Memoirs on Differential Equations and Mathematical Physics
Volume 44, 2008, 23–44
Jamel Benameur and Ridha Selmi
ANISOTROPIC ROTATING MHD SYSTEM
IN CRITICAL ANISOTROPIC SPACES
Abstract. The three-dimensional mixed (parabolic-hyperbolic) nonlinear magnetohydrodynamic system is investigated in the whole space R3 .
1
Uniqueness is proved in the anisotropic Sobolev space H 0, 2 . Existence and
1
uniqueness are proved in the anisotropic mixed Besov–Sobolev space B 0, 2 .
Asymptotic behavior is investigated as the Rossby number goes to zero.
Energy methods, Freidrichs scheme, compactness arguments, anisotropic
Littlewood–Paley theory, dispersive methods and Strichartz inequality are
used.
2000 Mathematics Subject Classification. 35A05, 35A07, 35B40.
Key words and phrases. Mixed (parabolic-hyperbolic) M HD system,
existence, uniqueness, critical spaces, Strichartz inequality, asymptotic behavior.
!
"
!
"
#
&
!
&
&
!
'
'
0, 12
B
!
%
&
$
$
"
0, 12
%
!
R3
H
&
!
"
'
!
%
"
!
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
25
1. Introduction
This paper deals with an incompressible mixed magnetohydrodynamic
system with anisotropic diffusion with the small in the limit Rossby number.
Namely, we consider the following system denoted by (M HDνεh ):

1
1

+
3

∂t u − νh ∆h u + u · ∇u − b · ∇b + ∂3 b + u × e3 = −∇p in R × R

ε
ε


1


∂t b − νh ∆h b + u · ∇b − b · ∇u + ∂3 u = 0 in R+ × R3
ε
div u = 0 in R+ × R3





div b = 0 in R+ × R3



(u, b)
3
|t=0 = (u0 , b0 ) in R ,
where the velocity field u, the induced magnetic perturbation b and p are
unknown functions of time t and the space variable x = (x1 , x2 , x3 ) =
(xh , x3 ), e3 is the third vector of the Cartesian coordinate system and νh is
a positive constant which represents both the cinematic viscosity and the
magnetic diffusivity. ∆h denotes the horizontal Laplace operator defined by
∆h = ∂12 + ∂22 and ε is a small positive parameter destined to go to zero. It
is clear that the system is hyperbolic with respect to the direction x3 called
the vertical direction. About the physical motivations, we refer the reader
to [3] and references therein.
If we denote by U = (u, b), then U is a solution of the following abstract
system:

∂t U + a2,h (D)U + Q(U, U ) + Lε (U ) = (−∇p, 0) in R+ × R3



div u = 0 in R+ × R3
(S ε )

