Na ie -S okes equa ions wi h delays on
unbounded domains
By Ma ´
ıa Jos´
e Ga ido-A ienza, & Ped o Ma ´
ın-Rubio †
Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa,
Apdo. de Co eos 1160,
41080-Se illa.
Spain
Some esul s on he exis ence and uniqueness o solu ions o Na ie -S okes equa-
ions when he domain is unbounded and he ex e nal o ce con ains some he edi-
a y cha ac e is ics a e p o ed o bo h he e olu iona y and he s a iona y cases.
Exponen ial s abili y o he s a iona y solu ion is also es ablished in dimension wo.
Keywo ds: Na ie -S okes equa ions, a iable and dis ibu ed delays,
unbounded domains.
1. In oduc ion and s a emen o he p oblem
The Na ie -S okes equa ions go e n he mo ion o usual luids like wa e , ai , oil,
e c. These equa ions ha e been he objec o nume ous wo ks since he i s pape o
Le ay was published in 1933 (see Cons an in & Foias 1988; Lions 1969; Temam 1979,
and he e e ences he ein), e en wi h unbounded domains, allowing he possibili y
o channel and mul i-channel lows among o he a ia ions (see o ins ance Rosa
1998; Temam 1979). On o he hand, delay e ec s ha e been p o ed o be use ul
in many physical and biological si ua ions. These si ua ions may appea when we
wan o con ol he sys em (in a ce ain sense) by applying a o ce which akes in o
accoun no only he p esen s a e o he sys em bu he his o y o he solu ion.
To ou knowledge, his has been a ely ea ed in he con ex o Na ie -S okes
equa ions (c . Ca aballo & Real 2001, 2003, 2004). To da e, we ha e no ound in he
li e a u e any wo k ha conside s he combina ion o delay e ms and unbounded
domains.
The aim o he pape is wo- old: i s ly we conside se e al si ua ions in which
he ex e nal o ce con ains some he edi a y ea u es and he domain is no bounded,
and p o e exis ence ( o dimension N= 2 and 3) and uniqueness (N= 2) o
solu ions. In a second pa o he pape he exis ence and uniqueness o a s a iona y
solu ion a e es ablished in dimensions 2 and 3,and, in he case N= 2 exponen ial
s abili y o he solu ion unde an addi ional assump ion.
Le Ω ⊂RN(N= 2 o 3) be an open se wi h bounda y Γ ha is no necessa ily
bounded bu sa is ies a Poinca ´e inequali y:
The e exis s λ1>0 such ha ZΩ|φ|2dx≤1
λ1ZΩ|∇φ|2dx, ∀φ∈H1
0(Ω).(1.1)
†E-mails: [email p o ec ed] ; [email p o ec ed]
A icle submi ed o Nonlinea Analysis TMA T
EX Pape
2M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
Conside he ollowing unc ional Na ie -S okes p oblem ( o u he de ails and
no a ions see Lions 1969 and Temam 1979):

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
∂u
∂ −ν∆u+PN
i=1 ui
∂u
∂xi
= ( )−∇p+g( , u ) in (0, T)×Ω,
di u= 0 in (0, T)×Ω,
u= 0 on (0, T)×Γ,
u(0, x) = u0(x), x ∈Ω,
u( , x) = φ( , x), ∈(−h, 0) x∈Ω,
whe e we assume ha T > 0 is gi en, ν > 0 is he kinema ic iscosi y, uis he
eloci y ield o he luid, p he p essu e, u0 he ini ial eloci y ield, a non-
delayed ex e nal o ce ield, gano he ex e nal o ce con aining some he edi a y
cha ac e is ic and φ he ini ial da um in he in e al o ime (−h, 0),whe e his a
posi i e ixed numbe .
To s a , we conside he ollowing usual abs ac spaces:
V=nu∈(C∞
0(Ω))N: di u= 0o,
H= he closu e o Vin (L2(Ω))Nwi h he no m |·|,and inne p oduc (·,·)
whe e o u, ∈(L2(Ω))N,
(u, ) =
N
X
j=1 ZΩ
uj(x) j(x)dx,
V= he closu e o Vin (H1
0(Ω))Nwi h he no m [ hanks o (1.1)] k·k associa ed
o he inne p oduc ((·,·)),whe e o u, ∈(H1
0(Ω))N,
((u, )) =
N
X
i,j=1 ZΩ
∂uj
∂xi
∂ j
∂xi
dx.
