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

Подождите немного. Документ загружается.

Basic calculus results 445

Proof The proof is fully elementary. Take D>Cand consider n ∈ N such

that ε

n+1

satisﬁes

−Dε

≤ 1 − Cε

For all s ∈ [0,t− ε

], we have

a(s + ε

) ≤

1 − Cε

a(s)+

Cε

1 − Cε

(ε

+ b(s)) .

Therefore,

−Dt

a(t) − a(0) =



k=0

−D (k+1) ε

a((k +1)ε

) − e

−Dkε

a(kε

)

≤



k=0

−Dkε



−Dε

1 − Cε

− 1



a(kε

)



k=0

−D (k+1) ε

Cε

1 − Cε

(ε

+ b(kε

))

≤

1 − Cε

(ε

+ b(t))



−Ds

ds ≤

D(1 − Cε

)

(ε

+ b(t)).

We get the ﬁnal estimate by letting n go to ∞ and then D go to C. 

FOURIER AND LAPLACE ANALYSIS

B.1 Fourier transform

There are several possible conventions for the deﬁnition of a Fourier transform,

depending on where the number 2π shows up.

We adopt the following one. For all v ∈ L

) its Fourier transform F v = v

is the continuous and bounded function deﬁned by

F v(η)=v(η):=



−iη·x

v(x)dx

for all η ∈ R

For any d-uple α =(α

, ···,α

) ∈ N

, we adopt the shortcut

∂

= ∂

···∂

for diﬀerential operators of order |α| :=



k=1

, either in the original variables

(x) or in the frequency variables (η).

For convenience, we state the following standard results.

Theorem B.1

The Fourier transform F restricted to the Schwartz class S (R

) is an

automorphism.

By duality, it can be also deﬁned on S



), where it is still an automor-

phism.

For al l v ∈ S (R

) and all d-uple α, we have

F [∂

v](η)=(iη)

v(η)

for all η ∈ R

. And this formula extends to S



) in the sense that for

v ∈ S



), the distribution F [∂

v] is v times the polynomial function

η → iη

(Plancherel’s identity) For al l v ∈ S (R

v

)

=(2π)

d/2

v

)

By density, F extends to an automorphism of L

), which still satisﬁes

Plancherel’s identity.

Laplace transform 447

(Inversion formula) If both v and v are integrable, one recovers v from

v through the inversion formula

v(x)=(2π)

−d



iη·x

v(ξ)dξ.

Another useful result is the following, of which the easy part is the direct one.

Theorem B.2 (Paley–Wiener) If v ∈ D (R

) then v extends to an analytic

function V on C

. Furthermore, if K = Supp v, for all p ∈ N, there exists C

0 so that

|V (ζ)|≤

(1 + |ζ|)

exp



max

x∈K

(x · Im ζ)



for all ζ ∈ C

. Conversely, if V is an analytic function on C

satisfying the above

estimate for some convex compact set K, there exists v ∈ D (R

) with support

included in K such that v = V |

The ‘dual’ result also holds true.

Theorem B.3 (Paley–Wiener–Schwartz) If v is a compactly supported distrib-

ution then v extends to an analytic function V on C

through the formula

V (ζ)=v, e

− iζ·

.

Furthermore, if K = Supp v, there exist p ∈ N and C

> 0 so that

|V (ζ)|≤C

(1 + |ζ|)

exp



max

x∈K

(x · Im ζ)



for all ζ ∈ C

. Conversely, if V is an analytic function on C

satisfying the above

estimate for some convex compact set K, there exists v ∈E



) with support

included in K such that v = V |

B.2 Laplace transform

The Laplace transform applies to functions of one variable t ∈ (0, +∞).

Deﬁnition B.1 If f is a measurable function of t ∈ (0, +∞) and if there exists

a ∈ R so that t → e

− at

f(t) is square-integrable, the Laplace transform of f is

the function L[f], holomorphic in the half-plane {τ ;Reτ>a}, deﬁned by

L[f](τ)=



+∞

f(t)e

− τt

dt.

The Laplace transform may be interpreted in terms of a Fourier transform.

Indeed, for all γ>aand δ ∈ R,

L[f](γ + iδ)=



(gf)(δ) ,

448 Fourier and Laplace analysis

where g(t)=1

(t)e

− γt

. Therefore, the theorem of Paley–Wiener has a coun-

terpart in Laplace transforms theory. This is the main result we need regarding

Laplace transforms in this book.

