Alinhac S. Hyperbolic Partial Differential Equations

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7.5 Klainerman’s Method 121

Here Z

means a product of k Lorentz ﬁelds (either ∂

, ∂

or S, H

,or

), and the bracket is the commutator C deﬁned by

Cu =[Z

,L]u = Z

(Lu) − L(Z

u).

To understand the structure of the operator C,wehavetonoteanumber

of elementary facts:

i) If P =Σ

|α|≤p

(y)∂

and Q =Σ

|α|≤q

(y)∂

are diﬀerential opera-

tors (in some y ∈ R

variable) of orders p and q, respectively, then

[P, Q] ≡ PQ−QP is a diﬀerential operator of order at most p + q −1,

since the terms of order exactly p+q in PQand in QP cancel, because

they are the same. Here, Z

is of order k, L is of order 2, hence C is

of order at most k +1.

ii) We can write symbolically a formula analogous to Leibniz formula,

(ab)=Σ

p+q=k

a)(Z

b),

where the righthand side means a sum of products of p ﬁelds applied

to a times q ﬁelds applied to b, these ﬁelds being taken among the

original ﬁelds of the product Z

iii) Denoting by ∂

a product of l ordinary derivatives, we have

[Z, ∂

]=[Z, ∂]∂

l−1

+ ···+ ∂

[Z, ∂]∂

l−1−p

+ ···+ ∂

l−1

[Z, ∂].

But, as we saw in Chapter 5, [Z, ∂] is either zero or a constant times ∂.

Hence, omitting irrelevant constants, we can write symbolically

[Z, ∂

]=Σ∂

iv) By induction on k,weprovenow

,∂

]=Σ∂

k−1

In fact, if Z

k+1

= Z

k+1

,∂

]=Z

∂

−∂

= Z

[Σ∂

k−1

]+[Z

,∂

=Σ∂

v) Finally, putting together the above remarks, we obtain

,a∂

]=Σ

1≤p,p+q≤k

a)∂

+ a∂

k−1

122 Chapter 7 Variable Coeﬃcient Wave Equations and Systems

We have proved the following lemma.

Lemma 7.10. The commutator C is a linear combination, with irrelevant

numerical coeﬃcients, of terms (Z

a)∂Z

u, 1 ≤ r ≤ k, p+r ≤ k +1,where

a stands for one of the coeﬃcients a

Step 2. Using an energy inequality. We assume now

Z, k≤(n+2)/2

(0) < ∞,

a condition that can be checked on the Cauchy data of u using the equa-

tion Lu = 0. This condition is certainly true in particular when u

are smooth with compact support (see also Exercise 11 of Chapter 6).

Applying then an energy inequality for L to the equation L(Z

u)=−Cu,

and summing the results for all products Z

, we get a control of

Z, k≤(n+2)/2

(t)

1/2

by a righthand side which contains only the time integral of ||Cu||

.Ifwe

assume for all coeﬃcients a = a



+∞

Z, p≤(n+2)/2

||Z

a(·,t)||

∞

dt < ∞,

the Gronwall lemma will ensure that, for some C independent of t,

Z, k≤(n+2)/2

(t) ≤ C.

Step 3. Using Klainerman’s inequality. In the situation of step 2, we

have a control of Σ||(∂Z

u)(·,t)||

. But the above commutation formula

,∂

]=Σ∂

k−1

, applied for l =1,gives

Σ|Z

(∂u)|≤Σ

p≤k

|∂Z

u|.

Hence Klainerman’s inequality (See Theorem 5.12) yields

|∂u|(x, t) ≤ C(1 + r + t)

−(n−1)/2

<r−t>

−1/2

which was our goal.

Of course, there are many variations of this argument: In particular cases

the commutator C may contains only terms of a special structure; It is pos-

sible to use, instead of the standard energy inequality for L,someimproved

or conformal inequality, etc.

7.6 Existence of Smooth Solutions 123

7.6 Existence of Smooth Solutions

Theorem 7.11. Consider as in Section 7.3 the Cauchy problem for a

symmetric hyperbolic system with C

∞

coeﬃcients

Lu = S∂

u +ΣA

∂

u + Bu = f, u(x, 0) = u

(x).

Let D be a compact domain of determination with base Σ

satisfying the

assumptions of Theorem 7.8, and assume u

∈ C

(Σ

) and f ∈ C

(D)

be given, k ≥ 4+[n/2]. Then there exists a unique solution u ∈ C

(D),

σ = k − [n/2] − 3.

Proof:

Step 1. Smoothing operators. We ﬁrst start with some preliminary consid-

erations about smoothing operators. Fix φ ∈ C

∞

) a nonnegative even

function φ with



φdx = 1, vanishing for |x|≥1; the family of functions



(x)=

−n

φ(x/) is an “approximation of the identity,” since





(x)dx =



φ(y)dy =1

and φ



vanishes for |x|≥. We deﬁne an operator C



v = φ



∗v, C



v(x)=





(x − y)v(y)dy.

