Triviality of the 2D stochastic Allen-Cahn equation

0.17 MB·English

by  Martin Hairer

Triviality of the 2D stochastic Allen-Cahn equation
January17,2012
2
1 M. Hairer1,M.D. Ryser2, and H.Weber3
0
2
1 UniversityofWarwick,Email:[email protected]

n 2 DukeUniversity,Email:[email protected]
a 3 UniversityofWarwick,Email:[email protected]
J

5
1 Abstract

We consider the stochastic Allen-Cahn equation driven by mollified space-time
]
R white noise. We show that, as the mollifier is removed, the solutions converge
P weaklyto0,independentlyoftheinitialcondition.Iftheintensityofthenoisesi-
. multaneouslyconvergesto0atasufficientlyfastrate,thenthesolutionsconverge
h
t tothoseofthedeterministicequation. Atthecriticalrate,thelimitingsolutionis
a
stilldeterministic,butitexhibitsanadditionaldampingterm.
m
[
1 Introduction

1
v Weconsider thefollowingevolution equation onthetwo-dimensional torusT2:
9
8 du = ∆u+u u3 dt+σdW , u(0) = u0 . (Φ)
0 −
3 (cid:0) (cid:1)
Here u0 is a suitably regular initial condition, σ a positive constant, and W an
.
1 L2(T2)-valuedcylindricalWienerprocessdefinedonaprobabilityspace(Ω, ,P).
0
F
2 Inotherwords,atleastataformallevel, dW isspace-time whitenoise.
dt
1 Thisequationandvariantsthereofhavealonghistory. Thedeterministicpartof
:
v theequationistheL2gradientflowoftheGinzburg-Landau freeenergyfunctional
i
X
1
r u(x)2 +V(u(x)) dx ,
a ZT2(cid:16)2|∇ | (cid:17)
with the potential energy V given by the standard double-well function V(u) =
1(u2 1)2, see [LG50]. This provides a phenomenological model for the evolu-
4 −
tionofanorderparameterdescribingphasecoexistenceinasystemwithoutpreser-
vation of mass. At large scales, the dynamic of phase boundaries is known to
converge tothemeancurvature flow[AC79,ESS92,Ilm93].
Thenoiseterm σdW accounts forthermalfluctuations atpositive temperature.
On a formal level the choice of space-time white noise is natural, because it satis-
fies the right fluctuation-dissipation relation. Atleast for finite-dimensional gradi-
ent flows it is natural to take the bilinear form that determines the mechanism of
INTRODUCTION 2
energy dissipation as covariance of the noise, as this guarantees the invariance of
the right Gibbs measure under the dynamics. Naively extending this observation
tothecurrentinfinitedimensional contextyields(Φ).
White noise driven equations such as (Φ) are known to be ill-posed in space-
dimension d 2[Wal86,DPZ92]. Actually, thelinearised version of(Φ)(simply
≥
remove the term u3) admits only distribution-valued solutions for d 2. For any
κ > 0 these solutions take values in the Sobolev space H2−2d−κ, bu≥t they do not
2−d
take values in H 2 . In general, it is impossible to apply nonlinear functions to
elements of these spaces and the standard approach to construct solutions of (Φ)
[DPZ92,Hai09]fails.
Inthepresentarticle,weintroduceacutoffatspatiallengthsoforderεandwe
studythelimitasε 0forfinitenoisestrength for(Φ). Moreprecisely, weset
→
W (t)= e β (t), ε >0 ,
ε k k
X
|k|≤1/ǫ
where e is the Fourier basis on T2, and β are complex Brownian
motion{stkh}akt∈aZre2 i.i.d.exceptfortherealitycondit{ionk}β¯k∈Z=2 β . Wethusconsider
k −k
du = ∆u +u u3 dt+σ(ε)dW , u (0) = u0 , (Φ )
ε ε ε− ε ε ε ε
(cid:0) (cid:1)
andstudytheweaklimitofu asε 0.
ε
→
The main result of this article can loosely be formulated as follows (a precise
statementwillbegiveninTheorems2.1and2.2below):
Theorem 1.1. Let σ be bounded and such that lim σ2(ε)log1/ε = λ2
ε→0
∈
[0,+ ]. Ifλ2 = + ,thenu convergesweaklyto0,inprobability. Otherwise,it
ε
∞ ∞
converges weaklyinprobability tothesolution w of
λ
∂ w = ∆w 3 λ2 1 w w3 , w (0) = u0 . (Ψ )
t λ λ− 8π − λ− λ λ λ
(cid:0) (cid:1)
Remark 1.2. The result for constant σ was conjectured in [RNT11], based on
numericalsimulations.
