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

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Facts about the characteristic case 175

ii) the generalized eigenvalue problem (6.1.25) does not have a non-trivial

solution.

Under these assumptions, we know that the stable subspace E

−

(τ,η) admits

a unique limit E

−

(0,η)asτ tends to zero, keeping a positive real part, and

this limit is described in (6.1.26). And since the boundary matrix has the form

B =(0,B

), we have:

Proposition 6.6 Assume properties i) and ii) above. Then the uniform Kreiss–

Lopatinskii condition is satisﬁed in the neighbourhood of (0,η) if, and only if,

=kerB

⊕ H

(η). (6.1.29)

6.1.6 Ohkubo’s case

This is the case where we shall be able to construct the dissipative symmetrizer

in the next section. Recall that Ohkubo considers Friedrichs-symmetric operators

with a

≡ 0

n−m

.Thuswehave

A(ξ)=



(η)

(η) ξ



(η)

= a

(η), (a

)

= a

The important point is the following.

Proposition 6.7 In Ohkubo’s case, it holds for η ∈ R

d−1

\{0} that

kera

(η)=R((a

)

−1

(η)),

or equivalently

ker(a

(η)(a

)

−1

)=R(a

(η)).

In particular, the subspaces kera

(η) and R(a

(η)) have dimension p.

Proof We already have the inclusion

R((a

)

−1

(η)) ⊂ kera

(η), (6.1.30)

from (6.1.6).

Let us make ξ =(η, 0) with η =0. Then kerA(ξ)=kera

(η) × kera

(η).

Since 0 is an eigenvalue of multiplicity m whenever ξ = 0, we infer that

m =dimkera

(η)+dimkera

(η).

However, we also have

m =rka

(η)+dimkera

(η).

We obtain therefore

rk((a

)

−1

(η)) = rk a

(η)=dimkera

(η),

implying the equality in (6.1.30). 

176 The characteristic IBVP

Corollary 6.2 In Ohkubo’s case, the conclusion (6.1.29) of Proposition 6.6

applies.

Proof We have to check properties i) and ii).

When ξ

=0,wehave

kerA(ξ)={(ξ

X, −(a

)

−1

(η)X |X ∈ R

Also, when ξ

= 0, it holds that

kerA(ξ)=kera

(η) × kera

(η).

It follows easily that H(η)=R

× kera

(η). This space is of dimension m + p,

hence maximal.

On the other hand, let U be a solution of (6.1.25). Since H(η)

⊥

= {0}×

R(a

(η)), we have U =(0,a

(η)z). Then

A(η + ξ

)U =



(η)a

(η)z



We deduce that a

(η)a

(η)z = 0, whence U =0. 

Remarks We warn the reader that our assumption on the symbol does depend

on the choice of space co-ordinates in which {x

=0} is the boundary of the

domain. A consistent assumption is that there is a tangent vector W, such that

(η)=(η · W )a

for every η in R

d−1

. This allows us to choose co-ordinates in

which a

≡ 0

n−m

, while the boundary is still {x

=0}. In other words, the lower-

right block in the symbol needs to move along a one-dimensional space (notice

that a

is a generator of this line since it is a non-zero matrix.) It is clear from

(6.1.28) that the system of isotropic elasticity does not satisfy this assumption.

We leave the reader to check that the Maxwell system of electromagnetism, as

well as the system of acoustics, do. Since isotropic elasticity does not belong to

the Ohkubo’s class, the construction of the forthcoming section does not apply

to this system. Since the choice of Ohkubo’s assumption is suﬃcient but not

necessary, this does not simply anything about the solvability of the general

IBVP under (UKL) in elasticity. A more elaborate construction, speciﬁc to the

system of isotropic elasticity, is given in [141, 142], which proves the solvability

of IBVPs in space dimension two and three.

6.2 Construction of the symmetrizer; characteristic case

The solvability of the IBVP under (UKL) will be shown, as in the non-charac-

teristic case, with the help of a symmetrizer K(τ,η). We recall that we restrict

to Friedrichs symmetric operators L = ∂

+ A(∇

) that have a null eigenvalue of

Construction of the symmetrizer; characteristic case 177

constant multiplicity m, and such that A(ξ) has the form

A(η, ξ



(η)

(η) ξ



,η∈ R

d−1

,ξ

∈ R.

The symmetrizer will satisfy again the two requirements that Σ := KA

Hermitian and Re M(τ,η) ≥ c

(Re τ)I

, with the notations of Section 5.2. Also,

it will be homogeneous of degree zero and bounded.