I ollows ha V⊂H≡H0⊂V0,whe e he injec ions a e dense and con inuous.
We will use k·k∗ o he no m in V0and h·,·i o he duali y hV0, V i.Now we deno e
a(u, ) = ((u, )), and de ine he ilinea o m bon V×V×Vby
b(u, , w) =
N
X
i,j=1 ZΩ
ui
∂ j
∂xi
wjdx∀u, , w ∈V.
Le Xbe a Banach space. Gi en a unc ion u: (−h, T)→X, o each ∈(0, T)
we deno e by u he unc ion de ined on (−h, 0) by he ela ion u (s) = u( +s), s ∈
(−h, 0).
In o de o s a e he p oblem in he co ec amewo k, le us i s es ablish
sui able assump ions on he e m in which he delay is p esen .
In a gene al way, le Xand Ybe wo sepa able Banach spaces, and g: [0, T ]×
C0([−h, 0]; X)→Ysuch ha
(I) o all ξ∈C0([−h, 0]; X), he mapping ∈[0, T]→g( , ξ)∈Yis measu -
able,
A icle submi ed o Nonlinea Analysis TMA
Na ie -S okes equa ions wi h delays on unbounded domains 3
(II) o each ∈[0, T], g( , 0) = 0,
(III) he e exis s Lg>0 such ha ∀ ∈[0, T],∀ξ, η ∈C0([−h, 0]; X)
kg( , ξ)−g( , η)kY≤Lgkξ−ηkC0([−h,0];X),
(IV) he e exis s Cg>0 such ha ∀ ∈[0, T],∀u, ∈C0([−h, T]; X)
Z
0kg(s, us)−g(s, s)k2
Yds≤CgZ
−hku(s)− (s)k2
Xds.
Obse e ha (I)-(III) imply ha gi en u∈C0([−h, T]; X), he unc ion gu: ∈
[0, T]→Yde ined by gu( ) = g( , u )∀ ∈[0, T], is measu able (see Bensoussan e
al. 1992) and, in ac , belongs o L∞(0, T;Y). Then, hanks o (IV), he mapping
G:u∈C0([−h, T]; X)→gu∈L2(0, T;Y)
has a unique ex ension o a mapping e
Gwhich is uni o mly con inuous om L2(−h, T;X)
in o L2(0, T;Y). F om now on, we will deno e g( , u ) = e
G(u)( ) o each u∈
L2(−h, T;X), and hus, ∀ ∈[0, T],∀u, ∈L2(−h, T;X),we will ha e
Z
0kg(s, us)−g(s, s)k2
Yds≤CgZ
−hku(s)− (s)k2
Xds.
Wi h he con en ion abo e, assume ha u0∈H,φ∈L2(−h, 0; V) , ∈L2(0, T ;V0),
and he ollowing delay ope a o s:
g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))N
sa is ying hypo heses (I)-(IV) wi h X=V,Y= (L2(Ω))N,Lg1=L1and Cg1=C1,
and
g2: [0, T]×C0([−h, 0]; V)→V0
sa is ying hypo heses (I)-(IV) wi h X=V,Y=V0,Lg2=L2and Cg2=C2.
We a e in e es ed in he ollowing p oblem:
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To ind u∈L2(−h, T;V)∩L∞(0, T;H) such ha , o all ∈V,
d
d (u( ), ) + νa(u( ), ) + b(u( ), u( ), ) = h ( ), i+ (g1( , u ), )
+hg2( , u ), i,
u(0) = u0, u( ) = φ( ), ∈(−h, 0),
(1.2)
whe e he equa ion in (1.2) mus be unde s ood in he sense o D0(0, T ).
Rema k 1.1. Obse e ha he e ms in (1.2) a e well de ined. In pa icula , by
hypo heses (I)-(IV), i u∈L2(−h, T;V) he e m g1( , u )de ines a unc ion in
L2(0, T; (L2(Ω))N), and he e m g2( , u )de ines a unc ion in L2(0, T;V0). Thus
(see Lions 1969), i u∈L2(−h, T;V)∩L∞(0, T;H)sa is ies he equa ion in (1.2),u
is weakly con inuous om [0, T ]in o H, and he e o e he ini ial condi ion u(0) = u0
makes sense. O cou se, o N= 2, i he e exis s a solu ion u o he p oblem (1.2),
i hen belongs o he space C0([0, T ]; H).
