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

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Well-posedness of linearized problems 365

R; R

), ˙v

|z=0

belongs to H

; R

),and(∂

˙χ, ∇

˙χ) belongs to H

; R

and altogether they satisfy the estimate

γ ˙v

d−1

×R

×R)

+ ˙v

|z=0



)

+  ˙χ

m+1

)

≤ C

(µ)



f

d−1

×R

×R)

+ g

)



for γ ≥ γ

(µ) ≥ 1,whereµ is an upper bound for the H

norm of the coeﬃ-

cients; more precisely,

u − u



≤ µ, u

|z=0

− u

|z=0



≤ µ and (∂

χ − σ, ∇

χ)

≤ µ.

Proof As for Theorem 9.20 on standard BVP, it is in two steps: 1) the proof for

inﬁnitely smooth coeﬃcients and 2) the extension to H

coeﬃcients by passing

to the limit in regularized problems (to which the ﬁrst step applies).

Step 1) Assume here that u belongs to D(R

d−1

× R

× R; R

)and(∂

χ −

σ, ∇

χ) belongs to D(R

; R

We already know from Theorem 12.4 that v := e

−γt

˙v belongs to L

d−1

× R), v

|z=0

belongs to L

), and ψ := e

−γt

˙χ belongs to H

). Our aim

is to prove enough regularity on (v,ψ) to be allowed to use Theorem 12.2 and

thus obtain H

estimates on ( ˙v, ˙χ)intermsofµ. A ﬁrst idea is to proceed

by induction and use the same method as in the proof of Theorem 9.10. So,

assume that v belongs to H

d−1

× R

× R), v

|z=0

belongs to H

), and

ψ belongs to H

k+1

) for 0 ≤ k ≤ m − 1. We want to show that v belongs to

k+1

d−1

× R

× R), v

|z=0

belongs to H

k+1

), and ψ belongs to H

k+2

In fact, thanks to Proposition 2.3 applied with z instead of t, it is suﬃcient

to show that v belongs to L

; H

k+1

)). We are going to use several

ingredients: the characterization of H

k+1

functions given by Proposition 9.2;

the two-sided inequality provided by Proposition 9.3; the commutators estimates

given by Theorem 9.11; the energy estimate (12.3.38). We introduce a smoothing

operator R

in the (y, t) directions satisfying the requirements of Proposition

9.3 and deﬁne ϕ

:= R

(v) ∈ L

; H

+∞

)), ψ

:= R

(ψ) ∈ H

+∞

). By

Proposition 2.3 again, ϕ

belongs at least to H

d−1

× R

× R), and therefore

(by Remark 12.4) we can apply the energy estimate (12.3.38) to the pair (ϕ

,ψ

This gives for γ large enough (and in particular γ ≥ 1),

γ ϕ



×R

)

+ (ϕ

)



)

+ ψ



)



L



×R

)

+ B

((ϕ

)

,ψ

)

)

Now, Theorem 9.11 provides some bounds for the commutators [B

]and

], with (as usual) P

= ∂

− (A

)

−1

. Since (A

)

−1

is uniformly

366 Persistence of multidimensional shocks

bounded, this eventually gives



+∞

v

k,θ

+ v



k,θ

+ ψ

k+1,θ





+∞

v

k,θ

+ γ



+∞

v

m−1,θ



+∞

k,θ

v(x

)) dx

+ n

k,θ

,ψ)) + v



k−1,θ

+ ψ

k,θ

For γ large enough the ﬁrst term in the right-hand side can be absorbed in the

left-hand side. Therefore, applying Propostions 9.2 and 9.3, we get



+∞

v

k,θ

+ v



k,θ

+ ψ

k+1,θ



L

v

k+1

))

+ B

,ψ)

k+1

)

+ γ v

k+1

))

+ v



)

+ ψ

k+1

By assumption (since k ≤ m − 1), the right-hand side is ﬁnite, and of course is

independent of θ, Proposition 9.2 thus shows that v belongs to L

; H

k+1

))

