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

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Well-posedness in Sobolev spaces 55

Each term here above can be estimated by using the Cauchy–Schwarz inequality.

For the ﬁrst and last ones, we use uniform bounds in t of Σ

B(L

)

and

dΣ/dt

B(L

)

. For the middle term we use (2.1.7) and a uniform bound in t

of Re (Σ P ). This yields

Σ u, u≤C

( u

+ Lu

Hence, we have

α u(t)

≤ C

u(0)

+ C



( u(τ)

+ Lu(τ)

)dτ,

where C

:= Σ(0)

B(L

)

. We conclude by Gronwall’s Lemma that (2.1.8) holds

for s =0withC = C



exp(C



T ), C



:= max(C

)/α.

General case Let s be an arbitrary real number. For u ∈ C

([0,T]; H

) ∩

C ([0,T]; H

s+1

) the inequality previously derived for s = 0 applies to Λ

u and

yields

Λ

u(t)

≤ C



Λ

u(0)



L Λ

u(τ)

dτ



Writing

L Λ

u =Λ

Lu +[Λ

,P] u,

and observing that the commutator [Λ

,P]isoforders +1− 1=s (see

Appendix C), we complete the proof exactly as in the scalar case (Proposition

2.1). 

Remark 2.1 By reversing time, that is, changing t to T − t and P (t)to

−P (T − t) in Theorem 2.1, we also obtain the estimate

u(t)

≤ C



u(T )



Lu(τ)

dτ



, (2.1.9)

for u ∈ C

([0,T]; H

) ∩ C ([0,T]; H

s+1

The problem is now to construct symmetrizers. Except for Friedrichs-

symmetrizable systems, this is not an easy task. We shall conveniently use

symbolic calculus. Similarly as in Chapter 1, we denote

A(x, t, ξ):=



(x, t) , (x, t) ∈ R

× R

,ξ∈ R

which can be viewed up to a −i factor as the symbol of the principal part of the

operator P (t) deﬁned in (2.0.2).

56 Linear Cauchy problem with variable coeﬃcients

Deﬁnition 2.3 A symbolic symmetrizer associated with A(x, t, ξ) is a C

∞

mapping

S : R

× R

× (R

\{0}) → M

(C),

homogeneous degree 0 in its last variable ξ, bounded as well as all its derivatives

with respect to (x, t, ξ) on ξ =1, such that, for all (x, t, ξ)

S(x, t, ξ)=S(x, t, ξ)

∗

≥ βI, (2.1.10)

for some positive β, uniformly on sets of the form R

× [0,T] ×(R

\{0}) (T>

0), and

S(x, t, ξ) A(x, t, ξ)=A(x, t, ξ)

∗

S(x, t, ξ)

∗

. (2.1.11)

Of course, a Friedrichs-symmetrizable system admits an obvious ‘symbolic’

symmetrizer independent of ξ

S(x, t, ξ)=S

(x, t).

Note that, in general, a symbolic symmetrizer is not exactly a pseudo-

diﬀerential symbol, due to the singularity allowed at ξ = 0. However, truncating

about 0 does yield a pseudo-diﬀerential symbol in S

, which is unique modulo

−∞

(see Appendix C). This enables us to associate S with a family of pseudo-

diﬀerential operators



Σ(t) of order 0 modulo inﬁnitely smoothing operators. This

in turn will enable us to construct a functional symmetrizer Σ(t).

Remark 2.2 In the constant-coeﬃcient case, neither A(x, t, ξ)norS(x, t, ξ)

depend on (x, t), and it is elementary to construct a functional symmetrizer

based on S. This symmetrizer is of course independent of t and is just given by

Σ:=F

−1

S F

(where F denotes the usual Fourier transform). Then (2.1.6) holds with α = β

since we have

Σ u, v = S u, v

for all u, v ∈ L

. And (2.1.7) follows from (2.1.11), because of the relations

Σ Pu,v + u, Σ Pv = S F Pu,v + u, SF Pv

= S (−iA + B) u, v + u, S(−iA + B) v = (SB + B

S) u, v.

Theorem 2.2 Assuming that A(x, t, ξ) admits a symbolic symmetrizer S(x, t, ξ)

(according to Deﬁnition 2.3), then the family P (t) deﬁned in (2.0.2) admits a

functional symmetrizer Σ(t) (as in Deﬁnition 2.2).

