XX Cong eso de Ecuaciones Di e enciales y Aplicaciones
X Cong eso de Ma em´
a ica Aplicada
Se illa, 24-28 sep iemb e 2007
(pp. 1–8)
Well-posedness and asymp o ic beha iou o he
Boussinesq sys em in Rn.
E. J. Villamiza -Roa1, L. C. Fe ei a 2
1Escuela de Ma em´a icas, Uni e sidad indus ial de San ande , A.A. 678, Buca amanga-Colombia.
E-mails: [email p o ec ed].
2Dp o. de Ma em´a ica, Uni e sidade Fede al de Pe nambuco, Reci e-B azil. E-mail: [email p o ec ed].
Palab as cla e: Well-posedness, asymp o ic beha iou , Boussinesq sys em
Resumen
We analyze he well-posedness o he ini ial alue p oblem o a Con ec ion P o-
blem. Mild solu ions a e ob ained in he weak-Lp(Rn) spaces and he exis ence o
sel -simila solu ions is showed, while he only small sel -simila solu ion in he Lebes-
gue space Lp(Rn) is he null solu ion. The asymp o ic s abili y o solu ions is analyzed
and, as a consequence, a c i e ium o sel -simila i y pe sis ence a la ge imes is ob-
ained.
1. In oduc ion
We conside a iscous incomp essible luid illing he whole space Rn, n ≥2.Due o
he Boussinesq app oxima ion (Chand asekha [3]), densi y a ia ions a e neglec ed excep
in he g a i a ional e m (buoyancy e m) and hey a e assumed o be p opo ional o
empe a u e a ia ions. The ela ionship among he eloci y ield u(x, )∈Rn, he p essu e
p(x, )∈Rand he empe a u e θ(x, )∈R,can be desc ibed by he ollowing ini ial alue
p oblem
∂u
∂ +u∇u−ν∆u+1
ρ∇p=βθ + 1, x ∈Rn, > 0,(1)
∇ · u= 0, x ∈Rn, > 0,(2)
∂θ
∂ +u∇θ−χ∆θ=h, x ∈Rn, > 0,(3)
θ(x, 0) = θ0(x), u(x, 0) = u0(x), x ∈Rn,(4)
whe e ep esen s a g a i a ional ec o ield a x,h he e e ence empe a u e and 1
an ex e nal o ce. ρ, ν, β, χ a e posi i e physical cons an s which ep esen , espec i ely,
1
E.J. Villamiza -Roa, L. C. Fe ei a
he densi y, he kinema ic iscosi y, he coe icien o olume expansion and he he mal
conduc ance. Wi hou loss o gene ali y, we will assume he cons an s ρ, ν, β, χ o be one
and he e e ence empe a u e hand he ex e nal o ce 1 o be ze o. The ini ial da a u0
sa is ies he condi ion ∇ · u0= 0 in he dis ibu ional sense.
New aspec s o s udies on he Con ec ion P oblem (1)-(4) a e conside ed in his wo k,
in ac , we will s udy he sys em (1)-(4) in he whole space Rnin he amewo k o
weak−Lpspaces. Fi s ly we p esen esul s o well-posedness in hese spaces and make
some conside a ions a ound he well-posedness in he Lebesgue Spaces Lp(see [2]). On
he o he hand, we show some esul s abou he exis ence, uniqueness, he asymp o ic
s abili y and he sel -simila i y pe sis ence o solu ions o he P oblem (1)-(4) in weak−Lp
spaces. Mo eo e , as a consequence o esul s o asymp o ic s abili y, we will show ha
he only sel -simila solu ions in Lebesgue spaces Lpis he null solu ion, ein o cing he
need o mo e singula ini ial da a o allow he exis ence o sel -simila solu ions. These
sel -simila solu ions co espond, o ins ance, o homogeneous ini ial unc ions o deg ee
−1. Finally, om a physical s andpoin , ou analysis can be applied o se e al classes o
ex e nal o ces . In ac , we can ake as he g a i a ional ield
= (x) = −G∇xφ=Gx
|x|3∈L(n
2,∞)(Rn),
whe e Gis he g a i a ional cons an , in o de o show he exis ence o global solu ions
(u, θ) which a e cons uc ed in di e en unc ional spaces (see Theo em 3.5, Theo em 3.3
and Rema k 4.5). This case can be ega ded as an in e es ing physical case o he B´ena d
P oblem. Mo e de ails abou he physical and ma hema ical analysis o sys em (1)-(4) see
[2] and he e e ences he ein.