In Sec ion 2 we shall p o e exis ence o solu ions o (1.2) and he uniqueness
o solu ion o he p oblem in he case N= 2.In Sec ion 3, gene al si ua ions
con aining delayed e ms – a iable and dis ibu ed– a e conside ed. We inish wi h
he s udy o exis ence and uniqueness o a s a iona y solu ion and i s exponen ial
s abili y ( o N= 2) in Sec ion 4.
A icle submi ed o Nonlinea Analysis TMA
4M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
2. Exis ence o solu ions
In his sec ion we will p o e a gene al heo em on he exis ence o solu ions when
N= 2 o 3, and uniqueness i N= 2.
In he p oo o exis ence we will need he ollowing wo esul s:
Theo em 2.1. (c . [Ca aballo & Real 2001, Theo em A.1]) Le u0∈Rm,φ∈
L2(−h, 0; Rm), k ∈L2(0, T ;Rm),g: [0, T]×C0([−h, 0]; Rm)→Rmsa is ying
hypo heses (I)-(IV) wi h X=Y=Rm, and : [0, T]×Rm→Rma con inuous
unc ion such ha ( , 0) = 0 and o all n > 0 he e exis s Ln>0such ha
| ( , u)− ( , )|Rm≤Ln|u− |Rm,∀|u|Rm≤n, | |Rm≤n, ∀ ∈[0, T ].
Then:
a) Fo each ∗∈(0, T ] he e exis s a mos one solu ion o he p oblem
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To ind u∈L2(−h, ∗;Rm)∩C0([0, ∗]; Rm)such ha
u( ) = φ( ), ∈(−h, 0),
u( ) = u0+Z
0
(s, u(s)) ds+Z
0
g(s, us) ds+Z
0
k(s) ds∀ ∈[0, ∗].
(2.1)
b) The e exis s ∗∈(0, T]such ha he e exis s one (and only one)solu ion o he
p oblem (2.1).
c) Suppose ha he e exis s a cons an C > 0such ha i ∗∈(0, T ]is such
ha he e is a solu ion uo (2.1), hen max ∈[0, ∗]|u( )|Rm≤C. Then, unde his
addi ional assump ion, he e exis s a solu ion o p oblem (2.1) wi h ∗=T.
Theo em 2.2. (c . [Simon 2003, Co olla y 2.34]) Le Θbe a bounded open se o
Rd,and X⊂EBanach spaces wi h compac injec ion. Conside 1≤ < q ≤ ∞.
Suppose F⊂L (Θ; E)sa is ies
(i) ∀ω⊂⊂ Θ,sup ∈Fkτh − kL (ω;E)→0when h→0 [whe e τh is he
ansla ion: (τh )(x) = (x+h)],
(ii) Fis bounded in Lq(Θ; E)∩L1(Θ; X).
Then Fis p ecompac in L (Θ; E).
The main esul in his sec ion needs ex a no a ion o an addi ional condi ion
(V), which will be discussed in some de ail below in Rema k 2.5.
Deno e V(O) he same space as Vbu wi h an open se Oins ead o Ω,and
analogously de ine V(O) he closu e o V(O) in (H1
0(Ω))N.
Theo em 2.3. Le u0∈H,φ∈L2(−h, 0; V), ∈L2(0, T;V0), and assume ha
g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))Nand g2: [0, T]×C0([−h, 0]; V)→V0sa is y
hypo heses (I)-(IV) in hei co esponding spaces. Then:
a) I N= 2 and ν2> C2, he e exis s a mos one solu ion o p oblem (1.2).
b) I N∈ {2,3}and ν2> C2, he e exis s a solu ion o (1.2) i , in addi ion,
he ollowing assump ion (V) holds:
(V) I mcon e ges weakly o in L2(−h, T;V),weakly-s a in L∞(0, T;H),
and s ongly in L2(−h, T; (L2(O))N) o a bounded open se O ⊂ Ωwi h smoo h
bounda y, hen gi(·, m
·)con e ges weakly o gi(·, ·)in L2(0, T ;V(O)0) o i= 1,2.
