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

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The Kreiss–Lopatinski˘ı condition 105

For this reason, we admit only those solutions of (4.2.10) for which U(0) ∈

−

(τ,η). They take their values in E

−

(τ,η) and form a vector space of dimension

p.LetU be such a solution and u be the corresponding solution of Lu =0.

If, moreover, BU(0) = 0, then u satisﬁes the homogeneous boundary condition

Bu(y, 0,t) ≡ 0. At initial time,

u(y, x

, 0) = e

iη·y

U(x

)

belongs to every H¨older space C

k,α

(Ω), while the norm of u(·,t) grows exponen-

tially fast (like exp(tRe τ)) as t increases, provided U(0) = 0. Now, rescaling both

space and time variables yields the parametrized solution to the homogeneous

IBVP:

(x, t):=u(λx, λt),λ∈ (0, +∞).

As λ → +∞, the sequence



(·, 0)



grows at most polynomially in H¨older

spaces, with respect to λ, while the sequence



(t)



grows always exponentially

fast for given positive time. This shows that the mapping

u(·, 0) → u(·,t), (t>0),

if ever deﬁned, may not be continuous between H¨older spaces, even at the price

of a loss of derivatives. From the Principle of Uniform Boundedness, we conclude

that this map cannot actually be deﬁned.

This analysis shows that a necessary condition for well-posedness in H¨older

spaces is that E

−

(τ,η) ∩ kerB = {0} for every η ∈ R

d−1

and Re τ>0.

Deﬁnition 4.3 We say that the hyperbolic IBVP (4.1.1)–(4.1.3) satisﬁes the

Kreiss–Lopatinski˘ı condition (or brieﬂy the Lopatinski˘ı condition)if

−

(τ,η) ∩ kerB = {0}

for every η ∈ R

d−1

and Re τ>0.

Making η = 0, we see that this condition implies that the IBVP is normal in

the sense of Deﬁnition 4.1. Because of Lemma 4.1, the Lopatinski˘ı condition is

equivalent to saying that

= E

−

(τ,η) ⊕ kerB, ∀η ∈ R

d−1

, ∀Re τ>0.

Note that, contrary to the hyperbolicity assumption, this condition is not invari-

ant by time reversing. If one wishes to solve the backward IBVP, one needs to

consider the Laplace variable τ of negative real part, instead of positive ones.

Also, the number of boundary conditions must be equal to the number of negative

(instead of positive) eigenvalues of A

The names of Lopatinski˘ı and Kreiss have been associated with the stability

condition described above since their seminal works [121–123] and [103]. Hersh’s

article [83] is anterior, but the merit of Kreiss, as well as of Sakamoto [174]

in the context of higher-order scalar operators, was to understand the role of

106 Initial boundary value problem in a half-space with constant coeﬃcients

the uniform version of the Lopatinski˘ı condition in the well-posedness theory.

Lopatinski˘ı is much better known for his analysis of the elliptic boundary value

problems, but did contribute to the hyperbolic problem [122]. The algebraic

conditions that ensure the well-posedness in the elliptic theory resemble actually

very much those of the hyperbolic case. The fundamental paper by Agmon et al.

[2], for instance, manipulates the same kind of stable subspaces as we do here.

The Lopatinski˘ı condition is by nature the fact that the following linear

diﬀerential problem is well-posed in L

)whenx

∈ C

and F ∈ L

)

are given





= M





+ F (s),X(0) = x

An important instance of such a problem occurs in control theory, X being the

state and Y the adjoint state. See [72] for a discussion. In the context of initial

boundary value problems, the matrix of the ODE depends on parameters, say

the Laplace–Fourier frequencies, and we shall need some kind of uniformity of

this well-posedness. This important aspect will be discussed in Section 4.3.

4.2.2 Well-posedness in Sobolev spaces

The solutions considered in Section 4.2.1 are not square-integrable with respect

to y and therefore cannot be used directly in the study of the well-posedness in

Sobolev spaces. In order to prove that well-posedness in this context still requires

that the Kreiss–Lopatinski˘ı condition be fulﬁlled, we improve our construction.