2. Func ion Spaces and De ini ions
In his sec ion, we in oduce he unc ional spaces ele an o ou s udy o solu ions
ega ding he Cauchy p oblem o sys em (1)-(4). Fo each Lebesgue mensu able unc ion
de ined on Rn, he ea angemen ∗is de ined by
∗( ) = ´ın {s > 0 : m({x∈Rn:| (x)|> s})≤ }, > 0.
The Lo en z space L(p,q)≡L(p,q)(Rn) is he se o all such ha
k k(p,q)=


³p
qR∞
0[ 1
p ∗∗( )]qd / ´1
q, i 1 < p < ∞,1≤q < ∞,
sup >0 1
p ∗∗( ) , i 1 < p ≤ ∞,q=∞.
is ini e, whe e ∗∗( ) = 1
R
0 ∗(s)ds, o > 0.We obse e ha Lp=L(p,p).L(p,∞)a e
called he Ma cinkiewicz spaces o weak-Lpspaces. Mo eo e , L(p,q1)⊂Lp⊂L(p,q2)⊂
L(p,∞) o 0 < q1≤p≤q2≤ ∞. See [4].
P oposi ion 2.1 [4] (Gene alized Holde ’s inequali y). Le 1< p1,p2, < ∞. Le ∈
L(p1,q1)and g∈L(p2,q2)whe e 1
p1+1
p2<1, hen he p oduc h= g belongs o L( ,s)whe e
1
=1
p1+1
p2, and s≥1is any numbe such ha 1
q1+1
q2≥1
s. Mo eo e ,
khk( ,s)≤ 0k k(p1,q1)kgk(p2,q2),
2
Well-posedness and asymp o ic beha iou o he Boussinesq sys em
being 0 he conjuga e index o .
Le us ecall he Helmhol z decomposi ion L (Ω) = L
σ(Ω)⊕G (Ω),1< < ∞,whe e
G (Ω) = {∇p∈L (Ω) : p∈L
loc(Ω)}.P (o simply P) deno es he p ojec ion ope a o
om L on o L
σ.The S okes ope a o is deno ed by A (o simply Ao −P∆) and he
Laplace ope a o is deno ed by B (o simply Bo −∆). We know ha −A ,−B gene a e
uni o mly bounded holomo phic semig oups {e− A } ≥0,{e− B } ≥0o class C0in L
σand
L , espec i ely. Bo che s and Miyakawa [1] es ablished he ollowing Helmhol z decom-
posi ion o he Lo en z spaces. We can ex end P o a bounded ope a o on L( ,d)(Ω),
which we deno e by P ,d.Se L( ,d)
σ(Ω) = Range(P ,d) and G( ,d)(Ω) = Ke nel(P ,d).
Then, L( ,d)(Ω) = L( ,d)
σ(Ω) ⊕G( ,d)(Ω).Based on [1], −A, −Bgene a e uni o mly boun-
ded analy ic semig oups on L( ,d)
σ(Ω) and L( ,d)(Ω), espec i ely. Howe e , no ice ha hese
semig oups a e no s ongly con inuous a = 0 i d=∞,since in his case D ,∞(A) and
D ,∞(B) a e no dense in L( ,∞)
σand L( ,∞), espec i ely. We ecall ha in ou case Ω = Rn,
{e− B} ≥0is he hea semig oup gi en as he con olu ion wi h he Gauss-Weie s ass ke -
nel: G(x, ) = (4π )−n/2exp(−|x|2/(4 )).
