linear algebra and its nonlinear analysisstaff.ustc.edu.cn/~wangran/papers/exponential...

Nonlinear Analysis 132 (2016) 196–213

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Nonlinear Analysis

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Linear Algebra and its Applications 466 (2015) 102–116

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Linear Algebra and its Applications

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Inverse eigenvalue problem of Jacobi matrix

with mixed data

Ying Wei 1

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 January 2014Accepted 20 September 2014Available online 22 October 2014Submitted by Y. Wei

MSC:15A1815A57

Keywords:Jacobi matrixEigenvalueInverse problemSubmatrix

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from its eigenvalues, its leading principal submatrix and part of the eigenvalues of its submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.

© 2014 Published by Elsevier Inc.

E-mail address: [email protected] Tel.: +86 13914485239.

http://dx.doi.org/10.1016/j.laa.2014.09.0310024-3795/© 2014 Published by Elsevier Inc.

Exponential mixing for stochastic model of two-dimensionalsecond grade fluids

Ran Wanga, Jianliang Zhaia,∗, Tusheng Zhangb,a

a School of Mathematical Sciences, University of Science and Technology of China, Wu Wen Tsun KeyLaboratory of Mathematics, Chinese Academy of Science, Hefei, 230026, Chinab School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

a r t i c l e i n f o

Article history:Received 6 April 2015Accepted 11 November 2015Communicated by Enzo Mitidieri

MSC:primary 60H15secondary 35R6037L55

Keywords:Exponential mixingStochastic model for theincompressible second grade fluidsInvariant measureMarkov process

a b s t r a c t

In this paper, we establish the exponential mixing property of stochastic model forthe incompressible second grade fluids. The general criterion established by Odasso(2008) plays an important role.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we are concerned with the exponential mixing property of stochastic model for theincompressible second grade fluids which is a particular class of Non-Newtonian fluids. Let O be a connected,bounded open subset of R2 with boundary ∂O of class C3. We consider the equation

d(u− α∆u) +−ν∆u+ curl(u− α∆u)× u+∇P

dt = φ(u)dW, (1.1)

in O × (0,∞), under the following conditionsdiv u = 0 in O × (0,∞);u = 0 in ∂O × [0,∞);u(0) = x0 in O.

∗ Corresponding author.E-mail addresses: [email protected] (R. Wang), [email protected] (J. Zhai), [email protected]

(T. Zhang).

http://dx.doi.org/10.1016/j.na.2015.11.0090362-546X/© 2015 Elsevier Ltd. All rights reserved.

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http://dx.doi.org/10.1016/j.na.2015.11.009

R. Wang et al. / Nonlinear Analysis 132 (2016) 196–213 197

Here u := (u1, u2) and P represent the random velocity and modified pressure, respectively; curl denotes thecurl operator map;W is a cylindrical Wiener process on a Hilbert space U defined on a complete probabilityspace (Ω ,F , (Ft)t≥0,P), and φ(u)dW represents the external random force.

The interest in the investigation of the second grade fluids arises from the fact that it is an admissiblemodel of slow flow fluids, which contains a large class Non-Newtonian fluids such as industrial fluids,slurries, polymer melts, etc. Furthermore, the second grade fluid has general and pleasant properties such asboundedness, stability, and exponential decay, see [6]. It also has interesting connections with many otherfluid models, see [2,3,10–12,17,18] and references therein. For example, it can be taken as a generalization ofthe Navier–Stokes Equation. Indeed, Eq. (1.1) reduces to Navier–Stokes equation when α = 0. Furthermore,it was shown in [12] that the second grade fluids models are good approximations of the Navier–Stokesequation. Finally, we refer to [6–8,21] for a comprehensive theory of the second grade fluids.

The stochastic model of two-dimensional second grade fluids (1.1) has been recently studied in [14–16],where the authors obtained the existence and uniqueness of solutions and investigated the behavior of thesolutions as α→ 0. We mention that the martingale solutions of the system (1.1) driven by Levy noise arestudied in [9].

In this paper, we establish the exponential mixing property of stochastic models for the incompressiblesecond grade fluid driven by multiplicative, but possibly degenerate noise. The exponential mixingcharacterizes the long time behavior of the solutions of the stochastic partial differential equations. Moreprecisely, under reasonable conditions, we show that Eq. (1.1) has a unique invariant measure, and the lawof the solution converges to the invariant measure exponentially fast. We will apply the criterion establishedin [13] by Cyril Odasso. To this end, we need to prove the exponential integrability of certain energyfunctionals of the solutions, which is non-trivial.

This article is divided into four sections. In Section 2, we present some preliminaries. Section 3 is devotedto the formulation of the main result. The proof of our main result is given in Section 4.

Throughout this paper, we denote by C,C1, . . . any generic constant which may change from one line toanother. For a Hilbert space H, let Bb(H) be the space of bounded measurable functions on H, Cb(H) bethe space of bounded continuous functions. The space of probability measures on H is denoted byM(H).

2. Preliminaries

In this section, we will introduce some functional spaces and preliminary facts.For p, k ∈ N, let Lp(O) and W k,p(O) be the usual Lp and Sobolev spaces, and write W k,2(O) =: Hk(O).

