# Strong Pathwise Solutions Of The Stochastic Navier-stokes System

## Transcript Of Strong Pathwise Solutions Of The Stochastic Navier-stokes System

Advances in Diﬀerential Equations

Volume 14, Numbers 5-6 (2009), 567–600

STRONG PATHWISE SOLUTIONS OF THE STOCHASTIC NAVIER-STOKES SYSTEM

Nathan Glatt-Holtz Department of Mathematics, Indiana University, Bloomington, IN 47405

Mohammed Ziane Department of Mathematics, University of Southern California

Los Angeles, CA 90089

(Submitted by: Roger Temam)

Abstract. We consider the stochastic Navier-Stokes equations forced by a multiplicative white noise on a bounded domain in space dimensions two and three. We establish the local existence and uniqueness of strong or pathwise solutions when the initial data takes values in H1. In the two-dimensional case, we show that these solutions exist for all time. The proof is based on ﬁnite-dimensional approximations, decomposition into high and low modes and pairwise comparison techniques.

1. Introduction

In this article we study the Navier-Stokes equations in space dimension d = 2, 3, on a bounded domain M forced by a multiplicative white noise

∂tu + (u · ∇)u − νΔu + ∇p = f + g(u)W˙ ,

(1.1a)

div u = 0,

(1.1b)

u(0) = u0,

(1.1c)

u|M = 0.

(1.1d)

The system (1.1) describes the ﬂow of a viscous incompressible ﬂuid. Here u = (u1, . . . , ud), p and ν represent the velocity ﬁeld, the pressure and the coeﬃcient of kinematic viscosity respectively. The addition of the white noise driven terms to the basic governing equations is natural for both practical and theoretical applications. Such stochastically forced terms are used to account for numerical and empirical uncertainties and have been proposed as a model for turbulence.

Accepted for publication: December 2008. AMS Subject Classiﬁcations: 60H15, 35Q30, 76D03. Supported in part by the NSF grant DMS-0505974.

567

568

Nathan Glatt-Holtz and Mohammed Ziane

The mathematical literature for the stochastic Navier-Stokes equations is extensive and dates back to the early 1970’s with the work of Bensoussan and Temam [2]. For the study of well posedness, new diﬃculties related to compactness often arise due to the addition of the probabilistic parameter. For situations where continuous dependence on initial data remains open (for example in d = 3 when the initial data merely takes values in L2), it has proven fruitful to consider martingale solutions. Here, one constructs a probabilistic basis as part of the solution. For this context we refer the reader to the works of Viot [30], Cruzeiro [10], Capinski and Gatarek [6], Flandoli and Gatarek [15], and Mikulevicius and Rozovskii [25].

On the other hand, when working in spaces where continuous dependence on the initial data can be expected, existence of solutions can sometimes be established on a preordained probability space. Such solutions are referred to in the literature as “pathwise” solutions. In the two-dimensional setting, Da Prato and Zabczyk [12] and later Breckner [4] as well as Menaldi and Sritharan [21] established the existence of pathwise solutions where u takes values in L∞([0, T ], L2). On the other hand, Bensoussan and Frehse [3] have established local solutions in 3-d for the class Cβ([0, T ]; H2s) where 3/4 < s < 1 and β < 1 − s. The existence of pathwise, global solutions for the two-dimensional primitive equations of the ocean with multiplicative noise was recently established by Glatt-Holtz and Ziane in [17], for the case when u and its vertical gradient are initially in L2. In the works of Brzezniak and Peszat [5] and Mikulevicius and Rozovsky [23], the case of arbitrary space dimensions for local solutions evolving in Sobolev spaces of type W 1,p for p > d is addressed. Despite these extensive investigations, to the best of our knowledge, no one has addressed the case of local, pathwise, H1−valued (W 1,2) solutions for the 3-d Navier-Stokes equations with multiplicative noise.

As we are working at the intersection of two ﬁelds, we should note that the terminology may cause some confusion. In th literature for stochastic diﬀerential equations the term “weak solution” is sometimes used synonymously with the term “martingale solution” while the designation “strong solution” may be used for a “pathwise solution”. See the introductory text of Øksendal [26] for example. The former terminologies are avoided here because it is confusing in the context of partial diﬀerential equations. Indeed, from the partial diﬀerential equations point of view, strong solutions are solutions which are uniformly bounded in H1, while weak solutions are those which are merely bounded in L2. In this work we are therefore considering

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569

solutions which are strong in both the probabilistic and partial diﬀerential equations senses, which we shall call “strong pathwise solutions,” often dropping the pathwise designation when the context is clear.

The exposition is organized as follows. In the ﬁrst section we review the basic setting, deﬁning the relevant function spaces and introducing various notions of pathwise solutions. We then turn to the Galerkin scheme which we analyze by modifying a pairwise comparison technique [23]. Key estimates are achieved using decompositions into high and low modes. In this way we are able to extract a locally strongly convergent subsequence and surmount the diﬃcult issue of compactness. In the third section, we establish the existence and uniqueness of a local solution u evolving continuously in H1 up to a maximal existence time ξ. For samples where ξ is ﬁnite we show that, on the one hand, the L2 norm remains bounded and that on the other hand the H1 norm of the solution blows up. By showing that certain quantities are under control in the two-dimensional case we are able to use this later blow-up criteria to give the proof for the global existence of strong solutions in the two-dimensional case. In the ﬁnal section, we formulate and prove some abstract convergence results used in the proof of the main theorem. We believe these results to be more widely applicable for the study of well posedness of other non-linear stochastic partial diﬀerential equations and therefore hold independent interest.

