Bol. Soc. Esp. Ma . Apl.
no41(2007), 101–116
A REVIEW ON REPRODUCTIVITY AND TIME PERIODICITY
FOR INCOMPRESSIBLE FLUIDS
B. CLIMENT-EZQUERRA∗, F. GUILL´
EN-GONZ´
ALEZ∗
AND M. ROJAS-MEDAR†
∗Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa, Espa˜na. †Depa amen o de Ciencias B´asicas,
Uni e sidad del B´ıo-B´ıo, Chile.
[email p o ec ed] [email p o ec ed] [email p o ec ed]
Abs ac
In his a icle, ou aims is o e iew some o he esul s ha a e
cu en ly a ailable conce ning he exis ence, uniqueness and egula i y o
ep oduc i e and ime pe iodic solu ions o he Na ie -S okes equa ions
and some a ian s. By he way, we p esen some open p oblems.
Key wo ds: Rep oduc i e and ime pe iodic solu ions, Na ie -S okes ype
equa ions, egula i y o solu ions
AMS subjec classi ica ions: 35B10 35Q35 76D03
1 In oduc ion
We s udy some p oblems ela ed wi h ime pe iodic solu ions o models o
incomp essible luids.
We s a ecalling he main ideas o p o e he exis ence o ep oduc i e
weak solu ions (i.e. weak solu ions de ined in he ime in e al (0, T) aking he
same ini ial and inal alues in ime) o he Na ie -S okes equa ions and some
a ian s whe e hese ideas a e applicable, such as Boussinesq, mic opola and
magne o-mic opola models. This p oo elies on he ob en ion o ime pe iodic
Gale kin app oxima ions ia Le ay-Schaude poin ixed a gumen .
Mo eo e , in he case o 2Ddomains, using he uniqueness o weak solu ions,
he egula izing p ope y o he sys em and he exis ence o global egula
solu ions when da a a e egula , one has ha he pe iodic in ime weak
solu ions de ined as ex ension o ep oduc i e solu ions o he whole ime
in e al (0,+∞) will be egula solu ions. An ex ension o hese esul s o he
3Dcase is possible imposing small enough ex e nal o ce, using he so called
“weak/s ong uniqueness” and he global s ong solu ions o small enough da a
(see Sec ion 5).
Pa ially suppo ed by DGI-MEC (Spain), G an MTM2006–07932 and CGCI MECD-
DGU B azil/Spain G an 117/06
101
102 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda
Also, we s udy in Sec ion 4 some coupled models o eloci y and
p essu e dynamic a iables wi h ano he a iable whe e he maximum p inciple
holds, such as he gene alized Boussinesq model (wi h empe a u e-dependen
iscosi y) and a nema ic liquid c ys al model wi h a Ginzbu g-Landau
penaliza ion. In hese cases one has, hanks o an adequa e e o mula ion o
he p oblem by unca ion, exis ence o ep oduc i e weak solu ions as limi o
ime pe iodic Gale kin app oxima ions. I is impo an o ema k ha Gale kin
app oxima ions do no e i y he maximum p inciple bu hei limi does.
Finally, we will see ha , o hese models ela ed wi h he maximum
p inciple, he a gumen o p o e egula i y o ep oduc i e solu ions in he
Na ie -S okes amewo k (see Sec ion 5 below) a e no alid in gene al. The
pa icula case o gene alized Boussinesq model wi h Neumann bounda y
condi ion o he empe a u e can be sol ed wi h o he a gumen s, bu he
case o nema ic liquid c ys al model emains as an open p oblem.
2 Na ie -S okes equa ions
The mode n heo y o he Na ie -S okes equa ions began in he 1930s wi h
Le ay’s pionee ing wo k ([10]).
Le Ω ⊂IRd(d= 2 o 3) a bounded and egula enough domain illed by he
luid, and [0, T] he ime in e al. We deno e Q= (0, T)×Ω and Σ = (0, T)×∂Ω.
