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

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Weakly dissipative symmetrizer 195

Let us turn now to point ii).Wehave

Re (σa)=



(Re τ)Re a

−

∗

−

∗

+ m

∗

)

(ma

−

+ a

∗

m) −κRe a



Since a

are diagonal, the subscript indicating the sign of the real parts of the

diagonal entries, we have ±Re a

> 0

. Therefore, Re (σa) ≤−c

(Re τ)I

will

hold true locally with c

> 0, provided κ>0 and the following property holds

true

−

+ a

∗

m = O(Re τ). (7.2.31)

Since m and a

are smooth, this amounts to saying that ma

−

+ a

∗

m vanishes

along the boundary {Re τ =0}. Recalling that in this instance, a

∗

equals −a

−

we deduce the following equivalent form to (7.2.31):

[m, a

−

]=0, whenever Re τ =0. (7.2.32)

In the (generic) case where a

−

has simple eigenvalues, this just tells us that m

is diagonal for Re τ =0.

Summarizing the above analysis, we conclude that if (7.2.23) holds true, then

there exists a weakly dissipative symmetrizer σ in a domain V∩{Re τ>0}

where V is a neighbourhood of P

, σ being given by (7.2.29) and (7.2.30) with

κ>0 large enough. This allows us to state

Theorem 7.1 Let L = ∂



∂

be a constantly hyperbolic operator, such

that A

∈ GL

(R). Assume that n =2p where p is the number of characteristics

incoming in Ω={x

> 0} (so that there may exist elliptic frequency boundary

points).

Let B ∈ M

p×n

(R) have rank p, such that the Kreiss–Lopatinski˘ı condition

is fulﬁlled at every point (τ,η) =(0, 0) with Re τ ≥ 0, except possibly at some

elliptic frequency boundary points.

Assume ﬁnally that, at boundary points P where it vanishes, the Lopatinski˘ı

determinant ∆ satisﬁes (7.2.27), and that E

−

(P ) ∩ kerb is of dimension 1,its

Krylov space under A(P ) being equal to E

−

(P ).

Then there exists a smooth dissipative symmetrizer Σ(τ,η) for the homoge-

neous IBVP

Lu = f in Ω × (0,T),Bu=0on ∂Ω × (0,T).

Note that the smoothness of the construction and the robustness of the assump-

tions allow us to treat operators L and B with variable coeﬃcients, as well as

homogeneous IBVPs in general domains with smooth boundaries.

Corollary 7.1 Let Ω be either a half-space, or a bounded open domain of R

with a smooth boundary. Let L be a constantly hyperbolic operator with smooth

Rigorously speaking, we only need a bound by |Re τ|

1/2

. But since everything is smooth, this

amounts to having a bound by Re τ .

196 The homogeneous IBVP

coeﬃcients, ∂Ω being non-characteristic. Let B(x) ∈ M

p×n

(R) be a boundary

matrix of rank p everywhere on ∂Ω.IfΩ is unbounded, we assume that L and B

have constant coeﬃcients outside of some ball.

Assume that the Kreiss–Lopatinski˘ı condition is fulﬁlled at every point (τ,η; x)

(x ∈ ∂Ω, η ∈ T

∂Ω, Re τ ≥ 0 and (τ,η) =(0, 0)), except possibly at some elliptic

frequency boundary points.

Assume at last that at these vanishing points, the Lopatinski˘ı determinant

vanishes at ﬁrst order only (say (7.2.27)), that the (non-trivial) intersection

of kerB with E

−

(τ,η; x) is a line, whose Krylov subspace under A(τ,η; x) is

−

(τ,η; x) itself.

Then the homogeneous BVP

Lu = f in Ω × (0,T),Bu=0on ∂Ω × (0,T),u(·, 0) = 0 in Ω

is well-posed in L

, in the sense that for every f ∈ L

(Ω × (0,T)), there exists a

unique solution u ∈ L

(Ω × (0,T), which satisﬁes, furthermore



−2ηt

u(t)

dt ≤



−2ηt

f(t)

for every η>0.

