Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation

0.27 MB·

by  Mouhamadou Sy

Invariant measure and long time behavior of
regular solutions of the Benjamin-Ono equation
MouhamadouSY
6
1 UniversitédeCergy-Pontoise
0
LaboratoireAGMUMR 8088CNRS
2
2 av. AdolpheChauvin, 95302Cergy-PontoiseCedex, France
n
[email protected]
a
J

9
1 Abstract

] TheBenjamin-Onoequationdescribesthepropagationofinternalwavesina
P stratifiedfluid. Inthepresent work, westudylargetimedynamics ofitsregular
A solutionsviasomeprobabilisticpoint of view. Weprovetheexistenceof anin-
. variantmeasureconcentratedonC¥ (T)andestablishsomequalitativeproperties
h
ofthismeasure. Wethendeducearecurrencepropertyofregularsolutions. The
t
a approachusedinthispaperappliestootherequationswithinfinitelymanyconser-
m vationlaws,suchastheKdVandcubicSchrödingerequationsin1D.
[ Keywords:Benjamin-Onoequation,invariantmeasure,longtimebehavior,regu-

larsolutions,inviscidlimit
1
v
5
Contents
5
0
5 1 Introduction 2
0 1.1 Theproblemandstatementofthemainresult . . . . . . . . . . . . . 2
.
1 1.2 Notations,spacesandforces . . . . . . . . . . . . . . . . . . . . . . 4
0
6 2 Preliminaries 5
1
2.1 Conservationlaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
:
v 2.2 Deterministicestimates . . . . . . . . . . . . . . . . . . . . . . . . . 6
i 2.3 Probabilisticestimates . . . . . . . . . . . . . . . . . . . . . . . . . 8
X
r
3 WellposednessofthestochasticBOBequation 10
a
4 Invariantmeasuresfortheviscousproblem 14
5 InvariantmeasurefortheBOequation 16
5.1 SomeconvergenceresultsofstochasticBOBtoBO . . . . . . . . . . 16
5.2 Anaccumulationpointfortheviscousmeasures . . . . . . . . . . . . 17
6 Qualitativepropertiesofthemeasure 20
6.1 Absolutecontinuityofsomeobservablesw.r.t. totheLebesguemeasure 20
6.2 Aboutthedimensionofthemeasurem . . . . . . . . . . . . . . . . . 23
6.3 AGaussiandecaypropertyforthemeasurem . . . . . . . . . . . . . 27
1
M.SY 1 INTRODUCTION
A ProofofLemma3.3 28
B TheperiodicHilberttransform 30
1 Introduction
1.1 The problem and statementofthe mainresult
TheBenjamin-Ono(BO)equation
¶ u+H¶ 2u+u¶ u=0 (1.1)
t x x
describesthepropagationofinternalwavesinastratifiedfluid. TheoperatorH enter-
ing the equation is the Hilbert transform, which can be defined in Fourier setting as
themultiplierby isgn(seeAppendix). Weassumethatu(t,x)isareal-valuedfunc-
tion, t R and−x belongs to the torus T =R/2p Z. In this setting, existence and
+
uniquen∈essofsolutionholdinL2(T)(see[Mol08,MP12]). Inourworkweneedthe
wellposednessof the problemonly in Sobolevspaces Hs(T) with s 2, so we refer
thereaderto[ABFS89]wherewellposednesswasprovedfors> 3. ≥
2
Positioning in L2 := L2(T), the wellposedness of (1.1) generates a topological
dynamicalsystem (DS) (L2,f ), where f is the flow of the BO equation (1.1). One
t t
ofthegreatquestionsofthequalitativestudyofevolutionPDEistodescribethelong
timebehaviorofsuchDS.
GivenaBorelmeasurem onL2,wesaythatm isinvariantfortheDS(L2,f )iffor
t
anyBorelsetAofL2,wehave
m (f 1A)=m (A), t. (1.2)
t− ∀
Whensuchameasureexists,thetriple(L2,f ,m )iscalledmeasurabledynamicalsys-
t
tem(MDS).Ifinadditionm isfinite,thePoincarérecurrencetheoremimpliesthatthe
dynamicsisrecurrent,thatis, m -almosteveryorbitgoesbackinanyneighborhoodof
itsorigininfinitetime. TheVonNeumannandBirkhoffergodictheoremscanalsobe
appliedtogivefurtherinformationsonthelongtimebehaviorofthesystem.