div b = 0 in R+ × R3



U|t=0 = U0 in R3 ,
where the quadratic term, the linear perturbation and the viscous term are
respectively defined by
u · ∇u − b · ∇b
Q(U, U ) =
,
u · ∇b − b · ∇u
1 ∂3 b + u × e 3
1
ε
L (U ) = L(U ) =
∂3 u
ε
ε
and
a2,h (D)U = −νh ∆h U.
In the isotropic case, that is, when the global Laplace operator is taken
instead of the horizontal one, some isotropic magnetohydrodynamic systems
were studied by several authors ([1], [2], [11]). However, according to our
knowledge, the first paper dealing with the anisotropic case is due to the
authors in [3]. It deals with existence, uniqueness in H 0,s for s > 12 and
asymptotic behavior of the solution as ε → 0 for the same (M HDνεh ).
Nevertheless, in [10], the author studied the case of anisotropic pure fluid
26
J. Benameur and R. Selmi
and proved the uniqueness in the critical Sobolev space and both existence
and uniqueness in the anisotropic Besov–Sobolev space.
In this paper, we extend those results to the rotating (M HDνεh ) system,
which presents the difficulty to be coupled in a nonlinear way. In addition,
as the considered system is a perturbed one, it is quite natural to ask about
the asymptotic behavior of the solution as the Rossby number ε tends to
zero. We note that this perturbation presents the difficulty of being singu1
1
lar. Precisely, we establish uniqueness results in H 0, 2 (R3 ) and B 0, 2 (R3 ),
uniform local existence for arbitrary initial data and global existence for
1
small initial data in B 0, 2 (R3 ). Moreover, we establish a convergence result
as ε → 0.
1
Let us first say that H 0, 2 (R3 ) is the space of regularity L2 (R2 ) in xh
1
1
1
2
(R) in x3 . As
and H 2 (R) in x3 , and B 0, 2 (R3 ) is also L2 (R2 ) in xh but B2,1
1
3
2
in [7], for H (R ) in the case of Navier–Stokes equation, we say here that
1
1
H 0, 2 and B 0, 2 are critical spaces for the system (S ε ). This means that they
are invariant by the following scaling of (S ε ): if U (t, x) is a solution of (S ε )
with the data U0 (x), then Uλ (t, x) = λU (λ2 t, λx) is also a solution of (S ε )
with the data λU0 (λx).
The main idea is to use the structure of the convection operator together
with the incompressibility condition to compensate the lack of information
due to the incomplete diffusion operator that describes the anisotropy effect. This very fine analysis is performed with the help of Littlewood–Paley
1
decomposition in order to deal with scale invariant spaces such as H 0, 2
1
and B 0, 2 .
1
The uniqueness result in the anisotropic homogenous Sobolev space H 0, 2
is dealt with by the following theorem:
Theorem 1. The system (M HDνεh ) has at most one solution U ε such
1
0, 12
(R3 )) and ∇h U ε ∈ L2T (H 0, 2 (R3 )).
that U ε ∈ L∞
T (H
The proof of this theorem is partially based on a technical lemma inspired
from [10] and adapted here for the case of (M HDνεh ). By this lemma, we
1
establish a doubly logarithmic estimate for the H 0,− 2 norm of W ε , the
difference of two solutions, and we use Osgood lemma to finish the proof.
1
1
Though W ε belongs to H 0, 2 , it will be estimated in H 0,− 2 . This is due to
the fact that the equation satisfied by W ε is hyperbolic in the variable x3 .
1
As in [10], we are not able to establish existence in H 0, 2 but only unique1
2
ness. This is due to the noninclusion of H 0, 2 in L∞
v (Lh ). Such inclusion
3
0, 12
holds for B (R ) and plays an essential role in proving the existence result.
In order to state such result, we introduce Besov type spaces that take into
account Lebesgue regularity in time on the dyadic blocs. These spaces are
fp (B 0, 12 ) and defined as in [6] by
denoted for p ≥ 1 by L
T
X q
2 2 k∆vq ukLp([0,T ];L2 (R3 )) .
kuk fp 0, 21 =
LT (B
)
q∈Z
27
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
1
In the case of critical anisotropic Besov–Sobolev space B 0, 2 , existence and
uniqueness results are given by the following theorem:
1
Theorem 2. Let U0 = (u0 , b0 ) ∈ B 0, 2 (R3 ) be a divergence free vector
fields. There exists a positive time T such that for all ε > 0 there exists a
∞ (B 0, 21 (R3 )) with ∇ U ε ∈
g
unique solution U ε of (M HDνεh ), where U ε ∈ L
h
T
f2 (B 0, 12 (R3 )) and satisfies the following energy estimate:
L
T
kU ε k g
∞
LT
1
(B0, 2
(R3 ))
+
√
ν h k∇h U ε k f2
LT
1
(B0, 2
(R3 ))
≤
√
2kU0 k
1
B0, 2
.
(1)
Moreover, if the maximal time of existence T ∗ is finite, then
kU ε k g
∞
1
LT ∗ (B0, 2 (R3 ))
= +∞,
(2)
and if there exists a constant c such that kU0 k 0, 12 3 ≤ cνh , then the
(R )
B
solution is global.
We use Friedrichs’s scheme to prove global in time existence result in
1
L2 (R3 ). To establish global in time existence result in B 0, 2 (R3 ), we use
again Friedrichs’s scheme and Littlewood–Paley theory. A suitable rearrangement of the nonlinear term allows to apply a technical lemma due
to [10]. Then, absorption techniques yield an estimate of the approximate
solution. Using standard compactness argument, we finish the proof. To
1
prove local in time existence result in B 0, 2 (R3 ), we decompose the initial
data into low and high frequency parts. The low frequency part will be the
initial data of a linear problem and the high one will be the initial data of
the remainder which is nonlinear. For the former, classical arguments give
explicitly the result. For the latter, Littelwood–Paley theory, and especially
Bony decomposition, plays a crucial role for estimation of the nonlinear
part. The target is to establish an estimate where the norm of the solution
will be bounded by an expression that depends on the life span T of the
solution and the high frequency part of the initial data. Thus, since we are
looking for a local in time result, we can choose, in the appropriate order,
the cut-off integer N as big as needed and T as small as needed. This is the
idea behind the frequency decomposition of the initial data. We note that
the uniform life span of the solution will not depend, as usual, on the norm
of the initial data but only on its frequency repartition.
Concerning the asymptotic behavior of the solution as the Rossby number
ε tends to zero, we prove the following convergence result:
1
Theorem 3. Let U0 = (u0 , b0 ) ∈ B 0, 2 (R3 ) ∩ L2 (R3 ) be a divergence-free
vector field and (U ε ) the family of solutions given by Theorem 2. If we set
e ε = U ε − U ε , then
