Well-posedness and invariant measures for a class of stochastic 3D Navier-Stokes equations with damping driven by jump noise

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ScienceDirect J. Differential Equations 267 (2019) 5938–5975 www.elsevier.com/locate/jde

Well-posedness and invariant measures for a class of stochastic 3D Navier-Stokes equations with damping driven by jump noise ✩ Hongjun Gao a,c,∗ , Hui Liu b a Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University,

Nanjing 210023, PR China b School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, PR China c Key Laboratory of Ministry of Education for Virtual Geographic Environment, Jiangsu Center for Collaborative

Innovation in Geographical Information Resource Development and Application, Nanjing Normal University, Nanjing 210023, PR China Received 23 October 2018; revised 25 May 2019; accepted 18 June 2019 Available online 22 June 2019

Abstract A class of stochastic 3D Navier-Stokes equation with damping driven by a noise of Lévy type is considered in this paper. The existence and uniqueness of the solution for the problem (1.1)–(1.2) are proved for β > 2 with any α > 0 and α ≥ 14 as β = 2. The main result covered various types of SPDE such as stochastic 3D Navier-Stokes equations with damping, stochastic tamed 3D Navier-Stokes equations, stochastic threedimensional Brinkman-Forchheimer-extended Darcy model. By using the exponential stability of solutions, the existence of a unique invariant measures is proved. © 2019 Elsevier Inc. All rights reserved. MSC: 60H15; 35Q30; 76D05; 37A25; 60J75 Keywords: Key words Navier-Stokes equation; Damping; Jump noise; Invariant measures; Existence and uniqueness

✩ The first author is supported in part by a NSFC Grant No. 11531006 and PAPD of Jiangsu Higher Education Institutions. The second author is supported by the Natural Science Foundation of Shandong under Grant No. ZR2018QA002 and China Postdoctoral Science Foundation No. 2019M652350. * Corresponding author at: Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China. E-mail address: [email protected] (H. Gao).

https://doi.org/10.1016/j.jde.2019.06.015 0022-0396/© 2019 Elsevier Inc. All rights reserved.

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

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1. Introduction We concern with a class of stochastic 3D Navier-Stokes equation with damping driven by a noise of Lévy type in this paper. The damping describes various physical phenomenons such as friction effects or drag, and some dissipative mechanisms [16]. It is well known that the importance of such problems for climate modeling and physical fluid dynamics in [13,36]. In this paper, we consider the following a class of stochastic three-dimensional Navier-Stokes equations with damping driven by jump noise:  ⎧ ˜ dz), du − udt + (u · ∇)udt + αg(|u|)udt + ∇pdt = Z σ (t, u, z)η(dt, ⎪ ⎪ ⎪ ⎨ ∇ · u = 0, ⎪ u(t, x)|∂D = 0, ⎪ ⎪ ⎩ u(x, 0) = u0 (x),

(1.1)

here, the function g(s) is smooth and satisfies ⎧ β ⎪ N1 > 0, s ≥ 0, β ≥ 2, ⎨s − N1 ≤ g(s), 0 ≤ g(s), s ≥ 0, ⎪ ⎩ 0 ≤ g  (s) ≤ C1 + C2 s β−1 , s ≥ 0, C1 , C2 are non-negative constants.

(1.2)

Where D ⊂ R 3 is an bounded domain with smooth boundary ∂D, u = (u1 , u2 , u3 ) is the velocity, p is the pressure, β > 0 and α > 0 are two constants, and t ∈ [0, T ]. The given function u0 is the initial velocity. η˜ is the Lévy process defined on a complete probability space (, F , P ) and σ (t, u, z) is a measurable function satisfying certain condition in later. For the function g(·) which satisfied the condition (1.2), then g(|u|)u contain the tamed is defined by (4.2) and α|u|β u, and Darcy model. The instructions of the details in section 4. The deterministic three-dimensional Navier-Stokes equation with damping has been extensively investigated. For instance, Cai and Jiu have studied the existence and regularity of solutions for three-dimensional Navier-Stokes equations with damping [6], they obtained the global weak solution for β ≥ 1, the global strong solution for β ≥ 72 and that the strong solution was unique for any 72 ≤ β ≤ 5. Based on it, Song and Hou have considered the global attractor in [32] and [33]. By using Fourier splitting method, the L2 decay of weak solutions for 3D Navier-Stokes equations with damping was proved for β > 2 with any α > 0 in [19]. The existence and uniqueness of solutions for the 2-D stochastic Navier-Stokes equations with multiplicative Gaussian noise were obtained in [12,34]. The existence of martingale solutions of stochastic 3D Navier-Stokes equation with jump was studied in [11]. The existence of stationary weak solutions of stochastic 3D Navier-Stokes equations involving jumps was studied in [9]. Brze´zniak et al. studied the existence and uniqueness of the stochastic 2D Navier-Stokes equations driven by jump noise in [4]. Using Galerkin’s approximation and compactness method, Liu and Gao in [18] proved the existence of martingale solutions, and existence and uniqueness of strong solution and small time large deviation principle for the stochastic 3D Navier-Stokes equations with damping for β > 3 with any α > 0 and α ≥ 12 as β = 3. The existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space was proved, meanwhile, the existence and uniqueness of invariant measures for the corresponding transition semigroup were proved in [30]. By using weak convergence method,

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise was obtained in [29]. The existence and uniqueness of strong solutions to stochastic 3D tamed Navier-Stokes equations were proved, simultaneously, a small time large deviation principle for the solutions was proved in [28]. Existence and uniqueness of invariant measure for stochastic 2D Navier-Stokes equation with Lévy noise were proved in [10]. Romito and Xu proved that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise is uniquely ergodic in [31]. By using the exponential stability of solutions, the existence of a unique invariant measures for two-dimensional stochastic MHD system perturbed by Lévy noise was proved in [25]. A small time large deviation principle for the stochastic non-Newtonian fluids driven by multiplicative noise was proved [21]. To obtain well-posedness for a class of stochastic 3D Navier-Stokes equation with damping driven by jump noise, the main difficulty lies in dealing with the nonlinear term B(u, u) = P (u · t ∇u), gN (|u|β ) is defined by (4.3) and g(|u|) = |u|β . We use the estimate of 0 D |u|β |∇u|2 dxds t  for β ≥ 2 to control the estimate of 0 D |u|2 |∇u|2 dxds in [18,19]. We will prove the wellposedness for the stochastic tamed 3D Navier-Stokes equations for β > 2 with any α > 0 and α ≥ 14 as β = 2. We will prove the well-posedness for a class of stochastic 3D Navier-Stokes equation with damping driven by jump noise by using the monotonicity method [5,22–24]. This paper is organized as follows. In section 2, we recall some fundamental concepts related to Lévy process and some lemmas which are used in the sequel. In section 3, we will prove the existence and uniqueness of solution to the problem (1.1) for β > 2 with any α > 0 and α ≥ 14 as β = 2. In section 4, the main result can be applied to various types of PDE such as stochastic 3D Navier-Stokes equation with damping, stochastic tamed 3D Navier-Stokes equation, stochastic three-dimensional Brinkman-Forchheimer-extended Darcy model. In section 5, by using the exponential stability of solutions, the existence of a unique invariant measures is proved. 2. Preliminaries First, we introduce some definitions and basic properties of Lévy processes [2]. Let (, F, F , P ) be a filtered probability space, where F = (Ft )t≥0 is a filtration, (Z, Z) is a measurable space, and ν is a σ -finite positive measure on it. We denote the Borel σ -field on ¯ be a time homogeneous Poisa topological space X by B(X). Let η :  × B(R+ ) × Z → N son random measure with the intensity measure ν defined over the filtered probability space (, F, F , P ). We will denote by η˜ (dt, dz) = η (dt, dz) − dt ν(dz) the compensated Poisson random measure associated to η. C represents for a positive constant which its value may change in the different line. We assume that (H, | · |H ) is a Hilbert space [3]. We define a unique continuous linear operator I which associates to each progressively measurable process ξ : [0, T ] × Z ×  → H such that T  |ξ(r, z)|2H ν(dz)dr < ∞,

E

T > 0.

(2.1)

0 Z

Moreover, we define I (ξ ) is a H -valued adapted and cadl ` ag ` process such that for any random step process ξ satisfying the inequality (2.1) with a representation

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

ξ(r) =

n 

1(tj −1 ,tj ] (r)ξj ,

5941

r ≥ 0,

j =1

where {0 = t0 < t1 < ... < tn < ∞} is a finite partition of [ 0, ∞), and for any j , ξj is an H -valued Ftj −1 measurable random variable, then I (ξ )(t) =

n  

ξj (z)η(dz, ˜ (tj −1 ∧ t, tj ∧ t]), t ≥ 0.

(2.2)

j =1 Z

Then, for each progressively measurable process ξ : [0, T ] × Z ×  → H , we have t  I (ξ )(t) =

ξ(r, z)η(dr, ˜ dz),

t ≥ 0.

0 Z

The continuity of the operator I implies that t 

t  ξ(r, z)η(dr, ˜ dz)|2H

E|

=E

0 Z

|ξ(r, z)|2H ν(dz)dr,

t ≥ 0.

