Manusc ip submi ed o Websi e: h p://AIMsciences.o g
AIMS’ Jou nals
Volume XX, Numbe 0xx, XXXXXX 20xx pp. –
SOME NEW REGULARITY RESULTS OF PULLBACK
ATTRACTORS FOR 2D NAVIER-STOKES EQUATIONS WITH
DELAYS
Julia Ga c´
ıa-Luengo, Ped o Ma ´
ın-Rubio & Jos´
e Real∗
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
(Communica ed by XXXXX)
Abs ac . In his pape we s eng hen some esul s on he exis ence and
p ope ies o pullback a ac o s o a 2D Na ie -S okes model wi h ini e delay
o mula ed in [Ca aballo and Real, J. Di e en ial Equa ions 205 (2004), 271–
297]. Ac ually, we p o e ha unde sui able assump ions, pullback a ac o s
no only o ixed bounded se s bu also o a se o empe ed uni e ses do
exis . Mo eo e , hanks o egula i y esul s, he a ac ion om di e en
phase spaces also happens in C([−h, 0]; V). Finally, om compa ison esul s
o a ac o s, and unde an addi ional hypo hesis, we es ablish ha all hese
amilies o a ac o s a e in ac he same objec .
1. In oduc ion and s a emen o he p oblem. Le Ω ⊂R2be an open
bounded se wi h smoo h enough bounda y ∂Ω, and conside an a bi a y ini ial
ime τ∈R, and he ollowing unc ional Na ie -S okes p oblem:



















∂u
∂ −ν∆u+ (u· ∇)u+∇p= ( ) + g( , u ) in Ω ×(τ, ∞),
di u= 0 in Ω ×(τ, ∞),
u= 0 on ∂Ω×(τ, ∞),
u(x, τ) = uτ(x), x ∈Ω,
u(x, τ +s) = φ(x, s), x ∈Ω, s ∈(−h, 0),
(1)
whe e ν > 0 is he kinema ic iscosi y, u= (u1, u2) is he eloci y ield o he luid,
pis he p essu e, is a non-delayed ex e nal o ce ield, gis ano he ex e nal o ce
wi h some he edi a y cha ac e is ics, and uτand φ(x, s −τ) a e he ini ial da a
in τand (τ−h, τ) espec i ely, whe e h > 0 is he ime o memo y e ec . Fo
each ≥τ, we deno e by u he unc ion de ined a.e. on (−h, 0) by he ela ion
u (s) = u( +s), a.e. s∈(−h, 0).
2010 Ma hema ics Subjec Classi ica ion. P ima y: 35B41, 35Q30, 37L30.
Key wo ds and ph ases. 2D Na ie -S okes equa ions; delay e ms; pullback a ac o s.
∗Deceased on Janua y 27 h, 2012. The i s wo coau ho s dedica e his pape o he memo y
o hei coau ho Jos´e Real.
1
2 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
The impo ance o physical models o luid mechanic p oblems including delay
e ms is ela ed, o ins ance, o eal applica ions whe e de ices o con ol p ope -
ies o luids ( empe a u e, eloci y, e c.) a e inse ed in domains and make a local
in luence on he beha iou o he sys em (e.g., c . [13] o a wind- unnel model).
The s udy o Na ie -S okes models including delay e ms –exis ence, unique-
ness, s a iona y solu ions, exponen ial decay, and o he asymp o ic p ope ies such
as he exis ence o a ac o s– was ini ia ed in he e e ences [3,4,5], and a e
ha , many di e en ques ions, as dealing wi h unbounded domains, and models
( o ins ance in h ee dimensions o modi ied e ms) ha e been add essed (e.g., c .
[10,17,21,19,14,20,11,15,16] among o he s).
In he ecen pape [9], we ha e ea ed a elaxa ion on he assump ions o he
delay ope a o in ol ed, emo ing condi ions ela ed o he con ol o he L2no m
o he delay e ms (see assump ions (IV) and (V) below). Al hough his implies o
es ic he phase space o con inuous unc ions ins ead o squa e in eg able in ime,
he delay unc ions d i ing he delayed ime wi hin his heo y can be aken jus
measu able, wi hou any addi ional assump ion as con inui y no C1wi h bounded
de i a i e, as usual in he li e a u e.
Mo eo e , in [9] we we e also able o es ablish a ac ion in a highe no m
(namely, H1ins ead o L2) making a sha p use o egula iza ion o he equa ions in
dimension wo and by ene gy me hods. Rela ionships among a ac o s in di e en
me ics was success ully ca ied ou he e, oo.
Ou goal in his pape is o keep all usual condi ions o he delay ope a o (in-
cluding (IV) and (V)) and o compa e bo h kind o a ac o s, o bo h possibili ies
o phase spaces (con inuous in ime, o jus squa e in eg able in ime). Obse e
ha in he au onomous amewo k his issue would be almos immedia e since
one inclusion is clea by con inuous embedding, and he o he is ob ained a e an
elapsed ime as long as he memo y e ec . Howe e , in he non-au onomous case
( ha we a e dealing wi h) his is no he case a all. Using he heo y o a ac-
ion o uni e ses (c . [1,2,18]) we deal wi h di e en amilies and unde di e en
me ics. Namely, we conside uni e ses o ixed (in ime) bounded se s and also
ime-dependen amilies gi en by a empe ed condi ion when ime goes o −∞.
