For the Euler equations in a thin domain Q_ε = Ω×(0, ε), Ω a rectangle in R^2, with initial data in (W^(2,q)(Qε))^3, q > 3, bounded uniformly in ε, the classical solution is shown to exist on a time interval (0, T(ε)), where T(є) → +∞ as є → 0. We compare this solution with that of a system of limiting equations on Ω

Marsden, Jerrold E.

Ratiu, Tudor S.

Raugel, Geneviève

English

Caltech Authors - Main

The Euler Equations on Thin Domains
Jerrold E. Marsden
Control and Dynamical Systems, California Institute of Technology 107-81
Pasadena, CA 91125, USA
Tudor S. Ratiu
Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne
CH-1015 Lausanne, Switzerland
Geneviève Raugel
Université de Paris-Sud et CNRS, UMR 8628, Mathématiques, Bât. 425
F-91405 Orsay Cedex, France
International Conference on Differential Equations, Berlin, 1999
Edited by B. Fiedler, K. Gröger and J. Sprekels,
World Scientific, 2000, 1198–1203.
Abstract
For the Euler equations in a thin domain Qε = Ω×(0, ε), Ω a rectangle in R2, with initial
data in (W 2,q(Qε))3, q > 3, bounded uniformly in ε, the classical solution is shown to exist
on a time interval (0, T (ε)), where T (ε) → +∞ as ε → 0. We compare this solution with
that of a system of limiting equations on Ω.
1 Introduction
The Euler equations in three-dimensional thin domains arise in geophysical problems such as
atmospheric and ocean dynamics. For instance, in ocean dynamics, one has large scale fluid
problems with small dissipation and the fluid regions are thin compared to the horizontal length
scales. In this situation, one would like to know to what extent two dimensional models are
appropriate and how one might incorporate three dimensional effects into such a model (for
shallow-water problems, see [CaHo], [Ol], for example).
Such problems lead to interesting mathematical questions. While global existence and unique-
ness of classical solutions of the Euler equations in two-dimensional domains is well known ([Li],
[Wo], [Ka1], [Yu]), only local existence (and uniqueness) of solutions is known in the three-
dimensional case ([EbMa], [Ka2], [Te]). When the three-dimensional domain is thin in one
direction, one can hope to show that the time of existence of the solutions tends to infinity,
when the thickness ε of the domain goes to zero, and also, to compare these solutions to those
of appropriate two-dimensional systems. In [MaRaRa1] and [MaRaRa2], we show that such ex-
pected properties are indeed true, by using standard estimates for the Euler equations combined
with thin domains techniques as developed in [HaRa1], [HaRa2] and [RaSe1]. In particular, the
mean value operator in the thin direction (see [HaRa1], [HaRa2]) plays an important role.
In this paper we present some of these results, emphasizing estimates between the solutions of
the three-dimensional Euler equations and an asymptotic expansion in terms of two dimensional
vector fields. As expected, the order zero term in this expansion is the solution of the limiting
two-dimensional Euler equation. The proofs in [MaRaRa2] indicate that, in general, three-
dimensional effects beyond the two-dimensional Euler equations need to be included to obtain
control on the error estimates for this asymptotic expansion. Taking into account the first order
term in this expansion allows us, in contrast to [Ol], to choose initial data U(0) for which the
mean value of the derivative in the thin direction need not be small (Theorem 1 below). In this
sense, our data are not nearly two-dimensional. We cannot prove that the solutions of the three-
dimensional Euler equations exist globally, when the thickness ε > 0 is small but fixed. This
feature is in contrast with the properties of the Navier-Stokes equations on three-dimensional
thin domains, where existence of global solutions for large initial data is proved ([RaSe1], [RaSe2],
[TeZi] and [IfRa]). In [MaRaRa2], we focus on two types of thin domains Qε: cylindrical domains