Remark1.3. Theborderlinecaseλ = 0isparticularlyinterestingasitprovidesan
6
exampleofstochasticdamping: inthelimitasε 0,thestochasticforcingiscon-
→
vertedintoanadditionaldeterministicdampingterm, 3 λ2w ,totheAllen-Cahn
−8π λ
equation. Inparticular, ifλ2 > 8π,thezero-solution becomesglobally attracting.
3
Remark 1.4. Recently, there has been alot ofinterest in(Φ)inthe regime where
the noise is small [KORVE07, BBM10, CF11]. There, the authors studied (Φ) in
arbitrary space dimension on the level of large deviation theory. As in (Φ ) they
ε
consideramodifiedversionof(Φ)wherethenoisetermdW isreplacedbyanoise
term dW with a finite spatial correlation length ε. For this modified equation,
ε
solutions can be constructed in a standard way and a large deviation principle a`
la Freidlin-Wentzell can be obtained. One can then show that the rate functionals
INTRODUCTION 3
converge as ε 0. The large deviation principle however is not uniform in ε;
→
thisprocedure corresponds totakingtheamplitudeofthenoisemuchsmallerthan
ε. The results obtained in this article quantify how small the noise should be as
a function of ε in order for the solutions of (Φ) to be close to the deterministic
equation.
Remark 1.5. Webelieve that the weak convergence to 0 as ε 0actually holds
→
for a much larger class of potentials. Actually, one would expect it to be true
whenever lim V′′(u) = + . The proof given in this article does however
|u|→∞
∞
dependcrucially onthefactthatV(u) u4 forlargevaluesofu.
∼
Themaintoolsusedinourproofsareprovidedbythetheoryofstochasticquan-
tisation. Actually, inthecontext ofEuclidean Quantum FieldTheorythe question
ofexistenceoftheformalinvariantmeasureof(Φ)hasbeentreatedintheseventies
(see e.g. [GJ87]). Then, it had been observed that this measure, the so called Φ4
2
field, can be defined, but only if a logarithmically diverging lower order term is
subtracted. Thecorresponding stochastic dynamical system (i.e. the renormalised
version of (Φ)) has also been constructed [PW81, AR91, DPD03]. Note that al-
thoughthisrenormalised equation,
du = ∆u+u :u3: dt+σdW ,
−
(cid:0) (cid:1)
formally resembles (Φ) it does not have a natural interpretation as a phase field
model.
Ourmainargument isamodification oftheconstruction provided in[DPD03].
We present here a brief heuristic argument for the case σ 1. First, let C > 1
ε
≡
andaddandsubtractthetermC u to(Φ )toget
ε ε ε
du = ∆u (C 1)u u u2 C dt+dW . (1.1)
ε ε− ε− ε− ε ε − ε ε
(cid:0) (cid:0) (cid:1)(cid:1)
The key idea is to choose C in such a way that, for small values of ε, the term
ε
u u2 C isequal totheWick product :u3:withrespect tothe Gaussian struc-
ε ε − ε ε
ture(cid:0)given by(cid:1)the invariant measure of the linearised system (which itself depends
on C ). Since, given the results in [DPD03], one would expect :u3: to at least re-
ε ε
mainboundedasε 0,itisnotsurprisingthattheadditionalstrongdampingterm
→
C u causesthesolutiontovanishinthelimit.
ε ε
−
Acknowledgements
MH acknowledgesfinancial support by the EPSRC trough grant EP/D071593/1 and the
Royal Society through a Wolfson Research Merit Award. Both MH and HW were sup-
portedbytheLeverhulmeTrustthroughaPhilipLeverhulmePrize.MDRisgratefultoP.F.
Tupper and N. Nigam for fruitful discussions, and acknowledges financial support from
a Hydro-Que´bec Doctoral Fellowship. All three authors are grateful for the relaxed at-
mosphereat the Newton institute, where thiscollaborationwas initiated duringthe 2010
“StochasticPDEs”programme.