Mo eo e , we also imp o e some esul s p e iously ob ained in he li e a u e
(c . [5]) since we can deal wi h he phase space V×L2(−h, 0; V) and no only
H×L2(−h, 0; H).Finally, om compa ison esul s o a ac o s and unde an ad-
di ional assump ion, we es ablish ha all hese amilies o a ac o s a e in ac he
same objec .
The s uc u e o he pape is he ollowing. We con inue his sec ion wi h he
abs ac se ing o he p oblem, gene al de ini ions and some well-known esul s
on exis ence o weak and s ong solu ions and egula i y p ope ies. In Sec ion
2 we ecall he basic heo y o pullback a ac o s o non-au onomous dynamical
sys ems wi hin he amewo k o uni e ses, and compa ison esul s, when di e en
me ics a e in ol ed, a e also gi en. Sec ion 3 is de o ed o es ablish all possible
a ac o s o di e en phase-spaces bu aking in o accoun he L2no m in space.
Ou main esul s, es ablished in he highe no m H1(in space), a e gi en in Sec ion
REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 3
4. In hese wo las sec ions, ene gy me hods (in oduced in his con ex by Rosa
in [23]) a e used o p o e asymp o ic compac ness in he espec i e uni e ses. As
said be o e, ela ionships among all hese objec s a e ob ained.
To se ou p oblem in he abs ac amewo k, we conside he ollowing usual
unc ion spaces:
V=u∈(C∞
0(Ω))2: di u= 0,
H= he closu e o Vin (L2(Ω))2wi h he no m |·|, and inne p oduc (·,·), whe e
o u, ∈(L2(Ω))2,
(u, ) =
2
X
j=1 ZΩ
uj(x) j(x)dx,
V= he closu e o Vin (H1
0(Ω))2wi h he no m k·k associa ed o he inne p oduc
((·,·)), whe e o u, ∈(H1
0(Ω))2,
((u, )) =
2
X
i,j=1 ZΩ
∂uj
∂xi
∂ j
∂xi
dx.
We will use k · k∗ o he no m in V0and h·,·i o he duali y be ween V0and
V. We conside e e y elemen h∈Has an elemen o V0, gi en by he equali y
hh, i= (h, ) o all ∈V. I ollows ha V⊂H⊂V0, whe e he injec ions a e
dense and con inuous, and, in ac , compac .
Now, we de ine he ope a o A:V→V0as
hAu, i= ((u, )) ∀u, ∈V.
Le us deno e D(A) = {u∈V:Au ∈H}. By he egula i y o ∂Ω, one has ha
D(A)=(H2(Ω))2∩V, and Au =−P∆u o all u∈D(A) is he S okes ope a o
(Pis he o ho-p ojec o om (L2(Ω))2on o H). On D(A) we conside he no m
|·|D(A)de ined by |u|D(A)=|Au|. Obse e ha on D(A) he no ms k·k(H2(Ω))2and
|·|D(A)a e equi alen (see [6] o [25]), and D(A) is compac ly and densely injec ed
in V.
Le us de ine
b(u, , w) =
2
X
i,j=1 ZΩ
ui
∂ j
∂xi
wjdx,
o e e y unc ions u, , w : Ω →R2 o which he igh -hand side is well de ined.
In pa icula , bhas sense o all u, , w ∈V, and is a con inuous ilinea o m
on V×V×V.
Some use ul p ope ies conce ning b ha we will use in he nex sec ions a e he
ollowing (see [22] o [24]): b(u, , w) = −b(u, w, ) o all u, ,w∈V, which also
implies ha b(u, , ) = 0 o all u, ∈V. Mo eo e , he e exis s a cons an C1>0,
only dependen on Ω, such ha ( ecall ha we a e in dimension wo)
|b(u, , w)| ≤ C1|u|1/2|Au|1/2k k|w| ∀ u∈D(A), ∈V, w ∈H. (2)
Now, we es ablish some sui able spaces in o de o deal wi h he delay e m, and
some app op ia e assump ions on he e m in (1) con aining he delay.
Le us deno e CH=C([−h, 0]; H), wi h he no m |ϕ|CH= maxs∈[−h,0] |ϕ(s)|,
and L2
X=L2(−h, 0; X) o X=H,V. On he delay ope a o om (1), we
conside ha is well de ined as g:R×CH→(L2(Ω))2, and i sa is ies he ollowing
assump ions:
4 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
(I) o all ξ∈CH, he unc ion R3 7→ g( , ξ)∈(L2(Ω))2is measu able,
(II) g( , 0) = 0, o all ∈R,
(III) he e exis s Lg>0 such ha o all ∈R, and o all ξ,η∈CH,
|g( , ξ)−g( , η)| ≤ Lg|ξ−η|CH,
(IV) he e exis s Cg>0 such ha o all τ≤ , and o all u, ∈C([τ−h, ]; H),
Z
τ
|g(s, us)−g(s, s)|2ds ≤C2
gZ
τ−h
|u(s)− (s)|2ds.
