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AIMS’ Jou nals
Volume 00, Numbe 0, Xxxx XXXX pp. 000–000
EQUIVALENCE OF INVARIANT MEASURES AND
STATIONARY STATISTICAL SOLUTIONS FOR THE
AUTONOMOUS GLOBALLY MODIFIED NAVIER-STOKES
EQUATIONS
PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
Abs ac . A new p oo o exis ence o solu ions o he h ee dimensional
sys em o globally modi ied Na ie -S okes equa ions in oduced in [3] by Ca a-
ballo, Kloeden and Real is ob ained using a smoo he Gale kin scheme. This
is hen used o in es iga e he ela ionship be ween in a ian measu es and
s a is ical solu ions o his sys em in he case o empo ally independen o c-
ing e m. Indeed, we a e able o p o e ha a s a iona y s a is ical solu ion is
also an in a ian p obabili y measu e unde sui able assump ions.
1. In oduc ion. Le Ω ⊂R3be an open bounded se wi h egula bounda y Γ,
and conside he ollowing sys em o Na ie -S okes equa ions (NSE) on Ω wi h a
homogeneous Di ichle bounda y condi ion
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∂u
∂ −ν∆u+ (u· ∇)u+∇p= ( ) in (0,+∞)×Ω,
∇ · u= 0 in (0,+∞)×Ω,
u= 0 on (0,+∞)×Γ,
u(0, x) = u0(x), x ∈Ω,
whe e 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, and ( ) a gi en ex e nal o ce ield.
The e exis many modi ied e sions o Na ie -S okes equa ions due o Le ay
and o he s wi h molli ica ion (and/o cu o ) o he nonlinea e m as a way o
app oxima e he o iginal p oblem, see o ins ance he e iew pape o Cons an in
[1]. We also men ion he pape [5] by Flandoli and Maslowski wi h a global cu o
unc ion used in a 2-dimensional s ochas ic con ex .
In 2006, Ca aballo, Kloeden and Real (c . [3]) p oposed a 3-dimensional model
whe e he nonlinea e m included a cu o ac o FN(kuk) based on he no m o
he g adien o he solu ion in he whole domain. Namely, o N∈(0,+∞) he
Da e: 17 Sep embe 2008.
2000 Ma hema ics Subjec Classi ica ion. 35Q30, 35K90, 37L30.
Key wo ds and ph ases. Modi ied Na ie -S okes model, s a is ical solu ions, in a ian
measu es.
Pa ly suppo ed by Minis e io de Educaci´on y Ciencia p ojec MTM2005-01412.
1
2 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
unc ion FN: [0,+∞)→(0,1] is de ined by
FN( ) := min 1,N
, ∈[0,+∞).
They called he esul ing sys em
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∂u
∂ −ν∆u+FN(kuk) [(u· ∇)u] + ∇p= ( ) in (0,+∞)×Ω,
∇ · u= 0 in (0,+∞)×Ω,
u= 0 on (0,+∞)×Γ,
u(0, x) = u0(x), x ∈Ω,
(1)
he globally modi ied Na ie -S okes equa ions (GMNSE) and es ablished he well-
posedness o he model, in pa icula he absence o blow-up o solu ions, as well
as he exis ence o global V−a ac o s and ce ain ela ionships wi h he o iginal
NSE model. See also [7,8] o o he s udies and applica ions o he GMNSE as
well as he e iew pape [2].
Foias e al. [6] (see also he pape s ci ed he ein) ha e made a e y sys ema ic
s udy o in a ian measu es and s a is ical solu ions o he NSE models in 2 and
3 dimensions. This in es iga ion was con inued in [4] by Ca aballo, Kloeden and
Real o he GMNSE, who p o ed he exis ence o ime-a e aged measu es, hen o
in a ian measu es and inally ha any in a ian measu e is a s a iona y s a is ical
solu ion. We no e ha a ela ion be ween a amily o ime-a e age p obabili y
measu es and he pullback a ac o o a non-au onomous e sion o he NSE was
es ablished in [10].