(2.3)

0 Z

For fixed T > 0, the class of all progressively measurable processes ξ : [ 0, T ] × Z ×  → H satisfying the inequality (2.1) will be denoted by M 2 (0, T , L2 (Z, ν, H )). Let D be a bounded domain in R 3 with sufficiently smooth boundary and C0∞ (D, R 3 ) be the set of all smooth functions from D to R 3 with compact support. We define the usual function spaces V = {u ∈ (C0∞ (D, R 3 ))3 : divu = 0}, H = the closure of V in L2 (D), V = the closure of V in H01 (D). It is well known that H , V are separable Hilbert spaces and identify H with its dual H  , we have V → H → V  with dense and continuous injections, and V → H is compact. H and V endowed, respectively, with the inner products  (u, v) =

u · vdx, ∀u, v ∈ H, D

((u, v)) =

3  

∇ui · ∇vi dx, ∀u, v ∈ V ,

i=1 D 1

and norms | · |2 = (·, ·) 2 , || · ||2 = ((·, ·)). Let P be the orthogonal projection of L2 (D; R 3 ) to H .

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For u, v ∈ L2 (D; R 3 ), Au = −P u is the Stokes operator defined by Au, v = ((u, v)). B : V × V → V  is a bilinear operator defined by B(u, v), w = b(u, v, w), B(u) = B(u, u), where b(u, v, w) =

3  

ui

i,j =1 D

∂vj wj dx, ∂xi

and ·, · is the duality product between V and V  . Assumption 2.1. Assume that there exist nonnegative constants k, k0 , k1 , k2 , l1 , l2 , in which k2 is small enough and l2 ∈ (0, 2), and there exists function h ∈ L2p (Z, ν)(p ≥ 1) such that for all t ∈ [0, T ] and u, v ∈ V , we have (A1) |σ (t, u, z)|2L2 (Z,ν,H ) ≤ k0 + k1 |u|22 + k2 ||u||2 . (A2) |σ (t, u, z) − σ (t, v, z)|2L2 (Z,ν,H ) ≤ l1 |u − v|22 + l2 ||u − v||2 . 2p

2p

(A3) |σ (t, u, z)|L2 (Z,ν,H ) ≤ k(1 + |u|2 ), p > 1.

(A4) ||σ (t, u, z)||2p ≤ |h(z)|2p ||u||2p , for all z ∈ Z. Lemma 2.1 (Burkholder-Davis-Gundy inequality [7]). For any p ≥ 1, then there exists a positive constant Cp such that for any real-valued square integrable cadl ` ag ` martingale M with M(0)=0, and for every T > 0, Cp−1 E[M]T

p/2

p/2

≤ E( sup |M(t)|p ) ≤ Cp E[M]T , 0≤t≤T

where [M]T is called the quadratic variation of M. Lemma 2.2 (Itô formula [2]). Assume that E is a Hilbert space. Let X be a process defined by t Xt = X 0 +

t  a(s)ds +

0

f (s, z)η(ds, ˜ dz), t ≥ 0, 0 Z

where a is an E-valued progressively measurable process on the space (R+ × , B(R+ ) ⊗ F) t such that for any t ≥ 0, 0 a(s, w)ds < ∞, P -a.s. and f is a predictable process on E with t  E 0 Z |f (s, z)|2 ν(dz)ds < ∞. Let G be a separable Hilbert space. Let φ : E → G be a function of C 1 such that φ  is (p − 1)-Hölder continuous (for some p > 1). Then for any t > 0, we get P -a.s. t φ(Xt ) = φ(X0 ) +

+ 0 Z

t 

φ (Xs )(a(s))ds + 0

t 



[φ  (Xs− )f (s, z)]η(ds, ˜ dz)

0 Z

[φ(Xs− + f (s, z)) − φ(Xs− ) − φ  (Xs− )f (s, z)]η(ds, dz).

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

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3. A class of stochastic 3D Navier-Stokes equations with damping We now write (1.1) as follows in the abstract form:  du(t) = −(Au(t) + B(u(t)) + αg(|u(t)|)u(t))dt +

σ (t, u, z)η(dt, ˜ dz),

(3.1)

Z

u(x, 0) = ξ,

x ∈ D.

Definition 3.1. An V-valued cadl ` ag ` Ft -measurable process u(t) is said to be a strong solution of (3.1) if the following conditions are satisfied (1) u ∈ L2 (, L2 ([0, T ], H 2 )) ∩ L2 (, L∞ ([0, T ], V )); (2) For any t ∈ [0, T ] and F0 -measurable V-valued initial data ξ , the following equality holds P -a.s. t u(t) = ξ −

t Au(s)ds −

0

t B(u(s))ds − α

0

t  g(|u|)uds +

0

σ (s, u(s), z)η(ds, ˜ dz). 0 Z

The main result of this section is the following theorem. Theorem 3.2. Suppose that Assumption (2.1) and k2 sufficiently small hold and β > 2 with any α > 0 and α ≥ 14 as β = 2. Then for any V-valued F0 -measurable initial data ξ satisfying E||ξ ||2p < ∞ for p ≥ 1, there exists a unique strong solution u(t) to the problem (3.1) with the initial condition u(0) = ξ , and u ∈ L2 (, L2 ([0, T ], H 2 )) ∩ L2 (, L∞ ([0, T ], V )), it satisfies the following inequality T E( sup ||u(t)||

2p

+

0≤t≤T

T  ||u(s)||

2p−2

||∇u(s)|| ds +

0

0 D

T

 ||u(s)||

+

2p−2

0

||u(s)||2p−2 |u|β |∇u|2 dxds

2

g  (|u|)|u||∇u|2 dxds) ≤ C(E||ξ ||2p + 1).

(3.2)

D

For clarity, we denote F (u) = −(A(u) + B(u) + αg(|u|)u) = M(u) − αg(|u|)u. Remark. Inspired by [15], we have 4μβ ≥ 1. But if μ = 1, by using similarly method, we have the similarly result. Meanwhile, we can improve the result for the global well-posedness of a 3D MHD in porous media by using similarly method in [37]. We consider the Galerkin approximation of (3.1) and the proof of Theorem 3.2 is divided into four steps. Step 1: Suppose that {ei : i ∈ N} ⊂ D(A) = V ∩ H 2 is an orthonormal basis of H such that span{ei : i ∈ N } is dense in V . Denote Hn := span{e1 , ...en }. Let Pn : V  → Hn be defined by

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

Pn x =

n 

x, ei  ei . i=1

Then, Pn |H is the orthogonal projection of H onto Hn . For each finite n ∈ N , we can consider the following equation:  dun (t) = Pn F (un (t))dt +

Pn σ (t, un (t−), z)η(dt, ˜ dz),

(3.3)

Z

un (0) = Pn ξ. As we know from [1], (3.3) has a unique Hn -valued cadl ` ag ` strong solution un which is defined by t un (t) = Pn ξ +

t  Pn F (un (s))ds +

0

σn (s, un (s−), z)η(ds, ˜ dz),

t ∈ [0, T ].

0 Z

The following lemma provides the existence and uniqueness of approximate solutions and uniform estimate. This is the main preliminary step in the proof of Theorem 3.2. Lemma 3.3. Under the same assumptions as in Theorem 3.2, there exists a constant C such that T sup E( sup n≥1

0≤t≤T

|un (t)|22

+

T

β+2

un (s) ds +

|un (s)|β+2 ds) ≤ C(E|ξ |22 + 1).

2

0

(3.4)

0

Proof. Applying Itô formula to the process |un (t)|22 , we have t |un (t)|22

+2

t  ||un (s)|| ds + 2α

g(|un |)|un |2 dxds

2

0

0 D

t



= |Pn ξ |22 +

|σn (s, un (s−), z)|22 η(ds, dz) 0 Z

t



+2

(un (s−), σn (s, un (s−), z))η(ds, ˜ dz) 0 Z

= |Pn ξ |22 + I1 (t) + I2 (t). Denote t  I1 (t) =

|σn (s, un (s−), z)|22 η(ds, dz), 0 Z

(3.5)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5945

t  I2 (t) = 2

(un (s−), σn (s, un (s−), z))η(ds, ˜ dz). 0 Z

First, we fix a natural number R > 0 and consider the following stopping time τRn := inf{t ≥ 0 : |un (t)|22 ≥ R} ∧ T . Since the process un (t), t ∈ [0, T ] is adapted and cadl ` ag, ` we have that P {τRn < T } → 0 as R → ∞. First, we deal with the term of I1 (t) n t∧τ  R

sup

E

0≤s≤t∧τRn

I1 ≤ E

|σn (s, un (s−), z)|22 ν(dz)ds 0

Z n t∧τ  R

≤ k0 t + k1 E

n t∧τ  R

|un (s)|22 ds + k2 E 0

||un (s)||2 ds.

(3.6)

0

Applying the Burkholder-Davis-Gundy inequality and the Hölder’s inequality and the Young’s inequality, we have s  E

sup 0≤s≤t∧τRn

I2 ≤ 2E

sup 0≤s≤t∧τRn

|

(un (r−), σn (r, un (r−), z))η(dr, ˜ dz)| 0 Z

n t∧τ  R

≤ CE[ 0

≤ C[

1

|un (s)|22 |σn (s, un (s−), z)|22 ν(dz)ds] 2 Z

1 1 E sup |un (s)|22 ] 2 n 2C 0≤s≤t∧τ R

n t∧τ  R

× [2CE

1

(k0 + k1 |un (s)|22 + k2 ||un (s)||2 )ds] 2 0

1 ≤ E sup |un (s)|22 + 2C 2 k0 t + 2C 2 k1 E 2 0≤s≤t∧τ n R

n t∧τ  R

|un (s)|22 ds 0

n t∧τ  R

+ 2C 2 k2 E

||un (s)||2 ds. 0

Putting (3.6)–(3.7) into (3.5), we get

(3.7)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975 n t∧τ  R

1 E sup |un (s)|22 + 2E 2 0≤s≤t∧τ n

n t∧τ  R

||un (s)||2 ds + 2αE

R

β+2

|un (s)|β+2 ds

0

0

≤ E|ξ |22 + (2C 2 + 1)k0 t + (1 + 2C 2 +

n t∧τ  R

2αN1 )k1 E k1

|un (s)|22 ds 0

t∧τRn



+ (2C 2 + 1)k2 E

||un (s)||2 ds.