and spherical shells. The cylindrical case is easier because the two-dimensional Euler equations
obviously embed into the three-dimensional ones. Because of limited space, we only describe
the cylindrical case.
Consider the thin cylinder Qε = Ω × (0, ε), where ε is a small positive parameter, Ω =
[0, l1] × [0, l2] and l1, l2 are two positive fixed numbers. We take x1 and x2 to be periodic
variables with period l1 and l2 respectively and, correspondingly, we may regard Ω as a two
torus. Thus, the boundary ∂Qε of Qε is given by Γ0 ∪Γε where Γ0 = Ω×{0} and Γε = Ω×{ε}.
The vertical variable is denoted x3. If the velocity field of the fluid on the thin domain Qε and
the pressure are denoted by U = (U1, U2, U3) ≡ (U∗, U3) and P , respectively, the Euler equations
are given by
∂U
∂t
+ (U · ∇)U + ∇P = 0 ,
div U = 0 ,
U3 = 0 on Γ0 ∪ Γε ,
U is periodic in x1, x2 with periods l1, l2 ,
U( · , 0) = U0 ,
(1)
where the initial condition U0 belongs to the space
Xm,q(Qε) ≡ {U ∈ [Wm,q(Qε)]3 | divU = 0, U3 = 0 on Γ0 ∪ Γε,
U is periodic in x1, x2 with periods l1, l2, } ,
and where, for m ∈ N, q ≥ 1, Wm,q(Qε) denotes the Sobolev space on Qε, equipped with the
standard norm. Local existence theory shows that, if q > 1 and m > 1 + 3/q, there is a T∗ > 0
and a unique solution (U,P ) ∈ C0([0, T∗);Xm,q(Qε) × Wm+1,q(Qε)/R) of the Euler equations
(1). In the sequel, we shall assume that U0 ∈ X2,q(Qε) for q > 3.
In the next section, we describe the existence and comparison results for the solutions of
Equation (1) in the cylindrical case.
2 The thin cylindrical case
The mean value operator in the vertical direction is defined as follows. For any function f ∈
Lp(Qε), where p ≥ 1, let
(Mf)(x1, x2) =
1
ε
∫ ε
0
f(x1, x2, s) ds.
The mean value operator on vector fields is the bounded linear operator from (Lp(Qε))3 to
(Lp(Ω))3 defined by MU = M(U1, U2, U3) = (MU1,MU2, 0). Since the third component of MU
is zero, we can identify MU with a 2-dimensional vector field.
To find the limiting equations on Ω in a systematic way, we formally expand (U∗, U3, P ) in
a power series relative to the variable x3, that is,
U∗ = U∗0 + U
∗
1x3 +
1
2
U∗2x
2
3 + · · · ,
U3 =
1
2
U3,2x
2
3 + · · · ,
P0 = P0 + P1x3 +
1
2
P2x
2
3 + · · · .
(2)
Due to the boundary conditions on U3, the first two terms U3,0 and U3,1 vanish. Inserting (2)
into (1) and equating to zero every coefficient of xk3, k = 0, 1, yields the following equations,
where U∗i , U3,i and Pi are periodic in the variables x1,x2,
• Order 0
∂U∗0
∂t
+ U∗0 · ∇U∗0 = −gradP0 , divU∗0 = 0 , (3)
as well as the equation P1 = 0.
• Order 1
∂U∗1
∂t
+ (U∗1 · ∇)U∗0 + (U∗0 · ∇)U∗1 = 0 , (4)
as well as the equalities P2 = 0 and U3,2 = −divU∗1 .
Here, ∇ denotes the 2-dimensional gradient and div denotes the 2-dimensional divergence
operator. The equations (3) are the 2-dimensional Euler equations on Ω. As already noted,
one has global existence of solutions and estimates ([Ka1], [KaPo]). The system (4) for the
2-dimensional vector field U∗1 on Ω is linear and thus admits always global solutions. The vector
field U∗1 is the analogue of the director field in elasticity.
At order k, the equation for U∗k has the form
∂U∗k
∂t + (U
∗
k · ∇)U∗0 + (U∗0 · ∇)U∗k = “known
quantities”, where the “known quantities” have been determined in the previous steps. Then,
U3,k+1 and Pk+1 are given algebraically in terms of the previously computed quantities.
Below, we denote by U∗0 and U
∗
1 the solutions of the equations (3) and (4) with initial
data U∗0 = MU
0 and U∗1 = M(
∂U0
∂x3
) respectively. Since U0 belongs to X2,q(Qε), for q > 3,
(U∗0(t), U
∗
1(t)) belongs to C0([0,+∞); (W 2,q(Ω))2 × (W 1,q(Ω))2).
We can now state the main theorem for thin cylindrical domains, which is proved in [MaRaRa2].
The hypotheses will be as follows:
H1 Let q > 3, K0 > 0 be given. Suppose that γ and α are positive numbers satisfying:
0 ≤ γ < α < min
(
q − 3
q
,
(2q + 5)(q − 3)2 + 14(q − 3) + 1
q2(3q − 4)
)
.
H2 The following conditions on the initial data hold:
max
(∥∥MU0∥∥
W 2,q(Ω)
,
∥∥∥∥M ∂U0∂x3
∥∥∥∥
L∞(Ω)
)
≤ K0,
