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L2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION OF THE F-M EQUATIONS IN THE SPACE-PERIODIC CASE
1999,19(5):548-554
L 2 ASYMPTOTIC FOR THE EQUILIBRIUM
SOLUTION OF THE F-M EQUATIONS IN THE SPACE-PERIODIC CASE 1 He Shuhong ( 1-Ti*~) Q·u Chaoshun ( ~A[t
Abstract
F-M equations in the space-periodic case (n = 2) is considered. Under some assumptions on the external force, it can be shown that the weak solution of F-M equations with initial and boundary conditions in space-periodic case approaches the stationary solution of the system exponetially when time t goes to infinite. Key words
Weak solution, equilibrium solution, asymptotical stability
1991 MR Subject Classification
1
35Q30,76D99
Introduction
Let us assume that a viscous incompressible fluid fills a region Q within the earth. The state of the fluid can be described by the following functions: p = p( z , t) is the density of the fluid. v( z , t) = (VI(x, t), vz(x, t), V3( x, t» is the velocity of fluid which is at point x at time t. b(x,t) = (bl(x,t),b z(x,t),b3(x,t» is the magnetic field at point x at time t. p(x) , q(x) are the presure. We suppose that the fluid is homogeneous at time t = O(i.e,p(x,O) =const). incompressibility of the fluid,we know that p(x, t) =const, "Ix E n, "It ~ O. The differential system for the flow and magnetic field within the earth is[IJ:
vv
-;) + (v· \7)v = v6v ~ .
vb
J'l vt
1 1 -\7p- 2w x v + -(\7 x b) x b+ f(x), p W
= '\6b + \7 x (v x b) -
1 -\7q J.L
divv = 0, divb = 0 inn,
+ g(x),
By the
(1.1)
(1.2)
(1.3)
Where v , JL are respectivively constants of kinematical viscosity, magnetic permeability.A = 'fJ1J.L with electrical resistivity 'fJ, vector w represents the angular velocity, and f( x) , g( x )represent volume forces. 1 Received JuL28,1997; revised Dec.2,1998. Supported by the fund of the Yunnan Education Committe and the Applied Basic Research Foundation of Yunnan Province.
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In space periodic case, we supplement the system (1.1)-(1.3) with the following initial and boundary conditions:
v(x,O)
= vo(x),b(x,O) = bo(x),'v'x E 0, o
(1.4)
0
u(x+Lej,t)=v(X,t),-Ov(x+Lej,t)=-Ov(x,t), Xj Xj b(x+Lej,t)
o 0 = b(x,t)'ox.b(x+Lej,t) = ox.b(x,t). J
Where j
= 1,2, {ej }j=1,2
(1.5)
(1.6)
J
= (L1> L 2), 0 = (0, L 1) x
is an orthonormal basis of the space, L
(0,L 2 ) . In space periodic case,the domain 0 = (0, L 1 ) x (0, L 2 ) can be identified with the twodimensional torus S2. Physically,the two dimension case means that the region is a cylinder n x R, n c R 2 , and all the quantities are independent of X3, u and b being parallel to the OX1x2-plane. We also assume that the average on v and bon 0 vanish at all time
L
udx
= 0,
L
bdx
= 0.
(1.7)
For the solution of the F-M equations in the space-periodic case, Lu Yiping[7] has proved
"..,", "" = ( : )
E X. then the," exists a uuique weak solution
T. W I. 'IT > 0; Moreover,if Uo E W, then arc Hilbert spaces in the follwoing section. ..'. n
2
U
,,= ( : )
E e(O, T,
Xln
E C(O, T, W) n L 2(0, T, D(A)), where X, W
Some Function Spaces We denote by H;:'r(O) the space of restrications to 0 of periodic functions which belong
to H"'(Q)(Q ::> 0 is an open bounded set of R 2 ) . That is,H;:'r(O) is the space of u in H"'(O) such that the trace 'Yj"u on the corresponding faces of 0 are equal
Where r j = r n {Xj = O},rj +n = r n {Xj = Lj},r is the boundary of O,j u( x) E H;:'r (0 ). we can use the Fourier series expansion
u(x) '"
L
Uk
exp(21rik·
1,2. For
E)'
kEZ 2
with
ttk
= U_k (so that u is real) and L = (~, L;-)' k- L= k 1 • ~ + k2 • L;-' Then u E L 2(0) ~
Ilulli2(n) = 101· I: /ukl 2 < 00, where Inl is the Lebegue measure of O. kEZ 2
1£ E
H;er(O), s E R+ ~
L
(1 + IkI 2)"/ukI 2 <
00.
kEZ'
We denote by i 2(0 ), iI;::'r(O), iI;er(O) the space of function in L 2(0), H;:'r(O), H;er(O) respectively such that In u(x )dx = O.