As mentioned before, it is not possible to obtain an estimate of the full

trace of u on the boundary, in the same norm as the boundary data. What

we may reasonably expect is an estimate of the trace of A

u. For this reason, the

restriction of Σ to the kernel of B cannot be negative-deﬁnite since it vanishes on

kerA

. Hence, we ask only that this restriction be non-positive, while vanishing

only on kerA

. In other words, this restriction will be bounded from above by

−c

)

, with c

> 0. Since kerA

⊂ kerB, this condition will involve only

the vectors w ∈ C

n−m

such that (0,w)

∈ kerB. The space of these vectors is

simply kerB

The special form of A

and the fact that Σ is Hermitian immediately imply

that K has the form

K(τ,η)=



(τ,η)0

(τ,η) K

(τ,η)



where K

=: Σ

is Hermitian. We now list the objects we are looking for:

i) Find parametrized matrices K

(τ,η), K

(τ,η)andK

(τ,η), homogeneous

of degree zero and uniformly bounded on Re τ>0, τ ∈ R

d−1

ii) Such that Σ

(τ,η):=K

(τ,η)a

is Hermitian,

iii) The restriction of Σ

to kerB

must be less than −c

n−m

, with c

> 0

independent of (τ, η). In other words, we need a positive c

, independent

of (τ,η), such that

≤−c

n−m

+ c

That inequality may also be written

Σ ≤−c

)

+ c

iv) And ﬁnally,

Re M(τ,η) ≥ c

(Re τ)I

, ∀(τ,η), (6.2.31)

where

M(τ,η):=K(τ,η)(τI

+ iA(η, 0)) =



τK

+ iK

+ τK



Notice that we work only in the simpler and natural case a

≡ 0

n−m

. To our knowledge, the

systematic construction of the symmetrizer in a more general context is an open question.

178 The characteristic IBVP

We note that, since L is symmetric, the choice K ≡ I

ﬁlls three of the four

requirements. The remaining one, that Σ

restricted to kerB

is negative-deﬁnite,

would then be a dissipation assumption.

Our assumption, besides constant hyperbolicity, is the uniform Kreiss–

Lopatinski˘ı condition that E

−

(τ,η) ∩ kerB is trivial for every pair with Re τ ≥ 0,

η ∈ R

d−1

,(τ,η) = 0. The deﬁnition of E

−

(τ,η)forReτ =0,τ =0,isasusual

by continuous extension. Such an extension to points with Re τ =0,τ = 0 exists

and follows from the same arguments as in the non-characteristic case. The

continuous extension to points (0,η) was described in Proposition 6.6.

In the following, we shall use the following property, equivalent to (UKL), that

−

(τ,η) ∩ kerB

is trivial for every pair with Re τ ≥ 0, η ∈ R

d−1

,(τ,η) =0.

Steps 1, 2 and 3 of the proof of Theorem 5.1 work as well. Therefore, we

are led to construct the symmetrizer pointwise at interior points (Re τ>0) and

locally at boundary points (Re τ = 0). We shall split the latter case into two

subcases, according to whether τ =0orτ =0.

Interior points When (τ,η) is a ﬁxed interior point (Re τ>0), we only have

to build K(τ,η) satisfying ii),Σ

< 0onkerB

and Re M>0 at the sole point

(τ,η).

Following Step 4 of the non-characteristic case (see Chapter 4), we ﬁrst choose

a Hermitian matrix Σ

that satisﬁes Σ

< 0onkerB

and Re Σ

(τ,η) < 0.

Such a Σ

does exist, because of (UKL): (kerB

) ∩ e

−

(τ,η)={0}.

Next, we choose

=Σ

)

−1

= −

so that

M(τ,η)=



τK

0 −Σ



It remains to choose K

= k

, with k

∈ HDP

and >0 small enough. This

ensures Re M>0 and ends the construction in the case of an interior point.

Ordinary boundary points (Re τ =0, but τ =0) The above computation

suggests to rewrite M in the general form

M(τ,η)=



τK



−Σ







:= τK

+ iK

With this notation, we are searching a triplet (K



, Σ

) instead of

). The matrix Σ

is Hermitian. Matrices K

and Σ

are homogeneous

of degree zero, while K



has degree one.

