ON THE GROWTH OF SOBOLEV NORMS FOR THE
CUBIC SZEGŐ EQUATION
PATRICK GÉRARD AND SANDRINE GRELLIER
1. Introduction
The large time behavior of solutions to Hamiltonian partial differential equations is an important problem in mathematical physics. In
the case of finite dimensional Hamiltonian systems, many features of
the large time behavior of trajectories are described using the topology
of the phase space. For a given infinite dimensional systems, several
natural phase spaces, with different topologies, can be chosen, and the
large time properties may strongly depend on the choice of such topologies. For instance, it is known that the cubic defocusing Schrödinger
equation
i∂t u + ∆u = |u|2 u
posed on a Riemannian manifold M of dimension d = 1, 2, 3 with sufficiently uniform properties at infinity, defines a global flow on the
Sobolev spaces H s (M ) for every s ≥ 1 (see e.g. [?]). In this case, a
typical large time behavior of interest is the boundedness of trajectories. On the energy space H 1 (M ), the conservation of energy trivially
implies that all the trajectories are bounded. On the other hand, the
existence of unbounded trajectories in H s (M ) for s > 1 is a long standing question [?], naturally connected to weak turbulence. In [?], [?],
it was proved that dispersion properties imply a polynomial bound in
time on H s norms as time goes to infinity, but so far the optimality of this kind of bound has not been established. Let us mention
however that, for some special solutions of cubic NLS on R × T2 , Hani–
Pausader–Tzvetkov–Visciglia [?] succeeded in proving that H s norms
may not be bounded for s big enough, with a minoration of the type
1/2
ec(log log t)
,
on a sequence of times going to infinity. Natural model problems for
studying the unboundedness of Sobolev norms seem to be those for
which the calculation of solutions is the most explicit, namely integrable
Date: December 12, 2014.
1991 Mathematics Subject Classification. 35B15, 47B35, 37K15.
1
2
PATRICK GÉRARD AND S. GRELLIER
systems. Typical examples are the Korteweg de Vries equation [?],
[?] and the one dimensional cubic nonlinear Schrödinger defocusing
equation [?], [?]. However, in these cases, the set of conservation laws
is known to control the whole regularity of the solution, so that all the
trajectories of H s (M ) are bounded in H s (M ) for every nonnegative
integer s [?].
In order to investigate direct and inverse cascades between spacial
scales in one space dimension, Majda, Mc Laughlin, Tabak and their
collaborators introduced, in a series of papers starting with [?], — see
also Zakharov, Guyenne, Pushkarev, Dias [?] — a class of Hamiltonian
equations on T including
1
i∂t u − |D|α u = |u|2 u , D := ∂x , α > 0 ,
i
making numerical simulations suggesting weak turbulence effects for
some values of α. For α > 1, it is easy to prove that this equation
is globally well–posed on H s (T) for s ≥ α2 , and that the H α/2 norm
is uniformly bounded along the trajectories. Moreover, polynomial
bounds on the H s norms for big s can also be obtained in that case [?].
In the limit case α = 1, Equation (??) is a nonlinear (half–) wave
equation in one space dimension, which can be proved to be globally
well-posed on H s for s ≥ 21 (see [?]). However, compared to the case
α 6= 1, this equation is no more dispersive, which suggests that large
time transition to high frequencies may be facilitated. In [?] — see also
Pocovnicu [?], we proved that a Birkhoff normal form of this equation
near the origin is given at first order by the following cubic Szegő
equation,
(1)
(2)
i∂t u = Π(|u|2 u) ,
where Π denotes the Szegő projector on L2 (T),
c
Πu(k)
= 1k≥0 û(k) .
Notice that the range of Π is the closed subspace L2+ (T) consisting of
L2 functions on the unit circle which admit a holomorphic extension to
the unit disc. Equation (??) has been introduced in [?], where global
wellposedness was established on H+s (T) := H s (T) ∩ L2+ (T) for every
s ≥ 21 , as well as boundedness of the H 1/2 norm along every trajectory.
Moreover, we proved that this equation enjoys a Lax pair structure,
giving rise to integrability properties studied in [?] and [?], as well as
the following a priori estimate on H s norms of solutions,
(3)
ku(t)kH s ≤ Cs eCs t , s > 1 ,
ON THE GROWTH OF SOBOLEV NORMS
3
where Cs is a uniform constant depending only on a bound of the H s
norm of the initial data. It turns out that this estimate is almost
optimal, as shown by the following result.
Theorem 1.
(1) For every s > 21 , the set of initial data in H+s (T) leading to an
unbounded trajectory in H+s (T) is a dense Gδ –subset of H+s (T).