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ACTA MATHEMATICA SCIENTIA
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As described by Hale in [3],let
n, = closure of {u E C 1(0)2Idivu = 0,
L
udx
= O,Ujrj = Ulrj+n,j = 1,2} in
(L 2(O» 2,
. 1 2· = {U E Hper(O) IdlVU = OJ,
V
then (.i2(O» 2 = H" E9 s ; If let P be the orthogonal projection of Eq(1.1)""'(1.3) are equivalent to the system f)v f)t - vPl::,·v
(.i2(O» 2 onto n.,
-
= M 1 (v , b),
then
(2.1) (2.2)
· h vv W ere at
= vtv P V, vbvt = vtv Pb ' M1(v, b)
= P[-(v.V)v M2 (v; b) =
Let u = ( : ), A =
1 2w x v + -(V x b) x b + f(x)], PJ-L
P[V x (v x b) + g(x)].
(-V;" -A:")'
M(u) = [
;:~:::~ ] -
( ; : ).
F = ( ; : ) , then F-M equations ar written as du dt We equip
Hu
+ Au =
M(u)
+ F(x).
(2.3)
the usual inner product and norm of (L 2 (0 )2 (denoted by (.• ·).1, I) and
equip V with the inner product and norm of «Hter(O»2 (denoted by«·)), 11·11). Denote X = u, X ii., W = V x V, then we equip X with the following inner product and norm:
We equip W with the following inner product and norm:
2
where
3
«u, v» = j=li=l I: I: In ~ . ~dx. J
J
Similar with [4], A is a closed se~f-adjoint operator in X.D(A) = [H~er(Of x H1;er(O)2]n w. Let V· is the dual of V, then V C C V·, D(A) eWe X c W·. Obviously, A is a closed self-adjoint unbounded linear operator. Its inverse A-I also is a compact self-adjoint operator,
u,
He & QU: £2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION
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551
so A is a sectorial operator on X. Therefore,we can define fractional power space XO of X for any 0: E R and the powers AO of A. If f3 > 0:, then X{3 C XO, and the embedding is compact.
=
=
=
=
=
Define I. 10 lA o Ix. It is obviously that if 0: 0, then XO X, if 0: ~, then X t w. Below we give some lemmas Lemma 2.1[5] If u E D(A), then P6.u = 6.Pu. That is, the projector P commutes with
6. in
D(A). Lemma 2.2[6] Let b(u, v, w) = « u . '\7)v, w), then we have 1) b(~,v,w) = -b(u,w,v),'v'u,v,w E V, • 2) ('\7 x (a x b), c) = (b· '\7)a, c) - «a· '\7)b, c) = -«'\7 x c) x b, a). Lemma 2.3[4] Let b(u,v,w) = «u· '\7)v,w), then
lul~ Ilull ~ Ilvll ~1I6.vll~
Ib(u,v,w)l:s
C1
Iwl, 'I'll EV,v E H~er(Of n V, wE H", lul~ l6.ulllvlllwl, 'v'u E H1;er(O)Z n V, v E V, wE H", lulllvlllwl~ l6.wl~, 'I'll E H", v EV, wE Hger(Of n V, lul~llull~lIvlllwl~llwll~,'v'u,v,w EV.
3 The Equilibrium Solution of the F-M Equations in Space-Periodic Case We consider the equilibrium equation of F-M equations
Au
= M 1(u) + F(x) + B(u).
(3.1)
= (-2W
- (v . '\7)v + _.1 ('\7 x b) x b) ) . xv ) PI' ,B(u) '\7x(vxb) 0 The weak solution u' (in W) is defined by: 'Iv E W, satisfies
Where M1(u)
=(
[Au', v]
= [M1 (!t' ), v] + [F(x), v] + [B(u·). v].
. (3.2)
The main result of this section is to prove that equation (3.1) there exists a weak solution, that is Theorem 3.1 If F(x) E X, then there exists u· E W satisfies (3.2). To prove Theorem 3.1,we need two lemmas.
