Manusc ip submi ed o Websi e: h p://AIMsciences.o g
AIMS’ Jou nals
Volume 00, Numbe 0, Xxxx XXXX pp. 000–000
THREE DIMENSIONAL SYSTEM OF GLOBALLY MODIFIED
NAVIER-STOKES EQUATIONS WITH INFINITE DELAYS
P. Ma ´
ın-Rubio, A. M. M´
a quez-Du ´
an & J. 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 Aim Sciences)
Abs ac . Exis ence and uniqueness o solu ion o a globally modi ied e -
sion o Na ie -S okes equa ions con aining in ini e delay e ms a e es ablished.
Mo eo e , we also analyze he s a iona y p oblem and, unde sui able addi-
ional condi ions, we ob ain global exponen ial decay o he solu ions o he
e olu iona y p oblem o he s a iona y solu ion.
Keywo ds: Globally Modi ied Na ie -S okes Equa ions; in ini e delays.
Ma hema ics Subjec Classi ica ions (2000): 35K55, 35Q30, 34D23, 34K20
1. In oduc ion and s a emen o he p oblem. Le Ω ⊂R3be an open
bounded se wi h egula bounda y Γ, and le N∈(0,+∞) be ixed. Le us de ine
FN: [0,+∞)→(0,1] by
FN( ) := min 1,N
, ∈[0,+∞),
and conside he ollowing sys em o globally modi ied Na ie -S okes equa ions on
Ω wi h homogeneous Di ichle bounda y condi ion















∂u
∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) in (τ, T∗)×Ω,
∇ · u= 0 in (τ, T∗)×Ω,
u= 0 on (τ, T∗)×Γ,
u(τ, x) = u0(x), x ∈Ω,
(1)
whe e ν > 0 is he kinema ic iscosi y, u he eloci y ield o he luid, p he p es-
su e, τ∈Ran ini ial ime, u0 he ini ial eloci y ield, ( ) a gi en ex e nal o ce
ield, and T∗∈(τ, +∞] a gi en inal ime.
The sys em (1) is indeed a globally modi ied e sion o he Na ie -S okes sys-
em – he modi ying ac o FN(kuk) depends on he no m kuk=k∇uk(L2(Ω))3×3,
2000 Ma hema ics Subjec Classi ica ion. P ima y: 35K55; 35Q30; 34D23; 34K20.
Key wo ds and ph ases. Globally Modi ied Na ie -S okes Equa ions; in ini e delays.
Pa ially suppo ed by Minis e io de Ciencia e Inno aci´on (Spain), p ojec MTM2008-00088
and Jun a de Andaluc´ıa, g an P07-FQM-02468.
1
2 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
which in u n depends on ∇uo e he whole domain Ω and no jus a o nea he
poin x∈Ω unde conside a ion. Essen ially, i p e en s la ge g adien s domina ing
he dynamics and leading o explosions. I iola es he basic laws o mechanics,
bu ma hema ically he sys em (1) is a well de ined sys em o equa ions, jus like
he modi ied e sions o he Na ie -S okes equa ions o Le ay and o he s wi h o he
molli ica ions o he nonlinea e m, see he e iew pape [8]. I is wo h men ioning
ha a global cu o unc ion in ol ing he D(A1/4) no m o he wo dimensional
s ochas ic Na ie -S okes equa ions is used in [9], and a cu -o unc ion simila o
he one we will use he e was conside ed in [23].
The sys em (1) was in oduced and s udied in [1] (see also [2,14,15,16,3] and
he e iew pape [13]). Howe e , he e a e si ua ions in which he model is be e
desc ibed i some e ms con aining delays appea in he equa ions. These delays
may appea , o ins ance, when one wan s o con ol he sys em by applying a o ce
which akes in o accoun no only he p esen s a e bu he comple e his o y o he
solu ions. The e o e, in his pape we a e in e es ed in he case in which e ms
con aining in ini e delays appea . We conside he ollowing e sion:













∂u
∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) + g( , u ) in (τ, T∗)×Ω,
∇ · u= 0 in (τ, T∗)×Ω,
u= 0 on (τ, T∗)×Γ,
u(τ+s, x) = φ(s, x), s ∈(−∞,0], x ∈Ω,
(2)
whe e gis ano he ex e nal o ce con aining some he edi a y cha ac e is ic and φ
is a gi en unc ion de ined in he in e al (−∞,0].