(3.8)

0

By choosing sufficiently small constant k2 such that (2C 2 + 1)k2 < 1, we get n t∧τ  R

1 E sup |un (s)|22 + E 2 0≤s≤t∧τ n

n t∧τ  R

||un (s)||2 ds + 2αE

R

0

β+2

|un (s)|β+2 ds 0

2αN1 )k1 E ≤ E|ξ |22 + (2C 2 + 1)k0 t + (1 + 2C 2 + k1

n t∧τ  R

|un (s)|22 ds.

(3.9)

˜ |un (s)|β+2 ds) ≤ C(E|ξ |22 + 1).

(3.10)

sup 0

0≤s≤t∧τRn

Applying Gronwall lemma, there exists a constant C˜ such that n t∧τ  R

E(

sup

0≤s≤t∧τRn

|un (s)|22 +

n t∧τ  R

un (s)2 ds + 0

β+2

0

Recall that τRn ↑ T as R → ∞, and P {τRn < T } → 0 as R → ∞. By the Fatou lemma, we get T E( sup 0≤t≤T

|un (t)|22

+

T un (s) ds +

0

β+2

|un (s)|β+2 ds) ≤ C(E|ξ |22 + 1). 2 (3.11)

2

0

Lemma 3.4. Under the same assumptions as in Theorem 3.2 and p > 1, there exists a constant C such that

sup E( sup n≥1

0≤t≤T

T +

2p |un (t)|2

+

2p−2

|un (s)|2

||un (s)||2 ds

0 2p−2

|un (s)|2 0

T

β+2

2p

|un (s)|β+2 ds) ≤ C(E|ξ |2 + 1).

(3.12)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

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2p

Proof. Applying Itô formula to the process |un (t)|2 , we have 2p |un (t)|2

2p = |Pn ξ |2

t − 2p

2p−2

|un (s)|2

||un (s)||2 ds

0

t

2p−2 |un (s)|2

− 2pα 0

 g(|un |)|un |2 dxds D

t 

2p−2

|un (s)|2

+ 2p

(un (s−), σn (s, un (s−), z))η(ds, ˜ dz)

0 Z

t



2p

2p

(|un (s−) + σn (s, un (s−), z)|2 − |un (s−)|2

+ 0 Z

2(p−1)

− 2p|un (s−)|2

(un (s−), σn (s, un (s−), z)))η(ds, dz).

(3.13)

Taking the supremum over the interval [0, t] on the equality (3.13), we get sup s∈[0,t]

2p |un (s)|2

s + sup 2pα s∈[0,t]

s + sup 2p s∈[0,t]

2p−2 |un (r)|2

2p



≤ |Pn ξ |2 + 2p sup | s∈[0,t]

s∈[0,t]

 g(|un |)|un |2 dxdr D

s

2p−2

|un (r)|2

(un (r−), σn (r, un (r−), z))η(dr, ˜ dz)|

0 Z



+ sup |

||un (r)||2 dr

0

0

s

2p−2

|un (r)|2

2p

2p

(|un (r−) + σn (r, un (r−), z)|2 − |un (r−)|2 0 Z 2(p−1)

− 2p|un (r−)|2

(un (r−), σn (r, un (r−), z)))η(dr, dz)|

2p

= |Pn ξ |2 + I3 (t) + I4 (t). Applying the Burkholder-Davis-Gundy inequality and Young’s inequality, we have t  1 4(p−1) EI3 (t) ≤ 2pCE( |un (s)|2 |un (s)|22 |σn (s, un (s−), z)|22 ν(dz)ds) 2 0 Z

t 1 4p−2 ≤ 2pCE( |un (s)|2 (k0 + k1 |un (s)|22 + k2 ||un (s)||2 )ds) 2 0

(3.14)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

≤ 2pC(εE sup s∈[0,t]

≤ pCεE sup s∈[0,t]

pCk2 E + ε

2p 1 1 |un (s)|2 ) 2 ( E

t

2p−2

|un (s)|2

ε

1

(k0 + k1 |un (s)|22 + k2 ||un (s)||2 )ds) 2

0

2p |un (s)|2

pCk0 T pC(k0 + k1 ) + + E ε ε

t 0

t

2p−2

|un (s)|2

2p

sup |un (r)|2 ds

r∈[0,s]

||un (s)||2 ds.

(3.15)

0

By the Taylor formula, we get 2p

2p

2(p−1)

||x + h|2 − |x|2 − 2p|x|2

2(p−1)

(x, h)| ≤ Cp (|x|2

2p

|h|22 + |h|2 ),

for all x, h ∈ Hn . (3.16)

We have t 

2p

EI4 (t) ≤ E

|(|un (s−) + σn (s, un (s−), z)|2 0 Z 2p

2(p−1)

− |un (s−)|2 − 2p|un (s−)|2 t  ≤ Cp E

2p−2

(|un (s−)|2

(un (s−), σn (s, un (s−), z)))|η(ds, dz) 2p

|σn (s, un (s−), z)|22 + |σn (s, un (s−), z)|2 )ν(dz)ds

0 Z

t

2p

2(p−1)

(1 + |un (s)|2 + k2 |un (s)|2

≤ Cp kE

||un (s)||2 )ds

0

t

2p

≤ Cp kT + Cp kE

sup |un (r)|2 ds

0

t

r∈[0,s]

2(p−1)

|un (s)|2

+ Cp kk2 E

||un (s)||2 ds.

(3.17)

0

Putting (3.15) and (3.17) into (3.14), we have

E sup s∈[0,t]

2p |un (s)|2

t + 2pE

2p−2 |un (s)|2 ||un (s)||2 ds

t + 2pαE

0

2p−2

|un (s)|2 0

2p

2p

≤ |Pn ξ |2 + pCεE sup |un (s)|2 + s∈[0,t]

pCk0 T + Cp kT ε

β+2

|un (s)|β+2 ds

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

pC(k0 + k1 ) +( + Cp k + 2pαN1 )E ε

t

t

2p−2

|un (s)|2

2p

sup |un (r)|2 ds

0

pC + Cp k)E + k2 ( ε

5949

r∈[0,s]

||un (s)||2 ds.

(3.18)

0

By choosing ε and k2 sufficiently small such that k2 ( pC ε + Cp k) ≤ p, we may apply Gronwall’s lemma to obtain

E( sup 0≤t≤T

2p |un (t)|2

T +

2p−2 |un |2 ||un ||2 ds

T +

0

2p−2

|un |β+2 ds) ≤ C(E|ξ |2 + 1). (3.19)

2p−2

||un (s)||2 ds

|un |2

2p

β+2

0

Since the constant C is independent of n, we have T

2p

sup E( sup |un (t)|2 + n≥1

0≤t≤T

0

T

2p−2

+

|un (s)|2

|un (s)|2

2p

β+2

|un (s)|β+2 ds) ≤ C(E|ξ |2 + 1).

2

(3.20)

0

Lemma 3.5. Under the same assumptions as in Theorem 3.2, there exists a positive constant C such that T sup E( sup ||un (t)|| + n≥1

0≤t≤T

T  ||∇un (s)|| ds +

2

|un |β |∇un |2 dxds

2

0

t

0 D



+

g  (|un |)|un ||∇un |2 dxds) ≤ C(E||ξ ||22 + 1).

0 D

Proof. We apply Itô formula to ||un (t)||2 for t ∈ [0, T ], t ||un (t)||2 + 2

||∇un (s)||2 ds 0

t



t  g(|un |)|∇un | dxds + 2α

+ 2α

2

0 D

0 D

t ≤ |Pn ∇ξ |22 + 2

| un , B(un )|ds 0

g  (|un |)|un ||∇un |2 dxds

5950

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

t +2

(∇un (s−), ∇σn (s, un (s−), z))η(ds, ˜ dz) 0

t



+

[|∇(un (s−) + σn (s, un (s−), z))|22 − |∇un (s−)|22 0 Z

− 2(∇un (s−), ∇σn (s, un (s−), z))]η(ds, dz) = |Pn ∇ξ |22 +

7 

(3.21)

Ij (t).

j =5

Since 0<

2 < 1, β

for β > 2,

(3.22)

using Young’s inequality and (3.22) to estimate I5 (t), we deduce t I5 (t) ≤ 2

1 1 ( |un |22 + |un · ∇un |22 )ds 2 2

0

t ≤

t 

4

||∇un || ds + 0

t ≤

2− β4

|un |2 |∇un | β |∇un |

2

dxds

0 D

t   β−2 4 β 2 2− 4 β ||∇un || ds + [ (|un |2 |∇un | β ) 2 dx] β [ (|∇un | β ) β−2 dx] β ds 2

0

0

t ≤

D

t



||∇un || dx + ε

t |∇un | |un | dxds + C(ε)

2

0

D

2

|∇un (s)|22 ds.

