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

Подождите немного. Документ загружается.

NOTATIONS

The set of matrices with n rows and p columns, with entries in a ﬁeld K, is denoted

by M

n×p

(K). If p = n, we simply write M

(K). The latter is an algebra, whose

neutral elements under addition and multiplication are denoted by 0

and I

respectively. The space M

n×p

(K) may be identiﬁed to the set of linear maps

from K

to K

. The transpose matrix is written M

. The group of invertible

n × n matrices is GL

(K). If p =1,M

n×1

(K) is identiﬁed with K

Given two matrices M,N ∈ M

(C), their commutator MN − NM is denoted

by [M,N].

If K = C, the adjoint matrix is written M

∗

. It is equal to M

,whereM

denotes the conjugate of M. We equip C

and R

with the canonical Hermitian

norm

x =





| x

=(x

∗

1/2

This norm is associated to the scalar product

(x, y)=



= y

∗

The norm will sometimes be denoted |x|, especially when x is a space variable

or a frequency vector (used in Fourier transform.)

A complex square matrix U is unitary if U

∗

U = I

, or equivalently UU

∗

= I

The set U

of unitary matrices is a compact subgroup of GL

(C). Its intersection

with M

(R) is the set of real orthogonal matrices. The special orthogonal

group SO

is the subgroup deﬁned by the constraint det M =1.

As usual, M

n×p

M =sup

Mx

x

When the product makes sense, one knows that MN≤MN.Whenp =

n, M

≤M

.IfQ is a unitary

(for instance real orthogonal) matrix, one has Q = 1. More generally, the norm

is unitary invariant, which means that M = PMQ whenever P and Q are

unitary.

If M ∈ M

(C), the set of eigenvalues of M, denoted by SpM, is called the

spectrum of M. The largest modulus of eigenvalues of M is called the spectral

radius of M, and denoted by ρ(M). It is less than or equal to M,andsuch

xxii Notations

that

ρ(M) = lim

k→+∞

M



1/k

The following formula holds,

M

= ρ(M

∗

M)=ρ(MM

∗

Several other norms on M

norm, deﬁned by

M





j,k

| m

Since M

=Tr(M

∗

M)=Tr(MM

∗

), we have M≤M

A complex square matrix M is Hermitian if M

∗

= M. It is skew-Hermitian if

∗

= −M. The Hermitian n × n matrices form an R-vector space that we denote

by H

. The cone of positive-deﬁnite matrices in H

is denoted by HPD

.When

M is Hermitian, we have M = ρ(M). Every Hermitian matrix is diagonalizable

with real eigenvalues, its normalized eigenvectors forming an orthonormal basis.

The skew-Hermitian matrices with complex entries form an R-vector space that

we denote by Skew

.WeremarkthatM

(C)=H

⊕ Skew

and Skew

. The intersections of H

, HPD

and Skew

with the subspace M

(R)of

matrices with real entries are denoted by Sym

, SPD

and Alt

, respectively.

We have M

(R)=Sym

⊕ Alt

. Real symmetric matrices have real eigenvalues

and are diagonalizable in an orthogonal basis.

Given an n × n matrix M , one deﬁnes its exponential by

exp M =e

∞



k=0

which is a convergent series. The map t → exp(tM) is the unique solution of the

diﬀerential equation

= MA,

such that A(0) = I

. It solves equivalently the ODE

= AM.

The exponential behaves well with respect to conjugation, that is

exp(PMP

−1

)=P (exp M)P

−1

for all invertible matrix P . The eigenvalues of exp A are the exponentials of those

of A. In particular, ρ(exp A) is the exponential of the maximal real part Re λ,

as λ runs over SpA. The matrix exp(A + B) does not equal (exp A)(exp B)in

Notations xxiii

general, but it does when AB = BA. In particular, exp A is always invertible,

with inverse exp(−A). Other useful formulæ are

exp(M

)=(expM )

, exp M = exp M, exp(M

∗

)=(expM )

∗

The exponential of a Hermitian matrix is Hermitian, positive-deﬁnite. The map

exp : H

→ HPD

is actually an homeomorphism. The exponential of a skew-Hermitian matrix is

unitary.