In he case whe e he luid is subjec o he ac ion o a body o ce , he
Na ie -S okes equa ions can be w i en as ollows
∂u
∂ + (u· ∇)u−ν∆u+∇p= ,di u= 0 ,(1)
whe e u=u(x, ) is he eloci y ield e alua ed a he poin x∈Ω and a
ime ∈[0, T], p=p(x, ) is he p essu e ield and ν > 0 is he coe icien o
kinema ical iscosi y (which is aken cons an ). This sys em can be comple ed
wi h se e al bounda y condi ions. Fo simplici y, we ix he ollowing non-slip
bounda y condi ions:
u( , x) = 0,x∈∂Ω, > 0 (2)
Finally, supplemen a y condi ions in ime mus be conside ed. The mo e
classical is he ini ial condi ion:
u(0,x) = u0(x),x∈Ω (3)
O he possibili y is o change his ini ial condi ion by he ollowing ime-pe iodic
condi ion:
u(0,x) = u(T, x),x∈Ω.(4)
Ma hema ical p ope ies o sys em (1) ha e been deeply in es iga ed o e
he yea s and a e s ill he objec o p o ound esea ches.
We in oduce some space unc ions. Le V he ec o ial space o med by
all ields ∈C∞
0(Ω)dsa is ying ∇ · = 0. We conside he Hilbe spaces H
Rep oduc i i y and ime pe iodici y o incomp essible luids 103
( espec i ely V) as he closu e o Vin L2( espec i ely H1). Fu he mo e, one
has
H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω},
V={u∈H1;∇ · u= 0,u=0on ∂Ω}
We deno e L2
0(Ω) = ½p∈L2(Ω) : ZΩ
p dx = 0¾.
2.1 Main classical esul s o he ini ial-bounda y p oblem
De ini ion 1 Gi en u0∈Hand ∈L2(0, T;H−1(Ω)), i will said ha uis a
weak solu ion o he p oblem (1),(2), (3) in (0, T ), i
u∈L2(0, T;V)∩L∞(0, T;H),
and e i ies (3) and he a ia ional o mula ion
ZT
0ZΩn−u( ) ′( ) + ∇u( ) : ∇u( )−(u( )· ∇) ( )u( )− ( ) ( )odxd = 0,
o all ∈C1([0, T ]; H)∩C([0, T ]; V), wi h compac suppo con ained in (0, T).
In addi ion, i u0∈Vand ∈L2(0, T;L2(Ω)) any weak solu ion will be a
s ong solu ion i
u∈L2(0, T;H2∩V)∩L∞(0, T;V),u ∈L2(0, T ;H), p ∈L2(0, T;H1∩L2
0(Ω))
and e i ies he sys em (1) poin wise a.e. in (0, T)×Ω.
Rema k 1 The p e ious de ini ion can be ex end o he case o inal ime
T=∞changing he egula i y L2(0, T)by L2
loc(0,+∞).
The ollowing esul s hold.
Theo em 1 [22] Fo any u0∈Hand ∈L2(0, T;H−1(Ω)), he p oblem (1)-
(2) has (a leas ) a weak solu ion. I Ω⊂IR2, one has uniqueness o weak
solu ions.
Theo em 2 [22] Fo any u0∈Vand ∈L∞(0,∞;L2(Ω)), he p oblem (1)-
(2) has a unique s ong solu ion (u, p)local in ime, de ined in (0, T⋆)wi h
T⋆>0small enough. In ac , i a solu ion has he s ong egula i y, i
coincides wi h any weak solu ion associa ed wi h he same da a ( his p ope y is
called weak/s ong uniqueness). Mo eo e , his s ong solu ion is global in ime,
de ined in he whole ime in e al (0,∞)i ei he Ω⊂IR2o Ω⊂IR3and da a
(u0, )a e small enough in hei espec i e spaces V×L∞(0,∞;L2(Ω)).
