Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

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Physica D 398 (2019) 23–68

Contents lists available at ScienceDirect

Physica D journal homepage: www.elsevier.com/locate/physd

Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type G. Deugoué a,b , A. Ndongmo Ngana a , T. Tachim Medjo b ,

∗

a

Department of Mathematics and Computer Science, University of Dschang, P.O. BOX 67, Dschang, Cameroon

b

Department of Mathematics and Statistics, Florida International University, MMC, Miami, FL 33199, USA

highlights • • • • •

We introduce a stochastic nonlocal Cahn–Hilliard–Navier–Stokes model. Present a Galerkin approximation. Derive some a priori estimates. Prove the existence of a martingale weak solution. Pathwise uniqueness of the solution in 2D.

article

info

Article history: Received 10 December 2018 Received in revised form 15 May 2019 Accepted 28 May 2019 Available online 5 June 2019 Communicated by T. Sauer Keywords: Navier–Stokes equations Stochastic nonlocal Cahn–Hilliard equations Martingale solutions Galerkin approximation

a b s t r a c t In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system driven by a pure jump noise in both 2D and 3D bounded domains. Our goal is achieved by using the classical Faedo–Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada–Watanabe classical result to derive the existence of a strong solution. © 2019 Elsevier B.V. All rights reserved.

1. Introduction

The nonlocal Cahn–Hilliard–Navier–Stokes system describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier–Stokes equations for the fluid velocity u coupled with a convective nonlocal Cahn–Hilliard equation for the order parameter ϕ i.e. the relative concentration of one fluid or the difference of two concentrations. The aim of this paper is to study a stochastic nonlocal Cahn–Hilliard–Navier–Stokes system driven by a pure jump noise. More precisely, for a fixed time T > 0 and an open connected and bounded subset O of Rd , d = 2, 3, with a smooth boundary ∂ O, we consider the

∗ Corresponding author. E-mail address: [email protected] (T.T. Medjo). https://doi.org/10.1016/j.physd.2019.05.012 0167-2789/© 2019 Elsevier B.V. All rights reserved.

24

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

system

⎧ ⎪ ⎪ du(t) + [−ν ∆u(t) + (u(t).∇ )u(t) + ∇π]dt = µ(t)∇ϕ (t)dt + h(t , u(t))dt ⎪ ⎪ ∫ ⎪ ⎪ ⎪ ⎪ ⎪ + G(t , u(t − ); y)η˜ (dt , dy), ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎪ ⎪ div u(t) = 0, ⎪ ⎨ dϕ (t) + u(t).∇ϕ (t)dt = ∆µ(t)dt , ⎪ ⎪ ⎪ ⎪ ⎪µ(t) = aϕ (t) − J ∗ ϕ (t) + F ′ (ϕ (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ∂µ ⎪ ⎪ = 0 on ∂ O × (0, T ), u = 0, ⎪ ⎪ ⎪ ∂ξ ⎪ ⎪ ⎪ ⎩ u(0) = u0 , ϕ (0) = ϕ0 ,

(1.1)

