Stochastic partial differential equations

by Martina Hofmanová

TU Berlin, SS 2016

Table of contents

Info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Recommended literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Fluids in continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Lebesgue–Bochner spaces Lp(I;X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Spaces with zero divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Stochastic integration in infinite dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Pathwise vs. martingale solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.1 Pathwise uniqueness vs. uniqueness in law . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Preliminaries from functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.1 Spectral decomposition of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Fractional powers of some unbounded operators . . . . . . . . . . . . . . . . . . . . . . . 14

7 Stochastic Navier–Stokes equations - existence . . . . . . . . . . . . . . . . . . . . . . 15

7.1 Step 1: Faedo-Galerkin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Step 2: Uniform energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 Step 3: Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.3.1 Another uniform estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.3.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.4 Step 4: Skorokhod & passage to the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Transition semigroup and invariant measures . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Stochastic NSE - stationary martingale solutions . . . . . . . . . . . . . . . . . . . . . 30

Info

hofmanov@math.tu-berlin.de

http://page.math.tu-berlin.de/~hofmanov/

https://isis.tu-berlin.de/course/view.php?id=6893

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Recommended literature

• G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions, Encyclopedia Math.Appl., vol. 44, Cambridge University Press, 1992.

• I. Karatzas, S. E. Shreve: Brownian motion and stochastic calculus, Springer 1991.

• F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokesequations, Probab. Theory Related Fields 102 (3) (1995) 367-391.

• M. Hofmanová: Degenerate parabolic stochastic partial differential equations, Stoch. Pr.Ap. 123 (12) (2013) 4294-4336.

• A. Debussche: Ergodicity results for the stochastic Navier-Stokes equations: an introduction,Lecture Notes in Math., Springer, 2013.

• G. Da Prato, J. Zabczyk: Ergodicity for infinite dimensional systems, London MathematicalSociety Lecture Notes, 229, Cambridge University Press, 1996.

• M. Renardy, R. C. Rogers: An Introduction to Partial Differential Equations, Springer 2004.

• H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer2011.

1 Fluids in continuum mechanics

Fluid (liquid or gas) - substance that continually deforms (flows) under applied shear stress

Liquids form a free surface (that is, a surface not created by the container) while gases do not

Fluids display properties such as:

• not resisting deformation, or resisting it only slightly (viscosity), water vs. honig

• the ability to flow (also described as the ability to take on the shape of the container)

In contrast, solids can be subjected to shear stresses, they do not flow to take the shape of thecontainer.

Fluid dynamics - the natural science of fluids in motion

Aerodynamics - gases in motion

Hydrodynamics - liquids in motion

Applications

• calculating forces and moments on aircraft, eg Formula 1 racecars, computational fluiddynamics

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• determining the mass flow rate of petroleum through pipelines

(Secondary oil recovery stage)

• applications in biology and life sciences (blood flow, artificial heart before transplantationetc.)

• predicting weather patterns, global climate modelling, prediction of global warming

• understanding nebulae in interstellar space (interstellar cloud of dust, hydrogen, helium andother ionized gases)

(The Veil Nebula - small section of the expanding remains of a massive star that explodedabout 8 000 years ago)

• modelling fission weapon detonation (type of nuclear bomb)

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• some of its principles are even used in traffic engineering, where traffic is treated as acontinuous fluid, and crowd dynamics

(The Judge Harry Pregerson Interchange, LA)

⇒ Mathematical understanding needed for modelling and predictions!

Mathematical description

• a system of partial differential equations derived from basic physical principles = the systemof Navier–Stokes equations

• foundational axioms of fluid dynamics

conservation of mass - the mass is neither created nor destroyed

conservation of linear momentum (Newton’s Second Law of Motion) - product ofmass and velocity, force of objects in motion

− a heavy truck moving rapidly = large momentum, it takes long to make itmove or to stop it

− the total momentum is constant (eg Newton’s cradle)

conservation of energy (First Law of Thermodynamics) - total energy of an isolatedsystem remains constant

− energy can neither be created nor destroyed

− rather, it transforms from one form to another: chemical energy can be con-verted to kinetic energy in the explosion of a stick of dynamite

(Newton’s cradle - stainless stellballs, conservation of energy and momentum, but friction slows it down, also gravity acts,it is nto a closed system)

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Compressibility

• every fluid is compressible to some extent = changes in pressure or temperature causechanges in density

• often these changes negligible - incompressible fluid flow

liquids (water, oil) cannot be compressed much when you push down on them in anenclosed container

the molecules of liquids are tightly packed together and they can’t squeeze togetherany tighter

• compressible fluids - gases

when you push down on a gas (eg air) you can compress it within the container

the gas molecules are far apart from each other, so there is plenty of room for themto come closer together

Turbulence

• flow characterized by recirculation, eddies, and apparent randomness - motivation for sto-chasticity

• smoke rising from a cigarette

• flow over a golf ball - If the golf ball were smooth, the boundary layer flow over the frontof the sphere would be laminar at typical conditions. However, the boundary layer wouldseparate early, as the pressure gradient switched from favorable (pressure decreasing in theflow direction) to unfavorable (pressure increasing in the flow direction), creating a largeregion of low pressure behind the ball that creates high form drag. To prevent this fromhappening, the surface is dimpled to perturb the boundary layer and promote transitionto turbulence. This results in higher skin friction, but moves the point of boundary layerseparation further along, resulting in lower form drag and lower overall drag.

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• clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing(the blurring of images seen through the atmosphere); clear-air turbulence - a mass of aismoving at a particular speed meets another one moving at a different speed

• the oceanic and atmospheric mixed layers and intense oceanic currents, eddies - spiral shapes

(Oyashio and Kuroshio currents collide - phytoplankton becomes concentrated along theboundaries)

- The relation of the Navier–Stokes system to the phenomenon of hydrodynamic turbulence isregarded as one of the most fascinating problems in fluid mechanics.

Unsolved problems in physics and mathematics:

• physics: Is it possible to make a theoretical model to describe the statistics of a turbulentflow (in particular, its internal structures)?

• mathematics - one of the Millennium Prize Problems stated by the Clay MathematicsInstitute in 2000: Under what conditions do smooth solutions to the Navier–Stokes equationsexist?

2 Navier–Stokes equations

The Navier-Stokes system for incompressible viscous fluids reads as follows

∂tu+u·∇u+∇p= ν∆u+ f , div u=0, u|∂D=0, u(0)=u0, (1)

• x ∈ D ⊂ RN , N = 2, 3, bounded open domain with regular boundary, finite time intervalt∈ [0, T ]

• u: [0, T ]×D→RN fluid velocity (unknown)

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• p: [0, T ]×D→R denotes the pressure (unknown)

• f : [0, T ]×D→RN is an external force (given)

• u·∇u is the so called convective term, i.e. a quadratic nonlinearity of lower order

• ν > 0 is the viscosity (given)

• div,∇,∆ differential operators

Equation (1) is typically not satisfied in a strong sense, as equality of functions.

Derivatives exist only in distributional sense ⇒ weak solutions.

The equation is understood in the sense of distributions. Let ϕ∈ [Cc∞(D)]N=Cc∞(D) be divergence-free, i.e. divϕ=0. Test (1) against ϕ gives

⟨u(t), ϕ⟩+!

0

t

⟨u·∇u, ϕ⟩ds+!

0

t

⟨∇p, ϕ⟩ds= ⟨u0, ϕ⟩+ ν

!

0

t

⟨∆u, ϕ⟩ds+!

0

t

⟨f , ϕ⟩ds

and integration by parts

⟨u(t), ϕ⟩−!

0

t

⟨u⊗u,∇ϕ⟩ds−!

0

t

⟨p,divϕ⟩ds= ⟨u0, ϕ⟩− ν!

0

t

⟨∇u,∇ϕ⟩ds+!

0

t

⟨f , ϕ⟩ds

hence the pressure term vanishes and we obtain

⟨u(t), ϕ⟩ −!

