2D stochastic Navier–Stokes equations driven by jump noise

Nonlinear Analysis 79 (2013) 122–139

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2D stochastic Navier–Stokes equations driven by jump noise Zdzisław Brzeźniak a , Erika Hausenblas b , Jiahui Zhu c,d,∗ a

Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

b

Department of Mathematics and Information Technology, Montanuniversity Leoben, Franz Josef Str. 18, 8700 Leoben, Austria

c

Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

d

School of Finance, Zhejiang University of Finance & Economics, Hangzhou, 310018, China

article

info

Article history: Received 16 October 2012 Accepted 18 October 2012 Communicated by Enzo Mitidieri Keywords: Stochastic Navier–Stokes equations Lévy noise

abstract In the paper, we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of Lévy type. Our result is formulated in an abstract setting. This type of equation covers the stochastic 2D Navier–Stokes Equations, the 2D stochastic Magneto-Hydrodynamic Equations, the 2D stochastic Boussinesq Model for the Bénard Convection, the 2D stochastic Magnetic Bérnard Problem, the 3D stochastic Leray α -Model for the Navier–Stokes Equations and several stochastic Shell Models of turbulence. © 2013 Published by Elsevier Ltd

1. Introduction Stochastic Partial Differential Equations are used to model physical systems subjected to influence of internal, external or environmental noises or to describe systems that are too complex to be described deterministically, e.g. a flow of a chemical substance in a river subjected by wind and rain, an airflow around an airplane wing perturbed by the random state of the atmosphere and weather, a laser beam subjected to turbulent movement of the atmosphere, spread of an epidemic in some regions and the spatial spread of infectious diseases. SPDEs are also used in the physical sciences (e.g. in plasmas turbulence, physics of growth phenomena such as molecular beam epitaxy and fluid flow in porous media with applications to the production of semiconductors and to the oil industry) and biology (e.g. bacteria growth and DNA structure). Models related to the so called passive scalar equations have potential applications to the understanding of waste (e.g. nuclear) convection under the earth’s surface. The presence of noise leads to new and important phenomena. For example, the 2-dimensional Navier–Stokes equations with sufficiently degenerate noise have a unique invariant measure and hence exhibit ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data. Despite continuous efforts in the last thirty years such a property has so far not been found for the deterministic counterpart of these equations. This property could lead to profound understanding of the nature of turbulence. The aforementioned Navier–Stokes Equations (NSEs) are now a widely accepted model of fluid motion, see for instance the well known monograph by R. Temam [1]. The theory of NSEs is reasonable well understood. For instance, in the case of 2-dimensional domains, it is known since the pioneering works of Lions and Prodi in the 1960s (see for instance [2]) that the solutions exist for all times and are unique. In the 3-dimensional case it is known that the weak solutions exist for all times, see celebrated work of Leray [3], and that the strong solutions are unique. However, despite many efforts in the recent years the questions whether the weak solutions are unique or strong solutions exist for all times, remain unresolved, see for instance [4].

∗

Corresponding author at: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. E-mail addresses: [email protected] (Z. Brzeźniak), [email protected] (E. Hausenblas), [email protected] (J. Zhu).

0362-546X/$ – see front matter © 2013 Published by Elsevier Ltd doi:10.1016/j.na.2012.10.011

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

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The theory of stochastic Navier–Stokes equations apparently has its roots in the 1959 edition of the Landau and Lifshitz ‘‘Fluid Mechanics’’ [5].1 To our best knowledge, the first work on the stochastic NSEs (SNSEs) written from the mathematical point of view is a paper [6]. Later the motivation for the large deviations paper of Faris and Jona-Lasinio [7] was clearly the stochastic fluid dynamics as they wrote, roughly speaking, ‘‘the motion of a viscous incompressible fluid is described by the so called Navier–Stokes equations. However, these equations are only approximate. In particular, they take into account only the macroscopic nature of the fluid motion. Quantum effects and other sources of fluctuations are completely ignored and microscopic effects are only taken into account to a certain degree. It is therefore of utmost importance to understand those properties of solutions to the Navier–Stokes equations which persist with respect to perturbations, in particular small random perturbations’’. As we mentioned earlier, the study of stochastic Navier–Stokes equations has been initiated by Bensoussan and Temam in [6]. This paper was followed by articles Višik and A.I. Komeč [8], Fujita-Yashima [9], Brzeźniak, Capiński and Flandoli [10,11], Capiński and Cutland [12], Flandoli and Gatarek [13], and continued by several authors (for example Da Prato, Gatarek). These earlier papers were followed by a stream of papers studying deep mathematical properties like the existence of global solutions and of Markov selections for 3D SNSEs [14–16], the uniqueness of an invariant measure for 2D SNSEs with degenerate noise [17–19], the existence of solutions to stochastic Euler equations [20,21], the large deviation principle [22,23]. All these works were all concerned with Navier–Stokes equations perturbed by Gaussian noise, additive or multiplicative. In fact, according to [5], the mathematical correct setting for hydrodynamics are Navier–Stokes equations subject to random excitations. The idea of considering Navier–Stokes equations subject to random excitations with jumps is an old one. It has been a motivation for the work [24]. In a recent paper [25] by Dong and Xie the authors study the stochastic NSEs driven by a Poisson random measure whose Lévy measure is finite. This assumption implies that the jump times form a discrete subset of the real half line R+ and it is essentially the same as the deterministic one. The proof of the main result is based on the approximation of the Poisson random measure N by a sequence of Poisson random measures Nn whose Lévy measures are finite. Our approach is different. We consider the Galerkin approximation of the SNSEs and by employing the results from [24] we prove certain uniform estimates for the sequence {un } of the approximated problems. Then, as in [22,23,26], we use only the weak compactness and certain local monotonicity properties of the nonlinear Stokes operator to prove the existence and uniqueness of the solution. Note that our proof, contrary to the proof in the Appendix of [26] does not contain an argument based on approximating the Dirac delta measure by a sequence of test functions. Stochastic reaction diffusion equations driven by Lévy noise have been a subject of a recent paper [27] by the first two authors. However, our present paper does not rely on the compactness method used in that work. We believe that the compactness methods developed in [27] can be generalized so that the 3D Stochastic NSEs can be treated in this way. After this paper was practically finished we learned about a work by Fernando and Sritharan [28] in which similar questions were studied. Moreover, the first and the third authors together with W. Liu [29] have built upon this paper and generalized it in a twofold way. They considered general locally monotone operators and more general noise (including equations driven by general Lévy noise).

¯ we denote the set N ∪{+∞}. Whenever Notation 1.1. By N we denote the set of natural numbers, i.e. N = {0, 1, 2, . . .} and by N ¯ )-valued measurable functions we implicitly assume that the set is equipped with the trivial σ -field 2N (or we speak about N (or N ¯ 2N ). By R+ we will denote the interval [0, ∞) and by R∗ the set R \ {0}. If X is a topological space, then by B (X ) we will denote the Borel σ -field on X . By λ we will denote the Lebesgue measure on (R, B (R)). If (S , S ) is a measurable space then by S ⊗ B (R+ ) we will denote the product σ -field on S × R+ and by ν ⊗ λ the product measure of ν and the Lebesgue measure λ. Moreover, by M+ (S ) we denote the set of all positive measures on S, by MI (S × R+ ) we denote the family of all N-valued measures on (S × R+ , S ⊗ B (R+ )) and by MI (S × R+ ) the σ -field on MI (S × R+ ) generated by functions iB : MI (S × R+ ) ∋ µ → µ(B) ∈ N, B ∈ S . 2. Preliminaries Let (Z , Z) be a measurable space and let ν be a σ -finite positive measure on it. Suppose that P = (Ω , F , F, P) is a ¯ is a time homogeneous Poisson filtered probability space, where F = (Ft )t ≥0 is a filtration, and η : Ω × B (R+ ) × Z → N random measure with the intensity measure ν defined over the filtered probability space P. We will denote by η˜ = η − γ the compensated Poisson random measure associated to η, where the compensator γ is given by

B (R+ ) × Z ∋ (A, I ) → γ (A, I ) = ν(A)λ(I ) ∈ R+ . We assume that (H , | · |H ) is a Hilbert space. It is then known, see e.g. [30],2 that there exists a unique continuous linear operator I which associates to each progressively measurable process ξ : R+ × Z × Ω → H satisfying T





|ξ (r , z )|2H ν(dz ) dr < ∞,

E 0

T > 0.

