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

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Well-posedness of unsual type for BVPs of class WR 215

Main estimates

We now estimate ∆(τ,η)v

when (τ,η) ∈V. From (7.1.11), (8.4.15) and the

Young inequality, we have

∆(τ,η)v



≤ C|∆(τ,η)|



√

Re τ

(0)| +

Re τ

F





Thanks to (8.4.18), this yields

∆(τ,η)v



≤ C



Re τ

p(τ,η)F 

√

Re τ

|p(A

)

−1



. (8.4.21)

Merging (8.4.16) and (8.4.21), and using the uniformly dissipative Kreiss sym-

metrizer outside a neighbourhood of Λ, we conclude that the following estimate

holds uniformly in (τ,η):

p(τ,η)v

≤ C



Re τ

p(τ,η)F 

√

Re τ

|p(A

)

−1



. (8.4.22)

Likewise, (8.4.18) yields the uniform estimate

p(τ,η)v(0)

≤ C



√

Re τ

p(τ,η)F 

+ |p(A

)

−1



. (8.4.23)

Using, ﬁnally, Plancherel’s Formula, we derive our fundamental estimate



Ω×R

−2γt

P

u

dx dt +



∂Ω×R

−2γt

P

u

dx dt

≤ C





Ω×R

−2γt

P

)

−1

Lu

dx dt



∂Ω×R

−2γt

P

)

−1

Bu

dx dt



(8.4.24)

whenever γ is positive and u is smooth, compactly supported. This is clearly

weaker than (4.5.26), as expected. However, it has the nice feature that only

)

−1

Lu and P

)

−1

Bu, instead of Lu and γ

BU, are required to

be square-integrable. Therefore, there is some hope that such an estimate could

be useful in nonlinear problems when using a ﬁxed-point argument.

Remarks

Note that (A

)

−1

B is a projector, whose kernel is kerB. Therefore,

(8.4.24) is the same as (4.5.26), up to the presence of the operator P

everywhere. In other words, one passes from one estimate to the other by

changing the norm both in data and in output. We arrive at the strange

conclusion that a BVP of class WR

displays strong well-posedness in

some Hilbert space, though not in L

. This space is inhomogeneous in

216 A classiﬁcation of linear IBVPs

frequency, and in space-time variables as well, since x

and (y, t) play

diﬀerent roles.

When ∆ vanishes at the ﬁrst order only along Λ, the norms P

w are inter-

mediate between the L

-norm and the H

−1

-norm. Thus (8.4.24) implies,

in the case of constant coeﬃcients in a non-characteristic half-space, the

estimates ‘with loss of one derivative’ obtained by Coulombel [38,39,41,43].

Coulombel’s estimates also hold true with variable coeﬃcients, as well as

in some characteristic cases. We leave for a future study an extension of

our estimates in such contexts.

In the case of a BVP of class WR for the wave operator, a nicer analysis

can be made, see [191]. In a suitable range of parameters, the BVP can

be decomposed in a sequence of two hyperbolic BVP, each one satisfying

(UKL). The sequence of a priori estimates yields an estimate similar to

(8.4.24). This decomposition is robust to variation of coeﬃcients, and thus

can be employed in BVP with variable coeﬃcients.

8.4.3 The estimate for the adjoint BVP

We have seen that the adjoint BVP is of class WR too, with the Lopatinski˘ı

determinant

∆

∗

(θ, σ)=∆(−

θ, −σ).

The standard objects, when associated with the adjoint BVP, are indexed with

a subscript ∗, for instance E

−∗

+∗

,...The superscript ∗ is kept for denoting

adjoint operators. For instance, the adjoint π

∗

−

of the projection onto E

−

,of

kernel E

, is the projection onto E

⊥

, with kernel E

⊥

−

.Sincewehave

±∗





⊥

we ﬁnd the useful identities

∗

−

)

=(A

)

+∗

,π

∗

)

=(A

)

−∗

. (8.4.25)

Hereabove, and everywhere in the following, we always assume the relations

θ = −¯τ and σ = −η. Using (8.4.25), we deduce

˜p

∗

)

=(A

)

(π

−∗

∆π

+∗

from which we infer

˜p

∗

)

˜p

∗

∆(A

)

In the smoothing process, it is possible to keep track of this identity, in the

following way

∗

)

∗

δ(A

)

, (8.4.26)

Well-posedness of unsual type for BVPs of class WR 217

where δ(τ, η) is smooth, homogeneous of degree zero, and coincides with ∆

outside of a small neighbourhood of glancing points. Also, δ vanishes only

along Λ.