(2) For every positive integer M , for every s > 21 , there exists a
bounded subset B of C+∞ (T) := ∩s H+s (T) such that
ku(t)kH s
→ ∞ as |t| → ∞,
M
u0 ∈B (1 + |t|)
sup
where u denotes the solution of (??) with u(0) = u0 .
The above theorem calls for several comments.
• It is in fact possible to find the same dense Gδ –subset of initial
data leading to unbounded trajectories in H s for every s >
1
. However, we cannot display an explicit example of such
2
an initial datum. For example, we know that every rational
function with no pole in the closed unit disc gives rise to a
bounded trajectory in every H s (see [?]). This fact is in contrast
with the case of the cubic Szegő equation on the line, where
special examples of unbounded trajectories in H s , for every s >
1
, are displayed in [?].
2
• In the above result, the superpolynomial growth of Sobolev
norms is stated for families of solutions. In fact, we expect that
this superpolynomial growth holds for generic initial data, and
that this growth is indeed exponential, making estimate (??)
optimal. An explicit example of such a behaviour was recently
displayed by H. Xu in [?], for the special perturbation
Z 2
i∂t u = Π(|u| u) + α
u , α>0.
T
• A very natural question is of course whether Theorem ?? holds
for the half-wave equation (??) for α = 1.
The proof of Theorem ?? relies on explicit formulae for solutions of
Equation (??), which are non trivial consequences of the Lax pair structure and will be presented without proof in section ??. Then we will
sketch the proof of part 1) of the theorem, and give a complete proof
of part 2) in section ??.
4
PATRICK GÉRARD AND S. GRELLIER
2. Explicit formulae
In this section, we list the main formulae solving the Cauchy problem
for the cubic Szegő equation (??). The detailed proofs will appear in
a forthcoming paper.
First we introduce some additional notation. Given a positive integer
n, we set
Ωn := {s1 > s2 > · · · > sn > 0} ⊂ Rn .
Given a nonnegative integer d ≥ 0, we recall that a Blaschke product
of degree d is a rational function on C of the form
−iψ
Ψ(z) = e
d
Y
z − pj
, ψ ∈ T , pj ∈ D .
1 − pj z
j=1
Alternatively, Ψ can be written as
Ψ(z) = e−iψ
P (z)
,
z d P z1
where ψ ∈ T is called the angle of Ψ and P is a monic polynomial
of degree d with all its roots in D. Such polynomials are called Schur
polynomials. We denote by Bd the set of Blaschke products of degree
d. It is a classical result — see e.g. [?] — that Bd is diffeomorphic to
T × R2d .
Given a n-tuple (d1 , . . . , dn ) of nonnegative integers, we set
Sd1 ,...,dn := Ωn ×
n
Y
Bdr ,
r=1
endowed with the natural product topology. Given a sequence (dr )r≥1
(2)
of nonnegative integers, we denote by S(dr ) the set of pairs
∞
((sr )r≥1 , (Ψr )r≥1 ) ∈ R ×
∞
Y
Bdr
r=1
such that
∞
X
s1 > · · · > sn > · · · > 0 ,
(dr + 1)s2r < ∞ .
r=1
(2)
We also endow S(dr ) with the natural topology.
Finally, we denote by Sn the union of all Sd1 ,...,dn over all the n-tuples
(2)
(2)
(d1 , . . . , dn ), and by S∞ the union of all S(dr ) over all the sequences
(dr )r≥1 . Given (s, Ψ) ∈ Sn and z ∈ C, we define the matrix C(z) :=
ON THE GROWTH OF SOBOLEV NORMS
5
C(s, Ψ)(z) as follows. If n = 2q, the coefficients of C(s, Ψ)(z) are given
by
(4)
cjk (z) :=
s2j−1 − s2k zΨ2k (z)Ψ2j−1 (z)
, j, k = 1, . . . , q .
s22j−1 − s22k
If n = 2q − 1, we use the same formula as above, with s2q = 0.
Theorem 2. For every n ≥ 1, for every (s, Ψ) ∈ Sn , for every z ∈ D,
the matrix C(s, Ψ)(z) is invertible. We set
(5)
u(s, Ψ)(z) = hC(z)−1 (Ψ2j−1 (z))1≤j≤q , 1i,
where



1 := 


1
.
.
.
1


q
X

 , and hX, Y i :=
Xk Yk .


k=1
(2)
For every (s, Ψ) ∈ S∞ , the sequence (uq )q≥1 with
uq := u((s1 , . . . , s2q ), (Ψ1 , . . . , Ψ2q )),
1/2
is strongly convergent in H+ (S1 ). We denote its limit by u(s, Ψ).
