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

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Energy estimates 225

If one compares with the Cauchy problem studied in Chapter 2

∂

u − P (t) u = f, t>0 ,

u = g, t=0,

there are two intertwined diﬀerences: 1) the whole vector u is prescribed on the

‘boundary’ (t = 0) in the latter case, whereas in the former, only part of u is

prescribed on the ‘boundary’ (x

= 0) and 2) unlike ∂

− P (t), the operator

∂

u − P

)isnot hyperbolic in general, because L is not hyperbolic in the

direction x

. The gap between hyperbolic Initial Value Problems and Initial

Boundary Value Problems should now be clear to the reader.

9.1.1 Functional boundary symmetrizers

In what follows, the space R

is to be understood as the product of the boundary

of Ω by the real line in the time direction, i.e. R

= {(y,t); y ∈ R

d−1

,t∈ R }.

The delimiters ,  stand for the inner product on L

, dy dt).

Deﬁnition 9.1 A (functional) boundary symmetrizer for (9.1.5)(9.1.6) is a

family of C

mappings

: R

→ B(L

; R

))

→ R

)

for γ ≥ γ

≥ 1, with bounds for R

) and dR

/dx

that are uniform in both

γ ≥ γ

and x

≥ 0, such that

the operator R

) is self-adjoint,

the operator

Re ( R

)) :=

)+(R

))

∗

)

belongs to B(L

; R

)) and as such satisﬁes the lower bound

Re ( R

)) ≥ CγI, (9.1.10)

with C>0 independent of both x

≥ 0 and γ ≥ γ

there exist positive constants α and β so that

R

(0)u, u≥α u

− β Bu

(9.1.11)

for all u ∈ L

; R

) and all γ ≥ γ

To some extent, this deﬁnition resembles the one given in Chapter 2 (Deﬁni-

tion 2.2) for functional symmetrizers concerning the Cauchy problem. If one com-

pares the two deﬁnitions, the main novelties here are, besides the parameter γ,

the inequality (9.1.11), which is meant to deal with boundary terms in the energy

estimates, and also that a non-negative lower bound for Re ( R

)) is

requested: this is because the symmetrizer is designed to ﬁnd a lower bound for

the derivative of R

) u, u with respect to x

, eventually leading to a control

226 Variable-coeﬃcients initial boundary value problems

of the trace of u at x

= 0. (Observe the counterpart of this trace for the Cauchy

problem is merely the initial data, so that a rough bound Re ( Σ(t)P (t)) ≥−CI

suﬃces to show well-posedness in that case.)

In the case of Friedrichs symmetrizability, which is a property of the sole

operator L, we easily get boundary symmetrizers, provided that the boundary

matrix B is strictly dissipative: the deﬁnition of strict dissipativity given in

Chapter 3 for Initial Boundary Value Problems with constant coeﬃcients extends

in a straightforward way to variable coeﬃcients; we give it below for a general

domain Ω with outward unit normal vector ν.

Deﬁnition 9.2 Assume L = ∂



j=1

∂

is a Friedrichs-symmetrizable

operator, with Friedrichs symmetrizer S

(see Deﬁnition 2.1). The boundary

matrix B is called strictly dissipative if there exist α>0 and β>0 so that for

all (x, t) ∈ ∂Ω × R and all v ∈ R

;

(x, t) A(x, t, ν(x)) v ≥ α v

− β B(x, t)v

, (9.1.12)

where ν denotes the outward normal to ∂Ω.

In the case Ω = {x

> 0}, the strict dissipativity of B means

∗

(y, 0,t) A

(y, 0,t)v ≤−α v

+ β B(y, t)v

(9.1.13)

for all (y, t) ∈ R

and all v ∈ C

. If the inequality (9.1.13) is satisﬁed then the

operator

):u → R

) u := −S

(·,x

, ·) A

(·,x

, ·) u

deﬁnes a boundary symmetrizer according to Deﬁnition 9.1. As a matter of fact,

the inequality in (9.1.11) directly follows from (9.1.13) applied to v = u(y, t),

after integration in (y,t). And we easily compute

Re ( R

))u, u = γ S

(·,x

, ·) u, u

−

( ∂

d−1



j=1

∂

))u, u.