for χ ∈ D(R) URε = χ( |∇Rh | )U ε and U
R
R
1
1. ∀ α ∈ ]0, 4 [ , ε > 0 and R > 0 there exists Nα (ε, R) ∈ N such that
Nα (ε, R)
−−→
ε→0
+∞
28
J. Benameur and R. Selmi
and
sup
|p|≤Nα (ε,R)
2. ∀ η > 0,
k∆vq URε kL4T (L∞ ) = o(εα ), ε → 0.
eRε lim sup |∇h |1−η U
ε→0
−−−−−→
1
L2T (B0, 2 ) R→+∞
0.
The proof uses a Strichartz inequality and Fourier analysis. In fact,
dispersive effects are of great importance in the study of nonlinear partial
differential equations, since they yield decay estimates on waves in R3 .
The structure of this paper is as follows. The next section is devoted
to introduction of anisotropic Lebesgue spaces and anisotropic Littlewood–
Paley theory. In the third section, we prove Theorem 1. The fourth section
deals with the proof of Theorem 2. In the last section, we prove Theorem 3.
2. Notation and Technical Lemmas
2.1. Anisotropic Lebesgue spaces. Let us define anisotropic Lebesgue
spaces and recall some of their properties which are useful in the sequel.
Definition 1. We define Lph (Lrv ) to be the space Lp (Rx1 × Rx2 ; Lr (Rx3 ))
endowed with the norm
kf kLph (Lrv ) = kf (xh , ·)kLr (Rx3 ) Lp (Rx ×Rx ) .
1
Similarly,
norm
Lrv (Lph )
2
is the space Lr (Rx3 ; Lp (Rx1 × Rx2 )) endowed with the
kf kLrv (Lph ) = kf (·, x3 )kLp (Rx1 ×Rx2 ) Lr (R
x3 )
.
In the frame of anisotropic Lebesgue spaces, the Hölder inequality reads
kf gkLrv (Lph ) ≤ kf kLr0 (Lp0 ) kgkLr00 (Lp00 ) ,
v
h
v
h
where 1r = r10 + r100 and p1 = p10 + p100 .
Young’s convolution inequality takes the following form:
kf ? gkLrv (Lph ) ≤ kf kLr0 (Lp0 ) kgkLr00 (Lp00 ) ,
v
h
v
h
where 1 + r1 = r10 + r100 and 1 + p1 = p10 + p100 .
The following lemma will be useful in the sequel
Lemma 1. Let 1 ≤ p ≤ q and f : X1 × X2 → R be a function belonging
to Lp (X1 ; Lq (X2 )), where (X1 ; dµ1 ) and (X2 ; dµ2 ) are measurable spaces.
Then f ∈ Lq (X2 ; Lp (X1 )) and
kf kLq (X2 ;Lp (X1 )) ≤ kf kLp(X1 ;Lq (X2 )) .
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
29
2.2. Anisotropic Littlewood–Paley theory. The basic idea of Littlewood–Paley theory consists in a localization procedure in the frequency
space. The powerful point of this theory is that the derivatives and, more
generally, the Fourier multipliers act on distributions whose Fourier transform is supported in a ball or a ring in a very special way that we will return on. Such theory in its anisotropic form allows to introduce anisotropic
Sobolev and Besov spaces. To do so, we use an anisotropic dyadic decomposition of the frequency space. We begin by defining for any function a
the following operators of localization:
∆hj a = F −1 (ϕ(2−j |ξh |)F(a)) for j ∈ Z,
∆vq a = F −1 (ϑ(2−q |ξ3 |)F(a)) for q ∈ N,
∆v−1 a = F −1 (ϑ(|ξ3 |)F(a))
and
∆vq a = 0 for q ≤ −2.
The functions ϕ and ϑ represent a dyadic partition of unity in R; they are
regular non-negative functions and satisfy supp (ϑ) ⊂ B(0, 43 ), supp (ϕ) ⊂
C(0, 34 , 38 ). Moreover, for all t ∈ R,
X
ϑ(t) +
ϕ(2−q t) = 1.
q≥0
Furthermore, we define the operators Sqv and Sjh by
X
Sqv u =
∆vq0 u
q 0 ≤q−1
and
Sjh u =
X
∆hj0 u.
j 0 ≤j−1
In this way, we are considering a homogeneous decomposition in the horizontal variable and an inhomogeneous one in the vertical one. We define
respectively the corresponding Sobolev space and the mixed Besov–Sobolev
space by the following definitions:
Definition 2. Let s and s0 be two real numbers such that s < 1, u be a
tempered distribution and
X
12
0
kukH s,s0 =
22(js+qs ) k∆hj ∆vq uk2L2 .
j,q
The space H
s,s0
3
(R ) is the closure of D(R3 ) in the above semi-norm.
1
Definition 3. The anisotropic Besov space B 0, 2 is the closure of D(R3 )
in the following norm
12
Z
XZ
2
|ξ3 | |Fu(ξ)| dξ .
kuk 0, 12 =
B
q∈Z
ξh 2q−1 ≤|ξ3 |≤2q
30
J. Benameur and R. Selmi
The above norm is equivalent to the one defined by
X
2q/2 k∆vq ukL2 (R3 ) .
kuk 0, 21 =
B
q∈Z
The interest of this decomposition resides in the fact that any vertical derivative of a function localized in vertical frequencies of size 2q acts as a
multiplication by 2q .
The following lemma is an anisotropic Bernstein type inequality (see [10]).
Lemma 2. Let u be a function such that supp (F v u) ⊂ R2h × 2q C, where
C is a dyadic ring. Let p ≥ 1 and r ≥ r 0 ≥ 1 be real numbers. The following
holds:
2qk C −k kukLph (Lrv ) ≤ k∂xk3 ukLph (Lrv ) ≤ 2qk C k kukLph(Lrv ) ,
qk
2 C
−k
kukLrv (Lph ) ≤
k∂xk3 ukLrv (Lph )
kukLph (Lrv ) ≤ C2
q( r10
− 1r )
qk
k
≤ 2 C kukLrv (Lph ) ,
kukLp (Lrv0 )
(3)
(4)
(5)
h
and
1
1
kukLrv (Lph ) ≤ C2q( r0 − r ) kukLrv0 (Lp ) .
h
(6)
It is well known that the dyadic decomposition is useful to define the
product of two distributions. That is,
X
uv =
∆vq u · ∆vq0 v = Tv u + Tu v + R(u, v),
q∈Z,q 0 ∈Z
where
Tv u =
X
q 0 ≤q−2
Tu v =
X
q 0 ≤q−2
∆uq0 v · ∆vq u =
∆vq0 u · ∆vq v =
and
R(u, v) =
X
q
X
i∈{0,±1}
X
q
X
q
v
Sq−1
v · ∆vq u,
v
Sq−1
u · ∆vq v
∆vq u · ∆vq+i v.
The two first sums are said to be the paraproducts and the third sum is the
remainder. This is known to be the Bony’s decomposition in the vertical
variable (see [4], [5], [9]). In this framework, we have the following properties
∆vq (Sqv0 −1 u · ∆vq0 v) = 0 if |q − q 0 | ≥ 5
and
∆vq (Sqv0 +1 u · ∆vq0 v) = 0 if q 0 ≤ q − 4.
For the sake of simplification, we will denote by (aq ), P
(bq ) and (cq )Pgeneric
√
positive sequences (depending possibly on t) such that
aq ≤ 1,
bq ≤ 1
q∈Z
q∈Z
P 2
cq ≤ 1.
and
q∈Z
31
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
Notice that u belongs to H 0,s (R3 ) if and only if
k∆vq ukL2 ≤ C2−qs cq kukH 0,s ,
and that u belongs to B
0, 21
(7)
(R3 ) if and only if
k∆vq ukL2 ≤ C2−q/2 bq kuk
1
B0, 2
.
(8)
In the sequel, it will be useful to introduce mixed Besov–Sobolev type spaces
that take into account Lebesgue regularity in time on the dyadic blocs.
fp (B 0, 21 ) and defined as in [6]
Those spaces will be denoted, for p ≥ 1, by L
T
by
X
kuk fp 0, 21 =
2q/2 k∆vq ukLp([0,T ];L2 (R3 )) < +∞.
LT (B
)
q∈Z
Remark that we have
kuk
1
0,
2)
Lp
T (B
where
kukp p
1
LT (B0, 2 )
=
≤ kuk fp
1
LT (B0, 2 )
Zt
0
,
ku(t)kp 0, 1 dt.
B
2
3. Uniqueness
Following the ideas in [10], we establish Lemma 3 to prove uniqueness
1
in H 0, 2 .
Lemma 3. Let U = (u, b) and V = (v, c) be two divergence free vector
1
0, 12
), such that ∇h U and ∇h V in L2T (H 0, 2 ).
fields, which belong to L∞
T (H
1
0, 12
Let W = (w, β) ∈ L∞
) with ∇h W ∈ L2T (H 0, 2 ) be a solution of
T (H