The mapping
(s, Ψ) ∈
∞
[
1/2
(2)
Sn ∪ S∞
7−→ u(s, Ψ) ∈ H+ \ {0}
n=1
(2)
is bijective. Furthermore, its restriction to every S(d1 ,...,dn ) and to S(dr )
is a homeomorphism onto its range.
Finally, the solution at time t of equation (??) with initial data u0 =
u(s, Ψ) is u(s, Ψ(t)), where
r s2 t
r
Ψr (t) = ei(−1)
Ψr .
Let us briefly explain how the above nonlinear Fourier transform is
1/2
related to spectral analysis. If u ∈ H+ (S1 ), recall (see [?], [?]) that
the Hankel operator of symbol u is the operator Hu : L2+ (S1 ) → L2+ (S1 )
defined by
Hu (h) = Π(uh) .
It can be shown that Hu2 is a positive selfadjoint trace class operator.
If S is the shift operator defined by
Sh(z) = zh(z) ,
6
PATRICK GÉRARD AND S. GRELLIER
Hu satisfies
S ∗ Hu = Hu S = HS ∗ u .
We denote by Ku this new Hankel operator. Let us say that a positive real number s is a singular value associated to u if s2 is an
eigenvalue of Hu2 or Ku2 . The main point in Theorem ?? is that the
list s1 > · · · > sr > . . . is the list of singular values associated to
u = u(s, Ψ), and that the corresponding list Ψ1 , . . . , Ψr , . . . describes
the action of Hu and of Ku on the eigenspaces of Hu2 , Ku2 respectively, making more precise a theorem of Adamyan-Arov-Krein about
the structure of Schmidt pairs of Hankel operators [?]. In fact, denoting by uj the orthogonal projection of u onto ker(Hu2 − s22j−1 I) and by
u0k the orthogonal projection of u onto ker(Ku2 − s22k I), one can show
that uj 6= 0, u0k 6= 0, and
dim ker(Hu2 − s22j−1 I) = degΨ2j−1 + 1, Ψ2j−1 (z)Hu (uj )(z) = s2j−1 uj (z),
dim ker(Ku2 − s22k I) = degΨ2k + 1, Ku (u0k )(z) = s2k Ψ2k (z)u0k (z) .
Remark 1. These formulae have an interesting consequence in the
special case where the Blaschke products Ψr are all equal to 1. Indeed,
from formula (??), the Fourier coefficients of u are real, so that Hu and
Ku act as self adjoint operators on the real subspace of L2+ corresponding to real Fourier coefficients. Furthermore, from the above formulae,
these operators are positive.
As a consequence of Theorem ??, we get inverse spectral theorems
on Hankel operators, which generalize to singular values with arbitrary
multiplicity the ones we had proved in [?] and [?] for simple singular
values. Finally, the last assertion of Theorem ?? is a consequence of
the following Lax pair identities,
dHu
i
= [Bu , Hu ] , Bu (h) := Hu2 h − iΠ(|u|2 h) ,
dt
2
dKu
i
= [Cu , Ku ] , Cu (h) := Ku2 h − iΠ(|u|2 h) .
dt
2
Using Theorem ??, the proof of part 1) of Theorem ?? heavily relies on
a new phenomenon, which is the loss of continuity of the map (s, Ψ) 7→
u(s, Ψ) as three consecutive sr ’s are collapsing. More precisely, an
inspection of formula (??) yields the following elementary lemma.
Lemma 1. Given (s1 , . . . , sn , δ) ∈ Ωn+1 , (Ψ1 , . . . , Ψn ) ∈ Bd1 ×· · ·×Bdn ,
and ε > 0 small enough, the family
uε := u((s1 , . . . , sn , δ + ε, δ, δ − ε), (Ψ1 , . . . , Ψn , e−iϕ+ , e−iθ , e−iϕ− ))
ON THE GROWTH OF SOBOLEV NORMS
7
converge as ε tends to 0, in every H s space, to
u = u((s1 , . . . , sn , δ), (Ψ1 , . . . , Ψn , Ψ))
for some Ψ ∈ B1 , except if ϕ+ = ϕ− mod 2π. If ϕ+ = ϕ− mod 2π,
then uε is unbounded in H s for every s > 21 .
Using this lemma, we get the following long time instability result.
Proposition 1. For every u0 ∈ C+∞ (T), there exists a sequence (un0 )
converging to u0 in C+∞ (T) and a sequence (tn ) of times such that the
solution un of (??) with un (0) = un0 satisfies
∀s >
1
, kun (tn )kH s −→ ∞ .
n→∞
2
The first assertion of Theorem ?? follows from Proposition ?? and
a Baire category argument. This argument comes back to Hani [?],
who used it for finding solutions with unbounded Sobolev norms for
the totally resonant form of cubic NLS on T2 , as investigated by [?].