A icle submi ed o Nonlinea Analysis TMA
Na ie -S okes equa ions wi h delays on unbounded domains 5
P oo . a) Uniqueness o N= 2 ollows as he case wi h bounded domain gi en
in Theo em 2.1 in [Ca aballo & Real 2001] and i is ep oduced he e o he sake o
comple eness. I ν2> C2, le u, be wo solu ions o (1.2) and se w=u− . Then,
om he ene gy equali y, and he bound o he ilinea o m (see Ladyzhenskaya
1992), i ollows ha o all ∈(0, T)
|w( )|2+ 2νZ
0kw(s)k2ds=−2Z
0
b(w(s), u(s), w(s)) ds
+2 Z
0
(g1(s, us)−g1(s, s), w(s)) ds
+2 Z
0hg2(s, us)−g2(s, s), w(s)ids
≤21/2Z
0|w(s)|kw(s)kku(s)kds
+2 Z
0|g1(s, us)−g1(s, s)||w(s)|ds
+2 Z
0kg2(s, us)−g2(s, s)k∗||w(s)|| ds.
Then, om assump ion (IV), aking in o accoun ha w(s) = 0 o s∈(−h, 0),
and deno ing 2ε=ν−√C2>0, we ha e o all ∈(0, T )
|w( )|2+ 2νZ
0kw(s)k2ds≤1
2εZ
0|w(s)|2ku(s)k2ds+εZ
0kw(s)k2ds
+C1
εZ
0|w(s)|2ds+εZ
0kw(s)k2ds
+2pC2Z
0kw(s)k2ds,
and so,
|w( )|2+ 2εZ
0kw(s)k2ds≤1
2εZ
0|w(s)|2ku(s)k2ds+C1
εZ
0|w(s)|2ds,
om which uniqueness ollows hanks o he G onwall lemma: indeed, deno ing
C= max(2−1, C1)/ε, we ha e ha
d
d µ|w( )|2exp ½−CZ
0
(ku(s)k2+ 1)ds¾¶≤0.
b) Now o he exis ence, we assume N∈ {2,3},ν2> C2and ha condi ion
(V) holds.
We will de elop a p oo based on he unbounded case wi hou delays (see o
ins ance Temam 1979), and on he case wi h delays on bounded domains (Ca aballo
& Real 2001), bu wi h bo h di icul ies ea ed join ly.
A icle submi ed o Nonlinea Analysis TMA

6M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
Conside an o hono mal basis B={w1, . . . , wn, . . .} ⊂ V o Hsuch ha linea
combina ions o elemen s o Ba e dense in V.†[No ice one usually akes a special
basis, using he S okes ope a o . This is no alid he e since compac ness is los ,
and i will ha e in luence on he way one can ob ain a con e gen subsequence,
because s anda d es ima es on he de i a i es o he eloci y ields a e no alid
nei he .]
Le us deno e Vm=span[w1, . . . , wm], PH
Vm:H→Vm he p ojec o gi en by
PH
Vmu=Pm
j=1(u, wj)wj.We will also deno e PV
Vm:V→Vm he p ojec o gi en
by PV
Vmu=Pm
j=1(( , ˜wj)) ˜wj(whe e he sequence {˜w1, . . . , ˜wn}comes om he
G am-Schmid o hono maliza ion p ocess in V; his will be use ul since he ini ial
con e gence in (−h, 0) mus hold in V).
Finally, de ine um( ) = Pm
j=1 γmj( )wj, whe e
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um∈L2(−h, T;Vm)∩C0([0, T]; Vm)
d
d (um( ), wj) + νa(um( ), wj) + b(um( ), um( ), wj) = h ( ), wji+
+ (g1( , um
), wj) + hg2( , um
), wjiin D0(0, T),1≤j≤m,
um(0) = PH
Vmu0, um( ) = PV
Vmφ( ), ∈(−h, 0).
(2.2)
The p eceding is a sys em o o dina y unc ional di e en ial equa ions in he
unknown γm( ) = (γm1( ), ..., γmm( )). Exis ence and uniqueness o solu ion is ob-
ained by applying Theo em 2.1 s a ed abo e.
Obse e ha p oblem (2.2) has one solu ion de ined in an in e al [0, ∗] wi h
0< ∗≤T. Howe e , as usual, i can be deduced by he a p io i es ima es below,
we can se ∗=T.