Taking a lower bound for S

,sayσ>0, and an upper bound for ∂



∂

), say K, we clearly have for γ ≥ 2K/σ,

Re ( R

))u, u≥

γσ

u

which is (9.1.10) with C = σ/2. This shows that Friedrichs-symmetrizable oper-

ators together with strictly dissipative boundary matrices are endowed with

boundary symmetrizers. In what follows, we will consider boundary matrices

that are not necessarily dissipative, as in Chapter 4.

Energy estimates 227

Proposition 9.1 If there is a boundary symmetrizer for (9.1.5)(9.1.6), then

there exists γ

≥ 1 and c>0 so that for γ ≥ γ

and u ∈ D(Ω × R) we have



−2γt

u(x, t)

dx dt +



d−1

−2γt

u(y, 0,t)

dy dt

≤ c





−2γt

(Lu)(x, t)

dx dt+



d−1

−2γt

(Bu)(y, 0,t)

dy dt



(9.1.14)

and more generally,

γ u



))

+ u

(0)

)

≤ c



L

u



))

+ Bu

(0)

)



, (9.1.15)

where u

(x, t)=e

−γt

u(x, t), L

= L + γ,andH

) denotes the usual

Sobolev space of index s equipped with the γ-depending norm:

v

)





d−1



( γ

+ η

+ δ

)

|(F

(y,t)

v)(η,δ)|

dδ dη



1/2

= Λ

s,γ

v

)

(Note that by the deﬁnition of L

and u

, L

u

−γt

Lu.)

Proof The proof is very similar to that of Theorem 2.1 in Chapter 2 (also see

the proof of Theorem 2.13).

WebeginwiththeL

estimate (9.1.14). We readily have

R

u

, u

 =2ReR

)

−1

u

, u

 +2ReR

u

, u



+ 

u

, u

.

The ﬁrst and last terms can be bounded by below using uniform bounds for

(A

)

−1

, R



B(L

)

and dR

/dx



B(L

)

. The middle term is bounded by

below in view of (9.1.10). Hence there exist C

> 0andC

so that for all ε>0

and γ ≥ γ

R

u

, u

≥((2C − C

ε)γ − C

) u

)

)

−

4εγ

L

u

)

)

228 Variable-coeﬃcients initial boundary value problems

Integrating in x

and using (9.1.11) we get

α u

(0)

)

− β Bu

(0)

)

≤ ((C

ε − 2C)γ + C

) u



×R

)

4εγ

L

u



×R

)

Choosing, for instance, ε = C/C

, we ﬁnd that for γ ≥ 2C

u



×R

)

+ α u

(0)

)

≤

4Cγ

L

u



×R

)

+β Bu

(0)

)

This implies the L

estimate in (9.1.14) with c = max(C

/2C, β)/ min(C/4,α)

for γ ≥ max(2C

/C, 2C

/C) .

The proof of (9.1.15) makes use of pseudo-diﬀerential calculus with parameter

(see Section C.2). Indeed, applying the L

estimate obtained above

γ v

))

+ v(0)

)

≤ c



L

v

))

+ Bv(0)

)



to v =Λ

s,γ

u

, where the operator Λ

s,γ

acts on the variables (y, t), we readily

get

γ u



))

+ u

(0)

)

≤ c



L

s,γ

u



))

+ B Λ

s,γ

u

(0)

)



Now, using that {[ L

, Λ

s,γ

]}

γ≥1

is a family of pseudo-diﬀerential operators

of order 1 + s − 1=s,that{[ B, Λ

s,γ

]}

γ≥1

is a family of pseudo-diﬀerential

operators of order s − 1andv

s−1

 γ

−1

v

for all v we obtain a constant

≥ 2c so that

γ u



))

+ u

(0)

)

≤ c



L

u



))

u



))

+B u

(0)

)

u

(0)

)



For γ ≥

√

, the left-hand side absorbs the extra terms in the right-hand side,

and we obtain (9.1.15) with c =2c

. 