Notice that the previously quoted result in [?] relies on Hani’s theorem
in [?].
3. Superpolynomial growth of the Sobolev dynamics
In this section, we prove the super polynomial growth of the Sobolev
norms for solutions of the Szegő cubic equation. We recall the statement given in the introduction.
Theorem 3. For every positive integer M , every s > 21 , there exists a
bounded subset B of C+∞ (S1 ) such that
ku(t)kH s
→ ∞ as |t| → ∞
M
u0 ∈B (1 + |t|)
sup
where u(t) is the solution of (??) with u(0) = u0 .
Proof. This result is a consequence of the following proposition.
Proposition 2. For every N ≥ 2, every ε > 0, every ξ := (ξj )1≤j≤N ∈
RN , η := (ηk )1≤k≤N −1 ∈ RN −1 , such that
ξ1 > η1 > ξ2 > η2 > . . . ηN −1 > ξN ,
we consider
uε (z) = hCε−1 (z)(1), 1i
8
PATRICK GÉRARD AND S. GRELLIER
where Cε (z) = (cε,jk (z))1≤j,k≤N , with
1 + εξj − z(1 + εηk )
, 1≤k ≤N −1 ;
(1 + εξj )2 − (1 + εηk )2
1
cε,jN (z) :=
, 1≤j≤N .
1 + εξj
cε,jk (z) :=
Then, there exists (ξ, η) such that
C
(1) kuε kH s ≥
;
ε(N −1)(2s−1)
1
, · is bounded in C+∞ (S1 ) as ε → 0. Here
(2) The family uε 2ε
(uε (t), ·) denotes the solution of (??) with uε (0) = uε .
Theorem ?? is a direct consequence of this proposition by considering
M = [(N − 1)(2s − 1)] and the family of initial data
1
B := uε
,· , ε > 0 .
2ε
The proof of proposition ?? requires several step. The first step
C
consists in establishing that kuε kH s ≥ (N −1)(2s−1) for a dense open set
ε
of choices of (ξ, η). This is based on the following lemma.
C
Lemma 2. For a dense open set of (ξ, η), we have (uε )0 (1) ≥ 2(N −1) .
ε
Moreover, the poles of uε are simple,
(6)
uε (z) =
α0ε
+
N
−1
X
j=1
αjε
1 − pεj z
with α0ε , αjε > 0, 0 < pεj < 1, and pεj → 1 as ε → 0. Furthermore,
min1≤j≤N −1 (1 − pεj ) is equivalent to ε2(N −1) .
Let us first prove that the lemma implies that kuε kH s ≥
0
First, from (??) and the estimate on (uε ) (1),
0
(uε ) (1) =
N
−1
X
j=1
αjε pεj
C
≥ 2(N −1) .
ε 2
(1 − pj )
ε
C
ε(N −1)(2s−1)
.
ON THE GROWTH OF SOBOLEV NORMS
9
Hence, applying Cauchy-Schwarz, one gets
C
ε2(N −1)
≤
N
−1
X
j=1
αjε pεj
(1 − pεj )2
N
−1
X
≤
j=1
N
−1
X
≤
j=1
(αjε )2
(1 − pεj )2s+1
!1/2
(αjε )2
(1 − pεj )2s+1
!1/2
×
N
−1
X
j=1
×
(pεj )2
(1 − pεj )3−2s
C
ε(3−2s)(N −1)
!1/2
.
Eventually,
N
−1
X
j=1
(αjε )2
(1 − pεj )2s+1
!1/2
≥
C
ε(2s−1)(N −1)
.
It remains to observe that
kuε k2H s
≥C
X
2s
n
n≥0
N
−1
X
!2
αjε (pεj )n
j=1
≥C
N
−1
X
j=1
(αjε )2
.
(1 − pεj )2s+1
Proof. Let us turn to the proof of the lemma. We first estimate the
derivative of uε at z = 1. From the explicit formula of uε , we obtain
(7) (uε )0 (1) = hCε (1)−1 C˙ε Cε (1)−1 (1), 1i = hC˙ε Cε (1)−1 (1),t Cε (1)−1 (1)i
where C˙ε = (ċε,jk )1≤j,k≤N , with
ċε,jk :=
1 + εηk
, 1 ≤ k ≤ N −1 , ċε,jN := 0 , 1 ≤ j ≤ N .
(1 + εξj )2 − (1 + εηk )2
From the formula giving Cε , we have
!