NOTATIONS AND MAIN RESULT 4
2 Notations andMainResult
In order to formulate our results, we first introduce the class of Besov spaces that
we will work with. As in [DPD03] we choose to work in Besov spaces, because
they satisfy the right multiplicative inequalities (see LemmaA.2). Denote by (, )
· ·
the L2 inner product, and by e (x) = 1 eikx the corresponding orthonor-
k 2π k∈Z2
mal Fourier basis. Throughou(cid:8)t the article, we(cid:9)work with periodic Besov spaces
s (T2), where p,r 1 and s R. These spaces are defined as the closure of
Bp,r ≥ ∈
C∞(T2)underthenorm
∞
1/r
u := 2qrs ∆ u r , (2.1)
k kBps,r(T2) (cid:16)Xq=0 k q kLp(T2)(cid:17)
wherethe∆ aretheLittlewood-Paleyprojectionoperatorsgivenby∆ u= (e ,u)e
q 0 0 0
and
∆ u = (e ,u)e , q 1.
q k k
≥
2q−1X≤|k|<2q
Regarding the exponents appearing in these Besov spaces, we will restrict our-
selvesthroughout thisarticletoexponents p,randssuchthat
2
p 4 , r 1, < s < 0. (2.2)
≥ ≥ −7p
WenowreformulateTheorem1.1moreprecisely. Thecaseλ2 = + isgiven
∞
bythefollowing:
Theorem2.1. Assumeu0 s suchthat(2.2)holds. Thenforallε > 0andT >
∈ Bp,r
0,thereexistsauniquemildsolutionu . Ifσ(ε)isboundeduniformlyinεandsatis-
ε
fieslim σ2(ε)log(1/ε) = + then,forallδ (0,T),lim u =
ε→0 ∞ ∈ ε→0k εkC([δ,T];Bps,r)
0inprobability.
Ontheotherhand, whenσ2(ε)log(1/ε)converges toafinitelimit,wehave
Theorem2.2. Assumeu0 s suchthat(2.2)holds. Iflim σ2(ε)log(1/ε) =
∈ Bp,r ε→0
λ2 R then, for all δ (0,T), lim u w = 0 in probability,
∈ ∈ ε→0k ε− λkC([δ,T];Bps,r)
wherew istheuniquesolution to(Ψ ).
λ λ
Remark 2.3. If σ decays sufficiently fast, for example σ(ε) ετ for some τ >
∼
0, then the conclusion of Theorem 2.2 actually holds in the space of space-time
continuous functions.
To conclude this section, we introduce some concepts borrowed from the the-
ory of stochastic quantization. Since we are not concerned with the dynamics of
quantised fields, weonly introduce the notions necessary for the proof techniques
usedbelow, andrefer to[DPT07]forageneral introduction tothetopic. Consider
thelinearversionofequation (1.1),namely
dz = (∆z (C 1)z )dt+σ(ε)dW . (2.3)
ε ε ε ε ε
− −
TRIVIAL LIMIT FOR STRONG NOISE 5
For C > 1, this equation has a unique invariant measure on L2(T2), which we
ε
denotebyµ . Itisµ thatwillplaytheroleofthe“freefield”inthepresentarticle.
ε ε
Under µ , the kth Fourier component of z is a centred complex Gaussian ran-
ε ε
domvariablewithvariance σ2(ε) C 1+ k 2 −1. Furthermore,distinctFourier
2 ε− | |
components areindependent, exc(cid:0)eptforthereali(cid:1)tycondition z ( k) = z (k).
ε ε
−
Asaconsequence oftranslation invariance, onehastheidentity
2
1
D2 := φ (x)2µ (dφ )= φ 2 µ (dφ ) (2.4)
ε ZL2| ε | ε ε (cid:18)2π(cid:19) ZL2k εkL2 ε ε
1 σ2(ε)
= .
8π2 C 1+ k 2
X ε
|k|≤1/ε − | |
Wethendefine theWickpowersofanyfieldu withrespect totheGaussian struc-
ε
turegivenbyµ by
ε
:un:= DnH (u /D ) ,
ε ε n ε ε
where H denotes the nth Hermite polynomial. In this article, we will only ever
n
usetheWickpowersforn 3,forwhichonehastheidentities
≤
:u1: = u , :u2: = u2 D2 , :u3:= u3 3D2u . (2.5)
ε ε ε ε − ε ε ε − ε ε
From now on, whenever we use the notation :un:, (2.5) is what we refer to. For
ε
any two expressions A and B depending on ε, we will throughout this article use
the notation A . B to mean that there exists a constant C independent of ε (and
possibly ofother relevant parameters clear from the respective contexts) such that
A CB.
≤
3 Triviallimitforstrong noise
In this section, we provide the proof of Theorem 2.1. First, in Subsection 3.1,
we explain the “correct” choice of the renormalization constant C in (1.1). In
ε
Subsection 3.2, we then obtain bounds on the linearised equation, as well as its
Wick powers. Finally, in Subsection 3.3, weobtain abound on the remainder and
wecombinetheseresultsinordertoconclude.