Finally, le 1 < p < ∞,1< q < ∞and 1 ≤d≤ ∞.The ollowing no a ion is adop ed
o he no m o p oduc in Lo en z spaces L(p,d)
σ(Rn)×L(q,d)(Rn) :
°
°
°hu
θi°
°
°(p,q),d =kuk(p,d)+kθk(q,d).
I p=qand d=∞,we simply deno e his no m as
°
°
°hu
θi°
°
°(p,∞)=kuk(p,∞)+kθk(p,∞).
3. Resul s o well-posedness
We de ine he ope a o M:L(p,∞)
σ×L(q,∞)→L(p,∞)
σ×L(q,∞)by
Mhu
θi=h−P∆u
−∆θi.
Wi h he use o he semig oup {e− M } ≥0, he Cauchy p oblem (1)-(4) is con e ed o he
in eg al equa ion
hu( )
θ( )i=e− M hu0
θ0i−Z
0
e−( −s)M³h(u· ∇u)
(u· ∇θ)i−h(θ )
0i´ds, > 0,(5)
in L(p,∞)
σ(Rn)×L(q,∞)(Rn).The e m in (5)
−Z
0
e−( −s)Mh(u· ∇u)(s)
(u· ∇θ)(s)ids
will be called he bilinea ec o , and he e m in (5)
Z
0
e−( −s)P∆(θ )ds,
3
E.J. Villamiza -Roa, L. C. Fe ei a
we will called o coupling e m.
Le us emembe he ollowing L(p,d)−L( ,d)es ima es o he semig oup {e− M } ≥0
and gi e he p oo by comple eness.
Lemma 3.1 [2] Le 1≤d≤ ∞. Fo all (ϕ, φ)∈L(p,d)
σ(Rn)×L(q,d)(Rn),and all > 0,
he e exis s a cons an C(p, , s, q)such ha
°
°
°∇je− M hϕ
φi°
°
°( ,s),d ≤C −n
2(γ+j
n)°
°
°hϕ
φi°
°
°(p,q),d,
whe e γ= 1/p −1/ = 1/q −1/s, wi h 1< p ≤ < ∞and 1< q ≤s < ∞.
Nex , le us in oduce sui able ime dependen unc ional spaces in which we will need
o s udy he ini ial alue p oblem (1)-(4).
De ini ion 3.2 Le n < q < ∞and α= 1 −n/q. We de ine he spaces
E≡ {(u, θ) : (u, θ)∈BC((0,∞), L(n,∞)
σ×L(n,∞))},
Eq≡ {(u, θ)∈E: α/2(u, θ)∈BC((0,∞), L(q,∞)
σ×L(q,∞))},
Fq≡ {(u, θ) : u∈BC((0,∞); L(n,∞)
σ), α/2θ∈BC((0,∞); L(q,∞))},
which a e Banach spaces wi h he no ms in E, Eq, Fqde ined, espec i ely, as
°
°
°hu
θi°
°
°E= sup
>0°
°
°hu
θi°
°
°(n,∞),°
°
°hu
θi°
°
°Eq
=°
°
°hu
θi°
°
°E+ sup
>0
α/2°
°
°hu
θi°
°
°(q,∞),
°
°
°hu
θi°
°
°Fq
= sup
>0
kuk(n,∞)+ sup
>0
α/2kθk(q,∞).
Theo em 3.3 (i) Le n > 2a posi i e in ege numbe , (u0, θ0)be any pai in L(n,∞)
σ×
L(n,∞)and small enough wi h espec he ollowing no m
k kb= sup
>0
β
2k ( )k(b,∞)<∞, β = 2 −n
b, b > n
2.
