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Miscellaneous 45

Therefore Λ equals R

, and the symmetrizability has to be understood in the

usual sense. In the vacuum, H = B and E = D, for appropriate units. The

system is already in symmetric form. In a ‘linear’ material medium, which may be

anisotropic, H, E still are linear functions of B, D. For linear as well as non-linear

media, there is a stored electromagnetic energy density W (B, D), and E,H are

given by the following formulæ (see [37])

∂W

∂D

∂W

∂B

In the linear case, W is a quadratic form. The Maxwell system is compatible,

as long as we consider C

solutions, with the Poynting identity, which expresses

the conservation of energy

∂

W (B, D)+curl(E × H)=0. (1.5.55)

Let us consider the linearized system about some constant state (

D). The

former considerations show that if the matrix S := D

W (

D) is positive-

deﬁnite, then the linear Cauchy problem is L

-well-posed. We can actually relax

the convexity condition, with the following observation. The Maxwell system is

also compatible with the extra conservation law (herebelow, u := (B,D))

∂

(B × D)+div



∂W

∂B

⊗ B



+div



∂W

∂D

⊗ D



+ ∇(W −u ·∇

W )=0.

At the linearized level, we may consider a modiﬁed energy density

uSu +

det(X, B, D), where X is a given vector in R

. If there exists an X such

that

uSu +det(X, B, D) is positive-deﬁnite, then the linear system is Friedrichs

symmetrizable and the Cauchy problem is L

-well-posed. An obvious necessary

condition for such an X to exist is that

uSu > 0 whenever B × D =0 and

(B,D) = 0. At the non-linear level, the same procedure as the one imagined

by Dafermos in elastodynamics may be employed. The result is that the non-

linear Maxwell’s system is locally well-posed for smooth initial data and smooth

solutions, whenever W can be written as a convex function of B, D and B × D.

(See [21, 188].)

1.5.4 Splitting of the characteristic polynomial

We give in this section a property of the characteristic polynomial (X; ξ) →

det(XI

+ A(ξ)), when the operator L = ∂



∂

is constantly hyperbolic.

Let us begin with an abstract result.

Lemma 1.3 Let P (X; θ

, ···,θ

) be a homogeneous polynomial of degree n in

1+d variables, with real coeﬃcients. Assume that the coeﬃcient of X

is non-

zero. Assume also that for all θ in a non-void open subset O of R

, the polynomial

:= P (·,θ) has a root with multiplicity ≥ 2. Then P is reducible in R[X, θ].

Proof Let us denote by R := R[θ

, ···,θ

] the factorial ring of polynomials in

d variables θ and by k := R(θ

, ···,θ

) the ﬁeld of rational fractions in θ.We

46 Linear Cauchy Problem with Constant Coeﬃcients

ﬁrst consider P as an element of k[X]. Let us recall that k[X] is a Euclidean

ring, which has therefore a greatest common divisor (g.c.d.)

Let Q be the g.c.d of P and P



in k[X], a monic polynomial of X.Its

coeﬃcients, belonging to k, are rational fractions of θ. We denote by Z the

zero set of the product of denominators of these fractions; Z is a closed set with

empty interior.

When θ ∈O\Z (this is a non-void open set), Q

:= Q(·,θ) has a non-trivial

root, which means that either Q

≡ 0ord

◦

≥ 1. However, the condition Q

≡

0 deﬁnes a non-trivial algebraic manifold M (the intersection of the zero sets of

the coeﬃcients of Q), again a closed set with empty interior. Therefore, there

exists a θ for which d

◦

≥ 1, and consequently d

◦

Q ≥ 1.

Since Q divides P in k[X], we write P = QT , with T ∈ k[X]. Multiplying by

the l.c.m. of the denominators of all coeﬃcients of Q and T (a least common mul-

tiple (l.c.m.) and a g.c.d. do exist in the factorial ring R), we have g(θ)P = Q

where g ∈ A, Q

∈ R[X]and0<d

◦

<n. We recall that the contents of

a polynomial S ∈ R[X], denoted by c(S), is the g.c.d. of all its coeﬃcients.

From Gauss’ Lemma, c(Q

)=c(Q

)c(T

) and therefore g = c(Q

)c(T

), since

c(P ) = 1 by assumption. We conclude that P = Q

,whereQ

:= c(Q

)

−1

∈

R[X] and similarly R

∈ R[X]. Moreover, 0 <d

◦

<n, which shows that P is

reducible in R[X]=R[X, θ]. 

