Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes

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by  Ronghua Pan

GLOBAL CLASSICAL SOLUTIONS OF 3D VISCOUS MHD SYSTEM
WITHOUT MAGNETIC DIFFUSION ON PERIODIC BOXES
RONGHUAPAN,YI ZHOU,ANDYI ZHU
7
1
0
Abstract. In this paper, we study the global existence of classical solutions to the
2
three dimensional incompressible viscous magneto-hydrodynamical system without

n magnetic diffusion on periodic boxes, i.e., with periodic boundary conditions. We
a workinEuleriancoordinateandemployatime-weightedenergyestimatetoprovethe
J
global existence result, under the assumptions that the initial magnetic field is close

5 enough to an equilibrium state and theinitial data havesome symmetries.
2

]
P
A 1. Introduction
.
h The equations of viscous magnetohydrodynamics (MHD) model the motion of elec-
t
a trically conducting fluids interacting with magnetic fields. When the fluids are strongly
m collisional plasmas, or the resistivity due to collisions is extremely small, the diffusion
[ in magnetic field is often neglected [4, 8, 13]. When magnetic diffusion is missing, it

is extremely interesting to understand whether the fluid viscosity only could prevent
1
v singularity development from small smooth initial data in three dimensional physical
4 space, in view of the strongly nonlinear coupling between fluids and magnetic field.
6
Mathematically, it is also close in structure to the model of dynamics of certain com-
2
7 plex fluids, including hydrodynamics of viscoelastic fluids, c.f. [15, 16, 17, 18, 19]. To
0 this purpose, we investigate the global existence of smooth solutions to the following
.
1 initial boundary value problem
0
7 Bt+u·∇B = B ·∇u,
1 u +u·∇u−∆u+∇p = B ·∇B,
v: ∇t·u= ∇·B = 0, (1.1)
i
X u(0,x) = u (x), B(0,x) = B (x),
 0 0

r 

a with periodic boundary conditions
x ∈ [−π,π]3 = T3. (1.2)
here B = (B ,B ,B ) denotes the magnetic field, u = (u ,u ,u ) the fluid velocity,
1 2 3 1 2 3
p = q+ 1|B|2 where q denotes the scalar pressure of the fluid.
2
Extensively impressive progresses had been achieved in the past decades for MHD
systems. Indeed, according to the level of dissipations, there are roughly three different
layers of models: inviscid and non-resistive (no viscosity, no magnetic diffusion, hence
no dissipation); viscous and resistive (fully dissipative in fluids and in magnetic field);
Submitted: Nov. 1, 2016.
2010 Mathematics Subject Classification. 35Q35, 76D03, 76W05.
Key words and phrases. MHD System, Eulerian Coordinate, Global Regularity.
1
2 R.PAN,Y.ZHOU,ANDY.ZHU
and partially dissipative (only viscosity or magnetic diffusion presents). On one hand,
it is natural to expect global existence of classical solutions for viscous and resistive
MHD at least for small initial data, this has been confirmed in classical papers by
Duvaut and Lions [9] and by Sermange and Temam [22]. In 2008, Abidi and Paicu [1]
generalized these results to the inhomogeneous MHD system with initial data in the so-
called criticalspaces. Morerecently, CaoandWu[6], alsosee[7],provedtheglobalwell-
posednessforany datain H2(R2)with mixed partialviscosity and magnetic diffusionin
two dimensional MHD system. On the other hand, it is somehow striking that Bardos,
SulemandSulem[3]provedthattheinviscidandnon-resistiveMHDsystemalsoadmits
a unique global classical solution when the initial data is near a nontrivial equilibrium.
It seems that purely dispersion and some coupling of nonlinearity between fluids and
magnetic field are sufficient to maintain the regularity from initial data. Very recently,
the vanishing dissipation limit from fully dissipative MHD system to inviscid and non-
resistive MHD system has been justified by [10, 5, 24] under some structural conditions
between viscosity and magnetic diffusion coefficients. Therefore, it is not surprise why
the remaining case, partially dissipative MHD, attracts a lot of attentions in the recent
years. As documented in [6, 14], the inviscid and resistive 2D MHD system admits a
globalH1 weaksolution,buttheuniquenessofsuchsolutionwithhigherorderregularity
is still not known yet.