Ou goal is o es ablish he exis ence and uniqueness o solu ion o he abo e
p oblem and o s udy i s asymp o ic beha iou ( o a simila goal in Na ie -S okes
models wi h ini e delay c . [5,6,7], and o ODEs and PDEs wi h unbounded
delay e ms c . [4,18]).
The s uc u e o he pape is he ollowing: in Sec ion 2we ecall some spaces
use ul o he abs ac amewo k and some p ope ies and es ima es ela ed o
he ope a o s in ol ed in he model. In Sec ion 3 he exis ence and uniqueness o
solu ion o he (e olu iona y) p oblem is gi en. The s a iona y p oblem is ea ed
in Sec ion 4, whe e we also p o e ha unde adequa e assump ions, any solu ion o
he e olu iona y p oblem has an exponen ial decay owa d he s a iona y solu ion.
2. P elimina ies. To se ou p oblem in he abs ac amewo k, we conside he
ollowing usual abs ac spaces (see [17] and [21,22]):
V=nu∈(C∞
0(Ω))3: di u= 0o,
H= he closu e o Vin (L2(Ω))3wi h inne p oduc (·,·) and associa e no m |·| ,
whe e o u, ∈(L2(Ω))3,
(u, ) =
3
X
j=1 ZΩ
uj(x) j(x)dx,
3D-GMNS EQUATIONS WITH INFINITE DELAYS 3
V= he closu e o Vin (H1
0(Ω))3wi h scala p oduc ((·,·)) and associa e no m
k·k ,whe e o u, ∈(H1
0(Ω))3,
((u, )) =
3
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 pai ing be ween V0
and V. Finally, we will iden i y e e y u∈Hwi h he elemen u∈V0gi en by
h u, i= (u, )∀ ∈V.
I ollows ha V⊂H⊂V0,whe e he injec ions a e dense and compac .
We conside he linea con inuous ope a o A:V→V0de ined by
hAu, i= ((u, )) ∀u, ∈V. (3)
Deno ing D(A) = {u∈V:Au ∈H},wi h inne p oduc (u, )D(A)= (Au, A ),
hen, by he egula i y o Γ, D(A) = (H2(Ω))3∩V, and Au =−P∆u, ∀u∈D(A),
is he S okes ope a o (Pis he o ho-p ojec o om (L2(Ω))3on o H).
Le us deno e
λ1= in
∈V {0}
k k2
| |2>0,
he i s eigen alue o he S okes ope a o .
Now we de ine he ilinea o m bon V×V×Vby
b(u, , w) =
3
X
i,j=1 ZΩ
ui
∂ j
∂xi
wjdx, ∀u, , w ∈V,
and we deno e
bN(u, , w) = FN(k k)b(u, , w),∀u, , w ∈V.
The o m bNis linea in uand w, bu i is nonlinea in . E iden ly we ha e
bN(u, , ) = 0, o all u, ∈V. We will also make use o he ollowing inequali y
(see [21] and [10])
|b(u, , w)| ≤ 2−1|u|1/4kuk3/4k k|w|1/4kwk3/4,∀u, , w ∈V. (4)
In pa icula , his implies ha he e exis s a cons an C1>0 only dependen on Ω
(namely, C1= (2λ1/4
1)−1) such ha
|b(u, , w)| ≤ C1kukk kkwk,∀u, , w ∈V.
Thus by he de ini ion o FN,i we deno e
hBN(u, ), wi=bN(u, , w),∀u, , w ∈V,
we ha e
kBN(u, )k∗≤NC1kuk,∀u, ∈V. (5)
We ecall (see [21]) ha he e exis s a cons an C2>0 depending only on Ω
such ha
|b(u, , w)| ≤ C2kuk1/2|Au|1/2k k|w|,(6)
o all u∈D(A), ∈V, w ∈H, and
|b(u, , w)| ≤ C2kukk k|w|1/2kwk1/2,(7)
o all u, , w ∈V. (See [20] o he p oo o (7)).