Let W k,p0 (O) be the closure in W k,p(O) of C∞c (O) the space of infinitely differentiable functions withcompact support in O. We denote W k,20 (O) by Hk0 (O). We endow the Hilbert space H1

0 (O) with the scalarproduct

O∇u · ∇vdx =

2i=1

O

∂u

∂xi

∂v

∂xidx,

where ∇ is the gradient operator. The norm generated by this scalar product is equivalent to the usual normof W 1,2(O) in H1

0 (O).In what follows, we denote by X the space of R2-valued functions such that each component belongs to

the space X. Let

C :=u ∈ [C∞c (O)]2 such that div u = 0

,

V := closure of C in H1(O),H := closure of C in L2(O).

198 R. Wang et al. / Nonlinear Analysis 132 (2016) 196–213

On H, let (·, ·) and | · | be the inner product and the norm induced from L2(O). The inner product and thenorm of H1

0(O) are denoted respectively by ((·, ·)) and ∥ · ∥. We endow the space V with the norm generatedby the following scalar product

(u, v)V := (u, v) + α((u, v)), for any u, v ∈ V,

which is equivalent to ∥ · ∥. More precisely, we have

(c2P + α)−1∥v∥2V ≤ ∥v∥2 ≤ α−1∥v∥2V, for any v ∈ V, (2.2)

where cP is the constant from Poincare’s inequality.We also introduce the following space

W :=u ∈ V such that curl(u− α∆u) ∈ L2(O)

,

and endow it with the norm generated by the scalar product

(u, v)W := (u, v)V + (curl(u− α∆u), curl(v − α∆v)) . (2.3)

The following result states that the norm ∥ · ∥W induced by (·, ·)W is equivalent to the usual H3(O)-norm onW. This result can be found in [4,5,14].

Lemma 2.1. Set W :=v ∈ H3(O) such that div v = 0 and v|∂O = 0

. Then the following (algebraic and

topological) identity holds:

W = W.Moreover, there is a positive constant C such that

∥v∥2H3(O) ≤ C∥v∥2V + |curl(v − α∆v)|2

, for any v ∈W. (2.4)

From now on, we identify the space V with its dual space V∗ via the Riesz representation, and we havethe Gelfand triple

W ⊂ V ⊂W∗. (2.5)

We denote by ⟨f, v⟩ the action of any element f of W∗ on an element v ∈W. It is easy to see

(v, w)V = ⟨v, w⟩, ∀v ∈ V, w ∈W.

Note that the injection of W into V is compact. Thus, there exists a sequence ei : i = 1, 2, 3, . . . ofelements of W which forms an orthonormal basis in W. The elements of this sequence are the solutions ofthe eigenvalue problem

(v, ei)W = λi(v, ei)V, for any v ∈W, (2.6)

and the eigenvalues 0 < λi ↑ ∞. We have the following result from [4] about the regularity of the functionseii≥1.

Lemma 2.2. Let O be a bounded, simply-connected open subset of R2 with a boundary of class C3, then theeigenfunctions of (2.6) belong to H4(O).

Consider the following “generalized Stokes equation”:v − α∆v = f in O,div v = 0 in O,v = 0 on ∂O.

(2.7)

The following result can be found in [19,20], [16, Theorem 2.5] and [14, Theorem 2.2].


Lemma 2.3. Let O be a connected, bounded open subset of R2 with boundary ∂O of class Cl and let f be afunction in Hl, l ≥ 1. Then the system (2.7) admits a solution v ∈ Hl+2 ∩V. Moreover if f is in H, then vis unique and the following relations hold

(v, g)V = (f, g), for any g ∈ V,

and

∥v∥W ≤ K∥f∥V.

Define the Stokes operator by

Au := −P∆u, ∀u ∈ D(A) = H2(O) ∩ V, (2.8)

here we denote by P : L2(O)→ H the usual Helmholtz–Leray projector. It follows from Lemma 2.3 that theoperator (I +αA)−1 defines an isomorphism from Hl(O)∩H into Hl+2(O)∩V provided that ∂O is of classCl, l ≥ 1. Moreover, for any f ∈ Hl(O) ∩ V and g ∈ V, the following properties hold

((I + αA)−1f, g)V = (f, g),∥(I + αA)−1f∥W ≤ K∥f∥V.

Let

A := (I + αA)−1A.

Then A is a continuous linear operator from Hl(O) ∩ V onto itself for l ≥ 2, and A satisfies

( Au, v)V = (Au, v) = ((u, v)), for any u ∈W, v ∈ V.

Hence,

( Au, u)V = ∥u∥, for any u ∈W.

For any u, v, w ∈ C, let

b(u, v, w) :=2i,j=1

Oui∂vj∂xiwjdx.

Then for any smooth functions Φ, v and w, the following identity holds (see for instance [1,5]):

((curlΦ)× v, w) = b(v,Φ, w)− b(w,Φ, v).

Set

B(u, v) := curl(u− α∆u)× v, ∀u, v ∈W.

Now we recall the following lemma, which can be found in [14, Lemmas 2.3, 2.4], and also in [1,5].

Lemma 2.4. There exist positive constants K and Θ such that for any u, v, w ∈W,

|(curl(u− α∆u)× v, w)| ≤ K∥u∥H3 · ∥v∥V · ∥w∥W, (2.9)

and

|(curl(u− α∆u)× u,w)| ≤ Θ∥u∥2V · ∥w∥W. (2.10)


3. Formulation of the main result

In this section, we will state the precise assumptions on the coefficients and collect some preliminaryresults from [14,16].