1
1
Cε (1) =
,
2 + ε(ξj + ηk ) 1≤j≤N ;1≤k≤N −1 1 + εξj 1≤j≤N
which is a Cauchy matrix. Let us recall that a Cauchy matrix is a
1
matrix of the form
. Its determinant is given by
aj + b k
Q
Q
1
i<j (ai − aj )
k<l (bk − bl )
Q
(8)
det
=
.
aj + b k
j,k (aj + bk )
It allows to obtain
1
aj + b k
−1 !
jk
j+k λj µk
= (−1)
aj + bk jk
10
PATRICK GÉRARD AND S. GRELLIER
with
Q
λj = Q
+ bl )
l (ajQ
− ar )
+ bk )
l (al Q
− br )
− aj )
i<j (ai
r>j (aj
and
Q
µk = Q
i<k (bi
− bk )
r>k (bk
.
In the case of the matrix Cε (1), we have
Q
2N −1 (1 + εξj ) l (1 + ε(ξj + ηl )/2)
λj = N −1
, 1≤j≤N ,
ε
ξj0
and
2N
µk = N −2
ε
Q
Q
(1
+
ε(ξ
+
η
)/2)
(1 + εξi )
l
k
l
, k ≤ N − 1, µN = Q i
0
ηk (1 + εηk )
k (1 + εηk )
where we have set
Y
Y
Y
Y
(ηi − ηk ) (ηk − ηr ).
ξj0 :=
(ξi − ξj ) (ξj − ξr ), ηk0 :=
i<j
r>j
i<k
r>k
Eventually, if k ≤ N − 1,
−1
(Cε (1) (1)
k
k
= (−1) µk
N
X
j=1
(−1)j λj
2 + ε(ξj + ηk )
N
Y
2N −2 X
ξj + ηl
j (1 + εξj )
= (−1) µk N −1
(−1)
1+ε
;
ε
ξj0
2
j=1
l6=k
k
and
N
X
(−1)j λj
(Cε (1)−1 (1) N = (−1)N µN
1 + εξj
j=1
N
2N −1 X (−1)j Y
ξj + ηl
1+ε
= (−1) µN N −1
.
ε
ξj0
2
j=1
l
N
In order to simplify these quantities, we use the following identities.
Lemma 3. For any 0 ≤ p ≤ N − 1,
N
X
ξp
0
j j
(−1) 0 =
−1
ξj
j=1
if p < N − 1
if p = N − 1.
ON THE GROWTH OF SOBOLEV NORMS
11
p
P
j ξj
Proof. We view N
(−1)
as a rational function of ξN denoted by
j=1
ξj0
Qp (ξN ). Its poles are simple and equal to ξ1 , . . . , ξN −1 . Identifying the
residue at each of these poles, we get
Res(Qp ; ξN = ξr ) = Q
(−1)N +1 ξrp
(−1)r+1 ξrp
Q
+Q
=0.
i<r (ξi − ξr )
N −1≥j>r (ξr − ξj )
i<N,i6=r (ξi − ξr )
Hence Qp (ξN ) is in fact a polynomial. If p < N − 1, it tends to 0 as ξN
tends to ∞, therefore it is identically 0. If p = N − 1, it is a constant,
equal to its limit at infinity,
Qp (ξN ) =
N −1
(−1)N ξN
= −1 .
N −1
(−1)N −1 ξN
This completes the proof.
Expanding in powers of ε the above formula giving (Cε (1)−1 (1))k ,
and using Lemma ??, we infer
(9)
Cε (1)−1 (1) k = (−1)k+1 µk , k ≤ N .
Let us compute t Cε−1 (1)(1) in a similar way. From the formula, we
have
!
N
−1
k
N
X
(−1)
µ
(−1)
µ
k
N
t
Cε (1)−1 (1) j = (−1)j λj
+
2
+
ε(ξ
+
η
)
1
+
εξ
j
k
j
k=1
=: (−1)j λj S̃j (ε).
We expand in powers of ε and use again Lemma ??, changing N into
N − 1. We get
Q
ξ +η
N −1
2N −1 X (−1)k i6=j 1 + ε i 2 k
+ (−1)N + O(ε)
S̃j (ε) = N −2
0
ε
η
(1
+
εη
)
k
k
k=1
N −1
2N −1 X
N
= (−1) + O(ε) + N −2
(−1)k
ε
k=1
η p+q
(−1)q CNp −1 kp 0 + O(εN −1 )
εN −2
2 ηk
p+q=N −2
X
= −1 + O(ε).
As a consequence, we infer
(10)
t
Cε (1)−1 (1) j = (−1)j+1 λj (1 + O(ε)) .
!
12
PATRICK GÉRARD AND S. GRELLIER
Eventually, inserting (??) and (??) into the formula (??) of (uε )0 (1), it
gives
X
X
1 + εηk
(uε )0 (1) =
(−1)j+k λj µk (1 + O(ε))
2
2
(1 + εξj ) − (1 + εηk )
1≤j≤N 1≤k≤N −1
!
X
X
22(N −1)
(−1)j+k
= 2(N −1)
+ O(ε) .
