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 355

Here above α stands for a d-or(d + 1)-uple, (i.e. α =(α

,α

,...,α

d−1

)or

(α

,α

,...,α

), with α

∈ N), and |α| =



(called the length of α).

Theorem 12.2 Under the hypotheses of Theorem 12.1, assume, moreover, that

u − u

belongs to H

d−1

× R

× R; R

), (u − u)

|z=0

belongs to H

; R

)

and (∂

χ − σ, ∇

χ) belongs to H

; R

) for some integer m>(d +1)/2+1,

with

u − u



≤ µ, u

|z=0

− u

|z=0



≤ µ and (∂

χ − σ, ∇

χ)

≤ µ.

Then there exist γ

= γ

(ω, µ) ≥ 1 and C

= C

(ω, µ) > 0 such that, for all

γ ≥ γ

, for all ˙v ∈ D (R

d−1

× R

× R; R

) and all ˙χ ∈ D(R

; R),

γ ˙v

))

+ ˙v

|z=0



)

+  ˙χ

m+1

)

≤ C



L(u, dχ)˙v

))

+ B(u) · d˙χ + M (u, dχ) · ˙v

)



(12.3.29)

γ ˙v

d−1

×R

×R)

+ ˙v

|z=0



)

+  ˙χ

m+1

)

≤ C



L(u, dχ)˙v

d−1

×R

×R)

+ B(u) · d˙χ + M (u, dχ) · ˙v

)



(12.3.30)

The proof is analogous to the proof of Theorem 9.7 for standard BVP. It

starts with estimates of derivatives in the (y, t)-direction by using the L

estimate

(12.3.22) (in Theorem 12.1) together with commutator estimates in H

norms:

this yields (12.3.29). As a second step, estimates of derivatives in the z-direction

are obtained by diﬀerentiating the equality

∂

˙v = A

(u(y, z, t), dχ(y, t))

−1



L(u, dχ)˙v −

d−1



j=1

(u(y, z, t)) ∂

˙v



and by repeated use of Lemma 9.3: this eventually leads to (12.3.30). We omit

the (most technical) details and refer to [140], pp. 71–72, or [136], Section 4.6.

12.3.2 Adjoint BVP

In the previous section, we have performed a microlocal elimination of the

unknown front to derive energy estimates. Here we are going to use a more

algebraic approach, in order to deﬁne suitable adjoint problems.

Lemma 12.2 If b

, ..., b

are C

∞

mappings W → R

, with W a contractible

subset of R

, such that for all u ∈ W the family (b

(u),...,b

(u)) is independent,

356 Persistence of multidimensional shocks

there exists Q ∈ C

∞

(W ; GL

(R)) so that for all u ∈ W and all ξ ∈ R

Q(u)



j=1

(u)=(ξ

, ··· ,ξ

,,0, ··· , 0)

Proof If B(u) denotes the matrix with columns b

(u), ..., b

(u), the con-

tractibility of W enables us to deﬁne C

∞

mappings 

: W → M

1×n

(R)for

j ∈{d +1,...,n} such that for all u ∈ W ,(

d+1

(u),...,

(u)) is a basis of

kerB

(in other words, the vector bundle kerB

is trivializable, see [85] p. 97).

Then for all u ∈ W , the square matrix



(u),...,b

(u),

d+1

(u)

,...,

(u)



is invertible and its inverse Q(u) answers the question by construction. 

Applying Lemma 12.2 to b

(u)=f

j−1

) − f

j−1

−

), N =2n and W a

ball centred at u (of radius less than or equal to ρ say), we may rewrite the

boundary conditions in (12.3.20) as



∇ ˙χ



+ Q(u) M(u) · ˙v = Q(u) g.