(hence v belongs to H

k+1

d−1

× R

× R) by Proposition 2.3), v

belongs to

k+1

), and ψ belongs to H

k+2

). Therefore, the induction process shows

that v belongs to H

d−1

× R

× R), v

belongs to H

), and ψ belongs

to H

m+1

). The regularity obtained in this way for (v,ψ) is not suﬃcient yet

to apply Theorem 12.2 and get H

estimates. However, we can construct f

∈

D(R

d−1

× R

× R)andg

∈ D (R

d−1

× R) such that f

goes to f in H

d−1

× R)andg

goes to g in H

d−1

× R), and by the ﬁrst step the solution

,χ

)oftheBVP

= f

for z>0 ,B

,χ

)=g

at z =0

is smooth enough to satisfy the H

estimate (12.3.30): by linearity, this is also

true for the diﬀerences (v

− v



,χ

− χ



); hence the convergence of (v

)

ε>0

d−1

× R

× R),andof((v

)

ε>0

and (χ

)

ε>0

in H

d−1

× R); the

limit must be ( ˙v, ˙χ) since it is a solution of the same BVP, and by passing to the

limit in the estimate (12.3.30) applied to (v

,χ

) we get the estimate for ( ˙v, ˙χ).

Step 2): It is a matter of smoothing coeﬃcients and passing once more to the

limit. We omit the details, which are identical to those in the proof of Theorem

9.20 for the standard BVP. 

12.3.4 The IBVP with zero initial data

In this section we ﬁx T ∈ R and we denote by I

the half-line (−∞,T].

Theorem 12.6 Under the hypotheses (CH), (NC), (N) and (UKL)

of Theorem 12.1, for all Lipschitz-continuous u

:(y, z, t) ∈ R

d−1

× R

→ u

(y, z, t) ∈ R

and dχ :(y, t) ∈ R

d−1

× I

→ (∂

χ(y, t), ∇

χ(y, t)) ∈

Well-posedness of linearized problems 367

R × R

d−1

,with

u

− u



∞

d−1

×R

×I

)

≤ ρ and dχ − (σ, 0)

∞

d−1

×I

)

≤ ρ,

u



1,∞

d−1

×R

×R)

≤ ω and dχ

1,∞

d−1

×I

)

≤ ω,

for all f ∈ L

d−1

× R

× I

; R

) and all g ∈ L

d−1

× I

; R

) such that

|t<0

=0,g

|t<0

=0,

there exists a unique pair (˙v, ˙χ) ∈ L

d−1

× R

× I

; R

) × H

d−1

; R

) such that

L(u, dχ)˙v = f, B(u) · d˙χ + M(u, dχ) · ˙v

|z=0

= g, and ˙v

|t<0

=0, ˙χ

|t<0

=0.

Furthermore, ˙v

|z=0

=0belongs to L

d−1

× I

; R

), and there exist γ

≥ 1

and c>0 depending continuously on ω such that for all γ ≥ γ

γ e

−γt

˙v

d−1

×R

×I

)

+ e

−γt

˙v

|z=0



)

+ e

−γt

˙χ

d−1

×I

)

≤ c



e

−γt

f

d−1

×R

×I

)

+ e

−γt

g

d−1

×I

)



(12.3.39)

The proof is analogous to the proof of Theorem 9.18 for the ‘standard’ IBVP

with zero initial data: it relies on Theorem 12.4 and on the following support

theorem.

Theorem 12.7 In the framework of Theorem 12.4, if both f and g vanish for

t<t

then so do ˙v and ˙χ.

(To prove this result, proceed as for Theorem 9.13, using the energy estimate

(12.3.35) shown by Theorem 12.4; also see [140], p. 63–64.)

A reﬁned version of Theorem 12.6, with non-zero initial data but for

Friedrichs-symmetrizable systems, is proved by M´etivier in [136], Section 3.

For smoother coeﬃcients, we have the following.

Theorem 12.8 In the framework of Theorem 12.6, assume, moreover, that

u − u

belongs to H

d−1

× R

× I

; R

), (u − u)

|z=0

belongs to H

d−1

; R

) and (∂

χ − σ, ∇

χ) belongs to H

d−1

× I

; R

) for some integer m>

(d +1)/2+1, with (u − u

)

|t<τ

≡ 0, (∂

χ − σ, ∇

χ)

|t<τ

≡ 0 for some τ<T,

and

u − u



d−1

×R

×I

)

≤ µ, u

|z=0

− u

d−1

×I

)

≤ µ

(∂

χ − σ, ∇

χ)

d−1

×I

)

≤ µ.