Inthecaseofourconsideration,namelytheincompressibleMHDsystemwithpositive
viscosityandzeroresistivity, itisstillanopenproblemwhetherornotthereexistsglobal
classical solutions even in two dimensional space for generic smooth initial data. The
maindifficultyofstudyingtheseMHDsystemslies inthenon-resistivity ofthemagnetic
equation. Someinteresting resultshave beenobtained for smallsmooth solutions. For a
closely related modelin3D, theglobal well-posednesswas established by LinandZhang
[18], and a simpler proof was offered by Lin and Zhang [19]. With certain admissible
condition for initial data, Lin, Xu and Zhang [17] established the global existence in
2D for initial data close to an nontrivial equilibrium state and the 3D case was proved
by Xu and Zhang [25]. Later, Ren, Wu, Xiang and Zhang [20] removed the restriction
in 2D case (see Zhang [26] for a simplified proof). We also refer to another proof for
2D incompressible case by Hu and Lin [11]. Hu [12] further established some results
for 2D compressible MHD system. Very recently, under Lagrangian coordinate system,
Abidi and Zhang [2] proved the global well-posedness for 3D MHD system without the
admissible restriction. While all results down the line are about Cauchy problem, an
initial boundaryvalue problem for 2D case under Eulerian coordinate in a strip domain
R×(0,1)wasdonebyRen,XiangandZhang[21]recently. Andthe3DcaseonR2×(0,1)
for both compressible and incompressible fluids was considered by Tan and Wang [23]
under Lagrangian coordinate. In the three dimensional case, these inspiring results,
along with many innovative methods and estimates, made full use of partial dissipation
offered by viscosity, dispersion of waves on unbounded domain and the structure of
Lagrangian formulation (which contains one time derivative already and helps capture
the weak dissipation). It is then natural to explore the following two questions. Is it
possible to establish global existence of small smooth solutions on bounded domain,
where the dispersion effect is limited? Can one work with Eulerian coordinate where
the system takes a simpler form with the cost of the loss of one time derivative, thus
the loss of possible time decay?
GLOBAL CLASSICAL SOLUTIONS OF 3D MHD SYSTEM ON PERIODIC BOXES 3
Our main aim in this paper is to offer answers to these questions. Indeed, we will
establish the global existence of small smooth solutions to the 3D incompressible vis-
cous magneto-hydrodynamical system without resistivity on periodic boxes, under the
assumptions that the initial magnetic field is close enough to an equilibrium state and
theinitial datahavesomesymmetrystructure. We willalso avoid theuseofLagrangian
formulation. The advantage of the Eulerian coordinate is that, if successful, it would
be neat and simple.
To fix the idea, we adopt the following notations
x = (x ,x ), ∇ = (∂ ,∂ ), B = (B ,B )⊤,
h 1 2 h 1 2 h 1 2
and similar notations for other quantities without causing further confusions.
We assume that
u (x), B (x) are even periodic with respect to x ,
0,h 0,3 3
(1.3)
u (x), B (x) are odd periodic with respect to x ,
0,3 0,h 3
moreover
u dx = 0, B dx = α6= 0. (1.4)
Z 0 Z 0,3
T3 T3
Our main result can be stated as follows.
Theorem 1.1. Consider the 3D MHD system (1.1)-(1.2) with initial data satisfies the
conditions (1.3)-(1.4). Then, there exists a small constant ε > 0, only depending on α
such that the system (1.1) admits a global smooth solution provided that
ku k +k∇B k ≤ ε,
0 H2s+1 0 H2s
where s ≥ 5 is an integer.
Remark 1.2. Our methods can be used to other related models. Similar result for the
compressible system will be presented in a forthcoming paper.
Without loss of generality, we assume α = (2π)3, and following Lin and Zhang [18],
we let
B = b +e ,
0 0 3
where e = (0,0,1)⊤. Hence, we have
3
b dx = u dx = 0. (1.5)
Z 0 Z 0
T3 T3
Set B = b+e , we get the system of pair (u,b) as follows
3
b +u·∇b = b·∇u+∂ u,
t 3

u +u·∇u−∆u+∇p = b·∇b+∂ b, (1.6)
 t 3

∇·u= ∇·b = 0,

with initial data 
u(0,x) = u (x), b(0,x) = b (x).