Then, applying Lemma 9.4 in a ball W centred at u

=(u,σ,0,...,0) (in R

2n+d

)

to A = A

and B : u → Q(u) M(u)(=−Q(u) A

(u), of rank n independently

of u), we ﬁnd N, P and R in C

∞

(W ; M

n×2n

(R)) such that

=ker(Q(u) M(u)) ⊕ kerN (u) , R

=kerP (u) ⊕ kerR(u),

(u)=P (u)

Q(u)M(u)+R(u)

N(u), kerR(u)=(A

(u)ker(Q(u)M(u)))

⊥

for all u ∈ W . This material will serve for the deﬁnition of an adjoint version of

(12.3.20). Recalling that

L(u(y, z, t)) = A

(u(y, z, t)) ∂

d−1



j=1

(u(y, z, t)) ∂

+ A

(u(y, z, t)) ∂

and denoting

L(u(y, z, t))

∗

−(A

(u(y, z, t)))

∂

−

d−1



j=1

(u(y, z, t)))

∂

− (A

(u(y, z, t)))

∂

−∂

(u(y, z, t)))

−

d−1



j=1

∂

(u(y, z, t)))

− ∂

(u(y, z, t)))

we have for all smooth enough v and w,



z>0



( w

L(u)v − v

L(u)

∗

w)+



(u) v)

|z=0

=0,

Well-posedness of linearized problems 357

hence by deﬁnition of N, P and R,



z>0



( w

L(u)v − v

L(u)

∗

w)+



((P (u)w)

Q(u) M(u) v)

|z=0



((R(u)w)

N(u) v)

|z=0

=0. (12.3.31)

For v =˙v satisfying (12.3.20) together with some function ˙χ, the latter equality

equivalently reads



z>0



( w

f − v

L(u)

∗

w)+



((P (u)w)

Q(u) g )

|z=0



( −(∇ ˙χ)

(u)w +(N(u) v)

R(u)w)

|z=0

=0,

where P

(u):=πP(u), the ﬁrst d rows among the n rows in P (u). Finally, we

may integrate by parts and rewrite



( −(∇ ˙χ)

(u)w )

|z=0



(˙χ div

(t,y)

(u)w))

|z=0

provided that ˙χ is decaying suﬃciently fast.

The computation here above urges us to consider the ‘adjoint’ problem







L(u)

∗

w =0,z>0 ,

R(u)w =0, div

(t,y)

(u)w)=0,z =0,

(12.3.32)

which is to some extent ‘non-standard’, though less than (12.3.20): the boundary

operator in (12.3.32) is only (algebro)diﬀerential (instead of pseudo-diﬀerential

in (12.3.20)). A crucial step towards the well-posedness of (12.3.20) is the proof

of energy estimates for the adjoint BVP (12.3.32).

Theorem 12.3 Under the assumptions of Theorem 12.1, the adjoint BVP in

(12.3.32) meets the standard assumptions allowing L

energy estimates backward

in time, namely for all u in a neighbourhood of u

=(u,σ,0),

(CH

∗

) The coeﬃcient of ∂

in L(u)

∗

(i.e. the matrix A

(u)

) is non-singular

and the two blocks in L(u)

∗

are constantly hyperbolic in the t-direction;

(NC

∗

) The coeﬃcient of ∂

in L(u)

∗

is non-singular (i.e. the matrix A

(u)

non-singular);

(UKL

∗

) Denoting

∗

(u,η,τ)=(A

(u)

)

−1

(¯τ A

(u)

− i

d−1



j=1

(u)

358 Persistence of multidimensional shocks

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

d−1

× C with Re τ ≥ 0 and |τ|

η

=1,

W ≤C R(u)W  + (¯τ,−iη

) · P

(u) W  (12.3.33)

for all W ∈ E

∗

(u,η,τ)).