Matsuno[Mat84]derived(atleastformally)infinitelymanyconservationlawsfor
theBOequation(1.1),theyhavetheform
N
E (u)= u 2+R (u), n , (1.3)
n k kn n ∈ 2
where . standforthehomogenousSobolevnormofordernandR isalowerorder
n n
kk
term.
In [TV13, TV14, TV15, Den15, DTV15] the authors constructa sequence of in-
variantGaussiantypemeasures m for(L2,f )satisfyingthefollowing:
n t
{ }
m isconcentratedonHs(T),fors<n 1, ( )
• n −2 ∗
m (Hn 1/2(T))=0. ( )
n −
• ∗∗
Formally,m isdefinedasarenormalizationof
n
dm n(u)=e−En(u)du=e−Rn(u)e−kuk2ndu, (1.4)
2
M.SY 1 INTRODUCTION
whereE (u)andR (u)arethequantitiesgivenin(1.3).TheauthorsconstructaGaus-
n n
sian renormalization of the expression e−kuk2ndu on the concerned spaces and prove
thate Rn(u) is an integrabledensity. In viewof these results, a MDSfor (1.1) exists
−
inanySobolevspaceandthenitslargetimedynamicsisdescribedinthesenseofthe
theorems mentioned above. However, these results do not apply to infintely smooth
solutions,indeedby( )wehave
∗∗
m (C¥ (T))=0, foralln. (1.5)
n
InthepresentworkweconstructameasurabledynamicalsystemforBO(1.1)on
C¥ (T).OfcoursetheDiracmeasureat0isnotthedesiredmeasurealthoughitisin-
variantundertheflowofBO,becauseitgivesnoinformation. Moregenerally,toget
substantialinformationonthesystemwehavetoavoidtoosingularmeasures.Another
exampleofsuchmeasuresistheoneconcentratedonastationarysolution.Noticethat
themeasuresm discussedaboveverifythefollowing"consistency"property:everyset
n
offullm n-measureisdenseinH˙(n−1/2)−.However,anobstructiontotheconstruction
ofaninvariantGaussiantypemeasureinC¥ (T) isthenon-existenceofconservation
¥
lawcompatiblewiththeC -regularity.Inparticular,theapproachusedintheconstruc-
tionofthemeasuresm doesnotseemtoapply.
n
Another method allowing to construct invariant measures (a priori not of Gaus-
sian type) for PDE was introduced in [Kuk04, KS04] in the context of Euler and
Schrödingerequations. Itis basedon a fluctuation-dissipationargumentandconsists
inaddingtotheequationappropriatelynormalizeddampingandstochasticterms,con-
structinganinvariantmeasurefortheresultingproblem,andpassingtothelimit. The
idea inthis workisto developpthisapproachin thecontextofthe BO equation,and
combineitwiththestructureoftheconservationlawstoconstructanon-trivialinvari-
antmeasureconcentratedonC¥ (T).
Wefirstconsiderthestochastic-diffusionproblem(alsocalledstochasticBenjamin-
Ono-Burgers(BOB)equation)
¶ u+H¶ 2u+u¶ u=a¶ 2u+√ah , t>0, x T, (1.6)
t x x x ∈
where h is a stochastic forceand a (0,1) is a parameter. Then our problem(1.1)
is the limit as a 0 of the stochast∈ic-diffusion equation (1.6). After proving well-
→
posednessinsomeprobabilisticsenseofthe initialvalueproblemof (1.6), we prove
in Section4the existenceofinvariantmeasuresforthe latterandthatforsufficiently
regularinspacenoise,anyinvariantmeasureof(1.6)isconcentratedonC¥ (T).Pass-
ingtothelimitastheviscositygoestozero,weproveinTheorems5.3,6.1and6.3the
followingresult
Theorem 1.1. There is a probability measure m invariant under the flow of the BO
equation(1.1)definedonH3(T)andconcentratedontheinfinitelysmoothfunctions,
thatis
m (C¥ (T))=1.
Thismeasurem satisfiesthefollowingproperties:
1. Foranyintegernandanyrealnumber p 1,wehave
≥
0< u 2m (du)<¥ , 0< u pm (du)<¥ .