In ac , mul iplying in (2.2) by γmj( ) and summing in j, we ge o all ∈[0, ∗]
|um( )|2+ 2νZ
0kum(s)k2ds≤ |u0|2+ 2 Z
0h (s), um(s)ids
+2 Z
0
(g1(s, um
s), um(s)) ds
+2 Z
0hg2(s, um
s), um(s)ids,
and a guing in a simila manne as in he p oo o uniqueness in he 2-dimensional
case, we easily ge wo cons an s (depending on φ, ν, , g1, g2, h, T, bu no on m
no ∗)K1and K2such ha
sup
∈[0, ∗]|um( )|2≤K1,Z ∗
0kum(s)k2ds≤K2.(2.3)
So we can ake ∗=T, and ob ain ha {um}is bounded in L2(0, T;V)∩L∞(0, T;H),
so he e exis s a subsequence, elabelled he same, such ha
um* u in L2(0, T;V) weakly and in L∞(0, T;H) weak-s a as m→ ∞.(2.4)
†This can be ob ained as ollows: Vis sepa able (since i is a subse o (H1
0(Ω))N), and by
de ini ion Vis dense in Vand H, which implies ha Vis also dense in H. Thus, gi en a sequence
{ i}i≥1⊂Vdense in V, we may ake a sequence {wi
n}i,n≥1⊂ V which accumula es o e e y poin
i,and he e o e, linea combina ions o hese elemen s a e dense in Vand H. Since V⊂Ha e
ec o ial subspaces o (L2(Ω))N,linea (in)dependence is equi alen conside ed in any o hem,
whence Bis ob ained applying he G am-Schmid o hono maliza ion p ocess.
A icle submi ed o Nonlinea Analysis TMA
Na ie -S okes equa ions wi h delays on unbounded domains 7
Mo eo e , obse e ha um=PV
Vmφin (−h, 0) con e ges o φin L2(−h, 0; V), and,
in pa icula , hanks o (IV), g1(·, um) + g2(·, um) is bounded in L2(0, T;V0).
As we no iced be o e, when i is possible o choose a special basis (in bounded do-
mains i is so), s anda d es ima es on kd
d umkL2(0,T ;V0)( o N= 2,and L4/3(0, T;V0)
o N= 3) allow us o ob ain a compac ness esul : a subsequence umcon e ges o
uin L2(0, T;H).
He e we will ha e a simila esul bu no in a s aigh o wa d way, no on he
whole domain Ω.Ac ually, wha holds in his case is he ollowing:
Fo any bounded open se O ⊂ Ω he e exis s a subsequence (depending on Owhich
we elabel) sa is ying
um|O→u|Oin L2(0, T; (L2(O))N).(2.5)
Fo he sake o cla i y, we pos pone he p oo (we will use Theo em 2.2) o Lemma
2.4 below.
Now, le ψbe a con inuously di e en iable unc ion on [0, T ] wi h ψ(T) =
0.Conside equa ion (2.2) and a ixed elemen wjo B. Since (um(·), wj)ψ(·)∈
W1,1(0, T) (ac ually in H1(0, T) o N= 2,and W1,4/3(0, T) o N= 3) we ha e
−ZT
0
(um( ), ψ0( )wj)d +νZT
0
((um( ), wjψ( )))d
+ZT
0
b(um( ), um( ), wjψ( ))d = (um(0), wj)ψ(0) + ZT
0h ( ), wjψ( )id
+ZT
0
(g1( , um
), wjψ( ))d +ZT
0hg2( , um
), wjψ( )id .
Taking a diagonal subsequence, deno ed again um, ha sa is ies (2.5) o a
sequence o egula bounded open se s Oj⊂Ω ha con ain all suppo s o unc ions
wjo he basis, we may now pass o he limi , hanks o he weak con e gence in
(2.4) and condi ion (V) oo. Thus, we ob ain ( i s o any w∈ {w1, w2, . . .},and
by densi y o e e y w∈V):
−ZT
0
(u( ), ψ0( )w)d +νZT
0
((u( ), wψ( )))d
+ZT
0
b(u( ), u( ), wψ( ))d = (u0, w)ψ(0) + ZT
0h ( ), wψ( )id
+ZT
0
(g1( , u ), wψ( ))d +ZT
0hg2( , u ), wψ( )id . (2.6)
W i ing (2.6) o ψ∈ D(0, T ), u sa is ies (2.2) in he dis ibu ion sense.