Examples o ixed, a iable, and dis ibu ed delay ope a o s can be ound, o in-
s ance, in [3, Sec ion 3], [5, Sec ions 3.5 and 3.6], and [10, Sec ion 3], and we omi
hem he e jus o he sake o b e i y.
Obse e ha (I)−(III) imply ha gi en T > τ and u∈C([τ−h, T]; H), he
unc ion gu: [τ, T]→(L2(Ω))2de ined by gu( ) = g( , u ) o all ∈[τ, T ], is
measu able and, in ac , belongs o L∞(τ, T; (L2(Ω))2). Then, hanks o (IV), he
mapping
G:u∈C([τ−h, T]; H)→gu∈L2(τ, T; (L2(Ω))2)
has a unique ex ension o a mapping e
Gwhich is uni o mly con inuous om L2(τ−
h, T;H) in o L2(τ, T; (L2(Ω))2). F om now on, we will deno e g( , u ) = e
G(u)( )
o each u∈L2(τ−h, T ;H), and hus p ope y (IV) will also hold o all u,
∈L2(τ−h, T;H).
Assume ha uτ∈H,φ∈L2
H, and ∈L2
loc(R;V0).
De ini ion 1. A weak solu ion o (1) is a unc ion u ha belongs o L2(τ−h, T;H)
∩L2(τ, T;V)∩L∞(τ, T;H) o all T > τ, wi h u(τ) = uτand u( ) = φ( −τ) a.e.
∈(τ−h, τ), and such ha o all ∈V,
d
d (u( ), ) + νhAu( ), i+b(u( ), u( ), ) = h ( ), i+ (g( , u ), ),(3)
whe e he equa ion mus be unde s ood in he sense o D0(τ, ∞).
Rema k 1. I uis a weak solu ion o (1), hen om (3) we deduce ha o any
T > τ, one has u0∈L2(τ, T;V0), and so u∈C([τ, ∞); H), whence he ini ial da um
u(τ) = uτhas ull sense. Mo eo e , in his case he ollowing ene gy equali y holds:
|u( )|2+2νZ
s
ku( )k2d =|u(s)|2+2Z
sh ( ), u( )i+(g( , u ), u( ))d ∀τ≤s≤ .
A no ion o mo e egula solu ion is also sui able o p oblem (1).
De ini ion 2. A s ong solu ion o (1) is a weak solu ion uo (1) such ha u∈
L2(τ, T;D(A)) ∩L∞(τ, T;V) o all T > τ.
Rema k 2. I ∈L2
loc(R; (L2(Ω))2) and uis a s ong solu ion o (1), hen u0∈
L2(τ, T;H) o all T > τ, and so u∈C([τ, ∞); V). In his case he ollowing ene gy
equali y holds:
ku( )k2+ 2νZ
s
|Au( )|2d + 2 Z
s
b(u( ), u( ), Au( )) d
=ku(s)k2+ 2 Z
s
( ( ) + g( , u ), Au( )) d ∀τ≤s≤ . (4)
REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 5
Conce ning he exis ence and uniqueness o weak and s ong solu ions o (1),
we ha e he ollowing esul which can be p o ed simila ly as [3, Theo em 2.1] o
[4, Theo em 2.5] (see also [10, Theo em 2.3] o a mo e gene al case).
Theo em 1. Le us conside uτ∈H,φ∈L2
H, ∈L2
loc(R;V0), and g:R×CH→
(L2(Ω))2sa is ying (I)–(IV). Then, o each τ∈R, he e exis s a unique weak
solu ion u=u(·;τ, uτ, φ)o (1).
Mo eo e , i ∈L2
loc(R; (L2(Ω))2), hen
(a) u∈C([τ+ε, T]; V)∩L2(τ+ε, T;D(A)) o all T > τ +ε > τ.
(b) I uτ∈V,uis in ac a s ong solu ion o (1).
Be o e es ablishing he o iginal esul s abou he egula i y o pullback a ac o s,
we ecall he main exis ence esul s s udied in [5,17,19]. Fi s ly, in o de o do
ha , we emembe b ie ly he abs ac heo y on pullback a ac o s in he nex
sec ion.
2. Abs ac esul s on minimal pullback a ac o s. Now, we p esen a sum-
ma y o some esul s om [8] abou he exis ence o minimal pullback a ac o s
(see also [1,2,18]). In pa icula , we assume ha he p ocess Uis closed (see
De ini ion 3below).
Conside gi en a me ic space (X, dX), and le us deno e R2
d={( , τ)∈R2:τ≤ }.
