Annales de la Faculté des Sciences de Toulouse

Karim Boulabiar

Jamel Jabeur

Arens regularity of lattice-ordered rings

Chloé Mullaert

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
CHLOÉ MULLAERT
Remarks on the equatorial shallow water system
Tome XIX, no 1 (2010), p. 27-36.
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Annales de la Faculté des Sciences de Toulouse Vol. XIX, n◦ 1, 2010
pp. 27–36
Remarks on the equatorial shallow water system
Chloé Mullaert
(1)
RÉSUMÉ. — Cet article reprend les résultats démontrés par A. Dutrifoy,
A. Majda et S. Schochet dans [1] où ils obtiennent une estimation uniforme
du système ainsi que sa convergence vers une solution globale lorsque le
nombre de Froude tend vers zéro. Ensuite, un résultat de convergence sous
des hypothèses plus faibles sera démontré ainsi que le fait que le temps de
vie des solutions tend vers l’infini quand le nombre de Froude tend vers
zéro.
ABSTRACT. — This article recalls the results given by A. Dutrifoy, A. Ma-
jda and S. Schochet in [1] in which they prove an uniform estimate of the
system as well as the convergence to a global solution of the long wave
equations as the Froud number tends to zero. Then, we will prove the
convergence with weaker hypothesis and show that the life span of the
solutions tends to infinity as the Froud number tends to zero.
1. Introduction
When studying the climate of equatorial regions, the following mathe-
matical model of equatorial shallow water is used :
(ESW )

∂t−→v ε + −→v ε · ∇−→v ε + ε−1(y−→v ε⊥ + ∇hε) = 0
∂thε + −→v ε · ∇hε + hε div−→v ε + ε−1 div−→v ε = 0
(−→v ε, hε) (0) =
(−→v 0,ε, h0,ε)
where hε(x, y, t) is the height of the fluid, −→v ε(x, y, t) = (uε, vε) is the
horizontal velocity and −→v ε⊥ = (−vε, uε).
(∗) Reçu le 11/07/08, accepté le 13/03/09
(1) ENS Cachan, Laboratoire Jacques-Louis Lions
chloe.dousset@dptmaths.ens-cachan.fr
– 27 –
Chloé Mullaert
These equations are written in the spatial domain (x, y) ∈ T×R because
the longitude x is assumed to be periodic and under the assumption that the
positive parameter ε which represents the Froude number and the height
fluctuations tends to zero. More details about this model can be found in [2]
and [3]. Let us define
Uε = (−→v ε, hε) and L =
 0 −y ∂xy 0 ∂y
∂x ∂y 0

the penalisation matrix associated to (ESW ).
This paper extends results given by A. Dutrifoy, A. Majda and S. Scho-
chet in [1] in which they use the symmetric hyperbolic structure of a mod-
ified system to prove a uniform estimate in ε of the system on some time
interval [0, T ] as well as the convergence of the solutions (Uε) under the
assumption that the initial data is “well-prepared”. This notion has been
introduced by S. Klainerman and A. Majda in [5].
The Sobolev spaces in which the results are given are those introduced
in [1]. Let us define
Wn =
w such as ‖w‖̃ 2Wn = ∑
j+k+mn
‖yj∂kx∂my w‖
2
L2 is finite

