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 345

(Observe that unlike (12.2.18), the inequality (12.2.19) is not homogeneous, but

this will be harmless.)

The condition (UKL) is what we call the uniform Kreiss–Lopatinski˘ı condition

for the planar shock wave u of normal speed σ. As claimed above, (UKL) implies

(by just taking

U =0)

b(u, τ, iη) ≥1/C > 0,

hence also (12.2.18) by inserting

X = −

b(u, τ, iη)

∗

M(u, σ, 0)

b(u, τ, iη)

into (12.2.19).

In practice, the veriﬁcation of (UKL) is far from being straightforward.

Nevertheless, we claim it is mostly algebraic. The condition (UKL) is indeed

equivalent to the non-existence, for all (η, τ) in the compact set P

,ofnon-

trivial solutions in E

(A(u, η, τ)) × C to the algebraic system (12.2.17). This

property can be formulated as the absence of zeroes in P

of an analytic function

∆of(τ,η), depending smoothly on the shock wave u. And it can be shown

that the zero set of ∆ is contained in an algebraic manifold, say M .(See

Chapter 4 for theoretical explanations, and Chapter 15 for the example of gas

dynamics.) The analytical parts in the veriﬁcation of (UKL) are thus reduced to

the determination of the continuous extension of ∆ to purely imaginary values

of τ, and the elimination of fake zeroes of ∆ from P

∩ M . This is done in detail

in Chapter 15 for the gas dynamics (for a more analytical approach on the same

topic, see [92]). For more general systems, it is only known that small shocks are

stable, as was pointed out by M´etivier in [131] in the case n =2andprovedin

more generality in [133].

12.3 Well-posedness of linearized problems

12.3.1 Energy estimates for the BVP

In this section, we consider the linear BVP







, dχ)˙v

= f

,z>0 ,

B(u) · (d ˙χ, 0) + M(u, dχ) · (˙v

−

, ˙v

)=g, z =0.

It comes from (12.1.10) and (12.1.11) where we have sent the zeroth-order terms

in ˙v

and ˙χ to the (arbitrary) right-hand sides f

and g. (Recall that the good

unknowns ˙v

are merely related to ˙u

and ˙χ through the relation ˙v

=˙u

∓

˙χ∂

.) For simplicity, we just write this BVP as







L(u, dχ)˙v = f, z > 0 ,

B(u) · d˙χ + M(u, dχ) · ˙v = g, z =0,

(12.3.20)

346 Persistence of multidimensional shocks

where ˙v =(˙v

−

, ˙v

), the operator L(u, dχ) is to be understood as

L(u, dχ)=



−

, dχ)0

0 L

, dχ)



and B(u) · d˙χ stands for B(u) · (d ˙χ, 0) by a slight abuse of notation.

Theorem 12.1 We make the following main assumptions.

(CH) There exist open subsets of R

, say U

−

and U

containing, respec-

tively u

−

and u

, such that for all w ∈ U

the matrix A

(w) is

non-singular, and the operator A

(w) ∂



j=1

(w) ∂

is constantly

hyperbolic in the t-direction;

(NC) There exists σ>0 so that

) − f

−

)=σ ( f

) − f

−

))

and both matrices

,σ,0) = A

) − σA

)

are non-singular;

(N) The associated discontinuous solution of (12.1.1),

:(x, t) → u

for x

≷ σt,

is a Lax shock (according to Deﬁnition 12.1);

(UKL) There exists C>0 so that for all (η, τ) ∈ R

d−1

× C with Re τ ≥ 0

and |τ|

+ η

=1,

max( 

U, |

X|) ≤ C M(u

,σ,0)

U −

Xb(u,τ,iη)  (12.3.21)

for all (

X) ∈ E

(A(u,η,τ)) × C (with M, b and E

(A) deﬁned as in

previous sections in terms of the ﬂuxes f

and their Jacobian matrices

The conclusion is that there exists ρ>0 so that for all ω>0 there exist

C = C(ω) and γ

= γ

(ω) and for all compactly supported and Lipschitz-

continuous u

:(y, z, t) ∈ R

d−1

× R

× R → u

(y, z, t) ∈ R

and dχ :(y, t) ∈

d−1

× R → (∂

χ(y, t), ∇

χ(y, t)) ∈ R × R

d−1

,with

u

− u



∞

d−1

×R

×R)