By Rema k 1.1 i makes sense o wonde abou he alue a ime = 0.Now,
since (u( ), wj)ψ( )∈W1,4/3(0, T), o bo h N= 2 o 3,a guing as be o e, we ob-
ain an analogous exp ession o (2.6) wi h (u(0), w) ins ead o (u0, w).This implies
(u(0) −u0, w) = 0 o all w∈V, so u(0) = u0.
A icle submi ed o Nonlinea Analysis TMA
8M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
Fo he ollowing esul , le us obse e ha he cons an s appea ing in s an-
da d es ima es o he ilinea o m b– o a p oo see o ins ance Temam 1979,
lemmas 3.3, 3.4, 3.5 and Theo em 3.3, pp. 291 and o wa d– may be imp o ed (c .
Ladyzhenskaya 1992 o he case N= 2). Al hough i is no essen ial o make he
mos o hem o exis ence and uniqueness, since we will es ablish a s abili y esul
la e , we use hem in o de o ensu e i unde he minimal condi ions.
Lemma 2.4. Unde he assump ions o Theo em 2.3, he sequence umgi en in
(2.2) is p ecompac in he ollowing sense: suppose a bounded open se O ⊂ Ωis
gi en, hen he e exis s a subsequence depending on O,which we elabel, such ha
um|O→u|Oin L2(0, T; (L2(O))N),
whe e uis he weak limi gi en in (2.4).
P oo . We will adap he p oo o Theo em 9.4 [c . Simon 2003] o check ou si ua-
ion i s o he Theo em 2.2. Mo e exac ly, we claim i can be applied aking = 2,
q= +∞,Θ = (0, T).
Fo he se O ⊂ Ω le us make p ecise a echnical de ail: i O ⊂⊂ Ω one may
ob ain a ini e eco e ing o balls, deno ed ˜
O ⊂ Ω,which is bounded and open, and
hen X= (H1(˜
O))N⊂E= (L2(˜
O))Nwi h compac injec ion.
Howe e , o a gene al O ⊂ Ω he abo e commen may no be ue since O
and Ω can sha e pa o hei bounda ies. The compac injec ion om H1may
no hold o lack o egula i y on he bounda y (i is no imposed o Γ), howe e
i does in H1
0.One may hen use a unca ion a gumen (see o ins ance Rosa
1998): ix χ∈C1(R+) wi h χ(s) = 1 o s∈[0,1] and χ(s) = 0 o s≥4.Con-
side Oas in he s a emen , le R > 0 be such ha O ⊂ B(0, R) and deno e
˜
O= Ω ∩B(0,2R),and um,R(x) = um(x)χ(|x|2/R2).Again he compac ness holds
o X= (H1
0(˜
O))N⊂E= (L2(˜
O))Nwi h compac injec ion, and we conse e he
o iginal unc ions umon Ω ∩B(0, R).
Fo he sake o cla i y, we con inue he p oo di ec ly wi h umins ead o um,R.
Since condi ion (ii) in Theo em 2.2 is ob iously sa is ied by (2.3), we concen a e
on (i). Ac ually, we will p o e ha o he whole domain Ω he ollowing p ope y
holds:
sup
m∈Nkτhum−umkL2(0,T −h;(L2(Ω))N)→0 when h→0.
Conside h > 0 a bi a ily small. F om (2.2) we deduce o ( , +h)⊂(0, T)
ha
ZΩ
(um( +h)−u( ))wjdx+νZ +h
ZΩ∇um(s)·∇wjdxds+Z +h
b(um(s), um(s), wj)ds
=Z +h
h (s), wjids+Z +h
ZΩ
g1(s, um
s)wjdxds+Z +h
hg2(s, um
s), wjids.
A icle submi ed o Nonlinea Analysis TMA
Na ie -S okes equa ions wi h delays on unbounded domains 9
Mul iplying by γmj( +h)−γmj( ) and summing in jwe ob ain
ZΩ|um( +h)−u( )|2dx=−νZ +h
ZΩ∇um(s)·(∇um( +h)−∇um( ))dxds
+Z +h
b(um(s), um(s), um( +h)−um( ))ds
+Z +h
ZΩ
g1(s, um
s)(um( +h)−um( ))dxds
+Z +h
h (s) + g2(s, um
s), um( +h)−um( )ids.