max
(∥∥U0∥∥
W 2,q(Qε)
,
∥∥∥∥M ∂U0∂x3
∥∥∥∥
W 1,q(Ω)
)
≤ K0ε−γ .
Theorem 1. Assume that the hypotheses H1, H2 hold. Then there are positive numbers ε0 =
ε0(q,K0, γ, α) and δ = δ(q,K0, γ, α) and positive constants C1 = C1(q,K0, α, γ) and C̃ = C̃(K0)
such that, for ε ∈ [0, ε0], the Euler equations (1) on Qε have a unique solution (U(t), P (t)) ∈
C0([0, τ(ε)];X2,q(Qε) ×W 3,q(Qε)/R), where τ(ε) satisfies
τ(ε) ≥ C̃ log (log [− log ε]) .
Moreover, the following estimates hold for 0 ≤ t ≤ τ(ε):
‖U(t)‖W 2,q(Qε) ≤ 3K0ε−α (5)
and ∥∥∥∇(U − U∗0) (t)∥∥∥
L∞(Qε)
+
∥∥∥∥
(
∂U∗
∂x3
− U∗1
)
(t)
∥∥∥∥
L∞(Qε)
+
∥∥∥∥∂U3∂x3 (t)
∥∥∥∥
L∞(Qε)
≤ C1εδ. (6)
The proof of Theorem 1 relies on a contradiction argument, on a priori estimates involving
the quantity ‖∇U(t)‖L∞(Qε) similar to those of [BeKaMa], [Fe], [KaPo] and on the comparison
of the vector fields MU(t) with U∗0(t) and M(
∂U
∂x3
)(t) with U∗1(t) in the L
∞-norm on Ω. To
bound the terms (Id−M)U(t) and (Id−M)( ∂U∂x3 )(t) in the L
∞-norm on Qε, we use Poincaré
inequalities and exploit the divergence free condition of U(t).
Remarks 1. Theorem 1 is still true if the constant K0 in the hypothesis H2 is replaced by
K0f(ε), where f(ε) is a positive function of ε, which goes very slowly to infinity, as ε goes to
zero. For instance, we can choose f(ε) = log(log(log[− log ε])). The hypothesis H1 can also be
made weaker (see [MaRaRa2]).
2. If the initial data U0 are bounded in (W 3,p(Qε))3, p ≥ 2 for example, one shows that
‖U(t)‖W 3,p(Qε) satisfies the estimate (5) on the time interval [0, τ(ε)], by using Theorem 1 and
taking into account the terms of order 2 in the expansion (2). Of course, one obtains estimates,
similar to (6), involving U∗0(t), U
∗
1(t) and U
∗
2(t), where U
∗
2(t) is the vector field appearing in (2)
at order two and satisfying U∗2(0) = M(
∂2U0
∂x23
) (see [MaRaRa2]). The estimate of the derivatives
of order k ≥ 3 of U(t) requires the study of the terms in the expansion (2) up to order k − 1,
which becomes rather technical, when k increases.
3. In [MaRaRa1], we have described existence and comparison results in the spherical case, when
U0 is in (H3(Qε))3. To prove the existence of the solution U(t) on [0, τ(ε)], we took into account
the expansion (2) up to order two. [MaRaRa2] simplifies this part of the proof since now one
only needs to work with the first two vector fields U∗0 and U
∗
1.
4. Similar results to those of Theorem 1 hold if Qε = {(x1, x2, x3) | (x1, x2) ∈ Ω , 0 < x3 <
εg(x1, x2)}, where g(x1, x2) is a positive function, periodic in x1 and x2 of period l1 and l2
respectively. This case is studied in [Ol], who proved existence of a solution U(t) on long time
intervals and compared U(t) with a solution of the 2-dimensional Euler equations, assuming
that the data U0 are nearly 2-dimensional in Xm,2(Qε) for m ≥ 3. In the above situation, he
had to assume that, for 0 ≤ j ≤ 2, ‖MDj(∂U0∂x3 )‖Hm(Qε) ≤ δε
1/2, where δ is sufficiently small
and showed that ‖U(t)−U∗0(t)‖Hm(Qε) ≤ Cε1/2 on some time interval. Since, in our case, δ can
be large, we need to take into account the vector field U∗1.
5. In [MaRaRa2], Theorem 1 is generalized to the case of a thin spherical shell, that is, the
domain bounded by two spheres of radii r = 1 and r = 1 + ε. Again, we have to consider the
asymptotic expansion (2), where the variable x3 becomes ρ = r − 1. Thus, U∗0 is a solution
of the Euler equations on the sphere S2, while U∗1 is the solution of a “linearized” Euler-type
equation on S2. The geometry of Qε appears through the fact that now P0 = k(U∗0 , U
∗
0 ) and
P1 = k(U∗0 , U
∗
0 ) + 2k(U
∗
0 , U
∗
1 ), where k(·, ·) is the second fundamental form on S2.
6. In the context of rotating fluids, Babin, Mahalov and Nicolaenko [BaMaNi] showed existence
of solutions of the Euler equations with an additional Coriolis term, on a time interval [0, T (R)],
where T (R) goes to infinity as the Rossby number R goes to zero.
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The Euler Equations on Thin Domains