(2.1)

Z

1 Thanks to Sam Braunstein for this information. 2 In fact, the article [30] deals with more general spaces and more general Lévy measures, but here in this work, we only need the Hilbert space setting.

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Moreover, I (ξ ) is an H-valued adapted and càdlàg process such that for any random step process ξ satisfying the above condition (2.1) with a representation

ξ (r ) =

n 

1(tj−1 ,tj ] (r )ξj ,

r ≥ 0,

j=1

where {0 = t0 < t1 < · · · < tn < ∞} is a partition of [0, ∞), and for all j, ξj is an Ftj−1 measurable random variable, one has I (ξ )(t ) =

n  

  ξj (z ) η˜ dz , (tj−1 ∧ t , tj ∧ t ] ,

t ≥ 0.

(2.2)

Z

j =1

As usual in such cases we will write

 t 0

ξ (r , z ) η( ˜ dr , dz ) := I (ξ )(t ),

t ≥ 0.

Z

The continuity (more precisely, isometry in Hilbert spaces) of the operator I mentioned above means that

 t  

E 

0

Z

2  t  ξ (r , z ) η( ˜ dr , dz ) = E |ξ (r , z )|2H ν(dz ) dr , 0

H

t ≥ 0.

(2.3)

Z

The class of all progressively measurable processes ξ : R+ × Z × Ω → H satisfying the condition (2.1) will be denoted by M 2 (R+ , L2 (Z , ν, H )). If T > 0, the class of all progressively measurable processes ξ : [0, T ] × Z × Ω → H satisfying the condition (2.1) just for this one T , will be denoted by M 2 (0, T , L2 (Z , ν, H )). 3. 2D stochastic Navier–Stokes equations Let us consider an incompressible viscous fluid of a constant density (assumed to be equal to 1) and of a constant viscosity

ϱ > 0 enclosed in a region D ⊂ R2 and driven by an external time-dependent force f : R+ × D → R2 . We denote by u(t , x) ∈ R2 and p(t , x) ∈ R, respectively, the velocity and the pressure of the fluid at the point x ∈ D at time t ≥ 0. We assume that the time evolution of the velocity and pressure of the fluid is governed by the initial-boundary value problem associated with Navier–Stokes equations:

 ∂u     ∂ t − ϱ1u + (u · ∇)u + ∇ p = f div u = 0   u=0   u(·, 0) = u0

in D, in D, on ∂ D, in D.

(3.1)

We assume that D ⊂ R2 is an arbitrary (bounded or unbounded) domain with boundary ∂ D satisfying the cone property. We will use the standard mathematical framework of the NSEs, see e.g. [1]. The basic functional space is the Lebesgue space L2 (D) := L2 (D, R2 ) with scalar product

(u, v) =



(uj (x)vj (x)) dx D

j

and norm | · | = (·, ·)1/2 . We will also need the Sobolev space Hk,p (D) = H k,p (D, R2 ), k ∈ N and p ∈ [1, ∞) consisting of all u ∈ Lp (D, R2 ) whose weak derivatives up to order k belonging to Lp (D, R2 ) as well. It is known that Hk,p (D) is a separable Banach space with norm

 ∥u∥k,p :=

1/p



|Dα u(x)|p dx

.

|α|≤k D

Obviously Hk,2 (D), k ∈ N is a Hilbert space with a naturally defined scalar product. We will consider the weak solutions to problem (3.1) and for this we need a proper space of test functions. We take

V = V (D) := {φ ∈ C0∞ (D, R2 ) : div φ = 0 in D}. The closure of V in L2 (D), respectively in H1,2 (D) will be denoted by H, resp. V. The scalar products norms in those two spaces are those inherited from L2 (D), resp. H1,2 (D). The norm in H, resp. V, will be denoted in the whole paper by | · |, resp. by ∥ · ∥. We call D a Poincaré domain if there exists a real number λ1 > 0 such that



φ 2 dx ≤ D

1

λ1



|∇φ|2 dx, D

φ ∈ C0∞ (D, R2 ).

(3.2)

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125

The inequality (3.2) is called the Poincaré inequality and it can be shown that if D is bounded in some direction, i.e. there exists a vector b ∈ R2 such that supx∈D |(x, b)| < ∞, then D is a Poincaré domain. If D is a Poincaré domain, then the original norm on V is equivalent to the norm ∥ · ∥ induced by the scalar product

((u, v)) =

  2

∇ uj · ∇vj dx = (∇ u, ∇v),

u, v ∈ V .

(3.3)

D j =1

We next define a bilinear form a : V × V → R by a(u, v) := (∇ u, ∇v),

u, v ∈ V .

(3.4)

Since obviously the form a coincides with the ((·, ·)) scalar product in V, it is V-continuous, i.e. it satisfies |a(u, u)| ≤ C ∥u∥2 for some C > 0 and all u ∈ V. Hence, by the Riesz Lemma, there exists a unique linear operator A : V → V′ , where V′ is the dual of V, such that a(u, v) = ⟨Au, v⟩, for u, v ∈ V. Moreover, the form a is V-coercive, i.e. it satisfies a(u, u) ≥ α∥u∥2 for some3 α > 0 and all u ∈ V. Therefore, by means of the Lax–Milgram theorem, see e.g. Temam [31, Theorem II.2.1], the operator A : V → V′ is an isomorphism. Since V is densely and continuously embedded into H and H can be identified with its dual H′ , we have the following embeddings V⊂H∼ = H ′ ⊂ V′ .

(3.5)

The duality between V and V will be denoted by ⟨f , v⟩ for f ∈ V and v ∈ V. We assume that ′

′

⟨f , v⟩ = (f , v) whenever f ∈ H, v ∈ V.

(3.6)

Then we say that the spaces V, H and V form a Gelfand triple. Next we define an unbounded linear operator A in H as follows. ′



D(A) := {u ∈ V : Au ∈ H}, Au := Au, u ∈ D(A).

(3.7)

It is now well established that under some additional assumptions related to the regularity of the domain D, the space D(A) can be characterized in terms of Sobolev spaces. For example, see [32], where only the 3-dimensional case is studied. However, if D ⊂ R2 is a uniform C 2 -class Poincaré domain, the result are also valid in the 2-dimensional case. Let P : L2 (D) → H be the orthogonal projection. Then we have



D(A) := V ∩ H1,2 (D), Au := −P1u, u ∈ D(A).

(3.8)

It is also a classical result, see e.g. Cattabriga [33] or Temam [31, p. 56], that A is a non-negative self adjoint operator in H. Moreover, see p. 57 in [31], V = D(A1/2 ) and

(Au, u) = ∥u∥,

u ∈ V.

(3.9)

Let us recall a result of Fujiwara–Morimoto [34] that the projection P extends to a bounded linear projection in the space Lq (D), 1 < q < ∞. Remark 3.1.

1,2

(i) Let us denote by H0 (D) the closure of C0∞ (D, R2 ) in H1,2 (D). It can be shown that V is equal to the closure

1,2 H0

of V in (D). ⃗ denotes (ii) The characterization of the spaces H and V given in [1] holds true also when D is a Poincaré domain. Namely, if n the external normal vector field to ∂ D, then H⊥ = {u ∈ L2 (D) : u = grad p, for some p ∈ L2loc (D)},

⃗|∂ D = 0}, H = {u ∈ L2 (D) : div u = 0, u · n 1,2

V = {u ∈ H0 (D) : div u = 0}. (iii) If D is a bounded domain, then the operator A is invertible and its inverse A−1 is bounded, self-adjoint and compact in H. Hence the spectrum of A consists of an infinite sequence 0 < λ1 ≤ λ2 ≤ · · · , λm → ∞ of eigenvalues listed with their multiplicity, and there exists an orthogonal basis {wm }m≥1 of H consisting of eigenvectors of A : Awm = λm wm , m ∈ N. (iv) If D is a Poincaré domain, then the operator A is invertible, its inverse A−1 is bounded and

|Au|2 ≥ λ1 (Au, u) ≥ λ21 |u|2 ,

for all u ∈ D(A).

(3.10)

Moreover, the graph norm on D(A) is equivalent to the |A · |-norm and, see p. 57 in [31],

⟨Au, u⟩ = ((u, u)) = ∥u∥2 = |∇ u|2 , 3 Precisely for α = 1.

u ∈ D(A).