-P( ...!...(v x b') x b - (v· '\7)v') ) •. P I ' . • Define ( -P(V x (v x b')) mapping k( u)u' = A -1 Meu, u·), 'v'u' E W, then k(u) : W -> W is a bounded mapping,i.e there exists constant c such that /k(u)u'l* = IAh(u)u'l :S cllull·llu·ll. Proof 'v'w (W1,WZ) E W,[Ak(u)u',w] [M('u,tt'),w] (-P(p~('\7 x b') x b- (v' v)v'),wd + p~(-P('\7 x (v x b')),wz) By Lemma 2.1---2.3,we obtain Lemma 3.1
=
So that
If u. E W, let M(u,u')
=
=
=
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ACTA MATHEMATICA SCIENTIA
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IP(v. V')v*I_~ ::;
c21Ivll/lv*/I, IPV' x (v x b*)I_~ ::; c31Ivllllb*ll. Combining above, there exists constant csuch that IAk(u)u*l_ ~ = IA h( u)u*' ::; cllullllu*ll.
Similarly,we have
R" satisfies: continuous and ((v),v) < 0 for all v E aBo Then there exists v E B such that (I1) = O. Proof of Theorem 3.1 First we notice that equation (3.1)is iquivalent to u + k(u)u1B(u) A= A- 1F(x). Consider approximate equation Lemma
3.2(5)
Let B be a closed ball in R". Suppose : B
->
(3.3) Where P~ is the projector of Won the space spanned by
Wll"', W m
(the eigenvector of
A). Now we prove that for every m, equation (3.3) there exists solution
-m(u)
= U + P7~ (k(u)u) -
1
E P;; (W). Let
Um
1
PT~ (A- B(u) - P~ (A- F(x»), Vu E PT~' (W),
then (( -m (u), u))
= (( u + k(u)'u -
A -1 B(u) - A -1 F(x), P7~ (u)))
= ((u, u») + ((A- 1 M(u, u), u)) -
((A- 1 B(u), It))
-
((A- 1 F(:c). u»).
Similar to [6], we can show that VuE W,((A- 1M(u,u),u» = (M(u,u),u) = 0, so that,we get
((m(u), u» Choosing
= -11'u II~ ?
(2w
x v, v) + (F(x), u)
Eo
2
+ -2 111L11- + -/IFII=-l-' Eo ?
?
2
T > 0, then
sufficiently small such that 1 -
Eo
?
::; -llull~
1
Choosing a ball in P~(W) with radius R = 2 [2(1- T)EOr211F/I_~. By Lemma 3.2, there exists U m E P:(W) such that m(um ) = 0 and Ilumil ::; 2 [2(1- T)EOr~ IA-!FI. It implies that {u m } are uniformly bounded in W, so there exists subsequence {u~} such that U~I converge to
U
in W weakly. Because We X is compaet,so u~
->
u in X.
Below we prove that P~k(um)umconvergeto k(u)u in W weakly. Vh E W, we have
[A(P~ k(um)u m - k(u)u), h] = [A~(P~; k(um)u m - k(u)u), A ~ h]
=
[P: A-~Ak(um)um,A~h]-
= [M(um,um) - M(u m, u),h]
= [M(un.,um),P'~h]- [M(u.u),h]
[M(u;u),h]
+ [M(u m, u) -
M(u,u), h] + [M(um,um),P: h - h].
But
I[M(um,u m) - M(um,u),h]1
= I( 2-(V' x (bm p~
~ c41bm
-
+c6lbm
b»
x bm -
(v m . V')(vm
-
v), hd + 2-(V' p~
x (vm x (bm -
b)), h2)1
bl!lI bm - bll!lIh111I1bmll!lbml! + c5Ivml!lIvmll!lIh111Ivm- 'vltllvm- vii! -
bl!/Ibm
-
bll!/I'vm/lllh z/l!lhzl!.
By uniformly boundness of {-um} and
U
m ->
U
in X, we have
No.5
553
He & Qu: L 2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION
Similarly, we can prove that the second term also tend to 0 as m ---> the third term.
'Xi.