104 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda
2.2 On he ime-pe iodic weak solu ions
Theo em 3 [8] Fo any ∈L2(0, T;H−1(Ω)), he e exis s a weak solu ion o
(1)-(2) and (4), (i.e. he weak solu ion uhas he so-called ep oduc i e p ope y:
u(0, x) = u(T, x)).
No ice ha he ime pe iodic ex ension, e
u, o any weak ep oduc i e solu ion
u o he whole ime in e al (0,+∞) is a pe iodic weak solu ion o (1)-(2)
co esponding o he da a, e , de ined as he ime pe iodic ex ension o .
Main ideas o he p oo o Theo em 3
Le uk he unique app oxima e solu ion o he Gale kin ini ial-bounda y
p oblem o Na ie -S okes in he ini e-dimensional subspace Vk, spanned by he
i s kelemen s o he “spec al” basis o V(o hogonal in Vand o hono mal
in H), associa ed o a ini ial disc e e da a uk
0∈Vk.
Since V֒→H, he e exis s a Poinca ´e cons an c1>0 such ha
c1kukk2
L2≤ k∇ukk2
L2,
hus, om ene gy inequali y, we ha e
d
d kukk2
L2+c1kukk2
L2≤Ck k2
H−1,(5)
o equi alen ly d
d (ec1 kukk2
L2)≤C ec1 k k2
H−1.
In eg a ing om 0 o T, we ha e
ec1Tkuk(T)k2
L2≤ kuk(0)k2
L2+CZT
0
ec1 k ( )k2
H−1d . (6)
Now, we de ine he ope a o Lk: [0, T]→Rkas ollows
Lk( ) = (ck
1( ),...,ck
k( ))
whe e ck
i( ), i = 1,...,k, a e he coe icien s o he expansion o uk( ) in Vk.
No e ha
kLk( )kRk=kukkL2,
because we ha e choose he (o hono mal in L2) spec al basis in V.
We de ine he ope a o Φk:Rk→Rkas ollows: Gi en Lk
0∈Rk, we de ine
Φk(Lk
0) = Lk(T), whe e Lk( ) a e he coe icien s o he Gale kin solu ion wi h
ini ial alue wi h coe icien s Lk
0. I is easy o see ha Φkis con inuous and we
wan o p o e ha Φkhas a ixed poin .
Fo his, hanks o he Le ay-Schaude Theo em, i su ices o show ha o
all λ∈[0,1], he possible solu ions o he equa ion
Lk
0(λ) = λΦk(Lk
0(λ)),(7)
Rep oduc i i y and ime pe iodici y o incomp essible luids 105
a e bounded independen ly o λ.
Since Lk
0(0) = 0, i su ices o conside λ∈(0,1]. In his case, (7) is
equi alen o Φk(Lk
0(λ)) = 1
λLk
0(λ). Mo eo e , by he de ini ion o Φkand (6),
one ob ains
ec1T|| 1
λLk
0(λ)||2
Rk≤ kLk
0(λ)k2
Rk+cZT
0
ec1Tk ( )k2
H−1d ,
which implies
kLk
0(λ)k2
Rk≤cRT
0ec1Tk ( )k2
H−1d
ec1T−1=M,
o each λ∈(0,1]. This bound is independen o λ∈[0,1] and k. Consequen ly,
Le ay-Shaude Theo em implies he exis ence o a leas one ixed poin o Φk,
ha is he exis ence o ep oduc i e Gale kin solu ion.
Thus, since p e ious es ima es a e independen o k, one has he same
es ima es o hese ep oduc i e Gale kin solu ions.
Finally, he con e gence o a subsequence o a ep oduc i e solu ion o
(1),(2), (4) hold.
2.3 Rela ion be ween weak pe iodic solu ions and global solu ions
Assume : [0,+∞)→H−1(Ω) and T- ime pe iodic.