Corollary 1.2 Let P ∈ R(X, θ) be homogeneous with d

◦

P = d

◦

(X,θ)

P .Let

P =



l=1

its factorization into irreducible factors in R(X, θ), the P

s being pairwise dis-

tinct.

Then each P

has the following property: for an open dense subset of values

of θ in R

, the roots of P

(·,θ) are simple.

We now apply the corollary to the characteristic polynomial.

Proposition 1.7 Let the operator L := ∂



∂

be constantly hyperbolic.

Then the characteristic polynomial det(XI

+ A(ξ)) splits as a product



l=1

, (1.5.56)

where the P

s, normalized by P

(1, 0) = 1, satisfy

Each P

is a homogeneous polynomial of (X; ξ),

The P

s are irreducible, pairwise distinct,

For ξ ∈ R

\{0}, the roots of P

(·,ξ) are real and simple,

For ξ ∈ R

\{0} and l = k, P

(·,ξ) and P

(·,ξ) do not have a common root.

Miscellaneous 47

An example Let us consider Maxwell’s equations, where d =3,n =6and

A(ξ)=



J(ξ)

−J(ξ)0



,J(ξ)V := ξ × V.

An elementary computation gives:

det(XI

+ A(ξ)) = X

−|ξ|

)

Hence the splitting described in Proposition 1.7 corresponds to P

(X, ξ)=X,

(X, ξ)=X

−|ξ|

, q

= q

=2.

1.5.5 Dimensional restrictions for strictly hyperbolic systems

We begin with a matrix theorem, due to Lax [111] in the case n ≡ 2(mod4),

and to Friedland et al., [62] in the case n ≡ 3, 4, 5(mod8):

Theorem 1.11 Assume that n ≡ 2, 3, 4, 5, 6(mod8).LetV be a subspace of

(R) with the property that every non-zero element in V has its eigenvalues

real and pairwise distinct. Then dim V ≤ 2.

In terms of hyperbolic operators, this tells us that strictly hyperbolic operators

in space dimension d ≥ 3 can exist only if n ≡ 0, ±1 (mod 8) (assuming that the

operator really involves all the space variables). This explains why constantly

hyperbolic operators occur so frequently, as they exist in space dimension three

at every size n ≥ 4. The simplest examples are:

n =4. Linearized isentropic gas dynamics.

n =5. Linearized non-isentropic gas dynamics.

n =6. Maxwell’s equations. We know also of a non-equivalent example of this

size.

Proof We prove only the Lax case n ≡ 2 (mod 4). We argue by contradiction,

assuming that dim V ≥ 3. We label the eigenvalues in the increasing order:

(M) < ···<λ

(M),M=0

Every non-zero element M in V , having real and distinct eigenvalues, is asso-

ciated with ﬁnitely many (precisely 2

) unitary bases of R

, which depend

continuously on M . In other words, the set of (real) unitary eigenbases is a ﬁnite

covering (with 2

sheets) of V \{0

}. Since the base space is simply connected

(because d ≥ 3), the covering is trivial and a continuous map M →B(M )can

actually be deﬁned globally, where

B(M)={r

(M),...,r

(M)}

is an eigenbasis. By continuity, all the bases B(M) have the same orientation.

On the one hand, it holds that

(−M)=−λ

n−j+1

(M),j=1,...,n.

48 Linear Cauchy Problem with Constant Coeﬃcients

It follows that

(−M)=±r

n−j+1

(M).

By continuity, the sign ± above is constant and depends only on j, but not on

M.Denoteitρ

. Exchanging j with n − j + 1, we obtain

n−j+1

=1. (1.5.57)

On the other hand, we know that {r

(M),...,r

(M)} and

{ρ

(M),...,ρ

(M)} = {r

(−M),...,r

(−M)} have the same orientation.

Since n ≡ 2 (mod 4), the order reversal

(M),...,r

(M)} →{r

(M),...,r

(M)}

reverses the orientation. Therefore, it must hold that



j=1

= −1.

This, however, is incompatible with (1.5.57) when n is even. 