As a consequence, there exist ρ>0, C = C(ω) and γ

= γ

(ω) so that for

u − u



∞

≤ ρ and u − u

1,∞

≤ ω,

for all γ ≥ γ

and for all ˙w ∈ D (R

d−1

× R

× R; R

γ e

γt

˙w

d−1

×R

×R)

+ e

γt

˙w

|z=0



)

≤ C



e

γt

L(u)

∗

˙w

d−1

×R

×R)

+ e

γt

R(u)˙w

|z=0



)

+ e

γt

div

(t,y)

(u)˙w

|z=0

)

−1

)



Proof The ﬁrst two properties, (CH

∗

) and (NC

∗

) are trivial consequences of

(CH) and (NC). For the proof of (UKL

∗

), we ﬁrst need to check that

∗

(u,η,τ)) = (A

(u) E

(A(u,η,τ)))

⊥

, (12.3.34)

where

A(u,η,τ)=−(A

(u))

−1

( τ A

(u)+i

d−1



j=1

(u)).

(As already done before, we make a ‘subtle’ distinction between the notations

A and A

: the mappings A : R

× R

d−1

× C

→ M

2n×2n

(C)andA

d−1

× R

× R × R

d−1

× C

→ M

2n×2n

(y, z, t,η, τ) for all (y, z,t,η, τ) ∈ R

d−1

× R

× R × R

d−1

× C

The proof of (12.3.34) is classical and was already done in Chapter 4 but

we recall it for completeness. We ﬁrst observe the matrices A(u,η,τ)and

∗

(u,η,τ) are simultaneously hyperbolic, the latter A

∗

(u,η,τ) being conjugate

to −A(u,η,τ)

∗

: more precisely,

∗

(u,η,τ)=−(A

(u)

)

−1

∗

(u,η,τ) A

(u)

Hence the spaces E

∗

(u,η,τ)) and (A

(u) E

(A(u,η,τ)))

⊥

are both of the

same dimension, equal to n +1 since E

(A(u,η,τ)) is of dimension n − 1(by

the assumption (N)). Furthermore, when A(u,η,τ) is hyperbolic, a standard

ODE argument shows E

∗

(u,η,τ)) is a subspace of (A

(u) E

(A(u,η,τ)))

⊥

Indeed, take W = ψ(0) with ψ a solution of ψ



= A

∗

ψ tending to zero at +∞.

Well-posedness of linearized problems 359

For all U = φ(0) with φ a solution of φ



= Aφ,

ψ(x)

∗

φ(x)=(A

∗

ψ(x))

∗

φ(x)+ψ(x)

∗

Aφ(x)=0,

since (A

∗

)

∗

+ A

A = 0, hence if, moreover, φ tends to zero at +∞,

∗

U = lim

x→+∞

ψ(x)

∗

φ(x)=0.

This proves (12.3.34) when A(u,η,τ) is hyperbolic, in particular when Re τ>0.

By continuity, (12.3.34) is also true for Re τ =0.

Now, suppose W ∈ E

∗

) is such that

R(u)W =0 and (¯τ,−iη

) · P

(u) W =0.

We are going to show that P (u) W = 0, which will imply W = 0 (recall that by

construction kerR(u)andkerP (u) are supplementary). To prove P (u) W =0,

we compute Y

∗

P (u) W for an arbitrary Y ∈ C

. A straightforward reformula-

tion of the assumption (UKL) with our current notations shows there exists a

unique pair (

U) ∈ C × E

(A(u,η,τ)) such that

Y =

X (τ, iη,0,...,0)

+ Q(u) M(u)

Therefore,

∗

P (u) W =

X (¯τ,−iη

) · P

(u) W +(Q(u) M(u)

∗

P (u) W,

where the ﬁrst term is zero by assumption on W ; hence

∗

P (u) W = W

∗

P (u)

Q(u) M(u)

U = W

∗

(u)

U −W

∗

R(u)

N(u)

U =0,

since W belongs to (A

(A))

⊥

and R(u)W = 0. Therefore, the mapping

W ∈ E

∗

) → (R(u)W, (¯τ,−iη

) · P

(u) W ) ∈ C

× C

is one-to-one, thus also onto since dim E

∗

)=n + 1; the norm of the inverse

mapping is uniformly bounded on the compact set {(η, τ) ∈ R

d−1

× C ;Reτ ≥

0 , |τ |

+ η

=1}.