Proof The proof consists of a pseudo-diﬀerential extension of Remark 2.2

above. As mentioned above, S(·,t,·) can be associated with a pseudo-diﬀerential

operator of order 0,



Σ(t). We recall that the operator



Σ(t) is not necessarily self-

adjoint, even though the matrices



S(x, t, ξ) are Hermitian. But



Σ(t)

∗

diﬀers from

Well-posedness in Sobolev spaces 57



Σ(t) by an operator of order −1 (since they are both of order 0), see Appendix C.

As a ﬁrst step, let us deﬁne

Σ(t):=

(



Σ(t)+



Σ(t)

∗

By G˚arding’s inequality, there exists C

> 0 such that

Σ(t) u, u≥

u

− C

u

−1

for all t ∈ [0,T]andu ∈ L

). Now, noting that

u

−1

= Λ

−2

u, u,

we can change Σ(t)intoΣ(t)+C

−2

in order to have

Σ(t) u, u≥

u

This modiﬁcation does not alter the self-adjointness of Σ(t) and gives (2.1.6)

with α = β/2. Furthermore, Σ(t) P (t)+P (t)

∗

Σ(t) coincides with the operator

of symbol



S (−iA + B)+(−iA + B)



S =



SB + B



up to a remainder of order 0 + 1 − 1 = 0, see Appendix C. (To simplify notations,

we have omitted the dependence on the parameter t of the symbols.) Since

(



SB + B



S)(·,t,·) belongs to S

,Σ(t) P (t)+P (t)

∗

Σ(t)isoforder0and

therefore is a bounded operator on L

. 

As a consequence of Theorems 2.1 and 2.2, we have the following.

Corollary 2.1 If A(x, t, ξ)=



(x, t) admits a symbolic 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



where L = ∂



∂

− B.

As already noted this applies in particular to Friedrichs-symmetrizable sys-

tems, but not only. Another important class of hyperbolic systems that do admit

a symbolic symmetrizer is the one of constant multiplicity hyperbolic systems.

Theorem 2.3 We assume that the system (2.0.1) is constantly hyperbolic, that

is, the matrices A(x, t, ξ) are diagonalizable with real eigenvalues λ

, ···,λ

constant multiplicities on R

× R

× (R

\{0}). We also assume that these matri-

ces are independent of x for x≥R. Then they admit a symbolic symmetrizer.

Together with Theorem 2.2 this shows that constantly hyperbolic systems

are symmetrizable and thus enjoy H

estimates.

58 Linear Cauchy problem with variable coeﬃcients

Proof The proof is based on spectral projections associated with A(x, t, ξ),

which are well-deﬁned due to spectral separation. As a matter of fact, the

assumptions imply that the spectral gap |λ

(x, t, ξ) − λ

(x, t, ξ)| is bounded by

below for 1 ≤ j = k ≤ p,(x, t) ∈ R

× [0,T]andξ = 1. Let us deﬁne

ρ :=

min{|λ

(x, t, ξ) − λ

(x, t, ξ)|, 1 ≤ j = k ≤ p, (x, t) ∈ R

× [0,T], ξ =1}

and the projectors

(x, t, ξ):=

2 iπ



|λ−λ

(x,t,ξ)|=ρξ

(λI

− A(x, t, ξ))

−1

dλ

for 1 ≤ j ≤ p.SinceA and its eigenvalues λ

are homogeneous degree 1 in ξ,we

easily see by change of variables that Q

is homogeneous degree 0. Furthermore,

is independent of x for x≥R. Then we introduce

S(x, t, ξ):=



j=1

(x, t, ξ)

∗

(x, t, ξ).

By construction, the matrix S is Hermitian. Moreover, we have for any vector

v ∈ C

∗

Sv =



j=1

Q

v

≥ β v

where

β := min









j=1

Q

(x, t, ξ)v

; v =1, x≤R, 0 ≤ t ≤ T, ξ =1







> 0

since



= I

. This proves (2.1.10). And ﬁnally, since Q

A = AQ

= λ

because λ

is semisimple, we have

∗

SAv =



j=1

∗

for all v, w ∈ C

, and thus SA is Hermitian. 

2.1.3 Energy estimates for less-smooth coeﬃcients

The main purpose of this section is to obtain energy estimates for less-regular

operators in which we keep track of coeﬃcients (in the same spirit as in Proposi-

tion 2.2). This will be done in a framework preparing for later non-linear analysis.