3.1 Fixingtherenormalization constant
ForC >1,werewrite(Φ )as
ε ε
du = A u u (u2 C ) dt+σ(ε)dW , (3.1)
ε ε ε− ε ε − ε ε
(cid:0) (cid:1)
where the linear operator A is given by A = ∆ (C 1). Motivated by the
ε ε ε
− −
heuristic arguments provided in Section 1, the goal of this section is to determine
C insuchawaythatthenonlinear termu (u2 C )isequaltotheWickproduct
ε ε ε − ε
TRIVIAL LIMIT FOR STRONG NOISE 6
:u3:. It then follows from (2.4) and (2.5) that C is implicitly determined by the
ε ε
equation
3 σ2(ε)
C = 3D2 = . (3.2)
ε ε 8π2 C 1+ k 2
X ε
|k|≤1/ε − | |
Todescribethebehaviorofthesolutionto(3.2),weshallusethenotationA B
ε ε
∼
tomeanlim A /B = 1.
ε→0 ε ε
Lemma3.1. Foranyvaluesoftheparameters,equation(3.2)hasauniquesolution
C > 1. Ifσisuniformlyboundedandsuchthatlim σ2(ε)log(1/ε) = ,then
ε ε→0
∞
onehas
3 1
C σ2(ε)log . (3.3)
ε
∼ 4π ε
Inparticular, lim C = + .
ε→0 ε
∞
Beforeweproceedtotheproofofthisresult,westatethefollowingveryuseful
result:
Lemma3.2. Leta,R 1. Thenthereexistsaconstant C suchthatthebound
≥
1 R2 C R
πlog 1+ 1 , (3.4)
(cid:12)(cid:12)|kX|≤Ra+|k|2 − (cid:16) a (cid:17)(cid:12)(cid:12) ≤ √a(cid:16) ∧ √a(cid:17)
(cid:12) (cid:12)
holds. Here,thesumgoesoverelementsk Z2.
∈
Proof. The second expression on the left is nothing but dk , so wewant
|k|≤R a+|k|2
tobound thedifference between thesum andtheintegralR. Usingthemonotonicity
andpositivity ofthefunction x 1 andrestricting ourselves toonequadrant,
7→ a+x2
weseethatonehasthebounds
1 1 dk 1
.
a+ k 2 ≤ 4 Z a+ k 2 ≤ a+ k 2
|Xk|≤R | | |k|≤R | | |Xk|≤R | |
ki>0 ki≥0
Asaconsequence, therequired errorisboundedby
⌊R⌋ 1 4 R dx
4 +4 .
a+k2 ≤ a Z a+x2
Xk=0 0
Therequired bound followsatonce, using thefactthataandR arebounded away
from0byassumption.
ProofofLemma3.1. Sincetherighthandsidedecreases from downto0asthe
∞
left hand side grows from 1 to , it follows immediately that (3.2) always has a
∞
uniquesolution C > 1.
ε
TRIVIAL LIMIT FOR STRONG NOISE 7
Since,byLemma3.2,onehas 1 2πlog 1 andsincebyassump-
|k|≤1ε 1+|k|2 ∼ ε
tionσ2(ε)log 1 ,thereexistsPε suchthat
ε → ∞ 0
3 σ2(ε)
2 , (3.5)
8π2 1+ k 2 ≥
X
|k|≤1/ε | |
for all ε < ε . As a consequence, we have C 2 for such values of ε, and we
0 ε
≥
willusethisboundfromnowon. Ontheotherhand,ifweknowthatC 2,then
ε
≥
C isbounded fromabovebythelefthandsideof(3.5),sothat
ε
1
C Kσ2(ε)log , (3.6)
ε
≤ ε
forsomeconstant K andforεsmallenough.
ItnowfollowsfromLemma3.2that
3σ2(ε) 1
C = log 1+ +R , (3.7)
ε 8π (cid:18) ε2(C 1)(cid:19) ε
ε
−
forsomeremainderR whichisuniformlybounded asε 0. Since,by(3.6),the
ε
→
firsttermontherighthandsidegoesto ,thisshowsthatR isnegligiblein(3.7),
ε
∞
sothat
3σ2(ε) 1 3σ2(ε) 1
C log = log logC .