0

t

⟨u⊗u,∇ϕ⟩ds= ⟨u0, ϕ⟩ − ν!

0

t

⟨∇u,∇ϕ⟩ds+!

0

t

⟨f , ϕ⟩ds.

Important observation: working with div-free test function eliminates the pressure from theequation and simplifies the situation. The pressure can be then reconstructed a posteriori after aweak solution is constructed.

3 Function spaces

Lebesgue spaces Lp(D), p∈ [1,∞).

Sobolev spaces W k,p(D), k ∈N0, p∈ [1,∞].

Fractional Sobolev spacesWm,p(D), m!0, p∈ [1,∞], Sobolev-Slobodeckij spaces: let m= ⌊m⌋+θthen

∥f ∥Wm,p= ∥f ∥W ⌊m⌋,p+ sup|α|=⌊m⌋

"!

D

!

D

|Dαf(x)−Dαf(y)|p|x− y |θp+N

dxdy

#1

p

.

Sobolev spaces of functions vanishing at the boundary W01,2(D)=Cc

∞(D) ∥·∥W1,2.

Negative Sobolev spaces W−m,p(D)= (W0m,q(D))∗, where 1

p+ 1

q=1, p, q ∈ (1,∞).

3.1 Lebesgue–Bochner spaces Lp(I;X)

We are interested in functions v: I ⊂R→X, where I is an interval and X is a Banach space.

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Definition 1. A function v: I→X is called a simple function, if its range is finite, i.e. there existc1, ..., ck∈X and disjoint intervals I1, ..., Ik∈ I, such that v(t)=

$i=1k ci1Ii(t).

A function v:I→X is called strongly measurable if there exists a sequence of simple functions (vn)such that limn→∞ ∥vn(t)− v(t)∥X=0 for a.e. t∈ I.

Lemma 2. Let v be strongly measurable. Then ∥v(·)∥X:I→R is measurable in the Lebesgue sense.

Definition 3. A function v: I → X is Bochner integrable, if there exists a sequence of simplefunctions (vn) such that

• limn→∞ ∥vn(t)− v(t)∥X=0 for a.e. t∈ I,

• limn→∞%I∥vn(t)− v(t)∥X dt=0.

If J ⊆ I and v is Bochner integrable over I, then

!

Jv(t) dt= lim

n→∞

!

I1J(t) vn(t) dt= lim

n→∞

&

i=1

kn

cin|Iin∩ J |,

where the approximate simple functions (vn) were given by

vn=&

i=1

kn

cin 1Iin.

Lemma 4. If v is Bochner integrable over I, then

''''!

Iv(t) dt

''''X

≤!

I∥v(t)∥X dt.

Definition 5. Let X be a Banach space, p ∈ [1,∞], I ⊂R. We denote by Lp(I;X) the set of allstrongly measurable functions v: I→X such that

• p∈ [1,∞)!

I∥v(t)∥X

p dt <∞,

• p=∞

ess supt∈I∥v(t)∥X<∞.

Remark 6.

1. Lp(I ;X) are Banach spaces equipped with the norms

∥v∥Lp(I;X)=(!

I∥v(t)∥X

p dt

)1

p, p∈ [1,∞),

∥v∥L∞(I;X)= ess supt∈I

∥v(t)∥X.

2. If I is a bounded interval then Lp(I ;X)∈Lq(I ;X), 1" q" p.

3. Let p∈ [1,∞) and let X be separable. Then Lp(I ;X) is separable.

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3.2 Spaces with zero divergence

For p∈ [1,∞) we will consider spaces of the type

W0,div1,p (D) = v ∈ (W0

1,p(D))N; div v=0

which coincides with the closure

v ∈ (Cc∞(D))N; div v=0 ∥·∥W1,p.

For p=2 we set

L0,div2 (D)= v ∈ (Cc∞(D))N; div v=0 ∥·∥L2.

We want to write (L2(D))N =L0,div2 (D)⊗P and characterize the orthogonal complement P .

Theorem 7. It holds true that

L0,div2 (D)= v ∈ (L2(D))N; div v=0 inD′(D); γ(u)= 0, 1

P =(L0,div2 (D))⊥= v ∈ (L2(D))N; v=∇π ,π ∈W 1,2(D).

Helmholtz decomposition: v ∈ (L2(D))N can be decomposed into sum of a divergence-free (sole-noidal) vector field vdiv∈L0,div2 (D) and an irrotational (gradient) part vgrad∈ (Ldiv

2 (D))⊥

v= vdiv+ vgrad= vdiv+∇π.

Helmholtz projection: orthogonal projection onto div-free vector fields P : (L2(D))N→L0,div2 (D).

Note that P(∇π)=0.

4 Stochastic integration in infinite dimensions

Reference: G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions

Burkholder–Davis–Gundy’s inequality

Proposition 8. There exists a universal constant Cp< 0, 0< p<∞, such that

E sup0!s!t

''''!

0

s

ΦdW

''''H

p

"CpE(!

0

t

∥Φ∥L2(U ;H)2 ds

)p2.

Itô’s formula

Let Φ be an H-valued stochastically integrable process, let ϕ be an H-valued progressively mea-surable Bochner integrable process and X(0) an F0-measurable H-valued random variable. Thenthe following process

X(t)=X(0)+

!

0

t

ϕ(s) ds+

!

0

t

Φ(s) dW (s)

1. uhaszerotrace

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is well-defined. Assume that a function F : H → R and its derivatives F ′, F ′′ are uniformlycontinuous on bounded subsets of H.

Theorem 9. Under the above assumptions, P-a.s., for all t∈ [0, T ]

F (X(t))=F (X(0))+

!

0

t

⟨F ′(X(s)), ϕ(s)⟩H ds+!

0

t

⟨F ′(X(s)),Φ(s) dW (s)⟩H

+12

!

0

t

Tr(Φ(s)∗F ′′(X(s))Φ(s)) ds

where

Tr(Φ(s)∗F ′′(X(s))Φ(s))=&

k"1⟨F ′′(X(s))Φ(s)ek,Φ(s)ek⟩H .

Note that TrA=$

k"1 ⟨Aek, ek⟩H for A a bounded linear operator on H.

5 Pathwise vs. martingale solutions

Reference: I. Karatzas, S. E. Shreve: Brownian motion and stochastic calculus, Springer 1991.

Consider a stochastic differential equation (SDE)

dX = b(X) dt+σ(X) dW , X(0)=X0, (2)

whereW is anRd-valuedWiener process defined on a stochastic basis (Ω,F , (F t),P). For simplicitywe are in finite dimension so a solution X is an Rm-valued stochastic process. The coefficients b,σ are measurable functions

b:Rm→Rm, σ:Rm→Rm×d.

For SDEs we have to distinguish two notions of solution: probabilistically strong solution (path-wise solution) and probabilistically weak solution (martingale solution, solution to the martingaleproblem).

Definition 10. Let W be an Rd-valued Wiener process defined on (Ω,F , (F t),P) and let X0 beF0-measurable random variable. An Rm-valued stochastic process X is called a probabilisticallystrong (pathwise) solution to ( 2) with the initial condition X0 provided

• X is (F t)-adapted and has continuous trajectories

• X(0)=X0 a.s.

• P-a.s.!

0

t

|b(X(s))|+ |σ(X(s))|2ds<∞,

and

X(t)=X0+

!

0

t

b(X(s)) ds+

!

0

t

σ(X(s)) dW

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hold true a.s. for all t≥ 0.

Definition 11. Let Λ be a Borel probability measure on Rm. A triple ((Ω,F , (F t),P), X ,W ) iscalled a probabilistically weak (martingale) solution to ( 2) with the initial law Λ provided

• (Ω,F , (F t),P) is a stochastic basis

• W is an Rd-valued Wiener process on (Ω,F , (F t),P)

• X is an Rm-valued (F t)-adapted stochastic process with continuous trajectories

• P [X(0)]−1=Λ

• P-a.s.!

