In Section 10, we need to estimate the decay of the semigroup projected into the eigenspaces corresponding to the high frequencies of the Laplacian operator.
This estimation is not direct in the cases where the projectors on high-frequencies do not commute with A. The purpose of this Section is to prove results yielding quickly to estimates of the decay of the high-frequencies by using the above results Theorem A.3 and A.4. In particular, we generalized some results of Appendix B by showing that they hold uniformly with respect to cut-off frequency of the projection.
Notice that this type of decay estimates for the high-frequency part of the solutions of the damped wave equation is related to [BL13, Theorem 10], which shows that the eigenspaces corresponding to the high frequencies of the Laplacian operator are mainly preserved by the flow of the damped wave equation.
We use the notations of Appendix B. Since L is self-adjoint, positive and with compact resolvent, there exists an orthonormal basis (φk)k>0 of eigenfunctions of L.
We introduce the high-frequencies truncations Qn, that are the projectors on the space Span{φk, k>n}
Qnu= X
k>n
hu|φkiφk.
We also introduce the sequence of high-frequencies projections Qn = (Qn, Qn) onX.
We consider in QnH the operators
PQnBQn(µ) =QnPB(µ)Qn =−L−iµQnBQn+µ2Id .
and the projection of A on the high frequencies: that is, for any V ∈ QnX, QnAQn=
0 Id
−L −QnBQn .
We prove a generalization of the classical implication of Proposition B.2, which is uniform with respect to the high-frequencies projections.
Proposition C.1. — Assume that there exist a function M(µ), uniformly pos-itive, and µ0 > 0 such that, for any µ with |µ| > µ0, n ∈ N and u ∈ QnD(L), we have
(C.1) kukH 6 M(|µ|)
|µ| kPQnBQn(µ)ukH .
Then, there exists K >0such that, for all n ∈N and all |µ|>µ0,
(QnAQn−iµ)−1L(Q
nX) 6KM(|µ|).
Proof. — LetU = (u1, u2) andV = (v1, v2) be vectors ofQnX =Qn(D(L1/2)×H) such that (QnAQn−iµ)U =V. We have
(C.2)
u2−iµu1 =v1 in DL1/2 (−L+µ2Id)u1−iµQnBQnu1 =v2+QnBv1+iµv1 in H
We setw= (PQnBQn(µ))−1(v1). SincePQnBQn(µ) +L+iµQnBQn =µ2Id, we have w= 1
µ2 (PQnBQn(µ))−1(PQnBQn(µ)v1+Lv1+iµQnBQnv1)
= 1 µ2
v1+ (PQnBQn(µ))−1(Lv1) +iµ(PQnBQn(µ))−1(QnBQnv1) and so
(C.3) kwkH 6 1
µ2kv1kH + 1 µ2
(PQnBQn(µ))−1
L(D(L−1/2), H)kv1kD(L1/2)
+ 1 µ
(PQnBQn(µ))−1L(H)kBkL(H)kv1kH. Let us estimate k(PQnBQn(h))−1kL(H, D(L1/2)). We have
kuk2D(L1/2) =hLu|uiH
=D−PQnBQn(µ)u+µ2u−iµQnBQnu|uE
H
6O(µ2)kuk2H +kPQnBQn(µ)ukHkukH
6O(µ2)k(PQnBQn(µ))−1k2L(H,H)+k(PQnBQn(µ))−1kL(H,H)
kPQnBQn(µ)uk2H
where the above estimation are independent ofn. The estimate|||(PQnBQn(µ))−1|||L(H,H)
is given by Hypothesis (C.1), yielding
kuk2D(L1/2) 6 M(|µ|)2+M(|µ|)
|µ|
!
kPQnBQn(µ)uk2H. Using thatM is uniformly positive,M(µ)/µ =o(M(µ)2) and so
(PQnBQn(µ))−1
L(H, D(L1/2)) =O(M(|µ|)).
SincePQnBQn(µ) defined from D(L−1/2) in H is the adjoint ofPQnBQn(−µ) defined fromH inD(L1/2), we also have
(PQnBQn(µ))−1
L(D(L−1/2), H)=O(M(|µ|)). Coming back to (C.3), we obtain that
kwkH 6KM(|µ|)
µ2 kv1kD(L1/2). Considering (C.2), we have that
PQnBQn(µ)(u1−iµw) = v2+QnBv1
and thus, due to (C.1), that ku1−iµwkH 6O(M(|µ|)/µ)kVkX. Together with the above estimate for w, we obtain that ku1kH 6 O(M(|µ|)/µ)kVkX and, using the first equation of (C.2), thatku2kL2 6O(M(|µ|))kVkX.
It remains to estimate ku1kD(L1/2). To this end, we take the scalar product of second line of (C.2) with u1 and consider the real part to obtain
ku1k2D(L1/2)−µ2ku1k2H 6O(µ)kVkXku1kH
and thus, due to the above estimates,ku1k2D(L1/2)6O(M(|µ|)2)kVk2X. To obtain estimates as (C.1), it is convenient to generalize Proposition B.3 to the case where high-frequencies projections appear. In this way, we can use the classical observability estimate without projections to study the decay of the high-frequencies semigroup.
Proposition C.2. — We set
P(µ) =−L+µ2Id:=P0(µ).
Assume that there exist two positive functions f and g and µ0 > 0 such that, for any µ with |µ|>µ0 and any u∈D(L),
(C.4) kukH 6 f(µ)
µ kP(µ)ukH +g(µ)√ Bu
H . Then, for anyµ with |µ|>µ0, any n∈N and any u∈QnD(L),
(C.5) kukH 6 M(µ)
|µ| kPQnBQn(µ)ukH, where
(C.6) M(µ) = 3 maxf(µ), f(µ)2√ B2
L(H), g(µ)2 . Proof. — Let u∈QnD(L). We have
hPQnBQn(µ)u|ui=−hLu|ui+µ2hu|ui −iµhBQnu|Qnui. In particular, the imaginary part ofhPQnBQn(µ)u|ui isµhBQnu|Qnui and (C.7) ∀u∈QnH , hBQnu|Qnui=√
Bu2
H 6 1
|µ|kPQnBQn(µ)ukHkukH. We write
(C.8) kP(µ)ukH =
kPQnBQn(µ)u+iµQnBQnukH 6kPQnBQn(µ)ukH +|µ| kQnBQnukH . Then, we compute, for any u∈QnH
kQnBQnuk2H 6kBQnuk2H 6√ B2
L(H)
√
BQnu2
H
6√ B2
L(H)hBQnu|Qnui
and using (C.7), we obtain kQnBQnuk2H 6
√
B2
L(H)
|µ| kPQnBQn(µ)ukHkukH. Combining this result with (C.4), (C.7) and (C.8), we get
kukH 6 f(µ)
|µ| kPQnBQn(µ)ukH + f(µ)
q|µ|
√
B
L(H)kPQnBQn(µ)uk1/2H kuk1/2H
+ g(µ)
q|µ|kPQnBQn(µ)uk1/2H kuk1/2H .
We bound the sum on the right by three times the largest of the three terms.
Depending on which one is the largest one, we get three different bounds, which can
be gathered in (C.5) and (C.6).
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Manuscript received on 18th January 2019, revised on 7th November 2019,
accepted on 1st February 2020.
Recommended by Editor N. Anantharaman.
Published under license CC BY 4.0.
This journal is a member of Centre Mersenne.
Romain JOLY
Université Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, (France) romain.joly@univ-grenoble-alpes.fr Camille LAURENT
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France laurent@ann.jussieu.fr