1


∂t W + νh ∆h W + Q(W, W + 2U ) + ε L(W ) = (−∇p, 0)
div w = div β = 0


 W t=0 = (0, 0).
For all 0 < t < T , if kW k
1
H 0,− 2
≤ 1e , then
d
kW k2 0,− 1 ≤ Cf (t)kW k2 0,− 1 1−ln kW k2 0,− 1 ln 1−ln kW k2 0,− 1 ,
2
2
2
2
H
H
H
H
dt
where f is a time-locally integrable function defined by
f (t) = 1 + 2kU k2 0, 1 + 2kV k2 0, 1 1 + 2k∇h U k2 0, 1 + 2k∇h V k2 0, 1 .
H
2
H
2
H
2
H
Proof. We begin by noting that Q(W, W + 2U ) is explicitly given by
u · ∇w + w · ∇v − b · ∇β − β · ∇c
Q(W, W + 2U ) =
.
u · ∇β + w · ∇c − β · ∇u − c · ∇w
2
32
J. Benameur and R. Selmi
1
If we take the scalar product in H 0,− 2 , we obtain
X
1 d
kW k2 0,− 1 +νh k∇h W k2 0,− 1 ≤
2−q (∆vq Q(W, W + 2U )|∆vq W )L2 ,
2
2
H
H
2 dt
q≥−1
where
∆vq Q(W, W + 2U )|∆vq W
=
−
+
=
L2
v
∆q (u · ∇w)|∆uq w L2 + ∆vq (w · ∇v)|∆vq w L2 −
∆vq (b · ∇β)|∆vq w L2 − ∆vq (β · ∇c)|∆vq w L2 +
∆vq (u · ∇β)|∆vq β L2 + ∆vq (w · ∇c)|∆vq β L2 −
− ∆vq (β · ∇u)|∆vq β L2 − ∆vq (c · ∇w)|∆vq β L2 .
We mention that the above nonlinearities are of two types: those where two
variables are the same, for example (∆vq (u · ∇w)|∆uq w)L2 , and those where
the three variables are different like (∆vq (β · ∇c)|∆vq w)L2 . The former are
estimated in [10], for the latter it suffices to note, after applying Cauchy–
Schwarz or Hölder inequality, that kwk, kβk ≤ kW k and k∇h wk, k∇h βk ≤
k∇h W k. The same holds for U and V compared to their components (u, v)
and (b, c).
We return to the proof of the uniqueness result and suppose that U ε and
V are two solutions of (S ε ) with the same initial data, such that U ε and
1
0, 21
) with ∇h U ε and ∇h V ε belonging to L2loc (H 0, 2 ).
V ε belong to L∞
loc (H
1
0,− 2
We will prove that W ε = U ε − V ε is such that W ε = 0 in L∞
) with
T (H
ε
2
0, 21
∇h W = 0 in LT (H ).
W ε satisfies the following equation
1
∂t W ε + νh ∆h W ε + Qε (W ε , W ε + 2U ε ) + Lε (W ε ) = − ∇pε , 0 .
ε
ε
Lemma 3 implies that for all 0 < t < T if kW ε k
d
kW ε k2 0,− 1 ≤
2
H
dt
≤ Cf (t)kW ε k2
1
H 0,− 2
1 − ln kW ε k2
1
H 0,− 2
1
H 0,− 2
≤ 1e , then
ln 1 − ln kW ε k2
1
H 0,− 2
where f is a locally time-integrable function defined by
f (t) = 1 + 2kU ε k2 0, 1 + 2kV ε k2 0, 1 1 + 2k∇h U ε k2 0, 1 + 2k∇h V ε k2
H
H
2
2
H
,
1
H 0, 2
2
1
.
By the Osgood lemma, one infers the uniqueness in H 0, 2 .
1
To investigate uniqueness in B 0, 2 , note that W ε will be estimated in the
norm
X
kS0v W ε (t)k2L∞ L2 +
2−q k∆vq W ε (t)k2L2 +
v
h
q≥0
33
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
+ νh supx3
Zt
0
+ νh
X
q≥0
2
−q
v
S0 ∇h W ε (τ, ·, x3 )2 2 dτ +
L
h
Zt
0
∇h ∆vq W ε (τ )2 2 dτ.
L
The exact estimations are easy to obtain, since we use functions localized
in low vertical frequencies.
4. Proof of Existence Results
1
4.1. Proof of the existence result in B 0, 2 .
4.1.1. Global existence for small initial data. For a strictly positive integer
n, we define Friedrichs’s operators by
Jn (u) = F −1 (1B(0,n) Fu(ξ)),
and
Jnv (u) = F −1 (1{ξ,|ξ3 |≤n} Fu(ξ))
v
)(u).
Jen (u) = (Jn − J1/n
Let us consider the following approximate magnetohydrodynamic system
denoted (M HDνnh )

1

∂t u − νh ∆h Jen u + Jen (Jen u · ∇Jen u) − Jen (Jen b · ∇Jen b) + (Jen u × e3 )+


ε


X

1


+ ∂3 Jen b = ∇
∆−1 ∂i ∂j Jen (Jen ui Jen uj + Jen bi Jen bj )+


ε


i,j


X

1

−1

+ ∇
∆ ∂i (∂3 Jen b − Jen u × e3 )i ,
ε
i

1 e


e
e e
e
e e
e

J
∂
b
−
ν
∆
t
h h n b + Jn (Jn u · ∇Jn b) − Jn (Jn b · ∇Jn u) + ∂3 Jn u = 0,


ε



div u = 0,





div b = 0,


 U t=0 = Jen U 0 .
The above system is an ODE that can be rewritten in the following abstract
form:
∂t U = Fn (U ),
where U = (u, b) and the expression of Fn is given by the system (M HD nνh ).
1
Note that since U 0 ∈ B 0, 2 , U (0) = Jen U 0 belongs to L2 . Moreover, Fn is
a continuous function from L2 into L2 and the Cauchy–Lipschitz theorem
implies that (M HDνnh ) has a unique local solution Unε in C 1 ([0, Tn (ε)[, L2 ).
On the other hand, the fact that Jen is a projector implies that Jen Unε is also
a solution of (M HDνnh ). By uniqueness, it follows that
Jen Unε = Unε
34
J. Benameur and R. Selmi
and Unε is a solution of the following system also denoted (M HDνnh ):