≤ ρ and ∂

χ − σ

∞

)

+ ∇

χ

∞

)

≤ρ,

u



1,∞

d−1

×R

×R)

≤ ω and dχ

1,∞

)

≤ ω,

Well-posedness of linearized problems 347

for all γ ≥ γ

, for all ˙v ∈ D (R

d−1

× R

× R; R

) and all ˙χ ∈ D(R

; R),

γ e

−γt

˙v

d−1

×R

×R)

+ e

−γt

˙v

|z=0



)

+ e

−γt

˙χ

)

(12.3.22)

≤ C



e

−γt

L(u, dχ)˙v

d−1

×R

×R)

+ e

−γt

( B(u) · d˙χ + M(u, dχ) · ˙v

|z=0

)

)



Proof The method of proof is very similar to what has been done for standard

BVP (in Theorem 9.6). The main novelty is the additional unknown ˙χ, the deriv-

atives of which appear in the boundary terms: in the constant-coeﬃcients case

(i.e. actually for u

≡ u

and χ(y, t)=σt), after elimination of the unknown

˙χ, we are left with a BVP with pseudo-diﬀerential boundary conditions (see

(12.2.14)); for more general u

and χ the idea is still to eliminate ˙χ and treat

the reduced problem (almost) as a standard one, by means of a (generalized)

Kreiss’ symmetrizer of course, which will be possible after ‘para-linearizing’ the

equations. Before going into detail, let us introduce additional (!) convenient

notations.

We rewrite the diﬀerential operator L(u, dχ)as

L(u, dχ)=A

(u) ∂

d−1



j=1

(u) ∂

+ A

(u, dχ) ∂

with

(u)=



−

0 A

)



for all j ∈{0,...,d− 1} and

(u, dχ)=



−A

−

, dχ)0

0 A

, dχ)



with (recall)

(u, dχ)=A

(u) −

d−1



j=1

(∂

χ) A

(u) − (∂

χ) A

(u).

To simplify the writing we shall use the shortcut u for (u, dχ), or more precisely,

u will stand for the Lipschitz continuous mapping

u : R

d−1

× R

× R → R

× R

× R × R

d−1

(y, z, t) → ( u

−

(y, z, t),u

(y, z, t),∂

χ(y, t), ∇

χ(y, t)).

We introduce the further notation

:= L(u(y,z,t)) + γ A

(u(y, z, t)),

348 Persistence of multidimensional shocks

and, recalling that the operators in the boundary conditions in (12.3.20) are

deﬁned by

B(u) · d˙χ =(∂

˙χ)(f

) − f

−

)) +

d−1



j=1

(∂

˙χ)(f

) − f

−

)),

M(u, dχ) · ˙v = A

−

, dχ) · ˙v

−

− A

, dχ) · ˙v

we introduce the boundary operator

(˙v, ˙χ)= (γ ˙χ + ∂

˙χ)(f

) −f

−

))

|z=0

d−1



j=1

(∂

˙χ)(f

) −f

−

))

|z=0

+(A

−

, dχ) · ˙v

−

−A

, dχ) · ˙v

)

|z=0

With these notations, the energy estimate we want to prove equivalently reads

γ v



d−1

×R

×R)

+ (v

)

|z=0



)

+ χ



)

≤ C



L

v



d−1

×R

×R)

+ B

(v

, χ

)

)



with

v

:= e

−γt

˙v and χ

:= e

−γt

˙χ.

Para-linearization of the equations We ﬁrst observe that for u close

enough to u

and dχ close enough to (σ, 0) in the L

∞

norm, A

(u(y, z, t))

is non-singular for all (y, z, t). Hence the equality



= L

v

equivalently

reads

∂

v

− P

v

= A

(u(y, z, t))

−1



:= −A

(u(y, z, t))

−1

( A

(u(y, z, t)) ( γ + ∂

d−1



j=1

(u(y, z, t))∂

This induces us to introduce

(y, z, t,η, τ)=−A

(u(y, z, t))

−1



τ A

(u(y, z, t)) + i

d−1



j=1

(u(y, z, t))



One may observe, in particular, that for u = u

:= (u,σ,0), A

is related to the

symbol deﬁned in Section 12.2 by

(y, z, t,η, τ)=A(u,η,τ) for all (y,z,t).

