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)
Exis ence and Uniqueness o S ong Solu ions o he
Incomp essible Mic opola Fluid Equa ions in Domains o
R3
J. L. Bold ini 1, M. Du ´
an2, M.A. Rojas-Meda 3
1IMECC-UNICAMP, CP 6065, 13083-859, Campinas-SP, B azil. E-mails: [email p o ec ed].
2Facul ad de Ingenie ´ıa, Pon i icia Uni e sidad Ca ´olica de Chile, Casilla 306, San iago 22, Chile.
E-mail: [email p o ec ed].
3Uni e sidad del B´ıo-B´ıo Facul ad de Ciencias, Depa amen o de Ciencias B´asicas, Campus Fe nando
May, Casilla 447, Chill´an, Chile. E-mails: [email p o ec ed],
Palab as cla e: Mic opola luids, unbounded domains, hyd odynamics. exis ence o solu ions
Resumen
We conside he ini ial bounda y alue p oblem o he sys em o equa ions de-
sc ibing he nons a iona y low o an incomp essible mic opola luid in a domain Ω o
R3. Unde hypo heses ha a e simila o he Na ie -S okes equa ions ones, by using
an i e a i e scheme, we p o e he exis ence and uniqueness o s ong solu ion in Lp(Ω),
o p > 3.
1. In oduc ion
The objec i e o he p esen wo k is o s udy he exis ence o s ong solu ions o he
e olu ion equa ions o he mo ion o incomp essible mic opola (asymme ic) luids in
a bounded o unbounded domain Ω ⊂R3ha ing a compac C2-bounda y. Tha is, he
domains we a e conside ing include he he so called ex e io domains. To desc ibe hese
equa ions, le T > 0 and QT≡Ω×(0, T); hen he sys em we will s udy is he ollowing:













∂u
∂ + (u· ∇)u−(µ+µ )∆u+∇η= 2µ o w+ in QT,
di u= 0 in QT,
∂w
∂ + (u· ∇)w−(ca+cd)∆w+ 4µ w
−(c0+cd−ca)∇di w= 2µ o u+gin QT,
(1)
1
J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda
oge he wi h he ollowing bounda y and ini ial condi ions







u=0on ST,
w=0on ST,
u(x, 0) = u0(x) in Ω ,
w(x, 0) = w0(x) in Ω ,
(2)
whe e ST≡∂Ω×(0, T). The ec o - alued unc ions u= (u1, u2, u3),w= (w1, w2, w3)
and he scala unc ion ηdeno e espec i ely he eloci y, he angula eloci y o o a ion
o pa icles and he p essu e o he luid. The ec o - alued unc ions and gdeno e
espec i ely he ex e nal sou ces o linea and angula momen um. The posi i e cons an s
µ, µ , c0, caand cda e iscosi ies- ype coe icien s sa is ying he ollowing inequali y c0+
cd> ca.
Fo he de i a ion and physical discussion o equa ions (1)-(2) see Pe osyan [15],
Condi and Dalhe [1], E ingen [4], [5] and Lukaszewicz [9]. We obse e ha his model
o luid include as pa icula case he classical Na ie -S okes equa ions, which has been
widely s udied (see o ins ance he books by Ladyzhenskaya [6] o Temam [24], and he
e e ences he ein). In his case, since µ = 0, equa ions (1) and (2) decouple.
I is app op ia e o ecall ea lie wo ks on he ini ial- alue p oblems closely ela ed o
(1)-(2) in o de o cla i y he in ended con ibu ion o he p esen wo k.
Le us i s ly conside he si ua ion when Ω is a bounded egula domain. In his case,
Lukaszewicz [9] es ablished o a es ic ed class o ini ial da a, he exis ence o weak and
s ong global solu ions using, in bo h cases, an i e a i e linea ized scheme oge he wi h a
ixed poin esul . Fo ini ial da a simila o he case o he classical Na ie -S okes equa-
ions, by applying he spec al Gale kin me hod, Rojas-Meda & Bold ini [17] p o ed he
global exis ence and uniqueness o weak solu ions in he wo-dimensional case; exis ence
o he local and global in ime s ong solu ion was ob ained espec i ely by Rojas-Meda
in [18] and by O ega-To es and Rojas-Meda in [13]. O ega-To es and Rojas-Meda ,
ollowing he a gumen s gi en by Se in in [21], also conside ed he uniqueness o weak
solu ion in [14]. In [19], Rojas-Meda ob ained he con e gence a es associa ed o he
app oxima e solu ions cons uc ed by he Gale kin me hod. The exis ence o ep oduc i e
solu ion (so called pe iodic weak solu ion) o he p e ious sys em was p o ed in [17].