Na ie -S okes 2D
One has (see Theo em 1) uniqueness o weak solu ion o he ini ial-bounda y
p oblem (associa ed o any ini ial da a u0). Consequen ly, gi en a ep oduc i e
solu ion uassocia ed o u(0) = u(T) := u0, hen uis he (unique) solu ion o
he ini ial-bounda y p oblem associa ed o he ini ial da a u0, which is de ined
o all ime ∈(0,∞). Mo eo e , his solu ion is T-pe iodic, because in (T, 2T)
mus be equal o he ep oduc i e solu ion de ined as ¯
u( ) = u( −T) (which
e i ies u(T) = u(2T) = u0) and so on.
Finally, using egula i y o solu ion u o s ic ly posi i e imes (see [5]), i
is easy o p o e ha e e y pe iodic solu ion is egula .
Na ie -S okes 3D
Since uniqueness o weak solu ion is no known, i is possible ha he
ep oduc i e solu ion uand he global weak solu ion ˜
uassocia ed o he ini ial
da a u0:= u(0) = u(T) a e di e en in (0, T), al hough hey coincide locally in
ime, nea o he ini ial ime = 0.
2.4 Open p oblems
Na ie -S okes wi h la ge Reynolds numbe and a eac ion e m
adding ene gy
P e ious a gumen s o he p oo o ep oduc i e solu ions a e based on
(exponen ial) dec easing o ene gy ( hanks o dissipa i e e ms). Na u ally, he

106 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda
same a gumen , is applicable o models wi h ene gy s ic ly dec easing in ini e
ime. Bu his is no always possible. Fo ins ance, we conside he ollowing
Na ie -S okes sys em wi h la ge Reynolds numbe and a eac ion e m adding
ene gy:
∂ u−ε∆u−u+∇p= ,∇ · u= 0,
u(0) = u(T),u|Σ= 0.(8)
The ene gy inequali y is
∂ kuk2
L2+εk∇uk2
L2≤C(k k2
H−1+kuk2
L2).
Assuming εsmall enough such ha kuk2
L26< εk∇uk2
L2, he s ic ly dec easing
in ime o kuk2
L2is no clea . Consequen ly, he exis ence o ime-pe iodic weak
solu ions o (8) emains as an open p oblem.
Ex e io domains
Assume Ω is an ex e io domain whe e he Poinca ´e inequali y is no ue. Then,
o show he exis ence o ep oduc i e solu ions one could use he “embedding
domain echnique” oge he wi h he Gale kin Me hod, ob aining ep oduc i e
solu ions in a sequence o (bounded) unca ed domains, see o ins ance
[6, 18, 17]. Howe e , since Poinca ´e imbedding is no applicable, i is no clea
he con oll o he pass o he limi om unca ed domains o he whole domain.
Some pa ial esul s a e known. Fo example, he exis ence o s ong pe iodic
solu ions o he Na ie -S okes equa ions in he ollowing unbounded domains,
ei he Ω is he whole space Rno he hal -space Rn
+has been in es iga ed
by Kozono and Nakao [9] and Taniuchi [21] using he semig oup app oach.
By using po en ial heo y, Ma emon i [15] p o ed he exis ence o a unique
ime pe iodic solu ion on he whole space R3 o small ex e nal o ce. The
p oblem, in he hal -space R3
+,was conside ed in [16]. Kozono and Nakao [9],
making use o Lp−L es ima es o he semig oup gene a ed by he S okes
ope a o , cons uc ed ime-pe iodic solu ions o small ime-pe iodic o ces and
he s abili y o hese solu ions was conside ed in [21]. Yamazaki [23] analyzed
he same p oblem o [9] in Mo ey spaces.