Namely, following M´etivier [132], we assume the dependence of the matrices with

respect to (x, t) occurs via a known but not necessarily very smooth function

Well-posedness in Sobolev spaces 59

v(x, t). More precisely, we consider a family of operators

= ∂



(v(x, t)) ∂

associated with functions v ∈ L

∞

([0,T]; H

)) ∩ C ([0,T]; H

)) such that

∂

v ∈ L

∞

([0,T]; H

s−1

)) ∩ C ([0,T]; H

s−1

)). (We recall that H

stands for

the regular Sobolev space H

equipped with the weak topology.) Observe that

such functions belong to C

× [0,T]) as soon as s>

+ 1. As to the matrices

, they are assumed to be C

∞

functions of their argument v ∈ R

. We denote

A(v, ξ)=



(v) and extend in a straightforward way the deﬁnition of a

symbolic symmetrizer.

Deﬁnition 2.4 A symbolic symmetrizer associated with A(v, ξ) is a C

∞

mapping

S : R

× (R

\{0}) → M

(C),

homogeneous degree 0 in ξ such that

S(v, ξ)=S(v, ξ)

∗

> 0 and S(v, ξ) A(v, ξ)=A(v, ξ)

∗

S(v, ξ).

Theorem 2.4 Assume that A(v, ξ) admits a symbolic symmetrizer and take

T>0, s>

+1.If

v

∞

([0,T ];W

1,∞

))

≤ ω

and

sup



v

∞

([0,T ];H

))

, ∂

v

∞

([0,T ];H

s−1

))



≤ µ,

there exists K = K(ω) > 0 and C = C(µ), γ = γ(µ) so that for all u ∈

([0,T]; H

) ∩ C ([0,T]; H

m+1

+1 <m≤ s, we have

u(t)

≤ K e

γt

u(0)

+ C



γ(t−τ)

L

u(τ)

dτ.

Proof As mentioned before, the proof takes advantage of Bony’s para-

diﬀerential calculus.

TheideaisﬁrsttoreplaceL

by the para-diﬀerential operator

= ∂



(v)

∂

To estimate the error, assume ﬁrst that A

vanishes at 0. Then by Proposition

C.9 and Theorem C.12, we have

A

(v) u − T

(v)

u 

≤ C u

∞

A

(v)

≤ K(v

∞

) v

u

∞

for all u ∈ H

(→ L

∞

). Hence

P

u − L

u 

≤ K(v

∞

) v

∇u

∞

60 Linear Cauchy problem with variable coeﬃcients

pointwisely in time for all u ∈ C

([0,T]; H

) ∩ C ([0,T]; H

m+1

). By Sobolev

embedding this estimate merely implies

P

u − L

u 

≤ C(v

) u

In fact, the assumption A

(0) = 0 is superﬂuous. Indeed, denoting



(v)=

(v) − A

(0), we have

(v) ∂

− T

(v)

∂

=(A

(0) ∂

− T

(0)

∂

)+(



(v) ∂

− T



(v)

∂

where the ﬁrst term is a smoothing operator according to Theorem C.13

and

the second operator can be estimated as before.

Once we have the estimate for P

− L

, it suﬃces to show the result for the

operator P

instead of L

.Indeed,if

u(t)

≤ K e

γt

u(0)

+ C



γ(t−τ)

P

u(τ)

dτ.

Then

u(t)

≤ K e

γt

u(0)

+2C



γ(t−τ)



L

u(τ)

+ C(v

)

u(τ)



dτ,

which implies by Gronwall’s Lemma

u(t)

≤ K e



γt

u(0)

+ C





γ(t−τ)

L

u(τ)

dτ,

with γ = γ + CC(v

)

The next step is to use the symbolic symmetrizer S(v(x, t),ξ), which is

Lipschitz in x, to construct a functional symmetrizer for P

. The outline is

a para-diﬀerential version of the proof of Theorem 2.2. Deﬁne the symbol

r(x, t, ξ)=λ

(ξ) S(v(x, t),ξ). Up to a small-frequency cut-oﬀ, r(t)=r(·,t,·)

belongs to Γ

(see Deﬁnition C.5) and thus is associated with a para-diﬀerential

operator T

r(t)

, simply denoted by T

in what follows. The constants in the esti-

mates (C.4.45) of r and its derivatives depend boundedly on v

∞

([0,T ];W

1,∞

))

that is on ω. Furthermore, since r is everywhere positive-deﬁnite Hermitian we

know by G˚arding’s inequality (see Theorem C.18) there exist β = β(ω) > 0and

= K

(ω) > 0 such that

Re T

u, u≥β u

− K

u

m−1/2

for all u ∈ H

). Noting that

u

m−1/2

= Λ

2m−1

u, u,

In fact, we need here only a special, easy case of Theorem C.13 because A

(0) is constant.