Ou aim in his pape is o es ablish he in e se o he las esul in [4], i.e. ha
a s a iona y s a is ical solu ion o he GMNSE is an in a ian p obabili y measu e,
which we p o e unde sui able assump ions.
The s uc u e o he pape is as ollows. We ecall some p elimina ies on he
used unc ional spaces, de ini ions and exis ence esul s in Sec ion 2. In Sec ion
3we es ablish a new p oo o he exis ence o s ong solu ions o he GMNSE
wi h a di e en app oach om ha used in [3] since we need mo e egula i y on he
Gale kin app oxima ions. Es ima es in D(A) a e ob ained o hese solu ions o he
new Gale kin scheme in Sec ion 4. Then, in Sec ion 5we es ablish ou main esul ,
Theo em 15, p o ing unde addi ional assump ions ha a s a is ical solu ion is an
in a ian p obabili y measu e. Finally, o imp o e he cla i y o he pape , p oo s
o some echnical esul s used ea lie a e gi en in he Appendix.
2. P elimina ies. To se ou p oblem in he abs ac amewo k, we conside he
ollowing usual abs ac spaces (see Lions [9] and Temam [12,13]):
V=nu∈(C∞
0(Ω))3: di u= 0o,
H= he closu e o Vin (L2(Ω))3wi h inne p oduc (·,·) and associa ed no m |·| ,
whe e o u, ∈(L2(Ω))3,
(u, ) =
3
X
j=1 ZΩ
uj(x) j(x)dx,
STATISTICAL SOLUTIONS 3
V= he closu e o Vin (H1
0(Ω))3wi h inne p oduc ((·,·)) and associa ed no m
k·k ,whe e o u, ∈(H1
0(Ω))3,
((u, )) =
3
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 compac .
Finally, we will use k·k∗ o he no m in V0and h·,·i o he duali y pai ing be ween
Vand V0.
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. Mo eo e , by he p ope ies o b(see [11] o [12]),
he e exis s a cons an C1>0 only dependen on Ω such ha
|b(u, , w)| ≤ C1kukk k|w|1/4kwk3/4,∀u, , w ∈V, (2)
|b(u, , w)| ≤ C1|u|1/4kuk3/4k k|w|1/4kwk3/4,∀u, , w ∈V, (3)
|b(u, , w)| ≤ C1kukk kkwk,∀u, , w ∈V. (4)
Thus, i we deno e
hBN(u, ), wi=bN(u, , w),∀u, , w ∈V,
we ha e o example
kBN(u, )k∗≤NC1kuk,∀u, ∈V. (5)
We also conside he ope a o A:V→V0de ined by hAu, i= ((u, )).Deno ing
D(A) = (H2(Ω))3∩V, hen 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).
We ecall (see [12] and [11]) ha he e exis s a cons an C2>0 depending only
on Ω such ha
|b(u, , w)| ≤ C2|Au|k k|w|,∀u∈D(A), ∈V, w ∈H, (6)
|b(u, , w)| ≤ C2|u|1/4|Au|3/4k k|w|,∀u∈D(A), ∈V, w ∈H, (7)
|b(u, , w)| ≤ C2kuk1/2|Au|1/2k k|w|,∀u∈D(A), ∈V, w ∈H, (8)
|b(u, , w)| ≤ C2kukk k1/2|A |1/2|w|,∀u∈V, ∈D(A), w ∈H, (9)
|b(u, , w)| ≤ C2|u|k k1/2|A |1/2kwk,∀u∈H, ∈D(A), w ∈V. (10)
4 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
De ini ion 1. Le u0∈Hand ∈L2(0, T ; (L2(Ω))3) o all T > 0be gi en. A
weak solu ion o (1) is any u∈L2(0, T ;V) o all T > 0such ha
(u0( ) + νAu( ) + BN(u( ), u( )) = ( )in D0(0,+∞;V0),
u(0) = u0,
o equi alen ly
(u( ), w) + νZ
0
((u(s), w)) ds +Z
0
bN(u(s), u(s), w)ds
= (u0, w) + Z
0
( (s), w)ds, o all ≥0 and all w∈V.