(3.11)

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(v) If D is not a Poincaré domain, then ∥ · ∥ is only a seminorm on V. The same comment applies to the seminorm |A · | on D(A). Next, we define the following fundamental trilinear form: b(u, v, w) =

 u∇vw dx =

2   i,j=1

D

ui (x)Di v j (x)w j (x) dx, D

whenever u, v, w ∈ L1loc (D) are such that the integral on the right-hand side (RHS) exists. If u, v are such that the linear map b(u, v, ·) is continuous on V, the corresponding element of V′ will be denoted by  B(u, v). We2 will also denote, with a slight abuse of notation, B(u) = B(u, u). Note that if u, v ∈ H are such that (u∇)v = j uj Dj v ∈ L (D), then B(u, v) = P (u∇v). The following are some fundamental properties of the form b, see e.g. [1, Lemma 1.3, p. 163] and Temam [31]. There exists a constant C > 0 such that 1,2

b(u, v, v) = 0,

for all u ∈ V, v ∈ H0 (D),

(3.12)

1,2

b(u, v, w) = −b(u, w, v),

for all u ∈ V, v, w ∈ H0 (D),

and

 1/2 |u| |∇ u|1/2 |∇v|1/2 |Av|1/2 |w|, u ∈ V, v ∈ D(A), w ∈ H,    1/2 |u| |Au|1/2 |∇v||w|, u ∈ D(A), v ∈ V, w ∈ H, |b(u, v, w)| ≤ C |u||∇v||w|1/2 |Aw|1/2 , u ∈ H, v ∈ V, w ∈ D(A),    1/2 |u| |∇ u|1/2 |∇v||w|1/2 |∇w|1/2 , u, v, w ∈ V.

(3.13)

Also, from Temam [1, Lemma III.3.3], we have the following inequality 1/2

1/2

|v|L4 (D) ≤ 21/4 |v|L2 (D) |∇v|L2 (D) ,

v ∈ H10,2 (D).

(3.14)

By means of the Hölder inequality we can deduce the following inequality

|b(u, v, w)| ≤ |u|L4 (D) |∇v|L2 (D) |w|L4 (D) ,

1,2

u, v, w ∈ H0 (D).

(3.15)

Hence b is a bounded trilinear map from L4 (D) × V × L4 (D) to R. Moreover, we have following result which is fundamental for our purposes. Lemma 3.2. The trilinear map b : V × V × V → R has a unique extension to a bounded trilinear map from L4 (D)×(L4 (D)∩ H)× V to R. It follows from Lemma 3.2 that B maps L4 (D) ∩ H (and so V) into V′ and

|B(u)|V′ ≤ C1 |u|2L4 (D) ≤ 21/2 C1 |u||∇ u| ≤ C2 |u|2V ,

u ∈ V.

(3.16)

Using the above notation it is now customary to consider the following functional analytic version of problem (3.1)



du

+ ϱAu + B(u) = f (t ), dt u(0) = u0 .

t ≥ 0;

(3.17)

4. The abstract setting Let H be a separable Hilbert space and A be a (possibly unbounded) self-adjoint positive linear operator on H. We 1

shall denote the scalar product and the norm of H by (·, ·) and | · |, respectively. Set V = dom(A 2 ) equipped with norm 1

∥x∥ := |A 2 x|, x ∈ V . Let B : V × V → V ′ be a continuous mapping. Finally, let (Z , Z) be a measurable space, ν a σ -finite measure on Z , ν({0}) = 0, and σ : [0, ∞) × H → L2 (Z , ν; H ) a measurable function, satisfying certain condition specified later. Let η˜ be a compensated time homogeneous Poisson random measure on (Z , Z) with intensity ν . The aim of this paper is to study the abstract evolution equation given by du(t ) + Au(t ) dt + B(u(t ), u(t )) =



σ (t , u(t ), z ) η( ˜ dt , dz ),

(4.1)

Z

u(0) = ξ . Definition 4.1. An H-valued càdlàg F-adapted process u is called a solution of (4.1) if the following conditions are satisfied (S1) u(t , ω) ∈ V for almost all (t , ω) ∈ [0, T ] × Ω and E

T 0

∥u(t )∥2 dt < ∞;

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

127

(S2) the following equality holds for every t ∈ [0, T ], as an element of V ′ , P-a.s. u(t ) = ξ −

t



Au(s) ds −

t



B(u(s), u(s)) ds +

 t 0

0

0

σ (s, u(s), z ) η( ˜ ds, dz ).

(4.2)

Z

An alternative version of condition (S2) is to require that for any φ ∈ V

(u(t ), φ) = (ξ , φ) −

t



⟨Au(s), φ⟩ ds −



t

⟨B(u(s)), φ⟩ ds +

 t 0

0

0

(σ (s, u(s), z ), φ) η( ˜ ds, dz ),

t ≥ 0.

(4.3)

Z

Now we introduce the condition under which the existence and uniqueness can be proven. Condition 4.1. Assume that B : V × V → V ′ is a continuous bilinear mapping satisfying the following conditions: (B1) (skewsymmetricity of B)

⟨B(u1 , u2 ), u3 ⟩ = −⟨B(u1 , u3 ), u2 ⟩,

for u1 , u2 , u3 ∈ V

(4.4)

(B2) there exists a reflexive and separable Banach space space (Q, | · |Q ) and a constant a0 > 0 such that V ⊂ Q ⊂ H,

(4.5) for any v ∈ V ;

2 Q

|v| ≤ a0 |v|∥v∥,

(4.6)

(B3) there exists a constant C > 0 such that

|⟨B(u, v), w⟩| ≤ C |u|Q |v||w|Q ,

for all u, v, w ∈ V .

(4.7)

Remark 4.2. The above assumptions imposed on B have quite important consequences, see Remark 2.1 in [26]. For instance, for every k > 0, one can find a constant Ck > 0 such that

|⟨B(u, u) − B(v, v), u − v⟩| ≤ k∥u − v∥2 + Ck |u − v|2 |v|4Q ,

for all u, v ∈ V .

(4.8)

For the function σ we make the following assumptions. Condition 4.2. Suppose that there exist nonnegative constants K0 , K1 , K2 , L1 and L2 ∈ (0, 2), such that for all t ∈ [0, T ] and u ∈ V , and for any t ∈ [0, T ] and all u, v ∈ V

(σ 1) |σ (t , u)|2L2 (Z ,ν;H ) ≤ K0 + K1 |u|2 + K2 ∥u∥2 . (σ 2) |σ (t , u) − σ (t , v)|2L2 (Z ,ν;H ) ≤ L1 |u − v|2 + L2 ∥u − v∥2 . (σ 3) |σ (t , u)|4L4 (Z ,ν;H ) ≤ K (1 + |u|4 + ∥u∥4 ). Remark 4.3. Assumptions (σ 1) and (σ 2) are fairly standard assumptions when one considers the existence and uniqueness of SPDEs. Assumption (σ 3) is needed to show uniform estimates on the 4-th moment of the approximating solutions, see Proposition 4.5. We would like to point out that not only the 2D stochastic Navier–Stokes, but also several other models are satisfying these assumptions above. In particular, the 2D stochastic Magneto-Hydrodynamic Equations, the 2D stochastic Boussinesq Model for the Bénard Convection, the 2D stochastic Magnetic Bérnard Problem, the 3D stochastic Leray α -Model for the Navier Stokes equations and several stochastic Shell Models of turbulence satisfy these assumptions. For a nice description of the Equations we refer to [26, Section 2.1]. Theorem 4.4. Assume that Conditions 4.1 and 4.2 are satisfied. Then for any F0 -measurable H-valued function ξ satisfying E|ξ |4 < ∞, there exists a unique solution u = {u(t ) : 0 ≤ t < ∞} to Problem (4.1) such that for any T > 0

 E

sup |u(t )|4 +

0 ≤t ≤T



T

   ∥u(s)∥2 |u(s)|2 ds ≤ C E|ξ |4 + 1 .

0

For clarity, let us put in the following notation: F (u) := −Au − B(u, u),

u ∈ V.

From Remark 4.2, one can see that the following local monotonicity assumption holds:

⟨F (u) − F (v), u − v⟩ ≤ −(1 − k)∥u − v∥2 + Ck |v|4Q |u − v|2H .