4 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
Le Xbe a Banach space. The no a ion BX(a, ) will be used o deno e he open
ball o cen e aand adius in he space X. Gi en a unc ion u: (−∞, T∗)→X,
o each <T∗we deno e by u he unc ion de ined on (−∞,0) by he ela ion
u (s) = u( +s), s ∈(−∞,0).
One possibili y o deal wi h in ini e delays, and which we will use he e (c .
[18,11,12]), is o conside , o any γ > 0, he space :
Cγ(H) = ϕ∈C((−∞,0]; H) : ∃lim
s→−∞ eγsϕ(s)∈H,
which is a Banach space wi h he no m
kϕkγ:= sup
s∈(−∞,0]
eγs|ϕ(s)|.
In o de o s a e he p oblem in he co ec amewo k, le us i s es ablish some
ini ial assump ions on some e ms in he equa ion:
We will assume ha ∈L2(τ, T; (L2(Ω))3) o all T∈(τ, T∗]∩R.
Fo he e m g, in which he delay is p esen , we assume ha g: (τ, T∗)×
Cγ(H)→(L2(Ω))3sa is ies
(g1) Fo any ξ∈Cγ(H) he mapping (τ, T∗)3 7→ g( , ξ) is measu able,
(g2) g( , 0) = 0 o all ∈(τ, T∗),
(g3) he e exis s a cons an Lg>0 such ha o any ∈(τ, T∗) and all ξ, η ∈
Cγ(H),
|g( , ξ)−g( , η)| ≤ Lgkξ−ηkγ.
Rema k 1. (i) Condi ion (g2) is no eally a es ic ion, since o he wise, i |g(·,0)| ∈
L2(τ, T ) o all T∈(τ, T∗]∩R,we could ede ine ˆ
( ) = ( ) + g( , 0) and ˆg( , ·) =
g( , ·)−g( , 0).In his way he p oblem is exac ly he same and ˆ
and ˆgsa is y he
equi ed assump ions.
(ii) Condi ions (g2) and (g3) imply ha
|g( , ξ)| ≤ Lgkξkγ,
so ha |g(·, ξ)| ∈ L∞(τ, T∗).
We will deno e Pm he o hogonal p ojec o o Hon o he ec o space gene a ed
by he i s meigen unc ions o he S okes p oblem in Ω wi h homogeneous Di ichle
bounda y condi ions.
An example o ope a o sa is ying assump ions (g1)-(g3) is gi en he e.
Example 1. We conside he ope a o g: (τ, T∗)×Cγ(H)→(L2(Ω))3de ined as
ollows:
g( , ξ) := Z0
−∞
G( , s, ξ(s))ds ∀ ∈(τ, T∗),∀ξ∈Cγ(H),
whe e he unc ion G: (τ, T∗)×(−∞,0) ×R3→R3sa is ies he ollowing assump-
ions:
(a) G( , s, 0) = 0 o all ( , s)∈(τ, T∗)×(−∞,0).
(b) The e exis s a unc ion κ: (−∞,0) →(0,+∞) such ha
kG( , s, u)−G( , s, )kR3≤κ(s)ku− kR3
∀u, ∈R3,∀( , s)∈(τ, T∗),×(−∞,0),
3D-GMNS EQUATIONS WITH INFINITE DELAYS 5
(c) and he unc ion κsa is ies ha κ(·)e−(γ+)·∈L2((−∞,0)) o some  > 0.
Namely, he ope a o gde ines an elemen o (L2(Ω))3in he ollowing way:
g( , ξ)(x) = Z0
−∞
G( , s, ξ(s)(x))ds ∀x∈Ω.
We check now ha gsa is ies he assump ion (g3), and using (a) abo e, we
ob ain ha i is well de ined as a map wi h alues in (L2(Ω))3:
ZΩZ0
−∞
κ(s)kξ(s)(x)−η(s)(x)kR3ds2
dx
=ZΩZ0
−∞
κ(s)e−(γ+)se(γ+)skξ(s)(x)−η(s)(x)kR3ds2
dx
≤ZΩZ0
−∞
κ2(s)e−2(γ+)sdsZ0
−∞
e2(γ+)skξ(s)(x)−η(s)(x)k2
R3dsdx
=Z0
−∞
κ2(s)e−2(γ+)sdsZΩZ0
−∞
e2(γ+)skξ(s)(x)−η(s)(x)k2
R3dsdx
=CκZ0
−∞ ZΩ
e2(γ+)skξ(s)(x)−η(s)(x)k2
R3dxds
≤Cκ"sup
s∈(−∞,0]
e2γs ZΩ
kξ(s)(x)−η(s)(x)k2
R3dx#Z0
−∞
e2sds
=Cκkξ−ηk2
γ
1
2
=Lgkξ−ηk2
γ,
whe e we ha e deno ed Cκ=kκ(·)e−(γ+)·k2
L2(−∞,0) and Lg=Cκ/(2).
