on the solvability of stochastic navier-stokes equations ... · k. sakthivel (joint with s.s....
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On the Solvability of Stochastic Navier-StokesEquations with Lévy Noise
K. Sakthivel(joint with S.S. Sritharan, Naval Postgraduate School, Monterey, U.S.A.)
Department of MathematicsIndian Institute of Space Science & Technology(IIST)
Trivandrum, Kerala.
Winter School on Stochastic Analysis and Control of Fluid Flow,IISER, Trivandrum, Dec 3-20, 2012
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 1 / 37
Plan of the Talk
1 Navier-Stokes Equations with Lévy Noise
2 Formulation of a Martingale Problem
3 Moment Estimates
4 Tightness of Probability Measures
5 Main Issues in the Existence Result
6 Uniqueness of Martingale Solutions
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 2 / 37
Navier-Stokes Equations with Lévy NoiseConsider the Navier-Stokes model perturbed by the Gaussian andLévy type stochastic forces
du + (−ν∆u + u · ∇u +∇p)dt = gdt + σ(t ,u)dW
+∞∑
k=1
∫0<|zk |Z<1
φk(u(x , t−), zk
)πk (dt ,dzk ) (1)
+∞∑
k=1
∫|zk |Z≥1
ψk(u(x , t−), zk
)πk (dt ,dzk ) in O × (0,T )
with the incompressibility condition
∇ · u = 0 in O × (0,T ), (2)
the Dirichlet boundary and the initial conditions
u = 0 on ∂O × (0,T ), (3)u(x ,0) = u0(x) in O.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 3 / 37
Description of the Model
u = u(x , t) and p = p(x , t) denote the velocity and pressure fieldsg = g(x , t) : O × (0,T )→ Rn is a random external forceO ⊂ Rn,n = 2,3 is an open bounded domain with smoothboundary ∂OOne may also require the far-field conditionu(x , t)→ 0 as |x | → ∞ if O is unbounded.Parameter ν is the kinematic viscosityW (·) is a Hilbert-space valued Wiener process is independent ofthe Poisson random measure πk (dt ,dz), for all k = 1,2 · · · .
Motivation to Stochastic Forces: The noises chosen here may beconsidered to model exogeneous forces such as structural vibrations,magnetic fields and other environmental disturbances.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 4 / 37
Semi-martingale Problem Formulation
DefineH := v ∈ L2(O;Rd ) : ∇ · v = 0 v · n|∂O = 0 (4)
andV := v ∈ H1
0(O;Rd ) : ∇ · v = 0. (5)
Let PH : L2(O)→ H be the Helmholtz-Hodge(orthogonal) projection.Define the Stokes operator
A : D(A)→ H with Av = −PH∆v, (6)
where D(A) = v ∈ H10(O) ∩H2(O) : ∇ · v = 0.
The nonlinear operator
B : D(B) ⊂ H× V→ H with B(u,v) = PH(u · ∇v). (7)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 5 / 37
Semi-martingale Formulation continued...
Under the Helmholtz-Hodge projection PH : L2(O)→ H, the system
(1) can be written in semi-martingale formulation as
u(t) = u0 +
∫ t
0
(− νAu(s)− B(u(s)) + g(s) (8)
+∞∑
k=1
∫|zk |Z≥1
ψk(s,u(s), zk
)µk (dzk )
)ds + Mt in in Ω
u(0) = u0 in H
where B(u) = B(u,u), g ∈ L2(0,T ;V′) and the martingale
Mt =
∫ t
0σ(s,u(s))dW (s) +
∫ t
0
∞∑k=1
∫|zk |Z<1
φk(s,u(s−), zk
)πk (ds,dzk )
+
∫ t
0
∞∑k=1
∫|zk |Z≥1
ψk(s,u(s−), zk
)πk (ds,dzk ). (9)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 6 / 37
Assumptions on the Gaussian Noise Coefficient
Let Q be a positive, symmetric and trace class operator on H.Let (Ω,F ,P) be a probability space with filteration Ftt≥0.
DefinitionA stochastic process W(t) : 0 ≤ t ≤ T is said to be a H-valuedFt-adapted Wiener process with covariance operator Q if for eachnon-zero h ∈ H, |Q1/2h|−1(W(t),h) is a standard one-dimensionalWiener process and for each h ∈ H, (W(t),h) is a Ft−martingale.