Theorem B.4 (Paley–Wiener) If F is an holomorphic function in the right

half-plane {τ ;Reτ>a}, which is square-integrable on each vertical line

{τ ;Reτ = γ }, γ>a, and if there exists C>0 so that

sup

γ>a



+∞

−∞

|F (γ + iδ) |

dδ ≤ C,

then there exists f ∈ L

− at

dt}) (meaning that t → e

− at

f(t) is square-

integrable) such that F = L[f]. If, additionally, F is integrable on the vertical

line {τ ;Reτ = γ }, γ>a, we recover f through the inversion formula

f(t)=

2 iπ



Re τ=γ

F (τ )e

τt

ds.

B.3 Fourier–Laplace transform

Fourier and Laplace transformations can be extended to vector-valued functions

– in fact functions with values in Banach spaces. From a practical point of view,

this amounts to saying that for functions of several variables one may perform

Fourier or Laplace transformations in some variables only and keep the same

regularity/decay properties in the other variables.

More precisely, let u be a function of (x, t) ∈ R

× R

that belongs to

× R

− at

dt dx}) .

For almost all t ∈ R

, the function u(t):x ∈ R

→ u(x, t) is square-integrable

and thus admits a Fourier transform



u(t) ∈ L

) such that the function

(ξ,t) →



u(t)(ξ) belongs to

× R

− at

dt dξ}) .

Therefore, for almost all ξ ∈ R

, the function t ∈ R

→ e

− at



u(t)(ξ)issquare-

integrable and thus the Laplace transform

L[t →



u(t)(ξ)]

is well-deﬁned and holomorphic on the half-plane {τ ;Reτ>a}. The function

(ξ,τ) →L[t →



u(t)(ξ)](τ)

is what we call the Fourier–Laplace transform of u.

PSEUDO-/PARA-DIFFERENTIAL CALCULUS

The aim of this appendix is to facilitate the reading of the book for those who

are not familiar with para-diﬀerential or even pseudo-diﬀerential calculus. Just

a basic background on distributions theory and Fourier analysis is assumed.

As many textbooks deal with pseudo-diﬀerential calculus (see for instance

[7, 31, 87, 205]), we recall here only basic deﬁnitions and useful results for our

concern, mostly without proof. Para-diﬀerential calculus is much less widespread.

Originally developed by Bony [20] and Meyer [138], it has been used since then in

various contexts, in particular by G´erard and Rauch [68] for non-linear hyperbolic

equations and more recently by M´etivier and coworkers for hyperbolic initial

boundary value problems – see in particular the lectures notes [136]. Other

helpful references on para-diﬀerential calculus are [33, 88, 206]. We detail in this

appendix the most accessible part of the theory of para-diﬀerential operators,

among which we ﬁnd para-products. Some useful results are gathered together

with their complete (and most often elementary) proof, using the Littlewood–

Paley decomposition. The rest of the theory is presented heuristically, together

with a collection of results used elsewhere in the book. Additionally, we borrow

from [31] and [136] versions of pseudo-diﬀerential and para-diﬀerential calculus

with a parameter, which are needed for initial boundary value problems.

We use standard notations from diﬀerential calculus. To any d-uple α =

(α

, ···,α

) ∈ N

, we associate the diﬀerential operator of order |α| :=



k=1

∂

|α|

∂x

···∂x

When no confusion can occur this operator is simply denoted by ∂

.This

notation should not be mixed up with the one used throughout the book

∂

∂x

for α ∈{1, ···,d}. To avoid confusion, the indices lying in {1, ···,d} are here

preferably denoted by roman letters (typically j).

450 Pseudo-/para-diﬀerential calculus

C.1 Pseudo-diﬀerential calculus

C.1.1 Symbols and approximate symbols

Pseudo-diﬀerential operators are deﬁned through their symbol, which is a function

depending on x ∈ R

and on the ‘dual’ variable, or frequency, ξ ∈ R

.This

function is not polynomial in ξ, except for (standard) diﬀerential operators –

for instance the symbol of ∂

(i)

|α|

=(iξ

)

···(iξ

)

In the classical theory, symbols are scalar-valued (in C). As far as we are con-

cerned, matrix-valued symbols are also of interest. This extension costs nothing,

but some little care in Lemma C.1 below and in the handling of commutators.

From now on, we ﬁx some integers d and N that will be omitted in the

notations if no confusion can occur.

