Continuous dependence estimate for a degenerate parabolic–hyperbolic equation with Lévy noise

47 Pages·2016·0.78 MB·English

by  Ujjwal Koley

StochPDE:AnalComp(2017)5:145–191
DOI10.1007/s40072-016-0084-z
Continuous dependence estimate for a degenerate
parabolic–hyperbolic equation with Lévy noise
UjjwalKoley1 · AnantaK.Majee2 · GuyVallet3
Received:18May2016/Publishedonline:27October2016
©SpringerScience+BusinessMediaNewYork2016
Abstract In this article, we are concerned with a multidimensional degenerate
parabolic–hyperbolic equation driven by Lévy processes. Using bounded variation
(BV)estimatesforvanishingviscosityapproximations,wederiveanexplicitcontin-
uous dependence estimate on the nonlinearities of the entropy solutions under the
assumptionthatLévynoisedependsonlyonthesolution.Thisresultisusedtoshow
theerrorestimateforthestochasticvanishingviscositymethod.Inaddition,weestab-
lishfractionalBVestimateforvanishingviscosityapproximationsincasethenoise
coefficientsdependonboththesolutionandspatialvariable.
Keywords Degenerateparabolic–hyperbolicequation·Lévynoise·
Youngmeasures·Rateofconvergence·FractionalBVestimates
MathematicsSubjectClassification Primary60H15·Secondary35K65
B
GuyVallet
[email protected]
UjjwalKoley
[email protected]
AnantaK.Majee
[email protected]
1 CentreforApplicableMathematics,TataInstiuteofFundamentalResearch,P.O.Box6503,
GKVKPostOffice,Bangalore560065,India
2 MathematischesInstitut,UniversitätTübingen,AufderMorgenstelle10,72076Tübingen,
Germany
3 LMAPUMR-CNRS5142,IPRA,BP1155,64013PauCedex,France
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146 StochPDE:AnalComp(2017)5:145–191
1 Introduction
The last couple of decades have witnessed remarkable advances on the larger area
of stochastic partialdifferentialequations thatare driven by Lévy noise.An worthy
referenceonthissubjectis[28].However,verylittleisavailableonthespecificprob-
lemofdegenerateparabolic–hyperbolicequationwithLévynoise,andtherearestilla
numberofissueswaitingtobeexplored.Inthispaper,weaimatderivingcontinuous
dependence estimates based on nonlinearities for stochastic degenerate parabolic–
hyperbolicequationdrivenbymultiplicativeL(cid:2)évynoise.Aforma(cid:3)ldescriptionofour
problem requires a filtered probability space (cid:2),P,F,{Ft}t≥0 , and we are inter-
estedinanL2-valuedpredictableprocessu(t,·)whichsatisfiesthefollowingCauchy
problem
⎧
⎪⎨du(t,x)−divf(u(t,x))dt −(cid:3)A(u(t,x))dt
(cid:8)
=σ(u(t,x))dW(t)+ η(u(t,x);z)N(cid:9)(dz,dt), (x,t)∈(cid:6) , (1.1)
⎪⎩ E T
u(0,x)=u (x), x ∈Rd,
0
where(cid:6) =Rd×(0,T)withT >0fixed,u :Rd (cid:4)→Risthegiveninitialfunction,
T 0
f : R (cid:4)→ Rd isagiven(sufficientlysmooth)scalarvaluedfluxfunction(seeSect.2
forthecompletelistofassumptions),and A :R(cid:4)→Risagivennonlineardiffusion.
Regarding this, the basic assumption is that A(·) is nondecreasing with A(0) = 0.
Moreover,(1.1)isallowedtobestronglydegenerateinthesensethat A(cid:6)(·)isallowed
tobezeroonaninterval,see[13]. (cid:10)
W(t) is a cylindrical Wiener process: W(t) = k≥1ekβk(t) with (βk)k≥1 being
mutually independent real valued standard Wiener processes and (ek)k≥1 a com-
pleteorthonormalsysteminaseparableHilbertspaceH.Furthermore, N(cid:9)(dz,dt) =
N(dz,dt)−m(dz)dt,whereN isaPoissonrandommeasureon(E,E)withintensity
measurem(dz),where(E,E,m)isaσ-finitemeasurespace.Finally,u (cid:4)→σ(u)isan
H-valuedfunctionand(u,z)(cid:4)→η(u,z)isagivenrealvaluedfunctionsignifyingthe
multiplicativenatureofthenoise.