Let us assume that the Kreiss–Lopatinski˘ı condition fails at some point

(τ

,η

) with η

∈ R

d−1

and Re τ

> 0. We shall use an analytical tool described

in Section 4.6, called the Lopatinski˘ı determinant. This is a function ∆(τ,η),

which is analytic in η and holomorphic in τ, with the property that the Kreiss–

Lopatinski˘ı condition fails precisely at zeroes of ∆. Its construction will be

explained in Section 4.6.1.

As η moves around η

, the holomorphic function ∆(·,η) keeps as many zeroes

close to η

as the multiplicity of the root τ

of ∆(·,η

) (Rouch´e’s theorem

Since the multiplicities of zeroes are upper semicontinuous functions of η,we

may choose a zero (τ

,η

) such that, when η moves in the neighbourhood V of

,∆(·,η) has a unique root close to τ

, obviously with a constant multiplicity

m. This root, denoted by T (η), is analytic, because it is a simple root of

∂

m−1

∂τ

m−1

∆(·,η).

For each η ∈V, the space F (η):=kerB ∩ E

−

(T (η),η) is non-trivial and, as

above, we may assume that it has a constant dimension and is analytic in η.

A correct use of Rouch´e’s theorem requires that ∆(·,η

) do not vanish identically. This is proved

in Lemma 8.1.

The Kreiss–Lopatinski˘ı condition 107

Then we may choose a non-zero analytic vector ﬁeld X(η) in it. Taking X(η)as

an initial datum, (4.2.9) deﬁnes a function x

→ U(x

; η).

Now, each of the functions

(x, t) → exp(T (η)t + iη · y) U(x

; η)

solves Lu =0,aswellasBu(y, 0,t) = 0. To build an L

function with the same

property, we choose a non-zero scalar function φ ∈ D(V) and deﬁne

u(x, t):=



φ(η) exp(T (η)t + iη · y) U(x

; η)dη.

Plancherel’s Formula yields



Ω

|u(t)|

dx =(2π)

1−d



Ω

|φ(η)|

exp(2tRe T (η))|U(x

; η)|

dηdx

We see that the L

-norm of u(t) increases exponentially fast as t grows. Perform-

ing a rescaling as in Section 4.2.1, we ﬁnd a sequence of solutions (u

)

,such

that the L

-norm of u

(t) increases exponentially fast as λ → +∞ for any given

t>0, while the L

-norm of u

(0) is a constant times λ

−d/2

.EventheH

-norm

of u

(0) is polynomially bounded in λ for every s. We conclude that the IBVP

cannot be well-posed in Sobolev spaces, even at the price of a loss of derivatives.

Proposition 4.2 The Kreiss–Lopatinski˘ı condition is necessary for the well-

posedness of the IBVP in either H¨older or Sobolev spaces. When it fails, no

estimate can hold in such norms, even at the price of a loss of derivatives.

In other words, the failure of the Kreiss–Lopatinski˘ı condition implies a

Hadamard instability.

4.2.3 The characteristic case

When A

is singular, (4.2.9) is not an ODE any longer. To mimic the analysis

of Section 4.2.1, we need to extract from (4.2.9) an ODE. To do that, we ﬁrst

observe that, since A

is diagonalizable, C

is the direct sum of R(A

)andkerA

= R(A

) ⊕ kerA

Denoting by π the projector onto kerA

, along R(A

), we decompose U = r + k,

with k := πU, r := (I

− π)U. Then (4.2.9) is equivalent to

+(I

− π)(τI

+ iA(η))(r + k)=0, (4.2.12)

π(τI

+ iA(η))(r + k)=0. (4.2.13)

From Theorem 1.6, we know that the endomorphism πA(η)ofkerA

has a

real spectrum. Hence π(τI

+ iA(η)) is non-singular on kerA

. Therefore, we may

invert (4.2.13), as k = M(τ,η)r, with M(τ, η) ∈ L (R(A

); kerA

). Then (4.2.12)

108 Initial boundary value problem in a half-space with constant coeﬃcients

becomes an ODE,

= B(τ,η)r.

