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

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How energy estimates imply well-posedness 275

we obtain the same inequality for u

, hence (9.2.49) thanks to the boundedness

of (A

)

−1

. 2

Once we have Theorem 9.17 we can prove the L

-well-posedness of the IBVP

with zero initial data exactly as in the case of smooth coeﬃcients (Theorem 9.12):

as a matter of fact, the smoothness of coeﬃcients does not play any role in the

proof of the support theorem (Theorem 9.13), and thus the proof of Theorem 9.12

still works for Lipschitz coeﬃcients, hence the following well-posedness result for

the homogeneous IBVP.

Theorem 9.18 With the assumptions of Theorem 9.6, for all f ∈ L

d−1

× [0,T]) and g ∈ L

d−1

× [0,T]), for all Lipschitz-continuous

v : R

d−1

× R

× [0,T] → W ,

constant outside a compact subset and such that v

takes its values in W

there exists a unique u ∈ L

d−1

× R

× [0,T]) solution of the IBVP

u = f for x

> 0 ,t∈ (0,T) and B

= g, u

|t=0

=0.

Furthermore, u

belongs to L

d−1

× [0,T]) and there exist γ

> 0 and c>

0, depending only on v

1,∞

d−1

×R

×[0,T ]))

such that for all γ ≥ γ

γ e

−γt

u

d−1

×R

×[0,T ])

+ e

−γt



d−1

×[0,T ])

≤ c



e

−γt

f

d−1

×R

×[0,T ])

+ e

−γt

g

d−1

×[0,T ])



(9.2.51)

For the general IBVP we need timewise bounds, which are not easy to

obtain (see Section 9.2.3 regarding smooth coeﬃcients). When coeﬃcients are

just Lipschitz-continuous, the only known result is due to M´etivier [136] and

assumes Friedrichs symmetrizability.

Theorem 9.19 (M´etivier) With the assumptions of Theorem 9.6 and the addi-

tional one that the operator L is Friedrichs symmetrizable, for all f ∈ L

d−1

× [0,T]), g ∈ L

d−1

× [0,T]), u

∈ L

d−1

× R

), for all Lipschitz-

continuous

v : R

d−1

× R

× [0,T] → W ,

constant outside a compact subset and such that v

takes its values in W

the problem







u = f for x

> 0 ,t∈ (0,T) ,

= g for t ∈ (0,T) ,

|t=0

= u

for x

> 0 ,

(9.2.52)

admits a unique solution u ∈ L

d−1

× [0,T]), which is such that u

∂Ω×[0,T ]

∈

d−1

× [0,T]). Furthermore, u belongs to C ([0,T]; L

d−1

× R

)) and

276 Variable-coeﬃcients initial boundary value problems

satisﬁes an estimate of the form

u(T )

d−1

×R

)

u

d−1

×R

×[0,T ])

+ u



d−1

×[0,T ])

≤ c



u



d−1

×R

)

+ T f

d−1

×R

×[0,T ])

+ g

d−1

×[0,T ])



(9.2.53)

with c depending only on v

1,∞

d−1

×R

×[0,T ])

Proof There are basically three (unequal) steps: 1) improve Theorem 9.18 by

showing the solution of the homogeneous IBVP is continuous in time and satisﬁes

a reﬁned version of (9.2.51), namely

−2γT

u

|t=T



d−1

×R

)

+ γ e

−γt

u

d−1

×R

×[0,T ])

+ e

−γt



d−1

×[0,T ])

≤ c



e

−γt

f

d−1

×R

×[0,T ])

+ e

−γt

g

d−1

×[0,T ])



;

2) handle the IBVP with ‘regularized’ initial data u

∈ H

);

3) conclude by a density argument for initial data u

∈ L

Step 1) Recall that the solution given by Theorem 9.18 is u =˘u

|[0,T ]

,where

˘u is the solution of the BVP

˘u =

f, B

˘u

=˘g,

with

f and ˘g being extensions by zero for t<0ort>T,off and g, respectively.