(4.9)

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Proof of Theorem 4.4. The proof will be split into five steps, starting at step 0. Step. 0 Uniform a’priori estimates. Let {φi : i ∈ N} ⊂ D(A) be an orthonormal basis in H such that span{φi : i ∈ N} is dense in V . Let Πn denote the projection of V ′ onto Hn := span{φ1 , . . . , φn }. That is

Πn x =

n 

⟨x, φi ⟩φi .

i=1

Then Πn |H is the orthogonal projection of H onto Hn . For every n ∈ N, we consider the finite dimensional system of SDEs on Hn given by dun (t ) = Πn F (un (t )) dt +



Πn σ (t , un (t ), z ) η( ˜ dt , dz ),

t ≥ 0,

(4.10)

Z

un (0) = Πn ξ . We note that since Πn is a contraction of V ′ , we infer that under the Conditions 4.1 and 4.2, Fn = Πn F is locally Lipschitz and σn := Πn σ is globally Lipschitz. As we know from e.g. Albeverio, Brzeźniak and Wu [24], on the basis of Condition 4.2, Eq. (4.10) has a unique Hn -valued càdlàg local strong solution un . However, by the skew symmetricity of B the solution can be extended to any time interval [0, T ], T > 0. The proposition below is the key tool to prove the Theorem 4.4. Proposition 4.5. Let the assumptions be as in Theorem 4.4. Then there exists a constant C > 0 such that for p = 1, 2 we have

 sup E

sup |un (t )|

2p

∥un (s)∥ |un (s)| 2

+

t ∈[0,T ]

n

T



2p−2

 ds

  ≤ C E |ξ |2p + 1 .

0

Proof of Proposition 4.5. As mentioned above, it follows from Theorem 2.8 in [24] that Eq. (4.10) has a unique càdlàg global strong solution un in Hn . That means, for any n ∈ N there exists a unique solution on the interval [0, T ] satisfying un (t ) = Πn ξ +

t



Πn F (un (s)) ds +

 t

0

0

σn (s, un (s), z ) η( ˜ ds, dz ),

t ∈ [0, T ].

(4.11)

Z

Let un = (un (t ))t ∈[0,T ] be the unique solution to the Eq. (4.10), see for instance [24] and references therein. We begin by considering the case when p = 1. The Itô formula (B.2), see also for example Theorem II-5.1 in [35], applied to the function ϕ(x) = |x|2 , so that ∇ϕ(x) = 2x, Dij ϕ = 2δij and ϕ(x + y) − ϕ(x) − ⟨y, ∇ϕ(x)⟩ = ϕ(y) = |y|2 , and the skew-symmetricity of b, i.e. Condition 4.1-(2), give for 0≤t ≤T

|un (t )| + 2 2



t

∥un (s)∥ ds = |Πn ξ | + 2 2

2

0

 t 0

 t

|σn (s, un (s), z )|2 η(ds, dz ),

+ 0

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

0 ≤ t ≤ T.

(4.12)

Z

We proceed as done in [26, Appendix] and use Lemma A.1 from therein. Therefore, we fix a natural number R > 0 and put, for t ∈ [0, T ],

τRn := inf{t ∈ [0, T ] : |un (t )|2 ≥ R}, XR (t ) :=

sup 0≤s≤t ∧τRn

IR (t ) :=

sup

|un (s)|2 ,

YR (t ) :=

t ∧τRn



∥un (s)∥2 ds,

0

  s   s     2  (un (r −), σn (r , un (r ), z )) η( ˜ dr , dz ) + |σn (r , un (r ), z )|2 η(dr , dz ) . n 0

0≤s≤t ∧τR

Z

0

Z

We will handle the two terms in IR (t ) separately. Let us denote mn,R (t ) :=

t ∧τRn

 0



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

Z

Since the process un (t ), t ∈ [0, T ] is adapted and càdlàg, we see that limR→∞ R{τRn < T } = 0. Note that the process mn,R is a martingale. We may therefore apply first the Burkholder–Davis–Gundy inequality (see [36] or [37]), Condition 4.2, then

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

129

the Hölder inequality, and finally the Young inequality to get

 s E

sup

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

2

0≤s≤t ∧τRn



0

Z

t ∧τRn



 21 |un (s)| |σn (s, un (s), z )| ν(dz ) ds 2

≤ CE 0

2

Z

 12 

 ≤ C εE

|un (s)|

sup

2

sup s∈[0,t ∧τRn ]

CK0 t

≤

|un (s)|2 +

+ C ε EXR (t ) +

ε

K0 + K1 |un (s)| + K2 ∥un (s)∥

C

CK1

ε

2



ds

0 t ∧τRn

E

ε 

2



E

ε 

s∈[0,t ∧τnR ]

≤ CεE

 12

t ∧τRn



1



K0 + K1 |un (s)|2 + K2 ∥un (s)∥2 ds



0 t

E(XR (s)) ds +

CK2

ε

0

E(YR (t )),

t ∈ [0, T ].

(4.13)

Next, we will deal with the second term of I (t ). Taking into account that the process

 t

|σn (r , un (r ), z )|2 η(dr , dz )

t → 0

Z

has only positive jumps, we obtain

 s E

|σn (r , un (r ), z )|2 η(dr , dz )

sup 0≤s≤t ∧τRn

0

t ∧τRn



Z



|σn (r , un (r ), z )|2 η(dr , dz )

≤E 0

Z t ∧τRn





|σn (s, un (s), z )|2 ν(dz ) ds

=E 0

Z t ∧τRn

 ≤ K0 t + K1

E|un (r )|2 ds + K2 E

t ∧τRn



0

∥un (s)∥2 ds

0 t



E(XR (s)) ds + K2 E(YR (t )),

≤ K0 t + K1

(4.14)

0

where we also used Condition 4.2. By combining the last two inequalities, it follows that for t ∈ [0, T ]

 EI ( t ) ≤

CK0 t

ε





+ K0 t + C εE(XR (t )) +

CK1

ε



E(XR (t ) + 2YR (t )) ≤ 2C˜ + 2E∥ξ ∥2 + 2γ



E(XR (s)) ds +

+ K1

By choosing sufficiently small ε and K2 such that C ε <



t 0

1 2

CK2

ε

 + K2 E(YR (t )).

and ε 2 + K2 < 1, we have CK

t

E(XR (s))ds,

(4.15)

0

where C˜ = ε0 + K0 t and γ = C , K0 , K1 , K2 and T , such that CK t

CK1

ε

+ K1 . Thus, by the Gronwall Lemma there exists a constant Cˆ > 0, depending on

E [XR (t ) + 2YR (t )] ≤ Cˆ (E∥ξ ∥2 + 1),

t ∈ [0, T ].

Note that the latter equation is equivalent to

 E

sup s∈[0,t ∧τRn ]

|un (s)| + 2 2

t ∧τRn



 ∥un (s)∥ ds ≤ Cˆ (E∥ξ ∥2 + 1), 2

t ∈ [0, T ].

0

Recall that τRn ↑ T as R → ∞, P-a.s. and P{τRn < T } = 0. On the basis Fatou’s Lemma, we infer

 E

sup |un (s)| + 2 2

s∈[0,T ]



T



∥un (s)∥ ds ≤ Cˆ (E∥ξ ∥2 + 1). 2

0

130

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

Let us consider now the case p = 2. Without loss of generality, we may assume that for any given n, the process un is uniformly bounded in [0, T ]. Otherwise, we may get around this by introducing a sequence of stopping times as before. By 2p applying the finite dimensional Itô formula (see [35]) to the function | · |H and the process un yields

 t |un (s−)|2(p−1) (un (s−), |un (s)|2(p−1) ∥un (s)∥2V ds + 2p 0 Z 0  t  |un (s−) + σn (s, un (s), z )|2p σn (s, un (s), z )) η( ˜ ds, dz ) + 0 Z  − |un (s−)|2p − 2p|un (s−)|2(p−1) (un (s−), σn (s, un (s), z )) η(ds, dz ).

|un (t )|2p = |Πn ξ |2p − 2p

t



Here, again we have used the skew-symmetricity of B as well as the following fact

(Πn F (un ), un ) = −∥un ∥2V ,

un ∈ Hn .