Remark1.1 We will carry out our analysis under the structural assumption E =
O×R∗whereOisasubsetoftheEuclideanspace.ThemeasuremonEisdefinedas
λ×μwhereλisaRadonmeasureonOandμisso-calledLévymeasureonR∗.Such
anoisewouldbecalledanimpulsivewhitenoisewithjumppositionintensityλand
jumpsizeintensityμ.Wereferto[28]formoreonLévysheetandrelatedimpulsive
whitenoise.
Moreover, for each v ∈ L2(Rd), we consider the mapping σ(v) : H → L2(Rd)
(cid:10)definedbyσ(v)ek =gk(v(·)).Inparticular,wesupposethatgk ∈C(R)andG2(r)=
g2(r).
k≥1 k
TheEq.(1.1)couldbeviewedasastochasticperturbationofparabolic–hyperbolic
equation. Equations of this type model the phenomenon of convection-diffusion of
idealfluidsandthereforeariseinawidevarietyofimportantapplications,includingfor
instancetwoorthreephaseflowsinporousmedia[20]orsedimentation-consolidation
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StochPDE:AnalComp(2017)5:145–191 147
processes[12].Inthecaseσ =η= A=0,theEq.(1.1)becomesastandardconser-
vationlawsinRd.Fortheconservationlaws,thequestionofexistenceanduniqueness
ofsolutionswasfirstsettledinthepioneerpapersofKružkov[26]andVol’pert[32].
Inthecaseσ =η=0,theEq.(1.1)becomesadegenerateparabolic–hyperbolicequa-
tioninRd.Fordegenerateparabolic–hyperbolicequationsentropysolutionwerefirst
consideredbyVol’pertandHudajev[33],whileuniquenessofentropysolutionswas
firstprovedbyCarrillo[13].Anumberofauthorshavecontributedsincethen,andwe
mentiontheworksofAndreianovandMaliki[1],Cockburnetal.[16],Bendahmane
andKarlsen[5,6],Evjeetal.[21]andVallet[29].
1.1 Stochasticbalancelaws
Thestudyofstochasticbalancelawshassofarbeenlimitedmostlytoequationsofthe
type(1.1)with A=0.Infact,Kim[25]extendedtheKružkovwell-posednesstheory
toonedimensionalbalancelawsthataredrivenbyadditiveBrowniannoise,andVallet
andWittbold[31]tothemultidimensionalDirichletproblem.However,whenthenoise
isofmultiplicativenature,onecouldnotapplyastraightforwardKružkov’sdoubling
method to get uniqueness. This issue was settled by Feng and Nualart [23], who
establisheduniquenessofentropysolutionbyrecoveringadditionalinformationfrom
the vanishing viscosity method. The existence was proven using stochastic version
ofcompensatedcompactnessmethodanditwasvalidforonespatialdimension.To
overcome this problem, Debussche and Vovelle [17] introduced kinetic formulation
of such problems and as a result they were able to establish the well-posedness of
multidimensional stochastic balance law via kinetic approach. A number of authors
havecontributedsincethen,andwementiontheworksofBauzetetal.[2,3],Biswaset
al.[7,8].WealsomentionworksbyChenetal.[14],andBiswasetal.[9],wherewell-
posednessofentropysolutionisestablishedinLp∩BV,viaBVframework.Moreover,
theywereabletodevelopcontinuousdependencetheoryformultidimensionalbalance
lawsandasabyproducttheyderivedanexplicitconvergencerateoftheapproximate
solutionstotheunderlyingproblem.
1.2 Degeneratestochasticbalancelaws
Stochasticdegenerateparabolic–hyperbolicequationsareoneofthemostimportant
classesofnonlinearstochasticPDEs.Nonlinearityanddegeneracyaretwomainfea-
turesoftheseequationsandyieldsseveralstrikingphenomena.Infact,duetostrong
degeneracy, one cannot expect smooth solutions even if the initial data is smooth.