Furthermore, the ‘weak=strong argument’ in the proof of Theorem 9.17 shows

if R

is a regularization operator in the (y,t) directions,











:= R

−γt

˘u −→ e

−γt

˘u in L

d−1

× R

× R),

)

→ e

−γt

˘u

in L

d−1

× R),

:= R

−γt

)

−1

f → e

−γt

)

−1

f in L

d−1

× R

× R),

:= R

−γt

˘g → e

−γt

˘g in L

with u

∈ L

; H

+∞

)), F

∈ L

; H

+∞

)), g

∈ H

+∞

), and

additionally u

∈ H

d−1

× R

× R). Assume for a moment the reﬁned energy

estimate for the homogeneous IBVP, i.e. for all γ large enough, for all τ ∈ R and

all w ∈ H

d−1

× R

× R) such that w

|t<0

≡ 0,

−2γτ

w(τ)

d−1

×R

)

+ γ e

−γt

w

d−1

×R

×[0,τ])

+ e

−γt



d−1

×[0,τ])

≤ c





e

−γt

w

d−1

×R

×[0,τ])

+ e

−γt

w

d−1

×[0,τ])



(9.2.54)

How energy estimates imply well-posedness 277

Then, applying it to w =e

γt

− u



) and using Lemma 9.5 to deal with com-

mutators, we easily show (u

)

|t∈[0,T ]

is a Cauchy sequence in C ([0,T]; L

d−1

)) when ε goes to zero, hence has a limit in that Banach space, and that this

limit must be u by uniqueness of solutions. Additionally, by passing to the limit

in the reﬁned estimate (9.2.54) applied to e

γt

, we see it is also satisﬁed by u.

Therefore, step 1) will be complete when (9.2.54) is proved. By den-

sity of D(R

d−1

× R

× R)inH

d−1

× R

× R) → H

; L

d−1

× R)) →

C (R

; L

d−1

× R)) , it suﬃces to prove (9.2.54) for w ∈ D (R

d−1

× R

× R)

such that w

|t<0

≡ 0. (Any element of H

d−1

× R

× R) vanishing for t<0

can be approached by such a w: it suﬃces to choose a molliﬁer supported in

{t>0}.) Now, introducing a Friedrichs symmetrizer S

for L

, with

σI

≤ S

≤ σ

−1

for σ>0 (independent of v, which is bounded by assumption), denoting L

(γ + L

) and (as usual) w

−γt

w (in such a way that e

−γt

w = L

w

we have



w

∗

w

=2Re



w

∗

w



w

∗

(∂

) w

−2





w

∗

∂

w



w

∗

(∂



∂

)) w

+2Re



w

∗

w



w

∗

w

This implies after integration (using Cauchy–Schwarz and Young’s inequalities),

w

(t)

d−1

×R

)

≤ (γ

+ γ) w



d−1

×R

×[0,t])

γσ

L

w



d−1

×R

×[0,t])

+ δ ( w

)



d−1

×[0,t])

with γ

∂



∂

)

∞

d−1

×R

×(0,t))

and δ :=

S



∞

d−1

×R

×(0,t))

. And the L

norms of w

and its trace at x

are controlled by the inequality in (9.2.51). To be precise, we thus get that for

γ ≥ γ

w

(t)

d−1

×R

)

≤ (a + σ

−4

)

L

w



d−1

×R

×[0,t])

+ a B

( w

)



d−1

×[0,t])

278 Variable-coeﬃcients initial boundary value problems

with a = c max(2,δ). Going back to w itself, we see this inequality together with

(9.2.51) gives the reﬁned inequality in (9.2.54) with c



= c + a + σ

−4

Step 2) Assuming the initial data is better than square-integrable, say u

∈

d−1

× R

), we may solve the corresponding IBVP by the ‘naive’ approach.