368 Persistence of multidimensional shocks

Also assume that f belongs to H

d−1

× I

; R

) and g belongs H

d−1

; R

), with still

|t<0

=0,g

|t<0

=0.

Then ˙v belongs to H

d−1

× R

× I

; R

), ˙v

|z=0

belongs to H

d−1

; R

), ˙χ belongs H

m+1

d−1

× I

; R

), and they satisfy an estimate

˙v

d−1

×R

×[0,T ])

+ ˙v

|z=0



d−1

×[0,T ])

+  ˙χ

|z=0



m+1

d−1

×[0,T ])

≤ T f

d−1

×R

×[0,T ])

+ g

d−1

×[0,T ])

for C>0 depending only on and continously on µ.

The proof relies on Theorem 12.5 for the BVP: it is completely analogous to

the proof of Theorem 9.21 (in which the counterpart of Theorem 12.5 is Theorem

9.20) and is therefore omitted.

12.4 Resolution of non-linear IBVP

12.4.1 Planar reference shocks

The local-in-time existence (and uniqueness) of perturbed shocks front solutions

near uniformly stable planar Lax shocks was ﬁrst shown by Majda [124]. We will

give below a revisited, improved version of Majda’s theorem, which is due to

M´etivier and coworkers, as can be found (partially) in [140] (Section 4) and in

M´etivier’s lecture notes [136].

Before giving a precise statement, we simplify a little the problem by assuming

the normal speed of the reference, planar shock is σ = 0 (this just amounts

to making the change of frame x

→ x

− σt); in other words, we assume the

reference shock location is given by x

= χ(y,t) = 0. Then we use Remark 12.1

on the way of ﬁxing the unknown boundary (which is supposedly close to the

hyperplane {x

=0}). Hence, instead of (12.1.3) and (12.1.4), we are led to

consider the (slightly more complicated) BVP

d−1



j=0

) ∂

+ A

, dΨ

) ∂

=0, for z>0, (12.4.40)

d−1



j=0

( f

) − f

−

))∂

χ =(f

) − f

−

)) at z =0, (12.4.41)

(y, z, t)=±κz + ϕ(z) χ(y, t) , for z ≥ 0 , (12.4.42)

where ϕ ∈ D (R) is a cut-oﬀ function (equal to one on some interval [0,z

]), and

(v, ξ

,ξ

,...,ξ

d−1

,ξ

):=

(v, ξ

,ξ

,...,ξ

d−1

Resolution of non-linear IBVP 369

Clearly, a solution of (12.4.40)–(12.4.42) with

χ

∞

≤ η :=

2ϕ





∞

gives a solution of the original FBP, as formulated in Section 12.1.

To actually solve (12.4.40)–(12.4.42) with given initial data u

=(u

−

)

and χ

, we shall (unsurprisingly) need compatibility conditions, which (unsur-

prisingly) are uglier than for standard IBVP. The principle of their derivation is,

nevertheless, very simple, as we now explain.

On the one hand, thanks to Lemma 12.2, the jump conditions in (12.4.40)

can be rewritten equivalently as







∂

χ = q(u)

∇

χ = r(u)

0=s(u)

with





q(u)

r(u)

s(u)





= F (u):=Q(u

−

)(f

) − f

−

)),

where Q ∈ C

∞

(W ; GL

(R)), W being a neighbourhood of (u

−

). If both χ

and u are smooth enough, the ﬁrst equality (∂

χ = q(u

|z=0

)) implies by Fa´adi

Bruno’s formula,

∂

p+1

χ =



m=1



+···+i

,...,i

q ◦ u

|z=0

) · (∂

u,...,∂

|z=0

On the other hand, the interior equations in (12.4.40) can be rewritten in a

more compact way

(u) ∂

u +

d−1



j=0

(u) ∂

u + A

(u, dΨ) ∂

u =0,

with our usual blockwise deﬁnitions of A

for j ≤ d − 1, and a revisited deﬁnition

for A

(which would coincide with the former deﬁnition in the special case κ =1

and ϕ ≡ 1), namely

(u, dΨ) =



−

, dΨ

−

0 A

, dΨ

)



Since A

(u) is invertible, the interior equations are thus equivalent to

∂

u = −

d−1



j=0

(u))

−1

(u) ∂

u − (A

(u))

−1

(u, dΨ) ∂

which we may diﬀerentiate p times if u and Ψ (or equivalently χ) are smooth

enough. This yields

∂

p+1

u = −



=0









j=0

∂



(u, dΨ)) ∂

∂

p−

370 Persistence of multidimensional shocks

where (to save room and hopefully the reader’s nerves) we have used uniﬁed

notations, namely ∂

= ∂

, ∂

= ∂

, B

(u, dΨ) := (A

(u))

−1

(u)forj ≤ d − 1

and B

(u, dΨ) := (A

(u))

−1

(u, dΨ).