Existenceanduniquenessofsolutionsof (1.1)wasfirstsettledbyBauzetetal.[4](in
thecaseofη=0),andbyBiswasetal.[10](inthecaseofη(cid:9)=0).Infact,theyhave
extendedtheirpreviousworks([2]and[8],respectively)tothecontextofdegenerate
parabolic–hyperbolic problem inthe spiritof Carrillo’swork [13].The existence of
solutionisprovedbyusingavanishingviscositymethod,basedonthecompactness
proposedbythetheoryofYoungmeasures.Theuniquenessofthesolutionisobtained
viaKruzkov’sdoublingvariablemethod.WealsomentiontheworkofDebusscheet
al.[18],wheretheauthorshaveestablishedthewell-posednesstheoryforsolutionsof
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148 StochPDE:AnalComp(2017)5:145–191
theCauchyproblem (1.1)inanyspacedimension.Theyhaveadaptedthenotionof
kineticformulationandkineticsolutionwhichhasalreadybeenstudiedinthecaseof
hyperbolicscalarconservationlawsinbothdeterministic[27]andstochasticsetting
[17].
Independentlyofthesmoothnessoftheinitialdatau ,duetothepresenceofnonlin-
0
earfluxterm,degeneratediffusionterm,andanonlocalterminEq.(1.1),solutionsto
(1.1)arenotnecessarilysmoothandweaksolutionsmustbesought.Beforeintroduc-
ingtheconceptofweaksolutions,wefirstrecallthenotionofpredictableσ-field.Bya
predictableσ-fieldon[0,T]×(cid:2),denotedbyP ,wemeanthattheσ-fieldgenerated
T
bythesetsoftheform:{0}×Aand(s,t]×BforanyA∈F ;B ∈F , 0<s,t ≤T.
0 s
Thenotionofstochasticweaksolutionisdefinedasfollows:
Definition1.1 (Stochastic weak solution) A square integrable L2(Rd)-valued {F :
t
t ≥ 0}-predictablestochasticprocessu(t) = u(t,x)issaidtobeaweaksolutionto
ourproblem(1.1)provided
(i) u ∈(cid:11) L2((cid:2)(cid:8) ×(cid:6)T)and A(u)∈ L(cid:8)2(((cid:8)0,T)×(cid:2);H1(Rd)). (cid:12)
(ii) ∂ u − tσ(u(s,·))dW(s) − t η(u(s,·);z)N(cid:9)(dz,ds) ∈ L2((0,T) ×
∂t 0 0 E
(cid:2);H−1(Rd))inthesenseofdistribution.
(iii) Foralmosteveryt ∈ [0,T]andP−a.s,thefollowingvariationalformulation
holds:
(cid:13) (cid:11) (cid:14) (cid:14) (cid:14) (cid:12) (cid:15)
∂ t t
u− σ(u(s,·))dW(s)− η(u(s,·);z)N(cid:9)(dz,ds) ,v (cid:2) (cid:3)
∂t (cid:14) (cid:16)0 0 (cid:17)E H−1(Rd),H1(Rd)
+ ∇A(u(t,x))+ f(u(t,x)) .∇vdx =0, (1.2)
Rd
foranyv ∈ H1(Rd).
However,itiswell-knownthatweaksolutionsmaybediscontinuousandtheyare
notuniquelydeterminedbytheirinitialdata.Consequently,anadmissibleconditionso
calledentropysolutionmustbeimposedtosingleoutthephysicallycorrectsolution.
Sincethenotionofentropysolutionisbuiltaroundthesocalledentropyfluxtriple,
webeginwiththedefinitionofentropyfluxtriple.
Definition1.2 (Entropyfluxtriple)Atriplet(β,ζ,ν)iscalledanentropyfluxtriple
ifβ ∈C2(R)andβ ≥0,ζ =(ζ ,ζ ,...,ζ ):R(cid:4)→Rd isavectorvaluedfunction,
1 2 d
andν :R(cid:4)→Risascalarvaluedfunctionsuchthat
ζ(cid:6)(r)=β(cid:6)(r)f(cid:6)(r) and ν(cid:6)(r)=β(cid:6)(r)A(cid:6)(r).