The stochastic process W(t) : 0 ≤ t ≤ T is a H-valued Wienerprocess with covariance Q iff for arbitrary t , the process W(t) can beexpressed as W(t) =
∑∞k=1√λkβk (t)ek ,
βk (t), k ∈ N are independent one dimensional Brownian motionson (Ω,F ,P)
ek are the orthonormal basis functions of H. (Kallianpur andXiong, 1995)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 7 / 37
Assumptions on the Gaussian Noise Coefficient
Let LHS be the space of all bounded linear operators S : H→ H suchthat
∑∞k=1 |SQ1/2ek |2 <∞, where ek is an orthonormal basis in H.
The norm on LHS is given by tr(SQS∗) := ‖S‖2LHS.
The Gaussian noise coefficient σ satisfies[H1] For all t ∈ [0,T ], there is a positive constant N1 such that
‖σ(t ,u)− σ(t ,v)‖2LHS≤ N1|u− v|2, ∀u,v ∈ H. (10)
[H2] For all t ∈ [0,T ], there is a positive constant N2 satisfying thegrowth condition
‖σ(t ,u)‖2LHS≤ N2(1 + |u|2), ∀u ∈ H. (11)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 8 / 37
Jump Measure
Let Z be a separable Banach space. Let L(t)t≥0 be a Z−valued Lévyprocess with jump ∆L(t) := L(t)− L(t−) at t ≥ 0. Then
π([0, t ], Γ) = #s ∈ [0, t ] : ∆L(s) ∈ Γ, where Γ ∈ B(Z\0)
is the Poisson random measure or jump measure associated to theLévy process L(t) (Applebaum, 2002).
Compensated Poisson random measure:πk (dt ,dzk ) = πk (dt ,dzk )− dtµk (dzk ), for all k = 1,2 · · ·
Intensity measure or Lévy measure: µk (·) = E(πk (1, ·))
Compensator of the Lévy process L(t) : dtµk (dzk )
Lesbegue measure: dt .
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 9 / 37
Jump Measure
Compound Poisson process:( ∫ t0
∫|zk |Z≥1 ψk
(u(s−), zk
)πk (ds,dzk ), t ≥ 0, k ≥ 1
)Compensated Poisson integral:( ∫ t
0
∫|zk |Z<1 φk
(u(s−), zk
)πk (dt ,dzk ) t ≥ 0, k ≥ 1
).
The intensity measure µk (·) on Z satisfies µk (0) = 0, for allk = 1,2, · · · . Assume that µk (·) satisfies
∞∑k=1
∫Z
(1 ∧ |zk |2)µk (dzk ) < +∞
and∞∑
k=1
∫|zk |Z≥1
|zk |pµk (dzk ) < +∞, ∀p ≥ 1.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 10 / 37
Assumptions on the Jump Noise CoefficientsTaking the conditions of the Lévy measure into account, we state thefollowing assumptions:
[H3] For all t ∈ [0,T ], there is a positive constant N2 such that for allu,v ∈ H∞∑
k=1
∫|zk |Z<1
|φk (u, zk )− φk (v, zk )|2µk (dzk ) (12)
+∞∑
k=1
∫|zk |Z≥1
|ψk (u, zk )− ψk (v, zk )|2µk (dzk ) ≤ N2|u− v|2
[H4] For all t ∈ [0,T ], there is a positive constant N3 satisfiying∞∑
k=1
∫|zk |Z<1
|φk (u, zk )|pµk (dzk ) (13)
+∞∑
k=1
∫|zk |Z≥1
|ψk (u, zk )|pµk (dzk ) ≤ N3(1 + |u|p), ∀u ∈ H, p ≥ 1.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 11 / 37
Martingale Problem
Let Ω = D(0,T ;V′)J ∩ L∞(0,T ;H)w∗ ∩ L2(0,T ;V)w be the path spacewith ω ∈ Ω denoting a generic point in Ω, where D(.; .) is the class ofcàdlàg functions from [0,T ] into V′
Càdlàg functions are right continuous and have left limits at any pointt ∈ [0,T ]. Let F be the σ-algebra of Borel subsets of Ω.(Parthasarathy 1967, Ethier and Kurtz 1986)
Let ξ be the mapping from [0,T ]× Ω→ V′ defined by ξ(t , ω) := ω(t)and Ft = σξ(s, ω) : 0 ≤ s ≤ t for all t ∈ [0,T ].