3. Exis ence o solu ions. In his sec ion we es ablish exis ence o solu ion o
(2) by a compac ness me hod using a Faedo-Gale kin scheme.
De ini ion 1. A weak solu ion o (2) is a unc ion u∈C((−∞, T]; H)∩L2(τ, T ;V)
o all T∈(τ, T ∗]∩R, wi h uτ=φand such ha o all ∈V,
d
d (u( ), ) + ν((u( ), )) + bN(u( ), u( ), )=( ( ), )+(g( , u ), ),
in he sense o D0(τ, T ∗).
Rema k 2. I uis a solu ion o (2) in he sense gi en abo e, hen usa is ies an
ene gy equali y, namely:
|u( )|2+ 2νZ
s
ku( )k2d
=|u(s)|2+ 2 Z
s
[( ( ), u( )) + (g( , u ), u( ))] d ∀s, ∈[τ, T∗]∩R.
Fi s , we will p o e he uniqueness o weak solu ions o ou model in a simila
way as done in [20] o he model wi hou delay. We will only include he de ailed
es ima es which in ol e he delay e m.

6 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
Theo em 1. Unde he abo e assump ions, he e exis s a mos a weak solu ion u
o (2).
The p oo is simila o, bu a bi mo e complica ed han o he 2D-Na ie -S okes
equa ions and depends on he ollowing
Lemma 1. ([20]) Fo e e y u, ∈V, and each N > 0,
1. 0 ≤ kukFN(kuk)≤N,
2. |FN(kuk)−FN(k k)| ≤ 1
NFN(kuk)FN(k k)ku− k.
P oo o Theo em 1:Le u, be wo weak solu ions wi h he same ini ial condi ions
and se w= −u. Then, using he ene gy equali y, we ob ain
1
2
d
d |w|2+νkwk2+hNL(u, ), wi= (g( , )−g( , u ), w), ∈(τ, T∗),
whe e we ha e se hNL(u, ), wi=FN(kuk)b(u, u, w)−FN(k k)b( , , w).F om he
p ope ies o he ilinea o m bi easily ollows ha
hNL(u, ), wi=FN(kuk)b(w, u, w)+(FN(kuk)−FN(k k))b( , u, w)
+FN(k k)b( , w, w).
Now using Lemma 1, o mula (7) and Young’s inequali y (see [20] o he de ails)
he e exis s a cons an C3>0,which depends on C2and ν, such ha ,
| hNL(u, ), wi | ≤ νkwk2+C3N4|w|2.
Consequen ly, aking (g3) in o accoun , we ob ain
d
d |w( )|2≤2C3N4|w( )|2+ 2Lgkw kγ|w( )|, ∈(τ, T∗).
Obse e ha w(s) = 0 i s≤τ. The e o e, o ∈(τ, T∗) :
kw kγ= sup
θ≤0
eγθ|w( +θ)| ≤ sup
θ∈[τ− ,0]
|w( +θ)|.
Thus we ob ain
|w( )|2≤2C3N4Z
τ
|w(s)|2ds + 2LgZ
τ
sup
∈[τ,s]
|w( )||w(s)|ds
≤(2C3N4+ 2Lg)Z
τ
sup
∈[τ,s]
|w( )|2ds,
o any ∈[τ, T∗).
Now we deduce ha
sup
∈[τ, ]
|w( )|2≤(2C3N4+ 2Lg)Z
τ
sup
∈[τ,s]
|w( )|2ds,
o any ∈[τ, T∗),whence he G onwall lemma inishes he p oo .
