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

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Frequency boundary points 205

Finally, the equality of dimensions implies the equality of these subspaces. Letting

then γ → 0

, we complete the proof of the theorem. 

By symmetry, E

−

(iρ, η)isofrealtypewhen(ρ, ξ) ∈−Γ. In this case, we have

the formula

−

(iρ, η)=E



)

−1

(ρI

+ A(ξ))



In particular, we have

−

(−iρ, −η)=E

−

(iρ, η), (ρ, ξ) ∈ Γ.

This can be seen also as the consequence of the more general formula

−

(¯τ,−η)=E

−

(τ,η) η ∈ R

, Re τ>0. (8.2.4)

This shows that the formula above stands without any restriction on the fre-

quencies:

−

(−iρ, −η)=E

−

(iρ, η), (ρ, ξ) ∈ R × R

, (ρ, ξ) =(0, 0). (8.2.5)

Deﬁnition 8.1 The set of pairs (ρ, ξ) ∈ (R × R

) \{(0, 0)} such that A(iρ, η)

is diagonalizable with a pure imaginary spectrum, is called the hyperbolic set of

the boundary of the frequency domain. We denote the hyperbolic set by H.Itis

acone:

(y ∈ R

∗

,X∈H)=⇒ yX ∈H.

There is a slight abuse of words in this deﬁnition, as boundary points are of the

form (iρ, η), rather than (ρ, η), but this terminology is much easier to handle.

Hyperbolic boundary points are those for which the real matrix

B(ρ, η):=(A

)

−1

(ρI

+ A(η))

is R-diagonalizable. Theorem 8.1 tells us that

πΓ ⊂H, (8.2.6)

where Γ is the forward cone and π : R × R

→ R × R

d−1

is the projection where

the last entry is deleted:

π(ρ, η, ω)=(ρ, η).

We shall justify below the use of the word ‘hyperbolic’ for these boundary

frequencies; they are a candidate for the propagation phenomena for weakly

well-posed IBVPs.

8.2.2 On the continuation of E

−

(τ,η)

We digress here with a strange result, saying that the uniform continuity of

(τ,η) → E

−

(τ,η), proved in the constantly hyperbolic case (Lemma 4.5), fails in

general. More importantly, it even fails in the nice class of Friedrichs-symmetric

operators!

206 A classiﬁcation of linear IBVPs

Theorem 8.2 Let n ≥ 3 and 1 ≤ p ≤ n − 1 be given. Then there exist d ≥ 2

and a symmetric operator

L = ∂



α=1

∂

∈ Sym

(R),

with A

of signature (p, n − p) (p positive and n − p negative eigenvalues), such

that the map (τ,η) → E

−

(τ,η) does not admit a continuous extension to Re τ ≥

0, |τ|

+ |η|

=1.

Proof We choose

d =

n(n +1)

− 1,

and {A

,...,A

} a basis of

{S ∈ Sym

(R); TrS =0}.

Since max(p, n − p) ≥ 2, we may specify  = ±1and

)

−1





0 J



so that J is indeﬁnite.

From Theorem 8.1, we know that E

−

(τ,η) has a limit E

−

(iρ, η)atpoints

(ρ, η) ∈ πΓ. Our purpose will be to show that the map (ρ, η) → E

−

(iρ, η)does

not extend continuously to the boundary of πΓ. From Theorem 8.1, this amounts

to proving that the map

(ρ, ξ) → E



)

−1

(ρI

+ A(ξ))



does not extend continuously to the boundary of Γ.

We proceed ad absurdum. So let us assume that E



)

−1

(ρI

+ A(ξ))



continuous up to the boundary of Γ, but the origin of course. Identifying R × R

to Sym

(R) through

(ρ, ξ) → ρI

+ A(ξ),

it holds that

Γ=SPD

Thus we have supposed that S → E

((A

)

−1

S) is continuous up to ∂Γ but the

origin. Taking

S =



0Σ



this implies that the map Σ → E

(JΣ), which is well-deﬁned on SPD

n−1

extends continuously up to the boundary, including the origin. Since the closure

of SPD

n−1

is a closed cone, and since Σ → E

(JΣ) is positively homogeneous

Frequency boundary points 207

of degree zero, this map would have to be constant. Since J is indeﬁnite, this is

obviously false. 