Let A ∈ M

splits, in a unique way, as the

direct sum of three invariant subspaces, namely the stable, unstable and central

subspaces of A, denoted, respectively, E

(A), E

(A)andE

(A). Their invariance

properties read

(A)=E

(A),AE

(A)=E

(A)andAE

(A) ⊂ E

(A).

The stable invariant subspace is formed of vectors x such that (exp tA)x tends

to zero as t → +∞, and then the decay is exponentially fast. The unstable

subspace is formed of vectors x such that (exp tA)x tends to zero (exponentially

fast) as t →−∞. The central subspace consists of vectors such that (exp tA)x is

polynomially bounded on R. Since these spaces are invariant under A, this matrix

operates on each one as an endomorphism, say A

. The spectrum of A

(respectively, A

, A

) has negative (respectively, positive, zero) real part. The

union of these spectra is the whole spectrum of A, with the correct multiplicities.

Hence the dimension of E

(A) is the number of eigenvalues of A of negative real

part (these are called ‘stable eigenvalues’), counted with multiplicities. When

(A) is trivial, meaning that there is no pure imaginary eigenvalue, A is called

hyperbolic (in the sense of Dynamical Systems).

Dunford–Taylor formula. Let γ be a Jordan curve, oriented in the trigono-

metric way, disjoint from SpA.Letσ be the part of SpA that γ enclose. Then

the Cauchy integral

2iπ



(zI

− A)

−1

deﬁnes a projector (that is P

= P

) whose range and kernel are invariant under

A. (Moreover, A commutes with P

). The spectrum of the restriction of A to

the range of P

is exactly the part of the spectrum of A that belongs to σ.In

other words, R(P

) is the direct sum of the generalized eigenspaces associated

to those eigenvalues in σ.

More information about matrices and norms may be found in [187].

Functional spaces

GivenanopensubsetΩofR

, the set of inﬁnitely diﬀerentiable functions (with

values in C) that are bounded as well as all their derivatives on

Ω is denoted

xxiv Notations

by C

∞

(Ω). The set of compactly supported inﬁnitely diﬀerentiable functions

(also called test functions) is denoted by D (Ω). Its dual D



(Ω) is the space of

distributions. The derivation ∂

:= ∂/∂x

is a bounded linear operator on D (Ω).

Its adjoint is therefore bounded on D



(Ω). The distributional derivative, still

denoted by ∂

, is the adjoint of −∂

A multi-index α is a ﬁnite sequence (α

,...,α

) of natural integers. Its length

|α| is the sum



. The operator

∂

:= ∂

···∂

is a derivation of order |α|. We also use the notation

= ξ

···ξ

when ξ ∈ R

Given a C

function f :Ω→ C, the diﬀerential of f at point X is the linear

form

df(X):ξ → df (X)ξ :=



j=1

∂

f(X).

The map X → df(X) (that is the diﬀerential of f ) is a diﬀerential form. The

second diﬀerential, or Hessian of f at X is the bilinear form

f(X):(ξ, η) →



i,j=1

∂

f(X) .

We may deﬁne diﬀerentials of higher orders D

f,...

Given a Banach space E, the Lebesgue space of measurable functions u :Ω→

E whose pth power is integrable, is denoted by L

(Ω; E). When E = R or E = C,

we simply denote L

(Ω) if there is no ambiguity. The norm in L

(Ω; E)is

u





Ω

u(x)



1/p

If m ∈ N, the Sobolev space W

m,p

(Ω; E) is the set of functions in L

(Ω; E)whose

distributional derivatives up to order m belong to L

. Its norm is deﬁned by

u

m,p







|α|≤m

∂

u





1/p

If p =2andifE is a Hilbert space, W

m,2

(Ω; E) is a Hilbert space and is denoted

(Ω; E), or simply H

(Ω) if E = C or E = R or if there is no ambiguity.

Sobolev spaces of order s (instead of m) may be deﬁned for every real

number s. The simplest deﬁnition occurs when p =2,Ω=R

and E = C,where

) is isomorphic to a weighted space L

((1 + |ξ|

)

dξ) through the Fourier

transform. For a crash course on H

(Ω) (sometimes also denoted H

(Ω)), we

Notations xxv

refer the reader to Chapter II in [31]; for more details in more general situations,

see for instance the classical monograph by Adams [1]. The notation H

will

stand for the Sobolev space H

equipped with the weak topology instead of the

(strong) Hilbert topology.