A p ocess Uon Xis a mapping R2
d×X3( , τ, x)7→ U( , τ)x∈Xsuch ha
U(τ, τ)x=x o any (τ, x)∈R×X, and U( , )(U( , τ)x) = U( , τ)x o any
τ≤ ≤ and all x∈X.
De ini ion 3. Le Ube a p ocess on X.
(a) Uis said o be con inuous i o any pai τ≤ , he mapping U( , τ) : X→X
is con inuous.
(b) Uis said o be closed i o any τ≤ , and any sequence {xn} ⊂ X, i
xn→x∈Xand U( , τ)xn→y∈X, hen U( , τ)x=y.
Rema k 3. I is clea ha e e y con inuous p ocess is closed.
Le us deno e by P(X) he amily o all nonemp y subse s o X, and conside a
amily o nonemp y se s b
D0={D0( ) : ∈R}⊂P(X).
De ini ion 4. We say ha a p ocess Uon Xis pullback b
D0-asymp o ically compac
i o any ∈Rand any sequences {τn} ⊂ (−∞, ] and {xn} ⊂ Xsa is ying
τn→ −∞ and xn∈D0(τn) o all n, he sequence {U( , τn)xn}is ela i ely compac
in X.
Deno e
Λ( b
D0, ) =
s≤ [
τ≤s
U( , τ)D0(τ)
X
∀ ∈R,
whe e {· · · }Xis he closu e in X.
Gi en wo subse s o X,O1and O2, we deno e by dis X(O1,O2) he Hausdo
semi-dis ance in Xbe ween hem, de ined as
dis X(O1,O2) = sup
x∈O1
in
y∈O2
dX(x, y).
Le be gi en Da nonemp y class o amilies pa ame e ized in ime b
D={D( ) :
∈R}⊂P(X). The class Dwill be called a uni e se in P(X).

6 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
De ini ion 5. A p ocess Uon Xis said o be pullback D-asymp o ically compac
i i is pullback b
D-asymp o ically compac o any b
D∈ D.
I is said ha b
D0={D0( ) : ∈R}⊂P(X) is pullback D-abso bing o he
p ocess Uon Xi o any ∈Rand any b
D∈ D, he e exis s a τ0( , b
D)≤ such
ha
U( , τ)D(τ)⊂D0( )∀τ≤τ0( , b
D).
Wi h he abo e de ini ions, we may es ablish he main esul o his sec ion (c .
[8, Theo em 3.11]).
Theo em 2. Conside a closed p ocess U:R2
d×X→X, a uni e se Din P(X),
and a amily b
D0={D0( ) : ∈R}⊂P(X)which is pullback D-abso bing o U,
and assume also ha Uis pullback b
D0-asymp o ically compac .
Then, he amily AD={AD( ) : ∈R}de ined by AD( ) = Sb
D∈D Λ( b
D, )
X
, has
he ollowing p ope ies:
(a) o any ∈R, he se AD( )is a nonemp y compac subse o X, and AD( )⊂
Λ( b
D0, ),
(b) ADis pullback D-a ac ing, i.e., limτ→−∞ dis X(U( , τ)D(τ),AD( )) = 0 o
all b
D∈ D, and any ∈R,
(c) ADis in a ian , i.e., U( , τ)AD(τ) = AD( ) o all ( , τ)∈R2
d,
(d) i b
D0∈ D, hen AD( ) = Λ( b
D0, )⊂D0( )X o all ∈R.
The amily ADis minimal in he sense ha i b
C={C( ) : ∈R} ⊂ P(X)is a am-
ily o closed se s such ha o any b
D={D( ) : ∈R}∈D,lim
τ→−∞ dis X(U( , τ)D(τ),
C( )) = 0, hen AD( )⊂C( ).
Rema k 4. Unde he assump ions o Theo em 2, he amily ADis called he
minimal pullback D-a ac o o he p ocess U.
I AD∈ D, hen i is he unique amily o closed subse s in D ha sa is ies
(b)–(c).
A su icien condi ion o AD∈ D is o ha e ha b
D0∈ D, he se D0( ) is
closed o all ∈R, and he amily Dis inclusion-closed (i.e., i b
D∈ D, and
b
D0={D0( ) : ∈R}⊂P(X) wi h D0( )⊂D( ) o all , hen b
D0∈ D).
We will deno e by DF(X) he uni e se o ixed nonemp y bounded subse s o X,
i.e., he class o all amilies b
Do he o m b
D={D( ) = D: ∈R}wi h Da ixed
nonemp y bounded subse o X.
Now, i is easy o conclude he ollowing esul .
Co olla y 1. Unde he assump ions o Theo em 2, i he uni e se Dcon ains he
uni e se DF(X), hen bo h a ac o s, ADF(X)and AD, exis , and ADF(X)( )⊂
AD( ) o all ∈R.
Rema k 5. I can be p o ed (see [18]) ha , unde he assump ions o he p eceding
co olla y, i o some T∈R, he se ∪ ≤TD0( ) is a bounded subse o X, hen
ADF(X)( ) = AD( ) o all ≤T.