1

∂t uεn − νh ∆h uεn + Jen (uεn · ∇uεn ) − Jen (bεn · ∇bεn ) + ∂3 bεn +


ε


X

1 ε
−1
ε
ε

e

+ (un × e3 ) = ∇
Jn ∆ ∂i ∂j (ui,n uj,n + bεi,n bεj,n )+


ε


i,j



1 X −1


∆ ∂i (∂3 bεn − uεn × e3 )i ,
+ ∇
ε
(M HDνnh )
i

1

ε


∂t bn − νh ∆h bεn + Jen (uεn · ∇bεn ) − Jen (bεn · ∇uεn ) + ∂3 uεn = 0,


ε



div uεn = 0,





div bε = 0,


 ε n
Un t=0 = Jen U 0 .
The L2 energy estimate implies that
1 d
kU ε (t)k2L2 + νh k∇h Unε (t)k2L2 = 0.
2 dt n
So, one deduces the global existence in L2 .
1
To prove the existence result in B 0, 2 , we introduce the following lemma
due to [10].
Lemma 4. Let u and v be two vector fields defined on R3 such that
u(t) is divergence-free for all t ∈ [0,
]. There exists a real sequence (a q )
PT √
aq < 1 such that
satisfying aq = aq (u, v, T ) > 0 and
q∈Z
ZT
0
|(∆vq (u · ∇v)|∆vq v)L2 | dt ≤
≤ Caq 2−q k∇h uk f2
1
LT (B0, 2 )
1
2
+ kuk g
∞
LT
1
(B0, 2
)
kvk g
∞
1
LT (B0, 2 )
1
k∇h uk 2f2
LT
1
(B0, 2
1
)
k∇h vk f2
2
kvk g
∞
LT
1
LT (B0, 2 )
1
(B0, 2
)
+
3
k∇h vk 2f2
LT
1
(B0, 2
)
.
We apply the operator ∆vq and use the L2 energy estimate to obtain
1 d
∆v U ε (t)2 2 + νh ∇h ∆v U ε (t)2 2 ≤
q n
q n
L
L
2 dt
v ε
ε
v ε
≤ ∆q (un · ∇un )|∆q un L2 | + ∆vq (uεn · ∇bεn )|∆vq bεn L2 +
+ ∆vq (bεn · ∇bεn )|∆vq uεn L2 + ∆vq (bεn · ∇uεn )|∆vq bεn L2 . (9)
We note the following rearrangement:
∆vq (bεn · ∇bεn )|∆vq uεn L2 + ∆vq (bεn · ∇uεn )|∆vq bεn L2 =
−
= ∆vq (bεn · ∇(uεn + bεn ) |∆vq (uεn + bεn )
L2
− ∆vq (bεn · ∇uεn )|∆vq uεn L2 − ∆vq (bεn · ∇bεn )|∆vq bεn L2 .
(10)
35
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
Then lemma 4 leads to
2
k∆vq Unε k2L∞(L2 ) + 2νh ∇h ∆vq Unε L2 (L2 ) ≤
T
T
≤ k∆vq Un0 k2L2 + 2−q aq Cνh kUnε k g
∞
1
LT (B0, 2 )
k∇h Unε k2f2
Since a2 + b2 is equivalent to (a + b)2 , we deduce that
√
2 +
2q/2 k∆vq Unε kL∞
2νh 2q/2 k∇h ∆vq Unε kL2T (L2 ) ≤
T (L )
√
1
1√
1
2
k∇h Unε k f2
≤ 2q/2 2k∆vq Un0 kL2 + aq2 2C 2 kUnε k g
0, 1
∞
LT (B
2
1
.
1
.
LT (B0, 2 )
LT (B0, 2 )
)
We take the sum over q. Then, we reapply the same equivalence property
to infer that
kUnε k2g
∞
1
LT (B0, 2 )
+ 2νh k∇h Unε k2f2
≤
≤ 4kUn0 k2 0, 1 + CkUnε k g
∞
B
1
1
LT (B0, 2 )
1
LT (B0, 2 )
2
k∇h Unε k2f2
1
LT (B0, 2 )
.
(11)
Let U0 ∈ B 0, 2 be such that kU0 k 0, 12 < cνh , where c < C1 . Let Unε be the
B
regular solution of (M HDνnh ). If we set
n
o
Tn∗ = sup T > 0, kUnε k g
,
<
2cν
h
0, 1
∞
2
LT (B
)
then the estimate (11) implies that for all T such that 0 < T < Tn∗ ,
kUnε k g
∞
1
LT (B0, 2 )
Since the function T → kUnε k g
∞
≤ 2kU0 k
1
LT (B0, 2 )
1
B0, 2
< 2cνh .
is continuous, we obtain that Tn∗ =
+∞ and the sequence of global solutions (Unε )n∈N is such that Unε belongs to
f2 (R+ , B 0, 12 ). In particular, the facts
∞ (R+ , B 0, 21 ) and ∇ U ε belongs to L
g
L
h n
v ε
∞ 2
that S0 Un belongs to Lv Lh and (I − S0v )Unε belongs to L2 allow to deduce
that (Unε )n∈N is bounded in L∞ (R+ , L2loc ). By the system (M HDνnh ), one
−N
+
has that (∂t Unε )n∈N is bounded in L∞
loc (R , Hloc ), where N is a sufficiently
large integer. By Arzela–Ascoli theorem, there exists a subsequence denoted
also by (Unε )n such that
−N
+
Unε → U ε in L∞
loc (R , Hloc ).
Since (Unε )n is bounded in L∞ (R+ , L2loc ), an interpolation inequality implies
that
−σ
+
Unε → U ε in L∞
loc (R , Hloc ) ∀ σ > 0.
ρ
), classical product laws in
Since ∀ ρ < 21 , (Unε )n is bounded in L2loc (R+ , Hloc
Sobolev spaces imply that for σ < ρ
ρ−σ−3/2
Q(Unε , Unε ) → Q(U ε , U ε ) in L2loc (R+ , Hloc
In particular,
Q(Unε , Unε ) → Q(U ε , U ε ) in D0 .
).
36
J. Benameur and R. Selmi
Taking the limit in (M HDνnh ), we obtain the global solution for small initial
data.
4.1.2. Continuity in time of the solution. The equation verified by ∆vq U is
∂t ∆vq U ε = νh ∆h ∆vq U ε − ∆vq Qh (U ε , U ε ) − ∆vq Q3 (U ε , U ε ) − (∆vq ∇pε , 0).
The divergence-free condition implies that
divh (uε ⊗ uε ) − divh (bε ⊗ bε )
Qh (U ε , U ε ) =
divh (uε ⊗ bε ) − divh (bε ⊗ uε )
and
Q3 (U ε , U ε ) =
∂3 (uε ⊗ uε ) − ∂3 (bε ⊗ bε )
.
∂3 (uε ⊗ bε ) − ∂3 (bε ⊗ uε )
Note that νh ∆h ∆vq U ε − ∆vq Qh (U ε , U ε ) belongs to L2 ([0, T ], L2v (Ḣh−1 )) and
∆vq U ε belongs to L2 ([0, T ], L2v (Ḣh1 )) to deduce that
νh ∆h ∆vq U ε − ∆vq Qh (U ε , U ε )|∆vq U ε L2 (R3 ) ∈ L1 ([0, T ]).
(12)
Moreover, by the fact that U ε belongs to L2 ([0, T ], L4hL2v ) we have that
∆vq Q3 (U ε , U ε ) belongs to L1 ([0, T ], L2h L1v ). Furthermore, the fact that U ε
belongs to L∞ ([0, T ], L2h(L∞
v )) leads to
∆vq Q3 (U ε , U ε )|∆vq U ε L2 (R3 ) ∈ L1 ([0, T ]).
(13)
By (12) and (13), one obtains that ∂t k∆vq U ε k2L2 belongs to L1T and, in
particular for fixed q, k∆vq U ε k2L2 ∈ C([0, T ]). On the other hand, note that
t → ∆vq U ε (t) is weakly continuous. Finally, we obtain that t → ∆vq U ε (t) is
strongly continuous on [0, T ] with values in L2 .
∞ (B 0, 21 ), for ζ > 0 there exists N such that
g
Since U ε belongs to L
T
X
2 ≤ ζ.
2q/2 k∆vq U ε kL∞
T (L )
|q|≥N
Since ∆vq belongs to CT (L2 ), there exists δ > 0 such that for |t − t0 | ≤ δ one
has
X
2q/2 ∆vq (U ε (t) − U ε (t0 ))L2 ≤ ζ.
|q|≤N
0
Thus, for |t − t | ≤ δ one obtains that
v ε
∆q (U (t) − U ε (t0 ))
1
B0, 2
≤ ζ.
∞ (B 0, 21 ) be a solution of (M HD ε ). Then
g
Proposition 1. Let U ε ∈ L
νh
T
1
1. U ε ∈ Cb ([0, T ), B 0, 2 ).
1
2. The set {U ε (t), t ∈ [0, T ]} is relatively compact in B 0, 2 .
37
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
4.1.3. Local existence result for arbitrary initial data. To prove the local
existence result for arbitrary initial data, we decompose the initial data
into low and high frequency. Then we look for a local solution in the form
U = U1,N + U2,N ,
where U1,N is the solution of the linear problem below:

1


∂t V − νh ∆h V + ε L(V ) = (−∇p, 0),
div V = 0,


 V t=0 = SN U0 .
It follows that for all t
kU1,N (t)k2L2
+ 2νh
Zt
0
∇h U1,N (τ )2 2 dτ = kSN U0 k2 2 .
L
L
(14)
ε
In this way, U ε will be a solution of (M HDνεh ) if U2,N
is a solution of the
system below:

1
ε
ε

, W ) + Q(W, U1,N
)=
∂t W − νh ∆h W + Q(W, W ) + L(W ) + Q(U1,N



ε

ε
ε
= (−∇p, 0) − Q(U1,N , U1,N ),


div W = 0,


 W t=0 = (I − SN )U0 .
We use the Freidrichs scheme. Thus, the L2 energy estimate leads to
1 d
ε
ε
(t)k2L2 ≤
k∆vq U2,N
(t)k2L2 + νh k∇h ∆vq U2,N
2 dt
v
ε
ε
ε
ε
ε
ε
+
+ ∆q Q(U1,N
∆vq Q(U2,N
, U2,N
)|∆vq U2,N
, U2,N
)|∆vq U2,N
L2
L2
v
ε
ε
ε
+
+ ∆q Q(U2,N
, U1,N
)|∆vq U2,N
L2
v
(15)
+ ∆ Q(U ε , U ε )|∆v U ε
2 .
q
1,N
1,N
By Lemma 4, we can estimate directly the sums
Zt
0
and
Zt
0
v
ε
ε
ε
∆q Q(U2,N
dτ
, U2,N
)|∆vq U2,N
L2
v
ε
ε
ε
∆q Q(U1,N
dτ.
, U2,N
)|∆vq U2,N
L2
About the other terms, we note that
Zt
0
v
ε
ε
ε
≤
∆q Q(U2,N
, U1,N
)|∆vq U2,N
L2
q
2,N L
38
J. Benameur and R. Selmi
2 +
≤ ∆vq (uε2,N · ∇uε1,N )L1 (L2 ) k∆vq uε2,N kL∞
T (L )
T
v ε
2 +
+ ∆q (b2,N · ∇bε1,N )L1 (L2 ) k∆vq uε2,N kL∞
T (L )
T
v ε
ε
v
ε
2 +
+ ∆q (u2,N · ∇b1,N )L1 (L2 ) k∆q b2,N kL∞
T (L )
v εT
2 .
+ ∆q (b2,N · ∇uε1,N )L1 (L2 ) k∆vq bε2,N kL∞
T (L )
T
Then it follows that
Zt
0
v
ε
ε
ε
≤
∆q Q(U2,N
, U1,N
)|∆vq U2,N
L2
≤ 2−q/2 bq CT ∆vq (uε2,N · ∇uε1,N )L∞ (L2 ) kuε2,N k g
0, 1 +
2)
L∞
T
T (B
ε
ε
ε
−q/2
v
+2
bq CT ∆q (b2,N · ∇b1,N )L∞ (L2 ) ku2,N k g
0, 1 +
2)
L∞
T
T (B
+ 2−q/2 bq CT ∆vq (uε2,N · ∇bε1,N )L∞ (L2 ) kbε2,N k g
0, 1 +
2)
L∞
T
T (B
−q/2
v ε
ε
ε
+2
bq CT ∆q (b2,N · ∇u1,N ) L∞ (L2 ) kb2,N k g
∞
T
1
.
1
,
LT (B0, 2 )
Using the vertical Bony decomposition, one can write that
X
Sqv0 −1 uε2,N · ∆vq0 ∇uε1,N +
∆vq (uε2,N · ∇uε1,N ) = ∆vq
+∆vq
|q 0 −q|≤4
X
q 0 ≥q−N0
Sqv0 +1 ∇uε1,N · ∆vq0 uε2,N .
This leads to
v ε
∆q (u2,N · ∇uε1,N ) ∞ 2 ≤
LT (L )
X
v
ε
2 k∆ 0 ∇u
∞ +
≤
kSqv0 −1 uε2,N kL∞
q
1,N kL∞
T (L )
T (L )
|q 0 −q|≤4
X
+
q 0 ≥q−N
≤2
5N/2
0
v ε
∞ k∆ 0 u
2 ≤
kSqv0 +1 ∇uε1,N kL∞
q 2,N kL∞
T (L )
T (L )
kuε2,N k
+2
where beq =
P
q 0 ≥q−N
2(q−q
0
0
X
1
0,
2)
L∞
T (B
)/2
5N/2
|q 0 −q|≤4
2 +
k∆vq0 uε1,N kL∞
T (L )
−q/2 e
2 2
kuε1,N kL∞
bq kuε2,N k
T (L )
0,
2)
L∞
T (B
bq0 . beq belongs to l1 since bq0 does. Then,
4
applying the inequality ab ≤ 41 a4 + 43 b 3 , we deduce that
kuε1,N k
1
0,
2)
L∞
T (B
≤ ku0 k
1
B0, 2
.
39
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
The same holds for ∆vq (bε2,N ·∇bε1,N ), ∆vq (uε2,N ·∇bε1,N ) and ∆vq (bε2,N ·∇uε1,N ).
So, it follows that
v
ε
ε
ε
∆q Q(U2,N
k
) ∞ 2 ≤ 2−q/2 bq C25N/2 kU2,N
, U1,N
0, 1 .
LT (L )
L∞
T (B
2
)
Thus,
Zt
v
v ε
ε
ε
ε
∆q U2,N )L2 | ≤ 2−q aq C25N/2 T kU2,N
∆q Q(U2,N
k2
, U1,N
1
0,
2)
L∞
T (B
. (16)
0
To estimate
Rt
0
Zt
0
ε
ε
ε
|(∆vq Q(U1,N
, U1,N
)|∆vq U2,N
)L2 | dt, we proceed as follows:
v
ε
ε
ε
∆q Q(U1,N
≤
, U1,N
)|∆vq U2,N
L2
ε
ε
ε
2 ≤
k L∞
))L∞ (L2 ) k∆vq U2,N
, U1,N
≤ C2N T ∆vq (Q(U1,N
T (L )
T
ε
ε
∞ kU
k L∞
≤ C2N T 2−q/2 bq kU1,N
1,N k g
∞
T (L )
1
LT (B0, 2 )
ε
2 .
k L∞
≤ C2−q/2 bq 25N/2 T k∆vq U2,N
T (L )
This leads to
Zt
0
ε
2 ≤
k L∞
k∆vq U2,N
T (L )
v
ε
ε
ε
dt ≤
∆q Q(U1,N
, U1,N
)|∆vq U2,N
L2
ε
≤ C2−q aq 25N/2 T kU2,N
k2 ∞
1
LT (B0, 2 )
.
(17)
.
(18)
Finally, we integrate (15) and take the sum over q to obtain
ε
kU2,N
k2g
∞
1
LT (B0, 2 )
ε
+ νh k∇h U2,N
k2f2
≤
1
LT (B0, 2 )
ε
ε
≤ 4kU2,N
(0)k2 0, 1 + Ck∇h U2,N
k2f2
B
1
LT (B0, 2 )
2
+ CT (1 + 22N + 25N/2 )kU0 k
ε
kU2,N
kg
∞
1
B0, 2
Let N be a sufficiently big integer such that
ε
kU2,N
k2g
∞
νh
,
2
where c is sufficiently small. We consider T such that
1
CT 25N/2kU0 k 0, 21 ≤
B
2
and
ε
≤ cνh
kU2,N
kg
0, 1
∞
2
ε
kU2,N
(0)k
≤c
1
B0, 2
)
LT (B
to obtain by (18)
ε
kU2,N
kg
∞
1
LT (B0, 2 )
≤
cνh
.
2
1
LT (B0, 2 )
+
1
LT (B0, 2 )
40
J. Benameur and R. Selmi
ε
∞ (B 0, 21 ) for all ε > 0.
g
Then there exists T = T (N, U0 ) such that U2,N
∈L
T
To prove (2), we proceed by contraposition. To do so, we begin by noting
1
that Proposition 1 allows to deduce that U ε (tn ) has only one limit in B 0, 2
∗
as tn tends to T . One solves locally in time the (M HDνh ), where the
initial data is chosen to be U ε (T ∗ ) = limt→T ∗ U ε (t). By this way, the life
span of the solution can be extended and will be defined on [0, T ∗ + δ) and
so on. This contradicts the fact that T ∗ is finite.
Remark 1. The life span of the local solution depends only on the frequency repartition of the initial data and is not a function, as in classical
cases, of its norm kU0 k 0, 21 .
B
5. Proof of Convergence Results
The “linearized” equation associated to the system (S ε ) is