7.3 Surface waves of ﬁnite energy

When the Lopatinski˘ı condition fails at an elliptic point (iρ, η) of the boundary,

we know that there exists a non-trivial solution of the homogeneous BVP, of the

form

U(x, t)=e

i(ρt+η·y)

V (x

where V ∈ L

(0, +∞). This solution displays important properties. On the one

hand, it is a travelling wave in the direction η parallel to the boundary, with

velocity σ = −ρ/|η|. Contrary to waves associated with a zero (τ, η) of the

Lopatinski˘ı determinant ∆ with Re τ>0, this wave is not responsible for a

Hadamard instability. On the other hand, its energy density per unit surface

area, namely



∞

|V (x

is ﬁnite, and thus such waves can be used directly to construct exact solutions of

the homogeneous BVP. This makes a contrast with the case studied in the next

chapter, when ∆ vanishes on the boundary at non-elliptic points.

Because of these two properties, such special solutions are called (elementary)

surface waves. By extension, we also call any linear combination of elementary

ones a surface wave. At ﬁrst glance, one may think that since the vanishing of

∆ at some boundary point but not at interior points (Re τ>0) is highly non-

generic, the elementary surface waves are likely to form a single line, and there is

Surface waves of ﬁnite energy 197

no non-trivial linear combination. We may object to this observation with three

facts.

To begin with, the scale invariance of the BVP implies that U

(x, t):=

U(λx, λt) is again a solution. We can therefore construct (composite)

surface waves by the formula

u(x, t):=



∞

(x, t)dµ(λ),

where µ is a ﬁnite measure. If dµ = φ(λ)dλ with φ ∈ D (0, +∞), such a

solution belongs to the Schwartz class in the η direction. Thus we can

localize the wave in the direction of propagation. However, it remains a

travelling wave in that direction.

Next, many physically relevant systems are rotationally invariant, in the

sense that d = 3 and the orthogonal group O

(or perhaps only the

subgroup SO

of rotation) acts on both the independent variable x (in

a natural way), and on the dependent one u, leaving the PDEs unchanged.

Of course, the half-plane Ω is left invariant only by the subgroup SO

and invariance means that the action of SO

preserves the homogeneous

boundary condition. In such a case, SO

also acts on the set of elementary

surface waves, which is thus rather large. Let us denote by U

R,λ

the image

of U

by the action of the rotation R. This wave propagates in the direction

Rη.If(R, λ) → φ(R, λ) is a test function, we can deﬁne a surface wave by

the formula

(x, t):=



∞



R,λ

(x, t)φ(R, λ)dR dλ. (7.3.33)

Such a solution is now localized in every spatial direction, in the sense

that for each time t, it is square-integrable over the physical space Ω. We

warn the reader that this construction does not furnish all the ﬁnite energy

solutions of the homogeneous BVP, however. Thus it cannot be used to

solve the IBVP in a closed form.

Finally, it has been shown in [189] that for second-order BVPs that come

from a Lagrangian through the Euler–Lagrange equations, there exists

at least one elementary surface wave in every direction. This situation

contrasts with the general case, where the vanishing of ∆ at some elliptic

point but at no interior point does not persist under small variations of L

and/or B. Together with the scale invariance, this yields a set of elementary

waves U

parametrized by non-zero vectors η ∈ R

(more generally by

η ∈ R

d−1

). Thus a construction similar to (7.3.33) works, where the integral

is taken over R

A remarkable fact is that the decay of surface waves is slower than the decay

we observe in the Cauchy problem: The dispersion of the energy is aﬀected by the

boundary condition and by the possibility for waves to travel along the boundary.