Anentropyfluxtriple(β,ζ,ν)iscalledconvexifβ(cid:6)(cid:6)(s)≥0.
To define entropy solution(cid:8), w√e first define associated Kirchoff’s function of A,
denoted by G(x) as G(x) = x A(cid:6)(r)dr. With the help of a convex entropy flux
0
triple(β,ζ,ν),wepresentaformalderivationofentropyinequalities:
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StochPDE:AnalComp(2017)5:145–191 149
Forasmallpositivenumberε >0,assumethattheparabolicperturbation
duε(t(cid:14),x)−(cid:3)Aε(uε(t,x))dt −divx f(uε(t,x))dt =σ(uε(t,x))dW(t)
+ η(uε(t,x);z)N(cid:9)(dz,dt) (1.3)
E
of (1.1), with Aε(x) = A(x)+εx, has a unique weak solution uε(t,x). Note that
thisweaksolutionuε ∈ L2((0,T)×(cid:2);H1(Rd)).Thisenablesonetoderiveaweak
versionoftheItô-Lévyformula(asproposedin[10,18,22])forthesolutionsof (1.1).
Let (β,ζ,ν) be an entropy flux triple. Given a non-negative test function ψ ∈
C1,2([0,∞)×Rd),weapplygeneralisedversionoftheItô-Lévyformulatoyield,for
c
almosteveryT >0,
(cid:14) (cid:14)
β(uε(T,x))ψ(T,x)dx− β(uε(0,x))ψ(0,x)dx
Rd (cid:14) Rd (cid:14)
= β(uε(t,x))∂tψ(t,x)dxdt− ∇ψ(t,x)·ζ(uε(t,x))dxdt
(cid:6)(cid:18)T (cid:14) (cid:6)T
+ gk(uε(t,x))β(cid:6)(uε(t,x))ψ(t,x)dβk(t)dx
(cid:6)
k≥(cid:14)1 T
+ 1 G2(uε(t,x))β(cid:6)(cid:6)(uε(t,x))ψ(t,x)dxdt
2 (cid:6)
(cid:14) (cid:14)T (cid:14)
1 (cid:2) (cid:3)
+ η(uε(t,x);z)β(cid:6) uε(t,x)+λη(uε(t,x);z) ψ(t,x)dλN(cid:9)(dz,dt)dx
(cid:14)(cid:6)T (cid:14)E(cid:14)0
1 (cid:2)
+ (1−λ)η2(uε(t,x);z)β(cid:6)(cid:6) uε(t,x)
(cid:6)T E 0 (cid:3)
+λη(uε(t,x);z) ψ(t,x)dλm(dz)dxdt
(cid:14)
(cid:2) (cid:3)
− ε∇ψ(t,x)·∇β(uε(t,x))+εβ(cid:6)(cid:6)(uε(t,x))|∇uε(t,x)|2ψ(t,x) dxdt
(cid:14)(cid:6)T (cid:14)
− β(cid:6)(cid:6)(uε(t,x))|∇G(uε(t,x))|2ψ(t,x)dxdt+ ν(uε(t,x))(cid:3)ψ(t,x)dxdt.
(cid:6) (cid:6)
T T
Sinceβ andψ arenon-negativefunctions,weobtain
(cid:14) (cid:14) (cid:16)
0≤ β(uε(0,x))ψ(0,x)dx + β(uε(t,x))∂tψ(t,x)
Rd (cid:17) (cid:6)T
−∇ψ(t,x)·ζ(uε(t,x)) dxdt
(cid:14)
− β(cid:6)(cid:6)(uε(t,x))|∇G(uε(t,x))|2ψ(t,x)dxdt
(cid:14)(cid:6)T
+ ν(uε(t,x))(cid:3)ψ(t,x)dxdt +O(ε)
(cid:18)(cid:6)T(cid:14)
+ gk(uε(t,x))β(cid:6)(uε(t,x))ψ(t,x)dβk(t)dx
(cid:6)
k≥1 T
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150 StochPDE:AnalComp(2017)5:145–191
(cid:14)
1
+ G2(uε(t,x))β(cid:6)(cid:6)(uε(t,x))ψ(t,x)dxdt
2 (cid:6)
(cid:14) (cid:14)T (cid:14)
1 (cid:2)
+ η(uε(t,x);z)β(cid:6) uε(t,x)
(cid:6)T E 0 (cid:3)
+λη(uε(t,x);z) ψ(t,x)dλN(cid:9)(dz,dt)dx
(cid:14) (cid:14) (cid:14)
1 (cid:2)
+ (1−λ)η2(uε(t,x);z)β(cid:6)(cid:6) uε(t,x)
(cid:6)T E 0 (cid:3)
+λη(uε(t,x);z) ψ(t,x)dλm(dz)dxdt.