Below we estimate
From I\P:h - hll---> 0, we have [P,~;k(um)um - k(u)u,h]---> 0 as m ---> 'Xi, \lh E W. So u (the limit function of {u m } ) is the weak solution of equation (3.1). Below,we estimate u". Suppose u* E W is a solution of (3.1),i.e \I'lli E W
Multiplying u" in both side of (3.1) and integrating over [Au', u'] = [M 1(-u'), u']
n, we obtain
+ [B( lL'), u'] + [F( x), u'].
Notice [M1 (-u' ), u' ] = O,we have Ilu'll:S [2w x v',v'] + [F(x),'u']:S z~,llu'W + zl, 1F15. Where Al is the first eigenvalue of A. So that (AI - ~)111(IIZ :s ;; 1F15. Choosing e sufficiently small such that Al > j, then we have
>0
(3.5)
4
£2 Exponentially Stability for the Equilibrium Solution
In this section,we study L Z exponentially stability of equilibrium solution. Suppose u E G[O, 00, W] is a weak solution, u' E W is an equilibrium solution. Let w u - u", so w satisfies
=
dw + Aw -.M1(u) + MItu') - B(u) + B(-u')
dt
= O.
Taking the scalar product with w,we have
[~;,w] + [Aw,w] + [M 1 (u' ) -
M 1 (u),w] + [B(u') - B(u),w] = 0
I.e
~ dl~IZ + IIwll z :s I[M1 (u' ) -
M 1 (u), w] + [B(u') - B(u), w]l
:s I(w, V')v', WI) + ((v' . V')Wl, WI) + +(V' x b') x WZ,Wl)] + By Lemma 2.2,we have
2-[((V' x wz) x b, wd PfJ
~[(V' x (WI x b),wz) +
PJL
(V' x (v' x wz),wz)]I.
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ACTA MATHEMATICA SCIENTIA
=
=
Vo1.19
=
Where w (Wll wz) (v - v*, b - b*) u - u*. Below we estimate the terms on the right side of (4.1). I(wl . V')v*, wI)1 :S clOlwllllv*llwlli
~ clOlwlllv*lIwll :S c}~ Iwl zllv*lI z + ~lIwIl2.
For the second term,we have 1
pp, I(V' x b*)
X
cZ
<5
Wz, wdl ~ culwlllb*llIlwll:S 2~ IwlzlIb*W + 211wllz.
Similarly,for the third term,we have
1
cZ
-I(V' x (v* x wz),wz)1 ::; 21;Jwlzllv*lIz
w
.
.
u
.
<5
+ -lIwllz. 2
Combining the all above results, we get dlw[Z + 2(1- ~6.)lIwW < (io IIv~lIz + cilllv*W dt· 2 . <5 <5
+ ci 21Ib*W)lwl z. <5
By Poincare inequality,we obtain d]
IZ
3
2"
2
2 < ClO + Cil + clzllu*1I2lwl2 ~ + 2(1 _ -<5)>'llwI ill 2 <5 . . By Gronwall inequality,we have
Iw[Z
s 111I(0)l z exp{[~ lIu*1I 2 -
2(1-
~<5)>'I]t}.
If the equilibrium solution u* sufficiently small such that (4.2) then Iw(t)I Z converge to zero exponentially,so that the equilibrium solution is asymptotic stability. Notice (3.5),it is sufficient to choose F(x) E X such that
then (4.2) is true. Finally,combining the above results,we obtain the main result of this paper. Theorem 4.1
Suppose F(x) E X sufficiently small such that (4.3) is holds,then the
equilibrium solution of F-M equations is exponentially stability in X. References 1 Hide R. On planetary atomospheres and interiors. In: Reid W H ed. Mathematical Problem in the Geophysical Science 1. Amer Math Soc Providence RI, 1971.229-353 2 Henry D. Geometric theory of semilinear parabolic equations. Springer-verlag, 1981 3 Hale J K. Asymptotic behavior of dissipative system. Amer Math Soc Providence Ri,1988 4 Ternan R. Infintite dimensional dynamical system in mechanics and physics. Springer-verlag,1988 5 Constantin P, Foias C. Navier-Stokes equation. Univ Chicago Press, 1988 6 Qu Chaoshun, Son Shougen, Wang Ping. The solution of equations for the flow and the magnetic field within the earth. J Math Anal Appl, 1994,187(3):1003-1018 7 Lu Yiping. The existence of the solution and global attractor for F-M equations in the space-periodic case. Acta Math Appl Sinica, 1997,20(4):498-508
L 2 ASYMPTOTIC FOR THE EQUILIBRIUM
SOLUTION OF THE F-M EQUATIONS IN THE SPACE-PERIODIC CASE 1 He Shuhong ( 1-Ti*~) Q·u Chaoshun ( ~A[t
Abstract