0 0
ε
(ξ
j − ηk )ξj ηk
1≤j≤N 1≤k≤N −1
Considering
(−1)j+k
(ξj − ηk )ξj0 ηk0
1≤j,k≤N −1
X
as a function of ξN , the pole ηN −1 appears only once. Hence, this
meromorphic function does not vanish on an open dense set of (ξ, η).
This proves that
C
|(uε )0 (1)| ≥ 2(N −1)
ε
for an open dense set of choices of (ξ, η).
Next we prove the second statement of Lemma ??. Thanks to remark
??, operators Huε and Kuε act as positive self adjoint operators on
the subspace of L2+ corresponding to real Fourier coefficients. As a
consequence — see e.g. [?], Prop. 2.1 —, there exists a bounded
positive measure µ on [0, 1[ such that
Z
∀k ≥ 0 , ûε (k) =
tk dµ(t) .
[0,1[
On the other hand, Huε has only finitely many positive singular values,
hence is a finite rank operator. By the identity S ∗ Huε = Huε S, S ∗ acts
on the range of Huε , therefore there exits a non trivial linear relation
between the vectors (S ∗ )p uε , p ≥ 0. Equivalently, there exists non
trivial coefficients cp such that
X
∀k ≥ 0 ,
cp û(k + p) = 0 .
p
This precisely means that there exists a non trivial polynomial P such
that
Z
∀k ≥ 0 ,
tk P (t) dµ(t) = 0 .
[0,1[
By the Weierstrass theorem, µ is therefore a positive linear combination
of Dirac measures. This implies that the poles of uε are simple and that
uε (z) =
α0ε
+
N
−1
X
j=1
αjε
1 − pεj z
ON THE GROWTH OF SOBOLEV NORMS
13
with αjε > 0, 0 < pεj < 1. In particular (uε )0 (1) > 0 so that
(uε )0 (1) ≥
C
.
ε2(N −1)
Let us show that pεj → 1 as ε → 0. We know that the denominator of
uε (z) is
N
−1
Y
det Cε (z)
=
Pε (z) :=
(1 − pεj z),
det Cε (0)
j=1
therefore it suffices to prove that Pε (z) ' (1 − z)N −1 as ε tends to 0.
From the formula of Cε (z),
(2ε)N −1 det Cε (z) → (1 − z)N −1 det
1
ξj − ηk
!
,1
1≤j≤N ;1≤k≤N −1
and
(2ε)N −1 det Cε (0) → det
1
ξj − ηk
!
,1 ,
1≤j≤N ;1≤k≤N −1
so the claim is proved.
Finally, we establish that
min (1 − pεj ) ' ε2(N −1) .
1≤j≤N −1
Let us first prove that min1≤j≤N −1 (1 − pεj ) ≤ Cε2(N −1) . From the estimate of (uε )0 (1),
C
ε2N −1
≤ (uε )0 (1) =
≤
X
1
αjε pεj
(1 − pεj )2
X
min1≤j≤N −1 (1 − pεj )
αjε pεj
(1 − pεj )
≤
uε (1)
tr(Huε ) + tr(Kuε )
≤
ε
min1≤j≤N −1 (1 − pj )
min1≤j≤N −1 (1 − pεj )
≤
2N − 1 + O(ε)
.
min1≤j≤N −1 (1 − pεj )
14
PATRICK GÉRARD AND S. GRELLIER
It gives the bound from above for min1≤j≤N −1 (1 − pεj ). For the bound
from below, we use formula (??) to obtain
Y
det Cε (1)
(1 − pεj ) =
' εN −1 det Cε (1)
det
C
(0)
ε
j
!
1
1
' εN −1 det
,
2 + ε(ξj + ηk ) 1≤j≤N ;1≤k≤N −1 1 + εξj 1≤j≤N
' εN (N −1) .
On the other hand, as
−Pε0 (1)
=
X
pεj
j
Y
(1 − pεi ) '
i6=j
(1 − pεi )
εN (N −1)
'
,
min1≤j≤N (1 − pεj )
min1≤j≤N (1 − pεj )
Q
it suffices to bound −Pε0 (1) from above. Using formula (??) again,
d
(det Cε (z))|z=1
dz
N
−1 X
N
X
(−1)k+l (1 + εηk )
−
=
k=1 l=1
≤ ε
(1 + εξj )2 − (1 + εηk )2
det
1
2 + ε(ξj + ηm )
,
j6=l,m6=k
(N −2)2 −1
Eventually,
−Pε0 (1) ' −εN −1
d
(det Cε (z))|z=1 ≤ Cε(N −1)(N −2)
dz
so that
min (1 − pεj ) ≥ Cε2(N −1) .