Rema k 2. Obse e ha i u∈L2(0, T;V) o all T > 0and sa is ies he equa ion
u0( ) + νAu( ) + BN(u( ), u( )) = ( )in D0(0,+∞;V0),
hen, as a consequence o (5), u0( )∈L2(0, T;V0),and hence (see [13])u∈
C([0,+∞); H)and sa is ies he ene gy equali y
|u( )|2− |u(s)|2+ 2νZ
s
ku( )k2d = 2 Z
s
( ( ), u( )) d o all 0≤s≤ .
The ollowing esul was p o ed in [3].
Theo em 3. [c . [3, Th.7(a)]] Suppose ha ∈L2(0, T ; (L2(Ω))3) o all T > 0,
and le u0∈Vbe gi en. Then, he e exis s a unique weak solu ion uo (1), which
is in ac a s ong solu ion, i.e., i sa is ies
u∈C([0, T]; V)∩L2(0, T;D(A)) o all T > 0.
In he nex sec ion, o echnical easons, we ob ain a simila esul on exis ence
o solu ion (see P oposi ion 5and Rema k 6below), using a di e en Gale kin
app oxima ion om ha used in [3].
3. A smoo h Gale kin app oxima ion o GMNSE. By egula i y equi e-
men s o he manipula ions in he p oo o ou main esul (c . Theo em 15), we
need o in oduce a new Gale kin app oxima ion scheme o he GMNSE.
Ou i s s ep is o conside a amily o smoo he unc ions ha app oxima es
he unca ed e m FNappea ing in he GMNSE. This can be ca ied ou in he
classical way, by molli ica ion.
Le {ρm, m ≥1} ⊂ C∞(R) be a egula izing sequence in R, i.e., such ha
0≤ρm( )≤1, he suppo o ρmis included in he in e al [−1/m, 1/m],and
RRρm( )d = 1, o all m≥1.
Le us conside FNp olonged as FN( ) = 1 i ≤0,and deno e FN,m he
con olu ion ρm∗FN,i.e.,
FN,m( ) = ZR
ρm( −s)FN(s)ds ∀ ∈R.(11)
I is well known ha FN,m ∈C∞(R)∩L∞(R),and also FN,m( )→FN( ), as
m→+∞, uni o mly on bounded subse s o R.Mo eo e ,
0≤FN,m( )≤ kFNkL∞(R)= 1 ∀ ∈R,∀m≥1.
In ac , his las es ima e can be imp o ed. The ollowing esul conce ning he
unc ions FN,m holds.
STATISTICAL SOLUTIONS 5
Lemma 4. The unc ions FN,m de ined by (11) sa is y he ollowing inequali ies:
0≤ FN,m( )≤N+ 1 ∀ ≥0,∀m≥1.(12)
|F0
N,m( )| ≤ 1/N ∀ ∈R,∀m≥1.(13)
|F0
N,m( )| ≤ N+ 1
N, 2|F0
N,m( )| ≤ (N+ 1)2
N∀ ≥0,∀m≥1.(14)
F0
N,m( ) = 0 ∀ ≤1/m, ∀m > 2/N. (15)
I s p oo is gi en in he Appendix o a oid dis ac ing om he main low o
ideas he e.
Now, le us deno e BN,m( , ) he elemen o V0de ined by
hBN,m( , ), wi=FN,m(k k)b( , , w)∀u, w ∈V.
Conside he Gale kin app oxima ions o he GMNSE gi en by
u0
m+νAum+PmBN,m(um, um) = Pm , um(0) = Pmu0,(16)
whe e um=Pm
j=1 um,jφj,Aum=Pm
j=1 λjum,jφj. He e he λjand φja e he
co esponding eigen alues and eigen unc ions (o hono mal in H, o hogonal in V)
o he ope a o Aand Pmis he p ojec ion on o he subspace o Hspanned by
{φ1, . . . , φm}.
Now, we es ablish he main esul o his sec ion, which p o ides a new p oo o
he exis ence o solu ion gi en in Theo em 3.