Recen ly, Res´endiz and Rojas-Meda [16] ha e p o ed he exis ence o weak solu ion in
a smoo h ime dependen domain. By using and in e ac i e app oach Rojas-Meda and
O ega-To es [20] show he exis ence and uniqueness o he s ong solu ions in bounded
domains in he L2-con ex . The exis ence and uniqueness o pe iodic s ong solu ions was
done in [10] using he Gale kin me hod. Yamaguchi [25] also s udied he p oblem (1)-(2)in
bounded domains using he semig oup app oach in Lp, 1 < p < ∞; he shows he exis ence
o global s ong solu ions o small da a.
The case o unbounded domains Ω is less s udied. When Ω is an ex e io domain, o he
ela ed model o he magne o-mic opola luid, exis ence o a s a iona y weak solu ion was
s udied by Du ´an e al. in [2], while he exis ence o ep oduc i e solu ion was es ablished
in [3]. Fo wo-dimensional unbounded domains, one can look a he wo d by Lukaszewicz
an Sadowski [12].
In he p esen wo k, as we said p e iously, we a e in e es ed in he low o mic opola
luids in bounded ou unbounded domains o R3wi h compac C2-bounda ies. By using
an i e a i e p ocedu e we will p o e he exis ence and uniqueness o s ong solu ions in
2
Mic opola luids in domains o R3
Lp(Ω), o any p > 3. Speci ically, we will p o e he ollowing (local) exis ence esul o
s ong solu ions.
Theo em 1.1 Le Ω⊂R3ha e a non- oid egula bounda y ∂Ωin he sense o Solonniko
and le p > 3. Assume ha u0(x)∈W2−
2
p(Ω),u0|ST= 0,di u0= 0,w0(x)∈W2−
2
p
p(Ω),
w0|ST= 0, ,g∈Lp(QT).
Then he e exis s T1∈(0, T ]such ha p oblem (1)-(2) has a unique solu ion (u,w, η)
sa is ying u∈W2,1
p(QT1),∇η∈Lp(QT1),w∈W2,1
p(QT1).
In his s a emen , we used he classical no a ions o he Sobole - ype spaces Wk
p(Ω) and
W2,1
p(QT).
The p esen wo k is o ganized as ollows: in Sec ion 2 we ix he no a ions, and s a e
p elimina ies esul s ha will be use ul in he es o he pape . Mo e p ecisely, we s a e he
exis ence, he uniqueness and egula i y (a p io i es ima es) o wo linea p oblems closely
ela ed o (1)-(2). We also desc ibe in his sec ion he i e a i e scheme ha cons uc
he app oxima e solu ions. In Sec ion 3, we ob ain es ima es in se e al no ms o such
app oxima e solu ions. Finally, in Sec ion 4, we show ha he app oxima e he solu ions
con e ge o a s ong solu ion o ou o iginal p oblem.
We ema k ha , as i is usual in his kind o con ex o simpli y he no a ions, we
will deno e by c,C0,M0and so on gene ic ini e posi i e cons an s depending only on Ω
and he o he ixed pa ame e s o he p oblem (like he ini ial da a). Tha is, hey may
ha e di e en alues in di e en exp essions. In a ew poin s o emphasize he ac ha
he cons an s a e in ac di e en , we use C1, C2, ..., M1, M2.· · · and so on.
2. P elimina ies and i e a i e scheme
Fo any ∈(0, T], we will deno e Q = Ω ×(0, ). As p e iously said, we will use
classical no a ions o he Sobole - ype spaces; we will also use eely he s anda d esul s
o such spaces. He e we jus ecall ha he es ic ion o a unc ion in W2,1
p(QT) on
he hype plane = cons an belongs o ∀ ∈[0, T ] o he Slobode skii-Beso space
W2−
2
p
p(Ω) and depend con inuously on in he no m o W2−
2
p
p(Ω). Mo eo e , i holds ha
ku(·, )k
W
2−
2
p
p(Ω)
≤ ku(·,0)k
W
2−
2
p
p(Ω)
+bckukW2,1
p(QT),(3)
whe e he cons an bcdoes no depend on ∈[0, T]. Fo mo e de ails o he Slobode skii-
Beso space see [8], o ins ance.
Nex , we ecall some esul s associa ed o wo linea p oblems closely ela ed o (1)-(2).