8.2.3 Glancing points

Roughly speaking, glancing points are boundary points (ρ, η) in a neighbourhood

of which E

−

(τ,η) does not behave analytically, or at least at which the Implicit

Function Theorem does not apply. Since E

−

(τ,η) is the sum of generalized

eigenspaces of A(τ, η) associated to stable eigenvalues (Re µ<0), this means

that B(ρ, η) admits a real eigenvalue −ω, whose multiplicity does not persist

locally (crossing eigenvalues). In other words, an acceptable deﬁnition of glancing

points (ρ, η) ∈ (R × R

d−1

) \{(0, 0)} is that there exists an ω ∈ R such that

P (ρ, η, ω)=0,

∂P

∂ξ

(ρ, η, ω)=0, (8.2.7)

where P is the characteristic polynomial

P (X, ξ):=det(XI

+ A(ξ)).

When some irreducible factor occurs twice or more in P (see Proposition 1.7),

one should replace P by the product of its distinct irreducible factors, in this

deﬁnition.

We denote by G the set of glancing points. Elimination of ω in (8.2.7) yields

the result that G is contained in a real algebraic variety.

From the deﬁnition, G contains the apparent boundary of char(L) for an

observer sitting at inﬁnity in the ξ

-direction. It also contains the projection

of self-intersections of Γ. Self-intersections occur when L is not constantly

hyperbolic. It might happen that such a self-intersection projects within H,

showing that the construction of dissipative symmetrizers can be a diﬃcult task

even at hyperbolic points, in spite of the nice description given in Theorem 8.1.

Let (ρ, η) be a typical point of the apparent boundary of Γ from the ξ

direction. By typical,wemeanthatP (ρ, ξ) = 0, and the multiplicity m of ρ as a

root of P (·,ξ) is strictly less than the multiplicity M of ω as a root of P(ρ, η, ·).

Since A(ξ) is diagonalizable, the kernel of A(iρ, η) − iωI

, which equals that of

ρI

+ A(ξ), has dimension m.Sincem<M, the eigenvalue iω of A(iρ, η)isnot

semisimple; this matrix is not diagonalizable.

Recall that the maximal eigenvalue λ

(ξ)ofA(ξ) is a convex function that

is analytic in the constantly hyperbolic case. We have shown a kind of strict

convexity in Proposition 1.6. When this convexity is slightly stronger, say when

kerD

(ξ)=Rξ for every ξ ∈S

d−1

, then the above analysis applies to the

points (ρ = λ

(−ξ),ξ), and we deduce that on the boundary of H, the matrix

A(iρ, η) is not diagonalizable. Therefore, H is a connected component of the set

of pairs (ρ, η) ∈ (R × R

d−1

) \{(0, 0)} such that A(iρ, η) is diagonalizable.

Thesameargumentasaboveshowsthatif(ρ, η) is not a glancing point, then

every pure imaginary eigenvalue of A(iρ, η) is semisimple and locally analytic.

208 A classiﬁcation of linear IBVPs

In particular, E

−

(·, ·) admits an analytical extension in a small neighbourhood

of non-glancing points.

8.2.4 The Lopatinski˘ı determinant along the boundary

The Lopatinski˘ı determinant ∆(τ, η)maybedeﬁnedeverywherewehavean

−

(τ,η) deﬁned by continuity, and it is analytic whenever (τ, η) → E

−

(τ,η)has

this property. In particular, ∆ has an analytic extension to non-glancing points.

The noticable property of the restriction of ∆ to non-glancing points is that,

chosing an analytical basis of E

−

(iρ, η) that is made of real vectors when (ρ, η) ∈

H\G, ∆ becomes a real analytic function on this open subset, which we denote

D(ρ, η). In particular, its zero set in H\Gis an analytic submanifold. Section

4.6.2 shows that

−1

(0) ∩ (H\G)

is actually contained in a real algebraic variety. This will be of great importance

below.

Since D is homogeneous on the boundary, it will be appropriate to work on

the projective space P(R × R

d−1

). We shall denote by H and G the projectivized

objects obtained from H and G.

8.3 Weakly well-posed IBVPs of real type

Let (L

) ∈IB

be given. In particular, E

−

and ∆ are continuable up to the

boundary {Re τ =0;η ∈ R

d−1

} but at the origin (0, 0). Assume that the cor-

responding IBVP satisﬁes the Kreiss–Lopatinski˘ı condition, but not uniformly.

Speciﬁcally, we require that D

−1

(0) is contained in H \ G. In particular, since

−1

(0) is compact and H \ G is open, D

−1

(0) does not meet the boundary of

H \ G.

Examples If d = 2, the boundary points form a plane without its origin. The

projective object is a line, in which H \ G is a ﬁnite collection of open segments.

Our requirement is that the zeros of D belong to H \ G.SinceD is analytic,

these zeros are isolated.