The analogue of (8.4.24) for the adjoint problem is obviously



Ω×R

2γt

P

−γ∗

w

dx dt +



∂Ω×R

2γt

P

−γ∗

w

dx dt

≤ C





Ω×R

2γt

P

−γ∗

)

−T

∗

w

dx dt



∂Ω×R

2γt

P

−γ∗

)

−T

Cw

dx dt



(8.4.27)

At the level of the Laplace–Fourier transform, this amounts to saying that

|Re θ|p

∗

(θ, σ)ˆw

+ |p

∗

(θ, σ)ˆw(0)|

≤ C



|Re θ|

p

∗

(θ, σ)(A

)

−T



∗

w

(8.4.28)

+ |p

∗

(θ, σ)(A

)

−T

C ˆw(0)|

8.4.4 Existence result for the BVP

As in Section 4.5.5, we employ a duality argument. We do not give full details,

which essentially mimic those of Section 4.5. We content ourselves with explaining

what role our modiﬁed estimates play.

Let us ﬁx a γ>0. We give ourself a pair (f,g) with

)

−1

f ∈ L

)

−1

g ∈ L

. (8.4.29)

We denote by Y

the set of distributions of the form L

∗

w with the properties

that

−γ∗

)

−T

∗

w ∈ L

−γ

w ∈ L

−γ

,γ

Cw =0.

Admitting that (8.4.27) holds true on Y

, we deduce that the linear map L

∗

w →

w is well-deﬁned (a uniqueness property), with some continuity properties. We

then deﬁne a linear functional

(L

∗

w):=



Ω×R

(w, f)dxdt +



∂Ω×R

(Mw,g)dydt.

To begin with, we majorize the ﬁrst integral. Using the Plancherel formula, it

amounts to the same to deal with the integral of ( ˆw,

f). This has the eﬀect of

decoupling frequencies, so that we need an estimate of the integral with respect

218 A classiﬁcation of linear IBVPs

to x

only. To this end, we write



∞

(ˆw,

f)dx



)

ˆw, (A

)

−1





)

∗

ˆw

,p(A

)

−1



where the last equality makes use of (8.4.26). Using Cauchy–Schwarz, we infer

|(ˆw,

|≤C



∗

ˆw



p(A

)

−1

f

Applying now (8.4.28), we obtain

|(ˆw,

|≤

γ|δ|



∗

)

−T



∗



p(A

)

−1

f

Playing the same game with the second integral, and using the fact that p is

uniformly bounded, we obtain

|(ˆw,

+(M ˆw(0), ˆg)|

≤

|δ|



p(A

)

−1

f

√

|p(A

)

−1

ˆg|





∗

)

−T



∗



This shows that  extends continuously to the space of functions W such that

∗

)

−T

W ∈ L

(Re θ = −γ).

The dual of this space is, from (8.4.26), the space of functions U such that

U ∈ L

(Re τ = γ).

In other words, it is the set of Us such that

U ∈ L

. (8.4.30)

Therefore, there exists a u with property (8.4.30), such that

(L

∗

w)=



Ω×R

∗

w, u)dxdt.

Additionally, we have

P

u

≤ C





)

−1



√

P

)

−1

g



This u is the solution of the boundary value problem for t ∈ R. Note that the

duality method gives directly the dependency upon P

)

−1

g, instead of g,

conﬁrming the analysis done in Section 8.4.2.

8.4.5 Propagation property

There is an important diﬀerence between the classes WR and (UKL), as far as

the propagation (of support or of singularities) is concerned. For a uniformly

Well-posedness of unsual type for BVPs of class WR 219

stable IBVP, it can be proved that the signals propagate not faster than in the

pure Cauchy problem. A simple calculus, using energy inequality, shows that

in the strictly dissipative Friedrichs-symmetric case, the support of the solution

does not propagate faster than expected.

This is no longer true for an IBVP of class WR

, because of the fact that

the ‘boundary symbol’ p(iρ, η) vanishes along the set Λ, which is strictly included

in the forward characteristic cone in general.