F-M equations in the space-periodic case (n = 2) is considered. Under some assumptions on the external force, it can be shown that the weak solution of F-M equations with initial and boundary conditions in space-periodic case approaches the stationary solution of the system exponetially when time t goes to infinite. Key words
Weak solution, equilibrium solution, asymptotical stability
1991 MR Subject Classification
1
35Q30,76D99
Introduction
Let us assume that a viscous incompressible fluid fills a region Q within the earth. The state of the fluid can be described by the following functions: p = p( z , t) is the density of the fluid. v( z , t) = (VI(x, t), vz(x, t), V3( x, t» is the velocity of fluid which is at point x at time t. b(x,t) = (bl(x,t),b z(x,t),b3(x,t» is the magnetic field at point x at time t. p(x) , q(x) are the presure. We suppose that the fluid is homogeneous at time t = O(i.e,p(x,O) =const). incompressibility of the fluid,we know that p(x, t) =const, "Ix E n, "It ~ O. The differential system for the flow and magnetic field within the earth is[IJ:
vv
-;) + (v· \7)v = v6v ~ .
vb
J'l vt
1 1 -\7p- 2w x v + -(\7 x b) x b+ f(x), p W
= '\6b + \7 x (v x b) -
1 -\7q J.L
divv = 0, divb = 0 inn,
+ g(x),
By the
(1.1)
(1.2)
(1.3)
Where v , JL are respectivively constants of kinematical viscosity, magnetic permeability.A = 'fJ1J.L with electrical resistivity 'fJ, vector w represents the angular velocity, and f( x) , g( x )represent volume forces. 1 Received JuL28,1997; revised Dec.2,1998. Supported by the fund of the Yunnan Education Committe and the Applied Basic Research Foundation of Yunnan Province.
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He & QU: £2 ASYMPTOTIC FOR THE EQUILmRIUM SOLUTION
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In space periodic case, we supplement the system (1.1)-(1.3) with the following initial and boundary conditions:
v(x,O)
= vo(x),b(x,O) = bo(x),'v'x E 0, o
(1.4)
0
u(x+Lej,t)=v(X,t),-Ov(x+Lej,t)=-Ov(x,t), Xj Xj b(x+Lej,t)
o 0 = b(x,t)'ox.b(x+Lej,t) = ox.b(x,t). J
Where j
= 1,2, {ej }j=1,2
(1.5)
(1.6)
J
= (L1> L 2), 0 = (0, L 1) x
is an orthonormal basis of the space, L
(0,L 2 ) . In space periodic case,the domain 0 = (0, L 1 ) x (0, L 2 ) can be identified with the twodimensional torus S2. Physically,the two dimension case means that the region is a cylinder n x R, n c R 2 , and all the quantities are independent of X3, u and b being parallel to the OX1x2-plane. We also assume that the average on v and bon 0 vanish at all time
L
udx
= 0,
L
bdx
= 0.
(1.7)
For the solution of the F-M equations in the space-periodic case, Lu Yiping[7] has proved
"..,", "" = ( : )
E X. then the," exists a uuique weak solution
T. W I. 'IT > 0; Moreover,if Uo E W, then arc Hilbert spaces in the follwoing section. ..'. n
2
U
,,= ( : )
E e(O, T,
Xln
E C(O, T, W) n L 2(0, T, D(A)), where X, W
Some Function Spaces We denote by H;:'r(O) the space of restrications to 0 of periodic functions which belong
to H"'(Q)(Q ::> 0 is an open bounded set of R 2 ) . That is,H;:'r(O) is the space of u in H"'(O) such that the trace 'Yj"u on the corresponding faces of 0 are equal
Where r j = r n {Xj = O},rj +n = r n {Xj = Lj},r is the boundary of O,j u( x) E H;:'r (0 ). we can use the Fourier series expansion
u(x) '"
L
Uk
exp(21rik·
1,2. For
E)'
kEZ 2
with
ttk
= U_k (so that u is real) and L = (~, L;-)' k- L= k 1 • ~ + k2 • L;-' Then u E L 2(0) ~
Ilulli2(n) = 101· I: /ukl 2 < 00, where Inl is the Lebegue measure of O. kEZ 2
1£ E
H;er(O), s E R+ ~
L
(1 + IkI 2)"/ukI 2 <
00.
kEZ'
We denote by i 2(0 ), iI;::'r(O), iI;er(O) the space of function in L 2(0), H;:'r(O), H;er(O) respectively such that In u(x )dx = O.