The Euler Equations on Thin Domains
Jerrold E. Marsden
Control and Dynamical Systems, California Institute of Technology 107-81
Pasadena, CA 91125, USA
Tudor S. Ratiu
De´partement de Mathe´matiques, Ecole Polytechnique Fe´de´rale de Lausanne
CH-1015 Lausanne, Switzerland
Genevie`ve Raugel
Universite´ de Paris-Sud et CNRS, UMR 8628, Mathe´matiques, Baˆt. 425
F-91405 Orsay Cedex, France
International Conference on Diﬀerential Equations, Berlin, 1999
Edited by B. Fiedler, K. Gro¨ger and J. Sprekels,
World Scientiﬁc, 2000, 1198–1203.
Abstract
For the Euler equations in a thin domain Qε = Ω×(0, ε), Ω a rectangle in R2, with initial
data in (W 2,q(Qε))3, q > 3, bounded uniformly in ε, the classical solution is shown to exist
on a time interval (0, T (ε)), where T () → +∞ as  → 0. We compare this solution with
that of a system of limiting equations on Ω.
1 Introduction
The Euler equations in three-dimensional thin domains arise in geophysical problems such as
atmospheric and ocean dynamics. For instance, in ocean dynamics, one has large scale ﬂuid
problems with small dissipation and the ﬂuid regions are thin compared to the horizontal length
scales. In this situation, one would like to know to what extent two dimensional models are
appropriate and how one might incorporate three dimensional eﬀects into such a model (for
shallow-water problems, see [CaHo], [Ol], for example).
Such problems lead to interesting mathematical questions. While global existence and unique-
ness of classical solutions of the Euler equations in two-dimensional domains is well known ([Li],
[Wo], [Ka1], [Yu]), only local existence (and uniqueness) of solutions is known in the three-
dimensional case ([EbMa], [Ka2], [Te]). When the three-dimensional domain is thin in one
direction, one can hope to show that the time of existence of the solutions tends to inﬁnity,
when the thickness ε of the domain goes to zero, and also, to compare these solutions to those
of appropriate two-dimensional systems. In [MaRaRa1] and [MaRaRa2], we show that such ex-
pected properties are indeed true, by using standard estimates for the Euler equations combined
with thin domains techniques as developed in [HaRa1], [HaRa2] and [RaSe1]. In particular, the
mean value operator in the thin direction (see [HaRa1], [HaRa2]) plays an important role.
In this paper we present some of these results, emphasizing estimates between the solutions of
the three-dimensional Euler equations and an asymptotic expansion in terms of two dimensional
vector ﬁelds. As expected, the order zero term in this expansion is the solution of the limiting
two-dimensional Euler equation. The proofs in [MaRaRa2] indicate that, in general, three-
dimensional eﬀects beyond the two-dimensional Euler equations need to be included to obtain
control on the error estimates for this asymptotic expansion. Taking into account the ﬁrst order
term in this expansion allows us, in contrast to [Ol], to choose initial data U(0) for which the
mean value of the derivative in the thin direction need not be small (Theorem 1 below). In this
sense, our data are not nearly two-dimensional. We cannot prove that the solutions of the three-
dimensional Euler equations exist globally, when the thickness ε > 0 is small but ﬁxed. This
feature is in contrast with the properties of the Navier-Stokes equations on three-dimensional
thin domains, where existence of global solutions for large initial data is proved ([RaSe1], [RaSe2],
[TeZi] and [IfRa]). In [MaRaRa2], we focus on two types of thin domains Qε: cylindrical domains
and spherical shells. The cylindrical case is easier because the two-dimensional Euler equations
obviously embed into the three-dimensional ones. Because of limited space, we only describe
the cylindrical case.
Consider the thin cylinder Q = Ω × (0, ), where ε is a small positive parameter, Ω =