Namely, we can extend v and u

to the whole space R

(in such a way that the

extended functions are still Lipschitz-continuous and in H

, respectively, and

that the extended operator L

is still Friedrichs symmetrizable) and we consider

the Cauchy problem

=0,u

|t=0

= u

By Theorem 2.8 we know this problem admits a (unique) solution u

∈

C ([0,T]; L

d−1

× R

)), and thanks to Friedrichs symmetrizability, by Theo-

rem 2.9), u

even belongs to C ([0,T]; H

)). This additional regularity shows

u does have a L

trace on the hyperplane {x

=0}. Therefore, by Theorem 9.18,

the homogeneous IBVP

= f, Bu

= g − Bu

|t=0

=0,

admits a (unique) square-integrable solution u

, having a square-integrable trace

on {x

=0}. And of course, the sum u = u

+ u

solves the IBVP

u = f, Bu

= g, u

|t=0

= u

But obviously this construction of a solution to the general IBVP does not show

its uniqueness. Uniqueness will follow from the energy estimate, which requires

a little more work.

We ﬁrst observe the solution obtained here above is a ‘strong’ one (as the

sum of the smooth u

and the ‘strong’ solution u

of a homogeneous IBVP).

Therefore (thanks to Lemma 9.5 about commutators), it is suﬃcient to prove

the energy estimate

u

|t=T



u

+ u



≤ c



u

|t=0



+ T L

u

+B





(where these L

norms are, from left to right, on R

d−1

× R

, R

d−1

× R

× [0,T],

or R

d−1

× [0,T]) for u smooth enough, in H

say. This can be done by a duality

argument and step 1) applied backward to the adjoint problem

)

∗

z = f



= g



|t=T

=0, (9.2.55)

How energy estimates imply well-posedness 279

where C

is deﬁned as in the proof of Theorem 9.17. Indeed, revisiting the duality

identity (9.2.50) when t lies in the bounded interval [0,T]weget





−







u +



t=0

for all z satisfying (9.2.55) with f



∈ L

d−1

× R

× [0,T]) and g



∈ L

d−1

[0,T]) and all u ∈ H

d−1

× R

× [0,T]). Denoting for convenience (similarly

as in the proof of Theorem 9.17)

z

γ,T

:= e

γt

z

d−1

×R

×[0,T ])

|z|

γ,T

:= e

γt



d−1

×[0,T ])

we have by the Cauchy–Schwarz inequality,







−







≤z

γ,T

L

u

−γ,T

+ M



∞

|z|

γ,T

−γ,T

+ z

|t=0



u

|t=0



On the other hand, the counterpart of step 1) for the backward problem in

(9.2.55) implies that

z

|t=0



+ γ z

γ,T

+ |z|

γ,T



f





γ,T

+ |g



γ,T

hence for γ ≥ 1,

max(z

|t=0



, z

γ,T

, |z|

γ,T

) 

f





γ,T

1/2



γ,T

Together with the previous inequality, this yields, for γ ≥ 1,







−











f





γ,T

1/2



γ,T





L

u

−γ,T

+ |B

−γ,T

+u

|t=0





Since this holds true for any f



∈ e

−γt

d−1

× R

× [0,T]) and g



∈

d−1

× [0,T]), this implies

u

−γ,T





L

u

−γ,T

+ |B

−γ,T

+ u

|t=0





and

−γ,T



1/2



L

u

−γ,T

+ |B

−γ,T

+ u

|t=0





280 Variable-coeﬃcients initial boundary value problems

and a fortiori

γ u

−γ,T



L

u

−γ,T

+ |B

−γ,T

+ u

|t=0



−γ,T



L

u

−γ,T

+ |B

−γ,T

+ u

|t=0



for γ ≥ 1. Now, recall that by construction of N, at each point w ∈ W

, the square

matrix whose rows are those of N (w)andB(w) is non-singular. Therefore, there

exists a constant β>0 (depending only on v



∞

for v

with values in

) such that

|u|

−γ,T

≤β ( |N

−γ,T

+ |B

−γ,T

) 

L

u

−γ,T

+ |B

−γ,T

+u

|t=0



because of the bound for |N

−γ,T

. Combining this with the bound for u

−γ,T

here above and going back to more explicit notations, we thus get

γ e

−γt

u

d−1

×R

×[0,T ])