H3k kn H3k k
Z Z
3
M.SY 1 INTRODUCTION
2. There is an infinite sequence of independent preserved quantities whose laws
underm areabsolutelycontinuouswithrespecttotheLebesguemeasureonR.
3. Themeasurem isofatleasttwo-dimensionalnatureinthesensethatanycom-
pactsetofHausdorffdimensionsmallerthan2hasm -measure0.
In our context, the Poincaré recurrence theorem will translate into the following
result
Corollary1.2. For m -almost all w inC¥ (T), there is a sequence t increasing to
k
{ }
infinitysuchthat
lim S w w =0, n. (1.7)
k ¥ k tk − kn ∀
→
HereS denotestheflowoftheBenjamin-Onoequation(1.1)onH3(T).
t
Roughlyspeaking,itstates thatforsufficientlylargetime the regularsolutionsto
theBOequation(1.1)gobackneartheirinitialstates.
In the constructionof such a measure, we need the increasing Sobolevregularity
provided by the infinite sequence of conservation laws. It is the question of the se-
quence u 2 whichcomesfromthedispersivetermH¶ 2uoftheBOequation(1.1).
{k kn} x
The KdV and cubic 1D NLS equations have infinitely many conservation laws
whose form is similar to (1.3), this makesour appoachto apply for these equations.
RecallthataninfinitesequenceofinvariantGaussiantypemeasuresofincreasingreg-
ularitywasconstructedforKdVandcubic1DNLSin[Zhi01a,Zhi01b]. However,the
nonlocalnatureoftheBOequationmakesitmorecomplicatedthanKdVandcubic1D
NLSwhicharebetterunderstood.
Letusdiscussbrieflyanequationhavinginfinitelymanyconservationlawsbutnot
includinganincreasingSobolevregularity.Considerthenon-viscousBurgersequation
¶ u+u¶ u=0. (1.8)
t x
Itiseasytocheckthataninfinitesequenceofconservationlawsisgivenbythequan-
tities
L (u)= up, p 1. (1.9)
p
≥
Z
Ourapproachdoesnotapplyto(1.8)becauseofthelackofdispersion.
In [Sy16], the fluctuation-dissipation approach is used to construct a non-trivial
invariantmeasureconcentratedonH2 H1forthecubicKlein-Gordonequation
×
¶ 2u ∆u+u+u3=0, (1.10)
t −
consideredonaboundeddomainD R3 oronthetorusT3.
⊂
1.2 Notations, spaces andforces
LetAandBbetwopositivequantities,wewrite
A.B (1.11)
ifthereisauniversalconstantl 0suchthatA l B.
For a real number r, r+ (resp. r≥) denotes r+≤e (resp. r e ) where e is a positive
−
−
numbercloseenoughto0.
Z denotesthesetofnonzerointegers.
0
4
M.SY 2 PRELIMINARIES
H˙(T)= u L2(T): u(x)dx=0 .
T
H˙s(T)={ u∈ H˙(T): Dsu H˙(T) },Ds isthesthderivativeofu,wheres 0.
TheH˙s-no{rm∈isdenotedRby ∈. when}s>0andtheL2-normisdenotedby ≥. .
s
Fora functionalA(u), ¶ A=kkA and¶ 2A=A denotethefirstandsecondkdkerivative
u ′ u ′′
ofAw.r.t. u.
Thesequence e , n Z isdefinedby
n 0
{ ∈ }
sin(nx), for n>0,
e (x)= √p
n ( co√s(pnx), for n<0.
andformsanorthonormalbasisofH˙(T).
(Ω,F,P) is a complete probability space. F is a right-continuous filtration aug-
t
mented w.r.t. (F,P). Given numbers l R and a sequence of independentreal
n
standardBrownianmotions b (t) ada{pte}d⊂toF,wedefine
n t
{ }
z (t,x)= (cid:229) l b (t)e (x), (1.12)
n n n
n∈Z0
d
h (t,x)= z (t,x), (1.13)
dt
and
A = (cid:229) l 2n2s. (1.14)
s n
n∈Z0
2 Preliminaries
2.1 Conservationlaws
Foranenoughsmoothfunctionu,set
P = ¶ a u, ¶ a Hu a N ,
1 { x x | ∈ }
P2={(¶ xa 1Z1u)(¶ xa 2Z2u)|a i∈N, Zi∈{Id,H}}.