Given an initial datum r(0), it admits a unique solution r(x

), and k is deter-

mined by k(x

)=M(τ,η)r(x

). Our next result is

Lemma 4.3 For Re τ>0 and η ∈ R

d−1

, the matrix B(τ,η) does not have

pure imaginary eigenvalues. Consequently, the number of eigenvalues of positive

(respectively negative) real part does not depend on (τ,η). It equals the number

of negative (respectively positive) eigenvalues of A

Proof For λ to be an eigenvalue of B(τ,η), it is necessary and suﬃcient that

there exists an r ∈ R(A

), non-zero, and a k ∈ kerA

, such that

λA

r +(τI

+ iA(η))(r + k)=0.

This amounts to saying that

(τI

+ iA(η)+λA

)(r + k)=0.

Therefore one must have det(τI

+ iA(η)+λA

)=0. If λ is pure imaginary,

then τ is so, by hyperbolicity assumption. The rest of the proof is similar to that

of Lemma 4.1. 

We conclude from Lemma 4.3 that bounded solutions of (4.2.9) on R

actually decay exponentially fast at +∞, and form a vector space of dimension

p. They take values in a p-dimensional vector space E

−

(τ,η), again called the

stable subspace of (4.2.9). The space E

−

(τ,η) is made of sums r + M (τ,η)r, with

r in the stable subspace E

(B(τ,η)).

Mimicking Sections 4.2.1 and 4.2.2, we see that a necessary condition for

well-posedness in either H¨older or Sobolev spaces is again the Kreiss–Lopatinski˘ı

condition, which reads

=kerB ⊕ E

−

(τ,η), ∀η ∈ R

d−1

, ∀Re τ>0.

To ﬁnish this section, we notice the following property, whose meaning

is that the failure of the Kreiss–Lopatinski˘ı condition cannot come from the

characteristic nature of the boundary.

Proposition 4.3 For every η ∈ R

d−1

and Re τ>0, it holds that

−

(τ,η) ∩ kerA

= {0}.

Proof Let u = r + k belong to E

−

(τ,η). From (4.2.13), π(τI

+ iA(η))(r +

k) = 0 holds, that is u = r + M(τ,η)r.If,moreover,u ∈ kerA

,thenr =0and

therefore u =0. 

The uniform Kreiss–Lopatinski˘ı condition 109

4.3 The uniform Kreiss–Lopatinski˘ı condition

The symmetric, strictly dissipative case is very suggestive because its estimate

(3.2.19) is the kind that we need for iterative purposes: It measures the solution

u, its value at a given time T and its trace on the boundary, in the same norms

as the corresponding data, respectively f, u

and g.

Another nice feature of (3.2.19) is that it is invariant under the rescaling

(x, t, u) → (λx, λt, u). This transforms (f,g,u

)into(λf,g,u

)and(γ, T)into

(γ/λ, λT ). All ﬁve terms in (3.2.19) are multiplied by the same power of λ.This

immediately gives the following

Lemma 4.4 Assume that the (not necessarily symmetric) hyperbolic IBVP

(4.1.1)–(4.1.3) satisﬁes the a priori estimate (3.2.19) for every compactly sup-

ported, smooth u, for a given value of the parameter γ>0 and every time

T>0. Then the estimate holds for every parameter γ with the same constant

C. Similarly, if the estimate holds for a given time T>0 and every γ>0, then

it holds for every T,γ > 0 with the same constant C.

This suggests to introduce a stronger notion of well-posedness, which turns out

to be suitable for a generalization to variable-coeﬃcients IBVPs. It turns out to

be well-suited for non-homogeneous IBVPs too.

Deﬁnition 4.4 Let us consider a non-characteristic hyperbolic IBVP (4.1.1)–

(4.1.3) in the domain x

> 0,t>0. We say that this IBVP is strongly well-posed

in L

if the a priori estimate (3.2.19) holds for every smooth, rapidly decaying

(in x) u, and every value of γ,T > 0, with a ﬁxed constant C.

The presence of the parameter γ in (3.2.19) (see (4.3.14) below) provides some

ﬂexibility in a non-linear iteration. It can be adjusted to ensure contraction in

local-in-time problems.