+ e

−γt



d−1

×[0,T ])

≤ c



e

−γt

u

d−1

×R

×[0,T ])

+ e

−γt

u

d−1

×[0,T ])

+ u

|t=0





(9.2.56)

for all γ ≥ γ

, with γ

and c depending only on v

1,∞

. Finally, revisiting the

computation of Step 1) using the Friedrichs symmetrizer we easily ﬁnd that for

u smooth enough and γ large enough,

−2γt

u(t)

d−1

×R

)

 u(0)

d−1

×R

)

L

u

d−1

×R

×[0,t])

+ γ u

d−1

×R

×[0,t])

+ u



d−1

×[0,t])

Here above the last two terms are controlled by the previous estimate (9.2.56).

Therefore, the timewise estimate is a consequence of (9.2.56). Adding them

together and taking γ proportional to 1/T we eventually obtain (9.2.53).

Step 3): For square-integrable initial data, the existence of a solution to the

IBVP (9.2.52) readily follows from the density of H

in L

. Indeed, take a

sequence u

going to u

in L

d−1

× R

) and consider u

∈ C ([0,T]; L

d−1

))), the solution given by step 2) of

= f, Bu

= g, u

|t=0

= u

The energy estimate (9.2.53) shows that

u

(t) − u



(t)

d−1

×R

)

≤ c u

− u





d−1

×R

)

Hence (by the Cauchy criterion once more) u

is convergent in

C ([0,T]; L

d−1

× R

)), and the limit is a solution of (9.2.52), which

How energy estimates imply well-posedness 281

satisﬁes the energy estimate (9.2.53) (by passing to the limit in (9.2.53) applied

to u

). The uniqueness of this solution is a straightforward consequence of the

uniqueness part in Theorem 9.18. 

9.2.5 Coeﬃcients of limited regularity

This section aims at clarifying the intuitive idea that the more regular the

coeﬃcients and data, the more regular the solution. We begin with the Boundary

Value Problem, by completing Theorem 9.17.

Theorem 9.20 In the framework of Theorem 9.17, assume, moreover, that v

belongs to H

d−1

× R

× R) for some integer m>(d +1)/2+1, and that

belongs to H

d−1

× R). We also make the technical assumption that

v can be approached in H

d−1

× R

× R) by elements of D(R

d−1

× R

× R)

that take values in W and have traces at x

=0taking values in W

If the forcing term f belongs to to H

d−1

× R

× R) and the initial data

g belongs to H

d−1

× R) for all γ ≥ γ

, then the solution u of the BVP

in (9.2.48) is also in H

, as well as its trace on the boundary {x

=0},and

satisﬁes the energy estimate

γ u

d−1

×R

×R)

+ u



)

≤ C



f

d−1

×R

×R)

+ g

)



for all γ ≥ γ

, where γ

≥ γ

and C

> 0 depend continuously on the H

norms of v and v

(We recall that the norm u

is equivalent, independently of γ,toe

−γt

u

;

see Remark 9.9 for more details.)

Proof We begin with the special case when v is itself C

∞

: the coeﬃcients of

and B

being inﬁnitely smooth, the regularity of u and of its trace at x

thus follows from Theorem 9.9, while the energy estimate follows from Theorem

9.7 and Remark 9.11. In other words, the present theorem with the reinforced

assumption that v is C

∞

is an easy consequence of previous results.