Taking the supremum over the interval [0, t ] on both sides of the above equality leads to sup |un (s)|2p + sup 2p

s∈[0,t ]

s∈[0,t ]

s



|un (r )|2(p−1) ∥un (r )∥2V dr

0

 s     ≤ |Πn ξ |2p + 2p sup  |un (r −)|2(p−1) (un (r −), σn (r , un (r ), z )) η( ˜ dr , dz ) s∈[0,t ] 0 Z    s   |un (r −) + σn (un (r ), z ) |2p −|un (r −) |2p + sup  s∈[0,t ] 0 Z     2(p−1) − 2p|un (r −) | (un (s−), σn (r , un (r ), z )) η(dr , dz ). 

(4.16)

Put

 X (t ) := sup |un (s)|2p , t ∈ [0, T ];    s∈[0,t ]   t Y ( t ) := |un (s)|2p−2 ∥un (s)∥2 ds, t ∈ [0, T ],    0  and I (t ) := I1 (t ) + I2 (t ),

(4.17)

where

 s     2(p−1)  I1 (t ) := 2p sup  |un (r −)| (un (r −), σn (r , un (r ), z )) η( ˜ dr , dz ) s∈[0,t ] 0 Z    s   I2 (t ) := sup  |un (r −) + σn (r , un (r ), z ) |2p −|un (r −) |2p s∈[0,t ] 0 Z      2(p−1) − 2p|un (r −) | un (s−), σn (r , un (r ), z ) η(dr , dz ).  First, we apply the Burkholder–Davis–Gundy inequality. Then the Condition 4.2 is used. Next, the Young inequality allows us to write the term as a sum. Finally we used the fact that for p ≥ 2, |x|2p−2 ≤ 1 + |x|2p and |σn |2 ≤ |σ |2 . By doing so we get for t ∈ [0, T ]

EI1 (t ) ≤ 2pC E

 t  0

4(p−1)

|un (s)|

|un (s)|

≤ 2pC E

2

Z t



 12

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

4p−2



K0 + K1 |un (s)| + K2 ∥un (s)∥ 2

2



 12 ds

0



≤ 2pC ε E sup |un (s)|

2p

 21 

ε

s∈[0,t ]

≤

2pC 2

ε E sup |un (s)|2p + s∈[0,t ]

|un (s)|

E

2p−2



K0 + K1 |un (s)| + K2 ∥un (s)∥ 2

2



0 t ∧τN



pCK0 2ε

t



1

E 0

|un (s)|2p−2 ds +

2ε

t



2pCK1

|un (s)|2p ds

E 0

 12 ds

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

+

2pCK2

0

pCK0 T

≤

+ pC ε E sup |un (s)|2p +

ε +

|un (s)|2p−2 ∥un (s)∥2 ds

E

2ε

131

t



pC (K0 + K1 )

ε

s∈[0,t ]

|un (s)|2p−2 ∥un (s)∥2 ds.

E

ε

|un (s)|2p ds 0

t



pCK2

t

 E

0

Notice that by virtue of the Taylor formula, we have

   |x + h|2p − |x|2p − 2p|x|2(p−1) (x, h) ≤ Cp (|x|2(p−1) |h|2 + |h|)2p , where Cp =

p2 +p . 2

for all x, h ∈ Hn ,

(4.18)

In particular, for p = 1, (4.18) becomes an equality with Cp = 1:

   |x + h|2 − |x|2 − 2(x, h) = |h|2 ,

for all x, h ∈ Hn .

(4.19)

From (4.18) and Condition 4.2-(1), we find

EI2 (t ) ≤ E

 t 0

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

− 2p|un (s−) |2(p−1) (un (s−), σn (s, un (s), z ))| η(ds, dz )  t   ≤ Cp E |un (s−)|2(p−1) |σn (s, un (s), z )|2 + |σn (s, un (s), z )|p ν(dz ) ds 0 Z  t ≤ Cp K E (1 + |un (s)|2p + |un (s)|2(p−1) ∥un (s)∥2p ) ds 0  t  t 2p ≤ Cp KT + Cp K |un (s)| ds + Cp K |un (s)|2(p−1) ∥un (s)∥2 ds. 0

0

Therefore, for I (t ), we infer

EI ( t ) ≤



pCK0 T

ε





+ Cp KT

+ 

+ pC ε E(X (t )) +

pC (K0 + K1 )

ε

pCK2

ε

t



E(X (s)) ds

+ Cp K 0

 + Cp K E(Y (t )).

Choosing ε, K2 and K sufficiently small and employing Gronwall’s Lemma, we obtain

 E

sup |un (s)|2p +

s∈[0,T ]



T

 |un (s)|2p−2 ∥un (s)∥2 ds ≤ C (E|ξ |2p + 1).

(4.20)

0

Observe that the constant C depends on p, T , K0 , K1 , K2 and K , but is independent of n. For [0, T ] the process un is the unique solution to the Eq. (4.10). That is un satisfies un (t ) = Πn ξ +

t



Πn F (un (s)) ds + 0

 t 0

σn (un (s), z ) η( ˜ ds, dz ),

t ∈ [0, T ].

Z

The previous argument shows that

 E

sup |un (s)|2p +

s∈[0,T ]



T

 |un (s)|2p−2 ∥un (s)∥2 ds ≤ C (E|ξ |2p + 1).

0

Since the constant C is independent of n, we conclude that

 sup E n

sup |un (s)|2p +

s∈[0,T ]

which proves Proposition 4.5.

T

 0



 |un (s)|2p−2 ∥un (s)∥2 ds ≤ C (E|ξ |2p + 1),

132

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

Step. 1 Weak convergence of approximating sequences. Proposition 4.6. There exists a subsequence of {un : n ∈ N} (still denoted by using the same notation) of processes and elements u¯ ∈ L2 (Ω × [0, T ]; V ) ∩ L4 (Ω × [0, T ]; Q) ∩ L4 (Ω ; L∞ ([0, T ]; H ))

(4.21)

v ∈ L (Ω ; H ),

(4.22)

2

G ∈ L (Ω × [0, T ]; V ),

S ∈ L (Ω × [0, T ]; L (Z , ν; H )),

′

2

2

2

such that un ⇀ u¯

in L2 (Ω × [0, T ]; V ),

(4.23)

un ⇀ u¯

in L (Ω × [0, T ]; Q);

(4.24)

4

un (T ) ⇀ v un ⇀ u

in L (Ω ; H ); 2

(4.25)

(weakly star) in L (Ω ; L ([0, T ]; H )); ∞

4

(4.26)

Πn F (un ) ⇀ G in L (Ω × [0, T ]; V );

(4.27)

σn (un , ·) ⇀ S in L (Ω × [0, T ]; L (Z , ν; H )).

(4.28)

′

2

2

2

Proof. Applying Proposition 4.5 with p = 1 and p = 2 and using the condition |v|2Q ≤ a0 |v|∥v∥, we have

 sup E n

sup |un (t )| + 2

T



0 ≤t ≤T

∥un (t )∥ dt 2



< ∞,

(4.29)

0

and

 sup E n

sup |un (t )| + 4

0 ≤t ≤T

T



|un (t )|

4 Q dt



< ∞.

(4.30)

0 4

Since L2 (Ω × [0, T ]; V ) is a Hilbert space, L4 (Ω × [0, T ]; Q) ∼ = = [L 3 (Ω × [0, T ]; Q∗ )]∗ and L4 (Ω ; L∞ ([0, T ]; H )) ∼ (L4/3 (Ω ; L1 ([0, T ]; H )))∗ , by the Banach–Alaoglu Theorem and an easy argument based on the uniqueness of a limit, we can find a function u belonging to the intersection of the three spaces L2 (Ω × [0, T ]; V ), L4 (Ω × [0, T ]; Q) and L4 (Ω ; L∞ ([0, T ]; H )) such that the assertions (4.23), (4.24) and (4.26) in Proposition 4.6 hold. The claim (4.25) is satisfied for the same reason. It remains to show assertions (4.27) and (4.28). To prove claim (4.27), we take a function v ∈ L2 (Ω × [0, T ]; V ) and observe that T



|⟨Πn F (un (s)), v(s)⟩| ds ≤ E

E

T



0

∥un (s)∥ ∥v(s)∥ ds + E 0 T

 12  

∥un (s)∥ ds 2

E

0

T



|un (s)|2Q ∥v(s)∥ ds

+ C1 E 0

 12

∥un (s)∥ ds 2

E

 12

∥v(s)∥ ds 2

0 T

 

T

E

0

≤

|⟨B(un (s), v(s)), un (s)⟩| ds 0

  ≤

T



T

  ∥v∥L2 ([0,T ]×Ω ;V ) + C1 E

 12

|un (s)| ds 4 Q

0

∥v∥L2 ([0,T ]×Ω ;V ) ,

where we used the property of the operator B: |⟨B(u1 , u2 ), u3 ⟩| ≤ C1 |u1 |Q ∥u2 ∥ |u3 |Q . This follows from Condition 4.1 and the fact that V is continuously embedded into H. We also used the Cauchy–Schwartz inequality. Hence, we have

∥Πn F (un )∥

2 L2 (Ω ×[0,T ];V ′ )

T

 =

sup ∥v∥L2 ([0,T ]×Ω ;V ) ≤1

≤

0

T

 

|⟨Πn F (un (s), v(s))⟩| ds

E

 21

∥un (s)∥ ds 2

E

T

 

|un (s)| ds 4 Q

+ C1 E

0

 21

.