1

∂t U ε − νh ∆h U ε + L(U ε ) = 0 in Rt × R3x ,



ε

ε
in Rt × R3x ,
(LS ε ) div u = 0

div bε = 0
in Rt × R3x ,


 ε
U|t=0 = U0 (x)
in R3x .
In Fourier variables ξ ∈ R3 we obtain
1
∂t F(U ε ) + νh |ξh |2 F(U ε ) + A(ξ)F(U ε ) = 0.
ε
Hence, we are led to study the following family of operators
Z
a(ξ)
2
G ε : f 7−→
f (y)e−t(νh |ξh | +i ε )+i(x−y).ξ dξ dy,
R3y ×R3ξ
where the phase function a is such that
p
p
ξ3
ξ3
2
2
a(ξ) ∈ ± (1 + 1 + 4|ξ| ), ± (1 − 1 + 4|ξ| ) .
|ξ|
|ξ|
So it is almost stationary when ξ3 is almost equal to 0 as well as when |ξ|
goes to +∞.
For some 0 < r < min(R, R0 ), we define the domain Cr,R,R0 by
n
o
Cr,R,R0 = ξ ∈ R3 ; R0 ≥ |ξ3 | ≥ r and |ξh | ≤ R .
(19)
We consider a cut-off function ψ which is radial with respect to ξh and whose
value is 1 near Cr,R,R0 . First, to study the case where F(f ) is supported in
Cr,R,R0 , we write
G ε f (t, x) = K(t/ε, νt, ·) ? f (x),
where the kernel K is defined by
Z
2
K(t, τ, z) = ψ(ξ)eita(ξ)+iz.ξ−τ |ξh | dξ.
R3
41
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
As in [5], we recall the following property of K.
Lemma 5. For all r, R and R0 satisfying 0 < r < min(R, R0 ), there
exists a constant C(r, R, R0 ), such that
1
kK(t, τ, ·)kL∞(R3 ) ≤ C(r, R, R0 ) min 1, t− 2 .
Proof. The proof follows the lines of the stationary phase method. Using
the rotation invariance in ξh , we restrict ourself to the case z2 = 0. If we
denote α(ξ) = −∂ξ2 a(ξ), then
|α(ξ)| ≥ C(r, R, R0 )|ξ2 |,
where C is a strictly positive constant depending only on r, R and R 0 .
Next, for all ξ that belongs to Cr,R,R0 , we introduce the differential operator
L defined by
1
1 + iα(ξ)∂ξ2 .
L=
2
1 + tα (ξ)
This operator acts on the ξ2 variables and satisfies L(eita ) = eita . Integrating by parts, we obtain
Z
2
K(t, τ, z) = t L ψ(ξ)e−τ |ξh | eita(ξ)+iz.ξ dξ.
R3
Direct computation yields
t
2
L(ψ(ξ)e−τ |ξh | ) =
2
1
1 − tα2 ψ(ξ)e−τ |ξh | −
− i(∂ξ2 α)
2
1 + tα
(1 + tα2 )2
2
iα
∂ξ2 (ψ(ξ)e−τ |ξh | ).
−
2
1 + tα (ξ)
Finally, we use the fact that ξ is in a fixed annulus of R3 and ψ ∈ D(R3 ) to
infer that
Z
dξ2
.
kK(t, τ, ·)kL∞ ≤ C(r, R, R0 )
1 + tξ22
R
Let us denote by w ε the solution of the free linear system (P LF ε ) associated with (S ε )