P oposi ion 5. Unde he abo e no a ion, i ∈L∞(0, T;H) o all T > 0, he
scheme (16) is well de ined, i.e. o each u0∈V, he e exis s a smoo h amily (C1in
ime) o unc ions {um},solu ions o (16), ha a e de ined in (0,+∞).Mo eo e ,
he e exis s a subsequence o {um}m ha con e ges (in se e al senses) o he unique
solu ion o (1).
Rema k 6. The assump ion ∈L∞(0, T ;H)ins ead o L2(0, T; (L2(Ω))3)as in
Theo em 3is no essen ial, bu only o simplici y in he no a ion. Indeed, since
p ojec o ope a o s a e in ol ed in he es - unc ions space, he e is no es ic ion
in conside ing aking alues in Hins ead o (L2(Ω))3.On he o he hand, he
s onge equi emen in ime, L∞ins ead o L2,is also o he sake o con enience
in he calculus. Obse e ha ou inal goal in Sec ion 5is wi h an au onomous e m
.
P oo . [o P oposi ion 5] The exis ence and uniqueness o local solu ion o (16) is a
consequence o he Pica d Theo em, and he ac ha he local solu ion is a global
one de ined in he same in e al o ime as is in iew o he es ima es (19) and
(24) below.
A e ha , we will p oceed by he compac ness me hod: i s ly p o ing uni o m
es ima es o umin Hand hen in V. Obse e ha o he es ima es in Hwe only
need u0∈H.
Fix a alue T > 0 and le us deno e
| |∞=k kL∞(0,T ;H).

6 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
I is s anda d ha i we ake he inne p oduc o he Gale kin ODE (16) wi h
umand use b(um, um, um) = 0, we ha e
d
d |um|2+ 2νkumk2= 2( , um),(17)
and hen, using λ1|um|2≤ kumk2,we ob ain
d
d |um|2+νkumk2≤| |2
∞
νλ1
,(18)
and consequen ly
|um( )|2+νZ
0
kum(s)k2ds ≤ |u0|2+| |2
∞
νλ1
(19)
o all ≥0 in he in e al o de ini ion o um.The well posedness o umin (0, T)
ollows om G onwall’s lemma.
Now we ob ain uni o m es ima es in V o all umin (0, T ) using u0∈V. We ake
he inne p oduc o he Gale kin ODE (16) wi h Aumand ob ain
1
2
d
d kumk2+ν|Aum|2+bN,m(um, um, Aum) = ( , Aum).(20)
E iden ly,
|( , Aum)| ≤ ν
4|Aum|2+| |2
∞
ν
In addi ion, by (8), (12) and Young’s inequali y, one ob ains
|bN,m(um, um, Aum)| ≤ C2(N+ 1)kumk1/2|Aum|3/2
≤ν
4|Aum|2+C0
Nkumk2,
wi h C0
N=27(N+ 1)4C4
2
4ν3.
Thus (20) simpli ies o
d
d kumk2+ν|Aum|2≤2
ν| |2
∞+ 2C0
Nkumk2.(21)
F om (21) and he ac ha
kum(0)k=kPmu0k ≤ ku0k
by he choice o he basis {φj}o H, one easily concludes ha he sequence {um}
is bounded in C([0, T]; V) and in L2(0, T;D(A)) o all T > 0.
Then, obse e ha |bN,m(um, um, w)| ≤ (N+ 1)C2kumk1/2|Aum|1/2|w|, o any
w∈H, and hus, he sequence {PmBN,m(um, um)}is bounded in L2(0, T;H) o
all T > 0.
The e o e, om he equa ion u0
m=−νAum−PmBN,m(um, um)+Pm , one sees
ha he sequence {u0
m}is also bounded in L2(0, T;H).
Consequen ly, as D(A)⊂V⊂Hwi h compac injec ion, by Theo em 5.1 in
Chap e 1 o [9] he e exis s an elemen u∈L∞(0, T;V)∩L2(0, T;D(A)) o all
STATISTICAL SOLUTIONS 7
T > 0, and a subsequence {uµ}o {um}, such ha
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uµ→us ong in L2(0, T;V),
uµ→ua.e. in (0, T)×Ω,
uµ* u weak in L2(0, T;D(A)),
uµ
∗
* u weak-s a in L∞(0, T;V),
u0
µ* u0weak in L2(0, T;H),
(22)
o all T > 0.