Assume that W (s), s ∈ [0,∞) is an U -cylindrical Wiener process admitting the following representa-tion:

W =n

βnen,

where (en)n is a complete orthonormal system of U and (βn)n is a sequence of independent Brownian motions.Given Hilbert spaces H1, H2, we denote by L2(H1, H2) the space of all Hilbert–Schmidt operators from

H1 into H2, L(H1, H2) the space of bounded linear operators from H1 into H2.Let φ : V→ L2(U,V) be a given measurable mapping. We denote by PN the orthogonal projection from

V into the space Span(e1, . . . , eN ). Now we introduce the following conditions:

(H0) The mapping φ : V → L2(U,V) is bounded and Lipschitz, i.e., there exist constants R,Lφ ∈ (0,+∞)such that

R = supv∈V∥φ(v)∥2L2(U,V),

and

∥φ(v1)− φ(v2)∥L2(U,V) ≤ Lφ∥v1 − v2∥V, ∀v1, v2 ∈ V.

(H1) There exist a constant N ∈ N and a bounded measurable mapping g : V→ L(V, U) such that for anyv ∈ V

φ(v)g(v) = PN , (3.11)

and the viscosity constant ν satisfies

12Θ2

ν

c2P + α

2νc2P + α − 1−

K2L2φ

λ1

≥

1 + 2λ1

+ 2(c2P + α)λ1α2

K2R. (3.12)

Here K and Θ are the constants in Lemma 2.3 or in (2.10), cP is the Poincare constant, λ1 is the firsteigenvalue in (2.6).

Remark 1. Eq. (3.11) can be seen as a non degeneracy condition on the low modes, and (3.12) is a technicalcondition.

Now we recall the definition of solution for the problem (1.1) in [14].

Definition 3.1. A stochastic process u is called a solution of the system (1.1), if

(1) u(0) = x0;(2) u ∈ Lp(Ω ,F , P ;L∞([0,∞),W)), 2 ≤ p <∞;(3) For all t ≥ 0, u(t) is Ft-measurable;


(4) For any t ∈ (0,∞) and v ∈W, the following identity holds almost surely

(u(t)− u(0), v)V + t

0

ν((u(s), v)) +

curl(u(s)− α∆u(s))× u(s), v

ds =

t0

(φ(u(s))dW (s), v).

Using Galerkin approximation scheme for the system (1.1), Razafimandimby and Sango obtained thefollowing theorem (see Theorem 3.4 and Theorem 4.1 in [14]).

Theorem 3.2. Assume (H0) holds and x0 ∈W. Then

(1) the system (1.1) has a unique solution u,(2) the solution u admits a version which is continuous in V with respect to the strong topology and continuous

in W with respect to the weak topology.

Moreover, from the proof of Theorem 4.1 in [14], we have

Theorem 3.3. For any t ≥ 0 and any W-valued Ft-measurable functions x1, x2, let u1(t + s), s ≥ 0 andu2(t+s), s ≥ 0 be two solutions of the system (1.1) with conditions u1(t) = x1 and u2(t) = x2, respectively.Then, for any O ∈ Ft

Eσ(t+ s, t)∥u1(t+ s)− u2(t+ s)∥2V1O

≤ E

∥x1 − x2∥2V1O

+ C

t+st

Eσ(l, t)∥u1(l)− u2(l)∥2V1O

dl,

(3.13)

here σ(l, t) := exp− lt∥u2(s)∥2Wds

.

Remark 2. By (3.13), if u1(t) = u2(t) on O ∈ Ft, then u1(t+ ·) = u2(t+ ·) on O, P-a.s.

For a W-valued, Ft-measurable random variable Y , let u(t+ ·, t, Y ) be the unique solution of (1.1) on thetime interval [0,∞) with initial condition u(t, t, Y ) = Y . Define

Xx(t) := X(t,W, x) :=u(t, 0, x), x ∈W;x, x ∈ V \W.

(3.14)

Then we define the operators Pt : Bb(V)→ Bb(V) as

(Ptϕ)(x) := E[ϕ(Xx(t))], for any t ≥ 0.

Lemma 3.1. The process Xxx∈V in (3.14) is a Markov process on V, that is for every x ∈ V, ϕ ∈Cb(V), t, s > 0

E [ϕ(Xx(t+ s))|Ft] = (Psϕ)(Xx(t)), P-a.s. (3.15)

Proof. Notice that (3.15) holds when x ∈ V\W. Now we only need to prove this lemma in the case of x ∈W.

For any x ∈W, we have

Xx(t) = u(t, 0, x).

To prove (3.15), it is sufficient to prove that

E[ϕ(u(t+ s, 0, x))Z] = E[(Psϕ)(u(t, 0, x))Z]


for every bounded Ft-measurable random variable Z. By Theorem 3.2, we know that

u(t+ s, 0, x) = u(t+ s, t, u(t, 0, x)) and E(∥u(t, 0, x)∥2W) <∞.