In this paper, we recall the estimate result proved in [1].
Theorem 1.1. — Let us consider a family of initial data (Uε,0)0<εε0
bounded in W2n with n  2 by a constant M0. Then a positive time T
(depending only of the bound M0) exists such that the family (Uε)0<εε0of
associated solutions is defined on [0, T ] and remain uniformly bounded in
W2n on [0, T ].
The convergence result proved in [1] under the assumption
‖LUε,0‖W2n  Cε will be proved under the assumption that ‖LUε,0‖W2n
tends to zero as ε tends to zero.
Theorem 1.2. — Let us consider a family of initial data (Uε,0)0<εε0
converging in W2n to an element of Ker(L)∩W2n+1 with n  2. Then, for
every T > 0, the family (Uε)0<εε0 of associated solutions converge to the
limit of the initial data in C0
(
[0, T ] ,W2n
)
. Moreover, the life span T ∗ε of
the solution Uε tends to infinity as ε tends to zero.
– 28 –
Remarks on the equatorial shallow water system
In a first part, we shall briefly recall the method used by A. Dutrifoy, A.
Majda and S. Schochet in [1] to prove the uniform estimate. It consists in
a non linear change of unknowns which depends on the parameter ε. In a
second part, we shall prove Theorem 1.2 using an appropriated Ansatz on
the unknowns.
2. Symmetrization and stabilization of the system
The aim of this part is to symmetrize the system and to find derivation
matrices which preserve its structure i.e., to change (−→v ε, hε) into Vε solution
of an equation
∂tVε + B1(Vε)∂xVε + B2(Vε)∂yVε + ε−1NVε = 0
where B1(Vε), B2(Vε) are symmetric matrices and where N is a skew-
symmetric operator ie N∗ = −N such as [N, ∂x] = [N, D] with D a deriva-
tion matrix.
Indeed, for a symmetric system with an skewsymmetric penalisation, it is
easy to obtain an uniform L2 estimate and as we can derive the system with-
out changing its structure, we shall obtain uniform estimate in the Sobolev
spaces W2n. One can read [4] to see the advantages of an skewsymmetric
penalisation.
The following lemma describe the “raising” and “lowering” operators of
the harmonic oscillator Hamiltonian.
Lemma 2.1. — Given the operators
L+ =
1√
2
(∂y − y) , L− =
1√
2
(∂y + y) and H = L−L+ + L+L− = ∂2y − y2
then we have the relations :
[L+, L−] = I, L+L− =
1
2
(H + I) , [H, L+] = −2L+ and [H, L−] = 2L−
The following proposition, proved in [1], explains how to change the
unknowns of the system to symmetrize it and gives us derivation matrices
which preserve its structure.
– 29 –
Chloé Mullaert
Proposition 2.2. — The vector Vε defined by
Vε =

1√
2
(
−uε +
2
(√
1 + εhε − 1
)
ε
)
vε
1√
2
(
uε +
2
(√
1 + εhε − 1
)
ε
)

is a solution of
∂tVε + B1(Vε)∂xVε + B2(Vε)∂yVε + ε−1NVε = 0
where B1(Vε) and B2(Vε) are symmetric matrices whose coefficients are
linear combinations of the components of Vε and where
N =
 −∂x L− 0L+ 0 L−
0 L+ ∂x

is a skew-symmetric operator.
Moreover, if
D =
 H − 2 0 00 H 0
0 0 H + 2

where H = ∂2y −y2 is the harmonic oscillator, then [N, D] = [N, ∂xId] = 0.
This proposition leads the authors of [1] to the following definition of
adapted norms
‖W‖2W2n =
∑
k+2p2n
∥∥∂kxDpW∥∥2L2 .
Proposition 2.3. — The norms ‖ .‖W2n and ‖ .̃‖W2n are equivalent.
As the above proposition, the following one is proved in [1].
Proposition 2.4. — As Vε is solution of
∂tVε + B1(Vε)∂xVε + B2(Vε)∂yVε + ε−1NVε = 0
where B1(Vε) and B2(Vε) are symmetric matrices whose coefficients are lin-
ear combinations of the components of Vε, and where N is a skew-symmetric
matrix operator commuting with D, we have
– 30 –
Remarks on the equatorial shallow water system
d
dt
‖Vε‖2W2n  C ‖Vε‖
3
W2n .
This result leads to the theorem 1.1 proved in [1] thanks to the following
easy lemmas.
Lemma 2.5. — If f is a positive fonction which satisfies{
|f ′|  Cf3/2
f(0)  M0
then, f  2M0 on
[
0,
1
C
√
f(0)
]
.
Lemma 2.6. — The norms ‖Vε‖W2n and ‖Uε‖W2n are equivalents when
ε tends to zero.
Proof. — Using the Taylor formula at order two, we get
Vε =