0

t

|b(X(s))|+ |σ(X(s))|2ds<∞,

and

X(t) =X(0)+

!

0

t

b(X(s)) ds+

!

0

t

σ(X(s)) dW

hold true for all t≥ 0.

Remark 12. Existence of a pathwise solution implies existence of a martingale solution, not theother way around.

5.1 Pathwise uniqueness vs. uniqueness in law

Definition 13. We say that pathwise uniqueness holds true for ( 2) provided: let X, X be twosolutions to ( 2) defined on the same stochastic basis with the same Wiener process and assumethat X(0)= X(0) a.s. Then the processes X and X are indistinguishable, i.e.

P(X(t)= X(t) ∀t! 0)=1.

Definition 14. We say that uniqueness in law holds true for ( 2) provided: let

((Ω,F , (F t),P), X ,W ),**Ω, F ,

*Ft+, P+, X , W

+

be two solutions to ( 2) with the same initial law. Then the two processes X and X have the samelaw.

Remark 15. Yamada-Watanabe result:

1. existence of a martingale solution + pathwise uniqueness⇒ existence of a pathwise solution

2. pathwise uniqueness ⇒ uniqueness in law

A method of proving existence of a unique pathwise solution:

• construct a martingale solution (that’s often easier)

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• prove pathwise uniqueness

Example 16. Consider the 1D equation

X(t)=

!

0

t

sgn(X(s)) dW (s)

where

sgn(x) =,

1, x> 0,−1, x" 0.

If ((Ω, F , (F t), P), X, W ) is a martingale solution then X is a continuous square integrablemartingale with quadratic variation

⟨X ⟩t=!

0

t

sgn2(X(s)) ds= t

hence X is a Brownian motion due to Levy’s theorem and uniqueness in law holds true. On theother hand, ((Ω,F , (F t),P),−X,W ) is also a solution so pathwise uniqueness does not hold.

Moreover, this equation does not admit a pathwise solution!

Theorem 17. Let b,σ be globally Lipschitz continuous, i.e. there exists a constant K>0 such that

|b(x)− b(y)|+ |σ(x)−σ(y)|"K |x− y | ∀x, y ∈Rm.

Let (Ω, F , (F t), P) and a Wiener process W be given. Let X0 be an F0-measurable Rm-valuedrandom variable. Then there exists a unique pathwise solution to ( 2) with the initial condition X0.

Proof. Show convergence of Picard iterations. #

Theorem 18. Let b,σ be continuous with linear growth, i.e. there exists a constant K>0 such that

|b(x)|+ |σ(x)|"K(1+ |x|) ∀x∈Rm.

Let Λ be a given probabililty measure on Rm. Then there exists a martingale solution to ( 2) withthe initial law Λ.

Proof. The main steps are as follows:

• Approximate b,σ by Lipschitz continuous functions bn,σn.

• For fixed n one can apply the previous theorem to get existence of an approximate solutionXn.

• Prove tightness of the laws of Xn.

• Apply the Skorokhod representation theorem to get a new probability space and a sequenceof new random variables X

nhaving the same laws as Xn and converging a.s. to a random

variable X.

• Identify X as a solution to (2). #

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6 Preliminaries from functional analysis

References: H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,Springer 2011, Section 6.3 and 6.4.

In this section we collect several results from functional analysis that will be needed in the sequel.

Throughout this section, H is a separable Hilbert space over the scalar field R.

6.1 Spectral decomposition of compact operators

Definition 19. Let T :H→H be a linear operator. We say that T is compact and write T ∈K(H),if T (B)⊂H is relatively compact for every bounded set B⊂H.

Clearly, K(H)⊂L(H).

Definition 20. Let T ∈L(H). Then we define the spectrum of T as

σ(T )= λ∈R; T −λI is not invertible,

the point spectrum of T as

σp(T )= λ∈R; T −λI is not injective, one-to-one,

and the resolvent of T as

ρ(T )=R \σ(T ) = λ∈C; T −λI is bijective.

Clearly, σp(T )⊂σ(T ). Moreover

σp(T )= λ∈R; ∃h∈H such that (T −λI)h=0= λ∈R; Ker(T −λI)=/ 0,

which follows from

λ∈σp(T ) ⇒ ∃h1, h2∈H such that (T −λI)h1=(T −λI)h2 ⇒ (T −λI)(h1−h2)= 0,

by setting h=h1−h2. h is called eigenvector corresponding to λ.

The elements of σp(T ) are called eigenvalues and Ker(T − λI) is the corresponding eigenspace.The following lemma is a part of the result called the Fredholm alternative.

Lemma 21. Let T ∈K(H). Then Ker(T − I) is finite-dimensional.

Theorem 22. Let T ∈K(H) with dimH =∞. Then

1. 0∈σ(T ),

2. σ(T ) \ 0=σp(T ) \ 0,

3. one of the following cases holds:

• σ(T ) = 0,

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• σ(T ) \ 0 is a finite set,

• σ(T ) \ 0 is a sequence converging to 0.

Definition 23. T ∈L(H) is called self-adjoint if T ∗=T, that is

⟨Tg, h⟩= ⟨g, Th⟩ ∀g, h∈H.

The next statement is a fundamental result which states that every compact self-adjoint operatormay be diagonalized in some suitable basis.

Theorem 24. Let T be a compact self-adjoint operator. Then there exists a countable orthonormalbasis of H consisting of eigenvectors of T.

Let (en)n∈N denote the orthonormal basis of eigenvectors corresponding to T and let (λn)n∈N bethe associated sequence of eigenvalues. Then for every h∈H we have the decomposition

h=&

n∈N⟨h, en⟩en

and therefore

Th=T

"&

n∈N⟨h, en⟩en

#=&

n∈N⟨h, en⟩Ten=

&

n∈N⟨h, en⟩λnen.

6.2 Fractional powers of some unbounded operators

For simplicity we will only discuss the case of ∆ on L2(D) with zero Dirichlet boundary conditions.Let A:D(A)⊂H→H, be given by Au=−∆u with zero Dirichlet boundary conditions, i.e.

D(A)=W01,2(D)∩W 2,2(D).

Then A is unbounded, symmetric

⟨Au, v⟩=−⟨∆u, v⟩=−⟨u,∆v⟩= ⟨u,Av⟩ ∀u, v ∈D(A),

(strictly) positive

⟨Au, u⟩=−⟨∆u, u⟩= ∥∇u∥L22 ! 0 ∀u∈D(A),

where ⟨Au,u⟩=0 iff u=0 again due to the boundary conditions. Moreover, D(A)⊂L2(D) is denseand compact. Besides, let f ∈L2(D) and consider the problem

−∆u= f in D, u=0 in ∂D.

Then there exists a unique weak solution u∈W01,2(D), i.e.

⟨∇u,∇ϕ⟩= ⟨f , ϕ⟩ ∀ϕ∈W01,2(D),

and ∥u∥W 1,2"C∥f ∥L2. Define T :L2(D)→W01,2(D)⊂L2(D) by Tf = u. Then T is compact and

self-adjoint: let u, v ∈H and let u=Tf , v=Tg. Then

⟨Tf , v⟩= ⟨u,−∆v⟩= ⟨−∆u, v⟩= ⟨f , Tg⟩.

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Thus according to the spectral theorem for compact self-adjoint operators, Theorem 24, thereexists a countable orthonormal basis of L2(D) that consists of eigenvectors of T . Let σp(T )= (θn)be the corresponding sequence of eigenvalues. Note that 0 ∈/ σp(T ) since Ker T = 0 due to theboundary conditions: if u≡ 0 satisfies that Tf =u then necessarily f ≡ 0. Consequently, we get

Ten= θnen ⇒ θn−1en=Aen

and it follows that (λn= θn−1) are eigenvalues of A whereas (en) are the corresponding eigenvectors

forming the orthonormal basis of L2(D). Due to Lemma 21 we deduce that

0<λ1"λ2" ···

and the sequence is unbounded, i.e. λn→∞. Similarly as above, we may therefore write

Au=A

"&

n∈N⟨u, en⟩en

#=&

n∈N⟨u, en⟩Aen=

&

n∈N⟨u, en⟩λnen.