where ∂ξ denotes differentiation with respect to the unit outward normal of ∂ O. u, ϕ and π are unknown random fields defined on O × [0, T ] representing respectively the average velocity ∫ field, the order parameter and the pressure at each point of O × [0, T ]. h(t , u) stands for a deterministic external force. The term Y G(t , u(t − ); y)η˜ (dt , dy), where η˜ is a compensated time homogeneous Poisson random measure on a certain measurable space (Y , B(Y )), stands for a random force. Furthermore, u0 and ϕ0 are given non random initial velocity and order parameter respectively. We also should note that such a system is the nonlocal version of the well known Cahn–Hilliard–Navier–Stokes system, which has been investigated in [1–3] and references therein. We mention that the nonlocal term seems physically more appropriate than its approximation, i.e., when in place of aϕ − J ∗ ϕ there is −∆ϕ . We refer the reader to the basic papers [4–7], for more details. The system (1.1) is difficult to tackle because of the nonlinear coupling between µ and ϕ namely the term µ∇ϕ , the nonlocal term and the random external force. Even in two dimensions, the term µ∇ϕ can be less regular than the convective term (u.∇ )u (see [8]). In recent years, several articles have been devoted to the mathematical analysis of the local and nonlocal Cahn–Hilliard–Navier– Stokes system in the deterministic case, (that is G = 0). In the literature various authors have proved the existence of weak solutions, strong solutions results for both equations (see [8–12]) in dimensions 2 and 3. The existence of a unique strong solution and the existence of a V-attractor for a globally modified Cahn–Hilliard–Navier–Stokes system in a three-dimensional domain are established in [12]. The global existence of a weak solution is established in [8]. In the two-dimensional case, they showed that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition. Also in the deterministic case, the uniqueness is established in [9]. Stochastic partial differential equations are sometimes used to model physical system subjected to the influence of internal, external or environmental noises. To model turbulent fluids, mathematicians often use stochastic equations obtained from adding a noise term in the dynamical equations of the fluid. This approach is basically motivated by Reynold’s work which stipulates that the velocity of a fluid particle in turbulent regime is composed of slow (deterministic) and fast (stochastic) components. It is also pointed out in some recent articles [13] that some rigorous information on questions related to turbulence might be obtained from stochastic versions of the equations of fluid dynamics. Since the pioneering work of Bensoussan and Temam [14] on stochastic Navier–Stokes equations, the study of the stochastic local Cahn–Hilliard–Navier–Stokes system has been the object of intense investigations which have generated several important results. We refer, for instance, to [15–17]. It is worth mentioning that there are not many results available on the model considered in this article. The nonlocal term, the coupling operator and the random external force make the analysis of the proposed problem difficult and interesting as it involves tedious calculations. We prove the existence of a weak martingale solution for the problem (1.1) in both 2D and 3D bounded domains. The method of the proof closely follows the approach used in [18,19]. In [19], the authors investigated the existence of a weak martingale solution of nematic liquid crystal driven by a pure jump noise. In [18], Motyl proves the existence of a weak martingale solution for the stochastic Navier–Stokes equations driven by Lévy noise in unbounded 3D domain. The layout of the present paper is as follows. In Section 2, we introduce some functional spaces and useful operators needed throughout the work. In Section 3, we give the definition of the concept of solution of problem (1.1) and state our main result. We also state some compactness results and tightness criterion. In Section 4, we introduce our Galerkin approximation scheme for the problem (1.1) and obtain a priori estimates for the approximating solutions. We prove the crucial result of tightness of Galerkin’s solutions and apply the Skorokhod embedding theorem. We end this section with the proof of our main result. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada–Watanabe classical result to derive the existence of a strong solution in Section 5. 2. Functional setting and preliminaries 2.1. Basic definition, functional spaces and important embedding Let (Ω , F , F = (Ft )t ∈[0,T ] , P) be a given stochastic basis; that is, (Ω , F , P) is complete probability space and F is an increasing sub-σ algebras of F such that F0 contains every P-null subset of Ω . For any reflexive separable real Banach space X endowed with the norm ∥.∥X , for any p ≥ 1, Lp (0, T ; X ) is the space of X -valued measurable functions u defined on [0, T ] such that T

(∫ |u|Lp (0,T ;X ) =

∥u(t)∥ 0

p X dt

)1/p

< ∞.

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

25

For any r , p ≥ 1 we denote by Lp (Ω , P; Lr (0, T ; X )) the space of processes u = u(ω, x, t) with values in X defined on Ω × [0, T ] such that (1) u is measurable with respect to (ω, t) and, for each t, u is Ft measurable, (2) u(ω, t) ∈ X for almost all (ω, t) and

( (∫

T

∥u(t)∥

∥u∥Lp (Ω ,P;Lr (0,T ;X )) = E

r X dt

)p/r )1/p

< ∞,

0

where E denotes the mathematical expectation with respect to the probability measure P. The space Lp (Ω , P; Lr (0, T ; X )) so defined is a Banach space. (3) When r = ∞, we write