4.3.1 A necessary condition for strong well-posedness

Let v be the partial Fourier transform of u, with respect to y, the estimate

(3.2.19) is equivalent to

−2γT

v(T )

+ v

γ,T

(4.3.14)

≤ C



v(0)



−2γt



(

Lv)(t)

+ γ

Bv(t)





where

L = ∂

+ iA(η)+A

∂

. Since (4.3.14) reads



Φ(η)dη ≤ 0,

where Φ(η) depends only on the restriction v(η, ·, ·) (but does not depend on η-

derivatives), it decouples as parametrized inequalities Out[w] ≤ C In

[w] between

110 Initial boundary value problem in a half-space with constant coeﬃcients

measurements of the output and the input, for every η ∈ R

d−1

and every smooth

w(x

,t) with fast decay as x

→ +∞,where

Out[w]:=e

−2γT

w(T)



−2γt





+∞

|w|

+ |w(0,t)|



[w]:=w(0)



−2γt





+∞

+ |Bw(0,t)|



Hereabove, L

is obtained from

L by freezing η. We emphasize that the constant

C is the same as that in (4.3.14). In particular, it does not depend on η.Since

Out[w] is a sum of positive terms, the estimate (3.2.19) implies the inequality



−2γt

|w(0,t)|

dt ≤ C In

[w], (4.3.15)

for every η and smooth, fast decaying w. Let us now choose a complex number

τ with Re τ>0andavectorV ∈ E

−

(τ,η). We apply (4.3.15) to the space-

decaying function

w := e

tτ

exp(x

A(τ,η))V,

for which we have L

w ≡ 0. We then obtain

|V |

≤ C





|BV |





exp(2(Re τ − γ)t)dt



−1



+∞

|w|





Let us choose γ in the interval (0, Re τ). As T → +∞, the integral



exp(2(Re τ − γ)t)dt

tends to inﬁnity. Passing to the limit, we obtain

|V |

≤ C|BV |

In conclusion, we have found a necessary condition for strong well-posedness, in

the form

∀Re τ>0, ∀η ∈ R

d−1

, ∀V ∈ E

−

(τ,η), |V |

≤ C|BV |

, (4.3.16)

for some ﬁnite constant C, independent of (τ, η,v).

We point out that (4.3.16) implies the Kreiss–Lopatinski˘ı condition (that is,

BV =0andV ∈ E

−

(τ,η) imply V = 0). However, it is a stronger property, since

the Kreiss–Lopatinski˘ı condition is only equivalent to an estimate of the form

|V |≤C(τ,η)|BV |, ∀V ∈ E

−

(τ,η),

where the number C might not be uniformly bounded (though being an homo-

geneous function of (τ,η), of degree zero). Hence, (4.3.16) exactly means that

The uniform Kreiss–Lopatinski˘ı condition 111

the Kreiss–Lopatinski˘ı condition holds, with a uniform constant. This justiﬁes

the following

Deﬁnition 4.5 Let L be hyperbolic, A

be invertible. Given B ∈ M

p×n

(R),

we say that the IBVP (4.1.1)–(4.1.3) satisﬁes the uniform Kreiss–Lopatinski˘ı

condition (UKL) in the domain x

> 0,t>0,if

p equals the number of positive eigenvalues of A

there exists a number C>0, such that

|V |≤C|BV |, ∀η ∈ R

d−1

, ∀Re τ>0, ∀V ∈ E

−

(τ,η).

We point out that the inequality |V |≤C|BV | for a single pair (τ,η) already

implies that p is larger than or equal to the number of positive eigenvalues

of A

Remark We have shown above that (UKL) is a necessary condition for the

-well-posedness of the pure (namely f ≡ 0, u

≡ 0) boundary value problem.

If in addition, the operator L is Friedrichs symmetric (or symmetrizable as well),

it is rather easy (see [185], page 199–200) to show that (UKL) is also a suﬃcient

condition for the pure L

-well-posedness. Thus, it seems at ﬁrst glance that

the (UKL) condition concerns only the pure boundary value problem, though

the general IBVP decouples into this one, plus a pure Cauchy problem, via the

Duhamel formula. It is therefore fascinating that the (UKL) condition actually

enables us to prove the L

estimates for the complete IBVP, at least when L is

constantly hyperbolic and the boundary is not characteristic, as we shall see in

Chapter 5.