Hence, it is sufficient to prove that for every W-valued Ft-measurable random variable η

E[ϕ(u(t+ s, t, η))Z] = E[(Psϕ)(η)Z]. (3.16)

For any given ξn, ξ ∈W, by (3.13) and Gronwall’s inequality, we know that the strong convergence of ξnto ξ in V implies that (Psϕ)(ξn) converges to (Psϕ)(ξ). Hence, to prove (3.16), it is sufficient to prove itfor every random variable η of the form η :=

ki=1 η

i1Ai with ηi ∈W and Ai ∈ Ft. By Remark 2, we knowthat for any i = 1, . . . , k,

ϕ

u

t+ s, t,

ki=1ηi1Ai

= ϕ

ut+ s, t, ηi

on Ai, P-a.s. (3.17)

Hence, it is sufficient to prove (3.16) for every deterministic η ∈W.

Now the random variable u(t+ s, t, η) depends only on the increments of the Brownian motion betweent and t+ s, hence it is independent of Ft. Therefore

E[ϕ(u(t+ s, t, η))Z] = E[ϕ(u(t+ s, t, η))] · E[Z].

Since u(t+ s, t, η) has the same law of u(s, 0, η) (by uniqueness), we have

E[ϕ(u(t+ s, t, η))] = E[ϕ(u(s, 0, η))],

and

E[ϕ(u(t+ s, t, η))Z] = E[ϕ(u(s, 0, η))] · E[Z] = E[(Psϕ)(η)Z].

That is (3.16) and the proof is complete.

Remark 3. We need to extend u(·, 0, x)x∈W to the Markov process Xxx∈V, because the verification ofthe conditions (D1)–(D3) only can be obtained in terms of the V-norm (see Lemma 4.4 and Proposition 4.3).

The aim of this paper is to prove the following result.

Theorem 3.4. Assume that (H0) and (H1) hold. Then there exists a unique invariant probability measure µof (Pt)t∈R+ supported on W satisfying that

W∥u∥2Wµ(du) <∞, (3.18)

and there exist C, θ > 0 such that for any λ ∈M(W),

∥P∗t λ− µ∥VWass ≤ Ce−θt

1 +

W∥u∥2Wλ(du)

. (3.19)

Here ∥ · ∥VWass is the Wasserstein norm defined in (4.20) below.

4. Proof of the main result

This section is devoted to the proof of the main result. We first recall the general criterion establishedin [13].


Given a Polish space E, Lipb(E) will denote the space of all bounded, Lipschitz continuous functions onE. Set

∥ϕ∥L := |ϕ|∞ + Lϕ, ϕ ∈ Lipb(E),

here | · |∞ is the sup norm and Lϕ is the Lipschitz constant of ϕ. On the space of probability measuresM(E), consider the Wasserstein norm

∥µ∥EWass := supϕ∈Lipb(E),∥ϕ∥L≤1

E

ϕ(u)µ(du) , µ ∈M(E). (4.20)

Let (U, | · |U ) and (V, ∥ ·∥V) be the two Hilbert spaces introduced before. We consider a Markov process Υliving in V and depending measurably on a cylindrical Wiener process W on U . The map Υ can be writtenas

Υ(t) = Υ(t,W, x0),

where x0 is the initial value Υ(0,W, x0) = x0. We denote the distribution of Υ(·,W, x0) by D(Υ(·,W, x0)),and assume that D(Υ(·,W, x0)) is measurable with respect to x0. Let (Pt)t≥0 be the Markov transitionsemigroup associated with the Markov family (Υ(·,W, x0))x0∈V.

The basic idea behind the criterion in [13] is to construct an auxiliary process Υ(t,W, x0, x0), which is“close” to Υ(t,W, x0) and its law is absolutely continuous with respect to D(Υ(·,W, x0)). More precisely,suppose that there exists a function

Υ : [0,∞)× C([0,∞); R)N × V× V→ V,

satisfying the following conditions.

(A) For every x0, x0 ∈ V, Υ(·,W, x0, x0) is non-anticipative and measurable with respect to W . Moreover,

(Υ(t), Υ(t)) = (Υ(t,W, x0), Υ(t,W, x0, x0))

defines a homogeneous Markov process and its law D(Υ , Υ) is measurable with respect to (x0, x0).(B) There exist a positive measurable function H : V→ [0,+∞] and a positive constant γ such that for anyx0 ∈ V, t ≥ 0, β > 0 and any stopping time τ ≥ 0, there exist C0, Cβ > 0 satisfyingE

H(Υ(t,W, x0))

≤ e−γtH(x0) + C0,

Ee−βτH(Υ(τ,W, x0))1τ<∞

≤ H(x0) + Cβ ;

(C) There exist a cylindrical Wiener process W on U and a function h : V × V → U such that for any(t, x10, x20) ∈ [0,∞)× V× V, we have almost surely

Υ(t,W, x10, x20) = Υt,W +

·0hΥ(s,W, x10), Υ(s,W, x10, x20)

ds, x20

;

(D) For any x10, x20 ∈ V satisfy that

H(x10) +H(x20) ≤ 2C0, (4.21)


and any cylindrical Wiener processes W1,W2 on U , let

h(t) := hΥ(t,W1, x

10), Υ(t,W1, x

10, x

20),

then there exist γ0 > 0, Ci > 0, i = 1, 2, 3 such that(D1) for any t ≥ 0,

PΥ(t,W2, x

20)−Υ(t,W1, x

10)

V ≥ C1e−γ0t, Υ(·,W1, x

10, x

20) = Υ(·,W2, x

20) on [0, t]

≤ C1e

−γ0t;(D2) for any t0 ≥ 0 and any stopping time τ ≥ t0,

P τt0

|h(t)|2Udt ≥ C2e−γ0t0 , Υ(·,W1, x

10, x

20) = Υ(·,W2, x

20) on [0, τ ]

≤ C2e

−γ0t0 ;

(D3) there exists a constant p > 0 such that

P +∞

0|h(t)|2Udt ≤ C3

> p.