β

0 D

(3.23)

0

Taking the supremum and expectation over the interval [0, t] on both sides of the equality (3.21), we estimate the last two items, t  1 E sup I6 (s) ≤ 2 2E[ |∇un |22 |∇σn (s, un (s−), z)|22 ν(dz)ds] 2 √

0≤s≤t

0 Z

t   1 ≤ 2 2E[ |h(z)|2 ν(dz)|∇un |22 |∇un |2 dxds] 2 √

0 Z

D

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

1 ≤ E sup |∇un (s)|22 + C(|h(z)|2L2 (Z,ν) )E 2 0≤s≤t 1 ≤ E sup |∇un (s)|22 + C(|h(z)|2L2 (Z,ν) )E 2 0≤s≤t

5951

t  |∇un |2 dxds 0 D

t |∇un (s)|22 ds.

(3.24)

0

Similarly, we have t  E sup I7 (s) ≤ E

|∇σn (s, un (s−), z)|22 η(ds, dz)

0≤s≤t

0 Z

t  ≤E

 |h(z)|2 ν(dz)

0 Z

|∇un |2 dxds D

t ≤ C(|h|2L2 (Z,ν) )E

|∇un (s)|22 ds.

(3.25)

0

Substituting (3.23)–(3.25) into (3.21), we have 1 E sup ||un (s)||2 + E 2 0≤s≤t t

t ||∇un (s)||2 ds 0



+ (2α − ε)E

t  |un | |∇un | dxds + 2αE 2

β

0 D

g  (|un |)|un ||∇un |2 dxds

0 D

t ≤ E||ξ ||22 + (C + 2αN1 )E

|∇un (s)|22 ds.

(3.26)

0

Choosing ε sufficiently small and applying Gronwall’s lemma, we have T sup E( sup ||un (t)|| + n≥1

0≤t≤T

T  ||∇un (s)|| ds +

2

|un |β |∇un |2 dxds

2

0

t

0 D



+

g  (|un |)|un ||∇un |2 dxds) ≤ C(E||ξ ||22 + 1).

(3.27)

0 D 1 By Theorem 4.1 in [15], we have 1 − 2θ ≥ 0 and α − θ2 ≥ 0 for θ > 0. Then we can get the 1 above estimate easily for α ≥ 4 as β = 2. This completes the proof of Lemma 3.5. 2

5952

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

Lemma 3.6. Under the same assumptions as in Theorem 3.2 and p > 1, there exists a constant C such that T sup E( sup ||un (t)||

2p

+

||un (s)||2p−2 ||∇un (s)||2 ds

0≤t≤T

n≥1

0

T +

 ||un (s)||2p−2

0

g  (|un |)|un ||∇un |2 dxds

D

T  ||un (s)||2p−2 |un |β |∇un |2 dxds) ≤ C(E||ξ ||2p + 1).

+ 0 D

Proof. Applying Itô formula to ||un (t)||2p , we get t ||un (t)||

≤ ||Pn ξ ||

2p

2p

− 2p

||un (s)||2p−2 ||∇un (s)||2 ds 0

t  − 2pα

||un (s)||2p−2 g(|u|)|∇un |2 dxds 0 D

t − 2pα

 ||un (s)||

2p−2

0

g  (|un |)|un ||∇un |2 dxds

D

t + 2p

||un (s)||2p−2 | un , B(un )|ds 0

t  + 2p

||un (s)||2p−2 (∇un (s−), ∇σn (s, un (s−), z))η(ds, ˜ dz) 0 Z

t



+

(||un (s−) + σn (s, un (s−), z)||2p − ||un (s−)||2p 0 Z

− 2p||un (s−)||2(p−1) (∇un (s−), ∇σn (s, un (s−), z)))η(ds, dz). Taking the supremum over the interval [0, t] on the equality (3.28), we get s sup ||un (s)||

2p

s∈[0,t]

+ sup 2p s∈[0,t]

||un (r)||2p−2 ||∇un (r)||2 dr 0

(3.28)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5953

s  + sup 2pα s∈[0,t]

||un (r)||2p−2 g(|u|)|∇un |2 dxdr 0 D

s + sup 2pα s∈[0,t]

 ||un (r)||2p−2

0

g  (|un |)|un ||∇un |2 dxdr

D

s



≤ ||Pn ξ ||2p + 2p sup s∈[0,t]

||un (r)||2p−2 (∇un (r−), ∇σn (r, un (r−), z))η(dr, ˜ dz) 0 Z

s  + sup |

(||un (r−) + σn (r, un (r−), z)||2p

s∈[0,t]

0 Z

− ||un (r−)||2p − 2p||un (r−)||2(p−1) (∇un (r−), ∇σn (r, un (r−), z)))η(dr, dz)| s + sup 2p s∈[0,t]

||un (r)||2p−2 | un , B(un )|dr 0

= ||Pn ξ ||

2p

+ I8 (t) + I9 (t) + I10 (t).

(3.29)

Applying the Burkholder-Davis-Gundy inequality and Young’s inequality, we have t  1 EI8 (t) ≤ 2pCE( ||un (s)||4(p−1) ||un (s)||2 ||σn (s, un (s−), z)||2 ν(dz)ds) 2 0 Z

t  1 ≤ C|h(z)|L2 (Z,ν) E( ||un (s)||4p−2 |∇un |2 dxds) 2 0 D

t  1 ≤ C|h(z)|L2 (Z,ν) E( ||un (s)||2p ||un (s)||2p−2 |∇un |2 dxds) 2 0 D

1 ≤ E sup ||un (s)||2p + C(|h(z)|2L2 (Z,ν) )E 2 0≤s≤t 1 ≤ E sup ||un (s)||2p + C(|h(z)|2L2 (Z,ν) )E 2 0≤s≤t

t  ||un (s)||2p−2 |∇un |2 dxds 0 D

t ||un (s)||2p ds.

(3.30)

0

By the Taylor formula, we get 2p

2p

2(p−1)

||x + h|2 − |x|2 − 2p|x|2 For the third term, we have

2(p−1)

(x, h)| ≤ Cp (|x|2

2p

|h|22 + |h|2 ), ∀x, h ∈ Hn .

(3.31)

5954

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

t  EI9 (t) ≤ E

|(||un (s−) + σn (s, un (s−), z)||2p − ||un (s−)||2p 0 Z

− 2p||un (s−)||2(p−1) (∇un (s−), ∇σn (s, un (s−), z)))|η(ds, dz) s  (||un (r−)||2p−2 ||σn (r, un (r−), z)||2 + ||σn (r, un (r−), z)||2p )ν(dz)dr

≤ Cp E sup s∈[0,t]

0 Z

= E sup I91 + E sup I92 . s∈[0,t]

(3.32)

s∈[0,t]

By the Assumption 2.1, we have t  E sup s∈[0,t]

I91 ≤ C(|h|2L2 (Z,ν) )E

||un (s)||2p−2 |∇un |2 dxds 0 D

t ||un (s)||2p ds.

≤ C(|h|2L2 (Z,ν) )E

(3.33)

0

Applying Young’s inequality, we have 2p

E sup I92 ≤ C(|h|L2p (Z,ν) )E s∈[0,t]

t  ( |∇un |2 dx)p ds 0

D

t

2p

≤ C(|h|L2p (Z,ν) )E

||un (s)||2p ds.

(3.34)

0

Since 0<

2 < 1, β

for β > 2,

(3.35)

using Young’s inequality and (3.35) to estimate I10 (t), we deduce s I10 (t) = sup 2p s∈[0,t]

||un (r)||2p−2 | un , B(un )|dr 0

t ≤p

||un (s)||2p−2 (|un |22 + |un · ∇un |22 )ds 0

t ≤p

t ||un (s)||

2p−2

0

||∇un || ds + p

 ||un (s)||2p−2

2

0

4

2− β4

|un |2 |∇un | β |∇un | D

dxds

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5955

t ≤p

||un (s)||2p−2 ||∇un ||2 ds 0

t ||un (s)||

+p

2p−2

  β−2 4 β 2 2− 4 β 2 β β 2 [ (|un | |∇un | ) dx] [ (|∇un | β ) β−2 dx] β ds

0

D

D

t

t ||un (s)||2p−2 ||∇un ||2 ds + C

≤p 0

0

t

 ||un (s)||

+ε

||un (s)||2p ds

|∇un |2 |un |β dxds.

2p−2

0

(3.36)

D

Putting (3.30)–(3.36) into (3.29), we have t E sup ||un (s)||

2p

+ 2pE

||un (s)||2p−2 ||∇un (s)||2 ds

s∈[0,t]

0

t  ||un (s)||2p−2 |un |β |∇un |2 dxds

+ 2pαE 0 D

t

 ||un (s)||

+ 2pE

2p−2

0

g  (|un |)|un ||∇un |2 dxds

D

1 ≤ E||Pn ξ ||2p + E sup ||un (s)||2p 2 0≤s≤t t  + εE

||un (s)||2p−2 |un |β |∇un |2 dxds 0 D

t

t ||un (s)|| ds + pE

+ CE

||un (s)||2p−2 ||∇un (s)||2 ds.