( )1/p ∥u∥Lp (Ω ,P;L∞ (0,T ;X )) = Eess sup ∥u(t)∥pX < ∞. t ∈[0,T ]

Let O ⊂ R be a bounded domain with smooth boundary ∂ O, d = 2, 3. Let Cc∞ (O, Rd ) denote the space of all Rd –valued functions of class C ∞ with compact supports contained in O and let d

V := u ∈ Cc∞ (O, Rd ) : div u = 0 in O , Gdiv := the closure of V in L2 (O, Rd ), Vdiv := the closure of V in H 1 (O, Rd ),

{

}

where L2 (O, Rd ) is a Hilbert space with the scalar product given by (u, v )L2 :=

∫

u(x). v (x) dx, for all u, v ∈ L2 (O, Rd ) and H 1 (O, Rd ) stands for the Sobolev O

space of all u ∈ L2 (O, Rd ) for which there exist weak derivatives To simplify our notation, we set

∂u ∂ xi

∈ L2 (O, Rd ), i = 1, 2, . . . , d.

H = L2 ( O , R d ) . In the space Gdiv we consider the scalar product and the norm inherited from H and denote them by (., .) and |.| respectively, i.e. (u, v ) := (u, v )L2 , |u| = |u| = 2

2 H

∫

|u(x)|2 dx, u, v ∈ Gdiv . O

For the space Vdiv we consider the scalar product inherited from H 1 (O, Rd ), i.e., (u, v )Vdiv := (u, v )L2 + ((u, v )), where ((u, v )) := (∇ u, ∇v )L2 =

d ∫ ∑

O

j=1

∂ u ∂v . dx, for all u, v ∈ Vdiv , ∂ xj ∂ xj

(2.1)

and the norm induced by (., .)Vdiv is given by

∥u∥2Vdiv := |u|2 + ∥u∥2 , where ∥u∥2 := |∇ u|2H , for all u ∈ Vdiv . As in [20, Proposition 1.24] we can define a self-adjoint operator A : H 1 (O, Rd ) → (H 1 (O, Rd ))′ by

⟨Aϕ, ψ⟩ := ((ϕ, ψ )) :=

∫

∇ϕ . ∇ψ dx, ϕ, ψ ∈ H 1 (O, Rd ).

(2.2)

O

We will denote by A the Neumann Laplacian acting on Rd (d = 2, 3)-valued function, that is,

} ∂ψ := ψ ∈ H (O, R ) : = 0 on ∂ O , ∂ξ d (2.3) ∑ ∂ 2ψ Aψ := − , ψ ∈ HE2 (O). 2 ∂ xj j=1 { } It can be shown that HE2 (O) = ψ ∈ H 1 (O, Rd ) : Aψ ∈ H . It can also be shown that B = I + A is a define positive and self-adjoint operator in the Hilbert space H := L2 (O, Rd ). Let us notice that {

HE2 (O)

2

d

|Aψ|(H 1 (O,Rd ))′ ≤ ∥ψ∥H 1 (O,Rd ) .

(2.4) s/2

For any s ∈ R we denote by Vs = D(B 1/2

V := V1 = D(B ′

), the domain of the fractional power operator B

s/2

. It can be shown that

) = H (O, R ), d = 2, 3. 1

d

For every f ∈ V we denote by ⟨f ⟩ the average of f over O, that is ⟨f ⟩ = ⟨., .⟩ is the duality product.

1

|O |

⟨f , 1⟩ where |O| stands for the Lebesgue measure of O and

26

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

As in [9] we also need to introduce the Hilbert spaces V0 := {ϕ ∈ V : ⟨ϕ⟩ = 0} , V0′ := ϕ ∈ V ′ : ⟨ϕ⟩ = 0 .