The i s esul is p o ed in Solonniko [23] and is he ollowing:
Lemma 2.1 Le F(x, )∈Lp(QT)and u0(x)∈W2−
2
p
p(Ω) wi h u0|ST= 0 and di u0= 0,
hen he ollowing p oblem
u −(µ+µ )∆u+∇η=F,
di u= 0,
u|ST= 0,
u(0) = u0(x)
3
J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda
has a unique solu ion u∈W2,1
p(QT),η∈W1,0
p(QT)(ηis unique up o a cons an ,)
sa is ying
kukW2,1
p(QT1)+k∇ηkLp(QT1)≤K1(T1)(ku0k
W
2−
2
p
p(Ω)
+kFkLp(QT1)),
whe e K1(·)is an inc easing unc ion o T1∈(0, T ]
The ollowing esul is a special case o he esul o pa abolic sys em gi en in [22].
Lemma 2.2 Le G(x, )∈Lp(QT)and w0(x)∈W2−
2
p
p(Ω) wi h w0|ST= 0, hen he
ollowing p oblem
w −(ca+cd)∆w−(c0+cd−ca)∇di w+ 4µ w=G
w|ST= 0,
w(0) = w0(x)
has a unique solu ion w∈W2,1
p(QT), sa is ying
kwkW2,1
p(QT1)≤K2(T1)(kw0k
W
2−
2
p
p(Ω)
+kGkLp(QT1)),
whe e K2(·)is an inc easing unc ion o T1∈(0, T ].
I e a i e Scheme:
Nex , we desc ibe he i e a ion scheme used o cons uc app oxima e solu ions o ou
p oblem.
Take
u(0) =0,w(0) =0
and o k= 1,2,3, . . . ecu si ely ake {u(k), η(k)}and {w(k)} espec i ely as he solu ions
o p oblems
u(k)
−(µ+µ )4u(k)+∇η(k)= + 2µ o w(k−1) −(u(k−1) · ∇)u(k−1),
di u(k)= 0,
u(k)|ST= 0,
u(k)(0) = u0(x)
and
w(k)
−(ca+cd)4w(k)−(c0+cd−ca)∇di w(k)+ 4µ w(k)
=g+2µ o u(k−1) −(u(k−1) · ∇)w(k−1),
w(k)|ST= 0,
w(k)(0) = w0(x).
3. Es ima es o he app oxima e solu ions
To ob ain he equi ed es ima es o he sequence (uk, ηk,wk), we s a by de ining:
Φ(k)(T1) = ku(k)kW2,1
p(QT1)+kw(k)kW2,1
p(QT1)+k∇η(k)kLp(QT1),(4)
o 0 < T1≤T
Then, we can p o e he ollowing wo lemmas.
4
Mic opola luids in domains o R3
Lemma 3.1 The elemen s o he sequence {w(k)}sa is y o any T1∈(0, T ] he ollowing
es ima e:
k∇w(k−1)kLp(QT1)≤C(kw0k
W
2−
2
p
p(Ω)
+aT
1−a
ap
1Φ(k−1)(T1) + Tδ1Φ(k−1)(T1)),
whe e Cis independen o T1∈(0, T ]and
a=p−3
2p−3and δ1= (1 −1
p)(1 −3
p)(1 −a) + 1−a
p.
Rema k 3.2 Analogous esul is alid o {u(k)}.
Lemma 3.3 Le 0< T1≤1. Then, he e is a cons an α > 0such ha
k(u(k−1) · ∇)w(k−1)kLp(QT1)≤C[ku0k2
W
2−
2
p
2
p(Ω)
+kw0k2
W
2−
2
p
2
p(Ω)
+Tα(Φ(k−1)(T1))2].
whe e Cis independen o T1∈(0, T ].
Nex , we p o e he boundness o he sequence {u(k), η(k),w(k)}.
Lemma 3.4 Fo su icien ly small T1∈(0, T ], he sequence {u(k), η(k),w(k)}is bounded
in W2,1
p(QT1)×Lp(QT)×W2,1
p(QT1).