Now, and since i will be use ul below, we es ablish an abs ac esul (c . [8,
Theo em 3.15]) ha allows us o compa e wo a ac o s o a p ocess unde app o-
p ia e assump ions.
REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 7
Theo em 3. Le {(Xi, dXi)}i=1,2be wo me ic spaces such ha X1⊂X2wi h
con inuous injec ion, and o i= 1,2, le Dibe a uni e se in P(Xi), wi h D1⊂ D2.
Assume ha we ha e a map U ha ac s as a p ocess in bo h cases, i.e., U:R2
d×Xi→
Xi o i= 1,2is a p ocess.
Fo each ∈R, le us deno e
Ai( ) = [
b
Di∈Di
Λi(b
Di, )
Xi
i= 1,2,
whe e he subsc ip iin he symbol o he omega-limi se Λiis used o deno e he
dependence o he espec i e opology.
Then, A1( )⊂ A2( ) o all ∈R.
Suppose mo eo e ha he wo ollowing condi ions a e sa is ied:
(i) A1( )is a compac subse o X1 o all ∈R,
(ii) o any b
D2∈ D2and any ∈R, he e exis a amily b
D1∈ D1and a
∗
b
D1≤ (bo h possibly depending on and b
D2), such ha Uis pullback b
D1-
asymp o ically compac , and o any s≤ ∗
b
D1 he e exis s a τs≤ssuch ha
U(s, τ)D2(τ)⊂D1(s) o all τ≤τs.
Then, unde all he condi ions abo e, A1( ) = A2( ) o all ∈R.
Rema k 6. In he p eceding heo em, i ins ead o assump ion (ii) we conside he
ollowing condi ion:
(ii’) o any b
D2∈ D2and any sequence τn→ −∞, he e exis ano he amily
b
D1∈ D1and ano he sequence τ0
n→ −∞ wi h τ0
n≥τn o all n, such ha U
is pullback b
D1-asymp o ically compac , and
U(τ0
n, τn)D2(τn)⊂D1(τ0
n)∀n,
hen, wi h a simila p oo , one can ob ain ha he equali y A1( ) = A2( ) also holds
o all ∈R.
Obse e ha a su icien condi ion o (ii’) is ha he e exis s T > 0 such ha o
any b
D2∈ D2, he e exis s a b
D1∈ D1sa is ying ha Uis pullback b
D1-asymp o ically
compac , and U(τ+T, τ)D2(τ)⊂D1(τ+T) o all τ∈R.
3. P e ious esul s on p ocesses and pullback a ac o s in H.In his
sec ion we ecall some known esul s (c . [5,17,19]) on he exis ence o minimal
pullback a ac o s in he Hno m o sui able p ocesses associa ed o p oblem (1).
In o de o apply he heo y o he abo e sec ion, and ollowing [5,17,19], we
may conside he Banach space CH, and he Hilbe space M2
H=H×L2
Hwi h
associa ed no m
k(uτ, φ)k2
M2
H=|uτ|2+Z0
−h
|φ(s)|2ds o (uτ, φ)∈M2
H.
We can de ine wo p ocesses o p oblem (1).
P oposi ion 1. Assume ha ∈L2
loc(R;V0), and g:R×CH→(L2(Ω))2sa is ies
(I)–(IV). Then, he bi-pa ame ic amilies o mappings U( , τ) : CH→CHand
S( , τ) : M2
H→M2
Hgi en espec i ely by
U( , τ)φ=u (·;τ, φ(0), φ) o φ∈CH, τ ≤ , (5)
and
S( , τ)(uτ, φ)=(u( ;τ, uτ, φ), u (·;τ, uτ, φ)) o (uτ, φ)∈M2
H, τ ≤ , (6)
8 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
whe e uis he unique weak solu ion o (1), a e well de ined con inuous p ocesses on
CHand M2
H espec i ely.
P oo . The esul ollows om Theo em 1abo e, and om [5, Theo em 9].
Now, in o de o es ablish asymp o ic es ima es o he solu ions o (1), we impose
a i h assump ion on gand .
Deno e by λ1 he i s eigen alue o he S okes ope a o A.
(V) Assume ha νλ1> Cg, and ha he e exis s a alue η∈(0,2(νλ1−Cg)) such
ha o e e y u∈L2(τ−h, ;H),
Z
τ
eηs|g(s, us)|2ds ≤C2
gZ
τ−h
eηs|u(s)|2ds o any τ≤ , and
Z0
−∞
eηsk (s)k2
∗ds < ∞.
Lemma 1. Suppose ha ∈L2
loc(R;V0), and ha and g:R×CH→(L2(Ω))2
sa is y (I)–(V). Then, o any (uτ, φ)∈M2
H, he ollowing es ima e holds o he
solu ion u o (1) o all ≥τ:
|u( )|2≤e−η( −τ)max{1, Cg}k(uτ, φ)k2
M2
H+β−1e−η Z
τ
eηsk (s)k2
∗ds, (7)
whe e
β= 2ν−(η+ 2Cg)λ−1
1.(8)
P oo . By he ene gy equali y (see Rema k 1), and Young’s inequali y, we ha e
d
d |u( )|2+ 2νku( )k2
≤βku( )k2+β−1k ( )k2
∗+Cg|u( )|2+C−1
g|g( , u )|2,a.e. > τ.