Then, he e a e cons an s 0< τ =τ( )<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0)
such ha i °
°
°hu0
θ0i°
°
°(n,∞)< δ, he ini ial alue p oblem (1)-(4) has a global solu ion
(u( , x), θ( , x)) ∈Esa is ying (5), wi h
l´ım
→0(u( ), φ) = (u0, φ),l´ım
→0(θ( ), ϕ) = (θ0, ϕ),
o all φ∈L(n0,1)
σ(Rn), ϕ ∈L(n0,1)
σ(Rn).Mo eo e , i °
°
°hu
θi°
°
°E<2ε
1−τ, hen he solu ion is
unique.
(ii) I we assume ha (u0, θ0)∈(L(n,∞)
σ×L(p,∞)
σ)∩(L(n,∞)×L(p,∞))wi h 1< p0< n,
he e a e 0< δp≤δand 0< τp=τp( )≤τsuch ha i °
°
°hu0
θ0i°
°
°(p,∞)< δp, hen p e ious
solu ion (u, θ) e i ies ha (u, θ)∈BC((0,∞), L(p,∞)
σ×L(p,∞)).
4
Well-posedness and asymp o ic beha iou o he Boussinesq sys em
Theo em 3.4 (Regula iza ion) Unde he assump ions o Theo em 3.3, le n < q < ∞,
such ha 1
b+1
q>1
n. I k ( )kb= sup >0 β
2k ( )k(b,∞)is small enough, he e a e cons an s
0< τq( )<1and 0< δq≤δsuch ha i °
°
°hu0
θ0i°
°
°(n,∞)< δq, hen he solu ion (u, θ)o
Theo em 3.3 belongs o Eq.
In he case n > 2, he assump ion k kb<∞can be changed by he ollowing one:
sup >0k ( )k(n
2,∞)<∞. Indeed we will p o e he ollowing heo em:
Theo em 3.5 Le (u0, θ0)∈L(n,∞)
σ×L(n,∞)whe e n > 2and assume ha belongs o
BC((0,∞), L(n
2,∞)). I n < q < ∞and sup >0k k(n
2,∞)is su icien ly small, hen he e
a e cons an s 0< τ =τ( )<1,δ > 0and ε=ε(δ)>0(ε→0when δ→0) such
ha i °
°
°hu0
θ0i°
°
°(n,∞)< δ, hen he ini ial alue p oblem o (1)-(4) has a global solu ion
(u( , x), θ(x, )) ∈Fqsa is ying (5) oge he wi h
l´ım
→0(u( ), φ) = (u0, φ),l´ım
→0(θ( ), ϕ) = (θ0, ϕ),
o all φ∈L(n0,1)
σ(Rn), ϕ ∈L(n0,1)
σ(Rn).Mo eo e , i °
°
°hu
θi°
°
°Fq
≤2ε
1−τ, hen he solu ion
is unique in he space Fq.
Fu he mo e, i we assume ha (u0, θ0)∈(L(n,∞)
σ∩L(p,∞))×(L(n,∞)
σ∩L(p,∞)), wi h
q0< p0<n
2, he e a e 0< δp≤δand 0< τp=τp( )≤τsuch ha i °
°
°hu0
θ0i°
°
°(n,∞)< δp,
hen p e ious solu ion (u, θ)sa is ies (u, θ)∈BC((0,∞), L(p,∞)
σ×L(p,∞)).
3.1. Ske ch o P oo s o he well-posedness Theo ems
The p oo s o he well-possedness Theo ems ollows basically om he nex lemma in
a gene ic Banach space and lemmas 3.7, 3.8, 3.9 ( see [2]).
Lemma 3.6 Le Xbe a Banach space wi h no m k · kX,T:X→Xa linea con inuous
map wi h no m τ < 1and B:X×X→Xa con inuous bilinea map, ha is, he e
exis s a cons an K > 0such ha o all x1and x2in XkB(x1, x2)kX≤Kkx1kXkx2kX.
Then, o 0< ε < (1−τ)2
4Kand o any ec o y∈X,y6= 0, such ha kykX< ε, he e
exis s a solu ion x∈X o he equa ion x=y+B(x, x) + T(x)such ha kxkX≤2ε
1−τ.