Clearly,theaboveinequalityisstableunderthelimitε →0,ifthefamily{uε}ε>0has
p
L -typestability.Justasthedeterministicequations,theaboveinequalityprovides
loc
ustheentropycondition.Wenowintroducethenotionofstochasticentropysolution
asfollows:
Definition1.3 (Stochasticentropysolution)AsquareintegrableL2(Rd)-valued{F :
t
t ≥ 0}-predictable stochastic process u(t) = u(t,x) is called a stochastic entropy
solutionof (1.1)if
(i) ForeachT >0,
(cid:19) (cid:20)
G(u)∈ L2((0,T)×(cid:2);H1(Rd)), and sup E ||u(t,·)||2 <+∞.
2
0≤t≤T
(ii) Givenanon-negativetestfunctionψ ∈C1,2([0,∞)×Rd)andaconvexentropy
c
fluxtriple(β,ζ,ν),thefollowinginequalityholds:
(cid:14) (cid:16) (cid:17)
β(u(t,x))∂ ψ(t,x)+ν(u(t,x))(cid:3)ψ(t,x)−∇ψ(t,x)·ζ(u(t,x)) dxdt
t
(cid:6)T (cid:18)(cid:14)
+ g (u(t,x))β(cid:6)(u(t,x))ψ(t,x)dβ (t)dx
k k
(cid:6)
k≥(cid:14)1 T
+ 1 G2(u(t,x))β(cid:6)(cid:6)(u(t,x))ψ(t,x)dxdt
2 (cid:6)
(cid:14) (cid:14)T (cid:14)
1 (cid:2) (cid:3)
+ η(u(t,x);z)β(cid:6) u(t,x)+λη(u(t,x);z) ψ(t,x)dλN(cid:9)(dz,dt)dx
(cid:14)(cid:6)T (cid:14)E(cid:14)0
1 (cid:2)
+ (1−λ)η2(u(t,x);z)β(cid:6)(cid:6) u(t,x)
(cid:6)T E 0 (cid:3)
+λη(u(t,x);z) ψ(t,x)dλm(dz)dxdt
(cid:14)
≥ β(cid:6)(cid:6)(u(t,x))|∇G(u(t,x))|2ψ(t,x)dxdt
(cid:6)(cid:14)T
− β(u (x))ψ(0,x)dx, P−a.s. (1.4)
0
Rd
Due to nonlocal nature of the Itô–Lévy formula and the noise-noise interaction,
theDefinion1.3alonedoesnotgivetheL1-contractionprincipleinthesenseofaver-
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StochPDE:AnalComp(2017)5:145–191 151
age and hence the uniqueness. In fact, classical “doubling of variable” technique in
time variable does not work when one tries to compare directly two entropy solu-
tionsdefinedinthesenseofDefinion1.3.Toovercomethisproblem,theauthorsin
[4,10]usedamoredirectapproachbycomparingsolutionsoftworegularizedprob-
lems and subsequently sending the regularized parameter to zero, relying on “weak
compactness”oftheregularizedapproximations.
In order to successfully implement the direct approach, one needs to weaken the
notionofstochasticentropysolution,andsubsequentlyinstallthenotionofsocalled
generalizedentropysolution.