198 The homogeneous IBVP

Assume, for instance, that L is symmetrizable, and that the boundary condition

is conservative for the associated quadratic energy. Thus the total energy is

conserved. If we just have a Cauchy problem in R

× (0, +∞), the energy is

expected to propagate in the direction of the characteristic cone of L (a kind

of Huyghens principle, see, for instance, [184]). After a large time t, and if the

initial data is compactly supported, the solution concentrates in a corona C

B + tΛ, where Λ denotes the group velocities in every direction of R

,a(d − 1)-

dimensional manifold. Since the area of C

is of order t

d−1

, the energy density

typically decays like t

1−d

, meaning that the amplitude of the solution decays as

(1−d)/2

. If we have a conservative BVP instead, with surface waves, then the total

energy asymptotically splits into two parts. One part is associated with the bulk

waves, which are waves propagating away from the boundary. These waves obey

more or less the same description as in the Cauchy problem. The other part of the

energy is carried by surface waves and thus remains localized in a strip along the

boundary. Since we anticipate the same kind of dispersion as above, though along

the boundary instead of in Ω, we expect that the surface energy concentrates in

a domain B + tΛ

,whereΛ

denotes the group velocities of surface waves in

every direction of the boundary, a (d − 2)-dimensional manifold. Therefore the

amplitude of surface waves typically decays as t

1−d/2

, instead of t

(1−d)/2

.Thus

the decay of surface waves is weaker than that of bulk waves: the former are

roughly

√

t times larger than the latter for large time. We point out that this

eﬀect can be reinforced by inhomogeneities of the boundary. For instance, if d =3

and ∂Ω=G × R,whereG is a non-ﬂat curve, one frequently observes guided

waves in the direction of x

. A wave guided along an m-dimensional subspace

of the boundary typically decays as t

(1−m)/2

. For instance, a guided wave in our

three-dimensional space (thus m =(d − 1) − 1 = 1) does not decay at all!

Example: Rayleigh waves in elastodynamics The best known example of

surface waves arises in elastodynamics, where they are called Rayleigh waves.

They are responsible for the damage in earthquakes: the Earth is a half-space at

a local scale, and can be considered as an elastic medium. The vibrations of the

Earth obey exactly the Euler–Lagrange equation of the Lagrangian equal to the

diﬀerence of the kinetic and bulk energy. In particular, the boundary condition is

zero normal stress. As explained above, a signiﬁcant part of the elastic energy is

concentrated along the surface. This energy is formed of kinetic and deformation

energies, the latter being observable once the earthquake has gone away. When an

earthquake happens in a mountain range, a guided wave can form, which almost

does not decay, reinforcing the damages in a narrow strip. This phenomenon was

evoked `aproposofKob´e’s earthquake in 1995.

It is worth noting that the Rayleigh waves travel much slower than bulk

waves. The latter split into two families: compression waves, also called P-waves,

for which the medium vibrates in the direction of propagation, and shear waves,

called S-waves with perpendicular vibration. When the bulk energy is convex and

coercive over H

(Ω) (a more demanding property than coercivity over H

)),

Surface waves of ﬁnite energy 199

their velocities c

and c

satisfy 0 <c

; see [189]. The normal stress

of these waves does not vanish at the boundary and thus they can propagate

only away from the surface. It is a linear combination of P- and S-waves that

satisfy the boundary condition, and thus forms the Rayleigh waves. Amazingly,

this combination has the eﬀect of lowering the wave velocity. The square of

the Rayleigh velocity c

is the unique positive solution of the quartic equation

(see [185], Section 14.2 for details)



− 1





− 1



− 1



. (7.3.34)

Strangely enough, Equation (7.3.34) has two positive solutions if c

.How-

ever, this fact is meaningless because then we can establish that the IBVP

is Hadamard ill-posed: the Lopatinski˘ı condition fails at some interior point

(see [189]).