In order to define the fractional powers of A, let α> 0 and set

D(Aα) =

-u∈L2(D);

&

n∈Nλn2α |⟨u, en⟩|2<∞

..

On this set we define Aα by

Aαu=&

n∈N⟨u, en⟩λnα en,

and we equip D(Aα) with the Hilbertian norm

∥u∥D(Aα)= ∥Aαu∥L2="&

n∈Nλn2α |⟨u, en⟩|2

#1

2.

7 Stochastic Navier–Stokes equations - existenceReference: F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields 102 (3) (1995) 367-391.

We study the following system of SPDEs which describes the evolution of the velocity u of anincompressible viscous fluid

du+u·∇u+∇p= ν∆u+G(u) dW , div u=0, u|∂D=0, u(0)=u0, (3)

• W is a cylindrical Wiener process on a separable Hilbert space U defined on a stochasticbasis (Ω,F , (F t),P)

• the diffusion coefficient G satisfies suitable growth assumptions

Recall that testing (3) by a div-free test function ϕ eliminates the pressure. The same can beachieved by applying the Helmholtz projection P . We will make use of this fact and rewrite (3) inan abstract form. To this end, let

H =L0,div2 (D), V =W0,div

1,2 (D)

15

and denote by ∥·∥H and ⟨·, ·⟩ the norm and the inner product in H. Identifying H with its dualH∗ we obtain the Gelfand triplet

V ⊂H =∼H∗⊂V ∗

with continuous and dense embedding and we can denote to dual pairing between V and V ∗ by⟨·, ·⟩. Moreover, we set

D(A)= [W 2,2(D)]N ∩V

and define the linear operator

A:D(A)⊂H→H, Au=−νP∆u.

Since V =D*A1

2

+we may endow V with the norm ∥u∥V =

'''A1

2u'''H.

A is positive self-adjoint with compact resolvent, we denote by 0<λ1"λ2" ··· its eigenvalues, andby e1, e2, ... a corresponding complete orthonormal system in H of eigenvectors.

Remark that ∥·∥V2 !λ1∥·∥H2 .

Define the bilinear operator B:V ×V → (V ∩ [L2(D)]N)∗ as

⟨B(u, v), z⟩=!

Dz(x)·(u(x)·∇)v(x) dx.

Then

⟨B(u, v), v⟩=0, ⟨B(u, v), z⟩=−⟨B(u, z), v⟩.

B can be extended to a continuous operator

B:H ×H→D(A−α)

for certain α> 1.

And (3) rewrites as an abstract stochastic evolution equation

du+Audt+B(u, u) dt = G(u) dWu(0) = u0

(4)

Assume further

• u0∈H deterministic for simplicity

• G:H→L2(U;H) is Lipschitz continuous and ∥G(u)∥L2(U;H)2 "λ0∥u∥H2 + ρ for some λ0, ρ>0.

In particular, if (fk) is an orthonormal system in U, we assume that

G(u)fk= gk(x, u), u∈H,

where gk∈C(D×R) and

&

k"1|gk(x, ξ)|2"λ0|ξ |2+ ρ, x∈D, ξ ∈R,

&

k"1|gk(x, ξ)− gk(x, ζ)|2"λ0|ξ − ζ |2, x∈D, ξ , ζ ∈R. (5)

16

This implies

∥G(u)∥L2(U;H)2 =

&

k"1∥G(u)fk∥H

2 =&

k"1∥gk(x, u)∥H2 "λ0∥u∥H2 + ρ.

And the stochastic integral in (4) reads as

!

0

t

G(u) dW =&

k"1

!

0

t

G(u)fk dβk=&

k"1

!

0

t

gk(u) dβk.

Definition 25. Let a cylindrical Wiener process W on (Ω, F , (F t), P) and the initial conditionu0∈H be given. A progressively measurable process u:Ω× [0, T ]→H with a.e. trajectory in

u(·,ω)∈C([0, T ];D(A−α))∩L∞(0, T ;H)∩L2(0, T ;V )

is a pathwise solution to ( 4) provided a.s. the identity

⟨u(t), v⟩+!

0

t

⟨Au(s), v⟩ds+!

0

t

⟨B(u(s), u(s)), v⟩ds= ⟨u0, v⟩+/!

0

t

G(u(s)) dW , v

0

holds true for all t∈ [0, T ] and all v ∈D(Aα).

Definition 26. A triple ((Ω, F , (F t), P), u, W ) is called a martingale solution to ( 4) with theinitial condition u0 provided

• (Ω,F , (F t),P) is a stochastic basis

• W is a cylindrical Wiener process

• u:Ω× [0, T ]→H is (F t)-progressively measurable with a.e. trajectory in

u(·,ω)∈C([0, T ];D(A−α))∩L∞(0, T ;H)∩L2(0, T ;V )

• P-a.s. the identity

⟨u(t), v⟩+!

0

t

⟨Au(s), v⟩ds+!

0

t

⟨B(u(s), u(s)), v⟩ds= ⟨u0, v⟩+/!

0

t

G(u(s)) dW , v

0

holds true for all t∈ [0, T ] and all v ∈D(Aα).

Theorem 27. Under the above assumptions there exists a martingale solution to ( 4).

The proof of this result will be divided into several steps. We make use of the Faedo-Galerkinfinite dimensional approximation method, prove uniform estimates and then apply the stochasticcompactness method to pass to the limit.

TO DO LIST:

1. Prove existence for certain finite dimensional approximations un.

17

2. Establish the so-called energy estimate which holds true uniformly in n.

3. Deduce tightness of the laws of un on the space L2(0, T ;H)∩C([0, T ];D

1A−

β

2

2).

4. Identify the limit.

7.1 Step 1: Faedo-Galerkin approximation

The first step consists in constructing a sequence of approximate solutions solving suitable finitedimensional problems. Let Pn be the projection from H onto the finite dimensional space spannedby e1, ..., en. For a suitable α> N

2,

Pn:D*A−

α

2+→D

*Aα

2+, Pnx=

&

i=1

n

⟨x, ei⟩ei,

where ⟨·, ·⟩ denotes the duality pairing between D*A−α

2+and D

*Aα

2+. It holds

⟨Pnx, y⟩= ⟨x, Pny⟩, x, y ∈D*A−

α

2+.

We intend to consider a projected version of (4) and apply Theorem 17 to get existence of a pathwisesolution. Thus we need to modify the coefficient B to make it Lipschitz in PnH. Set

Bn(u, u)= χn(u)B(u, u), u∈PnH,

where χn:H→R given by χn(u)=Θn(|u|) with Θn:R→ [0, 1] of class C∞ such that χn(u)= 1 if|u|"n, χn(u) =0 if |u|>n+1.

Now the Faedo-Galerkin approximation is given for un∈PnH by

dun+Aundt+PnBn(un, un) dt = PnG(un) dWun(0) = Pnu0

(6)

Since APn=PnA which can be seen from

APnv=A

"&

i=1

n

⟨v, ei⟩ei

#=&

i=1

n

⟨v, ei⟩Aei=&

i=1

n

⟨v, ei⟩λiei,

PnAv=&

i=1

n

⟨Av, ei⟩ei=&

i=1

n

⟨v, Aei⟩ei=&

i=1

n

⟨v, ei⟩λiei.

Note that PnBn:PnH→PnH is indeed Lipschitz: let u, v ∈PnH then

∥PnBn(u, u)−PnBn(v, v)∥PnH= ∥Pn(Bn(u, u)−Bn(v, v))∥H

"∥Bn(u, u)−Bn(v, v)∥H= ∥χn(u)B(u, u)− χn(v)B(v, v)∥H

"∥χn(u)B(u, u− v)∥H+ ∥(χn(u)− χn(v))1∥u∥H!n+11∥v∥H!n+1B(u, v)∥H+ ∥χn(v)B(u− v, v)∥H

=I1+ I2+ I3.