Let us consider as an example the wave equation

∂

u = c

∆

with the boundary condition

∂u

∂ν

= γ

∂u

∂t

+ g,

with γ ∈ (0, 1/c) a constant, and ∂u/∂ν the normal derivative. This problem can

be recast at a ﬁrst-order system. We leave the reader to check that this BVP is

of class WR

, and to compute that Λ consists in the pairs (iρ, η) such that

|η|

= ρ



− γ



This reveals that signals propagate along the boundary at the velocity



1 − c

which is larger than c. In particular, c



tends to +∞ as γ tends to 1/c; recall

that the IBVP is not normal (and therefore ill-posed) for γ =1/c, and that it

does not satisfy at all the Kreiss–Lopatinski˘ı condition for γ>1/c.

VARIABLE-COEFFICIENTS INITIAL BOUNDARY

VALUE PROBLEMS

This chapter is the logical continuation of Chapter 4 on Initial Boundary Value

problems. We are concerned here with variable-coeﬃcient operators

L := ∂



j=1

(x, t) ∂

where x lies in a domain Ω strictly smaller than R

having a smooth boundary

∂Ω: we will not consider domains with corners or edges despite their physical

interest (e.g. in ﬂuid dynamics with the entrance of pipes), because the analysis

of the corresponding Initial Boundary Value Problems is not well understood up

to now (see [100, 153, 154, 175]).

Unless otherwise speciﬁed, the matrices A

(x, t) will be C

∞

functions of

(x, t), independent of (x, t) outside a compact subset of R

× R

. We assume

throughout this chapter that the operator L is hyperbolic in the direction of t,

which means in particular that the characteristic matrix

A(x, t, ξ)=



j=1

(x, t)

is diagonalizable in R for all ξ ∈ R

\{0}. In fact, we will mostly concentrate

on the more restricted class of constantly hyperbolic operators, for which by

deﬁnition the eigenvalues of A(x, t, ξ) are of constant multiplicities.

Boundary conditions are supposed to be encoded by a smooth matrix-

valued function B :(x, t) ∈ ∂Ω × (0, +∞) → B(x, t), the rank of B(x, t)being

prescribed by a frozen-coeﬃcients analysis: accordingly with Proposition 4.1,

the rank of B(x, t) must be equal to the number of incoming characteristics,

that is, the number of negative eigenvalues (counted with multiplicity) of the

characteristic matrix computed at (x, t, ξ = ν(x)), where ν(x) is the exterior

unit normal vector to ∂Ωatpointx.

Independently of the speciﬁc boundary conditions, we will thus require that

the number of negative eigenvalues of A(x, t, ν(x)) be constant, which is deﬁnitely

not innocent; unless A is independent of (x, t)andν is independent of x,

i.e. L has constant coeﬃcients and Ω is a half-space, as in Chapter 4. This

requirement on the number of negative eigenvalues of A(x, t, ν(x)) is (more

Variable-coeﬃcients initial boundary value problems 221

generally) fulﬁlled by non-characteristic problems in domains with connected

boundaries. (We warn the reader though, that it might be diﬃcult to ensure

the whole boundary is non-characteristic: see Section 11.1.1 for a more detailed

discussion in a quasilinear framework.) As a matter of fact, if the boundary ∂Ωis

everywhere non-characteristic, that is, if the matrix A(x, t, ν(x)) is non-singular

along ∂Ω × (0, +∞), and if additionally ∂Ω is connected, the hyperbolicity of

the operator L implies that the eigenvalues of A(x, t, ν(x)) are split into a

constant number of negative ones and a constant number of positive ones:

indeed, if A(x, t, ν(x)) is non-singular and only has real eigenvalues then of

course it has no eigenvalue on the imaginary axis; in the language of ODEs,

this means A(x, t, ν(x)) is a hyperbolic matrix for all (x, t) ∈ ∂Ω × (0, +∞)and

a connectedness argument then shows the dimension of its stable subspace must

be constant along ∂Ω × (0, +∞).

Denoting by p the number (assumed constant, no matter the restriction) of

incoming characteristics, we may suppose without loss of generality that B is

everywhere of maximal rank p,thatis,B(x, t) ∈ M

p×n

(R) for all (x, t) ∈ ∂Ω ×

(0, +∞), the p rows of B(x, t) being independent.