550
ACTA MATHEMATICA SCIENTIA
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As described by Hale in [3],let
n, = closure of {u E C 1(0)2Idivu = 0,
L
udx
= O,Ujrj = Ulrj+n,j = 1,2} in
(L 2(O» 2,
. 1 2· = {U E Hper(O) IdlVU = OJ,
V
then (.i2(O» 2 = H" E9 s ; If let P be the orthogonal projection of Eq(1.1)""'(1.3) are equivalent to the system f)v f)t - vPl::,·v
(.i2(O» 2 onto n.,
-
= M 1 (v , b),
then
(2.1) (2.2)
· h vv W ere at
= vtv P V, vbvt = vtv Pb ' M1(v, b)
= P[-(v.V)v M2 (v; b) =
Let u = ( : ), A =
1 2w x v + -(V x b) x b + f(x)], PJ-L
P[V x (v x b) + g(x)].
(-V;" -A:")'
M(u) = [
;:~:::~ ] -
( ; : ).
F = ( ; : ) , then F-M equations ar written as du dt We equip
Hu
+ Au =
M(u)
+ F(x).
(2.3)
the usual inner product and norm of (L 2 (0 )2 (denoted by (.• ·).1, I) and
equip V with the inner product and norm of «Hter(O»2 (denoted by«·)), 11·11). Denote X = u, X ii., W = V x V, then we equip X with the following inner product and norm:
We equip W with the following inner product and norm:
2
where
3
«u, v» = j=li=l I: I: In ~ . ~dx. J
J
Similar with [4], A is a closed se~f-adjoint operator in X.D(A) = [H~er(Of x H1;er(O)2]n w. Let V· is the dual of V, then V C C V·, D(A) eWe X c W·. Obviously, A is a closed self-adjoint unbounded linear operator. Its inverse A-I also is a compact self-adjoint operator,
u,
He & QU: £2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION
No.5
551
so A is a sectorial operator on X. Therefore,we can define fractional power space XO of X for any 0: E R and the powers AO of A. If f3 > 0:, then X{3 C XO, and the embedding is compact.
=
=
=
=
=
Define I. 10 lA o Ix. It is obviously that if 0: 0, then XO X, if 0: ~, then X t w. Below we give some lemmas Lemma 2.1[5] If u E D(A), then P6.u = 6.Pu. That is, the projector P commutes with
6. in
D(A). Lemma 2.2[6] Let b(u, v, w) = « u . '\7)v, w), then we have 1) b(~,v,w) = -b(u,w,v),'v'u,v,w E V, • 2) ('\7 x (a x b), c) = (b· '\7)a, c) - «a· '\7)b, c) = -«'\7 x c) x b, a). Lemma 2.3[4] Let b(u,v,w) = «u· '\7)v,w), then
lul~ Ilull ~ Ilvll ~1I6.vll~
Ib(u,v,w)l:s
C1
Iwl, 'I'll EV,v E H~er(Of n V, wE H", lul~ l6.ulllvlllwl, 'v'u E H1;er(O)Z n V, v E V, wE H", lulllvlllwl~ l6.wl~, 'I'll E H", v EV, wE Hger(Of n V, lul~llull~lIvlllwl~llwll~,'v'u,v,w EV.
3 The Equilibrium Solution of the F-M Equations in Space-Periodic Case We consider the equilibrium equation of F-M equations
Au
= M 1(u) + F(x) + B(u).
(3.1)
= (-2W
- (v . '\7)v + _.1 ('\7 x b) x b) ) . xv ) PI' ,B(u) '\7x(vxb) 0 The weak solution u' (in W) is defined by: 'Iv E W, satisfies
Where M1(u)
=(
[Au', v]
= [M1 (!t' ), v] + [F(x), v] + [B(u·). v].
. (3.2)
The main result of this section is to prove that equation (3.1) there exists a weak solution, that is Theorem 3.1 If F(x) E X, then there exists u· E W satisfies (3.2). To prove Theorem 3.1,we need two lemmas.