Note that if dim V = 2 (that is for strictly hyperbolic operators in two space

dimensions), there does not need to exist a continuously deﬁned eigenbasis on

V \{0

}, as this set is not simply connected. For instance, the system (1.5.36)

does not have this property: When following a loop around the origin in V ,the

eigenvectors are ﬂipped.

1.5.6 Realization of hyperbolic polynomial

Let p(X

,...,X

), a homogeneous polynomial of degree n,behyperbolic with

respect to a vector T ∈ R

d+1

in the sense of G˚arding (see Section 1.4.4).

Given a hyperbolic polynomial p of degree n in d + 1 variables, one may

always assume that T is the ﬁrst element e

of the canonical basis. A natural

question is whether p can be realized as p

for some hyperbolic operator L.The

case d = 1 is easy. Lax [110] conjectured that if d = 2, the answer is positive

and one can choose a Friedrichs-symmetrizable operator. This has been proved

recently by Lewis et al. [114], following a result by Helton and Vinnikov [82]. One

easily sees that the hyperbolic polynomial q(X):=X

− X

−···−X

cannot

be realized if d ≥ 3. However, it may happen that some power q



be realizable,

as in the case of Maxwell’s system, or Dirac systems. The fundamental question

whether every hyperbolicity cone can be realized as a forward cone for some

hyperbolic operator remains open so far.

Notice that two diﬀerent hyperbolic operators L and L



can yield the same

polynomial,

= p



This part of the proof would also work when n ≡ 3(mod4).

Miscellaneous 49

This happens at least when L



is obtained from L by a linear change of variables.

Then (A

)



= P

−1

P for some non-singular matrix. We can also make only

a linear combination of the equations, which yields an operator L



= S

∂



∂

,whereS

:= S

, which modiﬁes p

by a constant factor. Change

of co-ordinates should also be allowed.

It is an important problem to classify the hyperbolic operators up to such a

change. The characteristic cone is of course an invariant of this problem, but it

is not the only one. There actually exist non-equivalent operators that have the

same characteristic cone. A way to go forward, which has not been pushed so far,

is to consider the characteristic bundle, whose basis is the characteristic variety

(the projective set associated to the characteristic cone) and the ﬁbres are the

corresponding eigenﬁelds. This bundle is modiﬁed by the change of variables and

the combinations of equations, by its topology is not. Thus the Chern class of

the bundle is a more accurate invariant. When the characteristics have variable

multiplicities, the cone and the bundle are not smooth and the analysis becomes

more diﬃcult.

A similar problem, perhaps even more important is to classify within the

set of symmetrizable operator, since the physics usually provides a Friedrichs

symmetrizer, through an energy estimate or an entropy principle. Of course,

the characteristic bundle remains a crucial tool. But some other invariants may

appear, in particular in the case where L admits a linear velocity

λ(ξ)=V · ξ,

which can be brought to the case λ(ξ) ≡ 0. See the discussion in Section 6.1.2.

LINEAR CAUCHY PROBLEM WITH

VARIABLE COEFFICIENTS

The purpose of this chapter is to deal with variable-coeﬃcient generalizations of

the systems considered in Chapter 1. These are of the form

∂u

∂t



α=1

(x, t)

∂u

∂x

= B(x, t) u + f (x, t), (2.0.1)

where the n × n matrices A

and B depend ‘smoothly’ on (x, t). Such systems

may arise from constant-coeﬃcient ones by change of variable, which is useful

to show local uniqueness properties, see Section 2.2. Variable-coeﬃcient systems

also occur as the linearization of non-linear systems (see Chapter 10), in which

case the matrices A

and B may have a restricted regularity. In this chapter,

unless otherwise stated, it will be implicitly assumed that B and A

are C

∞

functions that are bounded as well as all their derivatives.

To be consistent with notations in Chapter 1 we may alternatively write

(2.0.1) as

∂

u = P (t) u + f,

where P (t) is the spatial diﬀerential operator

P (t):u → P (t) u := −



(·,t) ∂

u + B(·,t) u. (2.0.2)

Or, in short, (2.0.1) equivalently reads

Lu = f,

where L denotes the evolution operator

L : u → Lu := ∂

u − P (t) u. (2.0.3)

These notations being ﬁxed, we claim that Fourier analysis is not suﬃcient

to deal with the Cauchy problem for the variable-coeﬃcient operator L. We need

pseudo-diﬀerential calculus (in the variable x ∈ R

, the time t playing the role of

a parameter), and even para-diﬀerential calculus in the case of coeﬃcients with

a restricted regularity. For convenience, the results from this ﬁeld we shall use

are all collected in Appendix C.