3 Some a ian s o Na ie -S okes equa ions
We can apply he a gumen o ind ep oduc i e solu ions done o Na ie -S okes
in he p eceden Sec ion, o some a ian s:
3.1 Boussinesq equa ions
The Boussinesq sys em o hyd odynamics equa ions (see Joseph [7]) a ise om
ze o o de app oxima ion o he coupling be ween he Na ie -S okes equa ion
and he he modynamic equa ion. Such a ma hema ical model eads:
Rep oduc i i y and ime pe iodici y o incomp essible luids 107
Find he ield u:Q→R3, he scala unc ions (θ, p) : Q→R2which sa is y
he sys em o equa ions:
∂u
∂ −ν∆u+ (u· ∇)u+∇p=αθg+ in Q,
∇ · u= 0 in Q, (9)
∂θ
∂ −χ∆θ+ (u· ∇)θ= 0 in Q.
wi h ∂nθ= 0 on ∂Ω and RΩθ= 0
He e u, p, θ deno e he eloci y, he p essu e and he empe a u e,
espec i ely. gdeno es he g a i a ional ield, α > 0 is a cons an associa ed
o he coe icien o olume expansion and is a ield o ex e nal o ces. Again,
ν > 0 is he iscosi y coe icien . Finally, χ > 0 is he he mal conduc i i y
coe icien .
This sys em is comple ed wi h he bounda y condi ions ( o ins ance)
u|Σ= 0, θ|Σ= 0 and he ime-pe iodic condi ions u(0) = u(T), θ(0) = θ(T) in
Ω.
By aking uand θas es unc ion in he u-sys em and θ-equa ion o (9)
espec i ely, adding he esul ing equali ies conside ing an adequa e balance (in
o de o elimina e he e m ha con ains g), we ob ain
d
d kuk2
L2+βd
d kθk2
L2+νk∇uk2
L2+βχk∇θk2
L2≤ k k2
H−1,(10)
whe e βis a big enough numbe depending on αand kgkL∞. This oge he
wi h he Poinca ´e inequali y gi es an inequali y o ype (5). Indeed, i su ices o
conside a Gale kin app oxima ion o bo h a iables, eloci y and empe a u e,
and o ollow he p oo o Theo em 2.3, changing ukby (uk, θk).
Ano he bounda y condi ions a e possible: Neumann, mixed, e c, whene e
an inequali y o (u, θ) simila o (5) holds.
3.2 Mic opola equa ions
The equa ions ha desc ibes he mo ion o a incomp essible iscous and
mic opola luids in Qa e gi en by (see [12])
∂u
∂ + (u· ∇)u−(ν+ν )∆u+∇p= 2ν o w+ ,
di u= 0,(11)
∂w
∂ + (u· ∇)w−(ca+cd)∆w−(c0+cd−ca)∇di w+ 4ν w
= 2ν o u+g.
The unc ions u:Q→R3,w:Q→R3and p:Q→Rdeno e he line
eloci y, he angula eloci y (o o a ion o pa icles) and he p essu e o he
luid, espec i ely. The unc ions :Q→R3and g:Q→R3deno e ex e nal
108 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda
sou ces o linea and angula momen um, espec i ely. The posi i e cons an s
ν, ν , c0, caand cda e iscosi ies, such ha c0+cd> ca.
This sys em is comple ed wi h he bounda y condi ions u|Σ= 0, w|Σ= 0
( o ins ance) and he ime-pe iodic condi ions u(0) = u(T), w(0) = w(T) in
Ω.
By aking uand was es unc ion in he u-sys em and w-sys em o
(11) espec i ely, adding he esul ing equali ies, aking in o accoun ha
2ν ( o w,u)+2ν ( o u,w) = 4ν ( o u,w) and |∇u|2=| o u|2, we ob ain
d
d kuk2
L2+kwk2
L2+νk∇uk2
L2+ (ca+cd)k∇wk2
L2+ (c0+cd−ca)kdi wk2
L2
≤C(k k2
H−1+kgk2
H−1).
S a ing om his inequali y, he a gumen ollows as in p e ious sec ion.
3.3 O he models
O he luid models whe e one has exis ence o ep oduc i e solu ions a e:
magne ohyd odynamic model [14], Magne o-mic opola luid mo ion [20], a
con ec ion-di usion model desc ibing bina y alloy solidi ica ion p ocesses [3],
e c.