[0, l1] × [0, l2] and l1, l2 are two positive ﬁxed numbers. We take x1 and x2 to be periodic
variables with period l1 and l2 respectively and, correspondingly, we may regard Ω as a two
torus. Thus, the boundary ∂Qε of Qε is given by Γ0 ∪Γε where Γ0 = Ω×{0} and Γε = Ω×{ε}.
The vertical variable is denoted x3. If the velocity ﬁeld of the ﬂuid on the thin domain Qε and
the pressure are denoted by U = (U1, U2, U3) ≡ (U∗, U3) and P , respectively, the Euler equations
are given by
∂U
∂t
+ (U · ∇)U +∇P = 0 ,
div U = 0 ,
U3 = 0 on Γ0 ∪ Γε ,
U is periodic in x1, x2 with periods l1, l2 ,
U( · , 0) = U0 ,
(1)
where the initial condition U0 belongs to the space
Xm,q(Qε) ≡ {U ∈ [Wm,q(Qε)]3 | divU = 0, U3 = 0 on Γ0 ∪ Γε,
U is periodic in x1, x2 with periods l1, l2, } ,
and where, for m ∈ N, q ≥ 1, Wm,q(Qε) denotes the Sobolev space on Qε, equipped with the
standard norm. Local existence theory shows that, if q > 1 and m > 1 + 3/q, there is a T∗ > 0
and a unique solution (U,P ) ∈ C0([0, T∗);Xm,q(Qε) ×Wm+1,q(Qε)/R) of the Euler equations
(1). In the sequel, we shall assume that U0 ∈ X2,q(Qε) for q > 3.
In the next section, we describe the existence and comparison results for the solutions of
Equation (1) in the cylindrical case.
2 The thin cylindrical case
The mean value operator in the vertical direction is deﬁned as follows. For any function f ∈
Lp(Qε), where p ≥ 1, let
(Mf)(x1, x2) =
1
ε
∫ ε
0
f(x1, x2, s) ds.
The mean value operator on vector ﬁelds is the bounded linear operator from (Lp(Qε))3 to
(Lp(Ω))3 deﬁned by MU = M(U1, U2, U3) = (MU1,MU2, 0). Since the third component of MU
is zero, we can identify MU with a 2-dimensional vector ﬁeld.
To ﬁnd the limiting equations on Ω in a systematic way, we formally expand (U∗, U3, P ) in
a power series relative to the variable x3, that is,
U∗ = U∗0 + U
∗
1x3 +
1
2
U∗2x
2
3 + · · · ,
U3 =
1
2
U3,2x
2
3 + · · · ,
P0 = P0 + P1x3 +
1
2
P2x
2
3 + · · · .
(2)
Due to the boundary conditions on U3, the ﬁrst two terms U3,0 and U3,1 vanish. Inserting (2)
into (1) and equating to zero every coeﬃcient of xk3, k = 0, 1, yields the following equations,
where U∗i , U3,i and Pi are periodic in the variables x1,x2,
• Order 0
∂U∗0
∂t
+ U∗0 · ∇U∗0 = −gradP0 , divU∗0 = 0 , (3)
as well as the equation P1 = 0.
• Order 1
∂U∗1
∂t
+ (U∗1 · ∇)U∗0 + (U∗0 · ∇)U∗1 = 0 , (4)
as well as the equalities P2 = 0 and U3,2 = −divU∗1 .
Here, ∇ denotes the 2-dimensional gradient and div denotes the 2-dimensional divergence
operator. The equations (3) are the 2-dimensional Euler equations on Ω. As already noted,
one has global existence of solutions and estimates ([Ka1], [KaPo]). The system (4) for the
2-dimensional vector ﬁeld U∗1 on Ω is linear and thus admits always global solutions. The vector
ﬁeld U∗1 is the analogue of the director ﬁeld in elasticity.
At order k, the equation for U∗k has the form
∂U∗k
∂t + (U
∗
k · ∇)U∗0 + (U∗0 · ∇)U∗k = “known
quantities”, where the “known quantities” have been determined in the previous steps. Then,
U3,k+1 and Pk+1 are given algebraically in terms of the previously computed quantities.
Below, we denote by U∗0 and U
∗
1 the solutions of the equations (3) and (4) with initial
data U∗0 = MU0 and U
∗
1 = M(
∂U0
∂x3
) respectively. Since U0 belongs to X2,q(Qε), for q > 3,
(U∗0(t), U
∗
1(t)) belongs to C0([0,+∞); (W 2,q(Ω))2 × (W 1,q(Ω))2).
We can now state the main theorem for thin cylindrical domains, which is proved in [MaRaRa2].
The hypotheses will be as follows:
H1 Let q > 3, K0 > 0 be given. Suppose that γ and α are positive numbers satisfying:
0 ≤ γ < α < min
(
q − 3
q
,
(2q + 5)(q − 3)2 + 14(q − 3) + 1
q2(3q − 4)
)
.
H2 The following conditions on the initial data hold:
max
(∥∥MU0∥∥
W 2,q(Ω)
,
∥∥∥∥M∂U0∂x3
∥∥∥∥
L∞(Ω)
)
≤ K0,
max
(∥∥U0∥∥
W 2,q(Qε)
,
∥∥∥∥M∂U0∂x3
∥∥∥∥
W 1,q(Ω)
)
≤ K0ε−γ .
Theorem 1. Assume that the hypotheses H1, H2 hold. Then there are positive numbers ε0 =