Well-posedness in Sobolev spaces 61

we may modify the symbol r into

r = r + K

2m−1

and get the estimate

Re T



u, u≥β u

for the modiﬁed operator T



. We also have an upper bound (see Proposition

C.17)

|Re T



u, u| ≤ K

(ω) u

and T

∂

r

enjoys a similar estimate

|Re T

∂

r

u, u| ≤ C(ω, ∂

v

∞

×[0,T ])

) u

Observing that ∂

v

∞

×[0,T ])

is controlled by ∂

v

s−1

(by Sobolev

embedding) and thus by µ, we shall merely write this new constant

C(ω, ∂

v

∞

×[0,T ])

)=C(ω, µ). Finally, from the symmetry of the symbol

r(x, t, ξ) a(v(x, t),ξ) we also have an estimate

|Re (T



+ T

∗

r

)



(v)

∂

u, u| ≤ K

(ω) u

The end of the proof is similar to the proof of Proposition 2.2. We have

Re T



u, u =Re(T



+ T

∗

r

) P

u,u−Re (T



+ T

∗

r

)



(v)

∂

u,u

+Re T

∂

r

u, u.

Denoting Σ = T



+ T

∗

r

and using the Cauchy–Schwarz inequality for the inner

product Σ·, · we get from that identity

Σu, u≤ΣP

u, P

u + Σu, u +2K

(ω) u

+2C(ω, µ) u

Hence

Σu(t),u(t)≤Σu(0),u(0) +



ΣP

u(τ),P

u(τ)dτ



(1 + 2 (K

(ω)+C(ω, µ))/β(ω)) Σu(τ),u(τ )dτ.

Therefore by Gronwall’s Lemma, we have

Σu(t),u(t)≤e

γt

Σu(0),u(0) +



γ (t−τ )

ΣP

u(τ),P

u(τ)dτ

62 Linear Cauchy problem with variable coeﬃcients

for γ ≥ (1 + 2 (K

(ω)+C(ω, µ))/β(ω)). Coming back to the usual H

norm,

we get

u(t)

≤

(ω)

β(ω)



γt

u(0)



γ (t−τ )

P

u(τ)

dτ



The proof is complete once we note that any ‘constant’ depending on ω a fortiori

depends boundedly on µ (by Sobolev embedding). 

Remark 2.3 The only point where we have used the assumption m>

+1is

in the comparison between L

and P

. We shall see in the proof of Theorem 2.7

that we can bypass this assumption and still obtain energy estimates in the

Sobolev space of negative index H

−s

. Another possibility is to derive L

esti-

mates, under the only assumption that v be W

1,∞

in both time and space. This

is the purpose of the next theorem.

Theorem 2.5 Assume that A(v, ξ) admits a symbolic symmetrizer and that

v

1,∞

×[0,T ]))

≤ ω.

Then there exists K = K(ω) > 0 and γ = γ(ω) so that for all u ∈ C

([0,T]; L

) ∩

C ([0,T]; H

)

u(t)

≤ K



γt

u(0)



γ(t−τ)

L

u(τ)

dτ



. (2.1.12)

Proof We proceed exactly as in the proof of Theorem 2.4, just changing the way

of estimating P

− L

. Indeed, examining the second step of that proof shows

that it works for m = 1 as soon as we have bounds for v

∞

([0,T ];W

1,∞

))

and

∂

v

∞

([0,T ];L

∞

))

, which is the case by assumption. The estimate of P

− L

is given by Corollary C.4, namely

P

u − L

u

≤ C max

A

(v)

1,∞

u

≤



C v

1,∞

u



Remark 2.4 The estimate (2.1.12) here above applies in fact to any u ∈

× [0,T]) by a density argument. For, u ∈ H

× [0,t]) can be achieved

as the limit of a sequence u

∈ D (R

× [0,t]), in such a way that u

goes to u

in H

([0,t]; L

)) → C ([0,t]; L

)) and L

goes to L

u in L

× [0,t]).

Therefore, we can pass to the limit in the estimate (2.1.12) applied to u

, written

max

τ∈[0,t]



−γτ

u

(τ)



≤ K



u

(0)



−γτ

L

(τ)

dτ



Well-posedness in Sobolev spaces 63

Remark 2.5 If the operator L

is Friedrichs symmetrizable, we also have a H

estimate

u(t)

≤ K



γt

u(0)



γ(t−τ)

L

u(τ)

dτ



, (2.1.13)

for u ∈ C

([0,T]; H

)) ∩ C ([0,T]; H

)) and v ∈ W

1,∞

× (0,T)), with

K depending boundedly on v

∞

([0,T ];W

1,∞

)

and γ ≥ γ

depending boundedly

on v

1,∞

×[0,T ])