(cid:2) (cid:3)
Definition1.4 (GeneralizedEntropySolution)Asquareintegrable L2 R×(0,1) -
valued {F : t ≥ 0}-predictable stochastic process u(t) = u(t,x,α) is called a
t
generalizedstochasticentropysolutionof (1.1)if
(i) ForeachT >0,
(cid:19) (cid:20)
G(u)∈ L2((0,T)×(cid:2)×(0,1);H1(Rd)), and sup E ||u(t,·,·)||2 <∞.
2
0≤t≤T
(ii) Givenanon-negativetestfunctionψ ∈C1,2([0,∞)×Rd)andaconvexentropy
c
fluxtriple(β,ζ,ν),thefollowinginequalityholds:
(cid:14) (cid:14) (cid:16)
1
β(u(t,x,α))∂ ψ(t,x)+ν(u(t,x,α))(cid:3)ψ(t,x)
t
(cid:6)T 0 (cid:17)
−∇ψ(t,x)·ζ(u(t,x,α)) dαdxdt
(cid:14) (cid:14)
(cid:18) 1
+ g (u(t,x,α))β(cid:6)(u(t,x,α))ψ(t,x)dαdβ (t)dx
k k
k≥1 (cid:6)T 0
(cid:14) (cid:14)
1 1
+ G2(u(t,x,α))β(cid:6)(cid:6)(u(t,x,α))ψ(t,x)dαdxdt
(cid:14)2 (cid:6)(cid:14)T 0(cid:14) (cid:14)
1 1 (cid:2)
+ η(u(t,x,α);z)β(cid:6) u(t,x,α)
(cid:6)T 0 E 0 (cid:3)
+λη(u(t,x,α);z) ψ(t,x)dαdλN(cid:9)(dz,dt)dx
(cid:14) (cid:14) (cid:14) (cid:14)
1 1 (cid:2)
+ (1−λ)η2(u(t,x,α);z)β(cid:6)(cid:6) u(t,x,α)
(cid:6)T 0 E 0 (cid:3)
+λη(u(t,x,α);z) ψ(t,x)dαdλm(dz)dxdt
(cid:14) (cid:14)
1
≥ β(cid:6)(cid:6)(u(t,x,α))|∇G(u(t,x,α))|2ψ(t,x)dαdxdt
(cid:14)(cid:6)T 0
− β(u (x))ψ(0,x)dx, P−a.s. (1.5)
0
Rd
This definition rests on the possibility to pass to the limit by using the theory of
Youngmeasures.Wewillnotuseexplicitlythisdefinitioninthesequelsinceouraim
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152 StochPDE:AnalComp(2017)5:145–191
istousethewellposednesstheoryoftheentropysolutioninthesenseofDefinition
1.3 described in [4,10], where the authors have revisited [1,13,15]; but we will use
thecompactnessproposedbytheYoungmeasurestheory.
Infact,thiscompactnessargumentyieldsaconvergenceinlawofthesequenceof
approximationwherethenewprobabilityspaceproposedbythetheoremofSkorohod
is ((cid:2) × (0,1),dP × dα). This generalized solution can be seen as a “martingale
solution”andtheuniqueness resultdeveloped in[4,10]essentiallyusedaGyöngy–
Krylovtypeargumentbasedontheuniquenessofthelaws.
1.3 Scopeandoutlineofthepaper
Theabovediscussionsclearlyhighlightthelackofstabilityestimatesfortheentropy
solutionsofdegenerateparabolic–hyperbolicstochasticbalancelawsdrivenbyLévy
noise. In this paper, drawing preliminary motivation from [9,14,21], we intend to
developacontinuousdependencetheoryforstochasticentropysolutionwhichinturn
canbeusedtoderiveanerrorestimateforthevanishingviscositymethod.However,it
seemsdifficulttodevelopsuchatheorywithoutsecuringaBVestimateforstochastic
entropysolution.Asaresult,wefirstaddressthequestionofexistence,uniquenessof
stochasticBVentropysolutionin L2(Rd)∩BV(Rd)oftheproblem(1.1).
We need to mention that the BV(Rd) regularity needed in the sequel requires to
assumethatthespacevariable x isnotavariableofthefunctions g andη.Seefor
k
exemplein[9,14].