Energy estimates 229

Remark 9.1 By density of D in H

, the energy estimate (9.1.14) extends to

any u ∈ e

−γt

(

Ω × R) (which is suﬃcient to pass to the limit in the right-hand

side of (9.1.14) applied to D-approximations of u).

9.1.2 Local/global Kreiss’ symmetrizers

Kreiss’ work [103] has shown that strictly dissipative problems are not the only

ones to enjoy estimates of the kind (9.1.14). In other words, there are boundary

value problems that do admit boundary symmetrizers, even though they are

not strictly dissipative (in the sense of Deﬁnition 9.2). Such problems are those

that satisfy a stability condition based on a normal modes analysis: in fact, the

stability condition derived by Kreiss involves a reﬁned normal modes analysis,

in that it pays attention to the so-called neutral modes. (These terms will be

explained below.) This is in contrast with prior work – dating back to the 1940s

in gas dynamics – in which the role of neutral modes was not well understood.

For examples of problems that satisfy Kreiss’ stability condition without being

strictly dissipative, see for instance Chapter 13 on gas dynamics.

Kreiss introduced a tool known nowadays as a Kreiss symmetrizer, which

turned out to imply energy estimates without loss of derivatives. He performed

the proof of these estimates for constant coeﬃcients and claimed, advisedly,

that they were satisﬁed also in the case of variable coeﬃcients [103]. The

proof was later completed by several authors. The actual construction of Kreiss

symmetrizers, assuming Kreiss’ stability condition, is the bulk part of the original

paper [103]. This tour de force is explained in Chapter 5 – and also in Chapter 13

for the (linearized) Euler equations in the constant-coeﬃcients case. Its extension

to variable coeﬃcients does not contain major diﬃculties but deserves some

explanation.

Before that, let us pause and come back to the analysis of the Boundary Value

Problem (9.1.5)(9.1.6). We have decomposed the derivation of energy estimates

into the following steps:

i) the derivation of energy estimates from functional symmetrizers,

ii) the construction of functional symmetrizers from symbolic ones.

iii) the construction of symbolic symmetrizers.

The splitting between i) and ii) parallels Chapter 2 (on the Cauchy problem).

Step i) has been done in the previous section. Step ii) is the main purpose

of this section. If step iii) is rather easy for the Cauchy problem (assuming

only constant multiplicities of eigenvalues, see Theorem 2.3), it is a tough piece

for the Boundary Value Problem. However, once the work is done for constant

coeﬃcients (see [103] or Chapter 5 in this book), its extension to variable

coeﬃcients is not diﬃcult. Details will be given in Section 9.1.3.

Historical note Kreiss’ stability condition is often referred to as the (uniform)

Lopatinski˘ı–Kreiss–Sakamato condition, as both the Russian Lopatinski˘ıand

the Japanese Sakamato have contributed to the theory, independently of Kreiss:

230 Variable-coeﬃcients initial boundary value problems

even though Lopatinski˘ı is more famous for his work on elliptic boundary value

problems [123], he did contribute to the early developments of the hyperbolic

theory [122] and his name here is not misplaced (as seen in Chapter 4, it is more

speciﬁcally attached to the search for unstable modes by means of the so-called

Lopatinski˘ı determinant); and Sakamoto performed simultaneaously with Kreiss

a similar work on higher-order hyperbolic equations (see [171–174]).

We adopt here a presentation of Kreiss’ symmetrizers slightly diﬀerent from

Chapter 5, which is adapted to variable coeﬃcients and insists on local Kreiss’

symmetrizers.