The attentive reader will have noticed that (UKL

∗

) is simply the uniform

Kreiss–Lopatinski˘ı condition backward in time for the BVP (12.3.32). Indeed,

freezing the coeﬃcients in the principal part of L(u)

∗

and performing a Fourier–

Laplace transform in the direction (y, −t) the BVP (12.3.32) we get the ODE

problem











= A

∗

(u,η,τ)

W for z>0 ,

R(u)

W =0 and (¯τ,−iη

) · P

(u)

W =0 at z =0.

Thanks to the properties (CH

∗

), (NC

∗

) and (UKL

∗

), the same method of

proof as for the original BVP (12.3.20) (in Theorem 12.1), replacing there the

360 Persistence of multidimensional shocks

interior symbol A

by A

∗

(u(y, z, t),η,τ), and the boundary symbol Π



R(u(y, z, t))

(¯τ,−iη

) · P

(u(y, z, t))



shows the announced backward weighted estimate for the adjoint BVP (12.3.32):

the H

−1

norm in the right-hand side shows up because when the inequality

(12.3.33) in (UKL

∗

) is extended to all pairs (τ, η), it yields, for homogeneity

reasons,

W ≤C R(u)W  + λ

−1,γ

(δ, η) (¯τ,−iη

) · P

(u) W .



12.3.3 Well-posedness of the BVP

Theorem 12.4 Under the assumptions of Theorem 12.1, there exist ρ>0 and

= γ

(ω) so that for

u − u



∞

≤ ρ and u

1,∞

≤ ω,

for all γ ≥ γ

, for all f ∈ e

γt

d−1

× R

× R) and all g ∈ e

γt

d−1

R), there is one and only one solution (˙v, ˙χ) ∈ e

γt

d−1

× R

× R) ×

γt

1/2

d−1

× R) of the BVP in (12.3.20). Furthermore, ˙χ belongs in fact

to e

γt

d−1

× R), the trace of ˙v at z =0 belongs to e

γt

d−1

× R),and

(v

, χ

):=e

−γt

(˙v, ˙χ) enjoys the estimate

γ v



d−1

×R

×R)

+ (v

)

|z=0



)

+ χ



)

(12.3.35)

≤ C







d−1

×R

×R)

+ g



)



for some constant C = C(ω).

Note: the estimate (12.3.35) here above is simply another way of writing (12.3.22)

(in Theorem 12.1).

Proof Noting that (12.3.20) is equivalent to L

v



in the interior and





(γ + ∂

)χ

∇

χ





+ Q(u) M(u) · v

= Q(u) g

at the boundary z = 0, we may reformulate the conclusion of Theorem 12.4 as

follows: for γ large enough, for all f ∈ L

d−1

× R

× R) and all g ∈ L

d−1

R), there is one and only one (v, ψ) ∈ L

d−1

× R

× R) × H

1/2

d−1

× R)

Well-posedness of linearized problems 361

such that

v = f and





(γ + ∂

)ψ

∇





+ Q(u) M(u) · v

|z=0

= Q(u) g, (12.3.36)

and this solution satisﬁes an estimate of the form

γ v

d−1

×R

×R)

+ v

|z=0



)

+ ψ

)



f

d−1

×R

×R)

+ g

)

Existence of a weak solution Unsurprisingly, we are going to use the

adjoint BVP (12.3.32) introduced in the previous section, or more precisely the

equivalent problem

)

∗

γt

w =0,z>0 ,

R(u)e

γt

w =0, div

−γ

(u)e

γt

w)=0,z=0,

where the diﬀerential operator div

−γ

is deﬁned by

div

−γ

,...,p

d−1

)=(−γ + ∂

) p

d−1



j=1

∂

From this point of view the ‘dual’ energy estimate of Theorem 12.3 equivalently

reads

γ w

d−1

×R

×R)

+ w

|z=0



)

≤ C



(L

)

∗

w

d−1

×R

×R)

+ R(u)w

|z=0



)

+ div

−γ

(u)w

|z=0

)

−1

)



The resolution of the BVP in (12.3.36) follows the same lines as the proof of

Theorem 9.17. We consider the set

E := {w ∈ D (R

d−1

× R

× R); R(u)w

|z=0

=0, div

−γ

(u)w

|z=0

)=0}.