0

Since by Proposition 4.5 we have

 sup E n

0

T

∥un (s)∥2 ds + sup E n



T

|un (s)|4Q ds < ∞,

0

we infer that Πn F (un ) is bounded in L2 (Ω × [0, T ]; V ′ ). Hence, there exists G ∈ L2 (Ω × [0, T ]; V ′ ) such that Πn F (un ) converges weakly to G as n → ∞.

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

133

To prove claim (4.28), we observe that by Condition 4.2-(1) and by Proposition 4.6-(1) that T





n

0

n

Z

T



|σn (un (s), z )|2 ν(dz ) ds ≤ sup E

sup E

K0 + K1 |un (s)|2 + K2 ∥un (s)∥2 ds < ∞.





0

Hence, there exists an element S ∈ L (Ω × [0; T ]; L (Z , ν; H )) such that σn (un (·), ·) converges weakly to S as n → ∞. This completes the proof of Proposition 4.6.  2

2

Step. 2 As before let {φi : i ∈ N} ⊂ D(A) be an orthonormal basis in H such that the span of {φi : i} is dense in V . Let us fix g ∈ L2 (Ω × [0, T ]; R). Going back to Eq. (4.11), we have for every n ∈ N and i ≤ n T



g (s)(un (s), φi ) ds = E

E



T



g (s) (Πn ξ , φi ) +



Πn F (un (r )) dr , φi 0

0

0



s



  s

σn (r , un (r ), z ) η( ˜ dr , dz ), φi

+ 0

ds.

Z

T 

Since the linear map f → 0 Z f (s, z ) η( ˜ dz , ds) is continuous from L2 (Ω ; L2 (Z , ν; H )) to L2 (Ω ; H ) (in fact an isometry), it is continuous with respect to the weak topologies, see [38]. Therefore, in view of the weak convergences proved in Proposition 4.6 we infer that T



g (s)(¯u(s), φi ) ds = E

E

T



0



g (s) (ξ , φi ) +

s



0

G(r ) dr , φi



0

 s 

S (r , z ) η( ˜ dr , dz ), φi

+ 0



ds.

(4.31)

Z

Since g was an arbitrary element of L2 (Ω × [0, T ]; R) we deduce that t



u¯ (t ) = ξ +

G(s) ds +

 t

0

0

S (s, z ) η( ˜ ds, dz ),

dP ⊗ dt-a.e. on Ω × [0, T ].

(4.32)

Z

A similar consideration as in (4.32) leads to v=ξ+

T



G(s) ds + 0

T





0

S (s, z ) η( ˜ ds, dz ),

dP-a.s.

(4.33)

Z

Define next a V ′ -valued process u by t



u(t ) := ξ +

G(s)ds + 0

 t 0

S (s, z ) η( ˜ dz , ds),

t ∈ [0, T ] in V ′ .

Z

One can see that u is a V ′ -valued modification of the V -valued process u¯ ∈ L2 (Ω × [0, T ]; V ) and u(T ) = v,

dP-a.s.

(4.34)

It then follows from Theorem A.1 that u is an H-valued càdàg and (Ft )-adapted process, and for every t ∈ [0, T ], the following formula holds P-a.s.

|u(t )| = |ξ | + 2 2

2

t



⟨G(s), u¯ (s)⟩ ds + 2 0

 t 0

(u(s−), S (s, z )) η( ˜ ds, dz ) +

 t

Z

0

|S (s, z )|2 η(ds, dz ).

(4.35)

Z

Step. 3 The aim of this step is to identify the processes G and S in terms of the process u. To be precise, we will show that the following identities hold G(s, ω) = F (u(s, ω)),

for dP ⊗ dt-a.a. (s, ω) ∈ [0, T ] × Ω ,

(4.36)

S (s, ω, z ) = σ (s, ω, u(s, ω), z ) for dP ⊗ dt × ν -a.a. (s, ω, z ) ∈ [0, T ] × Ω × Z .

(4.37)

Let v be a progressively measurable process belonging to L2 (Ω × [0, T ]; V ) ∩ L4 (Ω × [0, T ]; Q) ∩ L4 (Ω ; L∞ [0, T ]; H ). Let us define a process r = {r (t ) : t ∈ [0, T ]} by r (t ) =

t



2Ck |v(s)|2Q + L1 ds,





t ∈ [0, T ],

(4.38)

0

with L1 being the constant from Condition 4.2 and a number k such that 0 < k < the following important consequence of (4.9) and (4.38) and Condition 4.2-(2):

2−L2 . 2

For the future use let us now note

134

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

  E −

T

T

e−r (t ) ⟨Πn F (un (t )) − Πn F (v(t )), un (t ) − v(t )⟩dt

0

0



T



e−r (t ) r ′ (t )|un (t ) − v(t )|2 dt + 2



+

 −r (t )

e 0

|σn (un (t ), z ) − σn (v(t ), z )| ν(dz )dt ≤ 0. 2

(4.39)

Z

In order to show inequality (4.39) it is enough to observe that r ′ (t ) = 2Ck |v(t )|2Q + L1 and

−(2Ck ∥v(t )∥4Q + L1 )|u − v|2 + 2⟨Πn F (u) − Πn F (v), u − v⟩ + |σn (u) − σn (v)|2     ≤ −(2Ck ∥v(t )∥4Q + L1 )|u − v|2 + 2 −(1 − k)∥u − v∥2V + Ck ∥v∥4Q |u − v|2H + L1 |u − v|2 + L2 ∥u − v∥2V ≤ 0. Since v = u(T ), by (4.25) in Proposition 4.6 we infer that un (T ) ⇀ u(T ) in L2 (Ω ; H ). Thus, since E|Πn ξ |2 ≤ E|ξ |2 , we infer that

E|u(T )|2 e−r (T ) − E|ξ |2 ≤ lim inf E|un (T )|2 e−r (T ) − E|Πn ξ |2 .





(4.40)

n→∞

Now, similarly to (4.35), by applying the Itô formula from Theorem A.1 to the process |u|2 e−r (·) we get

|u(T )|2 e−r (T ) = |ξ |2 −

T



e−r (t ) r ′ (t )|u(t )|2 dt + 2

0 T



T



e−r (t ) ⟨G(t ), u¯ (s)⟩ dt

0



−r (t )

+2

e 0

(u(t −), S (t , z )) η( ˜ dt , dz ) +

T



Z



0

e−r (t ) |S (t , z )|2 η(dt , dz ).

(4.41)

Z

Taking the expectation of both sides of (4.41) yields T



E|u(T )|2 e−r (T ) − E|ξ |2 = −E

e−r (t ) r ′ (t )|u(t )|2 dt + 2E

0

e−r (t ) ⟨G(t ), u¯ (s)⟩ dt

0 T



T





e−r (t ) |S (t , z )|2 ν(dz ) dt .

+E 0

(4.42)

Z

On the other hand, by the Itô formula applied to the Hn -valued process |un |2 e−r (·) get 2 −r (T )

|un (T )| e

= |Πn ξ | − 2

T



−r (t ) ′

r (t )|un (t )| dt + 2

e

2



0 T





0 T

Z



e−r (t ) |σn (un (t ), z )|2 η(dt , dz ).