The following criteria are obtained in [13, Theorem 2.1].

Theorem 4.1. Under the assumptions (A)–(D), there exists a unique stationary probability measure µ of(Pt)t∈R+ supported on u ∈ V : H(u) <∞, satisfying that

VH(u)dµ(u) <∞,

and there exist C, θ > 0 such that for any λ supported on u ∈ V : H(u) <∞,


1 +

VH(u)dλ(u)

.

We will prove Theorem 3.4 by using Theorem 4.1. In the rest part of this section, we will verify theassumptions (A)–(D) for the system (1.1) by constructing a suitable auxiliary process and taking the functionH(·) = ∥ · ∥2W.

4.1. The proof of Theorem 3.4

4.1.1. Some estimatesAs a part of the proof, we will prepare some estimates for the solutions of Eq. (1.1).For any M > 0, set

WM := Span(e1, . . . , eM ).

Let uM ∈WM be the Galerkin approximations of (1.1) satisfying that for any i ∈ 1, 2, . . . ,M

d(uM , ei)V + ν((uM , ei))dt+ b(uM , uM , ei)dt− αb(uM ,∆uM , ei)dt+ αb(ei,∆uM , uM )dt= (φ(uM ), ei)dW (t), (4.22)

where the notation (φ(u), ei) stands for the operator in L(U,R) defined by

[(φ(u), ei)] (h) := (φ(u)h, ei), ∀h ∈ U.

Then

∥(φ(u), ei)∥2L2(U,R) =∞j=1

(φ(u)ej , ei)2 ≤ ∥ei∥2H ·∞j=1∥φ(u)ej∥2H

≤ C∥ei∥2H ·∞j=1∥φ(u)ej∥2V ≤ C∥φ(u)∥2L2(U,V).


We have the following result for uM .

Lemma 4.1. Assume (H0) holds. There exist constants C1 and C2 only depending on ν, α, R and O suchthat, for any t ≥ 0, β > 0 and any stopping time τ ,

E∥uM (t)∥2W

≤ e− νt

c2P

+αxM0 2

W + C1 (4.23)

and

Ee−βτ

uM (τ)2

W Iτ<∞

≤xM0 2

W + C2/β. (4.24)

Proof. Applying Ito’s formula, we have

duM , ei

2V + 2

uM , ei

V

νuM , ei

+ b

uM , uM , ei

− αb

uM ,∆uM , ei

+ αb

ei,∆uM , uM

dt

= 2(uM , ei)V(φ(uM ), ei)dW (t) +(φ(uM ), ei)

2L2(U,R) dt.

Notice that ∥uM∥2V =Mi=1 λi

uM , ei

2V. Multiplying by λi and taking summation over i, we get

d∥uM∥2V + 2ν∥uM∥2dt = 2(φ(uM ), uM )dW (t) +Mi=1λi(φ(uM ), ei)

2L2(U,R) dt, (4.25)

here we used the fact that b(uM , uM , uM ) = 0.

Let G(uM (t)) be the operator in L2(U,W) defined as follows. For any h ∈ U , let G(uM (t))

(h) ∈W be

the unique solution of the following equation. G(uM (t))

(h)− α∆

G(uM (t))

(h)

=φ(uM (t))

(h) in O, G(uM (t))

(h) = 0 on ∂O.

(4.26)

By Lemma 2.3, Eq. (4.26) admits a unique solution. Moreover, G(uM (t))

(h), ei

V=φ(uM (t))

(h), ei

, ∀i ∈ 1, 2, . . . ,M, (4.27)

and there exists a positive constant K such that G(uM (t))

(h)

W≤ K

φ(uM (t))

(h)

V ,

which implies that G(uM (t))2

L2(U,W)≤ K2 φ(uM (t))

2L2(U,V) .

Hence, by (H0) and (2.6), we knowMi=1λi(φ(uM (t)), ei)

2L2(U,R) =

Mi=1λi

( G(uM (t)), ei)V

2

L2(U,R)

=Mi=1

1λi

( G(uM (t)), ei)W

2

L2(U,R)

≤ 1λ1

G(uM (t))2

L2(U,W)

≤ K2

λ1

φ(uM (t))2L2(U,V)

≤ K2R

λ1. (4.28)


Combining this with (4.25), we obtain

duM2

V + 2νuM2

dt ≤ 2φ(uM ), uM

dW (t) + K

2

λ1Rdt. (4.29)

Denote ∥v∥∗ := |curl(v − α∆v)| for any v ∈W. Next we estimateuM∗.

Set

Ψ(uM ) := −ν∆uM + curl(uM − α∆uM )× uM .