ε0(q,K0, γ, α) and δ = δ(q,K0, γ, α) and positive constants C1 = C1(q,K0, α, γ) and C˜ = C˜(K0)
such that, for ε ∈ [0, ε0], the Euler equations (1) on Qε have a unique solution (U(t), P (t)) ∈
C0([0, τ(ε)];X2,q(Qε)×W 3,q(Qε)/R), where τ(ε) satisﬁes
τ(ε) ≥ C˜ log (log [− log ε]) .
Moreover, the following estimates hold for 0 ≤ t ≤ τ(ε):
‖U(t)‖W 2,q(Qε) ≤ 3K0ε−α (5)
and ∥∥∥∇(U − U∗0) (t)∥∥∥
L∞(Qε)
+
∥∥∥∥
(
∂U∗
∂x3
− U∗1
)
(t)
∥∥∥∥
L∞(Qε)
+
∥∥∥∥∂U3∂x3 (t)
∥∥∥∥
L∞(Qε)
≤ C1εδ. (6)
The proof of Theorem 1 relies on a contradiction argument, on a priori estimates involving
the quantity ‖∇U(t)‖L∞(Qε) similar to those of [BeKaMa], [Fe], [KaPo] and on the comparison
of the vector ﬁelds MU(t) with U∗0(t) and M(
∂U
∂x3
)(t) with U∗1(t) in the L∞-norm on Ω. To
bound the terms (Id−M)U(t) and (Id−M)( ∂U∂x3 )(t) in the L∞-norm on Qε, we use Poincare´
inequalities and exploit the divergence free condition of U(t).
Remarks 1. Theorem 1 is still true if the constant K0 in the hypothesis H2 is replaced by
K0f(ε), where f(ε) is a positive function of ε, which goes very slowly to inﬁnity, as ε goes to
zero. For instance, we can choose f(ε) = log(log(log[− log ε])). The hypothesis H1 can also be
made weaker (see [MaRaRa2]).
2. If the initial data U0 are bounded in (W 3,p(Qε))3, p ≥ 2 for example, one shows that
‖U(t)‖W 3,p(Qε) satisﬁes the estimate (5) on the time interval [0, τ(ε)], by using Theorem 1 and
taking into account the terms of order 2 in the expansion (2). Of course, one obtains estimates,
similar to (6), involving U∗0(t), U
∗
1(t) and U
∗
2(t), where U
∗
2(t) is the vector ﬁeld appearing in (2)
at order two and satisfying U∗2(0) = M(
∂2U0
∂x23
) (see [MaRaRa2]). The estimate of the derivatives
of order k ≥ 3 of U(t) requires the study of the terms in the expansion (2) up to order k − 1,
which becomes rather technical, when k increases.
3. In [MaRaRa1], we have described existence and comparison results in the spherical case, when
U0 is in (H3(Qε))3. To prove the existence of the solution U(t) on [0, τ(ε)], we took into account
the expansion (2) up to order two. [MaRaRa2] simpliﬁes this part of the proof since now one
only needs to work with the ﬁrst two vector ﬁelds U∗0 and U
∗
1.
4. Similar results to those of Theorem 1 hold if Qε = {(x1, x2, x3) | (x1, x2) ∈ Ω , 0 < x3 <
εg(x1, x2)}, where g(x1, x2) is a positive function, periodic in x1 and x2 of period l1 and l2
respectively. This case is studied in [Ol], who proved existence of a solution U(t) on long time
intervals and compared U(t) with a solution of the 2-dimensional Euler equations, assuming
that the data U0 are nearly 2-dimensional in Xm,2(Qε) for m ≥ 3. In the above situation, he
had to assume that, for 0 ≤ j ≤ 2, ‖MDj(∂U0∂x3 )‖Hm(Qε) ≤ δε1/2, where δ is suﬃciently small
and showed that ‖U(t)−U∗0(t)‖Hm(Qε) ≤ Cε1/2 on some time interval. Since, in our case, δ can
be large, we need to take into account the vector ﬁeld U∗1.
5. In [MaRaRa2], Theorem 1 is generalized to the case of a thin spherical shell, that is, the
domain bounded by two spheres of radii r = 1 and r = 1 + ε. Again, we have to consider the
asymptotic expansion (2), where the variable x3 becomes ρ = r − 1. Thus, U∗0 is a solution
of the Euler equations on the sphere S2, while U∗1 is the solution of a “linearized” Euler-type
equation on S2. The geometry of Qε appears through the fact that now P0 = k(U∗0 , U∗0 ) and
P1 = k(U∗0 , U∗0 ) + 2k(U∗0 , U∗1 ), where k(·, ·) is the second fundamental form on S2.
6. In the context of rotating ﬂuids, Babin, Mahalov and Nicolaenko [BaMaNi] showed existence
of solutions of the Euler equations with an additional Coriolis term, on a time interval [0, T (R)],
where T (R) goes to inﬁnity as the Rossby number R goes to zero.
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Dispersive barotropic equations for stratiﬁed mesoscale ocean dynamics.