∂ wε − ν ∆ wε + 1 L(wε ) = f in R × R3 ,
t
h h
t
x
(P LF ε )
ε
wε (0) = w
3
in R .
0
x
Lemma 5 yields, in a standard way, the following Strichartz estimate (see [8]).
Corollary 1. For all constants r, R and R0 such that 0 < r < min(R, R0 ),
0
let Cr,R,R0 be the domain defined by (19). A constant C (r, R, R0 ) exists such
that if
supp (F(w0 )) ∪ supp (F(f )) ⊂ Cr,R,R0 ,
42
J. Benameur and R. Selmi
then the solution w ε of (P LF ε ) with the forcing term f and initial data w0
satisfies
1
kwε kL4 (R+ ,L∞ ) ≤ C 0 (r, R, R0 )ε 4 kw0 kL2 + kf kL1 (R+ ,L2 ) .
0
Notice that the constant C (r, R, R0 ) does not depend on ε.
Proof of Theorem 3. The first equation of the system (S ε ) can be rewritten
in the form
1
∂t U ε − νh ∆h U ε + L(U ε ) = F ε in Rt × R3x .
ε
Applying, consecutively, the operators χ( |∇Rh | ) and ∆vq , we obtain for all
R > 0 and q ∈ Z that
1
∂t ∆vq URε − νh ∆h ∆vq URε + L(∆vq URε ) = ∆vq FRε .
ε
By Corollary 1, we infer that
1
k∆vq URε kL4T (L∞ ) ≤ C 0 (2q+1 , R, 2q ) ε 4 k∆vq U0,R kL2 + k∆vq FRε kL1T (L2 ) .
To estimate the right hand side of the above inequality, notice that
k∆vq U0,R kL2 ≤ C(q, R).
(20)
If we denote by µ3 (C2q ,R,2q+1 ) the Lebesgue measure of the set C2q ,R,2q+1 ,
then the Plancherel formula and classical analysis imply that
12
ZT Z
F(∆v F ε )(τ, ξ)2 dξ
k∆vq FRε kL1T (L2 ) =
dτ ≤
q R
0
R3
≤ (2
q+1
+ R)
ZT 0
Z
C2q ,R,2q+1
F(U ε ⊗ U ε )(τ, ξ)2 dξ
≤ (2q+1 + R)T µ3 (C2q ,R,2q+1 )
≤ (2q+1 + R)T µ3 (C2q ,R,2q+1 )
≤ (2q+1 + R)T µ3 (C2q ,R,2q+1 )
Since the energy estimate
kU
ε
(t, ·)k2L2
+ 2νh
Zt
0
21 F(U ε ⊗ U ε )
21
21
21
dτ ≤
L∞ (L∞ )
≤
kU ε ⊗ U ε kL∞ (L1 ) ≤
kU ε k2L∞ (L2 ) .
k∇h U ε (τ, ·)k2L2 dτ ≤ kU0 k2L2
(21)
holds, it follows that
k∆vq FRε kL1T (L2 ) ≤ (2q+1 + R)T µ3 (C2q ,R,2q+1 )
By the inequalities (20) and (22), we infer that
12
1
k∆vq URε kL4T (L∞ ) ≤ (1 + T )C(q, R)ε 4 .
kU0 k2L2 .
(22)
43
Anisotropic Rotating MHD System in Critical Anisotropic Spaces
For α ∈]0, 14 [, R > 0 and ε > 0, we set
n
o
1
Nα,R (ε) = sup p ∈ Z; ∀ |q| ≤ p, (1 + T )C(q, R)ε 4 −α ≤ 1 .
It is clear that Nα,R (ε)
−−→
ε→0
+ ∞ and
sup
|q|≤Nα,R (ε)
k∆vq URε kL4T (L∞ ) ≤ εα .
(23)
Let η > 0 and R > 1. By the energy estimate (21) we have
|∇h |1−η U
eRε 2 2
LT
1
(B0, 2
)
≤R
≤ R−2η
ZT
0
−2η
ZT
0
∇ h U
eRε (t, ·)2 0, 1 dt ≤
B 2
p
∇ h U
e ε (t, ·)2 0, 1 dt ≤ R−2η 2/νh kU0 k2 1 .
0,
B 2
B
Consequently, it follows that
eε lim sup |∇h |1−η U
R
ε→0
−−−−−→
1
L2T (B0, 2 ) R→+∞
0.
The equations (23) and (24) finish the proof of Theorem 3.
2
(24)
References
1. J. Benameur, M. Ghazel and M. Majdoub, About MHD system with small parameter. Asymptot. Anal. 41 (2005), No. 1, 1–21.
2. J. Benameur, S. Ibrahim and M. Majdoub, Asymptotic study of a magnetohydrodynamic system. Differential Integral Equations 18 (2005), No. 3, 299–324.
3. J. Benameur, R. Selmi, Study of anisotropic MHD system in anisotropic Sobolev
spaces. Annales Mathématiques de la Faculté de Science de Toulouse (to appear).
4. J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations
aux derivees partielles non liéaires. (French) [Symbolic calculus and propagation of
singularities for nonlinear partial differential equations] Ann. Sci. Ecole Norm. Sup.
(4) 14 (1981), No. 2, 209–246.
5. J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Anisotropy and
dispersion in rotating fluids. Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. XIV, pp. 171-192, Paris, 1997/1998.
6. J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et
équations de Navier-Stokes. (French) [Flow of non-Lipschitz vector fields and NavierStokes equations] J. Differential Equations 121 (1995), No. 2, 314–328.
7. H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I. Arch. Rational
Mech. Anal. 16 (1964), 269–315.
8. J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation.
J. Funct. Anal. 133 (1995), No. 1, 50–68.
9. D. Iftimie, The resolution of the Navier-Stokes equations in anisotropic spaces. Rev.
Mat. Iberoamericana 15 (1999), No. 1, 1–36.
10. M-G. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques.
(French) [Anisotropic Navier-Stokes equation in critical spaces] Rev. Mat. Iberoamericana 21 (2005), No. 1, 179–235.
11. R. Selmi, Convergence results for MHD system. Int. J. Math. Math. Sci. 2006, Art.
ID 28704, 19 pp.
44
J. Benameur and R. Selmi
(Received 13.02.2008)
Authors’ addresses:
J. Benameur
Département de Mathématique
Faculté des Sciences de Bizerte
Jarzouna 7021, Bizerte
Tunisie
E-mail: jamel.BenAmeur@fsb.rnu.tn
R. Selmi
Département de Mathématiques et Informatique
Faculté des Sciences de Gabès
Cité Erriadh 6072 Zrig, Gabès
Tunisie
E-mail: Ridha.Selmi@isi.rnu.tn