0 0
And the property of initial data (1.3) will be hold, i.e.,
u (x), b (x) are even periodic with respect to x ,
h 3 3
(1.7)
u (x), b (x) are odd periodic with respect to x .
3 h 3
4 R.PAN,Y.ZHOU,ANDY.ZHU
Also, by the periodic boundary conditions (1.5) and system (1.6), we have
b dx = u dx = 0. (1.8)
Z Z
T3 T3
Remark 1.3. The property (1.7) will hold in the time evolution. Indeed, we can define
u¯(t,x),¯b(t,x) as follows
u¯ (t,x ,x ) =u (t,x ,−x ), u¯ (t,x ,x ) = −u(t,x ,−x ),
h h 3 h h 3 3 h 3 h 3
¯b (t,x ,x ) =−b (t,x ,−x ), ¯b (t,x ,x ) = b (t,x ,−x ).
h h 3 h h 3 3 h 3 3 h 3
Then, quantitiesu¯,¯b satisfy the same system (1.6)like u,b and also have the same initial
data. Hence, by uniqueness of solution, we obtain ¯b(t,x) = b(t,x) and u¯(t,x) = u(t,x).
Therefore, we see the property (1.7) persist.
In order to prove Theorem 1.1, we only need to consider the system (1.6) instead.
In this paper, we have to face the difficulties from bounded domain and the loss
of weak dissipation without using Lagrangian formulation. These challenges will be
overcome through a carefully designed weighted energy method with the help of some
observations to the structure of the system. One of the major observations is that
the time derivative of b is essentially quadratic terms plus a derivative term in the
good direction x where dissipation kicks in. Another observation is that although the
3
bounded domain pushes us away from possible dispersion of waves, it does compensate
us with Poincar´e inequality. However, the high space dimensions, the lack of magnetic
diffusion, and the strongly coupled nonlinearity of the problem, make the mathematical
analysis very challenging. Even with our carefully designed time-weighted energies,
there are still many dedicated technical issues. One of our main obstacles is to derive
thetimedissipativeestimate tothetermb·∇bwhichbehavesmostwildlyinthesystem.
Writing b·∇b= b ·∂ b+b ·∂ b, we notice that b ·∂ b contains one good quantity ∂ b
3 3 h h 3 3 3
can be estimated relatively easily due to dissipation in x direction. Hence, we focus
3
on the term b ·∇ b containing two bad terms. To overcome this difficulty, we make
h h
full use of the condition (1.7) and Poincar´e inequality in x direction. Thus the norm
3
of b can be controlled by the norm of ∂ b . This specific choice of estimate avoids
h 3 h
the presence of interaction between two wild quantities. Such an idea actually origins
from null condition in the theory of wave equations. However, we still have to come
across other difficulties in the estimate process. For example, we can not achieve the
uniform bound of all higher order norms that we wanted. Instead, we turn to control
the growth of such norms by the energy frame we construct in the next section. More
detailed decay estimates will also be presented in Section 2.
2. Energy estimate and the proof of main result
2.1. Preliminary. In this subsection, we first introduce a usefulproposition related to
Poincar´e inequality which plays an important role in our proof to the main theorem of
this paper.
Proposition 2.1. For any function f(x) ∈ Hk+1(T3), k ∈ N satisfying the following
condition
1 π
f(x ,x ) dx = 0, ∀ x ∈ T2, (2.1)
2π Z h 3 3 h
−π
GLOBAL CLASSICAL SOLUTIONS OF 3D MHD SYSTEM ON PERIODIC BOXES 5
it holds that
kfk . k∂ fk .
Hk(T3) 3 Hk(T3)
Proof. First, we can write
k π
kfk2 = |∂αf(x ,x )|2dx dx . (2.2)
Hk(T3) Z Z h 3 3 h
X T2 −π
|α|=0
Here, α = (α ,α ,α ) is a multi-index and ∂α = ∂α1∂α2∂α3.