• The operator C



is smoothing in the sense that it takes a locally

function into a C

∞

function.

• It is formally self-adjoint in L

since

(u, C



≡



u(x)



vdx =



u(x)φ



(x − y)¯v(y)dxdy



¯v(y)





(y − x)u(x)dx =(C



u, v)

• Using the Cauchy–Schwarz inequality, we have



v(x))

≤





(x − y)v

(y)dy,

||C



v||





v(x))

dx ≤



v(y)

dy = ||v||

Similarly, we have

||C



v||

∞

≤





(x − y)|v(y)|dy ≤||v||

∞

124 Chapter 7 Variable Coeﬃcient Wave Equations and Systems

• If v is continuous, C



v converges to v uniformly locally.

Finally, the following commutation lemma holds.

Lemma 7.12 (Commutation Lemma) There exists C such that, for

A, v ∈ C

), vanishing for |x|≥R,allj and all >0,

||[C



,A]∂

v ≡ C



A∂

v − AC



∂

v||

≤ C||∇A||

∞

||v||

Note that this lemma “gains” one derivative, the derivative originally

acting on v being transferred to A. The commutator can be explicitly

written





(x − y)[A(y) − A(x)]∂

v(y)dy.

Integrating by parts, this gives

−





(x−y)(∂

A)(y)v(y)dy +





−n−1

(∂

φ)



x − y





[A(y)−A(x)]v(y)dy.

The estimate for the ﬁrst integral is clear, so we concentrate on the second.

Since A(y) −A(x)=B(x, y) ·(x −y)with|B(x, y)| ≤ ||∇A||

∞

, the second

integral can be written





−n

(x, y)[z

(∂

φ)(z)]



z =

x − y





v(y)dy,

and the estimate follows. This proves the lemma. 

Step 2. Continuation of the proof of Theorem 7.11. After these prelimi-

naries, we start the actual proof of the theorem. The idea is to replace the

original Cauchy problem by the new problem

S∂

v +ΣA

∂



v + Bv = f, v(x, 0) = u

(x),

the operator C



being the one deﬁned above. We will show that this problem

has a solution u



, and that these solutions are bounded independently of 

in some C

(a) Suppose D ⊂⊂ B(0,R)×[0,T]: We choose extensions ˜u

∈ C

(B(0,R))

of u

and

f ∈ C

(B(0,R) × [O, T ]) of f;wealsoﬁxψ ∈ C

∞

(B(0,R))

being 1 in a neighborhood of D. We claim that there exists a solution



∈ C

× [0,T]) of the Cauchy problem

S∂



+ ψ(x)ΣA

∂



+ Bu



f, u



(x, 0) = ˜u

(x).

7.6 Existence of Smooth Solutions 125

In fact, the operator ∂



is, for ﬁxed ,abounded operator in C

,since

∂



v(x)=





−n−1

(∂

φ)



x − y





v(y)dy.

Hence the above new Cauchy problem can be viewed as a Cauchy problem

for an ODE in the t-variable for functions with values in C

(b) We proceed now to evaluate the tangential derivatives v

= ∂



for

|α| = l ≤ k − 1. To do this, we apply ∂

to the equation satisifed by u



and obtain



≡ S∂

+ ψΣA



∂

= F

where, for ﬁxed t, ||F

(·,t)||

≤ C + CΣ

|β|≤l

||∂



(·,t)||

ΣψA



∂

governing v

an energy inequality with ﬁxed constants indepen-

dent of .Infact,(, )beingtheL

scalar product in x,

,ψA



∂

)=− (∂



ψA

],v

)

= − (C



∂

(ψA

) − ([C



,ψA

]∂

)

− (ψA



∂

Hence

2(v

)=∂







+ R,

with |R(t)|≤C||v

(·,t)||

, thanks to the commutation lemma. Proceeding

as usual, we obtain, for a ﬁxed constant C, the standard energy inequality

for ﬁrst order systems

||v

(·,t)||

≤ C||v

(·, 0)||

+ C



||F

(·,s)||

ds.

(d) Summing all energy inequalities for all α, |α|≤k − 1, and using

theGronwallLemma,weobtainΣ||∂



≤ C. Using Sobolev lemma

(Lemma 5.11), this gives a control of the C

(s = k − 2 − [n/2]) norm

in x for ﬁxed t of u



(x, t), and using the system on u



, we ﬁnally get a

control of the C

norm in all variables of u



. Using Ascoli’s theorem, we

obtain a subsequence of u



, which we denote by u





,converginginC

s−1

to a

function u. Note that the assumption on k implies s − 1 ≥ 1. Writing







− u = C







− u)+C



u − u,

126 Chapter 7 Variable Coeﬃcient Wave Equations and Systems

we see that C







converges to u at least in C

, hence u is a solution of the

Cauchy problem

S∂

u + ψΣA

∂

u + Bu =

f, u(x, 0) = ˜u

(x).