If d ≥ 3, the zero set β := D

−1

(0) ∩ (H \ G) is generically a smooth real

analytic hypersurface in a real (d − 1)-projective space. For instance, if d =3,it

is a ﬁnite collection of loops. Note that the presence of such hypersurfaces is com-

patible with the (non-uniform) Lopatinski˘ı condition: Given a point (ρ

,η

)inβ,

where dD = 0, the complex zeros form a complex analytic, locally smooth, man-

ifold. More precisely, assume (∂D/∂ρ)(ρ

,η

) =0. Since ∆(τ,η)=D(−iτ, η),

zeros of ∆ locally satisfy

iτ + ρ

∼

(η − η

) ·∇

∂D/∂ρ

Together with the Implicit Function Theorem, this shows that the set of zeros

(τ,η) of ∆, with a real component η, is locally a manifold of real (projective)

Weakly well-posed IBVPs of real type 209

dimension d − 2. Thus it coincides with the set of pairs (iρ, η) such that (ρ, η) ∈ β.

In other words, ∆ does not vanish for Re τ>0when(τ,η) is close to (iρ

,η

)

and η is real. We point out that the situation is very diﬀerent when D vanishes

at some boundary point outside of H.

In [13], we have denoted WR the class described above. We summarize its

deﬁnition below.

Deﬁnition 8.2 We say that the IBVP associated to the pair (L, B) is of class

WR if the following properties are satisﬁed:

The operator L is constantly hyperbolic,

The boundary is non-characteristic (det A

=0),

The non-uniform Kreiss–Lopatinski˘ı condition is satisﬁed (∆ does not

vanish for Re τ>0 and η ∈ R

d−1

The Kreiss–Lopatinski˘ı condition is locally uniform at boundary points out

of H \ G,

TherealanalyticsetD

−1

(0) ∩ (H \ G) is non-void, and it holds that

(D =0)=⇒



∂D

∂ρ

=0



. (8.3.8)

In particular, this analytic set is smooth.

Because E

−

(τ,η) is always continuable at points of πΓ, provided L is hyper-

bolic, we might weaken the ﬁrst assumption above, at the price of replacing

H \ G by H

, the projectivization of πΓ. In order to keep our discussion clear,

we call WR

the class deﬁned by the list of conditions above, and by WR

the

weakened class. Likewise, WR

, WR

denote the subclasses of WR

in which

L is either Friedrichs symmetric or constantly hyperbolic.

The main point is that classes of the type WR are open, thus robust, in their

natural environment. For instance, IB

is naturally an open set in an R-vector

space. The fact that WR

is open is due to the following facts:

The uniform Kreiss–Lopatinski˘ı condition is robust, as seen in Section 8.1.

From compactness, if (L

) belongs to WR

, a small perturbation

(L, B) still satisﬁes the Lopatinski˘ı condition out of H \ G.

In H \ G, the Kreiss–Lopatinski˘ı condition fails only at zeros of D.When

−1

(0) is a smooth manifold, a small analytic perturbation of D yields a

small smooth perturbation of the zero set. There do not appear any new

connected components, neither along H \ G, nor in the interior Re τ>0.

Likewise, components of D

−1

(0) may not move towards the interior.

Therefore, pairs (L, B)inIB

, which are close to (L

), still belong to

210 A classiﬁcation of linear IBVPs

8.3.1 The adjoint problem of a BVP of class WR

Since existence theorems for linear BVPs rely upon the duality method, through

the Hahn–Banach and Riesz Theorems, we have to consider the adjoint BVP. We

recall that it is a backward BVP, meaning that we work in weighted spaces L

with θ<0, in contrast to spaces L

with γ>0 for the forward BVP. At the level

of the Laplace–Fourier analysis, this means that the relevant time frequencies θ

are those with Re θ<0. We recall (see Sections 4.4 and 4.6.4) that the adjoint

operator L

∗

and the dual boundary matrix C ∈ M

n−p,n

(R) are given by

∗

= −∂

−



)

∂

= C

N + M

for some matrices M,N of full rank.

Since A

is non-singular, because of Lemma 4.6 and the equality

dim E

−

(τ,η)+dimE

∗

−

(−¯τ,−η)=n,

we have

∗

−

(−¯τ,−η)=



−

(τ,η)



⊥

In particular, E

−

(iρ, η) is of real type if and only if E

∗

−

(iρ, −η) is of real type.

More precisely, it holds that

∗

= {(ρ, −η); (ρ, η) ∈H},

where H

∗

denotes the hyperbolic set associated to the adjoint BVP. We check

easily that the same relation holds true for the glancing sets

∗

= {(ρ, −η); (ρ, η) ∈G}.