Theorem 12.3 shows that for all w ∈ E ,

γ w

+ w

|z=0



≤

(L

)

∗

w

This allows the deﬁnition of a bounded linear form  on (L

)

∗

E by

((L

)

∗

w)=



z>0



f +



((P (u)w)

Q(u) g )

|z=0

362 Persistence of multidimensional shocks

Indeed, we have





z>0



f +



((P (u)w)

Q(u) g )

|z=0



≤f

w

+ P ◦u

∞

Q ◦ u

∞

g

w

|z=0







f

1/2

g



L

∗

v

Therefore, by the Hahn–Banach theorem,  extends to a continuous form on L

and by the Riesz theorem, there exists v ∈ L

such that

((L

)

∗

w)=



z>0



)

∗

We thus get in particular, by deﬁnition of ,



z>0



( v

)

∗

w − f

w )=0

for all w ∈ D (R

d−1

× (0, +∞) × R), hence L

v = f (in the sense of distribu-

tions). Consequently, using the identity (12.3.31), or more precisely its modiﬁed

version obtained with L

instead of L(u), we ﬁnd that



((P (u)w)

Q(u) g )

|z=0



((P (u)w)

Q(u) M(u) v )

|z=0

(12.3.37)

for all w ∈ E , and by extension this is true for all w ∈ H

d−1

× R

× R)such

that R(u)w = 0 and div

−γ

(u)w)=0.

In order to recover the boundary condition in (12.3.36), we ﬁrst use

a (standard) trace-lifting argument. For all ϕ ∈ H

1/2

d−1

× R; R

)there

exists Φ ∈ H

d−1

× R

× R; R

)sothatΦ

|z=0

= ϕ. Therefore, for all θ ∈

1/2

; R

) there exists w ∈ H

d−1

× R

× R; R

)sothatR(u) w

|z=0

and P (u) w

z=0

= θ: indeed, by construction

q :(y, z,t) →



P (u(y, z, t))

R(u(y, z, t))



−1

belongs to W

1,∞

× R

× R; GL

(R)), so it suﬃces to deﬁne w = q Φ, where

Φ is obtained by trace lifting from ϕ :(y, t) → (θ(y, t), 0,...,0). In particu-

lar, for all θ

∈ H

1/2

; R

n−d

) there exists w

∈ H

d−1

× R

× R; R

)so

that R(u)(w

)

|z=0

=0,P

(u)(w

)

|z=0

=0,andP

(u)(w

)

|z=0

= θ

,where

(u)=(I

− π) P (u) denotes the last (n − d)rowsinP (u). Applying

(12.3.37) to w = w

we get



(u)(g − M (u) v )

|z=0

=0,

Well-posedness of linearized problems 363

where naturally Q

(u)=(I

− π) Q(u). This implies

(u)(g − M (u) v )

|z=0

=0.

Similarly, for all θ

∈ H

1/2

; R

) there exists w

∈ H

d−1

× R

× R; R

)

so that R(u)(w

)

|z=0

=0,P

(u)(w

)

|z=0

= θ

,andP

(u)(w

)

|z=0

=0,and

if additionally div

−γ

= 0 we can apply (12.3.37) to w = w

, which gives



(u)(g − M (u) v )

|z=0

=0.

This implies the existence of ψ ∈ H

1/2

d−1

× R

× R) such that

(u)(g − M (u) v )

|z=0



(γ + ∂

)ψ

∇



thanks to the following simple result.