ε ∼ 8π (cid:18)ε2C (cid:19) 8π ε2 − ε
ε (cid:16) (cid:17)
SinceC isnegligible withrespectto 1 by(3.6),theclaimfollows.
ε ε2
3.2 Boundsonthelinearisedequation
Wesplit the solution to (3.1) into two parts by introducing the stochastic convolu-
tion
t
z (t):= σ(ε) e(t−s)AεdW (s) , (3.8)
ε ε
Z
−∞
andperformingthechangeofvariablesv (t) := u (t) z (t). Withthesenotations,
ε ε ε
−
v solves
ε
∂ v = A v v3+3v2z +3v :z2:+:z3: (Φaux)
t ε ε ε− ε ε ε ε ε ε ε
v (0) = u0 z ((cid:0)0). (cid:1)
ε ε
−
Wethussplittheoriginalproblem intotwoparts: first,weshowthatthestochastic
convolution converges to 0, then weshow that the remainder v also converges to
ε
0.
Byconstruction, thestochastic convolution (3.8)isastationary process andits
invariantmeasureisgivenbyµ . Wefirstestablishageneralestimateforitsrenor-
ε
malized powers :zn:, which will be useful for bounding v later on. Throughout
ε ε
this section, we assume that lim σ2(ε)log(1/ε) = and that C is given by
ε→0 ε
∞
(3.2). Wethenhave:
TRIVIAL LIMIT FOR STRONG NOISE 8
Lemma3.3. Letr,k,p 1,s <0. Then,foralln N,wehave
≥ ∈
lim E :zn: k = 0 . (3.9)
ε→0 k ε kBps,r
Proof. Following the calculations of the proof of [DPD03, Lemma 3.2], we see
that
kn
E :zn: k . γ 2 , (3.10)
k ε kBps,r k εkHβn
whereβ = 1+ rks and
n 2np
σ2(ε)
γ (x) = e (x) .
ε C 1+ k 2 k
X ε
|k|≤1/ε − | |
Since
σ4(ε)(1+ k 2)βn
γ 2 = | | ,
k εkHβn (C 1+ k 2)2
X ε
|k|≤1/ε − | |
and β < 1, the claim follows from the boundedness ofσ and the fact that C
n ε
→
.
∞
Corollary 3.4. Letn,p,r 1ands < 0. Then:zn: Lp([0,T]; s )P-a.s.,for
≥ ε ∈ Bp,r
allε> 0. Inparticular,
limE :zn: = 0. (3.11)
ε→0 k ε kLp(0,T;Bps,r)
Proof. This follows from the stationarity of z , Fubini’s theorem and Lemma3.3.
ε
Weestablish nowthemainresultofthissubsection.
Proposition 3.5. Consider the stochastic convolution z defined in (3.8) and let
ε
p,r 1,s < 0andT > 0. Then
≥
limE z = 0 . (3.12)
ε→0 k εkC([0,T];Bps,r)
Proof. Webeginbydecomposing thestochastic convolution intotwoparts,
t
z (t)= etAεz (0)+σ(ε) e(t−s)AεdW (s).
ε ε ε
Z
0
TheboundonthefirsttermfollowsfromLemma3.3andLemmaA.3,soitremains
tofocusonthesecondterm,whichwedenotehereafter asz¯ (t). Inordertobound
ε
it,weusethefactorization method, see[DPZ92,p.128], aswellas[Hai09,p. 47]
foramoredetailed presentation. Recallingthat
t π
(t s)α−1(s σ)−αds = ,
Z − − sinπα
σ
TRIVIAL LIMIT FOR STRONG NOISE 9
wefixα (0, 1)andrewritez¯ as
∈ 2 ε
sinπα t
z¯ (t)= e(t−s)AεY (s)(t s)α−1ds, (3.13)
ε ε
π Z −
0
where
s
Y (s) := σ(ε) (s σ)−αe(s−σ)AεdW (σ).
ε ε
Z −
0
Next,weintroduce themappingΓ : y Γ y definedby
ε ε
7→
sinπα t
Γ y(t) := e(t−s)Aεy(s)(t s)α−1ds,
ε
π Z −
0
and show that Γ : Lq([0,T]; s ) ([0,T]; s ) is a bounded mapping for
ε Bp,r → C Bp,r
q > 1/α. First, it is a consequence of the strong continuity of etAε that Γ y
ε
∈
([0,T]; s ) for all y ([0,T]; s ) such that y(0) = 0 [Hai09, p. 48]. Next,
C Bp,r ∈ C Bp,r
observe that s (t s)α−1 is in Lq¯([0,t]) for all q¯ [1,(1 α)−1), and hence
7→ − ∈ −
wecanuseHo¨lder’sinequality todeducethatforallq > 1,
α
sup Γ y(t) . y . (3.14)
t∈[0,T]k ε kBps,r k kLq([0,T];Bps,r)
A standard density argument allows us to conclude that Γ : Lq([0,T]; s )
ε Bp,r →
([0,T]; s )isindeedabounded mappingforq > 1/α.