Applying again Fa´a di Bruno’s formula to the derivatives ∂



(u, dΨ)), we

are led to the following formulation of compatibility conditions up to order s.

(CC

) A pair of initial data (u

,χ

) is said to be compatible up to order s if

the functions (u

,χ

) deﬁned inductively by











= q ◦ (u

)

|z=0

= −



j=0

, dΨ

) ∂

p+1



m=1





+···+



,...,

q ◦ (u

)

|z=0

) · (u



,...,u



)

|z=0

p+1

= −



j=1

, dΨ

) ∂

−



=1









j=1





k=1





+···+

=



,...,

) · (u



,...,u



) ∂

p−

(12.4.43)

with the (obvious) notations



=(u



, dΨ



) , Ψ



=(Ψ

−



, Ψ



) , Ψ



(y, z, t)=±κz + ϕ(z) χ



(y, t),

are such that, for all p ∈{0,...,s},











∇



m=1





+···+



,...,

r ◦ (u

)

|z=0

) · (u



,...,u



)

|z=0



m=1





+···+



,...,

s ◦ (u

)

|z=0

) · (u



,...,u



)

|z=0

(12.4.44)

Theorem 12.9 Under the assumptions of Theorem 12.1 with σ =0, there exists

ρ>0 so that for all u

=(u

−

) ∈ u + H

m+1/2

d−1

× R

; R

) and all χ

∈

m+1/2

d−1

) with m an integer greater than (d +1)/2+1, compatible up to

order m − 1, and such that

u

− u



∞

d−1

×R

)

≤ ρ and ∇



∞

d−1

)

≤ ρ,

Resolution of non-linear IBVP 371

there exists T>0 and a solution (u, χ)=(u

−

,χ) of (12.4.40)–(12.4.42)

such that

= u

and χ = χ

at t =0,

with u ∈ u

+ H

d−1

× R

× [0,T]; R

) taking values in U := U

−

× U

,with

κ =4ϕ





∞

χ



∞

,andχ ∈ H

d−1

× [0,T]) taking values in the ball of

radius η =2χ



∞

Proof It is very similar to the proof of Theorem 11.1 on the ‘standard’ IBVP. It

starts with the construction of an ‘approximate solution’, and uses an iterative

scheme to solve the IBVP with zero initial data (obtained by subtracting the

approximate solution): thanks to our knowledge of linear IBVP with zero initial

data (here Theorem 12.8), the iterative scheme can be shown to converge on

small enough time intervals.

On the one hand, the interior equations in (12.4.40) merely read

L(u, dΨ)u =0, with L(u

dΨ) := A

(u) ∂

d−1



j=1

(u) ∂

+ A

(u, dΨ) ∂

the matrices A

and A

being deﬁned as above. This induces us to consider the

iterative scheme

L(u

+ v

, dΨ

+dΦ

k+1

= −L(u

+ v

, dΨ

+dΦ

, (12.4.45)

where (u

, Ψ

) is an ‘approximate solution’ (to be speciﬁed in Lemma 12.4 below)

of the IBVP associated with (12.4.40)–(12.4.42) and the initial data (u

,χ

On the other hand, the boundary conditions (12.4.40) can be written as

J∇χ + F (u

|z=0

)=0, (12.4.46)

with J the constant n × d matrix consisting of the identity I

∈ M

n×n

(R)and

a zero block underneath, and F deﬁned as above (thanks to Lemma 12.2) by

F (u)=Q(u

−

)(f

) − f

−

)) for u ∈ W , a contractible open subset

of U . (In the following we shall assume, up to reducing it, that W is convex,

for example, a ball centered at u

.) By linearity of (12.4.46) in χ,weareledto

consider the induction formula

J∇χ

k+1

+dF ((u

+ v

)

|z=0

k+1

|z=0

(12.4.47)