The solu ion xis unique in he closed ball B2ε
1−τ:= B(0,2ε
1−τ)⊂X. Mo eo e , he solu ion
depends con inuously on yin he ollowing sense: I k˜ykX≤ε,˜x= ˜y+B(˜x, ˜x) + T(˜x),
and k˜xkX≤2ε
1−τ, hen kx−˜xkX≤1−τ
(1−τ)2−4Kε ky−˜ykX.
Lemma 3.7 I (u0, θ0)∈L(n,∞)
σ×L(n,∞).Then e− M hu0
θ0i∈E, wi h °
°
°e− M hu0
θ0i°
°
°E≤
C°
°
°hu0
θ0i°
°
°(n,∞)and e− M hu0
θ0i*hu0
θ0iwhen →0+,whe e he limi is aken in he
5

E.J. Villamiza -Roa, L. C. Fe ei a
weak-s a opology o he L(n,∞)
σ×L(n,∞).Mo eo e °
°
°e− M hu0
θ0i°
°
°Eq
≤C°
°
°hu0
θ0i°
°
°(n,∞),and
i (u0, θ0)∈(L(p,∞)
σ×L(p,∞)) hen °
°
°e− M hu0
θ0i°
°
°(p,∞)≤C°
°
°hu0
θ0i°
°
°(p,∞).
Lemma 3.8 Le n, b be as in he Theo em 3.3 and T(θ) = R
0e( −s)P∆(θ )(s)ds. Then
kT(θ)k(n,∞)≤Ck kbsup
>0
kθk(n,∞),kT(θ)k(p,∞)≤Ck kbsup
>0
kθk(p,∞).
Mo eo e , i n, b sa is y he assump ions o Theo em 3.4, hen
kT(θ)kEq≤Ck kbsup
>0
α
2kθk(q,∞).
Lemma 3.9 I 1< p < q < ∞ hen o all φ∈L(p,1)(Rn)hold:
s1
2(n
p−n
q+1)k∇e−sM φk(q,1) ≤Ckφk(p,1), s1
2(n
p−n
q)ke−sM φk(q,1) ≤Ckφk(p,1),
Z∞
0
s1
2(n
p−n
q)−1
2k∇e−sM φk(q,1)ds ≤Ckφk(p,1),Z∞
0
s1
2(n
p−n
q)−1ke−sM φk(q,1)ds ≤Ckφk(p,1).
4. Sel -Simila i y
Assuming ha ( , x) = λ2 (λ2 , λx) is smoo h and ha (u( , x), θ( , x)) is a smoo h
solu ion o he con ec ion p oblem (1)-(4), i is s aigh o wa d o check ha (u, θ)λ( , x) =
λ(u(λ2 , λx), θ(λ2 , λx)) is also a solu ion o he Sys em (1)-(4). In ac , we can look o
pa icula solu ions o he Sys em (1)-(4) sa is ying
(u( , x), θ( , x)) = (u( , x), θ( , x))λ( , x),(6)
o any > 0, x∈Rnand λ > 0.These solu ions a e called sel -simila solu ions o he
sys em and i is clea ha aking →0+, o mally in (6), (u(0, x), θ(0, x)) should be a
homogeneous unc ion o deg ee −1. This ema k gi es he hin ha a sui able space o ind
sel -simila solu ions should be one con aining homogeneous unc ions wi h ha exponen .
The space L(n,∞)is he only weak-Lpspace such ha |x|−1∈L(p,∞). Mo eo e , in case
ha such a sel -simila solu ion exis s, i s no m is in a ian by he scaling ans o ma ion,
(u( , x), θ( , x)) →(u( , x), θ( , x))λ=λ(u(λ2 , λx), θ(λ2 , λx)).