Additional requirements are to be imported from Chapter 4. Not only must

the rank of B coincide with the dimension of the stable subspace E

(A)ofA but

we should have the normality condition:

=kerB(x, t) ⊕ E

(A(x, t, ν(x))) for all (x, t) ∈ ∂Ω × (0, +∞) . (9.0.1)

If we were to consider possibly characteristic boundaries, we should also require

that

kerA(x, t, ν(x)) ⊂ kerB(x, t) for all (x, t) ∈ ∂Ω × (0, +∞) .

But we will concentrate on non-characteristic problems.

Finally, we will need an assumption speciﬁc to variable coeﬃcients: we ask

that the kernel of B admit a smooth basis, that is, a family of C

∞

vector-valued

functions (e

p+1

,...,e

) such that

Span (e

p+1

(x, t),...,e

(x, t)) = kerB(x, t)

for all (x, t) ∈ ∂Ω × (0, +∞). A standard result in diﬀerential topology [85]

(p. 97), saying that any vector bundle over a contractible manifold is trivial,

implies the existence of such a smooth basis for some particular boundaries ∂Ω:

for instance, a hyperplane is contractible. For non-contractible boundaries ∂Ω,

the existence of a smooth basis is a non-trivial assumption.

Our aim is to solve Initial Boundary Value Problems (IBVP) of the form

(Lu)(x, t)=f(x, t),x∈ Ω ,t>0 , (9.0.2)

(Bu)(x, t)=g(x, t),x∈ ∂Ω ,t>0 , (9.0.3)

u(x, 0) = u

(x),x∈ Ω , (9.0.4)

222 Variable-coeﬃcients initial boundary value problems

where the source term f, the boundary data g and the initial condition u

are

given in a Sobolev space H

, s ≥ 0. We may also add a zeroth-order operator to

As for the Cauchy problem, well-posedness crucially relies on energy esti-

mates. As for the Cauchy problem, the basic tools for deriving energy estimates

are symmetrizers. However, symmetrizers for IBVPs are much more compli-

cated than for the Cauchy problem: they were ﬁrst introduced by Kreiss in

his celebrated paper [103] (see also [160]); up to now, the main reference is

the (unfortunately depleted) book by Chazarain and Piriou [31], where Kreiss’

symmetrizers are constructed in detail and used to show the well-posedness of

IBVPs. For constant-coeﬃcient problems, Chapter 5 of this book gives a new

construction of Kreiss’ symmetrizers, and Chapter 4 shows how to use them for

the well-posedness theory.

The purpose of this chapter is to answer the two main questions: 1) how

do Kreiss’ symmetrizers associated with frozen coeﬃcients systems yield energy

estimates for variable coeﬃcients? and 2) how do energy estimates imply the

well-posedness of Initial Boundary Value Problems?

As for the Cauchy problem, we shall at ﬁrst deal with inﬁnitely smooth

coeﬃcients. In this case, the answers to questions 1) and 2) make use of pseudo-

diﬀerential calculus with parameter. They are contained in Chapter VII of [31].

Our presentation is diﬀerent but makes use of the same arguments.

Coeﬃcients with poorer regularity (arising, for instance, in the resolution

of quasilinear problems) are trickier to deal with, and will be considered sep-

arately. The main reference on this topic is the (unpublished) PhD thesis of

Mokrane [140], which takes advantage of para-diﬀerential calculus. We shall give

a presentation here that parallels the smooth coeﬃcients theory, and point out

the special features related to poor regularity.

In any case, we will only consider non-characteristic problems, for the analysis

of variable-coeﬃcient characteristic problems is still in its infancy: apart from the

seminal paper by Majda and Osher [127], most results known concern dissipa-

tive boundary conditions for Friedrichs-symmetrizable systems; as far as linear

problems are concerned, the main references are the papers by Rauch [163,164]

(see Chapter 11 for further references concerning quasilinear problems).

Another deliberate choice by us is to consider ﬁrst the simplest case of a

planar boundary and explain afterwards how to deal with arbitrary domains

through co-ordinate charts.

9.1 Energy estimates

As announced above, we proceed gradually and ﬁrst consider a half-space

Ω, which we assume to be Ω = {x ∈ R

; x

> 0 } without loss of generality.