2p

0

0

Choosing sufficiently small ε and applying Gronwall lemma, we have T E( sup ||un (t)||

2p

+

0≤t≤T

||un (s)||2p−2 ||∇un (s)||2 ds 0

T

 ||un (s)||2p−2

+ 0

D

g  (|un |)|un ||∇un |2 dxds

(3.37)

5956

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

T  +

||un (s)||2p−2 |un |β |∇un |2 dxds) ≤ C(E||ξ ||2p + 1). 0 D

Since the constant C is independent of n, we have T sup E( sup ||un (t)||

2p

n≥1

+

0≤t≤T

0

T +

||un (s)||2p−2 ||∇un (s)||2 ds

 ||un (s)||2p−2

0

T

g  (|un |)|un ||∇un |2 dxds

D



+

||un (s)||2p−2 |un |β |∇un |2 dxds) ≤ C(E||ξ ||2p + 1).

(3.38)

0 D 1 By Theorem 4.1 in [15], we have 1 − 2θ ≥ 0 and α − θ2 ≥ 0 for θ > 0. Then we can get the above estimate easily for α ≥ 14 as β = 2. This completes the proof of Lemma 3.6. 2

Step 2: Weak convergence of approximating sequences. Lemma 3.7. There exists a subsequence of processes, still denoted by {un, n ≥ 1} and process u ∈ L2 ([0, T ] × , H 2 ) ∩ Lβ+2 ([0, T ] × , Lβ+2 ) ∩ L2p (, L∞ ([0, T ], V )), g(|u|)u ∈ Ls ([0, T ] × , Ls (D)), s = (β + 2)∗ = L2 ([0, T ] × , L2 (Z, ν, H )), such that (i) (ii) (iii) (iv) (v) (vi)

β+2 β+1 ,

M ∈ L2 ([0, T ] × , H −2 ), S ∈

un → u weakly in L2 ([0, T ] × , H 2 ), un → u weakly in Lβ+2 ([0, T ] × , Lβ+2 ), un is weak star converging to u in L2p (, L∞ ([0, T ], V )), Pn g(|un |)un → g(|u|)u weakly in Ls ([0, T ] × , Ls (D)), Pn M(un ) → M weakly in L2 ([0, T ] × , H −2 ), σn (s, un , ·) → S weakly in L2 ([0, T ] × , L2 (Z, v, H )).

Proof. By Lemma 3.3, Lemma 3.4, Lemma 3.5 and Lemma 3.6, it is easy to see that (i), (ii) and (iii) are satisfied. Since (1.2) and the inequality (3.4), we have T |Pn g(|un |)un |Ls ([0,T ]×,Ls (D)) ≤ C(E

β+2

|un (s)|β+2 ds + 1) 0

≤ C(E|ξ |22 + 1),

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5957

2 here, s = β+2 β+1 . By Lemma 3.3, we get un → u weakly in C([0, T ] × , H ) ∩ L ([0, T ] × , V ) ∩ β+2 β+2 ([0, T ] × , L ). By the similar method in Lemma 3.4 [18], we get L

|g(|un |)un − g(|u|)u| ≤ |g(|un |)||un − u| + |g  (|ξ |)||u||un − u| ≤ C(1 + |un |β + |u|β )|un − u|,

()

here, ξ between un and u, and C depends on C1 , C2 in (1.2). For any ψ such that Pn ψ → ψ in L2 , then | Pn ψg(|un |)un − g(|u|)u, ψ| = | g(|un |)un , Pn ψ − g(|u|)u, ψ| = | g(|un |)un , Pn ψ − ψ + g(|un |)un − g(|u|)u, ψ| β+1

≤ C(|un |β+2 + 1)|Pn ψ − ψ| β+2 + | g(|un |)un − g(|u|)u, ψ| β+1

β+1 ≤ C(|un |β+2

+ 1)|Pn ψ − ψ|2 + | g(|un |)un − g(|u|)u, ψ|.

Then, by Lemma 3.3 and Lemma 3.5, (i) in Lemma 3.7, () and Young inequality, we get that g(|un |)un → g(|u|)u weakly in Ls ([0, T ] ×, Ls (D)). We complete the proof of (iv). Moreover, for v ∈ L2 ( × [0, T ], H 2 ), we have T ||Pn M(un )||2L2 ([0,T ]×,H −2 )

=

sup

| Pn M(un (s)), v(s)|ds

E

||v||2 2

≤1 L ([0,T ]×,H 2 )

0

T ≤

sup ||v||2 2

T + CE

≤1

T

||un (s)|| ds) (E

((E

L ([0,T ]×,H 2 )

1 2

2

0

1

1

||v(s)||2 ds) 2 0

3

|un (s)|22 ||un (s)|| 2 ||v(s)||ds) 0

T ≤

sup

1

||un (s)||2 ds) 2 ||v||L2 ([0,T ]×,H 2 )

((E

||v||2 2 ≤1 L ([0,T ]×,H 2 )

0

T + C(E

1 2

0 1

T

||un (s)||2 ds) 2 + C(E

≤ (E 0

1

|un (s)|2 ||un (s)||3 ds) 2 0

T

1 2

T

||un (s)|| ds) + C(E 2

0

1

||v(s)||2 ds) 2 ) 0

T

≤ (E

T

|un (s)|2 ||un (s)|| ds) (E 3

1

(|un (s)|22 + ||un (s)||6 )ds) 2 . 0

(3.39)

5958

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

By the Lemma 3.3–Lemma 3.6, we have T ||un (s)|| ds + E

sup E n≥1

T (|un (s)|22 + ||un (s)||6 )ds < ∞.

2

0

(3.40)

0

2

We complete the proof of (v).

By the Assumption 2.1 and Lemma 3.6, T  |σn (s, un (s−), z)|22 ν(dz)ds

sup E n≥1

0 Z

T (k0 + k1 |un (s)|22 + k2 ||un (s)||2 )ds < ∞.

≤ sup E n≥1

(3.41)

0

We complete the proof of (vi). Step 3: Define a V  -valued process u by t u(t) = ξ +

t  F (s)ds +

0

S(s, z)η(dz, ˜ ds).

(3.42)

0 Z

By Lemma 3.7, we can see that u is a V  -valued modification of the V -valued process u, ˜ i.e. u = u˜ dt × P -a.e. in V . In (3.42), we have to prove that ds ⊗ dP a.s. on T , one has F (s) = F (u(s, ω)),

S(s) = σ (s, u(s, ω), z).

To establish these relations, we use the same idea as in [4]. Let χ ={v ∈ L2 ([0, T ] × , H 2 ) ∩ L∞ ([0, T ] × , Lβ+2 ) ∩ L2 (, L∞ ([0, T ], V ))}. For every t ∈ [0, T ], let t r(t) =

(2cε ||v(s)||4 + l1 )ds,

t ∈ [0, T ],

(3.43)

0

and F (un (s)) = −Aun (s) − B(un (s)) − αg(|un (s)|)un (s) = M(un (s)) − αg(|un (s)|)un (s). By (3.43) and Assumption 2.1, we have the following inequalities

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5959

−(2cε ||v(t)||4 + l1 )|un − v|22 + 2 Pn M(un ) − Pn M(v), un − v + |σn (t, un (t−), z) − σn (t, v(t), z)|22 ≤ −(2cε ||v(t)||4 + l1 )|un − v|22 + 2[−(1 − ε)||un − v||2 + cε |un − v|22 ||v||4 ] + [l1 |un − v|22 + l2 ||un − v||2 ] ≤ 0. So, we deduce that T E[−

e−r(t) r  (t)|un (t) − v(t)|22 dt

0

T +2

e−r(t) Pn M(un (t)) − Pn M(v(t)), un (t) − v(t)dt

0

T  +

e−r(t) |σn (t, un (t−), z) − σn (t, v(t), z)|22 v(dz)dt] ≤ 0.

(3.44)

0 Z

By Lemma 3.7 and applying the Itô formula from [14,17] for the process e−r(t) |u(t)|22 , we have

e

−r(T )

T |u(T )|22

= |ξ |22

−

e

−r(s) 

r

T (s)|u(s)|22 ds

+2

0

T



+2

e−r(s) F (s), u(s)ds

0

e−r(s) (u(s−), S(s, z))η(dz, ˜ ds)

0 Z

T  +

e−r(s) |S(s, z)|22 η(ds, dz).

(3.45)

0 Z

Taking the expectation of (3.45), we get

Ee

−r(T )

T |u(T )|22

− E|ξ |22

= −E

e

−r(s) 

r

T (s)|u(s)|22 ds

+ 2E

0

T



+E

e−r(s) F (s), u(s)ds

0

e−r(s) |S(s, z)|22 ν(dz)ds.

0 Z

Applying Itô formula to the process e−r(t) |un (t)|22 , we have

(3.46)

5960

e

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

−r(T )

T |un (T )|22

= |Pn ξ |22

−

e

−r(s) 

r

T (s)|un (s)|22 ds

+2

0

T  +2

e−r(s) Pn F (un (s)), un (s)ds

0

e−r(s) (un (s−), σn (s, un (s−), z))η(dz, ˜ ds)

0 Z

T  +

e−r(s) |σn (s, un (s−), z)|22 η(ds, dz).