As long as τ does not approach zero, we may choose K

= I

and K



= ia

with >0 small enough. This would have worked well in the case of interior

Construction of the symmetrizer; characteristic case 179

points yet. Such a choice yields

Re M(τ,η)=



(Re τ)I

0Re(−Σ

) − (Re τ )|τ|

−2



Let (τ

,η

) be given, with Re τ

≥ 0andτ

= 0. Following the non-characteristic

case, the uniform Kreiss–Lopatinski˘ı property ensures that there exists a Her-

mitian matrix Σ

(τ,η) deﬁned in a neighbourhood V of (τ

,η

), homogeneous of

degree zero and bounded, whose restriction to F

is negative-deﬁnite, and such

that Re (−Σ

) ≥ 2c

(Re τ)I

n−m

. Property iv) will then be satisﬁed provided

a

≤ c

|τ|

n−m

in V, which is true when  is small enough.

We emphasize that the above construction of Σ

needs the block-structure

property for the matrix A

(τ,η) near glancing points (see [134]). This property

is ensured by the important remark in Section 5.1 and the following identity,

coming from Schur’s complement formula (see [187]):

det(ωI

n−m

−A

(τ,η)) =

det(τI

+ A(iη, ω))

det a

, (6.2.32)

where the right-hand side is analytic near (τ

,η

,ω) for every ω.

Central points (τ =0) From now on, we shall focus our attention on the

vicinity of points of the form (0,η

) with η

= 0, where the previous analysis

does not apply.

In such a neighbourhood, the symmetrizer (actually an incomplete one, see

below) will be chosen as a linear combination of three simpler matrices:

K(τ,η)=K

(η) − λK

(η)+µK

III

(η),λ>0,µ>0.

The real parameters λ, µ will be chosen later. The role of λ is to ensure the

dissipativeness of the boundary condition, while the role of µ is to ensure that

Re M ≥ c

γI

with a positive c

. It turns out that it is possible to choose λ ﬁrst,

and then to adapt µ.

The three pieces are given by the formulæ

:= diag(0

)

−1

:= diag(a

)

−2

)

−2

+(a

)

−1

)

−1

where we drop the argument η for simplicity, and (recalling our notation τ =

γ + iρ)

III



−k

ρk



:= σ(a

)

−1

σ = i(a

)

−1

−(a

)

−1

Note that Σ := KA

has the form diag(0

, Σ

). We denote by Σ

,... the blocks

that correspond to K

,....Then

= a

, Σ

= a

)

−1

+(a

)

−1

, Σ

III

= ρσ.

180 The characteristic IBVP

These matrices are clearly Hermitian, thus Σ

=Σ

− λΣ

+ µΣ

III

is too.

We shall often work with variables (p, q)inC

n−m

,wherew = p + q and

p ∈ kera

, q ∈ R(a

). Because of the Kreiss–Lopatinski˘ı condition, written at

τ =0,wehaveC

n−m

=kerB

⊕ kera

. In other words, kerB

has an equation

of the form p = Dq,whereD(η) is some linear operator. Obviously, D depends

smoothly, thus boundedly, on η/|η|.OnkerB

, the norms |w| and |q| are

equivalent, uniformly in η.

As the Hermitian form w →|a

is positive-deﬁnite on kerB

, there exists

a (large enough) λ such that the form

w → w

∗

w − λw

∗

is negative-deﬁnite on kerB

, uniformly in η. Regardless of the value of µ,the

restriction of Σ

to kerB

will be uniformly negative-deﬁnite for ρ, hence τ ,small

enough.

We now turn to the study of the Hermitian form Q(u):=Re(u

∗

Mu), in

which we use repeatedly (6.1.6). It decomposes naturally into Q

− λQ

µQ

III

,whereλ has been ﬁxed yet and µ is still at our disposal. The formulæ for

each piece are:

(u) = Re (τq

∗

)

−1

p),

(u)=γ



|(a

)

−1

+ |a

)

−1



+Re(τq

∗

)

−2

w),

III

(u) = Re (w

∗

(−γa

v − ia

q + ρτw)) ,

where we notice that

∗

= i(q

∗

)

−1

− p

∗

)

−1

)(a

)

−1

. (6.2.33)

Let us establish some bounds, where the numbers c

> 0 may be taken

independently of η. On the one hand, we have

|λQ

(u)|≤c

|τ||p||q|,Q

(u) ≥ c



+ |p|



− c

|τ||w||q|. (6.2.34)

On the other hand, (6.2.33) yields

∗

(−γa

v − ia

q + ρτw)=iq

∗

)

−2

(−γa

v − ia

q + ρτw)

−iρτp

∗

)

−1

)

−1

= |(a

)