{

}

1 ′ We recall that A maps V onto V ′ and the restriction AN of A to V0 maps V0 onto V0′ isomorphically. Further, we denote by A− N : V0 → V0 −1 ′ the inverse map. As is well known, for every f ∈ V0 , AN f is the unique solution with zero mean value of the Neumann problem

⎧ ⎨−∆ϕ = f , in O ∂ϕ = 0 in ∂ O. ⎩ ∂ξ In addition, we have

⟨

−1 Aϕ, AN f = ⟨ϕ, f ⟩ , ∀ϕ ∈ V , ∀f ∈ V0′ ,

⟨

f , AN ϕ = ϕ, AN f =

⟩

−1

⟩

−1

⟨

⟩

∫

1 −1 ′ ∇ (A − N f ). ∇ (AN ϕ ) dx, ∀f , g ∈ V0 .

O

Let Av := ((v, .)), v ∈ Vdiv , where ((., .)) is defined in (2.1). We note that if v ∈ Vdiv , then Av ∈ Vdi′ v and

|Av|V ′ ≤ ∥v∥. div

Since for all v ∈ Vdiv ((u, v )) ≤ ∥u∥∥v∥ ≤ ∥u∥(∥v∥2 + |v|2 )1/2 = ∥u∥∥v∥Vdiv . We recall that the operator A is what we also call the Stokes operator (see for example [21] for its definition). In (2.2), ⟨., .⟩ denotes the dual pairing between V and V ′ . We will also denote by ⟨., .⟩ the dual pairing between Vdiv and Vdi′ v . As we are working on bounded domain, it is clear that V is dense in H, Vdiv is dense in Gdiv and the embedding are continuous and compact. We have V ↪→ H ∼ = G′div ↪→ Vdi′ v . = H ′ ↪→ V ′ , Vdiv ↪→ Gdiv ∼ j′1

j1

i′1

i1

For every m > 0, we define the following standard scale of Hilbert spaces J˜m := the closure of V in H m (O, Rd ). If m >

d 2

+ 1 then by the Sobolev embedding theorem, see [22],

H (O, Rd ) ↪→ H m−1 (O, Rd ) ↪→ Cb (O, Rd ) ↪→ L∞ (O, Rd ), m

(2.5)

where Cb (O, R ) denotes the space of R -valued continuous and bounded functions defined on O. Again if s > 2d + 1 then by the Sobolev embedding theorem, see [22], d

d

Vs ↪→ H s−1 (O, Rd ) ↪→ Cb (O, Rd ) ↪→ L∞ (O, Rd ).

(2.6)

Let us consider the following trilinear form, see for instance [23,24] b(u, v, w ) =

∫

(u .∇ )v.w dx, u ∈ Lp , v ∈ W 1,q , w ∈ Lκ (O, Rd ), where p, q, κ ∈ [1, ∞], O

satisfying

1 p

+

1 q

+

1

κ

≤ 1.

We will recall the fundamental properties of the form b that are valid for both bounded and unbounded domains. By the Sobolev embedding theorem, see Adams [22], and Hölder’s inequality, we obtain

|b(u, v, w )| ≤ c |u|1/2 ∥u∥1/2 |v|1/2 ∥v∥1/2 ∥w∥vdiv , in the case d = 2 u, v, w ∈ Vdiv ,

(2.7)

|b(u, v, w )| ≤ c |u|1/4 ∥u∥3/4 |v|1/4 ∥v∥3/4 ∥w∥, in the case d = 3 u, v, w ∈ Vdiv .

(2.8)

for some positive constant c. Thus the form b is continuous in Vdiv , see also [24]. Now we define a bilinear map B0 by B0 (u, v ) := b(u, v, .), then we infer that B0 (u, v ) ∈ Vdi′ v for all u, v ∈ Vdiv and the following inequalities hold

|B0 (u, v )|V ′ ≤ c |u|1/2 ∥u∥1/2 |v|1/2 ∥v∥1/2 , when d = 2 u, v ∈ Vdiv ,

(2.9)

|B0 (u, v )|V ′ ≤ c |u|1/4 ∥u∥3/4 |v|1/4 ∥v∥3/4 , when d = 3 u, v ∈ Vdiv .