Ou main esul is he ollowing
Theo em 2. Suppose ha ∈L2(τ, T; (L2(Ω))3) o all T∈(τ, T∗]∩R, g :
(τ, T∗)×Cγ(H)→(L2(Ω))3sa is ying he assump ions (g1)–(g3), and φ∈Cγ(H)
a e gi en, and ha 2γ > νλ1.Then, he e exis s a unique weak solu ion uo (2),
which in ac is a s ong solu ion in he sense ha
u∈C((τ, T ]; V)∩L2(τ+ε, T;D(A)),
3D-GMNS EQUATIONS WITH INFINITE DELAYS 7
o all 0< ε < T∗−τand any T∈(τ+ε, T∗]∩R.
Mo eo e , i φ(0) ∈V, hen usa is ies
u∈C([τ, T ]; V)∩L2(τ, T;D(A)),
o all T∈(τ, T∗]∩R.
P oo . We spli he p oo in se e al s eps.
S ep 1: A Gale kin scheme. Le us conside { j} ⊂ V, he o hono mal basis
o Ho all he eigen unc ions o he S okes ope a o . Deno e Vm= span[ 1, . . . , m]
and conside he p ojec o Pmu=Pm
j=1(u, j) j.
De ine also
um( ) =
m
X
j=1
αm,j( ) j
whe e he uppe sc ip mwill be used ins ead o (m) o sho since no con usion
is possible wi h powe s o u, and whe e he coe icien s αm,j a e equi ed o sa is y
he ollowing sys em:
d
d (um( ), j) + ν((um( ), j)) + bN(um( ), um( ), j)
= ( ( ), j)+(g( , um
), j),1≤j≤m, (8)
whe e he equa ions a e unde s ood in he sense o D0(τ, T∗),and he ini ial condi-
ions is um(τ+s) = Pmφ(s) o s∈(−∞,0].
The abo e sys em o o dina y unc ional di e en ial equa ions wi h in ini e delay
ul ills he condi ions o exis ence and uniqueness o local solu ion o [12, Th.1.1,
p.36].
Nex , we will deduce a p io i es ima es ha assu e ha he solu ions do exis
o all ime ∈[τ, T∗]∩R.
S ep 2: A p io i es ima es. Mul iplying (8) by umwe ob ain
1
2
d
d |um( )|2+νλ1
2|um( )|2+ν
2kum( )k2≤( ( ), um( )) + (g( , um
), um( ))
≤ | ( )||um( )|+Lgkum
kγ|um( )|
≤ν
4kum( )k2+| ( )|2
νλ1
+Lgkum
k2
γ.
Hence
|um( )|2+ν
2Z
τ
e−νλ1( −s)kum(s)k2ds
≤e−νλ1( −τ)|u(τ)|2+ 2 Z
τ
e−νλ1( −s)| (s)|2
νλ1
+Lgkum
sk2
γds. (9)
Fu he
kum
k2
γ≤max (sup
θ∈(−∞,τ− ]
e2γθ|φ(θ+ −τ)|2,sup
θ∈[τ− ,0] e2γθ−νλ1( −τ+θ)|u(τ)|2
+2e2γθ Z +θ
τ
e−νλ1( +θ−s)| (s)|2
νλ1
+Lgkum
sk2
γds).
8 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
On he one hand
sup
θ∈(−∞,τ− ]
eγθ|φ(θ+ −τ)|= sup
θ≤0
eγ(θ−( −τ))|φ(θ)|
=e−γ( −τ)kφkγ.
On he o he hand, as we a e assuming ha 2γ > νλ1,
sup
θ∈[τ− ,0]
e2γθ−νλ1( −τ+θ)|u(τ)|2≤e−νλ1( −τ)|u(τ)|2
and
sup
θ∈[τ− ,0]
e2γθ Z +θ
τ
e−νλ1( +θ−s)| (s)|2
νλ1
+Lgkum
sk2
γds
≤Z
τ
e−νλ1( −s)| (s)|2
νλ1
+Lgkum
sk2
γds.
Collec ing hese inequali ies we deduce
kum
k2
γ≤e−νλ1( −τ)kφk2
γ+ 2 Z
τ
e−νλ1( −s)| ( )|2
νλ1
+Lgkum
sk2
γds.
By he G onwall lemma we ha e
kum
k2
γ≤e−(νλ1−2Lg)( −τ)kφk2
γ+2
νλ1Z
τ
e−(λ1ν−2Lg)( −s)| (s)|2ds.