Well-posedness of linearized problems 349

In more generality, for all ﬁxed z, A

(y, z, t,η, τ = γ + iδ) can be regarded as

a symbol with parameter γ in the variables (y, t), which can be associated with

a para-diﬀerential operator with parameter T

. By Theorem C.20 we have the

error estimate

P

v − T

v

≤ C(ω) v

On the other hand, the form of B

suggests the introduction of the symbol

(belonging to Γ

(y, t, η, τ):= τ (f

(y, 0,t)) − f

−

(y, 0,t)))

+ i

d−1



j=1

(y, 0,t)) − f

−

(y, 0,t))) .

Compared to the notation b introduced in Section 12.2 we have in the special

case u = u

(y, t, η, τ)=b(u,τ,iη) for all (y, t).

Finally, we shall use the notation

(y, t):=



−

(y, 0,t), dχ)0

0 −A

(y, 0,t), dχ)



(= −A

(u(y, 0,t)) ).

By Theorem C.20 we have the error estimate

B

(v, ψ) − T

v − T

ψ

≤ C(ω)(ψ

+ v

−1

) ≤

C(ω)

( ψ

+ v

) .

Therefore, the searched energy estimate will be proved by absorption of the errors

in the left-hand side if we show its para-linearized version

γ v

d−1

×R

×R)

+ v

|z=0



)

+ ψ

)

(12.3.23)

≤ C



∂

v − T

v

d−1

×R

×R)

+ T

|z=0

+ T

ψ

)





Proof of the para-linearized energy estimate: elimination of the front

The assumption (UKL) implies, in particular, the existence of a constant c>0

such that (by homogeneity),

b(u

,τ,iη)

≥ c ( |τ |

+ η

) for all (η, τ) ∈ R

d−1

× C\{(0, 0)}

with Re τ ≥ 0 ,

350 Persistence of multidimensional shocks

and for all (Lipschitz) continuous u such that u −u

∞

≤ ρ small enough,

b

(y, t, η, τ)

≥

( |τ|

+ η

) for all (η, τ) ∈ R

d−1

× C\{(0, 0)}

with Re τ ≥ 0

and all (y, t) ∈ R

d−1

× R. Therefore,

(y, t, η, τ):=I

−

(y, t, η, τ) b

(y, t, η, τ)

∗

b

(y, t, η, τ)

(consistently with the notation introduced in Section 12.2) is well-deﬁned and

homogeneous degree 0 in (η, τ) and thus belongs to Γ

if u is Lipschitz continuous

with u − u



∞

≤ ρ, and by Theorem C.22, T

− T

is of order 0 + 1 −

1 = 0: this means there exists C depending only on ω = u − u



1,∞

such that

T

ψ − T

ψ

≤ C ψ

Since by deﬁnition Π

is identically zero, this implies

T

ψ

≤

ψ

for all smooth enough ψ. Also by Theorem C.22 we have another constant, still

denoted by C, depending only on ω such that

T

v − T

v

≤ C v

−1

≤

v

for all smooth enough v.

Proof of the para-linearized energy estimate: estimate of the front

The estimate b

(y, t, η, γ + iδ)≥

1,γ

(δ, η)andG˚arding’s inequality (in

Theorem C.23) show that

Re T

∗

ψ, ψ≥

ψ

Theorems C.21 and C.22 show that R

:= T

∗

− (T

)

∗

is an operator of

order at most one; hence

R

ψ, ψ≤C ψ

ψ

≤

ψ

Therefore, combining the two inequalities we get

ψ

≤



T

ψ

ψ



;

hence for γ ≥ 8C/c,

ψ

≤

T

ψ

Well-posedness of linearized problems 351

in which the right-hand side can be bounded in a trivial way:

T

ψ

≤ 2 T

ψ + T

|z=0



+2T

|z=0



≤ 2 T

ψ + T

|z=0



+2C v

|z=0



for a new C depending on ω.