2. The abstract functional analytic setting

We begin by reviewing some basic function spaces associated with (1.1). In what follows d is the spatial dimension, the physical cases d = 2, 3, being the focus of our attention below. For simplicity, we assume that the boundary ∂M is smooth. Let

V := {φ ∈ (C0∞(M))d : ∇ · φ = 0},

(2.1)

and

H := clL2(M)V = {u ∈ L2(M)d : ∇ · u = 0, u · n = 0}.

(2.2)

Here, n is the outer pointing unit normal to ∂M. On H we take the L2

inner product and norm

(u, v) := u · vdM, |u| := (u, u).

M

(2.3)

The Leray-Hopf projector, PH , is deﬁned as the orthogonal projection of L2(M)d onto H. Deﬁne also

V := clH1(M)V = {u ∈ H01(M)d : ∇ · u = 0}.

(2.4)

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On this set we use the H1 norm and inner products

((u, v)) := ∇u · ∇vdM,

M

u := ((u, u)).

(2.5)

Note that, due to the Dirichlet boundary condition (cf. (1.1d)), the Poincar´e

inequality,

|u| ≤ C u , ∀u ∈ V,

(2.6)

holds, justifying (2.5) as a norm. Take V to be the dual of V , relative to H

with the pairing notated by ·, · .

We next deﬁne the Stokes operator A. A is understood as a bounded

linear map from V to V via Au, v = ((u, v)) u, v ∈ V. A can be extended

to an unbounded operator from H to H according to Au = −PH Δu with the domain D(A) = H2(M)∩V . By applying the theory of symmetric, compact, operators for A−1, one can prove the existence of an orthonormal basis {ek} for H of eigenfunctions of A. Here, the associated eigenvalues {λk} form an unbounded, increasing sequence 0 < λ1 < λ2 ≤ . . . ≤ λn ≤ λn+1 ≤ . . . . We shall also make use of the fractional powers of A. For u ∈ H, we denote

uk = (u, ek). Given α > 0, take

D(Aα) = u ∈ H : λ2kα|uk|2 < ∞ ,

k

(2.7)

and deﬁne Aαu = k λαk ukek, u ∈ D(Aα). We equip D(Aα) with the norm |u|2α := |Aαu|2 = k λ2kα|uk|2. Deﬁne Hn = span{e1, . . . , en} and take Pn to be the projection from H onto this space. Let Qn = I − Pn. The following extension of the Poincar´e inequality will be used for the estimates below.

Lemma 2.1. Suppose that α1 < α2. For any u ∈ D(Aα2), |Qnu|α1 ≤ λαn1−α2 |Qnu|α2 , |Pnu|α2 ≤ λαn2−α1 |Pnu|α1 .

(2.8) (2.9)

Proof. Working from the deﬁnitions,

|Q u|2 ≤ ∞ λk2(α2−α1) λ2α1 |u |2 = 1 |Q u|2 .

n α1 k=n+1 λn2(α2−α1) k k

λn2(α2−α1) n α2

(2.10)

Similarly,

n

|Pnu|2α2 ≤ λn2(α2−α1) λ2kα1 |uk|2 = λn2(α2−α1)|Pnu|2α1 .

k=1

(2.11)

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The non-linear portion of (1.1) is given by

B(u, v) := PH (u · ∇)v = PH (uj∂jv), u, v ∈ V.

(2.12)

Here and below, we occasionally make use of the Einstein convention of summing repeated indices from 1 to d. For notational convenience we will sometimes write B(u) := B(u, u). For d = 2, 3, the non-linear functional B can be shown to be well deﬁned as a map from V × V to V according to

B(u, v), w := (u · ∇)v · wdM = uj∂jvkwkdM.

M

M

We shall need the following classical facts concerning B.

(2.13)

Lemma 2.2. (i) B is continuous from V × V to V with

B(u, v), v = 0,

(2.14)

and

⎧

⎪⎨|u|1/2 u 1/2 v |w|1/2 w 1/2

| B(u, v), w | ≤ C ⎪⎩|uu|1/2v u|w1|/12/2vw w1/2

in d = 2, in d = 3, in d = 3,

(2.15)

for all u, v, w ∈ V . (ii) B is also continuous from V × D(A) to H. If u ∈ V , v ∈ D(A), and

w ∈ H, then

|(B(u, v), w)| ≤ C

|u|1/2 u 1/2 v 1/2|Av|1/2|w| u v 1/2|Av|1/2|w|

in d = 2, in d = 3.

(2.16)

(iii) If u ∈ D(A), then B(u) ∈ V, and B(u) 2 ≤ C u |Au|3 + |u|1/2|Au|7/2 in d = 2, 3.

(2.17)

Proof. The items (i) and (ii) are classical and are easily established using H¨older’s inequality and the Sobolev embedding theorem (see [29] or [8]). For item (iii), ﬁx u ∈ V. We have

B(u) 2 ≤ |∂m(uj∂juk)∂m(ul∂luk)| dM.

M

We prove the case d = 3; the case d = 2 is similar. We have

|φ|L∞ ≤ C|Aφ|3/4|φ|1/4, φ ∈ D(A). This estimate and the embedding of H1 in L6 implies

B(u) 2 ≤ C(|∇u|3L6 u + |Au||∇u|2L6 |u|L6 + |u|2L∞ |Au|2) ≤ C(|Au|3 u + |Au|7/2|u|1/2).