MakinguseofthecrucialBVestimate,weprovideacontinuousdependenceesti-
mateanderrorestimateforthevanishingviscositymethodprovidedinitialdataliesin
u ∈ L2(Rd)∩BV(Rd).Finally,weturnourdiscussionstomoregeneraldegenerate
0
stochastic balance laws driven by Lévy processes, namely when the functions σ,η
appear in the Lévy noise has explicit dependency on the spatial position x as well.
Inviewofthediscussionsin[9,14],inthiscasewecannotexpectBVestimates,but
instead a fractional BV estimate is expected. However, that does not prevent us to
provideanexistenceproofformoregeneralclassofequationsin L2(Rd).
The rest of the paper is organized as follows: we describe technical framework
andstatethemainresultsinSect.2.InSect.3,wederiveuniformspatialBVbound
forviscoussolutions.Usingthisbound,weestablishwellposednessofBVentropy
solutionoftheCauchyproblem(1.1).Section4isdevotedonderivingthecontinuous
dependence estimate on nonlinearities, while Sect. 5 deals with the error estimates.
Finally,inSect.6,weestablishafractionalBVestimateforalargerclassofdegenerate
stochasticbalancelaws.
2 Technicalframeworkandstatementofthemainresults
Throughoutthispaper,weusetheletterC todenotevariousgenericconstants.There
are situations where constant may change from line to line, but the notation is kept
unchanged so long as it does not impact central idea. Moreover, for any separable
HilbertspaceH,wedenotebyNw2(0,T,H),theHilbertspaceofallthepredictableH-
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StochPDE:AnalComp(2017)5:145–191 153
(cid:11)(cid:8) (cid:12)
valuedprocessesusuchthatE T (cid:14)u(cid:14)2 <+∞.Furthermore,wedenote BV(Rd)
0 H
asthesetofintegrablefunctionswithboundedvariationonRdendowedwiththenorm
|u|BV(Rd) = (cid:14)u(cid:14)L1(Rd) +TVx(u),where TVx isthetotalvariation ofu defined on
Rd.Theprimaryaimofthispaperistoderivecontinuousdependenceestimatesfor
theentropysolutionsoftheCauchyproblem(1.1),andwedosounderthefollowing
assumptions:
A.1 Theinitialfunctionu isaF measurablerandomvariablesatisfying
0 0
(cid:11) (cid:12)
E (cid:14)u0(cid:14)22+|u0|BV(Rd) <+∞.
A.2 A : R → Risanon-decreasingLipschitzcontinuousfunctionwith A(0) = 0.
(cid:6)(cid:6)
Furthermore,weassumethat A isbounded.
A.3 f = (f , f ,..., f ) : R → Rd is a Lipschitz continuous function with
1 2 d
f (0)=0,forall1≤k ≤d.
k
A.4 Thespace E isoftheformO×R∗andtheBorelmeasuremonE hastheform
λ×μ,whereλisaRadonmeasureonO andμisso-calledonedimensional
Lévymeasure.
A.5 Weassumethatg (0)=0,forallk ≥1.Moreover,thereexistspositiveconstant
k
K >0suchthat
(cid:18)(cid:21) (cid:21)
(cid:21)g (u)−g (v)(cid:21)2 ≤ K|u−v|2, and G2(u)
k k
k≥1
(cid:18)
= g2(u)≤ K |u|2, forall u,v ∈R.
k
k≥1
A.6 There exist positive constants λ∗ ∈ (0,1), C > 0 and h(z) ∈ L2(E,m) with
0≤h(z)≤1suchthatforallu,v ∈R; z ∈ E
(cid:21) (cid:21)
(cid:21)η(u;z)−η(v;z)(cid:21)≤λ∗|u−v|h(z), and |η(u,z)|≤C|u|h(z).
Moreover,weassumethatη(0,z)=0,forallz ∈ E.
Remark2.1 Weremarkthat,onecanaccommodatepolynomiallygrowingfluxfunc-
tionasaresultoftherequirementthattheentropysolutionssatisfyLp boundsforall
p ≥2.ThisinturnforcestochooseinitialdatathatareinLp,forall p.However,we
havechosentoworkwiththeassumptions A.1and A.3.Theassumption A.6isnatural
inthecontextofLévynoisewiththeexceptionofλ∗ ∈(0,1),whichisnecessaryfor
theuniqueness.Finally,theassumptionsA.1–A.6collectivelyensuresexistenceand
uniquenessofstochasticentropysolution,andthecontinuousdependenceestimateas
well.