For more general v, a natural idea is to regularize v and pass (once more) to

the limit. Assume v

∈ D (R

d−1

× R

× R) is going to v in H

d−1

× R

× R)

as ε goes to zero. Our technical assumption ensures that we can choose v

staying

in the set of validity of the main assumptions (CH, NC, N, UKL). So we can

apply the ﬁrst step to the smooth v

for all ε>0. Therefore, there exists a unique

solution u

to the BVP

= f on {x

> 0},B

= g on {x

=0}, (9.2.57)

282 Variable-coeﬃcients initial boundary value problems

and u

belongs to H

d−1

× R

× R) while (u

)

belongs to H

d−1

R) for all γ greater than a number γ

independent of ε, with the inequality

γ u



d−1

×R

×R)

+ (u

)



)

≤ C



f

d−1

×R

×R)

+ g

)



for C

independent of ε. Consequently, there exist u ∈ H

d−1

× R

× R)

and u

∈ H

) such that u

converges weakly in H

d−1

× R

× R)and

strongly (up to extracting a subsequence) in e

γt

d−1

× R

× R)tou, while

)

converges weakly in H

) and strongly in e

γt

)tou

To conclude we now need to prove that u = u

and u

= u

.Since

− u) converges weakly to zero in H

m−1

d−1

× R

× R) and thus

strongly (up to extracting a subsequence) in e

γt

d−1

× R

× R), Theorem

9.8 implies that (u

)

converges strongly to u

in e

γt

d−1

× R

× R),

hence u

= u

. By passing to the limit in the approximate BVP (9.2.57) we

thus see that u

solves the same problem as u: by uniqueness this proves that

u = u

and u

= u

Finally, we obtain the H

estimate for u by taking the lim inf of the estimate

we have for u

. 

We have a similar result for the Initial Boundary Value Problem with zero

initial data.

Theorem 9.21 In the framework of Theorem 9.18, we assume, moreover, that

v belongs to H

d−1

× R

× [0,T]) for some integer m>(d +1)/2+1,and

more precisely that

v =˘v

|t∈[0,T ]

with ˘v ∈ H

d−1

× R

× R) , ˘v

|t<τ

≡ 0

for some τ<T, and additionally we suppose ˘v is the limit in H

of ˘v

∈

D(R

d−1

× R

× R) taking values in W, such that (˘v

)

takes values in W

If f ∈ H

d−1

× R

× [0,T]) and g ∈ H

d−1

× [0,T]) are such that

∂

f =0, ∂

g =0at t =0 for all j ∈{0,...,m− 1}, then the solution u of

the IBVP

Lu = f for x

> 0 ,t∈ (0,T) and Bu

= g, u

|t=0

=0,

is also in H

, as well as its trace on the boundary {x

=0}, and satisﬁes ∂

u =

0 at t =0for all j ∈{0,...,m− 1}, together with the energy estimate

u

d−1

×R

×[0,T ])

+ u



d−1

×[0,T ])

≤ C



T f

d−1

×R

×[0,T ])

+ g

d−1

×[0,T ])



where C

> 0 depends only on the H

norm of ˘v.

How energy estimates imply well-posedness 283

Proof The assumptions on f and g allow us to extend them, respectively, into

f ∈ H

d−1

× R

× R)and˘g ∈ H

d−1

× R), both vanishing for t<0. By

Theorem 9.20 (applied to the extended operator L

˘v

), the corresponding BVP

admits a unique solution ˘u ∈ H

d−1

× R

× R), whose trace at x

= 0 is in

d−1

× R). Furthermore, by Theorem 9.13 ˘u vanishes for t<0, (and ˘u

|t≤T

does not depend on

|t>T

and ˘g

|t>T

). Therefore, u := ˘u

|t∈[0,T ]

is a solution of the

homogeneous IVBP with forcing term f and boundary data g, and it belongs to

d−1

× R

× [0,T]), while its trace at x

= 0 belongs to H

d−1

× [0,T]).

The uniqueness of this solution follows from the uniqueness part in Theorem 9.18

(on the L

well-posedness). It remains to show the ‘localized’ H

estimate. In

fact, it will be a straightforward consequence of the estimate

γ ˘u

d−1

×R

×(−∞,T ])

+ ˘u



d−1

×(−∞,T ])



L

˘v

˘u

d−1

×R

×(−∞,T ])

+ B

˘v

˘u

d−1

×(−∞,T ])

(9.2.58)

and the fact that ˘u vanishes for t<0 and coincides with u on [0,T]. As regards

the proof of the estimate (9.2.58), it can be deduced from the localized L

estimate (9.2.51) by the same method as in the proof of Theorem 9.7, with

(−∞,T] as the time interval instead of R. Indeed, the problem noted in Remark

9.10 about the bounded interval [0,T] does not arise for the half-line (−∞,T].