+ 0

e−r (t ) ⟨un (t ), Πn F (un (t ))⟩ dt

0

e−r (t ) (un (t −), σn (un (t ), z )) η( ˜ dt , dz )

+2 

T

(4.43)

Z

Thus by taking the expectation of both sides of (4.43) and using twice an identity |x|2 = 2[(u, v) − |y|2 ] + |x − y|2 we get 2 −r ( T )

E|un (T )| e

  − E|Πn ξ | = E −

T

2





e−r (t ) r ′ (t ) 2(un (t ), v(t )) − |v(t )|2 dt

(4.44)

0 T

 +2





e−r (t ) ⟨Πn F (un (t )) − Πn F (v(t )), v(t )⟩ + ⟨Πn F (v(t )), un (t )⟩ dt

0 T





−r (t )

+

e 0



2(σn (un (t ), z ), σn (v(t ), z )) − |σn (v(t ), z )|

2



 ν(dz ) dt

Z

  +E −

T

e−r (t ) r ′ (t )|un (t ) − v(t )|2 dt

0 T

 +2

e−r (t ) ⟨Πn F (un (t )) − Πn F (v(t )), un (t ) − v(t )⟩ dt

0 T







−r (t )

+

e 0

Z

|σn (un (t ), z ) − σn (v(t ), z )| ν(dz ) dt . 2

(4.45)

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

135

Thus, because of (4.39) we infer that

  E|un (T )|2 e−r (T ) − E|Πn ξ |2 ≤ E −

T





e−r (t ) r ′ (t ) 2(un (t ), v(t )) − |v(t )|2 dt

0



T

+2





e−r (t ) ⟨Πn F (un (t )) − Πn F (v(t )), v(t )⟩ + ⟨Πn F (v(t )), un (t )⟩ dt

0 T





+

e 0

−r (t )



2(σn (un (t ), z ), σn (v(t ), z )) − |σn (v(t ), z )|

2



 ν(dz ) dt .

Z

Note that, by the weak convergence (4.23) in Proposition 4.6, the first term on the right hand-side of (4.44) converges as n → ∞ to T

 E

  −e−r (t ) r ′ (t ) 2(¯u(t ), v(t )) − |v(t )|2 dt .

(4.46)

0

On the other hand, by using the weak convergence (4.27) of {Πn F (un )}n∈N in Proposition 4.6, we find T

 E

e−r (t ) ⟨Πn F (un (t )), v(t )⟩ dt → E

T



0

e−r (t ) ⟨G(t ), v(t )⟩ dt ,

n → ∞.

(4.47)

0

Hence, by the Lebesgue dominated convergence theorem we infer that T

 E

−e−r (t ) ⟨Πn F (v(t )), v(t )⟩dt → E

0

T



−e−r (t ) ⟨F (v(t )), v(t )⟩ dt ,

n → ∞.

(4.48)

0

From the Cauchy–Schwarz inequality, we observe that T

 E





e−r (t ) ⟨Πn F (v(t )), un (t )⟩ − ⟨F (v(t )), u¯ (t )⟩ dt

0 T

 =E





e−r (t ) ⟨Πn F (v(t )) − F (v(t )), un (t )⟩ + ⟨F (v(t )), un (t ) − u¯ (t )⟩ dt

0

  ≤

T

e

E

−r ( t )

∥Πn F (v(t )) − F (v(t ))∥

2 V′

 12  

 dt

−r (t )

∥un (t )∥ dt 2

0

T



e

E

0

 21

T

e−r (t ) ⟨F (v(t )), un (t ) − u¯ (t )⟩ dt

−E 0



 

T

≤ sup E n

 21     E e−r (t ) ∥un (t )∥2 dt

0 T



 12

T

e

−r (t )

∥Πn F (v(t )) − F (v(t ))∥ dt 2 V′

0

e−r (t ) ⟨F (v(t )), un (t ) − u¯ (t )⟩ dt .

+E 0

Hence by using Proposition 4.5 and applying again the Lebesgue dominated convergence theorem and the convergence (4.23) of un in Proposition 4.6, we have T

 E





e−r (t ) ⟨Πn F (v(t )), un (t )⟩ − ⟨F (v(t )), u¯ (t )⟩ dt → 0,

as n → ∞.

(4.49)

0

Combining (4.47)–(4.49), we see that the second term on the right-hand side of (4.44) converges to T

 2





e−r (t ) ⟨G(t ) − F (v(t )), v(t )⟩ + ⟨F (v(t )), u¯ (t )⟩ dt

(4.50)

0

as n → ∞. What concerns is the last term on the right-hand side of (4.44) we first note that T







0



e−r (t ) 2(σn (t , un (t ), z ), σn (t , v(t ), z )) − |σn (t , v(t ), z )|2 ν(dz ) dt

E Z

T





e−r (t ) 2(σn (t , un (t ), z ), σ (t , v(t ), z )) ν(dz ) dt + E

=E 0

Z

 0

T



 −e−r (t ) 2 σn (t , un (t ), z ), Z

136

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

 σ (t , v(t ), z ) − σn (t , v(t ), z ) ν(dz ) dt + E



T

0 T





Z

−e−r (t ) |σn (t , v(t ), z )|2 ν(dz ) dt    T

e−r (t ) 2(σn (t , un (t ), z ), σ (t , v(t ), z )) ν(dz ) dt + 2C E

≤E 0



|σ (t , v(t ), z )

0

Z

 21 − σn (t , v(t ), z )| ν(dz ) dt 2 H

T





−e−r (t ) |σn (t , v(t ), z )|2 ν(dz ) dt ,

+E 0

Z

Z

where

 

T

 21



C := sup E n

e 0

−2r (t )

|σn (t , un (t ), z )| ν(dz ) dt 2 H

.

Z

Then applying the weak convergence (4.28) of {σn (un , ·) : n ∈ N} in the Proposition 4.6 to the first term and the Lebesgue Dominated Convergence Theorem to the second and third terms on the right-hand side of the above inequality, we deduce that T







0



e−r (t ) 2(σn (t , un (t ), z ), σn (t , v(t ), z )) − |σn (t , v(t ), z )|2 ν(dz ) dt

E Z

converges, as n → ∞, to T





e−r (t ) ((S (t , z ), σ (t , v(t ), z )) − |σ (t , v(t ), z )|2 ) ν(dz ) dt .

E 0

(4.51)

Z

Therefore, in view of (4.40), (4.46), (4.50) and (4.51) we conclude that 2 −r ( T )

E|u(T )| e

  − E|ξ | ≤ E −

T

2





e−r (t ) r ′ (t ) 2(¯u(t ), v(t )) − |v(t )|2 dt

0 T

 +2





e−r (t ) ⟨v(t ), G(t ) − F (v(t ))⟩ + ⟨F (v(t )), u¯ (t )⟩ dt

0 T





−r (t )

+

e 0



2(S (t , z ), σ (t , v(t ), z )) − |σ (t , v(t ), z )|

2



 ν(dz ) dt .

(4.52)

Z

Employing equality (4.42) and inequality (4.52), we infer that T

 −E

−r (t ) ′

r (t )|¯u(t ) − v(t )| dt + 2E

e

2

0

T



e−r (t ) ⟨¯u(t ) − v(t ), G(t ) − F (v(t ))⟩ dt

0 T





e−r (t ) |S (t , z ) − σ (t , v(t ), z )|2 ν(dz ) dt ≤ 0.

+E 0

(4.53)

Z

So far we have proved that (4.53) holds for every F-progressively measurable process v ∈ L2 (Ω × [0, T ]; V ) ∩ L4 (Ω × [0, T ]; Q) ∩ L4 (Ω ; L∞ [0, T ]; H ). In particular, it holds for u¯ = v . Hence, by inserting u¯ for v in (4.53) we infer that S (·, ·) = σ (·, u¯ (·), ·) in L2 (Ω × [0, T ]; L2 (Z , ν; H )). Now using (4.53) with v replaced by vε := u¯ − εw, w ∈ L∞ (Ω × [0, T ]; V ) and ε > 0 gives that 2

T



−ε E

−r (t ) ′

e

r (t )|w(t )| dt + 2ε E 2

T



0

e−r (t ) ⟨w(t ), G(t ) − F (u(t ) − εw(t ))⟩ dt ≤ 0.

0

Dividing both sides of the above inequality by ε yields

−εE

T



e−r (t ) r ′ (t )|w(t )|2 dt + 2E

0

T



e−r (t ) ⟨w(t ), G(t ) − F (u(t ) − εw(t ))⟩ dt ≤ 0.