(2.10)

div

div

Moreover, the mapping B0 : Vdiv × Vdiv → Vdi′ v is bilinear and continuous. The operator b also has the following properties, see Temam [24], Lemma II. 1. 3, b(u, v, w ) = −b(u; w, v ), u, v, w ∈ Vdiv . In particular, b(u, v, v ) = 0, u, v ∈ Vdiv .

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

27

Hence

⟨B0 (u, v ), w⟩ = − ⟨B0 (u, w), v⟩ , u, v, w ∈ Vdiv and

⟨B0 (u, v ), v⟩ = 0, u, v ∈ Vdiv .

(2.11) ′

Also as in [23] we can prove that the map B0 : Vdiv × Vdiv → Vdiv is locally Lipschitz continuous. 2.2. Some assumptions (H1 ) We assume that h is a measurable continuous mapping from (0, T ) × Gdiv into Vdi′ v and it satisfies: for a.e. every t ∈ (0, T ) and for all v ∈ Gdiv ,

|h(t , v )|2V ′ ≤ lh (1 + |v|2 ), div

for some positive constant lh . (H2 ) η˜ is a compensated time homogeneous Poisson random measure on a measurable space (Y , B(Y )) over the stochastic basis (Ω , F , F, P) with a σ -finite intensity measure ν1 . (See for instance [25, Section 3] or [19, Appendix A], for definitions and more details). (H3 ) G : [0, T ] × Gdiv × Y → Gdiv is a measurable function and there exists a constant lg such that

∫

|G(t , u; y) − G(t , v; y)|2 ν1 (dy) ≤ lg |u − v|2 , u, v ∈ Gdiv , t ∈ [0, T ],

(2.12)

Y

and for each p ≥ 1, there exists a constant C˜ p such that

∫

|G(t , v; y)|p ν1 (dy) ≤ C˜ p (1 + |v|p ), v ∈ Gdiv , t ∈ [0, T ].

(2.13)

Y

(H4 ) (H5 ) (H6 ) (H7 )

As in [26] (see also [8]) our assumption on ∫ the kernel J, the potential F are the following: J ∈ W 1,1 (Rd ; R), J(x) = J(−x) and a(x) = O J(x − y)dy ≥ 0 a.e., in O. F ∈ C 2 (R) and there exists c0 > 0 such that F ′′ (s) + a(x) ≥ c0 , ∀s ∈ R, a.e., x ∈ O. r There exist c3 > 0, c4 ≥ 0 and r ∈ (1, 2] such that |F ′ (s)| ≤ c3 |F (s)| + c4 , ∀s ∈ R. 1 There exist c5 > 2 |J |L1 (Rd ) and c6 ∈ R such that F (s) ≥ c5 s2 − c6 , ∀s ∈ R.

Remark 2.1. The requirement of assumption (H4 ) is standard for the nonlocal Cahn–Hilliard equation (see also [26] for a different hypothesis). Remark 2.2. Assumption (H5 ) implies that the potential F is a quadratic perturbation of a (strictly) convex function. In fact, if we set a∗ := |a|L∞ , then F can be represented as F (s) = Q (s) −

a∗ 2

s2 ,

(2.14)

with Q ∈ C 2 (R) strictly convex, since Q ′′ ≥ c0 in O. Hereafter, for any (u, ϕ ) ∈ Vdiv × H, we set E (ϕ ) =

1

∫ ∫

4

O

J(x − y)(ϕ (x) − ϕ (y))2 dxdy + O

∫

F (ϕ (x))dx,

O

and Etot (u, ϕ ) = |u|2 + 2E (ϕ ).