The proof is most elementary. Denoting by S

(v) a Friedrichs symmetrizer,

we see that for v ∈ W

1,∞

, the estimate in (2.1.13) is equivalent to

u(t)

≤





γt

u(0)



γ(t−τ)





u(τ )

dτ



where u(x, t)=



(v(x, t))u(x, t)and



= ∂





A(v(x, t))

∂

, with



A(v):=



(v)

−1

(v) A

(v)



(v)

−1

being symmetric. Therefore, we can assume with no loss of generality (just

dropping the tildas) that the matrices A

(v) are symmetric. Thanks to this

property we easily ﬁnd that, for u smooth enough such that L

u = f ,

u

)

=2Reu, f,



∂

u

)

+2Re



α,β

∂

u, ∂

(v)) ∂

u  =2Re



∂

u, ∂

f ,

hence by taking the sum and using the Cauchy–Schwarz inequality,

u

)

≤ γ u

)

+ f

)

with γ := 1 + max

α,β

∂

(v))

∞

×(0,T ))

. After integration we get

(2.1.13) with K =1. 

2.1.4 How energy estimates imply well-posedness

The energy estimate in (2.1.8) easily implies uniqueness by linearity, for smooth

enough solutions. The existence of a solution u ∈ C ([0,T]; H

s−1

) for initial data

u(0) ∈ H

can be obtained by a duality argument, applying the energy estimate

in (2.1.8) to −s and the adjoint operator L

∗

. The proof that u actually lies in

C ([0,T]; H

) uses molliﬁers, smooth solutions and their uniqueness. The whole

result can be stated as follows, keeping the same compressed notations as in

(2.0.2) and (2.0.3).

Theorem 2.6 We assume the system (2.0.1) has C

∞

coeﬃcients and is sym-

metrizable in the sense that the operator P (t) admits a functional symmetrizer

64 Linear Cauchy problem with variable coeﬃcients

Σ(t). We take T>0, s ∈ R and f ∈ L

(0,T; H

)), g ∈ H

). Then there

exists a unique u ∈ C ([0,T]; H

)) solution of the Cauchy problem

∂

u − P (t) u = f, u(0) = g.

Furthermore, there exists C>0 (independent of u!) such that for all t ∈ [0,T]

u(t)

≤ C



g



f(τ)

dτ



. (2.1.14)

Finally, if f ∈ C

∞

([0,T]; H

+∞

)) and g ∈ H

+∞

) then u ∈

∞

([0,T]; H

+∞

)).

Remark 2.6 Of course the equation ∂

u − P (t) u = f is to be understood in

the sense of distributions when s is low enough.

Remark 2.7 If the bounds associated with the symmetrizer Σ(t)fort ∈ [0,T]

are independent of T , and if the source term f is given in L

; H

)) then

the solution u is global and belongs to L

; H

)) too. If additionally, s = k

is a positive integer and if f belongs to H

k−1

× R

), then u belongs to

× R

). This is due to the following simple observation – which will also

be used in the context of Boundary Value Problems (Chapter 9).

Proposition 2.3 As soon as the coeﬃcients A

and B of the operator

P (t)= −



(·,t) ∂

+ B(·,t)

are C

∞

functions of (x, t) on R

× R

,anyu ∈ L

; H

)) such that ∂

u −

P (t) u belongs to H

k−1

× R

), with k a positive integer, actually belongs to

× R

Proof If u ∈ L

; H

)) we can show indeed that for all m ∈ N and all

d-uple α such that m + |α|≤k, ∂

∂

u belongs to L

× R

). This is trivial

if m = 0, because of the identiﬁcation L

; L

)) = L

× R

). The rest

of the proof works by ﬁnite induction on m, using the decomposition

∂

m+1

∂

u = ∂

∂

(∂

u − P (t) u)+∂

∂

(P (t) u),

with (∂

u − P (t) u) ∈ H

k−1

× R

)andP (t) u ∈ L

; H

k−1

)). 

Proof of Theorem 2.6

Uniqueness As mentioned above, the uniqueness is easy to show. By linearity,

it is suﬃcient to show that the only solution in C ([0,T]; H

)) for f ≡ 0and

g ≡ 0isu ≡ 0. But if u ∈ C ([0,T]; H

)) satisﬁes (2.0.1) then necessarily u

belongs to C

([0,T]; H

s−1

)). So we can apply Theorem 2.1 to the index

s − 1. For u(0) = g ≡ 0andLu = f ≡ 0 this gives u ≡ 0.