,

which contains both (UKL

) and the ellipticity of the symbol b is the following.

(UKL) there exists C>0 so that for all (η, τ) ∈ P

max( 

U, |

X|) ≤ C M(u, σ, 0)

U −

Xb(u, τ, iη)  (12.2.19)

for all (

X) ∈ E

(A(u, η, τ)) × C .