Deﬁnition C.1 For any real number m, we deﬁne the set S

of functions

a ∈ C

∞

× R

; C

N×N

) such that for all d-uples α and β there exists C

α,β

> 0

so that

∂

∂

a(x, ξ) ≤C

α,β

(1 + ξ)

m−|β|

. (C.1.1)

Symbols belonging to S

are said to be of order m. The set of symbols of all

orders is

−∞

Basic examples

Diﬀerential symbols. Functions of the form

a(x, ξ)=



|α|≤m

(x)(iξ)

where all the coeﬃcients a

are C

∞

and bounded, as well as all their derivatives,

belong to S

‘Homogeneous’ functions. A function a ∈ C

∞

× R

\{0})thatis

bounded as well as all its derivatives in x and homogeneous degree m in ξ is

‘almost’ a symbol of order m. This means that it becomes a symbol provided

that we remove the singularity at ξ = 0. As a matter of fact, considering a C

∞

function χ vanishing in a neighbourhood of 0 and such that χ(ξ)=1forξ≥1,

we have the result that

a(x, ξ)=χ(ξ) a(x, ξ)

belongs to S

. Any other symbol constructed in this way diﬀers from a by a

symbol in S

−∞

. For convenience we shall denote

the set of such functions a.

Pseudo-diﬀerential calculus 451

Sobolev symbols Some special symbols are extensively used in the theory,

which we refer to as Sobolev symbols since they are naturally involved in Sobolev

norms. Denoting

(ξ):=(1+ξ

)

s/2

it is easily seen that λ

is a symbol of order s. The important point is that the

Sobolev space H

can be equipped with the norm

u 

= λ

u 

Additionally, the following result will be of interest in the proof of Theorem

C.4.

Lemma C.1 For al l a ∈ S

, respectively a ∈

, such that a(x, ξ) is Hermitian

and uniformly positive-deﬁnite for (x, ξ) ∈ R

× R

, respectively, for (x, ξ) ∈

× R

\{0}, there exists b ∈ S

, respectively b ∈

, such that b(x, ξ)

∗

b(x, ξ)=

a(x, ξ).

Proof The proof proceeds in the same way in both cases (a ∈ S

or a ∈

By assumption, a(x, ξ) lies in a bounded subset of the cone of Hermitian positive-

deﬁnite matrices and thus the set of eigenvalues of a(x, ξ) is included in some real

interval [α, β] ⊂ (0, +∞). In particular, there exists a positively oriented contour

Γ lying in C\(−∞, 0] that is symmetric with respect to the real axis and contains

[α, β] in its interior. Therefore, considering the holomorphic complex square root

√

in C\(−∞, 0], the Dunford–Taylor integral

b(x, ξ):=

2iπ



√

z ( z − a(x, ξ))

−1

answers the question. As a matter of fact, by the symmetry of Γ, b(x, ξ)is

obviously Hermitian, and

∗

b =

−1

4π





√



( z − a )

−1

( z



− a )

−1

dz dz



for another contour Γ



enjoying the same properties as Γ and containing it in its

interior. By the well-known resolvent equation, we thus have

∗

b =

(2iπ)





√



( z − a )

−1

− ( z



− a )

−1



− z

dz dz



On the one hand, for z



∈ Γ



, the function z →

√

z/(z



− z) is holomorphic in the

interior of Γ and thus



√



− z

dz =0.

452 Pseudo-/para-diﬀerential calculus

On the other hand, by Cauchy’s formula we have

√

z =

2iπ





√



− z



for z ∈ Γ. This eventually proves that

∗

b =

2iπ



z ( z − a )

−1

dz = a.

In view of the smoothness of the mapping (z,a) → ( z − a )

−1

, it is clear by the

chain rule and Lebesgue’s theorem that b is as smooth as a. It is also easily shown

by induction that b satisﬁes the estimate (C.1.1) with m =0,ifa belongs to S

If a belongs to

instead, it is obvious that b is also homogeneous degree 0 in

ξ. 

C.1.2 Deﬁnition of pseudo-diﬀerential operators

The introduction of pseudo-diﬀerential operators is based on the following obser-

vation. If a ∈ S

is polynomial in ξ, like in the ﬁrst basic example given above,

it is naturally associated with the diﬀerential operator

Op(a)=



|α|≤m

(x) ∂

in the sense that

(Op(a)u)(x)=F

−1

(a(x, ·) u)

for all u ∈ S and x ∈ R

. But this formula can be used to deﬁne operators

associated with more general symbols. This is the purpose of the following.

Proposition C.1 Let a be a symbol of order m. Then there exists a continuous

linear operator on S , denoted by Op(a), such that

(Op(a)u)(x)=

(2π)



ix·ξ

a(x, ξ) u(ξ)dξ (C.1.2)

for all u ∈ S . Furthermore, the mapping a → Op(a) is one-to-one.