Also, as uµcon e ges o in L2(0, T ;V) o all T > 0, we can assume, possibly
ex ac ing a subsequence, ha
kuµ( )k → ku( )ka.e. in (0,+∞).
Then, by he boundedness o {uµ}in C([0, T]; V) and he ac ha FNis con in-
uous and FN,m con e ges o FNuni o mly on bounded subse s o Ras m→+∞,
one ob ains
FN,m(kuµ( )k)→FN(ku( )k) a.e. in (0,+∞).(23)
F om (22) and (23) we can ake limi s in (16) exac ly as in [3], and we ob ain ha
uis a solu ion o (1).
Rema k 7. I we assume ha is also de ined o nega i e ime, o ins ance,
∈L∞(−T, T;H), he abo e solu ions uma e also well de ined o all (−T, T).
This will be used below -in he p oo o Theo em 15- o ensu e ha we can go
backwa ds in ime on each solu ion umo he Gale kin scheme.
This is due o (17), whence we ha e
d
d |um|2+ 2νλm|um|2≥ −| |2
∞− |um|2,
whe e | |∞deno es now k kL∞(−T,T ;H).The e o e, in eg a ing be ween ∈(−T, 0)
and 0we deduce
|um( )|2≤ |u0|2+| || |2
∞+(1+2νλm)Z0
|um(s)|2ds (24)
o all < 0in he in e al o de ini ion o um. Thus, by (19), (24) and again
G onwall’s lemma, umis de ined in all (−T, T ).
Rema k 8. Obse e ha , since he sequence {u0
m}is bounded in L2(0, T ;H) o
any T > 0, he sequence umas a sequence o unc ions om [0, T ]in o His
equicon inuous. Then, since {um}is bounded in C([0, T ]; V)and V⊂Hwi h
compac injec ion, we can asse ha om any subsequence o {um}we can ex ac
a subsequence ha con e ges in C([0, T]; H), and aking in o accoun he p e ious
a gumen s o he exis ence o solu ion uo (1), and he uniqueness o such a so-
lu ion (c . Theo em 3), one has ha um→uin C([0, T ]; H),as m→+∞, o all
T > 0.
4. Es ima es in D(A) o he Gale kin app oxima ions. In his sec ion we
go u he in de eloping new es ima es o he solu ions o (16), bu now wi h
ini ial da a u0∈D(A),and assuming ha ∈W1,∞(R;H).[The mo e es ic-
i e hypo hesis o ∈W1,∞(R;H),which we will use om he e on ins ead o
W1,∞((−T, T); H) o all T, is only o he sake o b e i y in he s a emen s.]
8 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
P oposi ion 9. Wi h he no a ion o Sec ion 3, i ∈W1,∞(R;H), he e exis s a
posi i e cons an ˜
M(N)
, independen o u0, and m, such ha
|Aum( )| ≤ ˜
M(N)
(1 + |Au0|)∀ ≥0, o all m > 2/N, u0∈D(A).(25)
P oo . Now le us deno e |·|∞=k·kL∞(R;H).F om (18) we ha e
d
d eνλ1 |um( )|2≤| |2
∞
νλ1
eνλ1 , ≥0,
and in eg a ing one ob ains
|um( )|2≤ |u0|2e−νλ1 +| |2
∞
ν2λ2
1
o all ≥0.(26)
Now, i we use (20) and ake in o accoun ha λ1kum( )k2≤ |Aum( )|2and ha ,
by (7) and (12), |bN,m(um( ), um( ), Aum( ))| ≤ (N+ 1)C2|um( )|1/4|Aum( )|7/4,
hen we can ob ain he inequali y
d
d kum( )k2+νλ1kum( )k2≤2
ν| |2
∞+C(N)
1|um( )|2,(27)
wi h C(N)
1=(N+ 1)8C8
277
29ν7.