-P( ...!...(v x b') x b - (v· '\7)v') ) •. P I ' . • Define ( -P(V x (v x b')) mapping k( u)u' = A -1 Meu, u·), 'v'u' E W, then k(u) : W -> W is a bounded mapping,i.e there exists constant c such that /k(u)u'l* = IAh(u)u'l :S cllull·llu·ll. Proof 'v'w (W1,WZ) E W,[Ak(u)u',w] [M('u,tt'),w] (-P(p~('\7 x b') x b- (v' v)v'),wd + p~(-P('\7 x (v x b')),wz) By Lemma 2.1---2.3,we obtain Lemma 3.1
=
So that
If u. E W, let M(u,u')
=
=
=
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ACTA MATHEMATICA SCIENTIA
Vol.19
IP(v. V')v*I_~ ::;
c21Ivll/lv*/I, IPV' x (v x b*)I_~ ::; c31Ivllllb*ll. Combining above, there exists constant csuch that IAk(u)u*l_ ~ = IA h( u)u*' ::; cllullllu*ll.
Similarly,we have
R" satisfies: continuous and ((v),v) < 0 for all v E aBo Then there exists v E B such that (I1) = O. Proof of Theorem 3.1 First we notice that equation (3.1)is iquivalent to u + k(u)u1B(u) A= A- 1F(x). Consider approximate equation Lemma
3.2(5)
Let B be a closed ball in R". Suppose : B
->
(3.3) Where P~ is the projector of Won the space spanned by
Wll"', W m
(the eigenvector of
A). Now we prove that for every m, equation (3.3) there exists solution
-m(u)
= U + P7~ (k(u)u) -
1
E P;; (W). Let
Um
1
PT~ (A- B(u) - P~ (A- F(x»), Vu E PT~' (W),
then (( -m (u), u))
= (( u + k(u)'u -
A -1 B(u) - A -1 F(x), P7~ (u)))
= ((u, u») + ((A- 1 M(u, u), u)) -
((A- 1 B(u), It))
-
((A- 1 F(:c). u»).
Similar to [6], we can show that VuE W,((A- 1M(u,u),u» = (M(u,u),u) = 0, so that,we get
((m(u), u» Choosing
= -11'u II~ ?
(2w
x v, v) + (F(x), u)
Eo
2
+ -2 111L11- + -/IFII=-l-' Eo ?
?
2
T > 0, then
sufficiently small such that 1 -
Eo
?
::; -llull~
1
Choosing a ball in P~(W) with radius R = 2 [2(1- T)EOr211F/I_~. By Lemma 3.2, there exists U m E P:(W) such that m(um ) = 0 and Ilumil ::; 2 [2(1- T)EOr~ IA-!FI. It implies that {u m } are uniformly bounded in W, so there exists subsequence {u~} such that U~I converge to
U
in W weakly. Because We X is compaet,so u~
->
u in X.
Below we prove that P~k(um)umconvergeto k(u)u in W weakly. Vh E W, we have
[A(P~ k(um)u m - k(u)u), h] = [A~(P~; k(um)u m - k(u)u), A ~ h]
=
[P: A-~Ak(um)um,A~h]-
= [M(um,um) - M(u m, u),h]
= [M(un.,um),P'~h]- [M(u.u),h]
[M(u;u),h]
+ [M(u m, u) -
M(u,u), h] + [M(um,um),P: h - h].
But
I[M(um,u m) - M(um,u),h]1
= I( 2-(V' x (bm p~
~ c41bm
-
+c6lbm
b»
x bm -
(v m . V')(vm
-
v), hd + 2-(V' p~
x (vm x (bm -
b)), h2)1
bl!lI bm - bll!lIh111I1bmll!lbml! + c5Ivml!lIvmll!lIh111Ivm- 'vltllvm- vii! -
bl!/Ibm
-
bll!/I'vm/lllh z/l!lhzl!.
By uniformly boundness of {-um} and
U
m ->
U
in X, we have
No.5
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He & Qu: L 2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION
Similarly, we can prove that the second term also tend to 0 as m ---> the third term.
'Xi.