Then the measure P such that P u−1 = P is the law of the processesu, which is defined on (Ω, F , Ft ).
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 12 / 37
Generator of the ProcessLet u(t) be the Itô-Lévy process defined on a complete probabilityspace (Ω,Ft ,P) with transition semigroup Tt . Then the formalgenerator L of u(t) defined on functions f (·) : H→ R by
L f = limt↓0
Tt f − ft
for each f ∈ D(L ),
where D(L ) := f : H→ R such that limt↓0Tt f−f
t exists . Forf ∈ D(L ), the formal generator L f is given by (Applebaum 2002,Peszat and Zabczyk 2007)
L f (u) = −⟨νAu + B(u)− g,
∂f∂u⟩
+12
tr(σ(t ,u)Qσ∗(t ,u)
∂2f∂u2
)(14)
+∞∑
k=1
∫|zk |Z<1
f (u + φk (u, zk ))− f (u)−
⟨φk (u, zk ),
∂f∂u⟩µk (dzk )
+∞∑
k=1
∫|zk |Z≥1
f (u + ψk (u, zk ))− f (u)
µk (dzk ), ∀u ∈ D(A).
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 13 / 37
Martingale Problem
DefinitionLet L f (·) be the generator as defined in (14). Then given an initialmeasure P on H, a solution to L f -martingale problem is a probabilitymeasure P : B(Ω)→ [0,1] on (Ω, F , Ft ) such that Pξ(0) = u0 = 1and the process
Mft := f (ξ(t))− f (ξ(0))−
∫ t
0L f (ξ(s))ds, with f ∈ D(L )
is a R-valued locally square integrable (Ω, F , Ft ,P)-local càdlàgmartingale (Stroock and Varadhan, 1969).
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 14 / 37
Existence of Martingale Solutions
Theorem (Existence Result)
Let O be an open domain in Rd ,d = 2,3. Then for a given initialprobability measure P on H with
∫H |x |
2dP(x) <∞, there exists amartingale solution to the equation (8).
Metivier, 1988 : The existence of a solution to the martingale problemis equivalent to that of a weak solution to the stochastic differentialequations.
Yamada-Watanabe, 1971: The existence of a weak solution togetherwith the pathwise uniqueness property imply the uniqueness in law.
Pathwise Uniqueness: if U(t) = u(t),u0,Q, πk ; t ≥ 0, k ≥ 1 andU′(t) = u′(t),u′0,Q′, π′k ; t ≥ 0, k ≥ 1 are any two solutions defined ona same probability space (Ω,F ,Ft ,P), then u0 = u′0,Q = Q′ andπk = π′k imply PU(t) = U′(t); t ≥ 0, k ≥ 1 = 1.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 15 / 37
Uniqueness of Martingale Solutions
Theorem (Pathwise Uniqueness)
Let u,v ∈ Ω be the two paths defined on a same probability space(Ω,F ,Ft ,P) with same Q-Wiener process W and Poisson measureπk , k = 1,2 · · · satisfying the system (8). Then there exist positiveconstants Cν and C such that
E(|u(t)− v(t)|2 exp
− Cν
∫ t
0‖v(s)‖4/(4−d)ds
)(15)
≤ exp(CT )E|u(0)− v(0)|2.