Notations In the whole chapter, we denote by C

= {τ ∈ C ;Reτ ≥ 0 } the

closed right-half complex plane. Furthermore, in this section and in the next one,

we use the following shortcuts:

the physical space–time set is

Y :=

Ω × R = {(y, x

,t); y ∈ R

d−1

∈ R

,t∈ R },

and its boundary is ∂Y = ∂Ω × R = {(y, 0,t); y ∈ R

d−1

,t∈ R };

the ‘frequency’ set is

P := (C

× R

d−1

)\(0, 0) = {(τ, η) =(0, 0) ; τ ∈ C

,η∈ R

d−1

and its intersection with the unit sphere is P

={(τ,η)∈P; |τ |

+ η

=1};

the whole space–time–frequency set is

X := {X =(y, x

,t,η,τ);(y,x

,t) ∈ Y , (τ, η) ∈ P }

and X

= X

∩ X

with

:= {X =(y, 0,t,η,τ);(y, 0,t) ∈ ∂Y , (τ,η) ∈ P },

:= {X =(y, x

,t,η,τ);(y,x

,t) ∈ Y , (τ, η) ∈ P

Finally, for all X ∈ X, we denote

A(X)=−(A

(x, t))

−1

( τI

+ iA(x, t, η)).

Deﬁnition 9.3 A Kreiss’ symmetrizer for (9.1.5)(9.1.6) at some point X

∈ X

is a C

∞

matrix-valued function r in some neighbourhood V of X in X, which is

associated with another C

∞

matrix-valued function T , and such that

i) the matrix r(X) is Hermitian and T (X) is invertible for all X ∈ V ,

ii) there exists C>0 independent of X ∈ V so that

Re ( r(X) T (X)

−1

A(X)T (X)) ≥ CγI

, (9.1.16)

iii) and additionally, if X

∈ X

, there exist α>0 and β>0 independent of

X ∈ V so that

r(X) ≥ αI

− β (B(y, t)T (X))

∗

B(y, t)T(X) . (9.1.17)

Energy estimates 231

Remark 9.2 The existence of a Kreiss symmetrizer at points in X\X

obviously independent of the boundary conditions!

We focus here on how Kreiss’ symmetrizers generate boundary functional

symmetrizers (and thus the estimates in (9.1.14)). This part of the analysis was

not performed by Kreiss in [103]. It appeared later, in particular in [31], even

though not really stated in this way.

Theorem 9.1 Assume the matrices A

and B are C

∞

functions of (y, x

,t) ∈

Y and (y,t) ∈ ∂Y, respectively, and that they are constant outside a compact set.

If (9.1.5)(9.1.6) admits a Kreiss’ symmetrizer at any point of X

then it admits

a (functional) boundary symmetrizer.

Proof Our ﬁrst aim is to construct a global symmetrizer R(X) deﬁned for all

X ∈ X and such that for some positive constant



Re ( R(X) A(X)) ≥



CγI

,X∈ X , (9.1.18)

and for other positive constants α and



R(X) ≥ αI

−



βB(y,t)

B(y, t) ,X∈ X

. (9.1.19)

Note that these inequalities look very much like microlocal versions of (9.1.10)

and (9.1.11), respectively, with R

) of symbol R(·,x

, ·, ·, ·,γ). So a little

pseudo-diﬀerential calculus with parameter γ will enable us to conclude.

Suppose the matrices A

and B are constant outside the open ball B(0; M).

By homogeneity in (τ,η) ∈ P and in (x, t) ∈ Y\B(0; M ) of the inequalities in

(9.1.18) and (9.1.19), it is suﬃcient to construct R in K := X ∩ (

B(0; M) × P

)

and then extend it by

R(x, t, η, τ)=R



(x, t)

(x, t)

(η, τ)

(η, τ)



for all (x, t, η, τ) ∈ (Y\B(0; M)) × P.

Now, the compact set K can be covered by a ﬁnite number of neighbourhoods

,ofpointsinX\X

together with a ﬁnite number of neighbourhoods V

of points in X

, on which we have well-deﬁned Hermitian matrices r

, r

,and

invertible matrices T

, T

satisfying the requirements of Deﬁnition 9.3. Let us

introduce a partition of unity associated with the covering {V

, V

},thatis

functions ϕ

∈ D (V

;[0, 1]) and ϕ

∈ D (V

;[0, 1]) satisfying the identity



≡ 1 .