Observe that for ‘constant-coeﬃcient symbols’, that is, symbols independent

of x, (C.1.2) reduces to the same formula as for diﬀerential operators

Op(a)u = F

−1

(a u) .

In short, for symbols a depending only on ξ, we have by deﬁnition

Op(a)=F

−1

a F .

Deﬁnition C.2 The set of pseudo-diﬀerential operators of order m is

OPS

:= {Op(a); a ∈ S

}⊂B(S ) .

Pseudo-diﬀerential calculus 453

For a ∈ S

, the operator Op(a) is called a pseudo-diﬀerential operator of order

m and symbol a.

It is more subtle to show that pseudo-diﬀerential operators extend to oper-

ators on S



. By a standard duality argument, this amounts to showing the

following.

Theorem C.1 The adjoint of a pseudo-diﬀerential operator of order m is a

pseudo-diﬀerential operator of order m. Furthermore, the symbol of the adjoint

operator Op(a)

∗

diﬀers from a

∗

(where a

∗

(x, ξ):=a(x, ξ)

∗

merely in the sense

of matrices) by a symbol of order m − 1, which means that

(Op(a))

∗

− Op(a

∗

) ∈ OPS

m−1

(C.1.3)

for all a ∈ S

The proof of this theorem is a very ﬁne piece of analysis. The interested reader

may refer to [7, 87, 205].

C.1.3 Basic properties of pseudo-diﬀerential operators

The ﬁrst important property of pseudo-diﬀerential operators is the following.

Theorem C.2 Let P be a pseudo-diﬀerential operator of order m, extended to



by the formula

Pu, φ = u, P

∗

φ 

for all u ∈ S



and φ ∈ S .Then,foralls ∈ R, P belongs to B(H

; H

s−m

A straightforward example. For all s ∈ R,letΛ

denote the pseudo-

diﬀerential operator of symbol λ

as deﬁned in Section C.1.1. Then for all real

numbers s we have

u

= Λ

u

Therefore, we have for all m ∈ R

Λ

u

s−m

= Λ

s−m

u

= Λ

u

= u

In fact, using the operators Λ

and Λ

−m

, all cases of Theorem C.2 can be

deduced from the case m = s = 0 and the other following basic result.

Theorem C.3 If P and Q are pseudo-diﬀerential operators of order m and n,

respectively, then

i) the composed operator PQ is a pseudo-diﬀerential operator of order m + n,

and its symbol diﬀers from the product of symbols by a lower-order term,

which means that

Op(a)Op(b) − Op(ab) ∈ OPS

m+n−1

(C.1.4)

for all a ∈ S

and b ∈ S

454 Pseudo-/para-diﬀerential calculus

ii) if one of the operators is scalar-valued, the commutator

[P, Q]:=PQ− QP

is of order m + n − 1 (and its symbol diﬀers from the Poisson bracket of

symbols

{a, b} :=



∂a

∂ξ

∂b

∂x

−

∂a

∂x

∂b

∂ξ

by a lower-order term).

The proofs of Theorems C.2 and C.3, as well as more complete results, can

be found in the textbooks quoted above [7,31,87,205]. Note that apart from the

bracketed statement, ii) is a trivial consequence i).

Finally, other important results are the G˚arding inequality, which relates

the positivity of an operator (up to a lower-order error) to the positivity of

its symbol, and the sharp form of G˚arding’s inequality, which applies to non-

negative symbols. We begin with the standard form of G˚arding’s inequality (for

matrix-valued symbols) and its elementary proof.

Theorem C.4 (G˚arding inequality) If A is a pseudo-diﬀerential operator of

symbol a ∈ S

,orA is associated with a ∈

by a low-frequency cut-oﬀ, such

that for some positive α

a(x, ξ)+a(x, ξ)

∗

≥ αλ

(ξ) I

(in the sense of Hermitian matrices) for all x ∈ R

and ξ large, then there

exists C so that

Re Au , u ≥

u

m/2

− C u

m/2−1

(C.1.5)

for all u ∈ H

m/2

Proof Replacing A by Λ

−m/2

A Λ

−m/2

, we can suppose without loss of gener-

ality that m =0.

• We have Re Au , u  =Re

(A + A

∗

)u, u, and by (C.1.3) we know

that

(A + A

∗

) − Op(a + a

∗

) ∈ OPS

−1

Therefore, there exists c>0sothat

(A + A

∗

)u, u≥Op(a + a

∗

)u, u−c u

−1

u

≥Op(a + a

∗

)u, u−

u

−

u

−1