= −J ∇χ

+dF ((u

+ v

)

|z=0

− F ((u

+ v

)

|z=0

with χ

=Φ

|z=0

, χ

=Φ

|z=0

Remark 12.5 As pointed out by M´etivier [136], it is possible to make a change

of unknowns u → u so that the boundary conditions (12.4.46) become linear

also in u. Indeed, on the manifold

M := {(u

−

,ξ) ∈ W × R

; ξ≤ρ and Jξ + F (u

−

)=0}

372 Persistence of multidimensional shocks

the mapping (u

−

,ξ) → (u

−

:= F (u

−

), u

:= u



ξ := ξ) is (for ρ small

enough) a local diﬀeomorphism (whose diﬀerential at (u

−

,ξ) is the mapping

(˙u

−

, ˙u

ξ) → (−Q(u

−

) A

−

,ξ) · ˙u

−

, ˙u

ξ)). So, up to reducing W and ρ,

we may assume it is a diﬀeomorphism on M . Then (12.4.46) is equivalent for

−

) ∈ W and ∇χ≤ρ to

J∇χ +Πu =0,

with u := (u

−

, u

) and Π merely the projection operator Π : (u

−

, u

) ∈ R

→

u

−

∈ R

. This way of rewriting the boundary conditions (12.4.46) would simplify

the iterative scheme (12.4.47) into

J∇χ

k+1

+Πv

k+1

|z=0

= −J ∇χ

− Πu

|z=0

However, we refrain from using this simpliﬁcation because it is too speciﬁc to

Lax shocks.

Before studying the iterative scheme (12.4.45) and (12.4.47) we must specify

the approximate solution (u

, Ψ

Lemma 12.4 Under the assumptions of Theorem 12.9, choose ρ

> 0 so that

w − u



≤ ρ

implies w ∈ W ⊂ U .

There exists T

> 0 and u

∈ u + H

m+1

d−1

× R

× R), taking values in U ,

∈ H

m+1

) with both u

− u and χ

vanishing for |t|≥2T

, such that

)

|t=0

= u

, (χ

)

|t=0

= χ

u

(y, z, t) − u

(y, z)≤

, χ

(y, t) − χ

(y)≤

for all (y, z, t) ∈ R

d−1

× R

× [−T

] and additionally, f

:= −L(u

, dΨ

with

=(Ψ

−

, Ψ

) , Ψ

(y, z, t)=±κz + ϕ(z) χ

(y, t) , (12.4.48)

and g

:= −J ∇χ

− F (u

|z=0

) are such that

∂

≡ 0, ∂

≡ 0 at t =0 for all p ∈{0,...,m− 1}.

Furthermore, f

belongs to H

d−1

× R

× R), g

belongs to H

),and

both vanish for |t|≥2T

Proof Similarly as in Lemma 11.1 we can construct u

∈ H

m+1/2−i

d−1

× R)andχ

∈ H

m+1/2−i

) satisfying (12.4.43) for all i ∈{1,...,m− 1}.

Then, by trace lifting (see, for instance, [1], pp. 216–217), we ﬁnd u

∈ u +

m+1

d−1

× R

× R)andχ

∈ H

m+1

) such that

u

− u

m+1

d−1

×R

×R)

 u

− u

m+1/2

d−1

×R

)

χ



m+1

)

 χ



m+1/2

d−1

)

Resolution of non-linear IBVP 373

and

∂

)

|t=0

= u

,∂

(χ

)

|t=0

= χ

for all i ∈{0,...,m− 1}.

By Sobolev embeddings we ﬁnd T

> 0sothat

u

(x, t) − u

(x)≤

and χ

(y, t) − χ

(y)≤

for |t|≤T

. Furthermore, if ϕ

∈ D (R) is a cut-oﬀ function such that ϕ

(t)=1

for |t|≤T

and ϕ

(t)=0for|t|≤2T

, we may replace u

by (1 − ϕ

) u + ϕ

which takes values in the convex set W . Similarly, we may replace χ

by ϕ

and deﬁne Ψ

by (12.4.48). That f

:= −L(u

, dΨ

belongs to H

d−1

× R) follows from Proposition C.11 and Theorem C.12: observe that f

−L(u

, dΨ

)(u

− u) and use that u

− u and dΨ

both belong to H

d−1

× R). Similarly, we ﬁnd that g

:= −J ∇χ

− F (u

|z=0

) belongs to H

)

because ∇χ

and u

|z=0

− u do so and F (u) = 0. Finally, the time derivatives

of f

and g

vanish at t = 0 up to order m − 1 thanks to the compatibility

conditions in (12.4.43) and (12.4.44) with s ≤ m − 1. 