18

Now, we estimate

I12"!

D1∥u∥H!n+1|u·∇(u− v)|2dx$n

!

D1∥u∥L∞!n+1|u·∇(u− v)|2dx

$n∥u− v∥W 1,22 $n ∥u− v∥H2 = ∥u− v∥PnH2

and similarly for I3. For I2 we use the fact that

∥1∥u∥H!n+11∥v∥H!n+1B(u, v)∥H2 =!

D1∥u∥H!n+11∥v∥H!n+1|u·∇v |2dx

$n!

D1∥u∥L∞!n+11∥v∥W1,2!n+1|u·∇v |2dx" (n+1)3$n 1

together with the Lipschitz continuity of χn that

I2$n ∥u− v∥H= ∥u− v∥PnH.

And therefore we get

∥PnBn(u, u)−PnBn(v, v)∥PnH$n ∥u− v∥PnH

which is the desired Lipschitz continuity.

The Lipschitz continuity of PnG:PnH→L2(U;PnH) follows from (5): let u, v ∈PnH then

∥PnG(u)−PnG(v)∥L2(U;PnH)2 =

&

k"1∥Pn(G(u)fk−G(v)fk)∥PnH2

"&

k"1∥gk(u)− gk(v)∥H2 "λ0∥u− v∥H2 =λ0∥u− v∥PnH2 .

In fact due to our assumptions, even G:H→L2(U;H) is Lipschitz.

Hence Theorem 17 applies and yields the existence of a unique pathwise solution un to (6).

7.2 Step 2: Uniform energy estimates

As the next step, we will derive estimates uniform in n. We apply the Itô formula to (6) and tothe function F :PnH→R, u 8→ ∥u∥H

p , p! 2. Since

F ′(u) = p∥u∥Hp−2u∈PnH, F ′′(u)= p(p− 1)∥u∥H

p−2 Id∈L(PnH;PnH),

this yields

∥un(t)∥Hp = ∥Pnu0∥H

p +

!

0

t

⟨F ′(un), dun⟩+12

!

0

t

Tr(Pn(G(un))∗F ′′(un)(PnG(un))) ds,

where

(PnG(un))∗F ′′(un)(PnG(un)):U→PnH→PnH→U

19

so the trace is taken in U and given by

TrΦ=&

k"1⟨Φfk, fk⟩.

Hence

Tr((PnG(un))∗F ′′(un)(PnG(un))) =&

k"1⟨(PnG(un))∗F ′′(un)(PnG(un))fk, fk⟩

=&

k"1⟨F ′′(un)PnG(un)fk, PnG(un)fk⟩

=p(p− 1)∥un∥Hp−2&

k"1⟨PnG(un)fk, PnG(un)fk⟩= p(p− 1)∥un∥H

p−2&

k"1∥PnG(un)fk∥H2

"p(p− 1)∥un∥Hp−2&

k"1∥gk(un)∥H2 " p(p− 1)∥un∥H

p−2(λ0∥un∥H2 + ρ)

"p(p− 1)λ0∥un∥Hp +C

We obtain

∥un(t)∥Hp " ∥Pnu0∥Hp − p

!

0

t

∥un∥Hp−2⟨un, Aun⟩ds− p

!

0

t

∥un∥Hp−2⟨un, PnBn(un, un)⟩ ds

+p

!

0

t

∥un∥Hp−2⟨un,G(un) dW ⟩+ 1

2p(p− 1)

!

0

t

∥un∥Hp ds+Ct.

Now using the key cancellation property for the convective term (due to div un=0)

⟨un, PnBn(un, un)⟩= ⟨un, Bn(un, un)⟩= χn(un)⟨un, B(un, un)⟩=0

together with

⟨un, Aun⟩= ∥un∥V2

we obtain

∥un(t)∥Hp + p

!

0

t

∥un∥Hp−2∥un∥V2 ds" ∥Pnu0∥Hp + p

!

0

t

∥un∥Hp−2⟨un,PnG(un) dW ⟩

+12p(p− 1)

!

0

t

∥un(s)∥Hp ds+Ct.

Taking expectation the stochastic integral vanishes

E∥un(t)∥Hp + pE

!

0

t

∥un∥Hp−2∥un∥V2 ds"E∥Pnu0∥H

p +12p(p− 1)

!

0

t

E∥un(s)∥Hp ds+Ct

and Gronwall lemma yields

E∥un(t)∥Hp "C(1+E∥Pnu0∥H

p )"Ct(1+E∥u0∥Hp ) (7)

as well as

E

!

0

t

∥un∥Hp−2∥un∥V2 ds"Ct(1+E∥u0∥H

p ) (8)

20

uniformly in n.

Now, we take first supt and then expectation. For the stochastic integral we have

E sup0!t!T

3333!

0

t

∥un∥Hp−2⟨un,G(un) dW ⟩

3333=E sup0!t!T

33333&

k"1

!

0

t

∥un∥Hp−2⟨un,gk(un)⟩dβk

33333

"CE

"!

0

T

∥un∥H2p−4&

k"1⟨un,gk(un)⟩2ds

#1

2 "CE

"!

0

T

∥un∥H2p−2&

k"1∥gk(un)∥H2 ds

#1

2

"CE

(!

0

T

∥un∥H2p−2(λ0∥un∥H2 + ρ) ds

)1

2 "CE

"

sup0!t!T

∥un(t)∥Hp!

0

T

∥un∥Hp−2(λ0∥un∥H2 + ρ) ds

#1

2

"12E sup0!t!T

∥un(t)∥Hp +CE

!

0

T

∥un∥Hp−2(λ0∥un∥H2 + ρ) ds

"12E sup0!t!T

∥un(t)∥Hp +CE

!

0

T

∥un∥Hp ds+CT.

Putting all together we deduce

E sup0!t!T

∥un(t)∥Hp + p

!

0

T

∥un∥Hp−2∥un∥V2 ds" ∥Pnu0∥Hp

+12E sup0!t!T

∥un(t)∥Hp +CE

!

0

T

∥un∥Hp ds+CT

hence due to (7)

E sup0!t!T

∥un(t)∥Hp "CT(1+E∥u0∥H

p ). (9)

In particular, it follows from (8) with p=2 that

E

!

0

T

∥un∥V2 ds"CT(1+E∥u0∥H2 ).

7.3 Step 3: Tightness

Let us define the path spaces

X u=L2(0, T ;H)∩C([0, T ];D

1A−

β

2

2), XW =C([0, T ];U0)

and denote by µun the law of un on X u, by µW the law of W on XW and by µn their joint law.Our aim is to show that the set of laws (µn) on X =Xu⊗XW is tight.

To this end, we first need to derive further estimates uniform in n. They are all based on the energyestimate from the previosu section.

21

7.3.1 Another uniform estimate

Now we establish a uniform estimate in Wα,2

(0, T ;D

1A−

β

2

2). We have

un(t) =Pnu0−!

0

t

Aun(s) ds−!

0

t

PnBn(un(s), un(s)) ds+

!

0

t

PnG(un(s)) dW (s)

=Jn1+Jn

2(t) + ···+ Jn4(t),

where

E∥Jn1∥H2 "C,

E∥Jn2∥W 1,2(0,T ;V ∗)2 "C.

To estimate the stochastic integral, we make use of the following Lemma.

Lemma 28. Let p! 2, α< 1

2, be given. Then for any progressively measurable process

f ∈Lp(Ω× [0, T ];L2(U;H)),

it holds that!

0

·f(s) dW (s)∈Lp(Ω;Wα,p(0, T ;H))

and there exists a constant C =C(p,α)> 0 independent of f such that

E

''''!

0

·f(s) dW (s)

''''Wα,p(0,T ;H)

p

"CE

!

0

T

∥f(t)∥L2(U;H)p dt.

Consequently,

E∥Jn4∥Wα,2(0,T ;H)2 "CE

!

0

T

∥PnG(un(t))∥L2(U;H)2 dt

"CE

!