Subs i u ing he bound (26) o |um( )|2in he di e en ial inequali y (27) gi es
d
d kum( )k2+νλ1kum( )k2≤C(N)
1|u0|2e−νλ1 +| |2
∞
ν 2 + C(N)
1
νλ2
1!.
In eg a ing his inequali y we deduce ha
kum( )k2≤(ku0k2+C(N)
1 |u0|2)e−νλ1 +| |2
∞
ν2λ1 2 + C(N)
1
νλ2
1!,∀ ≥0.(28)
On he o he hand, in eg a ing (21) be ween and + 1, we ob ain in pa icula
νZ +1
|Aum(s)|2ds ≤2
ν| |2
∞+ 2C0
NZ +1
kum(s)k2ds +kum( )k2∀ ≥0,
and hen, by (28), one ob ains
Z +1
|Aum(s)|2ds ≤1+2C0
N
ν(ku0k2+C(N)
1( + 1)|u0|2)e−νλ1 (29)
+| |2
∞
ν2"2 + 1+2C0
N
νλ1 2 + C(N)
1
νλ2
1!# ∀ ≥0∀m≥1.
Suppose now ha m > 2/N. F om (16) and (15), we ob ain o he second
de i a i e o um,
u00
m( ) = −νAu0
m( )−F0
N,m(kum( )k)((u0
m( ), um( )))
kum( )kχOm( )PmB(um( ), um( ))
−FN,m(kum( )k)PmB(u0
m( ), um( ))
−FN,m(kum( )k)PmB(um( ), u0
m( )) + Pm 0( ),(30)
whe e
Om={ ∈[0,+∞) : kum( )k>1/m}
and χOm( ) is he indica o unc ion o he se Om.
STATISTICAL SOLUTIONS 9
Mul iplying in (30) by u0
m,we ob ain
1
2
d
d |u0
m( )|2+νku0
m( )k2
=−F0
N,m(kum( )k)((u0
m( ), um( )))
kum( )kχOm( )b(um( ), um( ), u0
m( ))
−FN,m(kum( )k)b(u0
m( ), um( ), u0
m( )) + ( 0( ), u0
m( )) ≥0.(31)
F om (2), (14) and Young’s inequali y, we ha e
|F0
N,m(kum( )k)((u0
m( ), um( )))
kum( )kχOm( )b(um( ), um( ), u0
m( ))|
≤2C1(N+ 1)2
Nku0
m( )k|u0
m( )|1/4ku0
m( )k3/4
=2C1(N+ 1)2
NC1|u0
m( )|1/4ku0
m( )k7/4
≤ν
2ku0
m( )k2+7
4ν7
25C2
1(N+ 1)2
N8
|u0
m( )|2.(32)
By (3), (12) and Young’s inequali y again
|2FN,m(kum( )k)b(u0
m( ), um( ), u0
m( ))|
≤2(N+ 1)C1|u0
m( )|1/2ku0
m( )k3/2
≤ν
2ku0
m( )k2+27
2ν3(N+ 1)4C4
1|u0
m( )|2.(33)
Thus, i we deno e
L(N)
1= 1 + 1
27
2ν7C2
1(N+ 1)2
N8
+27
ν3(N+ 1)4C4
1,
om (31), (32) and (33) we easily ob ain
d
d |u0
m( )|2≤L(N)
1|u0
m( )|2+| 0|2
∞∀ ≥0∀m > 2/N. (34)
I we in eg a e his inequali y be ween s∈[ , + 1] and + 1,we ha e
|u0
m( + 1)|2≤ |u0
m(s)|2
+L(N)
1Z +1
s
|u0
m( )|2d +| 0|2
∞∀0≤ ≤s≤ + 1,∀m > 2/N.