1≤j≤N −1
This completes the proof of part 1 of Proposition ??.
We are left with proving the second part of Proposition ??. From
1
Theorem ??, at time t = 2ε
1
, z = hC˜ε (z)−1 (Ψεodd ), 1i
uε
2ε
ε
where Ψεodd := (e−iψj (1/2ε) )1≤j≤N with ψjε (t) := (1 + εξj )2 t. Hence
Ψεodd = e−i/2ε (1 + O(ε)) e−iξj 1≤j≤N ,
1
1 + εξj
j6=l
!
ON THE GROWTH OF SOBOLEV NORMS
15
and, for 1 ≤ j ≤ N ,
1 + εξj − z(1 + εηk )e−i(ξj −ηk ) (1 + O(ε))
C˜ε (z)jk =
, 1 ≤ k ≤ N − 1,
(1 + εξj )2 − (1 + εηk )2
1
.
C˜ε (z)jN =
1 + εξj
Let P̃ε (z) =
det C˜ε (z)
.
det C˜ε (0)
From Theorem ??, P̃ε (z) has (N − 1) roots
outside the closed unit disc D. Our aim is to prove that, for a suitable
choice of (ξ, η), the distance of these roots to the boundary of D is
bounded from below. Define
P (ξ, η)(z) := lim P̃ε (z) .
ε→0
Observe that
P (ξ, η)(z) = c(ξ, η) lim εN −1 det C˜ε (z)
ε→0
!
1 − ze−i(ξj −ηk )
= c̃(ξ, η) det
,1 .
ξj − ηk
1≤j≤N ;1≤k≤N −1
Notice that the determinant in the right hand side is a polynomial in
z of degree N − 1 whose coefficient of z N −1 equals
!
!
−iξj Y
e
N −1
iηk
(−1)
,1 .
e
det
ξj − ηk 1≤j≤N ;1≤k≤N −1
k
With the choice ξ = ξ ∗ = (2π(N − j + 1))1≤j≤N , this determinant is
!
!
Y
1
(−1)N −1
, 1 6= 0 .
eiηk det
ξ
j − ηk 1≤j≤N ;1≤k≤N −1
k
The non vanishing of this determinant follows from developing with
respect to the last column, and in view of the explicit formula for the
Cauchy determinants. By continuity, this determinant remains non
zero in a neighborhood of ξ = ξ ∗ . Furthermore, Theorem ?? tells us
that the roots of P̃ε (z) are located outside the unit disc. Hence all the
roots of P (ξ, η) belong to {z, |z| ≥ 1}. Furthermore,
! N −1
Y
1
,1
(1 − zeiηk ).
P (ξ ∗ , η)(z) = det
ξj − ηk 1≤j≤N ;1≤k≤N −1
k=1
Hence, for a suitable choice of η so that ηk 6= ηl (2π) for k 6= l , P (ξ ∗ , η)
has only simple zeroes which belong to the unit circle.
16
PATRICK GÉRARD AND S. GRELLIER
Lemma 4. For any η such that P (ξ ∗ , η) has only simple zeroes on the
unit circle, there exists an open neighborhood of ξ ∗ such that, for every
ξ 6= ξ ∗ in this neighborhood, the zeroes of P (ξ, η) are all outside D.
Proof. Denote by
{zk (ξ, η); 1 ≤ k ≤ N − 1}
the simple zeroes of P (ξ, η), with zk (ξ ∗ , η) = e−iηk . The functions
ξ 7→ zk (ξ, η) are analytic and satisfy |zk (ξ, η)|2 ≥ 1 and |zk (ξ ∗ , η)|2 = 1.
In particular the quadratic form ξ 7→ Qk (ξ) associated to the Hessian of
the function ξ 7→ |zk (ξ, η)|2 is positive at any ξ ∗ . We want to prove that,
for any k, Qk is not identically 0 for η in a dense open set. It suffices
to prove that the Laplacian of ξ 7→ |zk (ξ, η)|2 , which coincides with the
trace of Qk , is not identically zero. Let us compute this Laplacian.
!
N
N
2
X
X
∂ 2 |zk |2
∂ 2 zk
∂zk
.
Re z k 2
=
+
2
∂ξ
2
∂ξ
∂ξ
j
j
j
|
∗
|
∗
|
∗
ξ=ξ
ξ=ξ
j=1
j=1
ξ=ξ
Differentiating the equation P (ξ, η)(zk (ξ)) = 0, we obtain
∂P
∂zk
∂ξ
= − ∂Pj
∂ξj
∂z
and
2
− ∂∂ξP2
2
∂ zk
=
∂ξj2
j
∂P
∂z
+2
∂ 2 P ∂P
∂ξj ∂z ∂ξj
∂P 2
∂z
−
∂P
∂ξj
2
∂2P
∂z 2
∂P 3
∂z
Introduce the following quantities.