Then we ob ain he ollowing es ima es: o any R > 0 and T∈(τ, T∗]∩R, he e
exis s a cons an C=C(τ, T, R),depending on some cons an s o he p oblem
(namely, λ1, ν, Lgand ), and on τ, T and R, such ha
kum
k2
γ≤C(τ, T, R)∀ ∈[τ, T],∀ kφkγ≤R, ∀m≥1.(10)
In pa icula , his implies ha
{um}is bounded in L∞(τ, T;H)∀T∈(τ, T∗]∩R.(11)
Now, i ollows om (9) and (10) ha
ν
2e−νλ1(T−τ)ZT
τ
kum(s)k2ds
≤ν
2ZT
τ
e−νλ1(T−s)kum(s)k2ds
≤ |u(τ)|2+ 2 ZT
τ
e−νλ1(T−s)| (s)|2
νλ1
+Lgkum
sk2
γds
≤R2+ 2 ZT
τ
e−νλ1(T−s)| (s)|2
νλ1
+LgC(τ, T, R)ds,
so ha we conclude he exis ence o ano he cons an ( elabelled he same) C(τ, T, R)
such ha
kumk2
L2(τ,T ;V)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(12)
Now, obse e ha (8) is equi alen o
dum
d =−νAum−PmBN(um, um) + Pm ( ) + Pmg( , um
).(13)
3D-GMNS EQUATIONS WITH INFINITE DELAYS 9
F om (5), (11), (12) and (13), by he choice o he basis one also deduces ha
k(um)0k2
L2(τ,T ;V0)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(14)
So, his implies he exis ence o a
u∈L∞(τ, T ;H)∩L2(τ, T;V) wi h u0∈L2(τ, T;V0),∀T∈(τ, T∗]∩R,
and a subsequence o {um}which con e ges weak-s a o uin L∞(τ, T;H),weakly
o uin L2(τ, T ;V),wi h {(um)0}con e ging weakly o u0in L2(τ, T;V0) o all
T∈(τ, T∗]∩R.
Obse e in pa icula ha u∈C([τ, T]; H) o all T∈(τ, T∗]∩R.
By a compac ness esul (c . [17, Ch.1,Th.5.1]), one can hen deduce ha a
subsequence in ac con e ges s ongly o uin L2(τ, T;H) and a.e. in (τ, T) wi h
alues in Hand a.e. in (τ, T)×Ω o all T∈(τ, T∗]∩R.
S ep 3: Some mo e a p io i es ima es. The es ima es ob ained abo e a e
no enough o pass o he limi and deduce ha uis a solu ion o (2). Namely, we
ha e wo main di icul ies. On he one hand, we need o pass o he limi in g(um),
his will be done in S ep 4, p o ing ha ac ually um
→u in Cγ(H).On o he
hand, he weak con e gence in L2(τ, T;V) is no enough o ensu e ha
kum( )k→ku( )k
o a leas
FN(kum( )k)→FN(ku( )k) o a.a. ,
which is needed o manage he nonlinea e m BN(um, um).
In o de o so ou his las ouble, we need o ind a s onge es ima e. We
p oceed now wi h ha . Take he inne p oduc o he Gale kin ODE (8) wi h Aum
and ob ain
1
2
d
d kum( )k2+ν|Aum( )|2+bN(um( ), um( ), Aum( ))
= ( ( ), Aum( )) + (g( , um
), Aum( )).(15)
Ob iously,
( ( ), Aum( )) ≤ | ( )||Aum( )| ≤ ν
8|Aum( )|+2
ν| ( )|2
and
|(g( , um
), Aum( ))| ≤ ν
8|Aum( )|2+2
ν|g( , um
)|2.
By (6), Lemma 1and Young’s inequali y, i ollows
|bN(um( ), um( ), Aum( ))| ≤ N
kum( )kC2kum( )k3/2|Aum( )|3/2
=NC2kum( )k1/2|Aum( )|3/2
≤ν
4|Aum( )|2+CNkum( )k2,
wi h CN=27(NC2)4
4ν3.