Recall (Theorem 4.2) that the adjoint BVP satisﬁes the Kreiss–Lopatinski˘ı

condition at point (−¯τ,−η) if and only if the original one does at point (τ,η).

As a matter of fact, we have seen (Theorem 4.4) that, if ∆(τ,η) is a Lopatinski˘ı

determinant for the direct BVP, then

(θ, σ) →

∆(−

θ, −σ), Re θ<0,σ∈ R

d−1

is a Lopatinski˘ı determinant for the dual BVP. This implies the following relation

between D and D

∗

, the latter being associated to the adjoint BVP:

∗

(ρ, −η)=D(ρ, η).

It immediately follows that the various classes WR,... are preserved when we

pass to the dual problem. In particular, we have

∗

)

−1

(0) = {(ρ, σ ;(ρ, −σ) ∈ D

−1

(0)}.

Well-posedness of unsual type for BVPs of class WR 211

8.4 Well-posedness of unsual type for BVPs of class WR

We give ourself a BVP of class WR

. The operator L is therefore constantly

hyperbolic, with A

invertible. We denote

Λ:={(iρ, η); (ρ, η) ∈ D

−1

(0)}.

This is the set of boundary points at which the Kreiss–Lopatinski˘ı condition fails.

We ﬁrst establish optimal a priori estimates for the boundary value problem.

Then we prove an existence result by duality.

8.4.1 A priori estimates (I)

As usual, we work at the level of the Laplace–Fourier transform v.Webegin

with the case of a homogeneous initial data u(0, ·) ≡ 0 and extend u by zero to

negative times.

We must estimate v and its trace along the boundary, using the equations

τv + iA(η)v + A

∂v

∂x

= L f, x

> 0, (8.4.9)

Bv(τ,η,0) = G(τ,η):=L g. (8.4.10)

As long as (τ,η) does not approach a point of Λ, the construction done in Chapter

5 applies, and we have a dissipative symbolic symmetrizer K(τ,η), which depends

smoothly on (τ,η), as well as of possibly additional parameters. This implies an

estimate of the form

Re τ



∞

|v|

+ |v(0)|

≤ C(τ,η)



Re τ



∞

|L f |

+ |G|



. (8.4.11)

The number C(τ, η) above is bounded away from Λ. However, it does blow up

as (τ,η) tends to a point of Λ, since its boundedness is equivalent to the non-

vanishing of ∆.

In order to obtain an estimate up to Λ, we use the splitting v = v

+ v

introduced in Chapter 7. We recall below formulæ (7.1.10) and (7.1.11) for a

given pair (τ,η) with Re τ>0:

)=−



+∞

−z)A

(z)dz,

)=e

(0) +



−z)A

(z)dz,

where F := (A

)

−1

L f .

Since we know that strong estimates hold away from Λ, we concentrate on

points (τ,η) in a neighbourhood V of some (τ

= iρ

,η

) ∈ Λ. We make use of

the following statement.

Lemma 8.3 Assume that L is a constantly hyperbolic operator. Then, given a

point (ρ

,η

) in πΓ, there exists a neighbourhood V of (τ

= iρ

,η

), in which

212 A classiﬁcation of linear IBVPs

the following bounds hold uniformly for z>0:

exp(−zA

(τ,η))≤Ce

−ωzRe τ

, (8.4.12)

exp(zA

(τ,η))≤Ce

−ωzRe τ

. (8.4.13)

Here c and ω are positive constants.

These bounds follow from three important facts:

The eigenvalues of A(τ,η) remain of constant multiplicities in V,and

therefore are smooth functions.

They are real along Re τ =0,

The imaginary parts of their derivatives ∂µ/∂τ(τ

,η

) do not vanish,

because this point is non-glancing.

Actually, the same statement holds true near every non-glancing point, since

non-real eigenvalues at (τ

,η

) are harmless.

Corollary 8.1 Under the assumptions of Lemma 8.3, it holds that

z → exp(−zA

)

)

≤ C(Re τ)

−1/r

, (8.4.14)

z → exp(zA

)

)

≤ C(Re τ)

−1/r

, (8.4.15)

uniformly in V.

Applying (8.4.12) to (7.1.10), with the help of a Young inequality for the

convolution L

∗ L

⊂ L

, we obtain the ﬁrst estimate, which is a strong one:

v



)

≤

Re τ

F



)

. (8.4.16)

In the following, we abreviate ·

)

=: ·

when no confusion is possible.