1 2 3 1 2 3
Notice the condition (2.1), we have, for multi-index α = (α ,α ,0):
1 2
1 π
∂αf(x ,x )dx = 0,
2π Z h 3 3
−π
Andfor multi-index α = (α ,α ,α ) whereα > 0, by theperiodicboundarycondition,
1 2 3 3
we also have
π π
∂αf(x ,x )dx = ∂α1∂α2∂α3−1f(x ,·) = 0.
Z−π h 3 3 1 2 3 h (cid:12)−π
(cid:12)
(cid:12)
Therefore, the average value of ∂αf(x ,·) in x direction over [−π,π] is zero. Hence,
h 3
applyingstandardPoincar´e inequality to ∂αf(x ,x )in the x direction, we have ∀x ∈
h 3 3 h
T2:
π π
|∂αf(x ,x )|2dx . |∂α∂ f(x ,x )|2dx .
Z h 3 3 Z 3 h 3 3
−π −π
According to the definition of kfk i.e. (2.2), we finally obtain
Hk(T3)
k π
kfk2 = |∂αf(x ,x )|2dx dx
Hk(T3) Z Z h 3 3 h
X T2 −π
|α|=0
k π
. |∂α∂ f(x ,x )|2dx dx
Z Z 3 h 3 3 h
X T2 −π
|α|=0
=k∂ fk2 .
3 Hk(T3)
(cid:3)
Now, let us introduce the energy frame that will enable us to achieve our desired
estimate. Based on our discussion in Section 1, we define some time-weighted energies
for the system (1.6). The energies below are defined on the domain R+×T3. For s ∈ N
6 R.PAN,Y.ZHOU,ANDY.ZHU
and 0< σ < 1, we set
E (t) = sup (1+τ)−σ(ku(τ)k2 +kb(τ)k2 )
0 H2s+1 H2s+1
0≤τ≤t
t
+ (1+τ)−1−σ(ku(τ)k2 +kb(τ)k2 ) dτ
Z H2s+1 H2s+1
0
t
+ (1+τ)−σ(ku(τ)k2 +k∂ b(τ)k2 )dτ,
Z H2s+2 3 H2s
0
G (t) = sup (1+τ)1−σ(k∂ u(τ)k2 +k∂ b(τ)k2 )
0 3 H2s 3 H2s
0≤τ≤t
t
+ (1+τ)1−σk∂ u(τ)k2 dτ,
Z 3 H2s+1
0
G (t) = sup (1+τ)3−σ(k∂ u(τ)k2 +k∂ b(τ)k2 )
1 3 H2s−2 3 H2s−2
0≤τ≤t
t
+ (1+τ)3−σk∂ u(τ)k2 dτ,
Z 3 H2s−1
0
E (t) = sup (1+τ)3−σku(τ)k2
1 H2s−2
0≤τ≤t
t
+ (1+τ)3−σ(ku(τ)k2 +k∂ b(τ)k2 )dτ,
Z H2s−1 3 H2s−3
0
e (t) = sup kb(τ)k2 . (2.3)
0 H2s
0≤τ≤t
In the following, we will successively derive the estimate of each energy stated above.
By (1.8) and Poincar´e inequality, we only need to consider the highest order norms in
each energy.
2.2. A priori estimate. First, we will deal with the highest order energy, i.e., E (t).
0
It shows that the highest order norm H2s+1(T3) of u(t,·) and b(t,·) will grow in the
time evolution.
Lemma 2.2. Assume that s ≥ 5 and the energies are defined as in (2.3), then we have
1/2 1/2 5/6 1/6 1/2 1/2
E (t) . E (0)+E (t)E (t)+E (t)e (t)+E (t)E (t)e (t)+E (t)e (t).
0 0 0 1 1 0 0 1 0 0 0
Proof. We divide the proof into two steps. Instead of deriving the estimate of E
0
directly, we shall first get the estimate of E (t) defined by
0,1
t
E (t), sup (1+τ)−σ(ku(τ)k2 +kb(τ)k2 )+ (1+τ)−σku(τ)k2 dτ
0,1 H2s+1 H2s+1 Z H2s+2
0≤τ≤t 0
t
+ (1+τ)−1−σ(ku(τ)k2 +kb(τ)k2 )dτ.