We have

duM , ei

V +

Ψ(uM ), ei

dt =

φ(uM ), ei

dW (t).

Since ei ∈ H4(O) (see Lemma 2.2), uM ∈ WM is in H4(O), which implies that Ψ(uM ) ∈ H1(O). ByLemma 2.3, there exists a unique solution vM ∈W satisfying

vM − α∆vM = Ψ(uM ) in O;vM = 0 on ∂O.

Moreover, vM , ei

V =

ΨuM, ei, ∀i ∈ 1, 2, . . . ,M.

Thus,

duM , ei

V +

vM , ei

V dt =

φ(uM ), ei

dW (t). (4.30)

By (2.6) and (4.27), we have

λiφuM, ei

= G uM , ei

W.

Multiplying λi to (4.30), we have

duM , ei

W +

vM , ei

W dt =

G uM , eiWdW (t).

Applying Ito’s formula, we have

duM , ei

2W + 2

uM , ei

W · (v

M , ei)Wdt

= 2uM , ei

W · G uM , ei

WdW (t) +

G uM , eiW

2

L2(U,R)dt.

Consequently, we have

d∥uM∥2W + 2vM , uM

W dt = 2

G uM , uMWdW (t) +

Mi=1

G uM , eiW

2

L2(U,R)dt.

By (2.3), we rewrite the above equation as follows

duM2

V +uM2

∗

+ 2

vM , uM

V +

curl(uM − α∆uM ), curl

vM − α∆vM

dt

= 2 G uM , uM

VdW (t) +

Mi=1

G uM , eiW

2

L2(U,R)dt

+ 2

curluM − α∆uM

, curl

G uM− α∆ G(uM )dW (t).


Using the definition of vM and G (see (4.27)), we obtain

duM2

V +uM2

∗

+ 2

ΨuM, uM

+curl

uM − α∆uM

, curl

ΨuMdt

= 2φuM, uM

dW (t) +

Mi=1λ2i(φ(uM ), ei)

2L2(U,R) dt+ 2

curl

uM − α∆uM

, curl

φuMdW (t).

Subtracting (4.25) from the above equation, we obtain

d∥uM∥2∗ + 2curl

uM − α∆uM

, curl

ΨuMdt =

Mi=1

λ2i − λi

(φ(uM ), ei)2L2(U,R) dt

+ 2curl

uM − α∆uM

, curl

φuMdW (t), (4.31)

here we used the fact 2ΨuM, uM

= 2ν

uM2.

Since

curlcurl

uM − α∆uM

× uM

=uM · ∇

curl

uM − α∆uM

,

we have curl

uM − α∆uM

, curl

curl

uM − α∆uM

× uM

= 0.

Hence curl

uM − α∆uM

, curl

ΨuM

=

curluM − α∆uM

, curl

−ν∆uM

= να

uM2∗ −ν

α

curl

uM − α∆uM

, curl uM

.

It follows from (4.31) that

duM2

∗ + 2να

uM2∗ dt−

2να

curl

uM − α∆uM

, curl uM

dt

=Mi=1

λ2i − λi

φ uM , ei2L2(U,R) dt+ 2

curl

uM − α∆uM

, curl

φuMdW (t). (4.32)

Using the fact that

|curl(u)|2 ≤ 2α∥u∥2V for any u ∈ V,

we have curluM − α∆uM

, curl uM

≤ 2α

uM (s)

V ·uM (s)

∗

≤ 12uM (s)

2∗ ds+ 2

α

uM (s)2

V . (4.33)

Using similar arguments as that for (4.28), we have

Mi=1

λ2i + λi

φ uM (s), ei2L2(U,R) ≤

1 + 1λ1

K2R. (4.34)

Combining (4.32)–(4.34), we have

duM2

∗ + να

uM2∗ dt ≤

4να2

uM2V dt+

1 + 1λ1

K2Rdt

+ 2

curluM − α∆uM

, curl

φuMdW (t). (4.35)


By (2.2), (4.29) and (4.35), we obtain

d∥uM∥2W + luM2

W dt ≤ l0Rdt+

2 +4c2P + α

α2

φuM, uM

dW (t)

+ 2

curluM − α∆uM

, curl

φuMdW (t), (4.36)

here l = νc2P

+α , l0 =

1 + 2λ1

+ 2(c2P+α)λ1α2

K2.

Applying chain rule to eltuM2

W and taking the expectation, we obtain

EuM (t)

2W

≤ e−lt

uM (0)2

W + l0Rl,

which is the desired inequality (4.23).

Let β > 0 and τ be a stopping time. Applying Ito’s formula to e−βt∥uM (t)∥2W, we have

de−βt

uM2W

+ e−βt(β + l)

uM2W dt ≤ e

−βtl0Rdt+

2 + 4(c2P + α)α2

e−βt

φuM, uM

dW (t)

+ 2e−βt

curluM − α∆uM

, curl

φuMdW (t).

This implies that, for any n ∈ N,

Ee−β(τ∧n)

uM (τ ∧ n)2

W

≤uM (0)

2W + l0R

β.

Letting n→∞, we obtain (4.24). The proof is complete.

Denote by u(·,W, x) the unique solution of (1.1) with initial value x. By using the similar arguments asin [14, Theorem 4.2], we have the following lemma, which will be used later.