Thus,
d
d eη |u( )|2+eη 2ν−β−(η+Cg)λ−1
1ku( )k2
≤eη β−1k ( )k2
∗+eη C−1
g|g( , u )|2,a.e. > τ,
and he e o e, in eg a ing abo e and using p ope y (V), we ob ain
eη |u( )|2+2ν−β−(η+Cg)λ−1
1Z
τ
eηsku(s)k2ds
≤eητ |uτ|2+β−1Z
τ
eηsk (s)k2
∗ds +CgZ
τ−h
eηs|u(s)|2ds
≤eητ max{1, Cg}k(uτ, φ)k2
M2
H+β−1Z
τ
eηsk (s)k2
∗ds +CgZ
τ
eηs|u(s)|2ds,
o all ≥τ, and om his las inequali y and (8), in pa icula we deduce (7).
A e he abo e esul , i u ns ou app op ia e he in oduc ion o he ollowing
empe ed uni e ses.
De ini ion 6. Fo any η > 0, we will deno e by Dη(CH) he class o all amilies o
nonemp y subse s b
D={D( ) : ∈R}⊂P(CH) such ha
lim
τ→−∞ eητ sup
ϕ∈D(τ)
|ϕ|2
CH= 0.
REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 9
Analogously, we will deno e by Dη(M2
H) he class o all amilies o nonemp y subse s
b
D={D( ) : ∈R}⊂P(M2
H) such ha
lim
τ→−∞ eητ sup
(w,ϕ)∈D(τ)
k(w, ϕ)k2
M2
H= 0.
Fu he mo e, acco dingly o he no a ion in oduced in he p e ious sec ion,
DF(CH) and DF(M2
H) will deno e he uni e ses o ixed bounded se s in CHand
M2
H espec i ely.
Rema k 7. (i) The choices o he abo e uni e ses a e igh and con enien o
keep, in he sense ha , on he one hand, M2
His mo e gene al as phase space
o he ini ial da a o p oblem (1). On he o he hand, he egula i y o
he solu ion o (1) (c . Theo em 1) makes ha , a e an elapsed ime h,
e e y solu ion is con inuous wi h alues on H. Indeed, as i was obse ed in
[5], in he case o he uni e ses o ixed bounded se s, pullback a ac o s in
bo h spaces do exis , and hey a e in insically ela ed h ough he canonical
embedding j:CH→M2
Hde ined by j(ϕ) = (ϕ(0), ϕ) (see Theo em 4below).
(ii) The uni e ses Dη(CH) and Dη(M2
H),which a e inclusion-closed, con ain e-
spec i ely he uni e ses DF(CH) and DF(M2
H).
Now, we ob ain pullback abso bing amilies o U:R2
d×CH→CHand S:
R2
d×M2
H→M2
H.
Co olla y 2. Unde he assump ions o Lemma 1, he amily b
D1,η ={D1,η( ) : ∈
R} ⊂ P(CH)de ined by D1,η( ) = BCH(0, η( )), he closed ball in CHo cen e
ze o and adius η( ), whe e
2
η( ) = 1 + β−1e−η( −h)Z
−∞
eηsk (s)k2
∗ds,
wi h βgi en by (8), is pullback Dη(CH)-abso bing o he p ocess Uon CHde ined
by (5) (and he e o e pullback DF(CH)-abso bing oo), and b
D1,η belongs o Dη(CH).
Besides, he amily b
D2,η ={D2,η( ) : ∈R} ⊂ P(M2
H)de ined by D2,η( ) =
BM2
H(0, Rη( )), he closed ball in M2
Ho cen e ze o and adius Rη( ), wi h
R2
η( ) = 1 + β−1(1 + heηh)e−η Z
−∞
eηsk (s)k2
∗ds,
is pullback Dη(M2
H)-abso bing o he p ocess Son M2
Hgi en by (6) (and hus also
pullback DF(M2
H)-abso bing), and b
D2,η belongs o Dη(M2
H).
Since i will be use ul in o de o compa e he pullback a ac o s de ined in he
spaces CHand M2
H, we conside he bi-pa ame ic amily o mappings e
U( , τ) :
M2
H→L2
Hde ined as
e
U( , τ)(uτ, φ) = u (·;τ, uτ, φ) o (uτ, φ)∈M2
H, τ ≤ .
Rema k 8. Obse e ha e
U( , τ) maps M2
Hin o CHi ≥τ+h, and he e o e we
can w i e
S( , τ)(uτ, φ) = j(e
U( , τ)(uτ, φ)) o (uτ, φ)∈M2
H, ≥τ+h,
whe e S(·,·) is gi en by (6).