In eg a ing now his las inequali y o sbe ween and + 1,we ob ain
|u0
m( + 1)|2≤(1 + L(N)
1)Z +1
|u0
m(s)|2ds +| 0|2
∞∀ ≥0,∀m > 2/N. (35)
Now, obse e ha by (16), (6) and (12),
|u0
m( )| ≤ ν|Aum( )|+|FN,m(kum( )k)B(um( ), um( ))|+| ( )|
≤[ν+ (N+ 1)C2]|Aum( )|+| |∞, ≥0,
and he e o e
Z +1
|u0
m(s)|2ds ≤2| |2
∞+ 2[ν+ (N+ 1)C2]2
×Z +1
|Aum(s)|2ds ∀ ≥0,∀m≥2/N. (36)
16 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
Then, since Pm∈ L(H) is sel -adjoin , om (57) and (58) we ob ain
ZH
Φ(S(m)
N(T) )dµ( )
=ZH
Φ(Pm )dµ( ) (59)
+ZT
0ZH
(PmGN,m(Pm )−PmGN( ), D (Φm)( , )) dµ( )d ∀T > 0.
Now, obse e ha PmGN,m(Pm )−PmGN( ) = PmBN( , )−PmBN,m(Pm , Pm ),
and hen, by (58) and since P∗
m=Pm,we ob ain om (59)
ZH
Φ(S(m)
N(T) )dµ( ) (60)
=ZH
Φ(Pm )dµ( )
+ZT
0ZH
(PmBN( , )−PmBN,m(Pm , Pm ), D (Φm)( , )) dµ( )d
=ZH
Φ(Pm )dµ( )
+ZT
0ZH
(BN( , )−BN,m(Pm , Pm ), D (Φm)( , )) dµ( )d ∀T > 0.
Now, we p o e ha as m→+∞, he las in eg al e m goes o ze o o all T > 0.
Obse e ha
|(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|
=|FN(k k)b( , , D (Φm)( , )) −FN,m(kPm k)b(Pm , Pm , D (Φm)( , ))|
≤ |FN(k k)b( −Pm, , D (Φm)( , ))|
+|FN(k k)−FN,m(k k)||b(Pm , , D (Φm)( , ))|
+|FN,m(k k)b(Pm , −Pm, D (Φm)( , ))|
+|FN,m(k k)−FN,m(kPm k)||b(Pm , Pm , D (Φm)( , ))|,
and he e o e, by (4), (12) and (13),
|(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|
≤NC1k −Pm kkD (Φm)( , ))k
+|FN(k k)−FN,m(k k)|C1k k2kD (Φm)( , ))k
+(N+ 1)C1k −Pm kkD (Φm)( , ))k
+C1
Nk −Pm kk k2kD (Φm)( , ))k.(61)
On he o he hand, easoning exac ly as on page 232 in [6], one ob ains
kD (Φm)( , ))k
=kΦ0(S(m)
N( ) ksup
w∈H,
|A−1/2w| ≤ 1
|A−1/2(D S(m)
N( ) )w| ∀ ≥0,∀ , w ∈H,

STATISTICAL SOLUTIONS 17
o any m≥1,and hen, aking in o accoun ha Φ ∈ T has de i a i e Φ0(·)
bounded on V, he e exis s a cons an CΦ>0 such ha
kD (Φm)( , ))k ≤ CΦsup
w∈H,
|A−1/2w| ≤ 1
|A−1/2(D S(m)
N( ) )w|(62)
∀ ≥0,∀ , w ∈H, o any m≥1.
Since BNis a bounded subse o D(A), om (50), P oposi ion 9and (62) we
deduce ha he e exis s a cons an C ,BNsuch ha
kD (Φm)( , ))k ≤ CΦeT C ,BN∀0≤ ≤T, ∀ ∈ BN,(63)
o any m > 2/N.
F om (61) and (63) we deduce ha o each T > 0 he e exis s a cons an CT>0,
such ha
|(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|
≤CT(k −Pm k+|FN(k k)−FN,m(k k)|)
o all ( , )∈[0, T ]× BN, o any m > 2/N.
Then, as µ(H BN) = 0,we ob ain ha o each T > 0,
ZT
0ZH
|(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|dµ( )d (64)
≤TCTZBN
(k −Pm k+|FN(k k)−FN,m(k k)|)dµ( )∀m > 2/N.