Lemma 12.3 For al l γ ≥ 1, the mapping

∇

: H

1/2

; R) → H

−1/2

; R

)

ψ →∇

ψ :=



(γ + ∂

)ψ

∇



has range

(ker div

−γ

)

⊥

= {v ∈ H

1/2

; R

); v, θ

−1/2

1/2

)

for all θ ∈ H

1/2

; R

); div

−γ

θ =0}.

Proof This is a Fourier-transform exercise. Indeed, the range of ∇

is obviously

a subset of (kerdiv

−γ

)

⊥

, which can also be written as



v ∈ H

1/2

; R

); v(δ, η) 



γ + iδ

iη



and for all v in this set we have

v(δ, η)=

(γ − iδ, −iη

) · v(δ, η)

+ δ

+ η



γ + iδ

iη



which equivalently means that

v = ∇



(−γ

+∆)

−1

div

−γ



where (−γ

+∆)

−1

div

−γ

v does belong to H

1/2

; R). 

Weak=strong argument As ‘usual’, we consider a mollifying operator R

in the (y,t)-edirections and consider v

:= R

v and ψ

:= R

ψ (where (v, ψ)

is the weak solution found here above), f

:= R

(u)

−1

f (where f is a given

source term in L

)andg

:= R

g (where g is a L

data on the boundary). These

364 Persistence of multidimensional shocks

regularized functions satisfy the following properties

∈ L

; H

+∞

× R; R

)) and f

×R;R

))

−−−−−−−−−−−−−−→ A

(u)

−1

∈ L

; H

+∞

× R; R

)) and v

×R;R

))

−−−−−−−−−−−−−−→ v,

)

|z=0

∈ H

+∞

× R; R

)and(v

)

|z=0

−1/2

×R;R

)

−−−−−−−−−−−→ v

|z=0

∈ H

+∞

× R; R)andψ

1/2

×R;R)

−−−−−−−−−→ ψ,

∈ H

+∞

× R; R

)andg

×R;R

)

−−−−−−−−−→ g.

Furthermore, we have (see Lemma 9.5)

](v)

×R;R

))

−−−−−−−−−−−−−−→ 0and[B

](v

|z=0

,ψ)

×R;R

)

−−−−−−−−−→ 0

(where the operator P

is such that f = L

v equivalently reads ∂

v − P

v =

(u)

−1

f). Now we easily see that ∂

= f

+ P

− [P

](v) belongs

to L

; L

× R; R

)), hence v

is in H

× R

× R). By Remark 12.4,

the energy estimate (12.3.22) (in Theorem 12.1) thus applies to the pair (v

,ψ

more precisely we have

γ v



+ (v

)

|z=0



+ ψ





∂

− P



+ B

,ψ

)

By linearity of the operators P

and B

, this inequality also applies to pairs

− v



,ψ

− ψ



), and together with the above limiting properties this shows

that (v

)

|z=0

is a Cauchy sequence in L

and ψ

is a Cauchy sequence in H

.By

uniqueness of limits in the sense of distributions, this implies that v

|z=0

is the

limit of (v

)

|z=0

in L

,andψ is the limit of ψ

in H

. Then, by passing to the

limit in the estimate for (v

,ψ

), we get

γ v

+ v

|z=0



+ ψ



∂

v − P

v

+ B

(v, ψ)

or equivalently,

γ v

+ v

|z=0



+ ψ



L

v

+ B

(v, ψ)

, (12.3.38)

which is another way of writing (12.3.35). This completes the proof of

Theorem 12.4. 

Now, when the coeﬃcients (u) and the data f, g enjoy more regularity, we

can prove more regularity on the solution. This is the purpose of the following

result (analogous to Theorem 9.20 for standard BVP).

Theorem 12.5 We still make the assumptions of Theorem 12.1, and assume,

moreover, (as in Theorem 12.2) that u − u

, (u − u)

|z=0

and (∂

χ − σ, ∇

χ) are

in H

for some integer m>(d +1)/2+1. Then the solution (˙v, ˙χ) of the BVP

(12.3.20) given by Theorem 12.4 is such that ˙v belongs to H

d−1

× R