(2.18) (2.19) (2.20)

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The stochastically driven term in (1.1) can be written formally in the

expansion

g(u)W˙ = gk(u)β˙k,

(2.21)

k

where βk are independent standard Brownian motions. To make this rigorous, we recall some deﬁnitions.

Deﬁnition 2.1. A stochastic basis S := (Ω, F , {Ft}t≥0, P, {βk}k≥1) consists of a probability space (Ω, F, P) equipped with a complete, right-continuous

ﬁltration, namely P(A) = 0 ⇒ A ∈ F0, Ft = ∩s>tFs and a sequence of mutually independent, standard, Brownian motions βk relative to this ﬁltration.

We also need to deﬁne a class of spaces for g = {gk}k≥1.

Deﬁnition 2.2. Suppose U is any (separable) Hilbert space. We deﬁne 2(U ) to be the set of all sequences h = {hk}k≥1 of elements in U so that

|h|22(U) := |hk|2U < ∞.

k

For any normed space Y , we say that h : Y × [0, T ] × Ω → Lipschitz with constant KY , if for all x, y ∈ Y

(2.22) 2(U ) is uniformly

|h(x, t, ω) − h(y, t, ω)| 2(U) ≤ KY |x − y|Y ,

and |h(x, t, ω)| 2(U) ≤ KY (1 + |x|Y ).

We denote the collection of all such mappings Lipu(Y, 2(U )).

(2.23) (2.24)

For the analysis below we shall assume that g = {gk} : Ω × [0, ∞) × H → 2(H),

(2.25)

and that

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))). (2.26)

We shall assume moreover that if u : [0, T ] × Ω → H is predictable,1 then so is g(u). Given an H-valued predictable process u ∈ L2(Ω; L2(0, T ; H)),

1For a given stochastic basis S, let Φ = Ω × [0, ∞) and take G to be the σ-algebra generated by sets of the form (s, t] × F, 0 ≤ s < t < ∞, F ∈ Fs; {0} × F, F ∈ F0. Recall that a U valued process u is called predictable (with respect to the stochastic basis S) if it is (Φ, G) − (U, B(U )) measurable.

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the series expansion (2.21) can be shown to be well deﬁned as a stochastic

integral and

τ

g(u)dW, v =

0

τ

τ

gk(u)dβk, v =

(gk(u), v)dβk, (2.27)

k0

k0

for all v ∈ H and stopping times τ . See [11] or [27] for detailed constructions.

In order to show that the conditions imposed above for g are not overly

restrictive we now consider some examples of stochastic forcing regimes sat-

isfying (2.26).

Example 2.1. (i) (Independently forced modes) Suppose (κk(t, ω)) is any sequence uniformly bounded in L∞([0, T ] × Ω). We force the modes inde-

pendently, deﬁning gk(v, t, ω) = κk(t, ω)(v, ek)ek. In this case the Lipschitz constants can be taken to be

KH = KV = KD(A) = sup |κk(t, ω)|.

ω,k,t

(2.28)

(ii) (Uniform forcing) Given a uniformly square summable sequence ak(t, ω) we can take gk(v, t, ω) = ak(t, ω)v, with

KH = KV = KD(A) =

sup ak(t, ω)2 1/2

t,ω k

as the Lipschitz constants.

(iii) (Additive noise) We can also include the case when the noise term does

not depend on the solution gk(v, t, ω) = gk(t, ω). Here,

KU := sup

t,ω

|gk(t, ω)|2U 1/2

k

for U = H, V, D(A) as desired.

With the above framework in place, we next give a variational deﬁnition for local pathwise solutions of the stochastic Navier-Stokes equations. Given a Hilbert space X, for p ∈ [1, ∞], we denote

Lploc([0, ∞); X) = Lp([0, T ]; X),

T >0

Cw([0, ∞); X) = {v ∈ L∞ loc([0, ∞); X) : (v, x) ∈ C([0, ∞); R), ∀x ∈ X}.

Deﬁnition 2.3. (Weak and Strong Pathwise Solutions) Let S be a ﬁxed stochastic basis. Assume that u0 is F0 measurable with u0 ∈ L2(Ω, V ). Suppose that f and g are V and 2(H) valued, predictable processes respectively with

f ∈ L2(Ω; L2([0, ∞); H)),

(2.29)

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Nathan Glatt-Holtz and Mohammed Ziane

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))).

(i) We say that the pair (u, τ ) is a local weak (pathwise) solution if τ is a strictly positive stopping time and u(· ∧ τ ) is a predictable process in V , with

u(· ∧ τ ) ∈ L2(Ω; Cw([0, ∞); H)), u11t≤τ ∈ L2(Ω; L2loc([0, ∞); V )), (2.30)

and so that for any t > 0

t∧τ

t∧τ

t∧τ

u(t ∧ τ ) +

(νAu + B(u)) dt = u(0) +

f dt +

g(u)dW, (2.31)

0

0

0

in V . This equality is equivalent to requiring that for all v ∈ V

t∧τ

u(t ∧ τ ), v +

νAu + B(u), v dt

(2.32)

0

t∧τ

∞ t∧τ

= u(0), v +

f, v dt +

gk(u), v dβk.