Other examples The simplest Lagrangian to which the results of [189] apply

u →



− c

|∇

)dx dt

with a scalar ﬁeld u(x, t). The corresponding IBVP is the wave equation with the

Neumann boundary condition. This is a borderline case, where the surface wave

has inﬁnite energy. It corresponds to the vanishing of the Lopatinski˘ı determinant

at the boundary of the elliptic region of boundary points. This is the exceptional

situation with n =2andp = 1 that we discussed above.

A more appealing example arises in the modelling of liquid–vapour phase

transitions, which can be roughly described as follows. We consider a ﬂuid

governed by the isothermal Euler equations, which express the conservation of

mass and momentum. (See Chapter 13 for the complete system.) We assume an

equation of state p = P (ρ)(ρ the density, p the pressure), where P is van der

Waals-like, in particular it is not monotone. Each interval where P is increasing

corresponds to a phase, typically either vapour or liquid. A phase transition may

be viewed as a sharp discontinuity between a liquid state and a vapour state.

Physically interesting phase transitions are subsonic, and therefore correspond

to undercompressive discontinuities (see Chapter 12 for this notion). In this

case, Rankine–Hugoniot conditions are not suﬃcient to select physically relevant

patterns: an additional algebraic relation is needed, which is called a kinetic

relation in the mathematical theory of undercompressive shocks (see [113]). One

way to derive this relation for phase boundaries was introduced by Slemrod [196]

and Truskinovsky [213], who pointed out that the apparent jump should in fact

correspond to a microscopical internal structure called a viscous-capillary proﬁle.

(See also the survey paper [54].) In [9], Benzoni-Gavage considered the zero-

viscosity limit of the viscosity-capillarity criterion of Slemrod and Truskinovsky,

and found surface waves of ﬁnite energy at the linearized level (see Chapter 12 for

200 The homogeneous IBVP

the stability analysis of free boundary problems). In the light of the more recent

work [189], that is not surprising. Indeed, the capillarity criterion gives as a

kinetic relation the dynamical analogue of the equality of chemical potentials

between phases at equilibrium; alternatively, it means that the total energy

(= kinetic energy T plus free energy F ) is conserved across the discontinuity.

In particular, the corresponding free boundary problem is time reversible and

can be viewed as the Euler–Lagrange system of the Lagrangian



(T −F )dx dt.

A CLASSIFICATION OF LINEAR IBVPs

We continue the analysis of the linear IBVP with constant coeﬃcients, in the

half-space

Ω={x ∈ R

; x

> 0}.

Assume for the sake of simplicity that the boundary {x

=0} is non-

characteristic:

det A

=0.

We are interested in this chapter in classifying IBVPs. Because practical applica-

tions involve pairs (L, B) with variable coeﬃcients, we are interested in notions

that are stable upon a small variation of either L, B or both. Thus our parameter

space IB will be that of pairs (L, B)whereL is hyperbolic and the IBVP is

normal, namely

−

(1, 0) ∩ kerB = {0}.

The structure of IB may be rough at some points, where the operator is not

strictly or constantly hyperbolic. We note, however, that a small perturbation

preserves strict hyperbolicity, because the unit sphere is compact. Since normal-

ity is an open condition, we deduce that normal pairs (L, B) with L strictly

hyperbolic are interior points of IB. We denote the corresponding class IB

Likewise, the set of pairs (L, B), where L is a symmetric operator, is a linear

space in which the set IB

(F for ‘Friedrichs symmetric’) of normal pairs is a

dense open subset.

We say that a property (P) is ‘robust’ if it deﬁnes an open subset in either IB

or IB

, and then we specify the context. Our goal is to identify robust classes,

in terms of the estimates available for the corresponding IBVPs. We essentially

follow [13, 186]. The reader interested in the transitions between robust classes

should go to [13].

Because the description of E

−

(τ,η) is not completely understood along the

boundary {Re τ =0,η∈ R

d−1

} for general hyperbolic operators, we may wish

sometimes to restrict our study within either the class IB

(L is constantly

hyperbolic) or IB

(L is Friedrichs symmetrizable).