Mo eo e , i is clea ha
U( , τ)φ=e
U( , τ)j(φ) o φ∈CH, ≥τ,
16 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
Then, since all Jna e non-inc easing, we deduce ha o all n≥n(kδ)
Jn( n)−J( ∗)≤Jn(˜
kδ)−J( ∗)
≤ |Jn(˜
kδ)−J( ∗)|
≤ |Jn(˜
kδ)−J(˜
kδ)|+|J(˜
kδ)−J( ∗)|< δ.
The e o e, as δ > 0 is a bi a y, we ob ain ha
lim sup
n→∞
Jn( n)≤J( ∗),
and consequen ly, again by (15) and (16),
lim sup
n→∞
kun( n)k≤ku( ∗)k,
which combined wi h (20) and (17) allows us o claim ha un( n)→u( ∗) s ongly
in V, in con adic ion wi h (19). Thus, (18) is p o ed as desi ed.
As an immedia e consequence o he p e ious lemma, we ha e he ollowing esul .
Co olla y 3. Unde he assump ions o Lemma 4, i holds:
(a) Fo any ˜
h∈[0, h], he p ocess U:R2
d×C˜
h,V
H→C˜
h,V
His pullback D˜
h,V
η(CH)-
asymp o ically compac .
(b) The p ocess S:R2
d×M2
V→M2
Vis pullback DV
η(M2
H)-asymp o ically compac .
We es ablish now he ollowing esul abou he exis ence o minimal pullback
a ac o s o he p ocess Uon C˜
h,V
H, which can be p o ed in a same way as [9,
Theo em 5.1].
Theo em 5. Assume ha ∈L2
loc(R; (L2(Ω))2), and ha and g:R×CH→
(L2(Ω))2sa is y (I)–(V). Then, o any ˜
h∈[0, h], he p ocess Uon C˜
h,V
Hpossesses
a minimal pullback D˜
h,V
η(CH)-a ac o AD˜
h,V
η(CH), a minimal pullback D˜
h,V
F(CH)-
a ac o AD˜
h,V
F(CH), and a minimal pullback DF(C˜
h,V
H)-a ac o ADF(C˜
h,V
H). Be-
sides, he ollowing ela ions hold:
ADF(C˜
h,V
H)( )⊂ AD˜
h,V
F(CH)( )
⊂ ADF(CH)( )
⊂ AD˜
h,V
η(CH)( ) = ADη(CH)( )
⊂CV∀ ∈R,(21)
and o any amily b
D∈ Dη(CH),
lim
τ→−∞ dis CV(U( , τ)D(τ),ADη(CH)( )) = 0 ∀ ∈R.
Finally, i mo eo e sa is ies
sup
s≤0e−ηs Zs
−∞
eηθ| (θ)|2dθ<∞,(22)
hen all a ac o s in (21) coincide, and his amily is empe ed in CV, in he sense
ha
lim
→−∞ eη sup
∈ADη(CH)( )
k k2
CV= 0,
whe e k kCV= maxs∈[−h,0] k (s)k o any ∈CV.

REGULARITY RESULTS OF PULLBACK ATTRACTORS FOR NSE WITH DELAYS 17
Rema k 13. Obse e ha , unde he assump ions o Theo em 5, one has ha
AD˜
h,V
η(CH)≡ ADh,V
η(CH) o any ˜
h∈[0, h], i.e., he pullback a ac o AD˜
h,V
η(CH)is
independen o ˜
h.
Ac ually, i also sa is ies (22), hen AD˜
h,V
F(CH)≡ ADh,V
F(CH), and ADF(C˜
h,V
H)≡
ADF(Ch,V
H).
Rema k 14. Unde he assump ions o Theo em 5, since b
D1,η,h belongs o Dh,V
η(CH),
and he se D1,η,h( ) is closed in Ch,V
H o all ∈R, om Rema ks 4and 10, and
he equali y in (21), we deduce ha ADη(CH)belongs o Dh,V
η(CH).
In ac , i in addi ion sa is ies (22), hen, o each T∈R, he se {ADη(CH)( ) :
≤T}is bounded in Ch,V
H.
We a e also able o ob ain he exis ence o minimal pullback a ac o s o he
p ocess Son M2
V.
Theo em 6. Suppose ha ∈L2
loc(R; (L2(Ω))2), and ha and g:R×CH→
(L2(Ω))2sa is y (I)–(V). Then, he e exis he minimal pullback DF(M2
V)-a ac o
ADF(M2
V), and he minimal pullback DV
η(M2
H)-a ac o ADV
η(M2
H) o he p ocess S
on M2
V, and he ollowing ela ions hold:
ADF(M2
V)( )⊂ ADF(M2
H)( )⊂ ADη(M2
H)( ) = ADV
η(M2
H)( )∀ ∈R.(23)
In pa icula , o any amily b
D∈ Dη(M2
H),
lim
τ→−∞ dis M2
V(S( , τ)D(τ),ADη(M2
H)( )) = 0 ∀ ∈R.(24)
Finally, i also sa is ies (22), hen
ADF(M2
V)( ) = ADF(M2
H)( ) = ADη(M2
H)( ) = ADV
η(M2
H)( )∀ ∈R,
and his amily is empe ed in M2
V, i.e.,
lim
→−∞ eη sup
(w,ϕ)∈ADη(M2
H)( )
k(w, ϕ)k2
M2
V= 0.(25)
P oo . The exis ence o ADF(M2
V)and ADV
η(M2
H)is a di ec consequence o Theo em
2, Co olla y 1, P oposi ion 2, P oposi ion 3, and Co olla y 3.