Z H2s+1 H2s+1
0
(2.4)
Step 1
Applying ∇2s+1 derivative on the system (1.6). Then, taking inner product with
∇2s+1b for the first equation of system (1.6) and taking inner product with ∇2s+1u for
GLOBAL CLASSICAL SOLUTIONS OF 3D MHD SYSTEM ON PERIODIC BOXES 7
the second equation of system (1.6). Adding them up and multiplying the time weight
(1+t)−σ, we get
1 d σ
(1+t)−σ(kuk2 +kbk2 )+ (1+t)−1−σ(kuk2 +kbk2 )
2dt H˙2s+1 H˙2s+1 2 H˙2s+1 H˙2s+1 (2.5)
+(1+t)−σkuk2 = I +I +I +I ,
H˙2s+2 1 2 3 4
where,
I =−(1+t)−σ u·∇∇2s+1u∇2s+1u+u·∇∇2s+1b ∇2s+1b dx
1 Z
T3
+(1+t)−σ b·∇∇2s+1b ∇2s+1u+b·∇∇2s+1u∇2s+1b dx
Z
T3
+(1+t)−σ ∇2s+1∂ u∇2s+1b+∇2s+1∂ b ∇2s+1u dx,
Z 3 3
T3
2s+1
I =−(1+t)−σ ∇ku·∇∇2s+1−ku∇2s+1u dx,
2 Z
X T3
k=1
s
I =(1+t)−σ (∇kb·∇∇2s+1−ku−∇ku·∇∇2s+1−kb)∇2s+1b dx
3 Z
X T3
k=1
2s+1
+(1+t)−σ (∇kb·∇∇2s+1−ku−∇ku·∇∇2s+1−kb)∇2s+1b dx,
Z
X T3
k=s+1
s
I =−(1+t)−σ ∇kb·∇∇2s+1−kb ∇2s+1u dx
4 Z
X T3
k=1
2s+1
−(1+t)−σ ∇kb·∇∇2s+1−kb ∇2s+1u dx.
Z
X T3
k=s+1
We shall estimate each term on the right hand side of (2.5). First, for the term I ,
1
using integration by parts and the divergence free condition, we have
I = 0. (2.6)
1
The main idea for the next estimates is that we will carefully derive the bound of
each term so that it can be controlled by the combination of energies defined in (2.3).
By H¨older inequality and Sobolev imbedding theorem, we have
|I2|.(1+t)−σkukWs+1,∞kuk2H2s+1
.(1+t)−σkuk kuk2
Hs+3 H2s+1
.(1+t)−σkukH2s−1kuk2H2s+1.
provided that s ≥ 4. Hence,
t t
Z |I2(τ)| dτ . sup (1+τ)−σkuk2H2s+1Z kukH2s−1 dτ . E0(t)E11/2(t). (2.7)
0 0≤τ≤t 0
8 R.PAN,Y.ZHOU,ANDY.ZHU
Similarly, for the first part of I (we denote the first term on the right hand as I
3 3,1
and the second term as I ), we see that
3,2
|I3,1|.(1+t)−σ(kbkWs,∞kukH2s+1kbkH2s+1 +kukWs,∞kbk2H2s+1)
.(1+t)−σ(kbk kuk kbk +kuk kbk2 )
Hs+2 H2s+1 H2s+1 Hs+2 H2s+1
.(1+t)−σ(kbkH2skukH2s+1kbkH2s+1 +kukH2s−1kbk2H2s+1).
provided that s ≥ 3. And for the second part of I , we have
3
|I3,2|.(1+t)−σ(kbk2H2s+1kukWs+1,∞ +kukH2s+1kbkWs+1,∞kbkH2s+1)
.(1+t)−σ(kbk2 kuk +kuk kbk kbk )
H2s+1 Hs+3 H2s+1 Hs+3 H2s+1
.(1+t)−σ(kbk2H2s+1kukH2s−1 +kukH2s+1kbkH2skbkH2s+1).
provided that s ≥ 4. Hence, combining I and I together and using H¨older inequal-
3,1 3,2
ity, we get
t
|I (τ)| dτ
Z 3
0
t 1 t 1
.0≤suτp≤tkbkH2s(cid:16)Z0 (1+τ)−1−σkbk2H2s+1 dτ(cid:17)2(cid:16)Z0 (1+τ)1−σkuk2H2s+1 dτ(cid:17)2 (2.8)
t
+ sup (1+τ)−σkbk2H2s+1Z kukH2s−1dτ.