4. P oo o Theo em 1.1
Se ing u(n,s)( ) = u(n+s)( )−u(n)( ), η(n,s)=η(n+s)−η(n)and w(n,s)=w(n+s)−w(n),
we ha e
u(n,s) −(µ+µ )4u(n,s)+∇η(n,s)=F(n,s),
di u(n,s)= 0,
u(n,s)|ST= 0,
u(n,s)(0) = 0,
(5)
whe e
F(n,s)= 2µ o w(n−1,s)−(u(n−1,s)· ∇)u(n+s−1) −(u(n−1) · ∇)u(n−1,s).(6)
Also
w(n,s)
−(ca+cd)4w(n,s)−(c0+cd−ca)∇di w(n,s)+ 4µ w(n,s)=G(n,s),
w(n,s)|ST= 0,
w(n,s)(0) = 0,
(7)
whe e
G(n,s)= 2µ o u(n−1,s)−(u(n+s−1) · ∇)w(n−1,s)−(u(n−1,s)· ∇)w(n−1).(8)
5

J.L. Bold ini, M. Du ´an, M.A. Rojas-Meda
We hen a e able o p o e ha
kF(n,s)kp
Lp(Q )≤cZ
0
ku(n−1,s)kp
W2,1
p(Qτ)dτ + (ku0k
W
2−
2
p
p(Ω)
+bcku(n−1+s)(τ)kW2,1
p(Q ))pZ
0bcpku(n−1,s)kp
W2,1
p(Qτ)dτ
+(ku0k
W
2−
2
p
p(Ω)
(9)
+bcku(n−1+s)(τ)kW2,1
p(Q ))pZ
0bcpku(n−1,s)kp
W2,1
p(Qτ)dτ.
kG(n,s)kp
Lp(Q )≤c(k∇u(n−1,s)kp
Lp(Q )+k(u(n−1,s)· ∇)w(n−1)kp
Lp(Q )
+k(u(n+s−1) · ∇)w(n−1,s)kp
Lp(Q )
≤cZ
0
ku(n−1,s)kp
W2,1
p(Qτ)dτ +c(kw0k
W
2−
2
p
p(Ω)
+bckw(n−1)(τ)kW2,1
p(Q ))pZ
0
ku(n−1,s)kp
W2,1
p(Qτ)dτ (10)
+c(ku0k
W
2−
2
p
p(Ω)
+bcku(n+s−1)kW2,1
p(Q ))pZ
0
kw(n−1,s)kp
W2,1
p(Qτ)dτ.
F om es ima es (9)-(10) and Lemma 3.4, we conclude ha o ∈[0, T1] and p > 3, i
we call
Ψ(n,s)( ) = ku(n,s)kW2,1
p(Q )+kw(n,s)kW2,1
p(Q )+k∇η(n,s)kLp(Q ),(11)
we hen ha e
Ψ(n,s)( )≤cµZ
0
Ψ(n−1,s)(τ)p¶1
p
.
The e o e, hΨ(n,s)( )ip≤cpZ
0hΨ(n−1,s)(τ)ip
dτ, (12)
and consequen ly Ψ(n,s)( )→0 as n→ ∞,∀ ∈[0, T1].
In pa icula , since W2,1
p(QT1) and Lp(QT1) a e Banach spaces, he e exis u,w∈
W2,1
p(QT1) and η∈Lp(QT1) such ha
un→us ongly in W2,1
p(QT1),
wn→ws ongly in W2,1
p(QT1),
ηn→ηs ongly in Lp(QT1).
The nex s ep is o ake he limi as n→+∞in he app oxima e equa ions in he
i e a i e scheme. Howe e , once he abo e con e gences ha e been es ablished, his is
s anda d and we ob ain ha u,w, η is a s ong solu ion o he p oblem (1)-(2).
6
Mic opola luids in domains o R3
We need only o conside he uniqueness o he solu ion in o de o comple e he p oo
o Theo em. Fo his, suppose ha he e exis s ano he solu ion u1,w1, η1o (1) and (2)
wi h he same egula i y as s a ed in he heo em. Then, de ine
U=u1−u, W =w1−w, P =η1−η,
and obse e ha hese auxilia y unc ions e i y a se o equa ions simila o (5)-(7).
Repea ing he a gumen s used o ob ain (12), using he known egula i y o he solu ions,
we ge o θ( ) = kUkp
W2,1
p(Q )+kWkp
W2,1
p(Q )+kPkp
Lp(Q )an inequali y o he ollowing
ype
θ( )≤cZ
0
θ(τ)dτ
which by G onwall’s inequali y implies ha U= 0, W= 0, P= 0 and hus he uniqueness
o ou s ong solu ions.
Acknowledgmen s
Du ing his esea ch J.L. Bold ini was pa ially suppo ed by CGCI MECD-DGU
B azil/Spain G an 2137-05-4; M.A. Rojas-Meda was pa ially suppo ed by D.G.E.S. and
M.C. y T. (Spain) G an BFM2003-06446-C02-01 and CGCI MECD-DGU B azil/Spain
G an 2137-05-4. These au ho s a e g a e ul o such suppo .
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8