(3.47)

0 Z

Taking the expectation of (3.47), we get E(e−r(T ) |un (T )|22 ) T = E|Pn ξ |22

+ E[−

e−r(s) r  (s)(2(un (s), v(s)) − |v(s)|22 )ds

0

T +2

e−r(s) ( Pn F (un (s)) − Pn F (v(s)), v(s) + Pn F (v(s)), un (s))ds

0

T 

e−r(s) (2(σn (s, un (s−), z), σn (v(s), z)) − |σn (s, v(s), z)|22 )ν(dz)ds]

+ 0 Z

T + E[−

e−r(s) r  (s)|un (s) − v(s)|22 ds

0

T +2

e−r(s) Pn F (un (s)) − Pn F (v(s)), un (s) − v(s)ds

0

T



+

e−r(s) |σn (s, un (s−), z) − σn (s, v(s), z)|2 ν(dz)ds].

0 Z

We infer that E(e−r(T ) |un (T )|22 ) − E|Pn ξ |22 T ≤ E[− 0

e−r(s) r  (s)(2(un (s), v(s)) − |v(s)|22 )ds

(3.48)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

T +2

5961

e−r(s) ( Pn F (un (s)) − Pn F (v(s)), v(s) + Pn F (v(s)), un (s))ds

0

T − 2α

e−r(s) Pn g(|un (s)|)un (s) − Pn g(|v(s)|)v(s), un (s) − v(s)ds

0

T 

e−r(s) (2(σn (s, un (s−), z), σn (s, v(s), z)) − |σn (s, v(s), z)|22 )ν(dz)ds]

+ 0 Z

T ≤ E[−

e−r(s) r  (s)(2(un (s), v(s)) − |v(s)|22 )ds

0

T +2

e−r(s) ( Pn F (un (s)) − Pn F (v(s)), v(s) + Pn F (v(s)), un (s))ds

0

T  +

e−r(s) (2(σn (s, un (s−), z), σn (s, v(s), z)) − |σn (s, v(s), z)|22 )ν(dz)ds].

(3.49)

0 Z

By Lemma 3.7, we have un (T )  u(T ) in L2 (, H ). Since E|Pn ξ |2 ≤ E|ξ |2 , we get E(e−r(T ) |u(T )|22 ) − E|ξ |22 ≤ lim inf E(e−r(T ) |un (T )|22 ) − E|Pn ξ |22 . n→∞

By lower semi-continuity property of weak convergence, we have E(e−r(T ) |u(T )|22 − |ξ |22 ) ≤ lim inf E(e−r(T ) |un (T )|22 − |Pn ξ |22 ) n→∞

T ≤ lim inf E[− n→∞

e−r(s) r  (s)(2(un (s), v(s)) − |v(s)|22 )ds

0

T +2

e−r(s) ( Pn F (un (s)) − Pn F (v(s)), v(s) + Pn F (v(s)), un (s))ds

0

T  +

e−r(s) (2(σn (s, un (s−), z), σn (s, v(s), z)) − |σn (s, v(s), z)|22 )ν(dz)ds]

0 Z

T ˜ v(s)))ds = E[ e−r(s) r  (s)(|v(s)|22 − 2(u(s), 0

(3.50)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

T

e−r(s) ( F (u(s)) ˜ − F (v(s)), v(s) + F (v(s)), u(s))ds ˜

+2 0

T



e−r(s) (2(S(s, z), σ (s, v(s), z)) − |σ (s, v(s), z)|22 )ν(dz)ds].

+

(3.51)

0 Z

By virtue of (3.46) and (3.51), we get T E

e−r(s) [−r  (s)|u˜ − v(s)|22 + 2 u(s) ˜ − v(s), F (s) − F (v(s))

0



+

|S(s, z) − σ (s, v(s), z)|22 ν(dz)]ds ≤ 0.

(3.52)

Z

Taking u˜ = v, we get S(·, ·) = σ (·, u(·), ˜ ·). If we note vε = u˜ − εw, ω ∈ L∞ ( × [0, T ], V ) and ε > 0, we get T ε(−εE

e−r(s) r  (s)|ω(s)|22 ds + 2E

0

T

ω(s), F (s) − F (u(s) − εω(s))ds) ≤ 0. 0

Dividing by ε on both sides of the above inequality, we get T −εE

e

−r(s) 

r

T (s)|ω(s)|22 ds

0

+ 2E

ω(s), F (s) − F (u(s) − εω(s))ds ≤ 0.

(3.53)

0

By the Lebesgue Dominated convergence theorem, we get as ε ↓ 0, T

T

ω(s), F (s) − F (u(s) − εω(s))ds → E

E 0

ω(s), F (s) − F (u(s))ds.

(3.54)

0

Let ε ↓ 0 on (3.53), we have T

ω(s), F (s) − F (u(s))ds ≤ 0.

2E

(3.55)

0

Since ω is arbitrary, we get F (·) = F (u(·)). Thus the existence has been proved. Step 4: We will prove the uniqueness of the solution to problem (3.1). Assume that u1 and u2 are two solutions to problem (3.1) and ω(t) = u1 − u2 . Let us define the stopping time τN := inf{t ≥ 0, |u1 (t)|22 ≥ N } ∧ inf{t ≥ 0, |u2 (t)|22 ≥ N } ∧ T .

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5963

Let t φ(t) = exp(−a

||u2 ||4 ds),

t ∈ [0, T ],

0

where a is a positive constant. Applying the Itô formula to φ(t ∧ τN )|ω(t ∧ τN )|22 , we get φ(t ∧ τN )|ω(t ∧ τN )|22 t∧τ  N

≤ −a

t∧τ  N

φ(s)||u2 ||

4

0

|ω(s)|22 ds

−2

φ(s) Au1 − Au2 , ω(s)ds 0

t∧τ  N

+2

|φ(s) B(u1 (s)) − B(u2 (s)), ω(s)|ds 0 t∧τ  N

− 2α

φ(s) g(|u1 |)u1 (s) − g(|u2 |)u2 (s), ω(s)ds 0

t∧τ  N

+

φ(s)|σ (s, u1 (s), z) − σ (s, u2 (s), z)|22 ν(dz)ds 0

Z t∧τ  N

+2

φ(s)(ω(s), σ (s, u1 (s), z) − σ (s, u2 (s), z))η(dz, ˜ ds). 0

(3.56)

Z

We have B(u1 (s)) − B(u2 (s)) = B(u1 (s), ω(s)) + B(ω(s), u2 (s)), s ∈ [0, T ]. So

B(u1 (s)) − B(u2 (s)), ω(s) = B(ω(s), u2 (s)), ω(s), and hence 1

3

|2 B(u1 (s)) − B(u2 (s)), ω(s)| ≤ C|ω|22 ||ω|| 2 ||u2 ||, s ∈ [0, T ]. Therefore for every ε > 0, there exists Cε > 0 such that |2 B(u1 (s)) − B(u2 (s)), ω(s)| ≤ ε||ω||2 + Cε |ω|22 ||u2 ||4 , s ∈ [0, T ].

(3.57)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

By using (1.2), we have 2α g(|u1 (s)|)u1 (s) − g(|u2 (s)|)u2 (s), ω(s) ≥ 0.

(3.58)

Let a = Cε , we get −a|ω|22 ||u2 ||4 + Cε |ω|22 ||u2 ||4 = 0. Putting (3.57) and (3.58) into (3.56), we have t∧τ  N

φ(t ∧ τN )|ω(t

∧ τN )|22

+ (2 − ε)

φ(s)||ω(s)||2 ds 0

t∧τ  N

φ(s)|σ (s, u1 (s), z) − σ (s, u2 (s), z)|22 ν(dz)ds

≤ 0

Z t∧τ  N

φ(s)(ω(s), σ (s, u1 (s), z) − σ (s, u2 (s), z))η(dz, ˜ ds).

+2 0

(3.59)

Z

Taking expectation on both sides (3.59), we have t∧τ  N

Eφ(t ∧ τN )|ω(t

∧ τN )|22

+ (2 − ε)E

φ(s)||ω(s)||2 ds 0

t∧τ  N

φ(s)(l1 |ω(s)|22 + l2 ||ω(s)||2 )ds.

≤E

(3.60)

0

Choosing l2 < 2 and ε → 0, there exists a constant C1 > 0 such that t∧τ  N

Eφ(t ∧ τN )|ω(t

∧ τN )|22

+E

φ(s)||ω(s)||2 ds 0 t∧τ  N

≤ EC1

φ(s)|ω(s)|22 ds.

(3.61)

0

By the Gronwall lemma, we get the uniqueness of the solution. This completes the proof of Theorem 3.2.