Below we estimate
From I\P:h - hll---> 0, we have [P,~;k(um)um - k(u)u,h]---> 0 as m ---> 'Xi, \lh E W. So u (the limit function of {u m } ) is the weak solution of equation (3.1). Below,we estimate u". Suppose u* E W is a solution of (3.1),i.e \I'lli E W
Multiplying u" in both side of (3.1) and integrating over [Au', u'] = [M 1(-u'), u']
n, we obtain
+ [B( lL'), u'] + [F( x), u'].
Notice [M1 (-u' ), u' ] = O,we have Ilu'll:S [2w x v',v'] + [F(x),'u']:S z~,llu'W + zl, 1F15. Where Al is the first eigenvalue of A. So that (AI - ~)111(IIZ :s ;; 1F15. Choosing e sufficiently small such that Al > j, then we have
>0
(3.5)
4
£2 Exponentially Stability for the Equilibrium Solution
In this section,we study L Z exponentially stability of equilibrium solution. Suppose u E G[O, 00, W] is a weak solution, u' E W is an equilibrium solution. Let w u - u", so w satisfies
=
dw + Aw -.M1(u) + MItu') - B(u) + B(-u')
dt
= O.
Taking the scalar product with w,we have
[~;,w] + [Aw,w] + [M 1 (u' ) -
M 1 (u),w] + [B(u') - B(u),w] = 0
I.e
~ dl~IZ + IIwll z :s I[M1 (u' ) -
M 1 (u), w] + [B(u') - B(u), w]l
:s I(w, V')v', WI) + ((v' . V')Wl, WI) + +(V' x b') x WZ,Wl)] + By Lemma 2.2,we have
2-[((V' x wz) x b, wd PfJ
~[(V' x (WI x b),wz) +
PJL
(V' x (v' x wz),wz)]I.
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ACTA MATHEMATICA SCIENTIA
=
=
Vo1.19
=
Where w (Wll wz) (v - v*, b - b*) u - u*. Below we estimate the terms on the right side of (4.1). I(wl . V')v*, wI)1 :S clOlwllllv*llwlli
~ clOlwlllv*lIwll :S c}~ Iwl zllv*lI z + ~lIwIl2.
For the second term,we have 1
pp, I(V' x b*)
X
cZ
<5
Wz, wdl ~ culwlllb*llIlwll:S 2~ IwlzlIb*W + 211wllz.
Similarly,for the third term,we have
1
cZ
-I(V' x (v* x wz),wz)1 ::; 21;Jwlzllv*lIz
w
.
.
u
.
<5
+ -lIwllz. 2
Combining the all above results, we get dlw[Z + 2(1- ~6.)lIwW < (io IIv~lIz + cilllv*W dt· 2 . <5 <5
+ ci 21Ib*W)lwl z. <5
By Poincare inequality,we obtain d]
IZ
3
2"
2
2 < ClO + Cil + clzllu*1I2lwl2 ~ + 2(1 _ -<5)>'llwI ill 2 <5 . . By Gronwall inequality,we have
Iw[Z
s 111I(0)l z exp{[~ lIu*1I 2 -
2(1-
~<5)>'I]t}.
If the equilibrium solution u* sufficiently small such that (4.2) then Iw(t)I Z converge to zero exponentially,so that the equilibrium solution is asymptotic stability. Notice (3.5),it is sufficient to choose F(x) E X such that
then (4.2) is true. Finally,combining the above results,we obtain the main result of this paper. Theorem 4.1
Suppose F(x) E X sufficiently small such that (4.3) is holds,then the
equilibrium solution of F-M equations is exponentially stability in X. References 1 Hide R. On planetary atomospheres and interiors. In: Reid W H ed. Mathematical Problem in the Geophysical Science 1. Amer Math Soc Providence RI, 1971.229-353 2 Henry D. Geometric theory of semilinear parabolic equations. Springer-verlag, 1981 3 Hale J K. Asymptotic behavior of dissipative system. Amer Math Soc Providence Ri,1988 4 Ternan R. Infintite dimensional dynamical system in mechanics and physics. Springer-verlag,1988 5 Constantin P, Foias C. Navier-Stokes equation. Univ Chicago Press, 1988 6 Qu Chaoshun, Son Shougen, Wang Ping. The solution of equations for the flow and the magnetic field within the earth. J Math Anal Appl, 1994,187(3):1003-1018 7 Lu Yiping. The existence of the solution and global attractor for F-M equations in the space-periodic case. Acta Math Appl Sinica, 1997,20(4):498-508