Remark2.2 In view of the assumption A.5, for any v ∈ L2(Rd), σ(v) is a Hilbert-
Schmidt operator from the separable Hilbert space H to L2(Rd). Therefore, for
a given predictable process v ∈ L2((cid:2);L2(0,T;L2(Rd))), the stochastic integral
123
154 StochPDE:AnalComp(2017)5:145–191
(cid:8)
t (cid:4)→ tσ(v)dW(s)iswell-definedprocesstakingvaluesinaHilbertspace L2(Rd).
0
Moreover,thetrajectoriesofW areP-a.s.continuousinH ⊃H,where
0
⎧ ⎫
⎨ (cid:18) (cid:18)v2 ⎬
H := v = v e : k <+∞
0 ⎩ k k k2 ⎭
k≥1 k≥1
(cid:10) (cid:10)
v2
endowed with the norm (cid:14)v(cid:14)2 = k with v = v e . Furthermore, the
embeddingH(cid:15)→H isHilbeHr0t-Schmkid≥t1(ske2e[19]). k≥1 k k
0
Likeitsdeterministiccounterpart,existenceofentropysolutionlargelyrelatedto
the study of associated viscous problem. One can follow the argument presented in
[4,10,30]toensureexistenceofweaksolutionsfortheproblem(1.3).Moreprecisely,
wehavethefollowingpropositionfrom[4,10,30].
Proposition2.1 Let the assumptions A.1,A.3,A.5, and A.6 hold and A : R (cid:4)→ R
isnon-decreasingLipschitzcontinuousfunction.Then(cid:2),foran(cid:8)yε > 0,thereexistsa
u(cid:8)ni(cid:8)queweaksolutionuε ∈ N(cid:3)w2(0,T,H1(Rd))with∂t uε − 0tσ(uε(s,·))dW(s)−
t η(uε(s,·);z)N(cid:9)(dz,ds) ∈ L2((cid:2) × (0,T),H−1(Rd)), to the problem (1.3).
0 E
Moreover,uε ∈ L∞(0,T;L2((cid:2)×Rd))andthereexistsaconstantC >0,independent
ofε,suchthat
(cid:11)(cid:25) (cid:25) (cid:12) (cid:14) T (cid:11)(cid:25) (cid:25) (cid:12) (cid:14) T (cid:11)(cid:25) (cid:25) (cid:12)
sup E (cid:25)uε(t)(cid:25)2 +ε E (cid:25)∇uε(s)(cid:25)2 ds+ E (cid:25)∇G(uε(s))(cid:25)2 ds ≤C,
2 2 2
0≤t≤T 0 0
(2.1)
whereG istheassociatedKirchoff’sfunctionof A.
Wearenowinapositiontostatethemainresultsofthispaper.
Maintheorem (Continuousdependenceestimate)LettheassumptionsA.1–A.6hold
fortwosetsofgivendata(u , f,A,σ,η)and(v ,g,B,(cid:9)σ,(cid:9)η).Letu(t,x)beanyBV
0 0
entropy solution of (1.1) with initial data u (x) and v(s,y) be another BV entropy
0
solutionwithinitialdatav (y)andsatisfies
0
dv(s,y)−(cid:3)B(v(s,y))ds−divg(v(s,y))ds =(cid:9)σ(v(s,y))dW(s)
(cid:14)
+ (cid:9)η(v(s,y);z)N(cid:9)(dz,ds). (2.2)
E
Moreover,define
|g (ξ)−(cid:9)g (ξ)| (cid:18)
E (σ,(cid:9)σ):=sup k k ; E(σ,(cid:9)σ)2 := E (σ,(cid:9)σ)2,
k |ξ| k
ξ(cid:9)=0 k≥1
(cid:14) (cid:21) (cid:21)
(cid:21)η(u;z)−(cid:9)η(u;z)(cid:21)2
D(η,(cid:9)η):=sup m(dz),
u(cid:9)=0 E |u|2
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