202 A classiﬁcation of linear IBVPs

8.1 Some obvious robust classes

The ﬁrst robust class that we encountered in this book is, within IB

,that

of systems with a strictly dissipative boundary condition. See Section 3.2.

Within the class IB

, we may think of systems that satisfy the (UKL) con-

dition. As a matter of fact, the Lopatinski˘ı determinant ∆ is a continuous

function over the hemisphere

|τ|

+ |η|

=1, Re τ ≥ 0,η∈ R

d−1

It is not hard to extend Lemma 4.5 so that if L and B depend on parameters

, then ∆ can be deﬁned continuously in terms of (τ,η,). Assume now

that (UKL) holds true for  = 0. This means that the continuous function

|∆(·, ·, 0)| is bounded below by a positive number. Since the half-sphere is

compact, this remains true for small values of . Hence, nearby IBVPs of

which the operator is constantly hyperbolic satisfy (UKL).

Finally, the set of strongly unstable IBVPs is also a robust class. Let



) be a continuous family in IB,with(L

) strongly ill-posed.

This means that the Lopatinski˘ı determinant ∆

of (L

) vanishes at

some point (τ

,η

) with Re τ

> 0. In a neighbourhood V of (τ

,η

), E

−

depends continuously on L. Therefore, we may deﬁne in V×(−α, α)a

Lopatinski˘ı determinant ∆(·, ·,) that is continuous in , analytic in η

and holomorphic in τ .Wenowﬁxη = η

and let vary .For =0, the

holomorphic function f

(τ)=∆(τ,η

, 0) vanishes at τ

. Note that η

=0

since the IBVP is normal.

Lemma 8.1 f

is a non-trivial holomorphic function.

Proof. The analytic function F := ∆(·, ·, 0) is positively homogeneous. If

vanished identically, then F (τ,η) would vanish provided Re τ>0and

η ∈ R

+∗

. By continuity, it would also vanish for η = 0. But this contra-

dicts the normality of the IBVP. 2

It follows that τ

is an isolated zero of f

. Now, Rouch´e’s Theorem tells

us that holomorphic functions nearby f

do have roots near τ

. In particular,

for small enough values of , there exist a root τ()of∆(τ, η

,)=0,with

τ(0) = τ

and hence Re τ () > 0. Therefore, IBVPs with (L, B)nearby

) are strongly unstable.

8.2 Frequency boundary points

In this section, we study in more detail the structure of the ‘stable’ subspace

−

(τ,η)whenτ = iρ is pure imaginary. As we shall see below, this structure

depends highly on the region of the boundary set {Re τ =0,η∈ R

d−1

}.We

recall that in general, although E

−

(iρ

,η

) is invariant under A(iρ

,η

), it does

not equal the stable subspace of this matrix. By deﬁnition it is only the limit of

Frequency boundary points 203

the stable subspace E

−

(τ,η)when(τ,η) → (iρ

,η

) with Re τ>0. In particular,

it contains the stable subspace:

(A(iρ

,η

)) ⊂ E

−

(iρ

,η

). (8.2.1)

Likewise, we have

(A(iρ

,η

)) ∩ E

−

(iρ

,η

)={0}. (8.2.2)

8.2.1 Hyperbolic boundary points

We say that a subspace of C

is of real type if it admits a basis formed of vectors

in R

, or equivalently if it is the complexiﬁcation of some subspace of R

We recall that for a hyperbolic operator L = ∂

+ A(∇

), the characteristic

cone char(L) is the set deﬁned by the equation

(ρ, ξ) ∈ R × R

, det(ρI

+ A(ξ)) = 0.

In the complement of char(L), the connected component of (1, 0) has been

denoted Γ. It is a convex open cone, whose elements are the time-like vectors.

See Section 1.4 for details.