1√
2
(
−uε +
2
(√
1 + εhε − 1
)
ε
)
vε
1√
2
(
uε +
2
(√
1 + εhε − 1
)
ε
)
=

1√
2
(−uε + hε)
vε
1√
2
(uε + hε)
 (1 + G (εhε))
where G is a regular fonction such that G(0) = 0. Then we get the result
thanks to Faa di Bruno formula. 
3. Convergence and life span
In this part, we shall give a convergence result with weaker hypothesis
than in [1] and prove that the life span of the solutions tends to infinity as
ε tends to zero. The purpose of this section is to prove the second theorem.
In order to prove that Vε exists on a time interval which tends to R+ as
ε tends to zero, we compare it to V0 (y), the limit of the initial data, which
is, by definition, global. We make an Ansatz
Vε = V0 + εV1 + Vε
As the limit of the initial data V0 = (l0, v0, r0) belongs to Ker(N), we
have
– 31 –
Chloé Mullaert
−∂xl0 + L−v0 = 0
L+l0 + L−r0 = 0
L+v0 + ∂xr0 = 0
so v0 = [L+, L−] v0 = L+∂xl0 + L−∂xr0 = ∂x [L+l0 + L−r0] = 0.
Thus, ∂xr0 = 0 and ∂xl0 = 0 i.e. r0 and l0 are independant of x.
Then, we study the system verified by Ṽε = Vε − V0.
As Vε is solution of
∂tVε + B1(Vε)∂xVε + B2(Vε)∂yVε + ε−1NVε = 0
and V0 belongs to Ker(N), Ṽε is solution of
∂tṼε + B1(Vε)∂xṼε + B2(Vε)∂yṼε + ε−1NṼε = −B2(Vε)∂yV0.
Moreover, ‖ Ṽε(0)‖W2n tends to zero as ε tends to zero.
The problem is that zero is not solution of this equation because the
relation L+l0 + L−r0 = 0 implies that ∂y (r0 + l0) = y (l0 − r0) so
B2(V0)∂yV0 =
 01
4
(r0 + l0) ∂y (r0 + l0)
0
 =
 01
4
y
(
l20 − r20
)
0
 = 0 ·
That is the reason why we search solutions of
∂tWε + ε−1NWε =
 01
4
y
(
l20 − r20
)
0
 ·
If Wε =
 aεbε
cε
, then we want
∂t
 aεbε
cε
 + ε−1
 −∂xaε − L−bε−L+aε + L−cε
L+bε + ∂xcε
 =
 01
4y
(
l20 − r20
)
0
 ·
If we search a solution independant of t and x, then L−bε = L+bε = 0
so bε = 0 and the system becomes
– 32 –
Remarks on the equatorial shallow water system
−L+aε + L−cε =
1√
2
∂y (cε − aε) +
1√
2
y (cε + aε) =
ε
4
y
(
l20 − r20
)
·
We search aε and cε as fonctions of l20 − r20 , and we find that
cε = aε =
ε
4
√
2
(
l20 − r20
)
is the solution.
Moreover, the norm of this solution is controled by ε and Wε = εW1
with W1 belonging to W2n+1 solution of NW1 = B2(V0)∂yV0.
It remains to study the system satisfied by Vε = Ṽε − εW1.
As ‖ Ṽε(0)‖W2n tends to zero as ε tends to zero, so does ‖Vε(0)‖W2n .
Moreover, Vε is solution of
∂tVε + B1(Vε)∂xVε + B2(Vε)∂yVε + ε−1NVε = Fε
where Fε = −B2(Vε)∂yV0 − (εB2(Vε)∂yW1 + NW1)
= −B2(Vε)∂yV0 − (εB2(Vε)∂yW1 + B2(V0)∂yV0)
= −B2(Ṽε)∂yV0 − εB2(Vε)∂yW1.
In order to obtain a convergence in W2n, we apply the operator ∂kxDp,
with k + 2p  2n to the equation.
Given V k,pε = ∂
k
xD
pVε, we have
∂tV
k,p
ε + B1(Vε)∂xV
k,p
ε + B2(Vε)∂yV
k,p
ε + ε
−1NV k,pε = F
k,p
ε
with F k,pε = ∂kxD
pFε +
[
B1(Vε)∂x, ∂kxD
p
]
Vε +
[
B2(Vε)∂y, ∂kxD
p
]
Vε·
So, we have the estimate
1
2
d
dt
‖V k,pε ‖
2
L2  ‖V k,pε ‖L2 ‖F
k,p
ε ‖L2 +
1
2
‖divB(Vε)‖L∞ ‖V k,pε ‖
2
L2 ·
Moreover, ‖divB(Vε)‖L∞  C ‖Vε ‖H3
 C ‖Vε ‖H3 + C ‖V0 ‖H3 + εC ‖W1 ‖H3 ·
It remains to estimate F k,pε in L2.
The arguments given in the section 4 of [1] prove that, for n  2, we can
estimate
[
B1(Vε)∂x, ∂kxD
p
]
Vε and
[
B2(Vε)∂y, ∂kxD
p
]
Vε in L2 by
– 33 –
Chloé Mullaert
‖Vε ‖W2n ‖Vε ‖W2n  ‖Vε ‖W2n
(
‖Vε ‖W2n + ‖V0 ‖W2n + ε ‖W1 ‖W2n
)
·
It remains to estimate ∂kxD
pFε in L2 with Fε = −B2(Ṽε)∂yV0 −B2(Vε)∂yWε ·
This term is estimated by ‖ Ṽε ‖W2n ‖∂yV0 ‖W2n+ε ‖∂yW1 ‖W2n ‖Vε ‖W2n ·
As V0 belongs to W2n+1, we have
1
2
d
dt
‖Vε ‖2W2n  Aε ‖Vε ‖W2n + Bε ‖Vε ‖
2
W2n + C ‖Vε ‖
3
W2n
with Aε = C (ε ‖W1 ‖W2n ‖V0 ‖W2n+1 + ε ‖W1‖W2n+1 (‖V0‖W2n + ε ‖W1‖W2n))
 εC ′
and Bε = C (‖V0‖W2n+1 + ε ‖W1‖W2n+1)  C ′ ·
This can be written
d
dt
‖Vε ‖2W2n  R
(
‖Vε ‖3W2n + ‖Vε ‖
2
W2n + ε ‖Vε ‖W2n
)
·
To conclude, we use the following two lemmas.
Lemma 3.1. — If f is a positive fonction which verifies{
|f ′|  R
(
f3/2 + f + εf1/2
)
f(0) = f0
then the life span of f tends to infinity as ε and f0 tend to zero.
Moreover, for every T > 0, we have lim
ε,f0→0
‖f‖L∞[0,T ] = 0.
Proof. — Let us define Tε = Sup {M : ∀t ∈ [0,M ] , f(t)  1}.
On [0, Tε] we have |f ′|  R (2f + ε), so
f ′
2f + ε
=
[
1
2
log (2f + ε)
]′
 R.
Thus,
log (2 + ε) − log (2f0 + ε)  2RTε.
The left term tends to infinity as ε and f0 tend to zero. So does Tε.
– 34 –
Remarks on the equatorial shallow water system
For every T > 0 given, we have T  Tε when ε and f0 are small enough,
so for every t ∈ [0, T ],we have 2RT  log
(
2f(t) + ε
)
− log
(
2f0 + ε
)
·
This gives
f(t)  1
2
(
(2f0 + ε) exp(2RT ) − ε
)
= Cf0,ε
and Cf0,ε tends to zero as ε and f0 tend to zero. 
Lemma 3.2. — The norms ‖NVε‖W2n and ‖LUε‖W2n are equivalents
when ε tends to zero.
Proof. — As in the proof of the lemma 2.6, we have
LUε =
 −yvε + ∂xhεyuε + ∂yhε
∂xuε + ∂yvε