!
1
1
Djk :=
,1 ,
det
ξj − ηk
ξr − ηl r6=j,l6=k
X
Y
D :=
(−1)j+k Djk , ζk :=
(1 − ei(ηl −ηk ) )
j
l6=k
i(ηl −ηk )
alk := −
i(ηl −ηk )/2
e
e
=
.
1 − ei(ηl −ηk )
2i sin(ηl − ηk )/2
Notice that Re(alk ) = 12 .
Differentiating
P (ξ, η)(z) = det
1 − ze−i(ξj −ηk )
ξj − ηk
!
,1 ,
1≤j≤N ;1≤k≤N −1
ON THE GROWTH OF SOBOLEV NORMS
17
one gets, after some computations,
∂P
∂P
= i(−1)j+k Djk ζk ,
= −eiηk ζk D
∂ξj | ξ=ξ∗ ,z=zk
∂z | ξ=ξ∗ ,z=zk
∂ 2P
2i
j+k
= (−1) ζk Djk 1 −
∂ξj2 ξ=ξ∗ ,z=z
ξj − ηk
|
k
X ζk ei(ηl +ηk )
X
∂ P
2iηk
=
2
D
=
−2ζ
e
alk D ,
k
i(ηl −ηk )
∂z 2 | ξ=ξ∗ ,z=zk
1
−
e
l6=k
l6=k
2
and
!
X ei(ηl −ηk )
∂ 2P
1
+
= (−1)j+k ζk Djk eiηk i +
−i
i(ηl −ηk )
∂ξj ∂z | ξ=ξ∗ ,z=zk
ξj − ηk
1
−
e
l6=k
i(ηl −ηk )
X
1
e
+ ζk eiηk
(−1)j+l Djl
−i
ξj − ηl
1 − ei(ηl −ηk )
l6=k
!
!
X
X
1
1
(−1)j+l Djl
alk +
.
+i
+ ialk
= ζk eiηk (−1)j+k Djk i +
ξj − ηk
ξj − ηl
l6=k
l6=k
Hence, inserting these formulae to compute the Laplacian, we obtain
2
∂ 2 zk
iηk ∂ zk
Re z k 2
= Re e
∂ξj | ∗
∂ξj2 | ∗
ξ=ξ
ξ=ξ
= Ij + IIj + IIIj
with

Ij = Re eiηk
2
− ∂∂ξP2

 = (−1)j+k Djk ,
D
!
∂ 2 P ∂P
j
∂P
∂z
IIj = 2Re eiηk
∂ξj ∂z ∂ξj
∂P 2
∂z
 2
i Djk
i+

= 2Re
1
ξj −ηk
+i
P
P

1
k+l
a
+
(−1)
D
D
+
ia
jk jl ξj −ηl
lk
l6=k lk
l6=k

D2
P
2
Djk
(−2 − (N − 2)) − l6=k (−1)k+l Djk Djl
,
=
2
D

2 2 
!
P
∂P
∂ P
2
2
2
2
a
D
Djk
∂ξ
∂z
lk
j
jk


l6=k
IIIj = −Re eiηk
=
Re
=
(N
−
2)
.

∂P 3
D2
D2
∂z
18
PATRICK GÉRARD AND S. GRELLIER
Remark that
X
j+l
j+N
(−1) Djl = D − (−1)
det
l6=k
1
ξr − ηl
.
r6=j
Hence, it gives
!
N
N
X
X
X
∂ 2 |zk |2
D
D
D
jk jl
jk
=
(−1)k+l+1
+ (−1)j+k
2
2
∂ξ
2
D
D
j
|ξ=ξ∗
j=1
j=1
l
X
1
k+N 1
= (−1)
Djk det
D2 j
ξr − ηl r6=j
It remains to check that
(−1)
k+N
X
Djk det
j
1
ξr − ηl
r6=j
is not identically zero. This follows from the fact that the limit as ηk
tends to infinity of
X
1
k+N 2
(−1)
ηk
Djk det
ξr − ηl r6=j
j
equals
X
j
det
1
ξr − ηl
!2
,1
r6=j,l6=k
which is clearly not identically zero for a dense choice of (η, ξ).
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Université Paris-Sud XI, Laboratoire de Mathématiques d’Orsay,
CNRS, UMR 8628, et Institut Universitaire de France
E-mail address: Patrick.Gerard@math.u-psud.fr
Fédération Denis Poisson, MAPMO-UMR 6628, Département de Mathématiques,
Université d’Orleans, 45067 Orléans Cedex 2, France
E-mail address: Sandrine.Grellier@univ-orleans.fr