The constant of integration v

(0) must be determined from the boundary

condition:

(0) = B



+∞

−zA

(z)dz + G. (8.4.17)

Since we assume the Kreiss–Lopatinski˘ı condition at the interior points, (8.4.17)

together with v

(0) ∈ E

−

(τ,η) determine uniquely v

(0). We note, however, that

(0) does not remain bounded as (τ,η) tends to (τ

,η

), for general data F and

G. Introducing the eigenprojectors π

(τ,η)ontoE

(τ,η), we have v

= π

−

= π

v, and so on. The linear map v → (π

−

v, π

v) is bounded, uniformly in

V since E

−

and E

are transverse at (τ

,η

). Because of the Kreiss–Lopatinski˘ı

condition, the linear map

v →





Well-posedness of unsual type for BVPs of class WR 213

is non-singular. We emphasize that its inverse is not uniformly bounded as

(τ,η) → (τ

,η

), since

v →



(τ

,η



is singular. As a matter of fact, the Lopatinski˘ı determinant ∆(τ,η) equals, up to

a smooth and non-vanishing factor, the determinant of (B,π

(τ,η)). Therefore,

the matrix

q(τ,η):=∆(τ,η)



(τ,η)



−1

remains bounded as (τ,η) → (τ

,η

Since

∆(τ,η)v

(0) = q(τ,η)





+∞

−zA

(z) dz + G



we obtain the estimate

|∆(τ,η)v

(0)|≤C



√

Re τ

F



+ |G|



, (8.4.18)

where C is uniform in V.

Deﬁne now

˜p(τ,η)v := v

+∆(τ,η)v

that is

˜p(τ,η)=π

(τ,η)+∆(τ,η)π

−

(τ,η). (8.4.19)

This n × n matrix-valued symbol depends smoothly on (τ, η)inV and is homoge-

neous of degree zero. Deﬁnition 8.4.19 works also outside of V. The linear operator

˜p(τ,η) is non-singular, and its inverse is bounded except in a neighbourhood of

Λ. The symbol ˜p is continuous up to the boundary Re τ = 0, except for glancing

points, where it is even not bounded. Hence we smooth it out in a neighbourhood

of the glancing points, in such a way that the new symbol p(τ,η) fulﬁls the

following properties:

p coincides with ˜p, except in a small neighbourhood of glancing points; in

particular, they coincide in a neighbourhood of Λ,

(τ,η) → p(τ,η) is smooth and homogeneous of degree zero,

p(τ,η) is non-singular everywhere on Re τ ≥ 0 but along Λ.

We warn the reader that, when smoothing ˜p, we lose the holomorphy in τ.

Therefore, p is not the symbol of a unique pseudo-diﬀerential operator. Instead,

214 A classiﬁcation of linear IBVPs

we have a collection of ψDOs



∂

∇



of respective symbols

(ρ, η) → p

(ρ, η):=p(γ + iρ, η).

From homogeneity, the principal symbol of P

is p

, for every γ ≥ 0. In particular,

is of order zero and is microlocally elliptic at every pair ((ρ, η); v) but those

for which (τ,η) ∈ Λandv ∈ E

−

(τ,η) ∩ kerB. Its characteristic cone is precisely

Λ.

8.4.2 A priori estimates (II)

Improvement of (8.4.18)

Let us focus on the neighbourhood of (τ

,η

) ∈ Λ, since elsewhere the estimates

are those of the uniformly stable case (UKL). We therefore have p = π

+∆π

−

Going back to (8.4.17), we use the fact that (A

)

−1

is a right inverse of B

(see Section 4.4), and rewrite

(0) = B





+∞

−zA

(z)dz +(A

)

−1



Next, we decompose (A

)

−1

G,using1=p +(1− p)=p +(1−∆)π

−

(0) = B





+∞

−zA

(z)dz + p(A

)

−1

G +(1−∆)π

−

)

−1



Since v

(0) is in E

−

, this gives

(0) = v

+(1− ∆)π

−

)

−1

where v

∈ E

−

is the solution of

= B





+∞

−zA

(z)dz + p(A

)

−1



Following the same argument as above, v

satisﬁes an estimate of the form

(8.4.18), with G replaced by p(A

)

−1

Since ∆p

−1

=∆π

+ π

−

is uniformly bounded and (A

)

−1

is one-to-one,

we have a bound

|∆(τ,η)G|≤|p(A

)

−1

G|.

Gathering all these inequalities, we obtain the more accurate estimate

|∆(τ,η)v

(0)|≤C



√

Re τ

F



+ |p(A

)

−1



, (8.4.20)

where C is uniform in V.