on the solvability of stochastic navier-stokes equations ... · k. sakthivel (joint with s.s....

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


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.


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.


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)


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)


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)


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)


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 .


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.


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.


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 ).


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).


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).


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.


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 <∞.


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.


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 .


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)


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.


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 .


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.


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 .


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).


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.


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)


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.


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 ].


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.


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.


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.


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.


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.


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.


Thank You!


on the solvability of stochastic navier-stokes equations ... · k. sakthivel (joint with s.s....

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