0

T&

k"1∥gk(un(t))∥H2 dt"C

holds true for all α∈*0, 1

2

+. Besides, due to the Sobolev embedding, D

1Aβ

2

2⊂ (L∞(D))N for β> N

2hence

|⟨B(u, u), v⟩|=3333!

D(u·∇u )·vdx

3333" ∥u∥H∥u∥V ∥v∥L∞$ ∥u∥H∥u∥V'''A

β

2v'''H, u∈V , v ∈D

1Aβ

2

2

so

∥B(u, u)∥D

!A−β2

"$ ∥u∥H∥u∥V .

Thus

∥PnBn(un, un)∥L2

#0,T ;D

!A−β2

"$2 =

!

0

T

∥PnB(un, un)∥D

!A−β2

"2 dt"!

0

T

∥B(un, un)∥D

!A−β2

"2 dt

$!

0

T

∥un∥H2 ∥un∥V2 dt" sup0!t!T

∥un(t)∥H2!

0

T

∥un(t)∥V2 dt

22

and consequently

∥Jn3∥W 1,2

#0,T ;D

!A−β2

"$2 $ sup0!t!T

∥un(t)∥H2!

0

T

∥un(t)∥V2 dt

E∥Jn3∥W 1,2

#0,T ;D

!A−β2

"$$E

"sup

0!t!T∥un(t)∥H2

!

0

T

∥un(t)∥V2 dt#1

2

$E sup0!t!T

∥un(t)∥H2 +E

!

0

T

∥un(t)∥V2 dt"C.

Collecting all the above estimates, we deduce

E∥un∥Wα,2

#0,T ;D

!A−β2

"$"E∥Jn1∥Wα,2

#0,T ;D

!A−β2

"$+E∥Jn2∥Wα,2

#0,T ;D

!A−β2

"$

+E∥Jn3∥Wα,2

#0,T ;D

!A−β2

"$+E∥Jn4∥Wα,2

#0,T ;D

!A−β2

"$

$E∥Jn1∥H+E∥Jn2∥W1,2(0,T ;V ∗)2 +E∥Jn3∥

W 1,2

#0,T ;D

!A−β2

"$+E∥Jn4∥Wα,2(0,T ;H)$C,

where we used

D1A−

β

2

2⊂L∞(D)⊂L2(D),

D1Aβ

2

2⊂V ⊂H ⊂V ∗⊂D

1A−

β

2

2, W 1,2(0, T ;V ∗)⊂Wα,2

(0, T ;D

1A−

β

2

2),

W 1,2

(0, T ;D

1A−

β

2

2)⊂Wα,2

(0, T ;D

1A−

β

2

2),

Wα,2(0, T ;H)⊂Wα,2

(0, T ;D

1A−

β

2

2).

With this estimate in hand, we are able to prove the main result of this subsection.

7.3.2 Conclusion

Proposition 29. The set of laws (µn) is tight on X.

Proof. First of all we observe that the law µW is tight on XW since it is a Radon measure on aPolish space. Besides, the set of laws (µn) is tight on Xu which follows from the above estimates.Indeed, since the following embedding is compact

L2(0, T ;V )∩Wα,2

(0, T ;D

1A−

β

2

2)⊂L2(0, T ;H)

it holds that bounded sets in

L2(0, T ;V )∩Wα,2

(0, T ;D

1A−

β

2

2)

are compact in

L2(0, T ;H).

23

Therefore, let

BR=

-u∈L2(0, T ;V )∩Wα,2

(0, T ;D

1A−

β

2

2); ∥u∥L2(0,T ;V )+ ∥u∥

Wα,2

#0,T ;D

!A−β2

"$"R..

Then

µun(BRc )"P

(∥u∥L2(0,T ;V )>

R2

)+P

"∥u∥

Wα,2

#0,T ;D

!A−β2

"$>R2

#

" 2R

(E∥u∥L2(0,T ;V )+E∥u∥

Wα,2

#0,T ;D

!A−β2

"$)" CR.

This implies tightness in L2(0, T ;H).

In order to get tightness in C([0, T ];D

1A−

β

2

2). We argue similarly but use an improved estimate

for the stochastic integral Jn4 based again on Lemma 28 but using (9)

E∥Jn4∥Wα,p(0,T ;H)p "CE

!

0

T

∥PnG(un(t))∥L2(U;H)p dt

"CE

!

0

T"&

k"1∥gk(un(t))∥H2

#p

2dt"CE

!

0

T

(1+ ∥un(t)∥H2 )p

2 dt

"CT(1+E

!

0

T

∥un(t)∥Hp dt

)"C

and apply the following compact embedding result.

Theorem 30. If B1 ⊂ B are two Banach spaces with compact embedding, and real numbersα∈ (0, 1), p> 1 satisfy αp> 1, then the embedding

Wα,p(0, T ;B1)⊂C([0, T ]; B)

is compact. Similarly, if Banach spaces B1, ..., Bk are compactly embedded into B and the realnumbers α1, ...,αk∈ (0, 1) and p1, ..., pk> 1 satisfy αipi> 1 for all i=1, ..., k, then the embedding

Wα1,p1(0, T ;B1)+ ···+Wαk,pk(0, T ;Bk)⊂C([0, T ]; B)

is compact.

Altogether, we deduce that the set of laws (µun) is tight on X u. And accordingly, the set of laws(µn) is tight on X . Indeed, given ε> 0 there exists compacts Ku∈X u and KW ∈XW such that

µun(Ku)! 1− ε2, µW(KW)! 1− ε

2.

Due to Tichonoff’s theorem Ku × KW is compact in X and due to de Morgan’s law A ∩ B =(Ac∪Bc)c we get

µn(Ku×KW)=P(un∈Ku,W ∈KW)= 1−P(un∈/ Ku∪W ∈/ KW)

!1− (P(un∈/Ku)+P(W ∈/KW))! 1− ε. #

24

7.4 Step 4: Skorokhod & passage to the limit

Since the set of lwas (µn) is tight on X Skorokhod representation theorem applies and yields (alonga subsequence) that µn converges weakly to a law µ on X and the following holds true.

Proposition 31. There exists a probability space (Ω, F , P) with random variables*un, Wn

+,

n∈N, and (u, W ) such that

1. the law of*un, Wn

+under P coincides with µn,

2. the law of (u, W ) under P coincides with µ,

3.*un, Wn

+converges to (u, W ) in the topology of X P-a.s.

From the equality of laws we get the uniform estimates

E sup0!t!T

∥un(t)∥Hp "C, E

!

0

T

∥un(t)∥V2 dt"C,

hence from lower semicontinuity

E sup0!t!T

∥u(t)∥Hp "C, E

!

0

T

∥u(t)∥V2 dt"C,

and

un u in L2(Ω× [0, T ];V ).

Let*F tn+

and*F t

+, respectively, be the P-augmented canonical filtration generated by

*un, Wn

+

and (u, W ), respectively. For a Banach space X we define the restriction operator

ϱt:C([0, T ];X)→C([0, t];X), ϱtf = f |[0,t].

It is a continuous mapping.

Now, let ϕ∈ spane1, e2, ... and define the following processes:

Mn(t): =⟨un(t), ϕ⟩− ⟨Pnu0, ϕ⟩+!

0

t

⟨Aun, ϕ⟩ ds+!

0

t

⟨PnBn(un, un), ϕ⟩ds,

Mn(t): =⟨un(t), ϕ⟩− ⟨un(0), ϕ⟩+!

0

t

⟨A un, ϕ⟩ds+!

0

t

⟨PnBn(un, un), ϕ⟩ds,

M(t): =⟨u(t), ϕ⟩− ⟨u(0), ϕ⟩+!

0

t

⟨A u, ϕ⟩ds+!

0

t

⟨B(u, u), ϕ⟩ ds.

Our proof of existence of a martingale solution will be complete once we prove the following lemma.

Lemma 32. The process W is an*Ft+-cylindrical Wiener process in U with the expansion

W =&

k"1fkβk.