0

Observe that by (4.9) we have for ε > 0,

  |⟨εw(t ), F (u(t ) − εw(t )) − F (u(t ))⟩| ≤ C ε 2 ∥w(t )∥2 + |u(t )|4Q |w(t )|2H ,

t ∈ [0, T ].

(4.54)

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

137

Hence, on the basis of the Lebesgue Dominated convergence theorem, we have as ε ↓ 0, T

 E

e−r (t ) ⟨w(t ), G(t ) − F (u(t ) − εw(t ))⟩ dt → E

T



e−r (t ) ⟨w(t ), G(t ) − F (u(t ))⟩ dt .

0

0

Letting ε ↓ 0 on both sides of inequality (4.54) yields T

 2E

e−r (t ) ⟨w(t ), G(t ) − F (u(t ))⟩ dt ≤ 0.

0

Since w is arbitrary, this implies that process G(·) := F (u(·)) belongs to L2 (Ω × [0, T ]; V ′ ). Step. 4 We will show in this Step that the solution u to Problem (4.1) is unique. For this purpose we will use the ‘‘Schmalfuss trick’’, see [39]. Let L1 be the constant from Condition 4.2. Let us choose k > 0 such that k + L2 ≤ 2, and let be Ck the constant associated to k as in Remark 4.2-(2) and choose a > k. Assume that u1 and u2 are two solutions to Problem (4.1). Put w = u1 − u2 and note that w(0) = 0. Define an auxiliary process φ by

   t |u1 (s)|4Q ds , φ(t ) := exp −a

t ≥ 0.

0

Let us introduce the stopping time

τN := inf{t ≥ 0 : |u1 (t )| ≥ N } ∧ inf{t ≥ 0 : |u2 (t )| ≥ N }. Then the Itô Lemma gives



t ∧τN



t ∧τN

φ(t ∧ τN ) |w(t ∧ τN )| + a φ(s) ∥u1 (s)∥ |w(s)| ds + 2 φ(s)⟨Au1 − Au2 , w(s)⟩ ds 0 0  t ∧τN  t ∧τN  ≤ φ(s)⟨b(u1 (s), u1 (s)) − b(u2 (s), u2 (s)), w(s)⟩ ds + φ(s)|σ (u1 (s), z ) 0 0 Z  t ∧τN  φ(s) (w(s), σ (u1 (s), z ) − σ (u2 (s), z )) η( ˜ dz , ds). − σ (u2 (s), z )|2 ν(dz ) ds + 2 2

2

4 Q

0

Z

By taking expectation, Remark 4.2 and Condition 4.2 imply that

Eφ(t ∧ τN ) |w(t ∧ τN )| + a E t ∧τN



t ∧τN





t ∧τN

φ(s) |u1 (s)| |w(s)| ds + 2E φ(s)∥w(s)∥2 ds 0 0  t ∧τN     φ(s) k∥w(s)∥2 + Ck |w(s)|2 |u1 (s)|4Q ds + E φ(s) L1 |w(s)|2 + L2 ∥w(s)∥2 ds. 2

≤E

4 Q

2

0

0

Since k + L2 ≤ 2 and a > Ck , there exists a constant C1 such that

Eφ(t ∧ τN ) |w(t ∧ τN )|2 + C1 E

t ∧τN



φ(s) ∥u1 (s)∥4Q |w(s)|2 ds ≤ L1 E

0

The Gronwall Lemma concludes the proof.

t ∧τN



φ(s)|w(s)|2 ds.

0



Acknowledgments The support of Isaac Newton Institute for Mathematical Sciences in Cambridge is gratefully acknowledged where most of the work on this paper was done during the semester on ‘‘Stochastic Partial Differential Equations’’ in 2010. The authors would like to thank Paul Razafimandimby for a careful reading of the manuscript. The second author was supported by the Austrian Science Foundation (FWF), Grant number: P20705. The third author was supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO). Appendix A. A technical lemma Consider a V ′ -valued càdlàg process of the form X (t ) = X0 +

 0

t

ξ (s) ds +

 t 0

Z

G(s, z ) η( ˜ ds, dz ),

t ∈ [0, T ],

(A.1)

138

Z. Brzeźniak et al. / Nonlinear Analysis 79 (2013) 122–139

where ξ is a V ′ -valued process and G is an H-valued process. We shall state a conclusion followed from [40] that under some suitable measurability assumptions, the process X is in fact an H-valued adapted càdlàg process and a version of Itô formula also holds for X . Theorem A.1. Suppose that X0 ∈ L2 (Ω , F ; H ) and F ∈ L2 (Ω × [0, T ]; V ′ ) and G ∈ L2 (Ω × [0, T ]; L2 (Z , ν; H )) are both progressively measurable processes. Suppose that X is a V ′ -valued process given by (A.1) and there exists a V -valued process X¯ ∈ L2 (Ω × [0, T ]; V ) such that X = X¯ , dP ⊗ dt in V . Then X is an H-valued càdlàg Ft -adapted process (up to distinguishable) and t



|X (t )| = |X0 | + 2 2

2

⟨X¯ (s), ξ (s)⟩ ds + 2

 t 0

0

(X (s−), G(s, z )) η( ˜ dz , ds) +

 t 0

Z

|G(s, z )|2 η(dz , ds).

(A.2)

Z

Proof. This theorem  tisan immediately results from Gyöngy and Krylov [40] if we notice that the quadratic variation of the martingale h(t ) := 0 Z G(s, z ) η( ˜ dz , ds) is given by

 t

|G(t , z )|2 η(dz , ds). 

[ h] t = 0

Z

Remark A.2. (1) Note that since the Itô formula holds for the square of the H-norm, (A.2) can also be written in the following two forms t



|X (t )|2 = |X0 |2 + 2

⟨X¯ (s), ξ (s)⟩ ds + 2

 t

0

 t

(X (s−), G(s, z )) η( ˜ dz , ds) Z

  |X (s−) + G(s, z )|2 − |X (s−)|2 − 2(X (s−), G(s, z )) η(dz , ds),

+ 0

0

(A.3)

Z

or t



|X (t )|2 = |X0 |2 + 2

⟨X¯ (s), ξ (s)⟩ ds +

 t

0

 t



+ 0

0

  |X (s−) + G(s, z )|2 − |X (s−)|2 η( ˜ dz , ds) Z

 |X (s) + G(s, z )|2 − |X (s)|2 − 2(X (s), G(s, z )) ν(dz ) ds.

(A.4)

Z

(2) If the spaces V and H are finite dimensional spaces, above Itô formulas are in concert with the one by Ikeda–Watanabe [35]. Appendix B. Itô formula The main technical tool in our paper is the Itô formula. Let us first state here a special case of such from the Ph.D. Thesis of the 3rd named author, see [41, Theorem 3.5.3]. Later we will rewrite it in the form used in our paper. We stick to the notation from the current paper rather than [41]. Theorem B.1. Assume that E is a Hilbert space. Let X be a process given by t



a(s) ds +

Xt = X0 + 0

 t 0

f (s, z ) η( ˜ ds, dz ),

t ≥ 0,

(B.1)

Z

where a is an E-valued progressively measurable process on the space (R+ × Ω , B (R+ ) ⊗ F ) such that for all t ≥ t t  0, 0 ∥a(s, ω)∥ds < ∞, P-a.s. and f is a predictable process on E with E 0 Z ∥f (s, z )∥2 ν(dz ) ds < ∞, for each t > 0. Let G be a separable Hilbert space. Let φ : E → G be a function of class C 1 such that the first derivative φ ′ : E → L(E ; G) is (p − 1)-Hölder continuous. Then for every t > 0, we have P-a.s.

φ(Xt ) = φ(X0 ) +

t



 t   φ (Xs )(a(s)) ds + φ ′ (Xs− )(f (s, z )) η( ˜ ds, dz ) ′

0

0

Z

 t   + φ(Xs− + f (s, z )) − φ(Xs− ) − φ ′ (Xs− )(f (s, z )) η(dzds). 0

(B.2)

Z

References [1] R. Temam, Navier–Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. [2] J.L. Lions, G. Prodi, Un théorème d’existence et unicité dans les équations de Navier–Stokes en dimension 2, C. R. Acad. Sci. Paris 248 (1959) 3519–3521.

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