Well-posedness in Sobolev spaces 51

2.1 Well-posedness in Sobolev spaces

The main issue regarding well-posedness is the derivation of a priori energy

estimates of the form

u(t)

≤ C



u(0)



Lu(τ)

dτ



The scalar case (n = 1) merely corresponds to a transport operator L,whichis

much easier to deal with than the case of genuinely matrix-valued operators. For

clarity, we begin with this special case, which will serve as an introduction to the

powerful technique of symmetrizers for systems.

2.1.1 Energy estimates in the scalar case

Proposition 2.1 For a scalar operator

L = ∂

+ a ·∇−b,

where a(x, t) ∈ R

and b(x, t) ∈ R are smooth functions of (x, t), bounded as well

as their derivatives, we have the following a priori estimates. For all s ∈ R and

T>0, there exists C>0 so that for u ∈ C

([0,T]; H

) ∩ C ([0,T]; H

s+1

) we have

u(t)

≤ C



u(0)



Lu(τ)

dτ



Proof We start with the case s =0.Wehave

u

=2ReLu, u−2Re a ·∇u, u +2Rebu, u

=2ReLu, u +



(diva +2b) |u|

after integration by parts. Therefore, by the Cauchy–Schwarz inequality, we get

u

≤ (1 + diva +2b

∞

) u

+ Lu

u(t)

≤u(0)





Lu(τ)

+(1+diva +2b

∞

) u(τ)



dτ.

By Gronwall’s Lemma this implies

u(t)

≤ e

γt

u(0)



γ (t−τ )

Lu(τ)

dτ ∀t ≥ 0,

where γ ≥ 1+diva +2b

∞

. In particular, for all bounded interval [0,T],

there exists C

(namely, C

=exp(T (1 + diva +2b

∞

)) so that

u(t)

≤ C



u(0)



Lu(τ)

dτ



∀t ∈ [0,T].

52 Linear Cauchy problem with variable coeﬃcients

The general case s ∈ R actually follows from the special case s =0 almost

for free with the help of pseudo-diﬀerential calculus. Indeed, recall that the basic

pseudo-diﬀerential operator Λ

of symbol λ

(ξ):=(1+ξ

)

s/2

is of order s

(see Appendix C) and can be used to deﬁne the H

norm

u

= Λ

u

Now, if u ∈ C

([0,T]; H

) ∩ C ([0,T]; H

s+1

)thenΛ

u ∈ C

([0,T]; L

) ∩

C ([0,T]; H

) and therefore the inequality previously derived for s = 0 applies

to Λ

u. That is, we have

Λ

u(t)

≤ C



Λ

u(0)



L Λ

u(τ)

dτ



Furthermore, the commutator

[ L,Λ

]=[a ·∇, Λ

]

is of order 1 + s − 1=s (see Appendix C). This implies the existence of C

(T )

such that

[ L,Λ

] u(t) 

≤ C

(T ) u(t)

∀t ∈ [0,T].

Hence

Λ

u(t)

≤ C



Λ

u(0)



(Λ

Lu(τ)

+ C

(T )u(τ)

) dτ



Finally, by Gronwall’s Lemma we obtain

u(t)

≤ C

(T ) T



u(0)



Lu(τ)

)dτ





As should be clear from this proof, the crucial point is the L

estimate, relying

on the fact that the diﬀerential operator Re (a ·∇)=

((a ·∇)+(a ·∇)

∗

)is

bounded on L

. The symmetrizer’s technique described below aims at recovering

a similar property for non-scalar operators.

2.1.2 Symmetrizers and energy estimates

There is a special class of systems for which energy estimates are almost as

natural as for scalar equations. This is the class of Friedrichs-symmetrizable

systems, which fulﬁll the following generalization of Deﬁnition 2.1.

Deﬁnition 2.1 The system (2.0.1) is Friedrichs symmetrizable if there exists a

∞

mapping S

: R

× R

→ M

(R), bounded as well as its derivatives, such

that S

(x, t) is symmetric and uniformly positive-deﬁnite, and the matrices

(x, t)A

(x, t) are symmetric for all (x, t).