−1

+ ρ(ρ − iγ)|a

)

−1

+ O(γ|q||a

v|)+O(|ρτ ||q||w|),

whence

III

(u) ≥ c



|q|

+ ρ

|p|



− c

|q|(γ|a

v|+ |ρτ ||w|). (6.2.35)

Let us choose µ>0 large enough so that the quadratic form µc

+ Y

) −

XY is positive-deﬁnite. This allows us to absorb the bad terms −c

|ρ||p||q|

Construction of the symmetrizer; characteristic case 181

of Q

and −λQ

into the positive terms of µQ

III

. This results, with a slight

change of the positive numbers c

, in the following lower bound

Q(u) ≥ c



+ |p|



+ c



|q|

+ ρ

|p|



−c

|q|



|ρ||q|+ γ|w| + γ|a

v|+ ρ

|w|



Using Young’s inequality in the last three terms, we obtain

Q(u) ≥ c



γ(|a

+ |p|

)+ρ

|p|





− c



|q|

−



|w|

+ γ

+ ρ

|w|



It is now clear that there is a neighbourhood V of the origin, such that τ ∈V

implies

Q(u):=Re(u

∗

Mu) ≥ c



+ |p|



+ c



|q|

+ ρ

|p|



. (6.2.36)

Looking back at our results, we ﬁnd only one ﬂaw, which lies in (6.2.36):

The form Q dominates only a

v but not v itself. If a

is one-to-one, that is

n =3m, then our construction gives us the symmetrizer that we were looking for,

otherwise it does not. In the general case, let us denote by K

the symmetrizer

that we have deﬁned above, and Σ

, Q

the corresponding Hermitian forms. Our

ﬁnal symmetrizer will be of the form K = K

+ I

. We shall have

=Σ

+ a

,Q(u)=Q

(u)+γ|u|

For every choice of >0, the form Q satisﬁes our requirement

Q(u) ≥ c

γ|u|

for an appropriate c

> 0. At last, a small enough  does not hurt the negativeness

of the restriction of Σ

on kerB

. This completes the construction of a dissipative

symmetrizer. We leave the readers to convince themselves that the inequalities

satisﬁed by Σ

and Q are uniform in η.

THE HOMOGENEOUS IBVP

We continue the analysis of strong L

-well-posedness that we treated in the

previous chapters. However, we weaken our requirements by considering only

the homogeneous IBVP:

(Lu)(x, t)=f(x, t),x

,t> 0,y∈ R

d−1

, (7.0.1)

Bu(y, 0,t)=0,t>0,y∈ R

d−1

, (7.0.2)

u(x, 0) = u

(x),x

> 0,y∈ R

d−1

. (7.0.3)

The boundary condition is therefore Bu(t, y, 0) = 0, instead of Bu = g with a

general data g. An important consequence is that (7.0.1)–(7.0.3) is a semigroup

problem: At a formal level, the solution should read

u(t)=S



t−s

f(s)ds,

where (S

)

t≥0

is a semigroup on some Banach space. What we expect is that

this is a continuous semigroup on each Sobolev space H

(Ω). However, it is

not so simple to follow the standard strategy of semigroup theory, and we shall

merely use the same tools as in the non-homogeneous IBVP: Fourier–Laplace

transformation, dissipative symmetrizer, Lopatinski˘ı condition, Paley–Wiener

Theorem and duality. The main diﬀerence with the non-homogeneous theory lies

in the estimate we ask for: since the input consists only in two terms, say u(·, 0)

and Lu, we do not require an estimate for the boundary value of the solution.

This has two important consequences. On the one hand, the Kreiss–Lopatinski˘ı

condition is weakened. On the other hand, strongly well-posed IBVPs in the

above sense may admit surface waves of ﬁnite energy.

When dealing with second-order IBVPs, an important case, which we do not

treat here, is when the pair (L, B) expresses the Euler–Lagrange equations of

some Lagrangian

L [u]:=





− W (∇



dx dt

over H

(Ω × R

). This special situation encompasses the examples of the wave

equation with the Neumann condition, and of linearized elastodynamics with zero

normal stress at the boundary. The theory of such problems is done in [189,190],

where it is shown that the Hille–Yosida Theorem can always be applied.

The homogeneous IBVP 183

The present chapter is thus devoted to the more general situation of ﬁrst-

order problems.