4 Rep oduc i i y and maximum p inciple
Gi en u:Q→R3such ha ∇ · u= 0 in Qand u·n= 0 on ∂Ω, we conside
he ( ep oduc i e) di usion-ad ec ion p oblem o he unknown c:Q→R(a
concen a ion):
∂ c−∆c+u· ∇c= 0, c|Σ=cΣ, c(0) = c(T),
whe e 0 < c ≤cΣ≤con Σ, o some cons an s cand c. In pa icula ,
∂ (c−c)−∆(c−c) + (u· ∇)(c−c) = 0 in Q.
Mul iplying by (c−c)+and in eg a ing in Ω (no ice ha (c−c)+= 0 on Σ),
one has d
d ZΩ
|(c−c)+|2+ZΩ
|∇(c−c)+|2≤0.
In eg a ing in ∈(0, T) and using he pe iodic condi ion c(0) = c(T), one
a i es a ZT
0
k∇(c−c)+k2
L2= 0.
Hence c≤cin Qhold. Simila ly c≥cin Qhold.
The e o e, one has he ollowing conclusion: The ep oduc i e solu ion
conse e he maximum p inciple.
In he ollowing models, he maximum p inciple has an impo an ole.
Rep oduc i i y and ime pe iodici y o incomp essible luids 109
4.1 Gene alized Boussinesq sys em, wi h di usion depending on
empe a u e
When he iscosi y and hea conduc i i y a e empe a u e dependen un ions
in he Boussinesq sys em, one has he ollowing sys em:







∂ u− ∇ · (ν(θ)∇u) + (u· ∇)u+∇p=αθg+ ,
∇ · u= 0,
∂ θ− ∇ · (k(θ)∇θ) + (u· ∇)θ= 0,
(12)
whe e ν: IR →IR+and k: IR →IR+a e s ic ly posi i e con inuous unc ions
( he kinema ic iscosi y and he he mal conduc i i y espec i ely).
The p oblem is o ind a egula solu ion {u, θ, p}o (12) in Ω ×[0, T ],
oge he he ollowing bounda y Di ichle da a:
u= 0, θ =θ∂Ωon ∂Ω×[0, T),(13)
and ime-pe iodic condi ions:
u(0) = u(T), θ(0) = θ(T) in Ω.(14)
We de ine
θmin = min θ∂Ωθmax = max θ∂Ω.
Thanks o he maximum p inciple, one has θmin ≤θ≤θmax in Q. Then, he e
exis s νmin >0, kmin >0, νmax >0 and kmax >0 such ha
νmin ≤ν(s)≤νmax and kmin ≤k(s)≤kmax,∀s∈[θmin, θmax].
One can p o es he exis ence o ep oduc i e solu ion in he same way ha
in he classical Boussinesq case (see Sec ion 3.1), conside ing he equi alen
p oblem ha esul changing νby eνand kby ek, whe e
eν(θ) = 






ν(θmin) i θ < θmin,
ν(θ) i θmin ≤θ≤θmax,
ν(θmax) i θ > θmax,
ek(θ) = 






k(θmin) i θ < θmin,
k(θ) i θmin ≤θ≤θmax,
k(θmax) i θ > θmax.
4.2 Penalized Nema ic liquid c ys al model
We assume he ollowing nema ic liquid c ys al model in (0, T)×Ω, whe e
Ω⊂RN o N= 2 o 3 is an open bounded domain:
(∂ u+ (u· ∇)u−µ∆u+∇p=−λ∇ · (∇d ∇d),∇ · u= 0,
∂ d+ (u· ∇)d=γ(∆d− ε(d)).(15)
116 B. Climen -Ezque a, F. Guill´en-Gonz´alez y M. Rojas-Meda
[17] A.C. Mo e i, M.A. Rojas-Meda and M.D. Rojas Meda . Rep oduc i e
weak solu ions o gene alized Boussinesq models in ex e io domains. Ma .
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