Proof of the para-linearized energy estimate: the main estimate From

the two steps above (elimination and estimate of the front), it should now be

clear to the reader that (12.3.23) will follow (by absorption of the errors in the

left-hand side) from the estimate on the reduced para-linearized problem

γ v

d−1

×R

×R)

+ v

|z=0



)

(12.3.24)

≤ C



∂

v − T

v

d−1

×R

×R)

+ T

|z=0



)



Unsurprisingly, the proof of this estimate will be very similar to the proof of

Theorem 9.6: it will rely on the construction of a generalized Kreiss symmetrizer

for the reduced BVP



∂

v − T

v = F,

|z=0

= G.

Lemma 12.1 Under the assumptions of Theorem 12.1, let u =(u, dχ) be

Lipschitz continuous and close enough to u

=(u,σ,0) in the L

∞

norm (which

means more precisely that the bound ρ in the statement of Theorem 12.1 is small

enough). Then there exists a symbol

: X := {(y, z,t,η, τ) ∈ R

2d+1

× C ; z ≥ 0 , Re τ ≥ 0 }→HPD

X =(y, z,t,η, τ) →R

(X) ,

which belongs to Γ

and is homogeneous degree 0 in (η, τ), with an estimate

(y, 0,t,η,τ) ≥ αI

− β ((Π

)(y, t, η, τ))

∗

(Π

)(y, t, η, τ) (12.3.25)

for α>0 and β>0 depending only on the Lipschitz norm of u, and additionally

(X) A

(X) decomposes for all X ∈ X into a ﬁnite sum of the form

(X) A

(X)=



(X)

∗



γh

0,j

(X) 0

0 h

1,j

(X)



(X),

with P

∈ Γ

, homogeneous degree 0 in (η, τ) and



(X)

∗

(X) ≥

and h

0,j

∈ Γ

, homogeneous degree 0 in (η, τ) and Re (h

0,j

(X)) ≥ C

and h

1,j

∈ Γ

, homogeneous degree 1 in (η, τ) and

Re (h

1,j

(X)) ≥ C

γ,1

(η, δ) I

2n−k

352 Persistence of multidimensional shocks

We postpone the proof of this result and complete the proof of the estimate in

(12.3.24). The outline is the same as in the proof of Theorem 9.6. We consider

the family of operators

( T

+(T

)

∗

By construction, R

is a self-adjoint operator on L

and by Theorem C.21 and

Remark C.2, there exists C depending only on (the Lipschitz norm of u) ω so

that

(R

|z=0

− (T

|z=0



≤

v

|z=0



,dy dt)

for all smooth enough v. Hence, denoting by ·, · the scalar product on

, dy dt), the inequality in (12.3.25) together with the error estimates in

Theorems C.20 and C.22 and the G˚arding inequality in Theorem C.23 imply

R

|z=0

+ β Re T

(Π

)

∗

|z=0

≥

v

|z=0



,dy dt)

for γ large enough, hence by Theorems C.21 and C.22 again,

R

|z=0

 + β T

|z=0



,dy dt)

≥

v

|z=0



,dy dt)

up to increasing γ.

On the other hand, by Theorems C.21 and C.22, there exists C



depending

only on ω so that

Re R

v, v ≥



Re γT

0,j

+Re T

1,j

−C



v

with v

0,j

and v

1,j

the two blocks (taking values in C

and C

2n−k

, respectively)

of T

v, and by Theorem C.23 we have the inequalities

Re γT

0,j

≥γ

v

0,j



Re T

1,j

≥

v

1,j



1/2

≥ γ

v

1,j



Therefore, using once more the Theorems C.21, C.22, and C.23 (the latter being

applied to the degree 0 symbol



∗

), we get new constants C

and C



such

that

Re R

v, v≥( C

γ − C



) v

hence for large enough γ ≥ 2C





R

v, v dz ≥ γ

v

×R

,dy dt dz)

. (12.3.26)

Well-posedness of linearized problems 353

Now we prove the inequality in (12.3.24) by writing

R

v, v= 

v, v+2Re R

(∂

v −T

v) ,v+2Re R

v,v,

which implies after integration in z,

v

|z=0



)

− β Re T

(Π

)

∗

|z=0



≤ ( C

+ γ (θC

− C

))v

×R

)

4θγ

∂

v − T

v

×R

)

for some new constants C

and C

,andθ>0 arbitrary. Choosing θ = C

/(2C

we get

v

|z=0



)