232 Variable-coeﬃcients initial boundary value problems

Then we deﬁne a global Kreiss’ symmetrizer on K by

R(X):=



(X)(T

(X)

−1

)

∗

(X) T

(X)

−1



(X)(T

(X)

−1

)

∗

(X) T

(X)

−1

Note that R(X) reduces to



(X)(T

(X)

−1

)

∗

(X) T

(X)

−1

for X ∈ K ∩ X

. By construction, R(X) is always Hermitian.

Deﬁning C as the minimum of the constants C

and C

occurring in (9.1.16)

for r

and r

, we easily see that

Re ( R(X) A(X)) ≥ CγS(X) ,

where the Hermitian matrix

S(X):=



(X)(T

(X)

−1

)

∗

(X)

−1



(X)(T

(X)

−1

)

∗

(X)

−1

is uniformly bounded by below, say by σ>0 on the compact set K. Hence

(9.1.18) holds true with



C := σC.

Similarly, taking α, respectively β, the minimum of the α

, respectively, the

maximum of the β

in the estimates (9.1.17) for r

, we obtain that for all v ∈ C

and all X ∈ K ∩ X

∗

R(X) v ≥ αv

S(X) v − β B(y, t) v

This shows (9.1.19) with α := σα.

Once we have on hand the Hermitian matrices R(X) satisfying (9.1.18) and

(9.1.19), it is not diﬃcult to construct a boundary symmetrizer in the sense of

Deﬁnition 9.1. By a slight abuse of notation, let us simply write R(x

) for the

function

(y, t, η, δ, γ) →R(y, x

,t,η,τ = γ + iδ) .

This is a symbol in the variables (y, t) with parameter γ andofdegree0.

Therefore, by Theorem C.6i), {Op

(R(x

))}

γ≥1

is a family of operators of order

0. We claim that

):=



(R(x

)) + Op

(R(x

))

∗



deﬁnes a functional boundary symmetrizer. Indeed, by Theorem C.6ii), {R

)}

diﬀers from {Op

(R(x

))} by a family of order −1. By Remark C.2, this error can

be absorbed by the zero-order terms for γ large enough. Now, by the parameter

Energy estimates 233

version of G˚arding’s inequality (Theorem C.7), we deduce from (9.1.19) the

inequality

R

(0)u, u + β Bu

≥

α

u

for γ large enough. The case of (9.1.18) is a little trickier, and requires the sharp

G˚arding’s inequality (Theorem C.8, in which the smoothness of coeﬃcients is

crucial). Applying Theorem C.8 to the degree 1 symbol R(X) A(X) −



CγI

we infer from (9.1.18) that

Re R

)u, u≥



γ u

for γ ≥ γ

large enough. 

Remark 9.3 It is somewhat surprising that, on the one hand, a functional

boundary symmetrizer R

need only be deﬁned for γ large enough and, on

the other hand, (local) Kreiss symmetrizers are considered up to points where

γ =Reτ is zero. The reason is twofold, in relation to both the homogeneity

and compactness arguments invoked in the proof of Theorem 9.1. Indeed, the

construction of R

for γ ≥ γ

requires Kreiss symmetrizers at points (x, t, η, τ =

γ + iδ) with γ ≥ γ

but not only: in fact, Kreiss symmetrizers are needed on

a compact subset P

of the frequency set P containing all points of the form

(τ

,η

)=(τ,η)/(τ,η) with (τ, η) ∈ P and Re τ ≥ γ

; letting η go to inﬁnity,

we see that P

must contain P

and in particular points (τ

,η

) such that

Re τ

=0.