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5965

4. Example In this section, we give some examples to illustrate our result derived above. Example 4.1. Consider the following stochastic 3D Navier-Stokes equation with damping driven by jump noise  ⎧ ˜ dz), du − udt + (u · ∇)udt + α|u|β udt + ∇pdt = Z σ (t, u, z)η(dt, ⎪ ⎪ ⎪ ⎨ ∇ · u = 0, (x, t) ∈ D × (0, T ), (4.1) ⎪ u(x, 0) = u0 (x), x ∈ D, ⎪ ⎪ ⎩ u(t, x)|∂D = 0. Lemma 4.2 (Lemma 2.4 [32]). Let g(|u|) = |u|β , then (1) g(|u|)u is continuously differential in R 3 and the Jacobian matrix is defined by, for any u = (u1 , u2 , u3 ) in R 3 : ⎛ 2 ⎞ βu1 + |u|2 βu1 u2 βu1 u3 ⎜ ⎟ (g(|u|)u) = |u|β−2 ⎝ βu1 u2 βu22 + |u|2 βu2 u3 ⎠ . βu1 u3 βu2 u3 βu23 + |u|2 Moreover, (g(|u|)u) is positive definite and for any u, v, w ∈ R 3 : |((g(|u|)u) v) · w| ≤ c|u|β |v||w|, where c is a positive constant depending on β. (2) g(|u|)u is monotonic in R 3 , i.e., for any u, v ∈ R 3 : (g(|u|)u − g(|v|)v, u − v) ≥ 0. It is easy to check g(|u|) = |u|β satisfies (1.2). Similar as the discussion of [18–20], using the nonlinear structure and delicated analysis (since α is varied), we could have the following existence and uniqueness of the strong solution for problem (4.1) for β > 2 with any α > 0 and α ≥ 14 as β = 2. Theorem 4.3. Suppose that Assumption (2.1) holds and β > 2 with any α > 0 and α ≥ 14 as β = 2. Then for any V-valued F0 -measurable initial data ξ satisfying E||ξ ||2p < ∞ for p ≥ 1, there exists a unique strong solution u(t) to the problem (4.1) with the initial condition u(0) = ξ , and u(t) ∈ L2 (, L2 ([0, T ], H 2 )) ∩ L2 (, L∞ ([0, T ], V )), it satisfies the following inequality T E( sup ||u(t)||

2p

+

0≤t≤T

T ||u(s)||

2p−2

0

||∇u(s)|| ds +

||u(s)||2p−2 |∇|u|

2

β+2 2

0

T  ||u(s)||2p−2 |u|β |∇u|2 dxds) ≤ C(E||ξ ||2p + 1).

+ 0 D

|22 ds

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

Example 4.4. Introduction the following stochastic tamed 3D Navier-Stokes equation driven by jump noise  ⎧ ˜ dz), du − udt + (u · ∇)udt + αgN (|u|β )udt + ∇pdt = Z σ (t, u, z)η(dt, ⎪ ⎪ ⎪ ⎨ ∇ · u = 0, (x, t) ∈ D × (0, T ), ⎪ x ∈ D, ⎪ u(x, 0) = u0 (x), ⎪ ⎩ u(t, x)|∂D = 0,

(4.2)

here, N > 0 and the function gN (·) is smooth such that ⎧ ⎪ ⎨ gN (r) = 0, r ∈ [0, N ], gN (r) = r − N, r ≥ N + 1, ⎪ ⎩  (r) ≤ C, r ≥ 0. 0 ≤ gN

(4.3)

It is easy to check g(|u|) = gN (|u|β ) satisfies (1.2). If we used the method in [28–30], we need the condition β = 2 and α = 1, but we will prove the well-posedness for the problem (4.2) for β > 2 with any α > 0 and α ≥ 14 as β = 2. We now write (4.2) as follows in the abstract form:  du(t) = −(Au(t) + Bu(t) + αg(|u(t)|)u(t))dt +

σ (t, u, z)η(dz, ˜ dt),

(4.4)

Z

u(x, 0) = ξ,

x ∈ D.

Definition 4.5. An V-valued cadl ` ag ` Ft -measurable process u(t) is said to be a solution of (4.4) if the following conditions are satisfied (1) u(t) ∈ L2 (, L2 ([0, T ], H 2 )) ∩ L2 (, L∞ ([0, T ], V )); (2) For any t ∈ [0, T ] and F0 -measurable V-valued initial data ξ , the following equality holds P -a.s. t u(t) = ξ −

t Au(s)ds −

0

t B(u(s))ds − α

0

g(|u(s)|)u(s)ds 0

t  σ (s, u(s), z)η(dz, ˜ ds).

+ 0 Z

Lemma 4.6. For any u ∈ H 2 and g(|u|) = gN (|u|β ) is defined by (4.3), then we get β+2

− g(|u|)u, u ≤ −|u|β+2 + C · N |u|22 ,   β+2 4β β 2 |∇|u| 2 |2 ds + C · N ||u||2 . − g(|u|)u, uV ≤ − |u| |∇u| ds − 2 (β + 2) D

D

(4.5) (4.6)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5967

Proof. For the term (4.5), we have  − g(|u|)u, u = −

gN (|u|β )|u|2 ds D





|u|

≤−

β+2

ds + CN

D β+2 = −|u|β+2

|u|2 ds D

+C

· N |u|22 .

(4.7)

For the term (4.6), we have − g(|u|)u, uV = − ∇(g(|u|)u), ∇u =−

3  

 ∂i uk (gN (|u|β )∂i uk − gN (|u|β )∂i |u|β uk )ds

k,i=1 D



|∇u|2 gN (|u|β )ds −

≤− D



≤−



 gN (|u|β )|∇|u|

D

4β |u| |∇u| ds + C · N ||u|| − (β + 2)2 2

β

D

4β (β + 2)2

This completes the proof of Lemma 4.6.

 |∇|u|

2

β+2 2

|2 ds

β+2 2

|2 ds.

(4.8)

D

2

Inspired by [28–30], we can get the above theorem easily. Similar as the discussion of the above Theorem 3.2, using the nonlinear structure and delicated analysis, we could have the following existence and uniqueness of the strong solution for problem (4.4) for β > 2 with any α > 0 and α ≥ 14 as β = 2. The main result of this section is the following theorem. Theorem 4.7. Suppose that Assumption 2.1 holds and β > 2 with any α > 0 and α ≥ 14 as β = 2. Then for any V-valued F0 -measurable initial data ξ satisfying E||ξ ||2p < ∞ for p ≥ 1, there exists a unique strong solution u to the problem (4.4) with the initial condition u(0) = ξ , and u(t) ∈ L2 (, L2 ([0, T ], H 2 )) ∩ L2 (, L∞ ([0, T ], V )), it satisfies the following inequality T E( sup ||u(t)||

2p

+

0≤t≤T

T ||u(s)||

2p−2

0

||∇u(s)|| ds +

||u(s)||2p−2 |∇|u|

2

β+2 2

|22 ds

0

T  +

||u(s)||2p−2 |u|β |∇u|2 dxds) ≤ C(E||ξ ||2p + 1).

(4.9)

0 D

Example 4.8. Consider the following stochastic three-dimensional Brinkman-Forchheimerextended Darcy model driven by jump noise

5968

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

 ⎧ ˜ dz), du − udt + (u · ∇)udt + a|u|2γ udt + b|u|2β udt + ∇pdt = Z σ (t, u, z)η(dt, ⎪ ⎪ ⎪ ⎨ ∇ · u = 0, (x, t) ∈ D × (0, T ), ⎪ u(x, 0) = u0 (x), x ∈ D, ⎪ ⎪ ⎩ u(t, x)|∂D = 0, (4.10) where, γ > β ≥ 0 are constants, a > 0 and b ∈ R. It is easy to check g(|u|) = α1 (a|u|2γ + b|u|2β ) satisfies (1.2). By the Theorem 1.2, in [26], we can get the above similar theorem easily. 5. Invariant measures 5.1. Exponential stability We consider the following a class of stochastic three-dimensional Navier-Stokes equations with damping driven by a additive noise of Lévy type  du(t) = −(Au(t) + Bu(t) + αg(|u(t)|)u(t))dt +

σ (t, z)η(dz, ˜ dt),

(5.1)

Z

u(x, 0) = ξ. Let

(5.2)



4 Z |σ (t, z)|2 λ(dz) = C1

< ∞. If λ(·) is a finite measure, then we get



1

1

|σ (t, z)|22 λ(dz) ≤ [λ(Z)] 2 C12 < ∞. Z

Inspired by [10,25], if λ(·) is a σ -finite measure, then there exists a measurable subset Um of Z  such that Um → Z and λ(Um ) < ∞. Let U c |σ (t, z)|22 λ(dz) → 0 as m → ∞, t ∈ [0, T ]. This m  condition is satisfied if λ(z) < ∞. Let Um |σ (t, z)|42 λ(dz) = C2 < ∞, then we have 

1

1

|σ (t, z)|22 λ(dz) ≤ [λ(Um )] 2 C22 < ∞. Um

Denote by 0 < λ1 ≤ λ2 ≤ · · · the eigenvalues of A and by e1 , e2 , · · · the corresponding complete orthonormal system of these eigenvectors. By using Poincaré’s inequality, we get λ1 |u|22 ≤ ||u||2 . As in Lemma 2.1 of [27,35], we have the following lemma. Lemma 5.1. Let E(t) be an increasing progressively measurable process with E(0) > 0 a.s. t 1 Let M(t) be a càdlàg local martingale with M(0) = 0. If Y(t) = 0 E(s) dM(s) converges a.s. to a finite limit as t → ∞, then limt→∞ sup |M(s)|.

M ∗ (t) E(t)

= 0 a.s. on the set {E(∞) = ∞}, where M ∗ =

0≤s≤t

Next, we will show the càdlàg local martingale.

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

5969

  Lemma 5.2. Let limt→∞ Z |σ (t, z)|42 λ(dz) = M1 < ∞ and limt→∞ Z |σ (t, z)|22 λ(dz) = M2 < ∗ ∞. Then we have limt→∞ M t (t) = 0 a.s., t  M(t) =

[|σ (s−, z)|22 + 2(σ (s−, z), u(s−))]η(dz, ˜ ds).

(5.3)

0 Z

Proof. Let E(t) = t + ε, for any ε > 0 and Y(t) =

t

1 0 s+ε dM(s).