The Schwartz space of rapidly decreasing functions S (R

) will simply be

denoted by S when no confusion can occur as concerns the space dimension. And

similarly, its dual space, consisting of temperate distributions, will be denoted

by S



Other tools

We have collected in the appendix various additional tools, ranging from standard

calculus and Fourier–Laplace analysis to pseudo-diﬀerential and para-diﬀerential

calculus: we hope it will be helpful to the reader.

This page intentionally left blank

PART I

THE LINEAR CAUCHY PROBLEM

This page intentionally left blank

LINEAR CAUCHY PROBLEM WITH

CONSTANT COEFFICIENTS

The general Cauchy problem

Let d ≥ 1 be the space dimension and x =(x

,...,x

) denote the space variable,

t being the time variable. The Cauchy problem that we consider in this section

is posed in the whole space R

, while t ranges on an interval, typically (0,T),

where T ≤ +∞.

A constant-coeﬃcient ﬁrst-order system is determined by d + 1 matrices

,...,A

,B given in M

(R), where n ≥ 1isthesize of the system. Then the

Cauchy problem consists in ﬁnding solutions u(x, t)of

∂u

∂t



α=1

∂u

∂x

= Bu + f, (1.0.1)

where f = f(x, t) and the initial datum u(·,t=0)=a are given in suitable

functional spaces. To shorten the notation, we shall rewrite equivalently

∂

u +



∂

u = Bu + f.

When f ≡ 0, the Cauchy problem is said to be homogeneous. A well-posedness

property holds for the homogeneous problem when, given a in a functional

space X, there exists one and only one solution u in C (0,T; Y ), for some other

functional space Y ,themap

X → C (0,T; Y )

a → u

being continuous. ‘Solution’ is understood here in the distributional sense. Exis-

tence and continuity imply X ⊂ Y , since the map a → u(0) must be continuous.

We use the general notation

→ Y

a → u(t).

Since a homogeneous system is, at a formal level, an autonomous diﬀeren-

tial equation with respect to time, we should like to have the semigroup

4 Linear Cauchy Problem with Constant Coeﬃcients

property

t+s

= S

◦ S

,s,t≥ 0,

this of course requires that Y = X. We then say that the homogeneous Cauchy

problem deﬁnes a continuous semigroup if for every initial data a ∈ X,there

exists a unique distributional solution of class C (R

; X). Note that the word

‘continuous’ relies on the continuity with respect to time of the solution, but not

on the continuity of t → S

in the operator norm. Semigroup theory actually tells

us that, if X is a Banach space, the continuity in the operator norm corresponds

to ordinary diﬀerential equations, a context that does not apply in PDEs.

When the homogeneous Cauchy problem deﬁnes a continuous semigroup on a

functional space X, we expect to solve the non-homogeneous one using Duhamel’s

formula:

u(t)=S

a +



t−s

f(s)ds, (1.0.2)

provided that at least f ∈ L

(0,T; X). For this reason, we focus on the homoge-

neous Cauchy problem and content ourselves in constructing the semigroup.

Before entering into the theory, let us remark that, since (1.0.1) writes

∂u

∂t

= Pu+ f,

where P is a diﬀerential operator of order less than or equal to one, the

order with respect to time of this evolution equation, the Cauchy–Kowalevska

theory applies. For instance, if f = 0, analytic initial data yield unique analytic

solutions. However, these solutions exist only on a short time interval (0,T

∗

(a)).

Since analytic data are unlikely in real life, and since local solutions are of little

interest, we shall not concern ourselves with this result.

1.1 Very weak well-posedness

We ﬁrst look at the necessary conditions for a very weak notion of well-posedness,

where X = S (R

) (the Schwartz class) and Y = S



), the set of tempered

distributions. Surprisingly, this analysis will provide us with a rather strong

necessary condition, sometimes called weak hyperbolicity

Let us assume that the homogeneous Cauchy problem is well-posed in this

context. Let a be a datum and u be the solution. From the equation

∂u

∂t



α=1

∂u

∂x

= Bu, (1.1.3)

Some authors call it simply hyperbolicity, and use the term strong hyperbolicity for the notion

that we shall call hyperbolicity. Thus, depending on the authors, there is either the weak and normal

hyperbolicities, or the normal and strong ones.