C Bp,r
Toconcludetheproof,weassumeforthemomentthatthereexistK > 0such
ε
that
sup E Y (t) K , limK = 0 . (3.15)
k ε kBps,r ≤ ε ε→0 ε
t∈[0,T]
From(3.15),itthenfollowsthat
1/q
E Y T sup E Y q . T1/qK , (3.16)
k εkLq([0,T];Bps,r) ≤ (cid:16) t∈[0,T] k εkBps,r(cid:17) ε
where the first inequality is due to Jensen’s inequality and Fubini’s theorem, and
thesecond inequality follows from (3.15)inconjunction withFernique’s theorem.
By (3.16), Y Lq([0,T]; s ) P-a.s. and hence z¯ = Γ Y ([0,T]; s )
ε ∈ Bp,r ε ε ε ∈ C Bp,r
P-a.s. Furthermore, itfollowsfrom(3.14)–(3.16)that
E sup z¯ (t) . E Y . K ,
t∈[0,T]k ε kBps,r k εkLq([0,T];Bps,r) ε
sothat z¯ 0inprobability, asrequired.
k εkC([0,T];Bps,r) →
It remains to establish (3.15). By definition of the Besov norm (2.1) and
Jensen’s inequality,
∞
1/r
E Y (t) 2qrsE ∆ Y (t) r . (3.17)
k ε kBps,r ≤ (cid:16)Xq=0 k q ε kLp(cid:17)
TRIVIAL LIMIT FOR STRONG NOISE 10
Asaconsequence, (3.15)followsifwecanshowthat
E ∆ Y (t) p K 2qpτ , (3.18)
k q ε kLp ≤ ε
forsomeτ < s andsomeK 0.
ε
| | →
Fix now q N. Thanks to Fubini’s theorem, the Gaussianity of ∆ Y (t), and
q ε
∈
theindependence ofitsdifferent Fouriercomponents,
p
E ∆ Y (t) p = E (Y (t),e )e (ξ) dξ
k q ε kLp ZT2 (cid:12)(cid:12)2q−1X≤|k|<2q ε k k (cid:12)(cid:12)
(cid:12) (cid:12)
2 p/2
. E (Y (t),e )e (ξ) dξ (3.19)
ε k k
ZT2(cid:16) (cid:12)(cid:12)2q−1X≤|k|<2q (cid:12)(cid:12) (cid:17)
(cid:12) (cid:12)
p/2
. E (Y (t),e ) 2 dξ.
ε k
ZT2(cid:16)2q−1X≤|k|<2q | | (cid:17)
Itoˆ’sisometryandthedefinitionofA yield
ε
2α−1 ∞
E (Y (t),e )2 σ2(ε) 2 C 1+ k 2 e−ττ−2αdτ
ε k ε
| | ≤ h (cid:16) − | | (cid:17)i Z0
2α−1
. σ2(ε) C 1+ k 2 , (3.20)
ε
− | |
(cid:16) (cid:17)
where the last inequality is due to 2α < 1. Inserting (3.20) back into (3.19) we
obtainthebound
p/2
1−2α
1
Ek∆qYε(t)kpLp . σp(ε) (cid:18)C 1+ k 2(cid:19) 
2q−1X≤|k|<2q ε− | |
 
p/2
22qτ 1
. σp(ε)  ,
(Cε−1)δ 2q−1X≤|k|<2q |k|2+2τ−4α−2δ
 
which is valid for all τ > 0 and all δ (0,1 2α). Since we can make both
∈ −
α and δ arbitrarily small, we can in particular choose them in such a way that
2α + δ < τ < s , so that the exponent is strictly greater than 2. This implies
| |
that the corresponding inverse power of k is summable over all k, so that (3.18)
| |
issatisfied.
3.3 Boundsontheremainder
First,weneedatechnical lemmaforthemapping ε,definedas
M
t 3
( εy)(t) := etAε u0 z (0) + e(t−τ)Aε a yl(τ):z3−l(τ):dτ ,
M − ε Z l ε
(cid:0) (cid:1) 0 Xl=0
(3.21)