Thus (15) simpli ies o
d
d kum( )k2+ν|Aum( )|2≤4
ν| ( )|2+4
ν|g( , um
)|2+ 2CNkum( )k2.(16)
16 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
Thus, i we ake
β=λ−1/2
1| |
ν−Lgλ−1
1
,
we ob ain ((Rmu, u)) ≥0∀u∈Vmsuch ha kuk=β.
Consequen ly by a co olla y o he B ouwe ’s ixed poin heo em (see [17, p.53]),
o each m≥1 he e exis um∈Vmsuch ha Rm(um)=0,wi h kumk ≤ β.
Obse e mo eo e ha Aum∈Vm,and he e o e
ν|Aum|2=−hBN(um, um), Aumi+ ( , Aum)+(g(um), Aum) (40)
≤ν
2|Aum|2− hBN(um, um), Aumi+| |2
ν+L2
gβ2
νλ1
.
Mo eo e , by (6) and Young’s inequali y,
|hBN(um, um), Aumi| ≤ ν
4|Aum|2+CNkumk2(41)
≤ν
4|Aum|2+CNβ2,
wi h CN=27(NC2)4
4ν3.
F om (40) and (41), we deduce ha he sequence {um}is bounded in D(A), and
consequen ly, by he compac injec ion o D(A) in V, we can ex ac a subsequence
{um0} ⊂ {um}, ha con e ges weakly in D(A) and s ongly in V o an elemen
u∗∈D(A). I is now s anda d o ake limi s in (38) and o ob ain ha u∗is an
s a iona y solu ion.
In o de o p o e he inal egula i y ema k, i is enough o ake in o accoun
ha e e y s a iona y solu ion u∗ o (34) is also a solu ion o (2), bu wi h ini ial
da a φ( ) = u∗ o ∈(−∞,0], and o cing e m +g(u∗). Thus, we can apply
Theo em 2.
Uniqueness Le us suppose ha u∗and eu∗a e wo s a iona y solu ions o (34).
Then,
νhAu∗−Aeu∗, i+hBN(u∗, u∗)−BN(eu∗,eu∗), i
= (g(u∗)−g(eu∗), ),∀ ∈V. (42)
Taking =u∗−eu∗and p oceeding as in (39) we ob ain om (42)
νku∗−eu∗k2≤ |FN(ku∗k)b(u∗−eu∗, u∗, u∗−eu∗)|
+|(FN(ku∗k)−FN(keu∗k))b(eu∗, u∗, u∗−eu∗)|
+λ−1
1Lgku∗−eu∗k2.(43)
F om his inequali y, aking in o accoun (4), ha b(eu∗, u∗, u∗−eu∗) = 0,and
he ac ha FN(ku∗k)≤1,we ob ain
νku∗−eu∗k2≤λ−1/4
1ku∗k ku∗−eu∗k2+λ−1
1Lgku∗−eu∗k2,
and he e o e, by he es ima e (36),
νku∗−eu∗k2≤ λ−1/4
1
λ−1/2
1| |
ν−λ−1
1Lg
+λ−1
1Lg!ku∗−eu∗k2.(44)

3D-GMNS EQUATIONS WITH INFINITE DELAYS 17
On he o he hand, i in (43) we use (4), and Lemma 1, i.e. ku∗kFN(ku∗k)≤N
and keu∗kFN(keu∗k)≤N, we ob ain
νku∗−eu∗k2≤Nλ−1/4
1ku∗−eu∗k2+λ−1
1Lgku∗−eu∗k2.(45)
F om (44) and (45) we deduce ha i (37) holds, hen u∗=eu∗.
Theo em 4. Assume ha he assump ions in Theo em 2wi h and gindependen
o ime and (37) hold. Then he e exis s a alue 0< λ < 2γsuch ha o he
solu ion u(·,0, φ)o (2) wi h τ= 0,T∗= +∞and φ∈Cγ(H), he ollowing
es ima es hold o all ≥0:
|u( , 0, φ)−u∗|2≤e−λ |φ(0) −u∗|2+Lg
2γ−λkφ−u∗k2
γ,(46)
ku (·,0, φ)−u∗k2
γ
≤max e−2γ kφ−u∗k2
γ, e−λ |φ(0) −u∗|2+Lg
2γ−λkφ−u∗k2
γ,(47)
whe e u∗is he unique s a iona y solu ion o (34) gi en by Theo em 3.
P oo . Fo sho deno e u( ) = u(·,0, φ).Le us also deno e w( ) = u( )−u∗.