As usual, strong well-posedness means that we have control of the solution

exactly in the same terms as for the data. Since the boundary condition does

not really involve a data (it is trivial), we do not require any control of the trace

of the solution along the boundary. Therefore, we say that the homogeneous

IBVP (7.0.1)–(7.0.3) is strongly well-posed in L

if for every u

∈ L

(Ω) and

f ∈ L

(0,T; L

(Ω)), there exists a unique solution u ∈ C ([0,T]; L

(Ω)) with the

estimate

−2γT

u(T )

+ γ



−2γt

u(t)

dt ≤ C



u





−2γt

f(t)



(7.0.4)

Hereabove, all norms are taken in L

(Ω).

BVP vs IBVP When the initial data u

vanishes identically, we may extend u

and f to negative times by zero, so that the extension ˜u is a solution of the BVP

for L˜u =

f on R × Ω, instead of (0, +∞) × Ω. We say that the homogeneous BVP

is strongly well-posed in L

if for every γ>0 and every f ∈ L

(0, +∞; L

(Ω)),

the IBVP with u

≡ 0 has a unique solution in L

(0, +∞; L

(Ω)), with the

estimate



−2γt

u(t)

dt ≤



−2γt

f(t)

dt, (7.0.5)

where C does not depend either on f or on γ.

We therefore say that our BVP is strongly well-posed if, for every u smooth

with compact support in space and time, the property Bu(y, 0,t) ≡ 0 implies an

estimate



−2γt

u(t)

dt ≤



−2γt

Lu(t)

dt. (7.0.6)

We have seen in Section 4.5.5 that the well-posedness of the BVP is a step

towards that of the IBVP, the passage from the former to the latter needing

an extra argument, due to Rauch. Because we do not estimate the trace of the

solution, Rauch’s argument does not work out in the homogeneous case. For this

reason, the well-posedness of the homogeneous IBVP remains an open question.

So far, the only cases where the IBVP is fully understood is the symmetric

dissipative one (see Section 3.1), and more generally the ‘variational’ case treated

in [189, 190].

Need for a modiﬁed Lopatinski˘ı condition When considering Friedrichs-

symmetric operators with classical dissipative symmetrizers (see Chapter 3),

we observed that the amount of dissipation needed to treat the homogeneous

IBVP was weaker than that required for the non-homogeneous problem. Since

184 The homogeneous IBVP

Chapter 3 stated suﬃcient but not necessary criteria for well-posedness, we

cannot draw a deﬁnitive conclusion from it. However, we anticipate that some

condition weaker than (UKL) would ensure the homogeneous well-posedness, at

least for constantly hyperbolic operators.

Before entering into the details, we make the following observations. On the

one hand, since our preliminary study of Section 4.2 dealt with the homogeneous

problem, we already know that the Kreiss–Lopatinski˘ı condition is necessary for

well-posedness. On the other hand, the well-posedness of the non-homogeneous

IBVP clearly implies that of the homogeneous one. Therefore, at least for

constantly hyperbolic operators, the truth for the homogeneous case must lie

somewhere between the Kreiss–Lopatinski˘ı condition and its uniform version.

7.1 Necessary conditions for strong well-posedness

In this section, we derive a necessary condition that we expect also to be

suﬃcient when the diﬀerential operator is constantly hyperbolic. The suﬃciency

would be ensured, as in the non-homogeneous case, by the existence of a Kreiss

symmetrizer, though a weakly dissipative one, instead of strongly dissipative. We

postpone its construction to Section 7.2.

For the sake of simplicity, we assume that the boundary is non-characteristic.

Also, it will be comfortable to consider only constantly hyperbolic operators.

This allows us to extend by continuity the stable subspace E

−

(τ,η) and the

Lopatinski˘ı determinant to boundary points (Re τ =0,(τ,η) =(0, 0)).

Our necessary condition will be derived from the stability of the homogeneous

BVP (instead of the corresponding IBVP). As a matter of fact, it is not hard to

see that the well-posedness with u

≡ 0 implies the well-posedness of the BVP.

At the level of the Laplace–Fourier transform, the estimate (7.0.6) amounts

to saying that

v

≤

Re τ

F 

, (7.1.7)

for every v ∈ L

) such that

F := v



−A(τ,η)v ∈ L

), Re τ>0,η∈ R

d−1

and

Bv(0) = 0.

Let us introduce the decomposition v = v

+ v

into a stable part v

∈

−

(τ,η) and an unstable one v

∈ E

(τ,η). Decomposing F := (A

)

−1

Lf,we

have

= A

(τ,η)v

+ F

, (7.1.8)

= A

(τ,η)v

+ F

, (7.1.9)