Theorem 8.1 Let (ρ, ξ) ∈ Γ be given, with ξ =: (η, ξ

). Then the limit E

−

(iρ, η)

of E

−

(τ,η



) as (τ, η



) → (iρ, η) with Re τ>0, exists and is given by the formula

−

(iρ, η)=E



)

−1

(ρI

+ A(ξ))



. (8.2.3)

In particular, E

−

(iρ, η) is of real type.

This result is by no means obvious because of:

Lemma 8.2 Under the assumption of Theorem 8.1, the matrix A(iρ, η) is

diagonalizable with pure imaginary eigenvalues.

In particular, E

−

(iρ, η)isnot the stable space of A(iρ, η), the latter reducing

to {0}. We begin with the proof of the lemma.

Proof It suﬃces to prove that B(ρ, η):=(ρI

+ A(η))(A

)

−1

is R-

diagonalizable. This is obvious if ρ =0andξ  e

Otherwise, we apply Theorem 1.5 to the pair (ρ, ξ): the matrix

(ρI

+ A(ξ))

−1







α,β

αβ

+(V · m)I





is diagonalizable over the reals whenever

M(R, V ):=



Rξ

Vρ



,R∈ M

(R),V ∈ R

204 A classiﬁcation of linear IBVPs

is non-singular and m ∈ R

. We observe that, since (ρ, ξ) is not parallel to (0,e

we can choose a pair (R, V ) ∈ GL

(R) × R

, such that

−1

e

=0 and V

−1

ξ = ρ.

The inequality tells us that M(R, V ) is non-singular, because of Schur’s comple-

ment formula

det M(R, V )=(detR)(ρ − V

−1

ξ).

Deﬁning m := R

−1

e

,wehaveV · m = 0, and therefore

M(R, V )







e



With this choice, the matrix



α,β

αβ

+(V · m)I

equals A

,andTheo-

rem 1.5 tells us that (ρI

+ A(ξ))

−1

is diagonalizable over the reals. Shifting

by ξ

and conjugating by A

, the same is true for B(ρ, η). 

We prove now Theorem 8.1.

Proof Let δ be the segment joining the point (1, 0) to (ρ, ξ)inR × R

.From

Proposition 1.6, every point of δ is in Γ. Therefore, Lemma 8.2 applies along δ.

Parametrizing δ by s ∈ [0, 1], we may decompose

= E



)

−1

(ρ

+ A(ξ

))



⊕ E



)

−1

(ρ

+ A(ξ

))



since the matrix under consideration has non-zero real eigenvalues. Hence the

stable and unstable subspaces extend analytically as invariant subspaces to a

small neighbourhood V of δ. The above splitting still holds in V. In particular,

we have

= E



)

−1

(−iτI

+ A(ξ))



⊕ E



)

−1

(−iτI

+ A(ξ))



whenever γ =Reτ is small and (ρ =Imτ,ξ) ∈ δ.

Since the space E



)

−1

((−iγ + ρ

+ A(ξ

))



is invariant under A(γ +

iρ

,η

), we may consider the restriction of the latter matrix to that subspace.

When γ>0, its spectrum σ(γ, ρ



,ξ



) consists of complex numbers whose real part

is non-zero. By continuity in the connected set {γ>0}∩V, we deduce that the

number of eigenvalues with positive (resp., negative) real part remains constant.

Therefore, it can be computed at points close to (0, 1, 0), with γ>0; for instance

at the point (γ,1, 0) with γ>0. We then have A(γ + i, 0) = −(γ + i)(A

)

−1

and



)

−1

(−iτI

+ A(ξ))



= E



(1 − iγ)(A

)

−1



= E

((A

)

−1

Therefore σ(γ,1, 0) consists only in numbers of negative real part. Thus, γ>0

implies that the elements of σ(γ, ρ, ξ) have negative real parts, whence



)

−1

(−iτI

+ A(ξ))



⊂ E

−

(τ,η).