By [2], we get

2  t   |σ (s−, z)|22 + 2(σ (s−, z), u(s−)) [Y, Y]t = η(dz, ds) (s + ε)2 0 Z

t  ≤2 0 Z

t



≤ 0 Z

t  + 0 Z

|σ (s−, z)|42 η(dz, ds) + 8 (s + ε)2

t 

|(σ (s−, z), u(s−))|2 η(dz, ds) (s + ε)2

0 Z

[2|σ (s, z)|42 + 8|σ (s, z)|22 |u(s)|22 ] λ(dz)ds (s + ε)2 [2|σ (s−, z)|42 + 8|σ (s−, z)|22 |u(s−)|22 ] η(dz, ˜ ds). (s + ε)2

(5.4)

Taking expectation on both sides of inequality (5.4), we get t  E[Y, Y]t ≤ 2 0 Z

|σ (s, z)|42 λ(dz)ds + 8 (s + ε)2

t 

|σ (s, z)|22 E|u(s)|22 λ(dz)ds. (s + ε)2

0 Z

(5.5)

Assume that λ1 denotes the first eigenvalue of the operator A. By applying Itô formula to the process |u(t)|22 , we get t E|u(t)|22

≤ E|ξ |22

− 2λ1

t  E|u(s)|22 ds

0

+

|σ (s, z)|22 λ(dz)ds.

(5.6)

0 Z

Applying Itô formula to the |u(t)|22 e2λ1 t and Poincaré inequality, we have

E|u(t)|22

≤ E|ξ |22 e−2λ1 t

t + 0

e−2λ1 (t−s)

 |σ (s, z)|22 λ(dz)ds. Z

(5.7)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

As t → ∞ in (5.5), we deduce ∞ E[Y, Y]∞ ≤ C 0

1 ds < ∞. (s + ε)2

So, [Y, Y]∞ < ∞ a.s. By the above Lemma 5.1, this completes the proof of Lemma 5.2.

(5.8)

2

Theorem 5.3 (Exponential stability). Assume that X with  and Y denote two solutions of (5.1) 2 initial values X0 and Y0 respectively. Let limt→∞ Z |σ (t, z)|22 λ(dz) = M2 < ∞. If 2M < 1, λ1 where λ1 denotes the first eigenvalue of the operator A, then we have limt→∞ |X(t) − Y (t)|2 = 0 a.s. Proof. Since X(t) and Y (t) denote two solutions of (5.1), we get t X(t) − Y (t) +

t A(X(s) − Y (s))ds +

0

(B(X(s)) − B(Y (s)))ds 0

t +α

(g(|X(s)|)X(s) − g(|Y (s)|)Y (s))ds = X0 − Y0 . 0

Let W (t) = X(t) − Y (t), applying the Itô formula to |W (t)|22 , we get t |W (t)|22

+2

t ||W (s)|| ds + 2

(B(X(s)) − B(Y (s)), W (s))ds

2

0

0

t + 2α

(g(|X(s)|)X(s) − g(|Y (s)|)Y (s), W (s))ds = |W (0)|22 , 0

where W (0) = X0 − Y0 . We have B(X(s)) − B(Y (s)) = B(X(s), W (s)) + B(W (s), Y (s)), s ∈ [0, T ]. So (B(X(s)) − B(Y (s)), W (s)) = (B(W (s), Y (s)), W (s)), and hence 1

3

|2(B(X(s)) − B(Y (s)), W (s))| ≤ C|W |22 ||W || 2 ||Y ||, s ∈ [0, T ]. Therefore for every ε > 0, there exists Cε > 0 such that

(5.9)

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

|2(B(X(s)) − B(Y (s)), W (s))| ≤ ε||W ||2 + Cε |W |22 ||Y ||4 , s ∈ [0, T ].

5971

(5.10)

By (1.2), we have 2α(g(|X(s)|)X(s) − g(|Y (s)|)Y (s), W (s)) ≥ 0.

(5.11)

Choosing ε sufficiently small and applying Poincaré inequality, putting (5.10) and (5.11) into (5.9), we have t |W (t)|22

+ λ1

t |W (s)|22 ds

≤ |W (0)|22

+ Cε

0

|W |22 ||Y ||4 ds.

(5.12)

0

By applying Gronwall’s inequality, we have t |W (t)|22

≤ |W (0)|22 exp(C

||Y ||4 ds − λ1 t).

(5.13)

0

Applying the Gagliardo-Nirenberg inequality, we deduce t

t ||Y || ds ≤

0

t |Y |22 ||∇Y ||2 ds

4

≤ sup

0

0
|Y (s)|22

||∇Y ||2 ds.

(5.14)

0

Using the Itô formula to |Y (t)|22 , we have t |Y (t)|22

+2

t  ||Y || ds 2

≤ |Y (0)|22

0

t

+

|σ (s−, z)|22 λ(dz)ds 0 Z



+

[|σ (s−, z)|22 + 2(σ (s−, z), Y (s−))]N˜ (ds, dz).

(5.15)

0 Z

Hence, we have |Y (t)|22 |Y (0)|22 1 ≤ + t t t

t  |σ (s−, z)|22 λ(dz)ds 0 Z

+

1 t

t 

[|σ (s−, z)|22 + 2(σ (s−, z), Y (s−))]N˜ (ds, dz).

0 Z

By the above Lemma 5.2, we have

(5.16)

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H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

1 lim t→∞ t

t 

[|σ (s−, z)|22 + 2(σ (s−, z), Y (s−))]N˜ (ds, dz) = 0.

(5.17)

0 Z

By the assumption of the theorem, we get t  |σ (s, z)|22 λ(dz)ds = M2 .

lim

t→∞

(5.18)

0 Z

Putting (5.17) and (5.18) into (5.16), we get limt→∞ Using the Itô formula to ||Y (t)||2 , we have T ||Y (t)|| +

|Y (t)|22 t

T  ||∇Y (s)|| ds +

2

≤ M2 .

|Y |β ||Y (s)||2 dxds

2

0

T  +

0 D

g  (|Y |)|Y ||∇Y |2 dxds ≤ C,

0 D

C represents for a generic positive constant which just dependents on initial value. Hence, we have t ||∇Y ||2 ds ≤ C.

(5.19)

0

Let

CM2 λ1

< 1, we have limt→∞ |X(t) − Y (t)|2 = 0. This completes the proof of Theorem 5.3.

2

5.2. Uniqueness of stationary measures Theorem 5.4 (Uniqueness of invariant measures). Assume that σ (t, x, z) = σ (x, z) and the above theorem is satisfied. Then there exists a unique stationary measure with support in V for the solution X(t, x, ω) for a class of stochastic 3D Navier-Stokes with damping driven by jump noise. Proof. Inspired by [8,25,27] and the above conclusion, it is easy to get existence of invariant measures. For uniqueness, let μ1 and μ2 be two probability measures on V that are stationary for prob lem (5.1). We will show that for any φ ∈ Cb (V ), V φ(z)dμ1 (z) = V φ(z)dμ2 (z). Assume that X ε denotes the solution of (5.1) with X(0) = ξ . By   μ1 and μ2 are invariant measures, then we have μ1 (B) = V Pξ (t, B)dμ1 (ξ ) and μ2 (B) = V Pξ (t, B)dμ2 (ξ ), where Pξ (t, B) = t ξ P {X ξ (t) ∈ B} and X ξ (0) = ξ . Define μt (B) = 1t 0 Pξ (t, B)ds for any B ∈ B(V ). Then we get  E(φ(X ξ (t))) = V φ(x)Pξ (t, dx). Applying Fubini’s theorem, we have

H. Gao, H. Liu / J. Differential Equations 267 (2019) 5938–5975

 μxT (dz)dμ1 (x) =

1 T

T  T 1 ( Px (t, dz)dμ1 (x))dt = μ1 (dz)dt = μ1 (dz). T 0

V



5973

0

V

y

Similarly, we get V μt (dz)dμ2 (x) = μ2 (dz). By applying Fubini’s theorem, Jensen’s inequality and stationarity measures to get, for any φ ∈ Cb (V ),  |

 φ(z)dμ1 (z) −

V

φ(z)dμ2 (z)| V

 

=| V V

=|

 

φ(z)μxT (dz)dμ1 (x) −

1 T

1 =| T

T

V V

    [ φ(z)Px (t, dz)dμ1 (x) − φ(z)Py (t, dz)dμ2 (y)]dt|

0

V V

T



V V

1 E(φ(X (t)))dμ1 (x)dt − T

E(φ(X y (t)))dμ2 (y)dt| 0 V

T     x [ E(φ(X (t)))dμ1 dμ2 − E(φ(X y (t)))dμ2 dμ1 ]dt| 0

1 ≤ T

T 

x

0 V

1 =| T

y

φ(z)μT (dz)dμ2 (y)|

V V

V V

  T E|φ(X x (t)) − φ(X y (t))|dtdμ1 (x)dμ2 (y). V V

(5.20)

0

Applying the exponential stability and the continuity of φ, we get |φ(X x (t)) − φ(X y (t))| → T 0 a.s. as t → ∞. Hence, we have T1 0 |φ(X x (t)) − φ(X y (t))|dt → 0 as T → ∞. By using dominated convergence theorem, we have 1 T

  T E|φ(X x (t)) − φ(X y (t))|dtdμ1 (x)dμ2 (y) −→ 0 as V V

T → ∞.

0

This completes the proof of Theorem 5.4.

2

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