and using the Taylor formula at order two, we get
NVε =
1√
2

∂xuε + ∂yvε + yvε −
∂xhε√
1 + εhε
2
(
yuε +
∂yhε√
1 + εhε
)
∂xuε + ∂yvε − yvε +
∂xhε√
1 + εhε

=
1√
2
 ∂xuε + ∂yvε + yvε − ∂xhε2 (yuε + ∂yhε)
∂xuε + ∂yvε − yvε + ∂xhε
 (1 + G̃(εhε))
where G̃ is a regular fonction such as G̃(0) = 0. Then we get the result
thanks to Faa di Bruno formula. 
– 35 –
Chloé Mullaert
Bibliography
[1] Dutrifoy (A), Majda (A.) and Schochet (S.). — A simple justification of the
singular limit for equatorial shallow water dynamics, Communications in Pure and
Applied Mathematics, 62, p. 305-443 (2008).
[2] Gallagher (I.) and Saint-Raymond (L.). — Mathematical study of the betaplane
model, Mémoires de la Société Mathématique de France (2007).
[3] Gallagher (I.) and Saint-Raymond (L.). — On the influence of the Earth’s rota-
tion on geophysical flows.Mémoires de la Société Mathématique de France (2006).
[4] Chemin (J.-Y.). — A propos d’un problème de pénalisation de type antisymétrique,
Journal de Mathématiques Pures et Appliquées, 76, p. 739-755 (1997).
[5] Klainerman (S.) and Majda (A.). — Singular limits of quasilinear hyperbolic sys-
tems with large parameters and the incompressible limit of compressible fluids,
Communications in Pure and Applied Mathematics, 34, p. 481-524 (1981).
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