We can now analyse the iterative scheme (12.4.45) and (12.4.47) with the

pair (u

, Ψ

) given by Lemma 12.4. We introduce the following shorcuts:

v := (v, Φ) ,f

:= −



t>0



L(u

+ v, dΨ

+dΦ)u

and g

:= (1

t>0

)(−J ∇χ

+dF ((u

+ v)

|z=0

) · v

|z=0

− F ((u

+ v)

|z=0

)).

Lemma 12.5 Under the assumptions of Lemma 12.4, there exists M>0

(depending on ρ

and η) so that for all T ∈ (0,T

], denoting by I

the half-line

(−∞,T], for all v ∈ H

d−1

× R

× I

) having a trace v

|z=0

∈ H

d−1

) and for all Φ ∈ H

m+1

d−1

× I

) with

|t<0

≡ 0 , Φ

|t<0

≡ 0,

and v

d−1

×R

×I

)

≤M, v

|z=0



d−1

×I

)

≤M, Φ

m+1

d−1

×I

)

≤ M, we have f

∈ H

d−1

× R

× I

) and g

∈ H

d−1

× I

) and there

exist continuous functions M → C

(M), M → C

(M) and T → ε(T ), the latter

going to zero as T goes to zero, so that

f



d−1

×R

×I

)

≤ C

(M) and g



d−1

×I

)

≤ TC

(M)+ε(T ).

We omit the proof, which is mostly similar to that of Lemma 11.3 (also,

see [140] pp. 94–96). 

From now on, we ﬁx M as in Lemma 12.5 and we show how to complete

the proof of Theorem 12.9. Setting v

≡ 0andΦ

≡ 0, thanks to (12.4.45)

and (12.4.47), Lemma 12.5 and Theorem 12.8, we can construct by induction a

374 Persistence of multidimensional shocks

sequence (v

, Φ

) such that, up to diminishing T ,











∈H

d−1

×R

×I

),v

|z=0

∈ H

d−1

×I

), Φ

∈H

m+1

d−1

×I

)

with v



d−1

×R

×I

)

≤ M, v

|z=0



d−1

×I

)

≤ M,

Φ



m+1

d−1

×I

)

≤ M, and v

|t<0

≡ 0 , Φ

|t<0

≡ 0 .

(12.4.49)

Then, thanks to Theorem 12.6, we can show (v

), (v

|z=0

) and (Φ

)are,for

T small enough, Cauchy sequences in L

d−1

× R

× I

), L

d−1

× I

)and

d−1

× I

), respectively. By standard weak compactness and interpolation

arguments, we conclude that their limits are in the appropriate Sobolev spaces

and solve the original problem. For more details, the reader may refer to Section

11.2.2, which is mostly similar. 2

For Friedrichs-symmetrizable systems, M´etivier has improved Theorem 12.9

as follows.

Theorem 12.10 In the framework of Theorem 12.9, assume, moreover, that the

operator A

(w) ∂



j=1

(w) ∂

is Friedrichs symmetrizable for w in neigh-

bourhoods of u

. Then there is a unique solution in CH

× H

m+1

× [0,T]),

with

:= {v ∈ D



d−1

× R

× [0,T]) ; ∂

v ∈ C ([0,T]; H

m−p

d−1

× R

))

p ≤ m },

even for data u

∈ H

d−1

× R

; R

), χ

∈ H

d−1

The proof is based on reﬁned energy estimates, see [136] (Section 4).

12.4.2 Compact shock fronts

To stay as close as possible to earlier notations, we consider now a compact

codimension one surface Σ

in R

, and denote by n the outward unit normal

vector to Σ

. We also consider initial data u

being smooth (in a sense to be

speciﬁed) outside Σ

and experiencing a jump discontinuity across Σ

compatible

(at least at zeroth order) with the Rankine-Hugoniot condition. This means that

for all y ∈ Σ

there exists σ(y) ∈ R such that



j=1

(y)[f

)](y)=σ(y)[f

)](y).

Here above,

(u)](y) = lim

ε0

( f

(y + εn(y))) − f

(u(y − εn(y))) ).