0

k=1 0

(ii) The pair (u, τ ) is a local strong (pathwise) solution if τ is strictly positive and u(· ∧ τ ) is a predictable process in H with

u(· ∧ τ ) ∈ L2(Ω; C([0, ∞); V )), u11t≤τ ∈ L2(Ω; L2loc([0, ∞); D(A))), (2.33)

and such that u satisﬁes (2.31) as an equation in H. (iii) Suppose that u is a predictable process in V and that ξ is a strictly

positive stopping time. The pair (u, ξ) is said to be a maximal (pathwise) strong solution, if there exists an increasing sequence τn with

τn ↑ ξ a.s.,

(2.34)

such that each pair (u, τn) is a local strong solution and so that

ξ

sup u 2 + |Au|2dt = ∞,

t≤ξ

0

on the set {ξ < ∞}. If, in addition

sup u 2 +

t∈[0,τn]

τn

|Au|2ds = n,

0

(2.35) (2.36)

on the set {ξ < ∞}, then we say that {τn} announces ξ.

Remark 2.1. (i) For the “pathwise” solutions we consider, the stochastic basis is given in advance. In particular, solutions corresponding to diﬀerent initial laws are shown to be driven by the same underlying Wiener process. This is in contrast to the theory of martingale solutions considered for many non-linear systems. In that case, the underlying probability space

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is constructed as part of the solution. See [11], chapter 8 or [24] and the

references in the introduction. Since the context is clear, we will drop the

“pathwise” designation for the remainder of the exposition.

(ii) If (u, τ ) is a local strong solution, then (2.33) implies that

τ

E sup u 2 + |Au|2ds < ∞.

t∈[0,τ ]

0

(2.37)

So far, we are not able to show that E u(t) 2 is ﬁnite for any ﬁxed (deterministic) t > 0. This is the case even in the two-dimensional case where we prove the existence of a global strong solution (cf. Proposition 4.2). (iii) Suppose that (u, τ ) is a local strong solution. By applying an inﬁnitedimensional version of the Itoˆ lemma (see [28] or [27]) one can show that on the interval [0, τ ], for any p ≥ 2, |u|p satisﬁes

d|u|p + pν u 2|u|p−2dt = p f, u |u|p−2dt + p2 ∞ |gk(u)|2|u|p−2dt

k=1

+ p(p 2− 2) ∞ gk(u), u 2|u|p−4dt + p ∞ gk(u), u |u|p−2dβk.

k=1

k=1

(2.38)

Note that the non-linear term B drops out due to the cancellation property. Similarly for u p, we have

d u p + pν|Au|2 u p−2dt

(2.39)

= p f − B(u), Au

u p−2dt + p ∞ 2

gk(u) 2 u p−2dt

k=1

p(p − 2) ∞

∞ 2 p−4

+2

gk(u), Au u dt + p gk(u), Au

k=1

k=1

u p−2dβk.

3. The Galerkin Scheme and Comparison Estimates

The ﬁrst step to prove the existence of a solution is to approximate the full equations with a sequence of ﬁnite-dimensional stochastic diﬀerential equations, the Galerkin systems.

Deﬁnition 3.1. An adapted process un in C([0, T ]; Hn) is a solution to the Galerkin system of order n if, for any v ∈ Hn,

∞

d un, v + νAun + B(un), v dt = f, v dt + gk(un), v dβk,

(3.1)

k=1

un(0), v = u0, v .

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Nathan Glatt-Holtz and Mohammed Ziane

We can also write (3.1) as an equation in Hn(∼= Rn)

∞

dun + (νAun + PnB(un))dt = Pnf dt + Pngk(un)dβk,

k=1

un(0) = Pnu0 := un0 .

(3.2)

The existence of solutions to (3.1) is classical and relies on a priori bounds that are established using the cancellation property (2.14). See [16] for detailed proofs. Uniqueness, which is not essential for our purposes, is established as below for the full inﬁnite-dimensional system.

We now proceed to establish the main result of the section. Note that the conditions established hereafter are precisely those needed to apply Lemma 5.1 in Proposition 4.2 below.

Proposition 3.1. Suppose that d = 2, 3 and let {un} be the sequence of solutions of (3.1). We assume that for some 0 < M˜ < ∞

u0 ≤ M˜ a.s.,

(3.3)

and that f ∈ L2(Ω; L2([0, T ]; H),

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))),

(3.4)

where the spaces for g and the associated Lipschitz constants used are given

as in Deﬁnition 2.2. Consider the collection of stopping times

TnM,T = τ ≤ T :

τ

1/2

sup un 2 + ν |Aun|2dt ≤ M + un0

t∈[0,τ ]

0

, (3.5)

and take TmM,n,T := TmM,T ∩ TnM,T . Then (i) For any T > 0 and M > 1

lim sup sup E sup um − un 2 + ν

n→∞ m>n τ ∈TmM,n,T

t∈[0,τ ]

τ

|A(um − un)|2dt

0

= 0.

(3.6)

(ii) Moreover, if for n ∈ N, S > 0 and a stopping time τ, if

An(τ, S) =

τ ∧S

sup un 2 + ν

|Aun|2dt > un0 2 + (M−1)2 ,

t∈[0,τ ∧S]

0

then

lim sup sup P(An(τ, S)) = 0.