Mo eo e , homogenous unc ions o any o de do no belong o any s ong Lpspace. All
o hese ac s ein o ce he idea ha weak-Lpspaces wi h he igh homogenei y a e he
mos ele an spaces o inding global non- i ial sel -simila solu ions o he Con ec ion
P oblem.
4.1. Decay Es ima es in weak −Lpand Lp
Theo em 4.1 Le (u0, θ0)as in he Theo em 3.4 and ≥pis ini e and sa is ies 1
p+1
q−
1
<1
nand 1
b+1
p−1
<2
n. Then he solu ion o he Theo em 3.4 sa is ies
(n
2p−n
2 )u∈BC((0,∞); L( ,∞)
σ)n, (n
2p−n
2 )θ∈BC((0,∞); L( ,∞)).
Mo eo e , his heo em is ue by elaxing he assump ions o n≥2, and e en subs i u ing
weak-Lpspaces by hei s onge coun e pa s.
6
Well-posedness and asymp o ic beha iou o he Boussinesq sys em
4.1.1. P oo o Theo em 4.1.
Le (u0, θ0)∈L(p,∞)
σ×L(p,∞). As a di ec consequence o Lemma 3.1 we ha e ha
sup >0
−p
2γ°
°
°e− M hu0
θ0i°
°
°( ,∞)≤C°
°
°hu0
θ0i°
°
°(p,∞).Now, by he second pa o Theo em
3.3, we al eady know ha i he ini ial da a (u0, θ0)∈(L(p,∞)
σ∩L(n,∞)
σ)×(L(p,∞)∩L(n,∞)),
hen he solu ion (u( ), θ( )) sa is ies sup >0(ku( )k(p,∞)+kθ( )k(p,∞))<∞.Thus, in o de
o conclude he p oo o he Theo em 4.1, we need a lemma whe e we es ima e he no m
sup >0
−p
2γk·k ,∞o bilinea ec o e m and he linea ope a o e m T(θ), using he
no m k · kEq+ sup >0k·kp,∞o he solu ion. Fo his, we p o e he ollowing lemmas.
Lemma 4.2 Le pand bas in he Theo em 4.1 and ≥p, hen
sup
>0
(n
2p−n
2 )kT(θ)( )k( ,∞)≤Ck kbsup
>0
kθk(p,∞).
Lemma 4.3 Le pas in he Theo em 4.1 and ≥p, hen
sup
>0
ρ
°
°
°Z
0
e−( −s)Mh(u1· ∇u2)
(u2· ∇θ1)i°
°
°( ,∞)≤Csup
>0°
°
°hu1
θ1i°
°
°(p,∞)sup
>0
α
2ku2k(q,∞),
whe e ρ= ( n
2p−n
2 ).
4.2. Sel -Simila Solu ion in he spaces L(n,∞).
The aim o his subsec ion is o desc ibe he p incipal esul s ela i e o he exis ence
and he uniqueness o sel -simila i y solu ions in he L(n,∞)-spaces.
Theo em 4.4 Le (u0, θ0)∈L(n,∞)
σ×L(n,∞). Assume ha u0, θ0a e homogeneous unc-
ions o deg ee −1, ha is, u0(λx) = λ−1u0(x), θ0(λx) = λ−1θ0(x) o all x∈Rn,x6= 0
and all λ > 0and as in Theo em 3.3 and Theo em 3.5, sa is ies he scale ela ion
( , x) = λ2 (λ2 , λx).Then, i °
°
°hu0
θ0i°
°
°(n,∞)< ε he solu ion (u, θ)gi en by Theo em
3.3 and Theo em 3.5 is sel -simila , i.e., u( , x) = λu(λ2 , λx), θ( , x) = λθ(λ2 , λx),
o all x∈Rn,x6= 0 and all λ > 0. Mo eo e , in case o Theo em 3.3, i he
ini ial da a is smalle °
°
°hu0
θ0i°
°
°(n,∞)< εq, he p e ious unique sel -simila solu ion becomes
egula ized.