The igh hand side may be bounded by
ν|∇um( +h)−∇um( )|Z +h
|∇um(s)|ds
+Z +h
GN(|um(s)|,kum(s)k,kum( +h)−um( )k) ds
+Z +h
|g1(s, um
s)||um( +h)−um( )|ds
+Z +h
(k (s)k∗+kg2(s, um
s)k∗)kum( +h)−um( )kds
whe e he ilinea o m bis bounded (depending on he dimension) by he unc ion
GN:R3→Rde ined as
GN(x, y, z) = ½2−1/2xyz i N= 2,
2−1x1/2y3/2zi N= 3.(2.7)
Thus, using (1.1) and (2.3), we ha e p o ed ha
ZΩ|um( +h)−um( )|2dx≤ kum( +h)−um( )kZ +h
Gm(s)ds
whe e he unc ion Gm:R→Ris de ined ( ecall de ini ion gi en in (2.7)) as
Gm(s)=(νkum(s)k+(2−1K1)1/2kum(s)k+k (s)k∗+kg2(s, um
s)k∗+λ−1/2
1|g1(s, um
s)|i N= 2,
νkum(s)k+2−1K1/4
1kum(s)k3/2+k (s)k∗+kg2(s, um
s)k∗+λ−1/2
1|g1(s, um
s)|i N= 3.
To inish he p oo , we will es ima e
kτhum−umk2
L2(0,T −h;(L2(Ω))N)=ZT−h
0ZΩ|τhum−um|2dxd
≤ZT−h
0kum( +h)−um( )kZ +h
Gm(s)dsd .
Fo he igh hand side, he Fubini heo em yields, using he unc ion
¯s=


0 i s≤0,
si 0 < s ≤T−h,
T−hi s > T −h,
A icle submi ed o Nonlinea Analysis TMA
16 M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
Then he e is a unique s a iona y solu ion u∗o (4.2) and e e y solu ion o (1.2)
con e ges o u∗exponen ially as as →+∞, ha is, he e exis wo posi i e
cons an s Cand λ, such ha o all u0∈Hand φ∈L2(−h, 0; V), he solu ion u
o (1.2) wi h ( )≡ sa is ies o all ≥0 :
|u( )−u∗|2≤Ce−λ ³|u0−u∗|2+kφ−u∗k2
L2(−h,0;V)´.(4.7)
P oo . Conside u he solu ion o (1.2) o ( )≡ , and deno e u∗∈V he s a-
iona y solu ion o (4.2), which exis ence and uniqueness is ensu ed by Theo em
4.1. Se w( ) = u( )−u∗, and obse e ha
d
d (w( ), )+ν((w( ), ))+b(u( ), u( ), )−b(u∗, u∗, ) = (G(u( −ρ( ))), )−(G(u∗), ).
Since
b(u( ), u( ), w( )) −b(u∗, u∗, w( ))
=b(u∗, w( ), u∗)−b(u( ), w( ), u( ))
=b(u∗, w( ), u∗)∓b(u( ), w( ), u∗)−b(u( ), w( ), u( ))
=−b(w( ), w( ), u∗)−b(u( ), w( ), w( ))
=−b(w( ), w( ), u∗),
we can ob ain he ollowing es ima ion (he e λand δa e ixed posi i e alues o be
de e mined la e on):
d
d (eλ |w( )|2) = λeλ |w( )|2+ eλ d
d |w( )|2
=λeλ |w( )|2+ 2eλ (−νkw( )k2−b(u( ), u( ), w( ))
+b(u∗, u∗, w( )) + (G(u( −ρ( ))) −G(u∗), w( )))
≤eλ ¡λ|w( )|2−2νkw( )k2+ 2b(w( ), w( ), u∗)
+2L1|w( −ρ( ))||w( )|)
≤λ−1
1eλ (λ+δL1−2νλ1)kw( )k2+ 2eλ |b(w( ), w( ), u∗)|
+(δλ1)−1L1eλ kw( −ρ( ))k2.(4.8)
Using again ha |b(w( ), w( ), u∗)| ≤ (2λ1)−1/2kw( )k2ku∗kand aking in o accoun
he es ima e we p o ed in S ep 3 o Theo em 4.1 o he s a iona y solu ion, ku∗k ≤
k k∗/(ν−λ−1
1L1),we ha e
|b(w( ), w( ), u∗)| ≤ (2λ1)−1/2k k∗
ν−λ−1
1L1kw( )k2.