GenericallywedefineP , n 3asthesetoffunctionsoftheform
n
≥
k k
p (u)=(cid:213) Z(p (u)), Z Id,H , (cid:229) j =n, p P . (2.1)
n i ji i∈{ } i ji ∈ ji
i=1 1
Toafunctionp (u)oftheform(2.1),weassociate
n
k
(cid:213)
p˜ (u)= p (u), (2.2)
n ji
i=1
andwedefinethequantities
n
S(p(u))= (cid:229) a ,
i
i=1
M(p(u))= maxa .
i
1 i n
≤≤
5
M.SY 2 PRELIMINARIES
In the presentwork we only need the conservationlaws of integerorder. We de-
scribeherethecorrespondingremaindertermsfollowing[TV14]:
(cid:229) (cid:229)
R (u)= c (p) p(u)+ c (p) p(u), (2.3)
n n n
p˜(u)p=(uu)¶∈xnP−13u¶xnu Z p(Su()∈p(Pu)j)=j=23n,−...j,2+n2+2 Z
M(p(u)) n 1
≤ −
wherec (p)aresomeconstants. Thefirstthreeconservationlawsofintegerorderare
n
E (u)= u2,
0
Z
3 1
E (u)= (¶ u)2+ u2H¶ u+ u4,
1 x x
4 8
Z Z Z
5
E (u)= (¶ 2u)2 (¶ u)2H¶ u+2¶ 2uH¶ u
2 x −4 x x x x
Z Z
5 (cid:0) (cid:1)
+ 5u2(¶ u)2+u2(H¶ u)2+2uH(¶ u)H(u¶ u)
x x x x
16
Z
5(cid:0) 5 1 (cid:1)
+ u4H(¶ u)+ u3H(u¶ u) + u6.
x x
32 24 48
Z (cid:18) (cid:19) Z
2.2 Deterministic estimates
Lemma2.1. Therearec ,c+>0suchthatforalluinHn(T)
−n n
1
u 2 c u 2n+2 E (u) 2 u 2+c+ u 2n+2. (2.4)
2k kn− −nk k ≤ n ≤ k kn nk k
Lemma2.2. Foralle >0,thereisCe >0suchthatforalluinHn+1(T)
¶ uEn(u,¶ x2u)≤(−2+e )kuk2n+1+Ce kuk(1+kuk)bn, (2.5)
whereb dependsonlyonn.
n
Remark 2.3. Sincethe L2-normis preservedby (1.1) we candeducefrom(2.4) and
theargumentsoftheproofofLemma2.2,byaddingappropriatepolynomialsof u ,
newconservationlawsE (u)andE˜ (u)satisfying k k
n∗ n
0 u 2 E (u), (2.6)
≤k kn≤ n∗
0 u 2 ¶ E˜ (u,u). (2.7)
≤k kn≤ u n
Inequalities (2.4) can be established using similar arguments as in the proof of
Lemma2.2.
ProofofLemma2.2. Taking into account of the properties of the Hilbert transform
such as continuityin Hs andLp (s 0, p ]1,¥ [), we can "remove"H and consider
≥ ∈
thefunctions
R1(u)= u¶ n 1u¶ nu, (2.8)
n x− x
Z
j
R2,j(u)= (cid:213) ¶ a iu, j=3,...,2n+2. (2.9)
n x
Z i=1
HereR1(u)representsthefirsttermof(2.3)andthesecondtermof(2.3)canbeesti-
n
matedconsideringthequantitiesR2,j(u). Set
n
R0= u 2. (2.10)
n k kn
6
M.SY 2 PRELIMINARIES
EstimatesconcerningR0:
n
¶ R0(u,¶ 2u)= 2 u 2 (2.11)
u n x − k kn+1
EstimatesconcerningR1:
n
¶ R1(u,¶ 2u)= ¶ 2u¶ n 1u¶ nu+ u¶ n+1u¶ nu+ u¶ n 1u¶ n+2u
u n x x x− x x x x− x
Z Z Z
= ¶ 2u¶ n 1u¶ nu ¶ u¶ n 1u¶ n+1u
x x− x − x x− x
Z Z
=(1)+(2).
Nowinterpolationinequalitiesimply
|(1)|,|(2)|≤C1kukdn1+1kuk3−d1 (2.12)
where
4n+3
d = <2.