If the initial data u(0) = v(0) = u0, then(i) For d = 2, the solution u is pathwise unique, that is,
u(t) = v(t),P-a.s.(ii) For d = 3, the solution u is pathwise unique under the additional
condition E∫ T
0 ‖v(s)‖4ds <∞.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 16 / 37
Quadratic Variation and Meyer Process
Let Mt ∈M 2loc(H). Then there exist increasing process JMKt and
Mt with M0= JMK0 = 0 such that
JMKt =
∫ t
0
∞∑k=1
∫|zk |Z<1
(φk ⊗ φk )πk (dzk ,ds) (16)
+
∫ t
0
∞∑k=1
∫|zk |Z≥1
(ψk ⊗ ψk )πk (dzk ,ds) +
∫ t
0σQσ∗(s,u(s))ds
and
Mt =
∫ t
0
∞∑k=1
∫|zk |Z<1
(φk ⊗ φk )µk (dzk ,ds) (17)
+
∫ t
0
∞∑k=1
∫|zk |Z≥1
(ψk ⊗ ψk )µk (dzk ,ds) +
∫ t
0σQσ∗(s,u(s))ds.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 17 / 37
Finite Dimensional Galerkin Approximation
Let e1,e2, · · · be the orthonormal basis in H included in V with eachei ∈ D(A), i = 1,2 · · · .
Let Πn be the orthogonal projection in V onto the spaceVn := spane1,e2, · · · ,en.
Then un(t) := Πnu(t) =∑n
i=1(u(t),ei)ei solves the following finitedimensional Navier-Stokes equations
dun(t) = (−νΠnAun(t)− ΠnB(un(t)) + Πng(t))dt
+n∑
k=1
∫|zk |Z≥1
ψnk(un(t), zk
)µk (dzk )dt + dMn
t (18)
where the local martingale Mnt is the finite dimensional approximations
obtained from σn = Πnσ,Wn = ΠnW, φnk = Πnφk and ψn
k = Πnψk .
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 18 / 37
A Priori Estimates
Since the càdlàg process un solves the system (18) in Vn with initialcondition Πnu0 and Vn ⊂ H ⊂ V′, the laws Pn of these finitedimensional approximations are defined on D(0,T ;V′) and
Mft = f (ξ(t))− f (ξ(0))−
∫ t
0L f (ξ(s))ds (19)
is a Hn-valued locally square integrable Pn-local càdlàg martingale.
TheoremLet g be in L2(0,T ;V′) and
( ∫H |x |
2dP(x))<∞. Assume that Mf
tdefined in (19) is a Hn-valued square integrable Pn-local càdlàgmartingale. Then
EPn |ξ(t)|2 + νEPn∫ t
0‖ξ(s)‖2ds ≤ C
(E|ξ(0)|2 +
1ν
∫ T
0‖g(t)‖2V′dt
). (20)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 19 / 37
Tightness of Measures
Radon probability measures Pn on a completely regular topologicalspace E is said to converge weakly to a Radon probability measure Pif
limn→∞
∫Ω
FdPn =
∫Ω
FdP, ∀F ∈ Cb(Ω).
Tightness condition needed for the Prokhorov-Varadarajan theorem isthat for every ε > 0 there exists a compact set Kε ⊂ E such thatsupn Pn(E\Kε) ≤ ε:
TheoremIf the bounded Radon measures Pn on a completely regulartopological space E satisfy supn Pn(E) <∞ and Pn are tight, then themeasures Pn are relatively weakly compact in the set of boundedpositive Radon measures.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 20 / 37
Tightness of Probability Measures continued...
Taking the path space Ω = D(0,T ;V′)J ∩ L∞(0,T ;H)w∗ ∩ L2(0,T ;V)winto account, we define
T1 := D(0,T ;V′)J , T2 := L∞(0,T ;H)w∗ ,
T3 := L2(0,T ;V)w , T4 := L2(0,T ;H)s.
Note that the spaces D(0,T ;V′)J , L∞(0,T ;H)w∗ and L2(0,T ;V)w arecompletely regular and continuously embedded in L2(0,T ;V′)w . LetT = T1 ∨ T2 ∨ T3 ∨ T4. The space Ω endowed with the supremumtopology T is a Lusin space (Metivier, [5]).
Theorem (Tightness of Pn)
The sequence of probability measures Pn defined on (Ω, Ft ) with thesupport in L∞(0,T ;H)w∗ ∩ L2(0,T ;V)w is tight on D(0,T ;V′)J .
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 21 / 37
Some Steps in the Main Result: Domain O is Bounded
For any f ∈ D(L ), we need to show that
EP(
Ψ(·)(Mft −Mf
s))
= 0, (21)
for s < t and Ψ ∈ Cb(Ω) is Fs measurable.