Restricting u to D, we obtain a solution of the original Cauchy problem,

and we already know that this solution is unique. 

The “ﬂaw” of this proof is this: For a symmetric hyperbolic system in

×R

, we have no direct control of the C

norm of the solution by the C

norms of the data. We are forced to use L

norms and energy inequalities

to control the solution, and this implies, via the Sobolev lemma, some loss

of C

regularity. An optimal result can only be obtained in the framework

of Sobolev spaces, using distribution theory, and this is the reason why we

made no attempt to optimize the loss of derivatives in the present result.

7.7 Geometrical Optics

We ﬁnish this chapter by presenting a method for obtaining explicit

approximate solutions of variable coeﬃcients equations. This method has

an extremely wide range of applications, far beyond hyperbolic equations

(local solvability of general equations, counterexamples, etc.). In the ﬁrst

two paragraphs we sketch the method in the general setting of any operator

P in R

. Only in Section 7.7.3 do we come back to speciﬁc use of it for

the hyperbolic Cauchy problem.

7.7.1 An Algebraic Computation

Let P (x, ∂

)=Σα

(x)∂

+ β

(x)∂

+ γ(x) be a second order diﬀerential

operator with C

∞

coeﬃcients on R

. The following lemma is obtained by

a straightforward computation.

Lemma 7.13. Let a, φ ∈ C

∞

) and τ ∈ C be given. Then

exp(−iτφ)P [a(x)exp(iτφ(x))] = −τ

a(x)p

(x, ∇φ(x))

+ iτ[(∂

)(x, ∇φ(x))∂

a(x)+a(x)q(x)] + P (a).

Here, p

(x, ξ)=Σα

(x)ξ

is the principal symbol of P ,and

q = Pφ− γφ.

A similar lemma for operators of order m is left as Exercise 11.

7.7 Geometrical Optics 127

7.7.2 Formal and Actual Geometrical Optics

We use the above Lemma 7.13 as follows: We think of τ as a big parameter

(say |τ|→+∞), and try to choose a and φ so that e

−iτ φ

P (ae

iτ φ

)isas

small as possible:

First, to cancel the biggest term (i.e. the coeﬃcient of τ

), we look, in the

region of interest, for a function φ (the “phase”) satisfying the so-called

“eikonal equation”

(x, ∇φ(x)) = 0.

This is a fully nonlinear ﬁrst order equation exactly of the type studied

in Chapter 3. This equation may not have any solution in general, and if it

has, φ is not necessarily chosen real (so that the exponential factor e

iτ φ

may

be big even if τ ∈ R). If p

is real, however, the procedure of Chapter 3

provides us with at least a local real solution φ.

Second, to cancel the second term (i.e. the coeﬃcient of τ), we choose a

(the “amplitude”) to be solution of the ﬁrst order linear equation

a ≡ (∂

)(x, ∇φ(x))∂a + qa =0.

This equation is called “transport equation.” We remark that the

principal part of L

is the projection on R

of the Hamiltonian ﬁeld of p

takenonthegraphof∇φ: these are precisely the objects which appear when

applying the method of Chapter 3 to construct φ. Again, the transport

equation need not have any solution in general, but it has if p

and φ are

real, according to Chapter 1.

Third, we can improve the procedure by trying formally

a(x)=a

(x)+

(x)

+ ···+

(x)

+ ··· .

Choosing successively (if possible)

=0,iL

+ Pa

=0,iL

+ Pa

=0,...

we obtain e

−iτ φ

P (ae

iτ φ

) ∼ 0, this symbol meaning that all coeﬃcients of

the various powers of τ vanish.

This is the formal approach to geometrical optics.

128 Chapter 7 Variable Coeﬃcient Wave Equations and Systems

To deﬁne an actual approximate solution u = ae

iτ φ

of Pu =0,wehave

now two choices

i) For some integer N,wetakesimply

a(x)=a

(x)+···+

(x)

Then, locally in x,

−iτ φ

P (ae

iτ φ

)=O(τ

−N

ii) We use the Borel lemma (see Exercise 14).

Lemma 7.14 (Borel Lemma) Let a

∈ C be a sequence of numbers.

Then there exists a function f ∈ C

∞

(R) such that

f(x) ∼ Σa

,x→ 0.

This last line (“asymptotically equivalent”) means that for all integers N ,

f(x) − Σ

0≤n≤N

= O(x

N+1

),x→ 0.