Rema k 4.5 [B´ena d P oblem] No e ha , in he case o Theo em 3.5, we can ake
as he g a i a ional ield = (x) = −G∇x(1
|x|) = Gx
|x|3∈L(n
2,∞)(Rn),whe e Gis
he g a i a ional cons an . This case can be ega ded as he B´ena d p oblem (see [3])
which co esponds o he in e es ing physical case. This conside a ion is also ue o
he modi ied Theo em 3.3, whe e we assume ∈BC([0,∞); L(n
2,∞)(Rn)),ins ead o
sup >0 β
2k ( )k(b,∞)<∞, and we sea ch solu ion in he space Eq.
7
E.J. Villamiza -Roa, L. C. Fe ei a
5. S abili y in L(n,∞).
We analyze he la ge ime beha io o solu ions o Sec ion 3. In sho , we will show
ha pe u ba ions o he ini ial da a a e negligible o la ge imes.
Theo em 5.1 Assume ha (u, θ)and ( , φ)a e solu ions o (1)-(4) as in he Theo em 3.3
co esponding o he ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞)
σ×L(n,∞), espec i ely.
Suppose ha
l´ım
→∞ °
°e ∆(θ0−φ0)°
°(n,∞)= l´ım
→∞ °
°e P∆(u0− 0)°
°(n,∞)= 0,
hen
l´ım
→∞ ku( )− ( )k(n,∞)= 0,l´ım
→∞ kθ( )−φ( )k(n,∞)= 0.
Mo eo e , assume (u, θ)and ( , φ)a e solu ions o (1)-(4) gi en by Theo em 3.4 co es-
ponding o ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞)
σ×L(n,∞)sa is ying ha
l´ım
→∞ α
2°
°e ∆(θ0−φ0)°
°(q,∞)= l´ım
→∞ °
°e P∆(u0− 0)°
°(q,∞)= 0,
hen
l´ım
→∞ α
2ku( )− ( )k(q,∞)= 0,l´ım
→∞ α
2kθ( )−φ( )k(q,∞)= 0.
Theo em 5.2 Assume ha (u, θ)and ( , φ)a e solu ions o (1)-(4) as in he Theo em 3.5
co esponding o he ini ial condi ions (u0, θ0)and ( 0, φ0)∈L(n,∞)
σ×L(n,∞), espec i ely.
Suppose ha l´ım →∞ α
2°
°e ∆(θ0−φ0)°
°(q,∞)= 0 and ha l´ım →∞ °
°e P∆(u0− 0)°
°(n,∞)=
0, hen
l´ım
→∞ ku( )− ( )k(n,∞)= 0,l´ım
→∞ α
2kθ( )−φ( )k(q,∞)= 0.
Co olla y 5.3 Le (u0, θ0)∈Ln
σ×Ln(Lebesgue space) be as in he Theo em 3.4. Then
he co esponding solu ion sa is ies l´ım →∞ ku( )kLn= 0,l´ım →∞ kθ( )kLn= 0.As a
consequence, he unique sel -simila solu ion in Ln(Rn)×Ln(Rn)is he null solu ion.
Ag adecimien os
E J Villamiza -Roa was suppo ed by COLCIENCIAS, Colombia, P oyec o COLCIENCIAS-
BID III e apa and UIS and L C F Fe ei a was suppo ed by CAPES, B azil.
Re e encias
[1] W Bo che s, and T Miyakawa, 1995 On s abili y o ex e io Na ie -S okes lows. Ac a Ma h. 147,
311-382.
[2] L C F Fe ei a, E J Villamiza -Roa, 2006 Well-posedness and asymp o ic beha iou o he con ec ion
p oblem in Rn,Nonlinea i y 19, 2169-2191.
[3] S Chand asekha , 1981 Hid odinamic and Hyd omagne ic S abili y. Do e , New Yo k.
[4] R O’Neil, 1963 Con olu ion ope a o s and L(p, q)spaces. Duke Ma h. J. 30, 129-142.
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