S→0 n τ ∈TnM,T

(3.7)

Volume 14, Numbers 5-6 (2009), 567–600

STRONG PATHWISE SOLUTIONS OF THE STOCHASTIC NAVIER-STOKES SYSTEM

Nathan Glatt-Holtz Department of Mathematics, Indiana University, Bloomington, IN 47405

Mohammed Ziane Department of Mathematics, University of Southern California

Los Angeles, CA 90089

(Submitted by: Roger Temam)

Abstract. We consider the stochastic Navier-Stokes equations forced by a multiplicative white noise on a bounded domain in space dimensions two and three. We establish the local existence and uniqueness of strong or pathwise solutions when the initial data takes values in H1. In the two-dimensional case, we show that these solutions exist for all time. The proof is based on ﬁnite-dimensional approximations, decomposition into high and low modes and pairwise comparison techniques.

1. Introduction

In this article we study the Navier-Stokes equations in space dimension d = 2, 3, on a bounded domain M forced by a multiplicative white noise

∂tu + (u · ∇)u − νΔu + ∇p = f + g(u)W˙ ,

(1.1a)

div u = 0,

(1.1b)

u(0) = u0,

(1.1c)

u|M = 0.

(1.1d)

The system (1.1) describes the ﬂow of a viscous incompressible ﬂuid. Here u = (u1, . . . , ud), p and ν represent the velocity ﬁeld, the pressure and the coeﬃcient of kinematic viscosity respectively. The addition of the white noise driven terms to the basic governing equations is natural for both practical and theoretical applications. Such stochastically forced terms are used to account for numerical and empirical uncertainties and have been proposed as a model for turbulence.

Accepted for publication: December 2008. AMS Subject Classiﬁcations: 60H15, 35Q30, 76D03. Supported in part by the NSF grant DMS-0505974.

567

568

Nathan Glatt-Holtz and Mohammed Ziane

The mathematical literature for the stochastic Navier-Stokes equations is extensive and dates back to the early 1970’s with the work of Bensoussan and Temam [2]. For the study of well posedness, new diﬃculties related to compactness often arise due to the addition of the probabilistic parameter. For situations where continuous dependence on initial data remains open (for example in d = 3 when the initial data merely takes values in L2), it has proven fruitful to consider martingale solutions. Here, one constructs a probabilistic basis as part of the solution. For this context we refer the reader to the works of Viot [30], Cruzeiro [10], Capinski and Gatarek [6], Flandoli and Gatarek [15], and Mikulevicius and Rozovskii [25].

On the other hand, when working in spaces where continuous dependence on the initial data can be expected, existence of solutions can sometimes be established on a preordained probability space. Such solutions are referred to in the literature as “pathwise” solutions. In the two-dimensional setting, Da Prato and Zabczyk [12] and later Breckner [4] as well as Menaldi and Sritharan [21] established the existence of pathwise solutions where u takes values in L∞([0, T ], L2). On the other hand, Bensoussan and Frehse [3] have established local solutions in 3-d for the class Cβ([0, T ]; H2s) where 3/4 < s < 1 and β < 1 − s. The existence of pathwise, global solutions for the two-dimensional primitive equations of the ocean with multiplicative noise was recently established by Glatt-Holtz and Ziane in [17], for the case when u and its vertical gradient are initially in L2. In the works of Brzezniak and Peszat [5] and Mikulevicius and Rozovsky [23], the case of arbitrary space dimensions for local solutions evolving in Sobolev spaces of type W 1,p for p > d is addressed. Despite these extensive investigations, to the best of our knowledge, no one has addressed the case of local, pathwise, H1−valued (W 1,2) solutions for the 3-d Navier-Stokes equations with multiplicative noise.

As we are working at the intersection of two ﬁelds, we should note that the terminology may cause some confusion. In th literature for stochastic diﬀerential equations the term “weak solution” is sometimes used synonymously with the term “martingale solution” while the designation “strong solution” may be used for a “pathwise solution”. See the introductory text of Øksendal [26] for example. The former terminologies are avoided here because it is confusing in the context of partial diﬀerential equations. Indeed, from the partial diﬀerential equations point of view, strong solutions are solutions which are uniformly bounded in H1, while weak solutions are those which are merely bounded in L2. In this work we are therefore considering

The Navier Stokes Equations with Stochastic Forcing

569

solutions which are strong in both the probabilistic and partial diﬀerential equations senses, which we shall call “strong pathwise solutions,” often dropping the pathwise designation when the context is clear.

The exposition is organized as follows. In the ﬁrst section we review the basic setting, deﬁning the relevant function spaces and introducing various notions of pathwise solutions. We then turn to the Galerkin scheme which we analyze by modifying a pairwise comparison technique [23]. Key estimates are achieved using decompositions into high and low modes. In this way we are able to extract a locally strongly convergent subsequence and surmount the diﬃcult issue of compactness. In the third section, we establish the existence and uniqueness of a local solution u evolving continuously in H1 up to a maximal existence time ξ. For samples where ξ is ﬁnite we show that, on the one hand, the L2 norm remains bounded and that on the other hand the H1 norm of the solution blows up. By showing that certain quantities are under control in the two-dimensional case we are able to use this later blow-up criteria to give the proof for the global existence of strong solutions in the two-dimensional case. In the ﬁnal section, we formulate and prove some abstract convergence results used in the proof of the main theorem. We believe these results to be more widely applicable for the study of well posedness of other non-linear stochastic partial diﬀerential equations and therefore hold independent interest.