In the present case, we use an extension of Borel lemma for the variable

1/τ → 0, x being a parameter. We thus obtain a C

∞

function a(x, 1/τ)

such that

−iτ φ

P (ae

iτ φ

) ∼ 0.

7.7.3 Parametrics for the Cauchy Problem

Let us assume now that the operator P is a variable coeﬃcients strictly

hyperbolic second order equation in R

× R

. We will apply the theory

outlined above to solve, in a neighborhood of the origin, say, the homoge-

neous Cauchy problem

Pu =0,u(x, 0) = u

(x), (∂

u)(x, 0) = u

(x).

Step 1. We cho ose two real “phase” functions φ



(x, t, ω)( = ±), depend-

ing on the parameter ω ∈ R

, |ω| = 1, solutions of the Cauchy problem

∂



(x, t, ω)+λ



(x, t, ∂



(x, t, ω)) = 0,φ



(x, 0,ω)=x ·ω.

Here, λ

−

<λ

are the characteristic speeds of the equation.

7.7 Geometrical Optics 129

Step 2. Taking τ ∈ R, we choose two “amplitudes” a



(x, t, ω, 1/τ)

satisfying all the transport equations and such that





x, 0,ω,



=1.

The possibility of such a choice is a consequence of the constructive proof

of the Borel lemma.

We obtain in this way two families of approximate solutions of Pu =0,



(x, t, τ, ω)=a





x, t, ω,



exp(iτφ



(x, t, ω)).

Step 3. The “trick” is now to take

ξ ∈ R

,τ= |ξ|,ω=

|ξ|

and sum the corresponding solutions by writing some integral with respect

to ξ. We thus choose two functions A

−

(x)andA

(x)(sayinS for sim-

plicity), and set

(x, t)=(2π)

−n





iτ φ



(x,t,ω)



(x, t, ω,

)



(ξ)(1 − χ(|ξ|))dξ.

Here, χ ∈ C

∞

(R) is a truncation function which is 1 in a neighborhood

of the origin, used only to avoid any trouble in the integral for ξ close to

the origin. The subscript a stands for “approximate.” Taking into account

the choices made for φ



and a



,weget

i) Pu

∈ C

∞

ii) u

(x, 0) − (A

−

+ A

)(x) ∈ C

∞

iii) (∂

)(x, 0) = (2π)

−n



ix.ξ

[−iτλ



(x, 0,ω)+(∂



)(x, 0,ω,

)]



(ξ)(1 − χ(|ξ|))dξ.

(i) results from the fact that, after applying P to the integrals, we get a

similar integral with a



replaced by a rapidly decaying function as

|ξ|→+∞. (ii) is just the Fourier inversion formula, and (iii) is a straight-

forward computation.

Claim 7.15. Given functions u

∈ C

∞

), we can choose the func-

tions A

−

and A

in such a way that, for x close to zero,

(·, 0) − u

∈ C

∞

, (∂

)(·, 0) − u

∈ C

∞

130 Chapter 7 Variable Coeﬃcient Wave Equations and Systems

The proof of this claim would require some insight into the theory of

pseudodiﬀerential operators, which is far beyond the scope of this book (the

interested reader may consult the book by S. Alinhac and P. G´erard [5]).

However, we can grasp the essential point if we assume that the “symbols”



= −iτλ



(x, 0,ω)+(∂



)



x, 0,ω,



appearing in the formula for ∂

(x, 0) do not depend on x.Inthiscase,

to prove the claim, we just have to choose

−

and

such that, for |ξ| big

enough,

−

=ˆu

−

+ s

=ˆu

This is possible, since by strict hyperbolicity λ

−

= λ

, which implies

−

= s

for |ξ| big enough. 

To summarize, we have obtained u

such that

∈ C

∞

(·, 0) − u

∈ C

∞

, (∂

)(·, 0) − u

∈ C

∞

We say that u

is a solution modulo C

∞

of the Cauchy problem

Pu =0,u(x, 0) = u

(x), (∂

u)(x, 0) = u

(x),

and the formula deﬁning u

is called a “parametrix” of the Cauchy

problem. Since we know (from Section 7.6) how to solve the Cauchy prob-

lem for P with C

∞

data, we see that the exact solution u of the Cauchy

problem is the sum of the approximate solution u

and a C

∞

function. It is

thus possible to solve (locally at least) the Cauchy problem for non-smooth

data, and to read from the formula deﬁning the explicit approximate solu-

tion u

the singularities (modulo C

∞

) of the true solution.

7.8 Exercises

1.(a) Consider, in R

× R

, the operator

L = ∂

+2∂

(∂

− ∂

Show that L is strictly hyperbolic with respect to t. Compute the energy

(T ) obtained by integrating (Lu)(∂

u)inastripS

, and observe that E

is not positive.

(b) Performing the change of variables

= x

− t, X

= x

,T= t, u(x

,t)=v(x

− t, x

,t),