25

The processes

M , M2−&

k"1

!

0

.

⟨gk(u), ϕ⟩2ds, Mβk−!

0

·⟨gk(u), ϕ⟩ ds

are*Ft+-martingales.

In order to prove this result, let us fix times s, t∈ [0, T ] such that s< t and let

h:Xu|[0,s]×XW |[0,s]→ [0, 1]

be a continuous function. Since

Mn(t)=&

k"1

!

0

t

⟨gk(un), ϕ⟩ dβk,

the processes

Mn, Mn2−&

k"1

!

0

.

⟨gk(un), ϕ⟩2ds, Mnβk−!

0

·⟨gk(un), ϕ⟩ ds

are (F t)-martingales and hence

E[h(ϱsun, ϱsW )(Mn(t)−Mn(s))] = 0,

E

4h(ϱsun, ϱsW )

"Mn

2(t)−Mn2(s)−

&

k"1

!

s

t

⟨gk(un), ϕ⟩2ds#5

=0,

E

6h(ϱsun, ϱsW )

(Mn(t)βk(t)−Mn(s)βk(s)−

!

s

t

⟨gk(un), ϕ⟩ ds)7

=0.

From the equality of laws we obtain the same three equalities for Mn:

E8h*ϱsun, ϱsWn

+*Mn(t)− Mn(s)

+9=0,

E

4h*ϱsun, ϱsWn

+"Mn

2(t)− Mn2(s)−

&

k"1

!

s

t

⟨gk(un), ϕ⟩2ds#5

=0,

E

6h*ϱsun, ϱsWn

+(Mn(t)βk

n(t)− Mn(s)βk

n(s)−

!

s

t

⟨gk(un), ϕ⟩ds)7

=0.

And finally we pass to the limit to obtain

E8h(ϱsu, ϱsW )

*M(t)− M(s)

+9=0,

E

4h*ϱsu, ϱsW

+"M2(t)− M2(s)−

&

k"1

!

s

t

⟨gk(u), ϕ⟩2ds#5

=0,

E

6h*ϱsu, ϱsW

+(M(t)βk(t)− M(s)βk(s)−

!

s

t

⟨gk(u), ϕ⟩ ds)7

=0,

which yields the claim.

26

In order to justify the passage to the limit, let us recall the Vitali convergence theorem.

Definition 33. Let (X,F , µ) be a finite measure space. A sequence of functions (fn)⊂L1(µ) iscalled uniformly integrable if for every ε> 0 there exists δ> 0 such that

3333!

Efn(x) dµ(x)

3333< ε for alln∈N andE ∈F such that µ(E)< δ.

Lemma 34. Let (X,F , µ) be a finite measure space. Let a sequence (fn)⊂L1(µ) be bounded inLp(µ) for some p> 1. Then it is uniformly integrable.

Proof. Let 1

p+ 1

q=1. Then we estimate by Holder inequality

3333!

X1E(x) fn(x) dµ(x)

3333"(!

X|fn(x)|pdµ(x)

)1

p

(!

X1E(x) dµ(x)

)1

q = ∥fn∥Lp (µ(E))1

q "C(µ(E))1

q

so choosing δ sufficiently small the claim follows. #

Theorem 35. Let (X, F , µ) be a finite measure space. Assume that the sequence of functions(fn) ⊂ L1(µ) is uniformly integrable and converges to some f : X → R a.e. Then f ∈ L1(µ) andfn→ f in L1(µ), i.e.

limn→∞

!

X|fn(x)− f(x)| dx=0.

8 Transition semigroup and invariant measures

Consider an SDE in H

dX = b(X)dt+σ(X)dW , X(0)=x∈H. (10)

Assume that the coefficients are Lipschitz continuous and consequently there exists a uniquepathwise solution. Let Xt

x denote the solution at time t starting from x at time 0. The dependenceof Xt

x on x is measurable. Then we can define the operators Pt:Bb(H)→Bb(H) by

(Ptϕ)(x)=E[ϕ(Xtx)].

Let Xt0,tη denote the unique solution to (10) starting at time t0 from an F t0-measurable initial

condition η. The next Lemma show the continuous dependence on the initial condition.

Lemma 36. For all T > 0 and p∈ [2,∞) it holds true that

E|Xs,tη −Xs,t

θ |p"Cp,TE|η− θ |p.

Corollary 37. Pt:Cb(H)→Cb(H).

Lemma 38. The equation ( 10) defines a Markov process in the sense that

E[ϕ(Xt+sx )|F t] = (Psϕ)(Xt

x) ∀ϕ∈Cb(H), ∀x∈H, ∀t, s > 0.

27

Proof. We have to prove that

E[ϕ(Xt+sx )Z] =E[(Psϕ)(Xt

x)Z] ∀Z ∈F t.

By uniqueness

Xt+sx =Xt,t+s

Xtx

P− a.s.,

where Xt0,tη denotes the unique solution to (10) starting at time t0 from an F t0-measurable initial

condition η. It is then sufficient to show that

E[ϕ(Xt,t+sη )Z] =E[(Psϕ)(η)Z]

for every F t-measurable random variable η. By approximation (one uses dominated convergenceand the fact that ηn→ η in H implies Ptϕ(ηn)→ Ptϕ(η) in R a.s.), it is enough to prove it forrandom variables η=

$i=1k ηi1Ai with ηi∈H and Ai∈F t. And it is enough to prove it for every

deterministic η. Now the random variable Xt,t+sη depends only on the increments of the Brownian

motion between t and t+ s, hence it is independent of F t. Therefore

E[ϕ(Xt,t+sη )Z] =E[ϕ(Xt,t+s

η )]E[Z].

Since Xt,t+sη has the same law as Xs

η by uniqueness, we have

E[ϕ(Xt,t+sη )Z] =E[ϕ(Xs

η)]E[Z] =Psϕ(η)E[Z] =E[Psϕ(η)Z]

and the proof is complete. #

Taking expectation we get, on the one hand,

E[E[ϕ(Xt+sx )|F t]] =E[ϕ(Xt+s

x )] = (Pt+sϕ)(x)

and on the other hand

E[(Psϕ)(Xtx)] = (Pt(Psϕ))(x).

Thus the semigroup property Pt+s=Pt Ps follows.

Corollary 39. The equation ( 10) defines a Feller Markov process. The semigroup (Pt)t"0 is calledFeller.

Let us now denote by µt,x the law of Xtx. Then

Ptϕ(x)=E[ϕ(Xtx)] =

!

Hϕ(y)µt,x(dy)

and denoting by ⟨·, ·⟩ the duality product between bounded Borel functions and probability mea-sures, we obtain

Ptϕ(x) = ⟨ϕ, µt,x⟩= ⟨Ptϕ, δx⟩.

Thus it follows that µt,x=Pt∗δx. More generally, if we consider a solution to (10) with the initial

condition X0 having the initial law µ, we have µt,X0=Pt∗µ.

28

Definition 40. We say that a probability measure µ on H is an invariant measure if

Pt∗µ= µ, ∀t! 0.

Then if a solution has the law µ at time s it is the case for all t! s. In fact, for such a solution itcan be shown by the Markov property that for all (t1, ..., tn) and τ >0, the law of (Xt1+τ , ...Xtn+τ)and (Xt1, ...Xtn) coincides. We say that the process X is stationary.

Now, we have all in hand to formulate the Krylov-Bogoliubov theorem.

Theorem 41. Assume that the semigroup (Pt) is Feller and that there exists a random variableX0, a sequence (tn) increasing to ∞ and a probability measure µ such that

1tn

!

0

tn

µs,X0ds→→→→→→→→→→∗µ.

Then µ is an invariant measure for (Pt).

Proof. Let µ0 = L(X0) be the law of X0 and take ϕ ∈ Cb(H). Then since µs,X0 = Ps∗µ0 and

Ptϕ∈Cb(H) by the Feller property, we have

⟨ϕ, Pt∗µ⟩= ⟨Ptϕ, µ⟩= limn→∞

/Ptϕ,

1tn

!