Furthermore, we assume ∂Ωisnon-characteristic, which means that A

(y, 0,t)

is invertible for all (y, t)=(x

,...,x

d−1

,t) ∈ R

d−1

× (0, +∞) (with an obvious

convention if d =1).

Energy estimates 223

The main purpose here is to derive energy estimates for the Boundary Value

Problem

(Lu)(y, x

,t)=f(y, x

,t),y∈ R

d−1

,t∈ R ,x

> 0 , (9.1.5)

(Bu)(y, 0,t)=g(y,t),y∈ R

d−1

,t∈ R , (9.1.6)

in which the coeﬃcients of L and B are supposed to be well-deﬁned and

smooth for all t ∈ R. We have in mind energy estimates that are weighted

in time, as in Theorem 2.13. This is why we introduce the new unknown

u

(x, t):=e

−γt

u(x, t) and new data



, g

in the same way. We observe that

(9.1.5)(9.1.6) is equivalent to

u



, Ω × R , (9.1.7)

Bu

= g

,∂Ω × R (9.1.8)

with

:= ∂

+ γ +



j=1

∂

The whole diﬃculty lies in the fact that the principal part L

= L of the operator

is in general not hyperbolic in the direction of x

. This is not speciﬁc to variable

coeﬃcients. For, the hyperbolicity of L in the direction of x

requires that, for

any ‘frequency’

ξ =(η, δ)=(η

,...,η

d−1

,δ) ∈ R

\{0} the polynomial

det( δI

d−1



j=1

a + ζA

)

only has real roots ζ. This holds true in the frequency region called hyperbolic

for obvious reasons, which is in general not the whole frequency space.

Example In gas dynamics (see Chapter 13), the polynomial above can be

explicitly factorized as

( δ +(η, ζ) · u )

d−1



( δ +(η, ζ) · u )

− c

(η

+ ζ

)



where u denotes the gas velocity and c the sound speed. If the ﬁrst factor only

has real roots, in fact the multiple one

ζ = −(δ + η ·

u)/u

where

u := (u

,...,u

d−1

) denotes the tangential part of the velocity u =(

u,u

the second factor has real roots if and only if

(δ + η ·

≥ (c

− u

) η

The term frequency here is used in a wide mathematical sense, and does not refer speciﬁcally to

time oscillations.

224 Variable-coeﬃcients initial boundary value problems

This inequality holds true everywhere for supersonic ﬂows (u

≥ c

). But for a

subsonic ﬂow (u

), the hyperbolic region of L in the direction x

is restricted

to the cone

{ξ =(δ, η); (δ + η ·

≥ (c

− u

) η

It is notable that, for points ξ =(δ, η) on the boundary of the cone, i.e. such

that

(δ + η ·

=(c

− u

) η

called glancing modes, the matrix

)

−1

( δI

d−1



j=1

) ,

which will turn out to play a crucial role in the analysis, is not diagonalizable.

Coming back to the abstract problem (9.1.5)(9.1.6) and considering the

special role played by the variable x

, we rewrite (9.1.7) in the equivalent form

∂

u

− P

) u

=(A

)

−1



, (9.1.9)

):=−(A

)

−1

( ∂

+ γ +

d−1



j=1

∂

) .

This notation emphasizes the dependence of P

on the parameter x

, but it

should be clear to the reader that P

) is a variable-coeﬃcient diﬀerential

operator in (y, t). More precisely, in the terminology recalled in the appendix

(section C.2), for all x

≥ 0, {P

)}

γ≥1

is a family of diﬀerential operators of

order 1 on {(y, t) ∈ R

d−1

× R}, their symbols being

a(y, t, η, δ, γ; x

):=−(A

(y, x

,t))

−1

((iδ + γ) I

+ iA(y, x

,t,η)),

where

A(y, x

,t,η)=

d−1



j=1

(y, x

,t) .

Consistently with Chapter 4, we will also use the shorter notation

A(x, t, η, τ):=a(y, t, η, δ, γ; x

where x =(y, x

)andτ = γ + iδ.

Dropping for a while the tildas and the γ subscript in (9.1.7)(9.1.8), we are

facing a problem of the form

∂

u − P

) u = f, x

> 0 ,

Bu = g, x

=0.