Equations d’Euler dans une coque sph´ erique mince,

Equations d’Euler dans une coque sphe´rique mince,

Groups of diﬀeomorphisms and the motion of an incompressible ﬂuid,

Justiﬁcation of the shallow-water limit for a rigid-lid ﬂow with bottom topography, Theoretical and Comput. Fluid Dynamics 9,

Navier-Stokes equations in thin 3D domains I: Global attractors and global regularity of solutions,

Navier-Stokes equations in three-dimensional thin domains with various boundary conditions,

Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions. Nonlinear partial diﬀerential equations and their applications,

Nonstationary ﬂows of an ideal incompressible ﬂuid.

Nonstationary ﬂows of viscous and ideal ﬂuids in R3,

On classical solutions of the two-dimensional non-stationary Euler equation,

On the blow-up of solutions of the 3-D Euler equations in a bounded domain

On the Euler equations of incompressible perfect ﬂuids

Partial diﬀerential equations on thin domains,

Reaction-diﬀusion equations on thin domains,

Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating ﬂuids.

Remarks on the breakdown of smooth solutions for the 3D Euler equations.

Some results on the Navier-Stokes equations in thin 3D domains,

The Euler equations on three-dimensional thin domains,

U¨ber einige Existenzprobleme der Hydrodynamik homogener unzusammendru¨ckbarer, reibungsloser Flu¨ssigkeiten und die Helmholtzschen Wirbelsa¨tze,

Uber einige Existenzprobleme der Hydrodynamik homogener unzusammendr¨ uckbarer, reibungsloser Fl¨ ussigkeiten und die Helmholtzschen Wirbels¨ atze,

Un th´ eor` eme sur l’existence du mouvement plan d’un ﬂuide parfait, homog` ene, incompressible, pendant un temps inﬁniment long,

Un the´ore`me sur l’existence du mouvement plan d’un fluide parfait, homoge`ne, incompressible, pendant un temps infiniment long,

https://authors.library.caltech.edu/19819/1/MaRaRa2000.pdf

The Euler Equations on Thin Domains

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