1
2(n+1)
Thenforsuitableb
1
|¶ uR1n(u,¶ x2u)|≤e kuk2n+1+Ce1kukb1. (2.13)
EstimatesconcerningR2,j:
n
j
¶ R2,j(u,¶ 2u)= (cid:213) ¶ a iu j=3,...,2n+2, (2.14)
u n x x
Z i=1
where(cid:229) j a =2n j+4andmax a n+1.
Wefoi=ll1owitwoc−omplementaryca1s≤eis≤:j i≤
Case1:max a n.Let(g )be jrealnumberssuchthat(cid:229) j 1 =1. Then
• the generaliz1e≤di≤Hjölid≤er formulaicombined with usual interpolait=io1ngiinequalities
shows:
j
|¶ uR2n,j(u,¶ x2u)|≤C(cid:213) kukli, (2.15)
1
wherel = 1 1 +a .Then
i 2−gi i
|¶ uR2n,j(u,¶ x2u)|≤C(cid:213) j kukn+n1+−1likuknn+l+i11.
1
Remarknowthat
(cid:229) j l =(cid:229) j 1 1 +a =2n+4 j,
i=1 i 1 (cid:18)2−gi i(cid:19) −2
then
(cid:229) j l i = 2n+4−2j <2.
n+1 n+1
i=1
Thusforsuitableb ,
2
|¶ uR2n,j(u,¶ x2u)|≤e kuk2n+1+Ce2kukb2. (2.16)
7
M.SY 2 PRELIMINARIES
Case2:a =n+1.Then(cid:229) j a =n j+3 n.Wehavethen
• 1 i=2 i − ≤
1
j 2
¶ R2,j(u,¶ 2u) u (cid:213) ¶ a iu2 . (2.17)
| u n x |≤k kn+1 Z 2 | x | !
Takeagain(g )suchthat(cid:229) j 1 =1. Then
i i=2gi
j
|¶ uR2n,j(u,¶ x2u)|≤kukn+1(cid:213) k¶ xa iukL2gi
2
≤kukn+1(cid:213) 2j kukli, l i= 21−21gi +a i,
≤kukn+1(cid:213) j kukn+n1+−1likuknn+l+i11.
2
Since(cid:229) j l =n+2 j n+1,wehave 1 (cid:229) j l <1andtheexistenceof
i=2 i −2 ≤ 2 n+1 i=2 i
asuitableb suchthat
3
|¶ uR2n,j(u,¶ x2u)|≤e kuk2n+1+Ce3kukb3. (2.18)
Combining(2.11),(2.13)and(2.18)withgoodchoiceofe ,wehavetheclaim.
2.3 Probabilisticestimates
ConsidertheinitialvalueproblemconcerningthestochasticBOBequation(1.6)
¶ u+H¶ 2u+u¶ u=a¶ 2u+√ah t>0,
t x x x (2.19)
u =u .
t=0 0
(cid:26) |
Wellposednessof(2.19)insomeprobabilisticsensewillbeestablishedinProposition
3.1. Recallthedefinitionofthefollowingconstants
A = (cid:229) m2sl 2,
s m
m∈Z0
which"measure"theregularityinspaceofthenoise.
Theorem2.4. Letn 2. SupposeA finite.Thereareq >0,g >0suchthatforany
n n n
randomfunctionuin≥Hn solutionof(2.19)issuedtou whichsatisfiesEE (u )<¥ ,
0 n 0
wehave
t t t
EE (u)+a E u 2 ds EE (u )+a A t+c E u 2ds+g E u (1+ u )qnds ,
n 0 k kn+1 ≤ n 0 n n 0 k kn n 0 k k k k
Z (cid:20) Z Z (cid:21)
(2.20)
wherec dependsonlyonn.
n
Proof. ApplyingtheItoformula(seeTheoremA.7.5andCorollaryA.7.6of[KS12]for
theItoformulainHilbertspaces)totheconservationlawE (u),wefind
n
8
M.SY 2 PRELIMINARIES
a
dE (u)=¶ E (u,du)+ (cid:229) l 2¶ 2E(u,e )dt, (2.21)
n u n 2 m u m
m∈Z0
where
du=( H¶ 2u u¶ u+a¶ 2u)dt+√a dz , z (t,x)= (cid:229) l e (x)b (t).