As k −Pm k → 0 as m→+∞and k −Pm k ≤ 2k k o any ∈ BN,and FN,m
con e ges o FNas m→+∞uni o mly on bounded subse s o R,i is immedia e
o m (64) ha
ZT
0ZH
|(BN( , )−BN,m(Pm , Pm ), D (Φm)( , ))|dµ( )d →0 as m→+∞,
and hen, om (60) we conclude ha
ZH
Φ(S(m)
N(T) )dµ( )−ZH
Φ(Pm )dµ( )→0 as m→+∞, (65)
o all T > 0.
Bu , aking again in o accoun ha µ(H BN) = 0, ha BNis a bounded subse
o D(A),and ha Φ is bounded on bounded subse s o H, we deduce om Rema k
8and (52) ha
ZH
Φ(S(m)
N(T) )dµ( )→ZV
Φ(SN(T) )dµ( ) as m→+∞, o all T > 0,
and
ZH
Φ(Pm )dµ( )→ZV
Φ( )dµ( ) as m→+∞,
and hen, by (65) we ha e (45), as desi ed.
18 PETER E. KLOEDEN, PEDRO MAR´
IN-RUBIO, AND JOS´
E REAL
Appendix: Es ima es on he unca ed unc ion. This sec ion is de o ed
o p o e some echnical es ima es conce ning he unc ions FN,m,de ined by (11),
namely hose o Lemma 4.
To s a , obse e ha
FN,m( ) = Z +1/m
−1/m
ρm( −τ)FN(τ)dτ.
Consequen ly, i N < −1/m, hen
FN,m( ) = Z +1/m
−1/m
ρm( −τ)N
τdτ
≤N
−1/m.
Thus,
•i N≥ −1/m, hen FN,m( )≤ ≤N+ 1/m, and
•i N < −1/m, hen
FN,m( )≤ N
−1/m
=N1 + 1
m( −1/m)
≤N1 + 1
mN =N+ 1/m.
Hence, (12) is p o ed.
Mo eo e , i is s aigh o wa d o check ha F0
N,m( )=(ρm∗F0
N)( ),whe e
F0
N( ) = −N
2χ(N,+∞)( ),and he e o e (13) also holds.
Fo (14), ela ed o |F0
N,m( )|and 2|F0
N,m( )|,we ha e o p oceed mo e ca e-
ully.
Obse e ha
|F0
N,m( )| ≤ Z +1/m
−1/m
ρm( −τ)|F0
N(τ)|dτ
≤Z +1/m
−1/m
ρm( −τ)N
τ2dτ,
and consequen ly, i > N + 1, hen
|F0
N,m( )| ≤ N
( −1/m)2
≤N
( −1)2.
Thus,
STATISTICAL SOLUTIONS 19
•i > N + 1,
|F0
N,m( )| ≤ N
( −1)2
=N
−11 + 1
−1
<1 + 1
N
=N+ 1
N,
and
2|F0
N,m( )| ≤ N 2
( −1)2
=N1 + 1
−12
< N 1 + 1
N2
=(N+ 1)2
N.
•i 0 ≤ ≤N+ 1, hen, by (13), one also has
|F0
N,m( )| ≤ N+ 1
N, 2|F0
N,m( )| ≤ (N+ 1)2
N.
Hence, combining all si ua ions one eco e s he inequali y in (14).
Finally, in o de o p o e (15) obse e ha i m > 2/N and ≤1/m, hen
FN,m( ) = Z +1/m
−1/m
ρm( −τ)FN(τ)dτ = 1,
whence he las esul ollows.
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20 PETER E. KLOEDEN, PEDRO MAR´
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E REAL
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E-mail add ess, Pe e E. Kloeden: [email p o ec ed]
E-mail add ess, Ped o Ma ´ın-Rubio: [email p o ec ed]
E-mail add ess, Jos´e Real: [email p o ec ed]
(Pe e E. Kloeden) Ins i u ¨
u Ma hema ik, Johann Wol gang Goe he-Uni e si ¨
a ,
D-60054 F ank u am Main, Ge many
(Ped o Ma ´ın-Rubio and Jos´e Real) Dp o. 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)