To achieve this, we can eventually use EPn(
Ψ(·)(Mft −Mf
s))
= 0,∀n,
the tightness of Pn in Ω and P(L∞(0,T ;H) ∩ L2(0,T ;V)) = 1.
However they are not sufficient to conclude (21)!!!
Note that M ft is not continuous on Ω.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 22 / 37
Tightness of Pn in L2(0,T ;H)
Note that the embedding V → H → V′ is compact.
We have the following compactness result.
Lemma (Metivier, 1988)
Let K be a subset of L2(0,T ;H) which is included in a compact set ofL2(0,T ;V′) and supu∈K
∫ T0 ‖u(t)‖2Vdt <∞. Then K ⊂ L2(0,T ;H) is
relatively compact.
The compactness result along with tightness of Pn inD(0,T ;V′),L2(0,T ;V), we get the tightness in L2(0,T ;H) with thestrong topology T4.
This proves that Pn are tight in the Lusin space Ω ∩ L2(0,T ;H)endowed with the supremum topology T .
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 23 / 37
Domain O ⊂ R2 is Bounded : Minty Stochastic Lemma
How to prove the continuity of M ft on Ω?
For 2D-case, we use the Minty Stochastic Lemma (Viot 1976, Metivier1988, Sritharan 2000).
Local monotonicity of the operator Θ(u) + λu, whereΘ(u) := νAu + B(u) and λ > 0
Lemma (Local Monotonicity)For a given ρ > 0 and p > d , let Br denote the ballBr = v ∈ V : ‖v‖Lp(O) ≤ ρ. Then for any u ∈ V,v ∈ Br andw = u− v, there exists a λ > 0 such that the operator Θ(u) + λu ismonotone in Br :
〈Θ(u)−Θ(v),w〉+ λ|w|2 ≥ ν
2‖w‖2, (22)
where λ denotes the constant Cp,d ,νρ2p/(p−d).
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 24 / 37
Brief Sketch of the Proof (Existence)
Define the image of Pn under the map ξ → (ξ,Θ(ξ)) asPn(S) := Pnω ∈ Ω; (ω,Θ(ω)) ∈ S, for S ∈ B(Ω× L2(0,T ;V′)w ).
On Ω := Ω× L2(0,T ;V′)w , consider the canonical right-continuousfiltration Gt and canonical processes ξ(t , ω, v) = ω(t) andχ(t , ω, v) = v(t).
The product measures Pn are tight on Ω and satisfy the following:[N1] Pn(ω,v) ∈ Ω : Θ(ω) = v = 1.
[N2] For every (ω,v) ∈ Ω and for any f ∈ D(L ), the process Mft on Ω
defined byMf
t (ω,v) := f (ξ(t , ω, v))− f (ξ(0, ω, v))−∫ t
0 L f (s, ξ(s, ω,v))ds is aR-valued locally square integrable (Ω, Gt , Pn)-local càdlàgmartingale.
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Brief Idea of Minty Stochastic Lemma
Lemma (Minty Stochastic Lemma)
Let Pn be the sequence of probability measures on Ω satisfying [N1]and [N2]. Assume that the measures Pn converge weakly to a measureP on Ω such that [N2] holds for P. Then [N1] also holds true for P, thatis, P(ω,v) ∈ Ω : Θ(ξ(ω,v)) = χ(ω,v) = 1.
Let ζ(·, ·, t) be the continuous function of the formζ(ω,v, t) =
∑ki=1 ϕi(ω,v, t)ei with ei ∈ V where ϕi(·, ·, t) are continuous
in Ω with paths in L2(0,T ).
For each given ζ(·, ·, t) and ρ(t) := 27ν3
∫ t0 ‖ζ(s)‖4L4(O)
ds, let us define
Ψ(ω,v) := 2∫ T
0e−ρ(t)〈χ(ω,v, t)−Θ(ζ(ω,v, t)), ξ(ω, t)− ζ(ω,v, t)〉dt
+
∫ T
0e−ρ(t)ρ(t)|ξ(ω, t)− ζ(ω,v, t)|2dt . (23)
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 26 / 37
Brief Idea of Minty Stochastic Lemma ContinuedBut in view of [N1] and Local Monotonicity Result, we get∫
ΩΨ(ω,v)d Pn(ω,v) ≥ 0.