0≤τ≤t 0
UsingGagliardo–NirenberginterpolationinequalityandH¨olderinequality,wecanbound
t t
1 2
(1+τ)1−σkuk2 dτ . (1+τ)3−σkuk2 3 (1+τ)−σkuk2 3 dτ
Z0 H2s+1 Z0 (cid:2) H2s−1(cid:3) (cid:2) H2s+2(cid:3) (2.9)
2/3 1/3
.E (t)E (t).
0 1
Thus, combining (2.8) with (2.9), we finally obtain the estimate of I
3
t
|I (τ)| dτ . E5/6(t)E1/6(t)e1/2(t)+E1/2(t)e (t). (2.10)
Z 3 0 1 0 1 0
0
Next, for the last term I , we use the same method as above and obtain
4
|I4| .(1+t)−σkbkH2s+1kbkWs+1,∞kukH2s+1
.(1+t)−σkbk kbk kuk ,
H2s+1 H2s H2s+1
provided that s ≥ 3. Hence,
t
5/6 1/6 1/2
|I (τ)| dτ . E (t)E (t)e (t). (2.11)
Z 4 0 1 0
0
Summing up the estimates for I ∼ I , i.e., (2.6), (2.7), (2.10) and (2.11), and
1 4
integrating (2.5) in time, we can get the estimate of E (t) which is defined in (2.4)
0,1
1/2 1/2 5/6 1/6 1/2
E (t) . E (0)+E (t)E (t)+E (t)e (t)+E (t)E (t)e (t). (2.12)
0,1 0 0 1 1 0 0 1 0
Here, we have used the Poincar´e inequality to consider the highest order norms only.
Step 2
GLOBAL CLASSICAL SOLUTIONS OF 3D MHD SYSTEM ON PERIODIC BOXES 9
Now, let us work for the remaining term in E (t). Applying ∇2s on the second
0
equation of system (1.6) and taking inner product with ∇2s∂ b, then multiplying the
3
time weight (1+t)−σ we get
(1+t)−σk∂ bk2 = I +I +I +I , (2.13)
3 H˙2s 5 6 7 8
where
I =(1+t)−σ ∇2s(u·∇u)∇2s∂ b dx−(1+t)−σ ∇2s∆u∇2s∂ b dx,
5 Z 3 Z 3
T3 T3
s
I =−(1+t)−σ ∇kb ·∇ ∇2s−kb ∇2s∂ b+∇kb ·∇ ∇2s−kb ∇2s∂ b dx
6 Z h h 3 3 3 3
X T3
k=0
2s
−(1+t)−σ ∇kb ·∇ ∇2s−kb ∇2s∂ b+∇kb ·∇ ∇2s−kb ∇2s∂ b dx,
Z h h 3 3 3 3
X T3
k=s+1
d
I = (1+t)−σ ∇2su∇2s∂ b dx+σ(1+t)−1−σ ∇2su∇2s∂ b dx,
7 dt Z 3 Z 3
T3 T3
I =(1+t)−σ ∇2s∂ u ∇2s∂ b dx.
8 Z 3 t
T3
Like the process in Step 1, we shall derive the estimate of each term on the right
hand side of (2.13). First, using H¨older inequality and Sobolev imbedding theorem, we
can bound I as follows:
5
|I | .(1+t)−σkuk kuk k∂ bk +(1+t)−σkuk k∂ bk
5 H2s+1 Hs+2 3 H2s H2s+2 3 H2s
.(1+t)−σkukH2s+1kbkH2s+1kukH2s−1 +(1+t)−σkukH2s+2k∂3bkH2s,
provided that s ≥ 3. Thus, we have
t t
|I (τ)| dτ . E (t)E1/2(t)+E1/2(t) (1+τ)−σk∂ bk2 dτ 1/2. (2.14)
Z 5 0 1 0,1 Z 3 H2s
0 (cid:0) 0 (cid:1)
Next, we turnto theestimate of I . Notice thatI is themostwild term inour proof,
6 6
due to the bad behaviour of b·∇b. Although we have already divided this term into
b ·∇ b and b ·∇ b two terms, the estimate for b ·∇ b is still nontrivial. Thanks to
h h 3 3 h h
the Proposition 2.1 we have proved in the beginning of this section, we can overcome
this problem using the following strategy.