In (23), he inclusions ollow om Co olla y 1, Theo em 3, and Rema k 11. The
equali y holds by applying Theo em 3and Rema k 6, using Theo em 1, Lemma 2,
Rema k 11, and Co olla y 3.
The pullback a ac ion esul (24) comes om Rema k 8, Lemma 2, and he
ac ha by he egula i y p ope y (a) in Theo em 1, o any b
D∈ Dη(M2
H) and
any τ < −h−1,
dis M2
V(S( , τ)D(τ),ADη(M2
H)( ))
= dis M2
V(S( , τ +h+ 1)(S(τ+h+ 1, τ)D(τ)),ADη(M2
H)( ))
= dis M2
V(S( , τ +h+ 1)(j(e
U(τ+h+ 1, τ)D(τ))),ADη(M2
H)( ))
= dis M2
V(S( , τ +h+ 1)(j(D(h+1)(τ))),ADV
η(M2
H)( )),
since i is clea ha he amily {j(D(h+1)(τ)) : τ∈R}belongs o DV
η(M2
H).
I mo eo e sa is ies (22), he equali y ADF(M2
H)( ) = ADη(M2
H)( ) ollows om
Rema k 5, and he equali y ADF(M2
V)( ) = ADF(M2
H)( ) is again a consequence o
18 J. GARC´
IA-LUENGO, P. MAR´
IN-RUBIO, AND J. REAL
Theo em 3, by using he second es ima e in (13), Rema k 11, and Co olla y 3, since
(22) is equi alen o
sup
s≤0Zs
s−1
| (θ)|2dθ < ∞.(26)
Las ly, he empe ed p ope y (25) comes om (22) (and he e o e (26)) and he
empe ed cha ac e o ρ2( ) de ined in Lemma 3.
Rema k 15. Unde he assump ions o Theo em 6, easoning analogously as in
Rema k 14, one has ha ADη(M2
H)belongs o DV
η(M2
H).
To conclude, we ela e he minimal pullback a ac o s ob ained in C˜
h,V
Hand M2
V
h ough he canonical injec ion j.
Theo em 7. Assume ha ∈L2
loc(R; (L2(Ω))2), and ha and g:R×CH→
(L2(Ω))2sa is y (I)–(V). Then, he ollowing ela ions hold:
j(ADF(Ch,V
H)( )) ⊂ ADF(M2
V)( )∀ ∈R,and (27)
j(AD˜
h,V
η(CH)( )) = ADV
η(M2
H)( )∀˜
h∈[0, h], ∈R.(28)
Ac ually, i also sa is ies (22), hen, o any ˜
h∈[0, h],
j(ADF(C˜
h,V
H)( )) = j(AD˜
h,V
F(CH)( )) = ADF(M2
V)( )∀ ∈R.(29)
P oo . In o de o p o e he inclusion in (27) we p oceed simila ly as in [19, Theo em
5], aking in o accoun ha he map jis con inuous om Ch,V
Hin o M2
V, and ha
j(DF(Ch,V
H)) ⊂ DF(M2
V).
The equali y in (28) is a consequence o p ope y (11) in Theo em 4, using he
equali ies (21) and (23).
Finally, he equali ies in (29) ollow om (28) and he known ac s ha , unde
he addi ional assump ion (22), all a ac o s in (21) and (23) coincide.
Acknowledgmen s. While inishing his pape among o he p ojec s, ou coau-
ho P o . Jos´e Real deceased. This wo k is dedica ed o his memo y, wi h ou
deepes and mos since e admi a ion, g a i ude, and lo e. He was P.M.-R. and
J.G.-L.’s PhD-ad iso , cle e ma hema ician wi h a sha p iew on p oblems, gen-
e ous and wonde ul colleague, and be e iend. He passed away oo soon, being
only 60 yea s old. We miss him deeply, bu he will s ay o e e in ou hea s.
This wo k has been pa ially suppo ed by Minis e io de Ciencia e Inno aci´on
(Spain) unde p ojec MTM2011-22411. J.G.-L. is a ellow o P og ama de FPU
del Minis e io de Educaci´on (Spain).
The au ho s hank one o he e e ees by his/he commen s, which led o im-
p o emen s in he p esen a ion o his pape .
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E-mail add ess, J. Ga c´ıa-Luengo: [email p o ec ed]
E-mail add ess, P. Ma ´ın-Rubio: [email p o ec ed]
E-mail add ess, J. Real: [email p o ec ed]