2. The abstract functional analytic setting

We begin by reviewing some basic function spaces associated with (1.1). In what follows d is the spatial dimension, the physical cases d = 2, 3, being the focus of our attention below. For simplicity, we assume that the boundary ∂M is smooth. Let

V := {φ ∈ (C0∞(M))d : ∇ · φ = 0},

(2.1)

and

H := clL2(M)V = {u ∈ L2(M)d : ∇ · u = 0, u · n = 0}.

(2.2)

Here, n is the outer pointing unit normal to ∂M. On H we take the L2

inner product and norm

(u, v) := u · vdM, |u| := (u, u).

M

(2.3)

The Leray-Hopf projector, PH , is deﬁned as the orthogonal projection of L2(M)d onto H. Deﬁne also

V := clH1(M)V = {u ∈ H01(M)d : ∇ · u = 0}.

(2.4)

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Nathan Glatt-Holtz and Mohammed Ziane

On this set we use the H1 norm and inner products

((u, v)) := ∇u · ∇vdM,

M

u := ((u, u)).

(2.5)

Note that, due to the Dirichlet boundary condition (cf. (1.1d)), the Poincar´e

inequality,

|u| ≤ C u , ∀u ∈ V,

(2.6)

holds, justifying (2.5) as a norm. Take V to be the dual of V , relative to H

with the pairing notated by ·, · .

We next deﬁne the Stokes operator A. A is understood as a bounded

linear map from V to V via Au, v = ((u, v)) u, v ∈ V. A can be extended

to an unbounded operator from H to H according to Au = −PH Δu with the domain D(A) = H2(M)∩V . By applying the theory of symmetric, compact, operators for A−1, one can prove the existence of an orthonormal basis {ek} for H of eigenfunctions of A. Here, the associated eigenvalues {λk} form an unbounded, increasing sequence 0 < λ1 < λ2 ≤ . . . ≤ λn ≤ λn+1 ≤ . . . . We shall also make use of the fractional powers of A. For u ∈ H, we denote

uk = (u, ek). Given α > 0, take

D(Aα) = u ∈ H : λ2kα|uk|2 < ∞ ,

k

(2.7)

and deﬁne Aαu = k λαk ukek, u ∈ D(Aα). We equip D(Aα) with the norm |u|2α := |Aαu|2 = k λ2kα|uk|2. Deﬁne Hn = span{e1, . . . , en} and take Pn to be the projection from H onto this space. Let Qn = I − Pn. The following extension of the Poincar´e inequality will be used for the estimates below.

Lemma 2.1. Suppose that α1 < α2. For any u ∈ D(Aα2), |Qnu|α1 ≤ λαn1−α2 |Qnu|α2 , |Pnu|α2 ≤ λαn2−α1 |Pnu|α1 .

(2.8) (2.9)

Proof. Working from the deﬁnitions,

|Q u|2 ≤ ∞ λk2(α2−α1) λ2α1 |u |2 = 1 |Q u|2 .

n α1 k=n+1 λn2(α2−α1) k k

λn2(α2−α1) n α2

(2.10)

Similarly,

n

|Pnu|2α2 ≤ λn2(α2−α1) λ2kα1 |uk|2 = λn2(α2−α1)|Pnu|2α1 .

k=1

(2.11)

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571

The non-linear portion of (1.1) is given by

B(u, v) := PH (u · ∇)v = PH (uj∂jv), u, v ∈ V.

(2.12)

Here and below, we occasionally make use of the Einstein convention of summing repeated indices from 1 to d. For notational convenience we will sometimes write B(u) := B(u, u). For d = 2, 3, the non-linear functional B can be shown to be well deﬁned as a map from V × V to V according to

B(u, v), w := (u · ∇)v · wdM = uj∂jvkwkdM.

M

M

We shall need the following classical facts concerning B.

(2.13)

Lemma 2.2. (i) B is continuous from V × V to V with

B(u, v), v = 0,

(2.14)

and

⎧

⎪⎨|u|1/2 u 1/2 v |w|1/2 w 1/2

| B(u, v), w | ≤ C ⎪⎩|uu|1/2v u|w1|/12/2vw w1/2

in d = 2, in d = 3, in d = 3,

(2.15)

for all u, v, w ∈ V . (ii) B is also continuous from V × D(A) to H. If u ∈ V , v ∈ D(A), and

w ∈ H, then

|(B(u, v), w)| ≤ C

|u|1/2 u 1/2 v 1/2|Av|1/2|w| u v 1/2|Av|1/2|w|

in d = 2, in d = 3.

(2.16)

(iii) If u ∈ D(A), then B(u) ∈ V, and B(u) 2 ≤ C u |Au|3 + |u|1/2|Au|7/2 in d = 2, 3.

(2.17)

Proof. The items (i) and (ii) are classical and are easily established using H¨older’s inequality and the Sobolev embedding theorem (see [29] or [8]). For item (iii), ﬁx u ∈ V. We have

B(u) 2 ≤ |∂m(uj∂juk)∂m(ul∂luk)| dM.

M

We prove the case d = 3; the case d = 2 is similar. We have

|φ|L∞ ≤ C|Aφ|3/4|φ|1/4, φ ∈ D(A). This estimate and the embedding of H1 in L6 implies

B(u) 2 ≤ C(|∇u|3L6 u + |Au||∇u|2L6 |u|L6 + |u|2L∞ |Au|2) ≤ C(|Au|3 u + |Au|7/2|u|1/2).