Lemma 4.2. The sequence of Galerkin approximations (uM )M≥1 satisfies

limM→∞

EuM (t)− u(t)

2V

= 0, t > 0,

and

limM→∞

E

T0

uM (t)− u(t)2

V dt

= 0.

By Lemma 4.1, we deduce that

uM (t)→ u(t,W, x) weakly in L2(Ω ,F ,P; W).

Furthermore, the following result holds.

Proposition 4.2. Assume (H0) holds. There exist constants C1, C2 only depending on ν, α, R and O suchthat, for any t ≥ 0, β > 0 and any stopping time τ

E∥u(t,W, x0)∥2W

≤ e− νt

c2p+α ∥x0∥2W + C1 (4.37)

and

Ee−βτ ∥u(τ,W, x0)∥2W Iτ<∞

≤ ∥x0∥2W + C2/β. (4.38)


Recall that l = νc2P

+α and l0 =

1 + 2λ1

+ 2(c2P+α)λ1α2

K2. Define the energy functional

Ef (t) := ∥f(t)∥2W + l2

t0∥f(s)∥2Wds.

Lemma 4.3. Assume (H0) holds. There exists κ0 > 0 such that for any κ ≤ κ0/2

Eexp

κ supt≥0

EuM(·,W,xM0 )(t)− l0Rt

≤ 2 exp

κxM0 2

W

. (4.39)

Proof. Set

A(t) := t

0

2 + 4(c2P + α)

α2

φuM, uM

dW (s) + 2

t0

curl

uM − α∆uM

, curl

φuMdW (s).

It is easy to verify that the quadratic process ⟨A⟩(t) satisfies that

⟨A⟩(t) ≤ cR t

0

uM (s)2

W ds.

Let κ0 := lcR and

Aκ0(t) := A(t)− κ02 ⟨A⟩(t).

By (4.36), we have

EuM (t) ≤ ∥uM (0)∥2W + l0Rt+Aκ0(t). (4.40)

Since eκ0Aκ0 is a positive supermartingale whose value is 1 at t = 0, we have

P

supt≥0Aκ0(t) ≥ y

≤ P

supt≥0

exp(κ0Aκ0(t)) ≥ exp(κ0y)≤ exp(−κ0y),

which implies that for any κ ≤ κ0/2,

Eeκ supt≥0Aκ0 (t)

= 1 + κ ∞

0eκyP

supt≥0Aκ0(t) ≥ y

dy ≤ 2. (4.41)

Combining (4.40) and (4.41), we get (4.39).

The proof is complete.

For ρ > 0, let uM ∈WM be the solution of the following SPDE

duM , ei

V + ν

uM , ei

dt+ b

uM , uM , ei

dt− αb

uM ,∆uM , ei

dt+ αb

ei,∆uM , uM

dt

=ρPN

uM − uM (t,W, x0)

, ei

V dt+φ(uM ), ei

dW (t) (4.42)

for any i ∈ 1, 2, . . . ,M, with initial value uM (0) = PM x0, x0 ∈W .Let

rM := uM − uM .

Lemma 4.4. Assume that (H0) and (H1) hold. There exist ϖ,κ1 > 0 such that

supt≥0

Eet ∥rM (t)∥2V +

t0es ∥rM (s)∥2V ds

ϖ≤ 2 ∥rM (0)∥2ϖV exp

κ1uM (0)

2W

. (4.43)


Proof. Note that

d(rM , ei)V + ν((rM , ei))dt+ (δB, ei)dt+ ρ(PNrM , ei)Vdt = (δφ, ei)dW (s),

here δB := BuM , uM

−B

uM , uM

and δφ := φ

uM− φ

uM.

Applying Ito’s formula to (rM , ei)2V, and remembering that ∥rM∥2V =Mi=1 λi(rM , ei)2V, we have

d∥rM∥2V + 2ν∥rM∥2dt+ 2(δB, rM )dt+ 2(ρPNrM , rM )Vdt

= 2(δφ, rM )dW (t) +Mi=1λi ∥(δφ, ei)∥2L2(U,R) dt.

By (2.10), we have

|2(δB, rM )| = |2(B(rM , rM , )uM )| ≤ 2Θ∥rM∥2V∥uM∥W ≤ ∥rM∥2V + Θ2∥rM∥2V∥uM∥2W.

By the similar arguments as that in the proof of (4.28), we haveMi=1λi ∥(δφ, ei)∥2L2(U,R) ≤

K2

λ1∥δφ∥2L2(U,V) ≤

K2L2φ

λ1∥rM∥2V.

Setting l1 := 2νc2P

+α −K2L2

φ

λ1− 1 > 0 and Λ1 := Θ2, we have

d∥rM∥2V +l1 − Λ1∥uM∥2W

∥rM∥2Vdt ≤ 2(δφ, rM )dW (t).

Set G1(t) := e−l1t+Λ1 t

0∥uM (s)∥2Wds. By the chain rule, we have

detG−1

1 (t)∥rM∥2V

+ etG−11 (t)∥rM∥2Vdt ≤ 2etG−1

1 (t)(δφ, rM )dW (t).