Notice the property (1.7) we easily know that in the x direction, the average value
3
of function b (x ,·) over [−π,π] equals zero. Thus, using the Proposition 2.1, H¨older
h h
inequality and Sobolev imbedding theorem we get
|I6| .(1+t)−σ(kbhkWs,∞kbkH2s+1k∂3bkH2s +kb3kWs,∞k∂3bk2H2s)
+(1+t)−σ(kbhkH2skbkWs,∞k∂3bkH2s +kb3kH2sk∂3bkWs−1,∞k∂3bkH2s)
.(1+t)−σ(k∂ b k kbk k∂ bk +kb k k∂ bk2 )
3 h Hs+2 H2s+1 3 H2s 3 Hs+2 3 H2s
+(1+t)−σ(k∂ b k kbk k∂ bk +kb k k∂ bk k∂ bk )
3 h H2s Hs+2 3 H2s 3 H2s 3 Hs+1 3 H2s
.(1+t)−σ(k∂3bkH2s−3kbkH2s+1k∂3bkH2s +kbkH2sk∂3bk2H2s),
10 R.PAN,Y.ZHOU,ANDY.ZHU
provided that s ≥ 5. Hence,
t
1/2 1/2
|I (τ)| dτ . E (t)E (t)+E (t)e (t). (2.15)
Z 6 0 1 0 0
0
For the next term I , using H¨older inequality, it is straightforward to see
7
t
| I (τ) dτ| . E (t). (2.16)
Z 7 0,1
0
For the last term I , using the firstequation of system (1.6) and integrating by parts,
8
we find
I =(1+t)−σ ∇2s∂ u∇2s(∂ u+b·∇u−u·∇b) dx
8 Z 3 3
T3
=−(1+t)−σ ∇2s+1∂ u∇2s−1(∂ u+b·∇u−u·∇b)dx.
Z 3 3
T3
Thus, by H¨older inequality and Sobolev imbedding theorem, we have
|I |.(1+t)−σ(kuk2 +kuk2 kbk ),
8 H2s+2 H2s+2 H2s
provided that s ≥ 2. Hence, we arrive at
t
1/2
|I (τ)| dτ . E (t)+E (t)e (t). (2.17)
Z 8 0,1 0 0
0
Summing up the estimates for I ∼ I , i.e., (2.14), (2.15), (2.16) and (2.17), and
5 8
integrating (2.13) in time, usingYoung inequality and Poincar´e inequality we can easily
bound
t
(1+τ)−σk∂ bk2 dτ . E +E (t)E1/2(t)+E (t)e1/2(t). (2.18)
Z 3 H2s 0,1 0 1 0 0
0
Thisgives theestimate forthelastterminE (t). Now, multiplying(2.12)bysuitable
0
large number and plus (2.18), we then complete the proof of this lemma.
(cid:3)
Next, we work with the lower order energies defined in (2.3), especially we want to
derive the decay estimate and get the uniform bound of lower order norms of magnetic
field.
Lemma 2.3. Assume that s ≥ 5 and the energies are defined as in (2.3), then we have
1/2 1/2 1/2 1/2 1/2
G (t).E (t)+G (t)E (t)+E (t)G (t)[G (t)+E (t)]
0 0 0 1 0 0 1 1
1/2 1/4 1/4 1/2
+E (t)G (t)G (t)e (t).
0 0 1 0
Proof. First, applying∇2s∂ derivative on thesystem (1.6). Then,taking innerproduct
3
with ∇2s∂ b for the firstequation of system (1.6) and taking inner productwith ∇2s∂ u
3 3
for the second equation of system (1.6). Summing them up and multiplying the time
weight (1+t)1−σ we obtain
6
1 d
(1+t)1−σ(k∂ uk2 +k∂ bk2 )+(1+t)1−σk∂ uk2 = J , (2.19)
2dt 3 H˙2s 3 H˙2s 3 H˙2s+1 i
X
i=1