+ γ

v

×R

)

≤ β Re T

(Π

)

∗

|z=0



∂

v − T

v

×R

)

for all γ ≥ 4C

. Finally, using again Theorems C.21 and C.22 we obtain for

γ large enough,

v

|z=0



)

+ γ

v

×R

)

≤ β T

|z=0



)

∂

v − T

v

×R

)

which can be rewritten as (12.3.24) with C =4max(β, C

/(2C

))/

min(α, C

). 2

Sketch of proof of Lemma 12.1 To construct R

the idea is (as usual)

to construct a local symmetrizer in the neighbourhood of each point X =

(y, z, t,η, τ) ∈ X with |τ |

+ η

= 1, to piece together these local symmetrizers

by a partition of unity technique and then extend the resulting mapping to the

whole set X by homogeneity in (η,τ).

By local symmetrizer at X

∈ X

we mean a matrix-valued function r

deﬁned

on a neighbourhood X ⊂ X

, associated with another matrix-valued function T ,

both being at least Lispchitz in (y,z, t)andC

∞

in (η, τ), such that

i) the matrix r(X) is Hermitian and T (X) is invertible for all X ∈ X ,

ii) the matrix Re ( r(X) T (X)

−1

(X)T (X) ) is block-diagonal, with blocks

(X)andh

(X) such that h

(X)/γ is C

∞

and

Re ( h

(X)) ≥ CγI

, Re ( h

(X)) ≥ CI

n−p

, (12.3.27)

for some C>0 independent of X ∈ X ,

354 Persistence of multidimensional shocks

iii) and additionally, if X ∈ X

(i.e. if z = 0) there exist α>0andβ>0

independent of X ∈ X ∩X

so that for all V ∈ C

∗

r(y,0,t,η,τ) V ≥ α V 

(12.3.28)

−β Π

(y, t, η, τ)M

(y, t)T (y,0,t,η,τ) V 

The construction of a local symmetrizer at a point X

∈ X

such that

u(y

,z,t)=u relies on the assumption (UKL) and more speciﬁcally on its

consequence

(UKL

) for all (τ, η) with Re τ ≥ 0and|τ |

+ η

= 1, for all V ∈

(A(u,η,τ)),

V ≤C Π(u

,τ,iη) M(u,σ,0)V .

Details are basically the same as for the standard BVP. More precisely, we may

consider the pair (η, τ) in the projection operator Π as parameters, and use

the block-diagonal structure of A and ΠM together with the ‘standard’ con-

struction of symmetrizers. This is made clear in the example of gas dynamics in

Chapter 15.

By continuity, (UKL

) also implies that for u close enough to u in the L

∞

norm,

V ≤

Π

(y, t, η, τ) M

(y, t, η, τ)V 

for all (y, t) ∈ R

, for all (τ, η) with Re τ ≥ 0and|τ|

+ η

= 1, and for all

V ∈ E

(y, t, η, τ)). Consequently, the construction of local symmetrizers is

in fact valid at all points X ∈ X

Then it suﬃces to piece together local symmetrizers as in the proof of

Theorem 9.1: R

is deﬁned on X

as a ﬁnite sum of terms of the form

P (X)

∗

r(X)P (X) with P (X)=ϕ(X)

1/2

T (X)

−1

, ϕ coming from a partition of

unity of X

(of which the lack of compactness in the (y,z,t)-directions is why we

require constant coeﬃcients outside a compact set), and ﬁnally R

is extended

by homogeneity of degree 0 in (η, τ). 2

Remark 12.4 By density of D in H

, the energy estimate (12.3.22) extends

to all pairs ( ˙v, ˙χ) ∈ e

γt

d−1

× R

× R) × e

γt

). (This is obviously

enough to pass to the limit in the right-hand side of (12.3.22) applied to

D-approximations of ( ˙v, ˙χ).)

The energy estimate (12.3.22) (in Theorem 12.1) can be used to derive higher-

order energy estimates, which will be useful to eventually deal with the full non-

linear problem. These higher-order estimates are as for standard BVP in terms

of H

norms, deﬁned by

w



|α|≤m

2(m−|α|)

e

−γt

∂

w