4.3.2 The characteristic IBVP

Recall that we assume kerA

⊂ kerB in the characteristic case (A

is singular.)

The best control of boundary terms that we expect is that of A

u. Control of

the components of u in the kernel of A

will at least involve higher-order norms

(norms of derivatives) of the data. Consequently, we must adapt our deﬁnition

as follows. We recall that γ

denotes the trace operator on the boundary (while

γ is some positive real number).

Deﬁnition 4.6 Consider a (possibly characteristic) hyperbolic IBVP (4.1.1)–

(4.1.3) in the domain x

> 0,t>0. We say that this IBVP is strongly L

-well-

posed if kerA

⊂ kerB and if, moreover, the quantity

−2γT

u(T )



−2γt





∂Ω

|γ

u(y, t)|

dy + γ



Ω

|u(x, t)|



is bounded by above by



u(0)



−2γt



(Lu)(t)

dt + γ

Bu(t)





112 Initial boundary value problem in a half-space with constant coeﬃcients

for every smooth, rapidly decaying (in x) u, and every value of γ, T > 0,witha

ﬁxed constant C.

As before, we obtain a necessary condition for strong well-posedness, in the form

of a (UKL) condition

∃C>0,



η ∈ R

d−1

, Re τ>0,V ∈E

−

(τ,η)



=⇒(|A

V |≤C|BV |). (4.3.17)

Let us note that (4.3.17) not only implies

−

(τ,η) ∩ kerB ⊂ kerA

In view of Proposition 4.3, this actually ensures the Kreiss–Lopatinski˘ı condition

−

(τ,η) ∩ kerB = {0},

though uniformity holds only

‘modulo kerA

’, according to (4.3.17).

4.3.3 An equivalent formulation of (UKL)

We shall prove later that the (UKL) condition is actually a necessary and suﬃ-

cient condition for well-posedness, at least for the important class of constantly

hyperbolic operators, in the non-characteristic case

. However, this condition, as

deﬁned above, does not give a practical tool when one faces a particular IBVP,

because the computation of the constant C(τ, η) is intricate, and it is not easy

to see whether it is upper bounded as (τ,η) varies. It turns out that there is a

much more explicit way to check (UKL) condition. To explain what is going on,

we begin with the following observation, which will be proved in Chapter 5.

We recall that the set G(n, p) of vector subspaces of dimension p in C

is a

compact diﬀerentiable manifold, called the Grassmannian manifold.Thisobject

is isomorphic to the homogeneous space (set of left cosets) GL

(R)\M

n×p

(R),

where M

n×p

(R) denotes the dense open set of M

n×p

(R) consisting in matrices of

full rank p.

Lemma 4.5 Assume that the operator L is constantly hyperbolic and the

boundary is non-characteristic. Then the map (τ,η) → E

−

(τ,η) (already deﬁned

for Re τ>0, valued in G(n, p)) admits a unique limit at every boundary point

(iρ, η) (with ρ ∈ R,η∈ R

d−1

), with the exception of the origin.

It is then natural to call E

−

(iρ, η) this limit. We emphasize that, in general,

−

(iρ, η) only contains, but need not be equal to, the stable subspace of the

In view of Proposition 4.3, we also have an estimate |V |≤C



|BV | on E

−

(τ,η), at least when

Re τ>0. However, we do not know whether E

−

(τ,η) ∩ kerA

is trivial at boundary points Re τ =0.

This left the possibility that C



= C



(τ,η) and that the estimate of V in terms of BV be non-uniform.

We should temper this sentence. By well-posedness, we mean strong L

-well-posedness of the

non-homogeneous BVP. We shall see in Chapter 8 that strong well-posedness may hold true in some

complicated space, when the Kreiss–Lopatinski˘ı condition is satisﬁed but not uniformly. In Chapter

7, we show that the IBVP with an homogeneous boundary condition needs a property weaker than

(UKL).

The uniform Kreiss–Lopatinski˘ı condition 113

diﬀerential equation A



+ i(ρI

+ A(η))U = 0, since the latter has dimension

less than or equal to p.