Conside ing equa ions (34) o u( ) and (35) o u∗,one has
d
d (w( ), ) + ν((w( ), )) + bN(u( ), u( ), )−bN(u∗, u∗, )=(g(u )−g(u∗), ),
o > 0, o any ∈V.
F om ene gy equali y and he Lipschi z condi ion on g, and in oducing an ex-
ponen ial e m eλ wi h a posi i e alue λ o be ixed la e on, we ob ain
d
d (eλ |w( )|2)≤eλ λ|w( )|2−2νkw( )k2
+2|bN(u( ), u( ), w( ))−bN(u∗, u∗, w( ))|+ 2Lgkw kγ|w( )|),
o > 0.
Reasoning as o (44) and (45), we ha e
|bN(u( ), u( ), w)−bN(u∗, u∗, w)| ≤ µku( )−u∗k2,
whe e
µ:= min (Nλ−1/4
1,λ−3/4
1| |
ν−λ−1
1Lg).
Hence, using a Young inequali y wi h δ > 0 o be ixed la e on, we conclude
ha
d
d (eλ |w( )|2)≤eλ (−2ν+λλ−1
1+ 2µ+δλ−1
1Lg)kw( )k2+Lg
δeλ kw k2
γ.
The e o e, in eg a ing om 0 o , we ha e
eλ |w( )|2≤ |w(0)|2+Lg
δZ
0
eλskwsk2
γds
+(−2ν+λλ−1
1+ 2µ+δλ−1
1Lg)Z
0
eλskw(s)k2ds. (48)
18 P. MAR´
IN-RUBIO, A. M. M´
ARQUEZ-DUR´
AN & J. REAL
In o de o con ol he e m R
0eλskwsk2
γds, we p oceed as ollows.
Z
0
eλs sup
θ≤0
e2γθ|w(s+θ)|2ds
=Z
0
eλs max{sup
θ≤−s
e2γθ|w(s+θ)|2,sup
θ∈[−s,0]
e2γθ|w(s+θ)|2}ds
=Z
0
max{e−(2γ−λ)skφ−u∗k2
γ,sup
θ∈[−s,0]
e(2γ−λ)θeλ(s+θ)|w(s+θ)|2}ds.
So, i λ≤2γ, using he abo e equali y in (48), we ob ain
eλ |w( )|2≤ |w(0)|2+Lg
δkφ−u∗k2
γZ
0
e(λ−2γ)sds
+−2ν+λλ−1
1+2µ+δλ−1
1Lg+Lg(λ1δ)−1Z
0
max
∈[0,s]eλ kw( )k2ds.
Obse e ha he (op imal) choice o δ= 1 makes ha δλ−1
1Lg+Lg(λ1δ)−1is
minimal and he coe icien o he las in eg al is nega i e wi h a sui able choice o
λ∈(0,2γ) by (37). So, we can omi his e m and deduce ha
eλ |w( )|2≤ |w(0)|2+Lg
2γ−λ(1 −e(λ−2γ) )kφ−u∗k2
γ,
whence (46) ollows.
Finally, (47) can be deduced in he ollowing way:
kw k2
γ= sup
θ≤0
e2γθ|w( +θ)|2
= max{sup
θ∈(−∞,− ]
e2γθ|φ( +θ)−u∗|2,max
θ∈[− ,0] e2γθ|w( +θ)|2}
= max{e−2γ kφ−u∗k2
γ,max
θ∈[− ,0] e2γθ|w( +θ)|2},
and he second e m can be es ima ed using (46) and ha e(2γ−λ)θ≤1.
Acknowledgemen s. The au ho s would like o hank he e e ees o hei in e -
es ing sugges ions and commen s.
This wo k was pa ially suppo ed by Spanish Minis e io de Ciencia e Inno aci´on,
P ojec MTM2008-00088, and he Conseje ´ıa de Inno aci´on, Ciencia y Emp esa
(Jun a de Andaluc´ıa), P oyec o de Excelencia P07-FQM-02468.
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E-mail add ess, P. Ma ´ın-Rubio: [email p o ec ed]
E-mail add ess, A. M. M´a quez-Du ´an: [email p o ec ed]
E-mail add ess, J. Real: [email p o ec ed]