Subs i u ing his las inequali y in o (4.8) i ollows ha
d
d (eλ |w( )|2)≤λ−1
1eλ µλ+δL1−2νλ1+(2λ1)1/2k k∗
ν−λ−1
1L1¶kw( )k2
+(δλ1)−1L1eλ kw( −ρ( ))k2,
A icle submi ed o Nonlinea Analysis TMA

Na ie -S okes equa ions wi h delays on unbounded domains 17
and so o all ∈[0, T]
eλ |w( )|2≤ |w(0)|2+ (δλ1)−1L1Z
0
eλskw(s−ρ(s))k2ds
+λ−1
1µλ+δL1−2νλ1+(2λ1)1/2k k∗
ν−λ−1
1L1¶Z
0
eλskw(s)k2ds.
We concen a e momen a ily in he delay e m on he igh hand side. Obse ing
ha he unc ion φ( ) := −ρ( ) is s ic ly inc easing, ha ρ akes alues on [0, h],
and so φ−1(η)≤η+h, we can apply he change o a iable η=s−ρ(s) = φ(s) :
Z
0
eλskw(s−ρ(s))k2ds=Z −ρ( )
−ρ(0)
eλτ−1(η)kw(η)k21
1−ρ0(φ−1(η)) dη
≤eλh
1−ρ∗Z
−h
eληkw(η)k2dη.
Combining he abo e wo inequali ies we ob ain
eλ |w( )|2≤ |w(0)|2+ (δλ1)−1L1
eλh
1−ρ∗Z
−h
eλskw(s)k2ds
+λ−1
1µλ+δL1−2νλ1+(2λ1)1/2k k∗
ν−λ−1
1L1¶Z
0
eλskw(s)k2ds.
Obse e he coe icien s o he in eg al R
0eλskw(s)k2ds. Le us no e ha δ∗=
(1 −ρ∗)−1/2is he minimum o he map δ7→ δ+ 1/(δ(1 −ρ∗)).Then, hanks o
(4.6) he e exis s λ > 0 small enough such ha
λ+δ∗L1−2νλ1+(2λ1)1/2k k∗
ν−λ−1
1L1
+L1eλh
δ∗(1 −ρ∗)≤0.
Thus, we deduce ha
eλ |u( )−u∗|2≤ |u0−u∗|2+λ−1
1L1eλh
1−ρ∗Z0
−h
eληkw(η)k2dη,
whence (4.7) is sa is ied wi h C= max ½1,λ−1
1L1eλh
1−ρ∗¾.
Rema k 4.3. The abo e esul has been gi en wi h g1as in Case 1 o Sec ion
3 o simpli y he no a ion. O he ypes o delay could be used, o example one
could conside an au onomous case wi h dis ibu ed delay by emo ing in Case 2
he dependence o Gon i s i s a iable.
Conclusions and inal commen s
Exis ence, uniqueness and s abili y esul s ha e been es ablished unde di e en
condi ions –essen ially iscosi y is asked o be la ge enough–. One may wonde
abou esul s unde weake assump ions, whe e uniqueness o s abili y may no be
A icle submi ed o Nonlinea Analysis TMA
18 M. J. Ga ido-A ienza, & P. Ma ´ın-Rubio
ensu ed. This leads us o conside addi ional concep s om he heo y o dynamical
sys ems, namely a ac o s, bo h he classical ( o wa d) one, and he ’pullback’ de -
ini ion ha is well-sui ed o non-au onomous sys ems (see Ca aballo & Real 2004
and Kloeden & Schmal uß 1997).
Acknowledgemen
This wo k was pa ially suppo ed by Minis e io de Ciencia y Tecnolog´ıa (Spain)
and FEDER P oyec o BFM2002-03068. The au ho s hanks P o esso Jos´e Real o
his ad ice on he subjec o his pape .
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A icle submi ed o Nonlinea Analysis TMA