− x − x x m m m
m∈Z0
SinceE (u)ispreservedbytheBOequation,wehave
n
¶ E (u, H¶ 2u u¶ u)=0.
u n − x − x
ByLemma2.2,wehave
¶ E (u,a¶ 2u) a u 2 +a P ( u ), (2.22)
u n x ≤− k kn+1 n k k
P is thepolynomialofLemma2.2.Followingthe argumentsofthe proofofLemma
n
2.2,weestablishthat
¶ 2E (u,e ) c m2n( u 2+Q ( u )), (2.23)
| u n m |≤ n k kn n k k
whereQn(r)=qnr(1+r)kn,qnandkndependonlyonn.Itremainstointegrate(2.21)
int,taketheexpectationthenthestochasticintegralvanishes,andtocombine(2.22)
with(2.23)togettheclaim.
Proposition2.5. Letubethesolutionof(2.19).
1. SupposethatEE (u)<¥ ,then
0
t
EE (u)+2a E u(s) 2ds=EE (u )+a A t. (2.24)
0 0 k k1 0 0 0
Z
2. Let p 1. SupposethatEEp(u )<¥ ,then
≥ 0 0
EEp(u) e pa tEEp(u )+ppAp. (2.25)
0 ≤ − 0 0 0
Proof. Theidentity(2.24)iseasilyprovenapplyingtheItoformulatotheconservation
lawE (u).Letusprove(2.25):
0
For p>1,weapplytheItoformulatoEp(u)tofind
0
dE0p(u)= pE0p−1(u)dE0(u)+a p(p2−1)E0p−2(u) (cid:229) l m2|E0′(u,em)|2dt.
m∈Z0
Takingtheexpectation,weget
t
EE0p(u)+E fa (u(s))ds=EE0p(u0),
0
Z
where
fa (u)=2pa E0p−1(u)kuk21−a pE0p−1(u)A0−a p(p2−1)E0p−2(u) (cid:229) l m2|E0′(u,em)|2.
m∈Z0
a Q
| {z }
9
M.SY 3 WELLPOSEDNESSOFTHESTOCHASTICBOBEQUATION
Nowweseeeasilythat
(cid:229) l 2 E (u,e )2 2A E (u).
m| 0′ m | ≤ 0 0
m∈Z0
ThenapplyingYounginequalitywefind
p2p
Q e Ep(u) Ap.
≥− 0 −e p 1 0
−
Ontheotherhand
pa E0p−1(u)kuk21≥pa E0p(u).
Choosinge = p,weseethat
fa (u) pa EEp(u) pp+1Apa .
≥ 0 − 0
Then
t
EEp(u)+pa EEp(u(s))ds EEp(u )+pp+1Apa t.
0 0 0 ≤ 0 0 0
Z
Gronwalllemmagivestheclaimedresult.
3 Wellposedness of the stochastic BOB equation
Proposition3.1. Lets 2. SupposethatA <¥ ,thentheproblem(2.19)isglobally
s
wellposedinH˙s(T)int≥hesensethatforallT >0
1. foranyrandomvariableu inH˙s(T)wehave,foralmostallw Ω,
0
∈
(a) (Existence)thereisu:=uw Λ (s):=C(0,T;H˙s) L2(0,T;H˙s+1) sat-
T
∈ ∩
isfyingthefollowingrelationinHs 2
−
t
u(t)=u (H¶ 2u+u¶ u a¶ 2u)ds+z (t) forallt [0,T], (3.1)
0− 0 x x − x ∈
Z
(b) (Uniqueness)ifu ,u Λ (s)aretwosolutionsinthesenseof(3.1)then
1 2 T
∈
u u on[0,T],
1 2
≡
2. theprocesssolutionu:(t,w ) uw (t)isprogressivelymeasurablew.r.t.s (u ,F),
0 t
7→
3. (Continuityw.r.t. theinitialdata)foru andu twosolutionsof(1.6)startingat
1 2
u andu respectively,wehavealmostsurely
1,0 2,0
lim u =u inΛ (s). (3.2)
1 2 T
u1,0→u2,0
InordertoprovetheexistenceresultinProposition3.1,wesplittheproblem(2.19)
asfollow:
Alinearstochasticproblem:
•
¶ tza +H¶ x2za =a¶ x2za +√ah t>0, (3.3)
za t=0=0.
(cid:26) |
10