In other words, we do have the inequality analogous to (9.1.32), namely

e

−γt

w

d−1

×(−∞,T ])

≤

e

−γt

∂

w

d−1

×(−∞,T ])

(9.2.59)

for all w ∈ H

d−1

× (−∞,T]). Indeed, the inequality (9.2.59) can be viewed

as a L

–L

convolution estimate, since we have

−γt

w(t) H(T −t)=



+∞

−∞

H(t − s)e

−γ(t−s)

−γs

∂

w(s) H(T −s)ds,

where H denotes the Heaviside function, and the L

norm of t → H(t)e

−γt

precisely 1/γ.

Note: The multiplicative constant hidden in the sign  in (9.2.58) depends a

priori on ˘v

. 

Finally, there is also a result for the general IBVP with coeﬃcients of limited

regularity, provided the operator is Friedrichs symmetrizable. The compatibility

conditions needed are still (CC

) (Section 9.2.3), with A

= A

and B = B

We say initial data u

, boundary data g and forcing term f are compatible up

to order k if (CC

) holds true for all p ∈{0,...,k}.

284 Variable-coeﬃcients initial boundary value problems

Theorem 9.22 In the framework of Theorem 9.19, assume, moreover, that v

belongs to

:= {v ∈ D



d−1

× R

× [0,T]);

∂

v ∈ C ([0,T]; H

m−p

d−1

× R

))p ≤ m }

for some integer m>(d +1)/2+1.

If f ∈ H

d−1

× R

× [0,T]), g ∈ H

d−1

× [0,T]),andu

∈

d−1

× R

) are compatible up to order m − 1, then the solution of

the IBVP (9.2.52) belongs to CH

This result is implicitly contained in the lecture notes of M´etivier [136]. The

conclusion is as in Rauch and Massey’s theorem, with slightly less regular data

instead of H

m+1/2

), and with much less regular coeﬃcients. The drawback

is the additional, Friedrichs-symmetrizability assumption (which is satisﬁed by

‘most’ physical systems anyway).

Proof The sketch of proof is roughly the same as for Theorem 9.19 (which gives

solutions in C ([0,T]; L

d−1

× R

)) for L

data): 1) revisit the IBVP with zero

initial data (Theorem 9.21) and show additional time-regularity of solutions; 2)

consider initial data (smoother than in the statement) u

∈ H

m+1/2

d−1

× R

and reduce the problem to zero initial data by substracting an ‘approximate

solution’ (whose counterpart in the proof of Theorem 9.19 is just the solution

of a Cauchy problem in the whole space); 3) conclude by a density argu-

ment for initial data u

∈ H

d−1

× R

), thanks to appropriate (new) energy

estimates.

Step 1) The key ingredient will be the reﬁned estimate



|α|≤m

2(m−|α|)

−2γτ

∂

w(τ)

d−1

×R

)

+ γ w

d−1

×R

×(−∞,τ])

+ w



d−1

×(−∞,τ])



L

w

d−1

×R

×(−∞,τ])

+ B

w

d−1

×(−∞,τ])

(9.2.60)

to be shown to be uniform for all γ large enough, for all τ>0 and all w ∈

D(R

d−1

× R

× R) vanishing for t<0. In the inequality (9.2.60) here above, the

simplifying notation  means less than or equal to a constant depending only on

the W

1,∞

norm and the H

norm of ˘v, an extension of v as in Theorem 9.21.

For simplicity we shall omit the ˘ sign. In addition, we introduce shortcuts for

the sets of integration

Ω × (−∞,τ]=R

d−1

× R

× (−∞,τ]

:= ∂Ω × (−∞,τ]=R

d−1

× (−∞,τ] .