(2.18) (2.19) (2.20)

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Nathan Glatt-Holtz and Mohammed Ziane

The stochastically driven term in (1.1) can be written formally in the

expansion

g(u)W˙ = gk(u)β˙k,

(2.21)

k

where βk are independent standard Brownian motions. To make this rigorous, we recall some deﬁnitions.

Deﬁnition 2.1. A stochastic basis S := (Ω, F , {Ft}t≥0, P, {βk}k≥1) consists of a probability space (Ω, F, P) equipped with a complete, right-continuous

ﬁltration, namely P(A) = 0 ⇒ A ∈ F0, Ft = ∩s>tFs and a sequence of mutually independent, standard, Brownian motions βk relative to this ﬁltration.

We also need to deﬁne a class of spaces for g = {gk}k≥1.

Deﬁnition 2.2. Suppose U is any (separable) Hilbert space. We deﬁne 2(U ) to be the set of all sequences h = {hk}k≥1 of elements in U so that

|h|22(U) := |hk|2U < ∞.

k

For any normed space Y , we say that h : Y × [0, T ] × Ω → Lipschitz with constant KY , if for all x, y ∈ Y

(2.22) 2(U ) is uniformly

|h(x, t, ω) − h(y, t, ω)| 2(U) ≤ KY |x − y|Y ,

and |h(x, t, ω)| 2(U) ≤ KY (1 + |x|Y ).

We denote the collection of all such mappings Lipu(Y, 2(U )).

(2.23) (2.24)

For the analysis below we shall assume that g = {gk} : Ω × [0, ∞) × H → 2(H),

(2.25)

and that

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))). (2.26)

We shall assume moreover that if u : [0, T ] × Ω → H is predictable,1 then so is g(u). Given an H-valued predictable process u ∈ L2(Ω; L2(0, T ; H)),

1For a given stochastic basis S, let Φ = Ω × [0, ∞) and take G to be the σ-algebra generated by sets of the form (s, t] × F, 0 ≤ s < t < ∞, F ∈ Fs; {0} × F, F ∈ F0. Recall that a U valued process u is called predictable (with respect to the stochastic basis S) if it is (Φ, G) − (U, B(U )) measurable.

The Navier Stokes Equations with Stochastic Forcing

573

the series expansion (2.21) can be shown to be well deﬁned as a stochastic

integral and

τ

g(u)dW, v =

0

τ

τ

gk(u)dβk, v =

(gk(u), v)dβk, (2.27)

k0

k0

for all v ∈ H and stopping times τ . See [11] or [27] for detailed constructions.

In order to show that the conditions imposed above for g are not overly

restrictive we now consider some examples of stochastic forcing regimes sat-

isfying (2.26).

Example 2.1. (i) (Independently forced modes) Suppose (κk(t, ω)) is any sequence uniformly bounded in L∞([0, T ] × Ω). We force the modes inde-

pendently, deﬁning gk(v, t, ω) = κk(t, ω)(v, ek)ek. In this case the Lipschitz constants can be taken to be

KH = KV = KD(A) = sup |κk(t, ω)|.

ω,k,t

(2.28)

(ii) (Uniform forcing) Given a uniformly square summable sequence ak(t, ω) we can take gk(v, t, ω) = ak(t, ω)v, with

KH = KV = KD(A) =

sup ak(t, ω)2 1/2

t,ω k

as the Lipschitz constants.

(iii) (Additive noise) We can also include the case when the noise term does

not depend on the solution gk(v, t, ω) = gk(t, ω). Here,

KU := sup

t,ω

|gk(t, ω)|2U 1/2

k

for U = H, V, D(A) as desired.

With the above framework in place, we next give a variational deﬁnition for local pathwise solutions of the stochastic Navier-Stokes equations. Given a Hilbert space X, for p ∈ [1, ∞], we denote

Lploc([0, ∞); X) = Lp([0, T ]; X),

T >0

Cw([0, ∞); X) = {v ∈ L∞ loc([0, ∞); X) : (v, x) ∈ C([0, ∞); R), ∀x ∈ X}.

Deﬁnition 2.3. (Weak and Strong Pathwise Solutions) Let S be a ﬁxed stochastic basis. Assume that u0 is F0 measurable with u0 ∈ L2(Ω, V ). Suppose that f and g are V and 2(H) valued, predictable processes respectively with

f ∈ L2(Ω; L2([0, ∞); H)),

(2.29)

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Nathan Glatt-Holtz and Mohammed Ziane

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))).

(i) We say that the pair (u, τ ) is a local weak (pathwise) solution if τ is a strictly positive stopping time and u(· ∧ τ ) is a predictable process in V , with

u(· ∧ τ ) ∈ L2(Ω; Cw([0, ∞); H)), u11t≤τ ∈ L2(Ω; L2loc([0, ∞); V )), (2.30)

and so that for any t > 0

t∧τ

t∧τ

t∧τ

u(t ∧ τ ) +

(νAu + B(u)) dt = u(0) +

f dt +

g(u)dW, (2.31)

0

0

0

in V . This equality is equivalent to requiring that for all v ∈ V

t∧τ

u(t ∧ τ ), v +

νAu + B(u), v dt

(2.32)

0

t∧τ

∞ t∧τ

= u(0), v +

f, v dt +

gk(u), v dβk.