Well-posedness in Sobolev spaces 53

Like scalar equations, Friedrichs-symmetrizable systems enjoy a priori esti-

mates that keep track of coeﬃcients. We give the L

estimate below, which is

proved elementarily and will be extensively used in the non-linear analysis of

Chapter 10. We postpone to the end of this section the more complete result in

, valid for any system admitting a symbolic symmetrizer (see Deﬁnition 2.3),

whose proof takes advantage of Bony’s para-diﬀerential calculus.

Proposition 2.2 Assume that (2.0.1) is Friedrichs symmetrizable, with a sym-

metrizer S

satisfying

βI

≤ S

≤ β

−1

in the sense of quadratic forms. We also assume that S

, A

, and their ﬁrst

derivatives are bounded, as well as B.Then,forallT>0 and u ∈ C ([0,T]; H

) ∩

([0,T]; L

) we have

u(t)

≤ e

γt

u(0)



γ (t−τ )

Lu(τ)

dτ ∀t ∈ [0,T],

(2.1.4)

where L is deﬁned in (2.0.3) and γ is chosen to be large enough, so that

β (γ − 1) ≥



∂



∂

)+S

B + B



∞

. (2.1.5)

Proof The proof is basically the same as the ﬁrst part of the proof of

Proposition 2.1. After integration by parts we get

S

u, u =2ReS

u, Lu + Ru , u,

where

R = ∂



∂

)+S

B + B

Of course, we have used here the symmetry of the matrices S

and S

Integrating in time and using the Cauchy–Schwarz inequality (for the inner

product S

·, ·) we arrive at

S

u(t) ,u(t) ≤S

u(0) ,u(0)  +



S

Lu(τ) ,Lu(τ )dτ



(1 + β

−1

R

∞

) S

u(τ) ,u(τ )dτ.

By Gronwall’s Lemma this implies

S

u(t) ,u(t) ≤e

γt

S

u(0) ,u(0)  +



γ (t−τ )

S

Lu(τ) ,Lu(τ )dτ,

54 Linear Cauchy problem with variable coeﬃcients

with γ ≥ (1 + β

−1

R

∞

). This yields the ﬁnal estimate after multiplication

by β. 

More generally, a priori estimates hold true for systems admitting a functional

symmetrizer, deﬁned as follows.

Deﬁnition 2.2 Given a family of ﬁrst-order (pseudo-)diﬀerential operators

{P (t)}

t≥0

acting on R

,afunctional symmetrizer is a C

mapping

Σ:R

→ B(L

; C

))

such that, for 0 ≤ t ≤ T ,

Σ(t)=Σ(t)

∗

≥ αI, (2.1.6)

for some positive α depending only on T ,and

Re (Σ(t)P (t)) :=

(Σ(t)P (t)+P (t)

∗

Σ(t)

∗

) ∈ B(L

)

) (2.1.7)

with a uniform bound on [0,T].

Example For a Friedrichs-symmetrizable system of symmetrizer S

,thesimple

multiplication operator Σ(t):u → Σ(t)u := S

(·,t)u is a functional symmetrizer.

As a matter of fact, (2.1.6) merely follows from the analogous property of

the matrices S

(x, t). And, because the matrices S

(x, t)A

(x, t)aresym-

metric, 2 Re (Σ(t)P (t)) reduces to the multiplication operator associated with

(



∂

)+S

B + B

∗

)(·,t).

Theorem 2.1 If a family of operators P (t) admits a functional symmetrizer,

then, for all s ∈ R and T>0, there exists C>0 so that for u ∈ C

([0,T]; H

) ∩

C ([0,T]; H

s+1

) we have

u(t)

≤ C



u(0)



Lu(τ)

dτ



, (2.1.8)

where L is deﬁned in (2.0.3).

Proof The proof is very much like the one of Proposition 2.1. The ﬁrst step is

elementary. It consists in showing the estimate in (2.1.8) for s = 0. The second

one infers the estimate for any s from the case s = 0 with the help of pseudo-

diﬀerential calculus.

Case s = 0 From (2.1.6) we know that

Σ(t) u(t) ,u(t) 

≥ α u(t)

To bound the left-hand side we write

Σ u, u =2ReΣ Lu, u +2ReΣ Pu,u + 

dΣ

u, u.