Let us assume that the IBVP deﬁned by the pair (L, B) satisﬁes (UKL)

condition. Then, by continuity, (4.3.16) still holds when Re τ = 0, which means

that E

−

(τ,η) ∩ kerB = {0} for these parameters too. Conversely, assume that

this intersection is trivial for every non-zero pair (τ,η) with Re τ ≥ 0. Then, for

every such pair, the number

c(τ,η) := sup



|V |

|BV |

; V ∈ E

−

(τ,η),V =0



is ﬁnite. The function (τ,η) → c is continuous and homogeneous of degree

zero. Since the hemisphere deﬁned by Re τ ≥ 0, η ∈ R

d−1

and |τ |

+ |η|

=1is

compact, we infer that this function is upper bounded. Hence, the IBVP satisﬁes

(UKL) condition. Finally,

Corollary 4.1 Let L be constantly hyperbolic and the boundary be non-

characteristic. Then the IBVP (4.1.1)–(4.1.3) satisﬁes the uniform Kreiss–

Lopatinski˘ı condition if, and only if, E

−

(τ,η) ∩ kerB = {0} for every non-zero

pair (τ,η) with Re τ ≥ 0 and η ∈ R

d−1

This corollary gives a practical tool for checking the (UKL) condition. The

main diﬃculty during calculations being the computation of E

−

(τ,η)whenReτ

vanishes.

Remark We warn the reader that the continuous extension of E

−

(τ,η)to

boundary points (Re τ = 0) may not exist for non-constantly hyperbolic oper-

ators (operators for which eigenvalues do cross each other). This happens

even within the class of Friedrichs-symmetric systems. See, however, the deep

analysis [135] by M´etivier and Zumbrun of Friedrichs-symmetric IBVPs with

characteristic ﬁelds of variables multiplicities, which works out in the case

of MHD.

4.3.4 Example: The dissipative symmetric case

To show the relevance of the notion of the uniform Kreiss–Lopatinski˘ı condition,

we prove:

Proposition 4.4 Let L be Friedrichs symmetric. If B is dissipative, then the

IBVP satisﬁes the Kreiss–Lopatinski˘ı condition. If B is strictly dissipative, the

IBVP satisﬁes (UKL).

Proof Let η ∈ R

d−1

and Re τ>0 be given. Let u be an element of E

−

(τ,η)

and U be the solution of the diﬀerential-algebraic Cauchy problem

(τI

+ iA(η))U + A

=0,U(0) = u.

114 Initial boundary value problem in a half-space with constant coeﬃcients

Then U decays exponentially fast at +∞.SinceL is Friedrichs symmetric, the

standard energy estimates gives

2(Re τ)



+∞

|U|

=(A

u, u).

If B is dissipative and if u ∈ kerB, we deduce that



+∞

|U|

≤ 0,

hence U ≡ 0andu = 0. Therefore E

−

(τ,η) ∩ kerB = {0}. This is the Kreiss–

Lopatinski˘ı condition.

If B is strictly dissipative, we know that there exist positive constants ε and

C such that the Hermitian form

w → ε|A

+(A

w, w) − C|Bw|

is non-positive. With u as above, we immediately obtain

ε|A

≤ C|Bu|

Since this inequality does not depend on (τ, η), the Kreiss–Lopatinski˘ı condition

is satisﬁed uniformly. 

It is not true in general that every IBVP satisfying (UKL) can be put

in a symmetric, strictly dissipative form. A crude reason is that if n ≥ 3

and d ≥ 2, most hyperbolic operators are not symmetrizable. However, when

Lu =(∂

− ∆

)u with u :Ω→ R

(hence L is diagonal) and the IBVP is coupled

through ﬁrst-order boundary conditions, Godunov et al. proved the converse of

Proposition 4.4; see, for instance, [71].

4.4 The adjoint IBVP

The existence of a solution of a general IBVP will be proved by a duality

argument

. Hence, we need to construct an adjoint IBVP. To a pair (L, B),

where L is a hyperbolic operator and B a boundary matrix, we shall associate a

pair (L



,C), such that whenever

Lu = f, L



v = F in x

> 0

and

Bu = g, Cv = h on x

=0,

In the present context of constant coeﬃcients and a half-space domain, the solution could

be constructed directly, as we did in the symmetric dissipative case. But we have in mind the

generalization to variable coeﬃcients.