Integrating the above inequality and taking expectation, we have

EetG−1

1 (t)∥rM (t)∥2V + t

0esG−1

1 (s)∥rM (s)∥2Vds≤ ∥rM (0)∥2V. (4.44)

By the Holder inequality, for any ϖ ∈ (0, 1),

E

et∥rM (t)∥2V +

t0es∥rM (s)∥2Vds

ϖ

≤

E(supt≥0G2ϖ

1 (t))

EetG−1

1 (t)∥rM (t)∥2V + t

0esG−1

1 (s)∥rM (s)∥2Vdsϖ

. (4.45)

Choosing ϖ > 0 sufficiently small, it follows from Lemma 4.3 and condition (H1) that

Esupt≥0G2ϖ

1 (t)

= Esupt≥0e−2ϖl1t+2Λ1ϖ

t0∥uM (s)∥2Wds

≤ E

supt≥0e

4ϖΛ1l

EuM (t)− l1l2Λ1

t

≤ Eexp

supt≥0

4ϖΛ1l

EuM (t)− l1l2Λ1

t

. (4.46)

Then (4.43) follows from (4.44)–(4.46). The proof is complete.

Recall Xx defined by (3.14). Consider the equation

d(u− α∆u) +−ν∆u+ curl(u− α∆u)× u

dt+ ρPN ((u−Xx0)− α∆(u−Xx0))dt = φ(u)dW, (4.47)


with initial value u(0) = x0 ∈ W. By [14, Theorems 3.4 and 4.1], Eq. (4.47) admits a unique solutionu = u(·,W, x0, x0).

By Lemma 4.2 and using similar arguments as that in the proof of Theorem 4.2 in [14], we have

Lemma 4.5. The sequence of approximations (uM )M≥1 satisfies that

limM→∞

E∥uM (t)− u(t)∥2V

= 0,

limM→∞

E

T0∥uM (t)− u(t)∥2Vdt

= 0.

Let

r(t) := u(t)− u(t).

Combining Lemmas 4.2, 4.4 and 4.5, and applying Fatou’s lemma, we have

Proposition 4.3. Assume that (H0) and (H1) hold. There exist ϖ > 0 and κ0 > 0 such that

E

et∥r(t)∥2V +

t0es∥r(s)∥2Vds

ϖ≤ 2∥r(0)∥2ϖV exp

κ0∥x0∥2W

. (4.48)

4.1.2. Completion of the proof for Theorem 3.4Now we verify the conditions (A)–(D) of Theorem 4.1 for the system (1.1).Recall X(t,W, x0) in (3.14) and u(t,W, x0, x0) in (4.47). Set

X(t,W, x0, x0) :=u(t,W, x0, x0), x0 ∈W;x0, x0 ∈ V \W,

(4.49)

and Υ(t), Υ(t)

:=X(t,W, x0), X(t,W, x0, x0)

.

The proof of Theorem 3.4. We shall prove that the process (Υ(t), Υ(t)) satisfies the conditions (A)–(D) ofTheorem 4.1.

(1) Condition (A) is a consequence of the well-posedness of the equations.(2) Set

H(·) := ∥ · ∥2W.

By Proposition 4.2, we obtain condition (B).(3) Let

h(u0, u1) := −g(u1)ρPN (u1 − u0).

Then condition (C) follows from (H1) and the constructions of (Υ(t), Υ(t)).(4) By the constructions of (Υ(t), Υ(t)), the following inequalities hold: there exist constants Ci > 0, i =

1, 2, 3 satisfied that


(i) for any t ≥ 0,PΥ(t,W2, x

20)−Υ(t,W1, x

10)

V ≥ C1e−γ0t, Υ(·,W1, x

10, x

20) = Υ(·,W2, x

20) on [0, t]

≤ P

∥r(t)∥V ≥ C1e

−γ0t

;(ii) for any t0 ≥ 0 and any stopping time τ ≥ t0,

P τt0

|h(t)|2Udt ≥ C2e−γ0t0 and Υ(·,W1, x

10, x

20) = Υ(·,W2, x

20) on [0, τ ]

≤ P

τt0

∥r(t)∥2Vdt ≥ C3e−γ0t0

;

(iii)

P +∞

0|h(t)|2Udt ≤ C4

≥ P

+∞

0∥r(t)∥2Vdt ≤ C5

.

Those inequalities, together with Proposition 4.3 and Chebyshev inequality, imply that the conditions(D1)–(D3) hold for all x10, x20 ∈W such that ∥x10∥2W + ∥x20∥2W ≤ 2C0.

By Theorem 4.1, we know that the equation (1.1) has a unique invariant probability measure µ supportedon W satisfying that

V∥u∥2Wµ(du) <∞, (4.50)

and there exist C, θ > 0 such that for any λ ∈M(V),


1 +

V∥u∥2Wλ(du)

. (4.51)

If the probability measure λ ∈M(V) satisfies thatV∥u∥2Wλ(du) < +∞,

the probability measure λ is supported on W. Consequently, the invariant measure µ is supported on W,and the exponential inequality (3.19) makes sense only for the measures λ ∈M(W).

The proof is complete.

Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions. Thiswork was supported by National Natural Science Foundation of China (NSFC) (Nos. 11431014, 11401557,and 11301498), and the Fundamental Research Funds for the Central Universities (Nos. WK0010000033,WK0010000038, and WK0010000048).

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