0

tn

Ps∗µ0ds

0= limn→∞

1tn

!

0

tn

⟨Ptϕ, Ps∗µ0⟩ ds

= limn→∞

1tn

!

0

tn

⟨ϕ, Pt+s∗ µ0⟩ ds= limn→∞

1tn

!

t

t+tn

⟨ϕ, Ps∗µ0⟩ ds

= limn→∞

1tn

6!

0

tn

⟨ϕ, Ps∗µ0⟩ds+!

tn

t+tn

⟨ϕ, Ps∗µ0⟩ ds−!

0

t

⟨ϕ, Ps∗µ0⟩ds7

=⟨ϕ, µ⟩+0− 0.

Indeed, |Psϕ(x)|" ∥ϕ∥∞ hence dominated convergence applies. #

Corollary 42. If for some random variable X0 and a sequence (tn), tn↑∞, the sequence

,1tn

!

0

tn

µs,X0ds; n∈N

:

is tight, then there exists an invariant measure.

Proof. Prokhorov thm: tightness implies sequential relative compactness. #

Let x∈H. Then the tightness of

1tn

!

0

tn

µs,xds (11)

implies in particular that (a compact set is in particular bounded)

∀ε> 0 ∃R> 0 ∀T ! 1 1T

!

0

T

P(|Xtx|!R) dt < ε. (12)

29

Remark 43. Note that (12) implies the tightness of (11) and hence existence of an invariantmeasure provided dimH <∞. If dimH =∞ then this is not true nto even in the deterministicsetting (there are counterexamples).

9 Stochastic NSE - stationary martingale solutions

In this section we prove that the stochastic NSE admits a martingale solution which is a stationarystochastic process in H.

Definition 44. A triple ((Ω,F , (F t),P), u,W ) is called a stationary martingale solution to ( 4)with the initial condition u0 provided

• (Ω,F , (F t),P) is a stochastic basis

• W is a cylindrical Wiener process

• u:Ω× [0, T ]→H is (F t)-progressively measurable with a.e. trajectory in

u(·,ω)∈C([0, T ];D(A−α))∩L∞(0, T ;H)∩L2(0, T ;V )

• u is a stationary process in H, that is, the joint law L(ut1+τ , ..., utn+τ) is independent ofτ > 0 for any choice of (t1, ..., tn), n∈N, such that ti, ti+ τ ∈ (0,∞) for i=1, ..., n

• P-a.s. the identity

⟨u(t), v⟩+!

0

t

⟨Au(s), v⟩ds+!

0

t

⟨B(u(s), u(s)), v⟩ds= ⟨u0, v⟩+/!

0

t

G(u(s)) dW , v

0

holds true for all t∈ [0, T ] and all v ∈D(Aα).

Assume that G:H→ L2(U;H) is Lipschitz continuous and ∥G(u)∥L2(U;H)2 " λ0∥u∥H2 + ρ for some

λ0, ρ> 0. In particular, if (fk) is an orthonormal system in U, we assume that

G(u)fk= gk(x, u), u∈H,

where gk∈C(D×R) and&

k"1|gk(x, ξ)|2"λ0|ξ |2+ ρ, x∈D, ξ ∈R,

&

k"1|gk(x, ξ)− gk(x, ζ)|2"λ0|ξ − ζ |2, x∈D, ξ , ζ ∈R. (13)

Theorem 45. Assume in addition that

2λ1>λ0

where λ1 is the first eigenvalue of A. Then there exists a stationary martingale solution.

We consider again the approximation scheme based on the Galerkin approximation. Namely, letun be the unique solution to

dun+Aundt+PnBn(un, un) dt = PnG(un) dWun(0) = 0

(14)

30

We need to get a uniform energy estimate which does not blow up as t→∞. To this end, we applyagain the Itô formula to (6) and to the function F :PnH→R, u 8→ ∥u∥H

p , p! 2. Since

F ′(u) = p∥u∥Hp−2u∈PnH, F ′′(u)= p(p− 1)∥u∥H

p−2 Id∈L(PnH;PnH),

this yields

∥un(t)∥Hp = ∥Pnu0∥H

p +

!

0

t

⟨F ′(un), dun⟩+12

!

0

t

Tr(Pn(G(un))∗F ′′(un)(PnG(un))) ds,

where

(PnG(un))∗F ′′(un)(PnG(un)):U→PnH→PnH→U

so the trace is taken in U and given by

TrΦ=&

k"1⟨Φfk, fk⟩.

Hence

Tr((PnG(un))∗F ′′(un)(PnG(un))) =&

k"1⟨(PnG(un))∗F ′′(un)(PnG(un))fk, fk⟩

=&

k"1⟨F ′′(un)PnG(un)fk, PnG(un)fk⟩

=p(p− 1)∥un∥Hp−2&

k"1⟨PnG(un)fk, PnG(un)fk⟩= p(p− 1)∥un∥H

p−2&

k"1∥PnG(un)fk∥H2

"p(p− 1)∥un∥Hp−2&

k"1∥gk(un)∥H2 " p(p− 1)∥un∥H

p−2(λ0∥un∥H2 + ρ)

We obtain

∥un(t)∥Hp " ∥Pnu0∥Hp − p

!

0

t

∥un∥Hp−2⟨un, Aun⟩ds− p

!

0

t

∥un∥Hp−2⟨un, PnBn(un, un)⟩ ds

+p

!

0

t

∥un∥Hp−2⟨un,G(un) dW ⟩+ 1

2p(p− 1)

!

0

t

∥un∥Hp−2(λ0∥un∥H2 + ρ) ds.

Now using the key cancellation property for the convective term (due to div un=0) together with

⟨un, Aun⟩= ∥un∥V2

we get

∥un(t)∥Hp + p

!

0

t

∥un∥Hp−2∥un∥V2 ds

"p!

0

t

∥un∥Hp−2⟨un,G(un) dW ⟩+ 1

2p(p− 1)

!

0

t

∥un∥Hp−2(λ0∥un∥H2 + ρ) ds.

Next, we use the fact that ∥u∥H " λ1∥u∥V and due to Young inequality ab " εar

r+ br

′

εr′/r r ′

forconjugate exponents p, p′ we also have (with r= p

p− 2 , r′= p

2)

∥u∥Hp−2" ε

p∥u∥H

p +1

ε(p−2)/22p.

31

So we obtain

∂tE∥un∥Hp +

(pλ1−

12p(p− 1)λ0−

ε2(p− 1)ρ

)E∥un∥H

p "Cp,ε,ρ

need to choose ε small enough and p! 2 such that

pλ1−12p(p− 1)λ0−

ε2(p− 1)ρ! 0.

For p=2 we have

pλ1−12p(p− 1)λ0=2λ1−λ0> 0

due to our assumptions hence choosing ε small enough does the job. Consequently, it holds

∂tE∥un∥Hp + cE∥un∥H

p "Cp,ε,ρ

which implies

E∥un(t)∥Hp "C ∀t! 0 ∀n∈N.

Therefore the process un is bounded in probability by Chebyshev inequality, namely,

∀ε> 0 ∃R> 0 ∀t! 0 P(∥un(t)∥H!R)< ε

thus

∀ε> 0 ∃R> 0 ∀T ! 1 1T

!

0

T

P(∥un(t)∥H!R) dt< ε.

Since un is finite-dimensional we can use Krylov-Bogoliubov theorem to deduce existence of aninvariant measure µn.

Take (Ω′,F ′, (F t′),P′) and F0

′ -random variables un,0 having the law µn and satisfying E∥un,0′ ∥Hp "

C. Solve the Galerkin system with this initial condition and denote the solution by un′ . These

solutions are stationary processes in PnH. Repeat the bounds from previous section to show that(un′ ,W ′) are tight on

Lloc2 (0,∞;H)∩Cloc

([0,∞);D

1A−

β

2

2)×Cloc([0,∞);U0).

Apply the Skorokhod representation theorem and proceed as before. The limit solution is sta-tionary.

32