9.1.3 Construction of local Kreiss’ symmetrizers

We keep the notations of the previous section, and discuss the construction of

(local) Kreiss’ symmetrizers at points of X

. As already noted, Kreiss’ sym-

metrizers at points X =(y, x

,η,τ) with x

> 0 do not depend on the boundary

conditions: their construction will not necessitate any tricky assumption other

than the constant hyperbolicity of the operator L. Otherwise, for points in X

,we

may distinguish between the case Re τ>0 (i.e. (τ,η) ∈

◦

) and the case Re τ =0,

the latter being much trickier than the former: at points of

:= {X =(y, 0,t,η,τ) ∈ X

;Reτ>0 },

the existence of Kreiss’ symmetrizers relies on the Lopatinski˘ı condition; at points

of X

it requires the (uniform) Kreiss–Lopatinski˘ı condition.

Before going into detail, let us introduce some additional material. In what

follows, we denote

X := {X =(y, x

,t,η,τ) ∈ X

;Reτ>0 }.

The hyperbolicity of the operator L ensures that the matrix A(X) is hyperbolic –

in the sense of ODEs – for all X ∈

X: this observation dates back to Hersh [83] and

234 Variable-coeﬃcients initial boundary value problems

has already been used in Chapter 4 for constant-coeﬃcient problems. Therefore,

for all X ∈

X, we may consider the stable subspace of A(X), which we denote by

−

(X). By Dunford–Taylor’s formula, E

−

is locally smooth. More precisely, if

V is a small enough open subset of

X, there exists a closed contour Γ enclosing

all eigenvalues of A(X) of negative real part for X ∈ V , and the formula

−

(X):=

2 iπ



( zI

−A(X))

−1

deﬁnes a projector onto the stable subspace E

−

(X) such that Ker P

−

(X)isthe

unstable subspace of A(X). Clearly, P

−

inherits the regularity of A:namely,it

is analytic in (η,τ)andC

∞

in (x, t). The projector P

−

will be our basic tool to

construct Kreiss’ symmetrizers at points of

Remark 9.4 The above representation of E

−

(X) shows its dimension is locally

constant and thus independent of X in the connected set

X. Observing that, at

X =(y, 0,t,0, 1), the matrix A(X) reduces to

A(y, 0,t,0, 1) = −(A

(y, 0,t))

−1

=(A(y, 0,t,ν(y, 0)))

−1

we see the stable subspace E

−

(y, 0,t,0, 1) of A(y, 0,t,0, 1) coincides with the

stable subspace E

(A(y, 0,t,ν(y, 0))). Consequently, if the rank p of B(y, t)is

known to be the dimension of E

(A(y, 0,t,ν(y, 0)), the dimension of E

−

(X) also

equals p for all X ∈

The Lopatinski˘ı condition

We call the Lopatinski˘ı condition at some point X ∈

the requirement that

the mapping B(y, t)

−

(X)

: E

−

(X) → R

be an isomorphism. This algebraic

condition is equivalent to an analytical condition on the homogeneous constant-

coeﬃcients problem, say (Π

(y,t)

), obtained by freezing the coeﬃcients at (y, 0,t)

in (9.1.5)(9.1.6) and by taking f = g = 0 as source terms. The mapping

B(y, t)

−

(X)

turns out to be an isomorphism if and only if the problem (Π

(y,t)

)

does not have any non-trivial solution with the following features: square integra-

bility in the direction orthogonal to the boundary; oscillations with wave vector η

in the direction of the boundary; exponential-type behaviour e

τ ·

in time. Would

they exist, such solutions would be called normal modes. The equivalence between

the algebraic condition and the analytical one is a straightforward consequence

of the deﬁnition of E

−

(X), the stable subspace of the hyperbolic matrix A(X).

For B(y, t)

−

(X)

: E

−

(X) → R

to be an isomorphism, an obvious

necessary condition is dim E

−

(X)=p, which is true as soon as p =

dim E

(A(y, 0,t,ν(y, 0)) (see Remark 9.4 above). Once dim E

−

(X)=p is

known, it suﬃces to check that B(y, t)

−

(X)

is one-to-one, a quantitative version

of this condition being

) There exists C>0 so that for all V ∈ E

−

(X), V ≤C B(y, t) V .