0

k=1 0

(ii) The pair (u, τ ) is a local strong (pathwise) solution if τ is strictly positive and u(· ∧ τ ) is a predictable process in H with

u(· ∧ τ ) ∈ L2(Ω; C([0, ∞); V )), u11t≤τ ∈ L2(Ω; L2loc([0, ∞); D(A))), (2.33)

and such that u satisﬁes (2.31) as an equation in H. (iii) Suppose that u is a predictable process in V and that ξ is a strictly

positive stopping time. The pair (u, ξ) is said to be a maximal (pathwise) strong solution, if there exists an increasing sequence τn with

τn ↑ ξ a.s.,

(2.34)

such that each pair (u, τn) is a local strong solution and so that

ξ

sup u 2 + |Au|2dt = ∞,

t≤ξ

0

on the set {ξ < ∞}. If, in addition

sup u 2 +

t∈[0,τn]

τn

|Au|2ds = n,

0

(2.35) (2.36)

on the set {ξ < ∞}, then we say that {τn} announces ξ.

Remark 2.1. (i) For the “pathwise” solutions we consider, the stochastic basis is given in advance. In particular, solutions corresponding to diﬀerent initial laws are shown to be driven by the same underlying Wiener process. This is in contrast to the theory of martingale solutions considered for many non-linear systems. In that case, the underlying probability space

The Navier Stokes Equations with Stochastic Forcing

575

is constructed as part of the solution. See [11], chapter 8 or [24] and the

references in the introduction. Since the context is clear, we will drop the

“pathwise” designation for the remainder of the exposition.

(ii) If (u, τ ) is a local strong solution, then (2.33) implies that

τ

E sup u 2 + |Au|2ds < ∞.

t∈[0,τ ]

0

(2.37)

So far, we are not able to show that E u(t) 2 is ﬁnite for any ﬁxed (deterministic) t > 0. This is the case even in the two-dimensional case where we prove the existence of a global strong solution (cf. Proposition 4.2). (iii) Suppose that (u, τ ) is a local strong solution. By applying an inﬁnitedimensional version of the Itoˆ lemma (see [28] or [27]) one can show that on the interval [0, τ ], for any p ≥ 2, |u|p satisﬁes

d|u|p + pν u 2|u|p−2dt = p f, u |u|p−2dt + p2 ∞ |gk(u)|2|u|p−2dt

k=1

+ p(p 2− 2) ∞ gk(u), u 2|u|p−4dt + p ∞ gk(u), u |u|p−2dβk.

k=1

k=1

(2.38)

Note that the non-linear term B drops out due to the cancellation property. Similarly for u p, we have

d u p + pν|Au|2 u p−2dt

(2.39)

= p f − B(u), Au

u p−2dt + p ∞ 2

gk(u) 2 u p−2dt

k=1

p(p − 2) ∞

∞ 2 p−4

+2

gk(u), Au u dt + p gk(u), Au

k=1

k=1

u p−2dβk.

3. The Galerkin Scheme and Comparison Estimates

The ﬁrst step to prove the existence of a solution is to approximate the full equations with a sequence of ﬁnite-dimensional stochastic diﬀerential equations, the Galerkin systems.

Deﬁnition 3.1. An adapted process un in C([0, T ]; Hn) is a solution to the Galerkin system of order n if, for any v ∈ Hn,

∞

d un, v + νAun + B(un), v dt = f, v dt + gk(un), v dβk,

(3.1)

k=1

un(0), v = u0, v .

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Nathan Glatt-Holtz and Mohammed Ziane

We can also write (3.1) as an equation in Hn(∼= Rn)

∞

dun + (νAun + PnB(un))dt = Pnf dt + Pngk(un)dβk,

k=1

un(0) = Pnu0 := un0 .

(3.2)

The existence of solutions to (3.1) is classical and relies on a priori bounds that are established using the cancellation property (2.14). See [16] for detailed proofs. Uniqueness, which is not essential for our purposes, is established as below for the full inﬁnite-dimensional system.

We now proceed to establish the main result of the section. Note that the conditions established hereafter are precisely those needed to apply Lemma 5.1 in Proposition 4.2 below.

Proposition 3.1. Suppose that d = 2, 3 and let {un} be the sequence of solutions of (3.1). We assume that for some 0 < M˜ < ∞

u0 ≤ M˜ a.s.,

(3.3)

and that f ∈ L2(Ω; L2([0, T ]; H),

g ∈ Lipu(H, 2(H)) ∩ Lipu(V, 2(V )) ∩ Lipu(D(A), 2(D(A))),

(3.4)

where the spaces for g and the associated Lipschitz constants used are given

as in Deﬁnition 2.2. Consider the collection of stopping times

TnM,T = τ ≤ T :

τ

1/2

sup un 2 + ν |Aun|2dt ≤ M + un0

t∈[0,τ ]

0

, (3.5)

and take TmM,n,T := TmM,T ∩ TnM,T . Then (i) For any T > 0 and M > 1

lim sup sup E sup um − un 2 + ν

n→∞ m>n τ ∈TmM,n,T

t∈[0,τ ]

τ

|A(um − un)|2dt

0

= 0.

(3.6)

(ii) Moreover, if for n ∈ N, S > 0 and a stopping time τ, if

An(τ, S) =

τ ∧S

sup un 2 + ν

|Aun|2dt > un0 2 + (M−1)2 ,

t∈[0,τ ∧S]

0

then

lim sup sup P(An(τ, S)) = 0.

S→0 n τ ∈TnM,T

(3.7)