We decompose Ψ into Ψ1 and Ψ2 as follows
Ψ1=2∫ T
0e−ρ(t)〈χ(t), ξ(t)〉dt +
∫ T
0e−ρ(t)ρ(t)|ξ(t)|2dt −
∫ T
0e−ρ(t)d [M]t
andΨ2 = −2
∫ T
0e−ρ(t)〈χ(t)−Θ(ζ(t)), ζ(t)〉dt
−2∫ T
0e−ρ(t)〈Θ(ζ(t)), ξ(t)〉dt +
∫ T
0e−ρ(t)ρ(t)|ζ(t)|2dt
−2∫ T
0e−ρ(t)ρ(t)〈ζ(t), ξ(t)〉dt +
∫ T
0e−ρ(t)d [M]t ,
where M is the local martingale. Then we will show that∫Ω
Ψ(ω,v)d P(ω,v) ≥ 0.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 27 / 37
Martingale Solution in 2D caseMinty Stochastic Lemma shows that Φ(Mf
t − Mfs) ∈ C(Ω), for any
Gs-measurable function Φ ∈ Cb(Ω).
Lemma
Let Ω be a Lusin space and Pn be the sequence of probabilitymeasures on Ω converging weakly to a measure P as n→∞. Letg ∈ C(Ω) and supn EPn
[|g|1+ε] ≤ C for some ε > 0. ThenEPn
(g)→ EP(g) as n→∞.
By moment estimates, we arrive at EPn |Mft |2 <∞ and hence
EPn |Mft |1+ε <∞, ε > 0
Eventually we have shown that
limn→∞
EPn(
Ψ(Mft −Mf
s))
= EP(
Ψ(Mft −Mf
s))
;
butlim
n→∞EPn(
Ψ(Mft −Mf
s))
= 0
establishes that P is a solution of the martingale problem.K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 28 / 37
Domain O ⊂ R3 is Bounded
By Ladyzhenskaya’s inequalities, the nonlinear term B(·) satisfies
‖B(v)‖V′(O) ≤ ‖v‖2L4(O) ≤ L|v|2−(d/2)‖v‖d/2, ∀v ∈ V, d = 2,3.
When d = 3, by moment estimates, it is clear that Θ(·) exists only inthe space L4/3(0,T ;V′).
It appears that Minty Stochastic Lemma does not hold for 3D-case.
So we prove the continuity of M ft on Ω in the supremum topology T as
follows:
LemmaLet f (u) := ϕ(〈e1,u〉, 〈e2,u〉, · · · , 〈em,u〉), u ∈ H be the tame functionwith ϕ(·) ∈ C∞0 (Rm) and ek ∈ D(A), k = 1, · · · ,m. If un → u in Ω forthe Lusin topology T , then Mf
t (un)→ Mft (u) on Ω,∀t ∈ [0,T ].
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 29 / 37
Domain O is Unbounded:Case of Non-compact Embeddings V → H → V′
In this case, tightness properties of Pn established on D(0,T ;V′) andL2(0,T ;H) are not longer valid. The measures Pn are tight in the weaktopology of H, namely Hw .
Theorem (Tightness of Pn)
The sequence of probability measures Pn on (Ω, Ft ) with the supportin L∞(0,T ;H) ∩ L2(0,T ;V) is tight on D(0,T ;Hw ).
Since we don’t have the compactness result, we can only establish thetightness in D(0,T ;Hw ) ∩ L2(0,T ;V) and L2(0,T ;Hw ). This is notsufficient to prove the continuity of (Mf
t −M fs)s≤t on Ω!!!! either in 2D or
3D case.
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Some Remarks on Unbounded Domain
If the noise terms are additive, that is, σ(t ,u) = σ(t),φk (u, zk ) = φk (zk ) and ψk (u, zk ) = ψk (zk ), k = 1,2, · · · , the proof ofMinty Stochastic Lemma, and hence the existence of martingalesolutions in 2D-case, still hold.
If the noise terms are multiplicative to get the stochastic Minty-Browdertechnique to work, we need to make stronger assumptions on theGaussian and Jump noise coefficients.
However, one can cut the unbounded domain O into a sequence ofbounded domains Oi , i = 1,2 · · · and construct martingale solutionsPi , i = 1,2, · · · for the SNSEs (1) in each of these bounded domains Oiand then show that in the limit the martingale solution P for O exists.
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Pathwise Uniqueness of SolutionsLet u,v ∈ Ω be the two paths defined on a same probability space(Ω,F ,Ft ,P). Define
w = u− v, σ = σ(t ,u)− σ(t ,v), φk = φk (u(t−), zk )− φk (v(t−), zk )
andψk = ψk (u(t−), zk )− ψk (v(t−), zk ), k = 1,2 · · · .
Then, we have the following semimartingale
w(t)=w(0)−∫ t
0
([Θ(u(s))−Θ(v(s))]−
∞∑k=1
∫|zk |Z≥1
ψkµk (dzk ))
ds + Mt ,
where Θ(u) = νAu + B(u) and
Mt=
∫ t
0σdW(s) +
∫ t
0
( ∞∑k=1
∫|zk |Z<1
φk πk (ds,dzk ) +∞∑
k=1
∫|zk |Z≥1
ψk πk (ds,dzk ))
ds.
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Pathwise Uniqueness of Solutions
By Ladyzhenskaya’s inequalities,
|〈B(u)− B(v),w〉| ≤ ‖w‖2L4(O)‖v‖ ≤ C|w|(4−d)/2‖w‖d/2‖v‖
≤ ν
2‖w‖2 + Cν‖v‖4/(4−d)|w|2, for d = 2,3.
Define ρ(t) := 2Cν
∫ t0 ‖v(s)‖4/(4−d)ds,d = 2,3. Apply Itô formula to
e−ρ(t)|w(t)|2 to get
e−ρ(t)|w(t)|2 + ν
∫ t
0e−ρ(s)‖w(s)‖2ds ≤ |w(0)|2
+(1 + CµN2)
∫ t
0e−ρ(s)|w(s)|2ds
+
∫ t
0e−ρ(s)d(trJMKs) + 2
∫ t
0e−ρ(s)(w(s),dMs).
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 33 / 37
Pathwise Uniqueness of Solutions
Let τm be the stopping time localizing the martingale. Then
E[e−ρ(t∧τm)|w(t ∧ τm)|2] ≤ E|w(0)|2 + C∫ t
0E[e−ρ(s∧τm)|w(s ∧ τm)|2]ds.
Using Chebychev’s inequality and energy estimates, we can argue thatτm → T as m→∞.
Thus the proof can be completed by Gronwall’s inequality.
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References I
D.Applebaum, Lévy Processes and Stochastic Calculus,Cambridge University Press, Second Edition, Cambridge, 2009.
Z.Dong and J.Zhai, Martingale solutions and Markov selection ofstochastic 3D Navier-Stokes equations with jump, J. DifferentialEquations, 250(2011), 2737-2778.
S. N. Ethier and T. G. Kurtz, Markov Processes Characterizationand Convergence, John Wiley and Sons, Inc., New York, 1986.
F.Flandoli and D.Gatarek, Martingale and stationary solutions forstochastic Navier-Stokes equations, Probab. Theory RelatedFields, 102 (1995), 367-391.
M.Metivier, Stochastic Partial Differential Equations in InfiniteDimensional Spaces, Scuola Normale Superiore, Pisa, 1988.
K. R. Parthasarathy, Probability Measures on Metric Spaces,Academic Press, New York, 1967.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 35 / 37
References II
S.Peszat and J.Zabczyk, Stochastic Partial Differential Equationswith Lévy Noise, Cambridge University Press, Cambridge, 2007.
S.S.Sritharan, Deterministic and stochastic control ofNavier-Stokes equation with linear, monotone, andhyperviscosities, Appl. Math. Optim., 41 (2000), 255-308.
D.Stroock and S.R.S.Varadhan, Multidimensional DiffusionProcesses, Springer-Verlag, New York, 1979.
T.Yamada and S.Watanabe, On the uniqueness of solutions ofstochastic differential equations, J. Math. Kyoto Univ., 11 (1971),155-167.
K.Sakthivel (IIST) Navier-Stokes Equations with Levy Noise IISER, Trivandrum 36 / 37
Thank You!
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