(2.15)

Now we rewrite Eqs. (1.1) as

⎧ du(t) + [ν Au(t) + B0 (u(t), u(t))]dt ⎪ ⎪ ∫ ⎪ ⎪ ⎨= µ(t)∇ϕ (t)dt + h(t , u(t))dt + G(t , u(t − ); y)η˜ (dt , dy) in V ′ , div Y ⎪ ′ ⎪dϕ (t) + [u(t).∇ϕ (t)]dt + Aµ(t)dt = 0 in V , µ(t) = aϕ (t) − J ∗ ϕ (t) + F ′ (ϕ (t)), ⎪ ⎪ ⎩ (u, ϕ )(0) = (u(0), ϕ (0)) = (u0 , ϕ0 ). 3. Statement of the main result Let Gw div denote the Hilbert space Gdiv endowed with the weak topology. Now, we introduce the concept of martingale solution to problem (2.16). Definition 3.1.

¯ , F¯ , F¯ , P¯ , u¯ , ϕ, A martingale solution of problem (2.16) is a system (Ω ¯ η¯ ), where

¯ , F¯ , F¯ , P¯ ) is a filtered probability space with a filtration F¯ = (F¯t )t ∈[0,T ] , (a) (Ω

(2.16)

28

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

¯ , F¯ , F¯ , P¯ ) with the intensity measure ν1 , (b) η¯ is a time homogeneous Poisson random measure on (Y , B(Y )) over (Ω ¯ ¯ (c) u¯ : Ω × [0, T ] → Vdiv is a progressively measurable process with P-a.e. paths 2 u¯ (ω, ¯ .) ∈ D([0, T ]; Gw div ) ∩ L (0, T ; Vdiv )

(3.1)

¯ -a.s. such that for all t ∈ [0, T ] and all v ∈ Vdiv the following identity holds P ∫ t∫ ∫ t ∫ t ⟨B0 (u¯ (s), u¯ (s)), v⟩ ds = − ⟨Au¯ (s), v⟩ ds + (v.∇ µ ¯ (s))ϕ¯ (s)dxds (u¯ (t), v ) +ν 0 O 0 0 ∫ t∫ ∫ t ⟨h(s, u¯ (s)), v⟩ ds + (G(s, u¯ (s− ); y), v )η¯ (ds, dy). + (u0 , v ) + 0

0

(3.2)

Y

¯ × [0, T ] → V is a progressively measurable process with P¯ -a.e. paths (d) ϕ¯ : Ω ϕ¯ (ω, ¯ .) ∈ C ([0, T ]; Hw ) ∩ L2 (0, T ; V )

(3.3)

¯ -a.s. such that for all t ∈ [0, T ] and all ψ ∈ V the following identity holds P (ϕ¯ (t), ψ ) +

t

∫

(∇ ρ¯ (., ϕ¯ (s)), ∇ψ )ds 0

∫ t∫

= (ϕ0 , ψ ) + (u¯ (s).∇ψ )ϕ¯ (s)dxds 0 O ∫ t∫ + (∇ J ∗ ϕ¯ (s)).∇ψ dxds, 0

(3.4)

O

¯ (s) = ρ¯ (x, ϕ¯ (s)) − J ∗ ϕ¯ (s). where ρ¯ (x, ϕ¯ (s)) = a(x)ϕ¯ (s) + F (ϕ¯ (s)), µ ′

The spaces D([0, T ]; Gw div ) and C ([0, T ]; Hw ) will be clearly defined below. Remark 3.1. It is immediate to see that the total mass is conserved. Indeed, choosing ψ = 1 in (3.4), we have (ϕ¯ (t), 1) = (ϕ0 , 1) for all t ≥ 0. Our main result is given by the following theorem: Theorem 3.1. We assume that the hypotheses (H1 )–(H7 ) are satisfied. Let d = 2, 3 and (u0 , ϕ0 ) ∈ Vdiv × H. Then system (2.16) has a weak ¯ , F¯ , F¯ , P¯ , u¯ , ϕ, martingale solution (Ω ¯ η¯ ) in the sense of Definition 3.1. Moreover, the solution satisfies the following estimates

¯ sup |¯u(t)|2 + E¯ E t ∈[0,T ]

T

∫ 0

¯ sup |ϕ¯ (t)|2 + E¯ E t ∈[0,T ]

∥¯u(s)∥2Vdiv ds < ∞ and

(3.5)

∥ϕ¯ (s)∥2V ds < ∞.

(3.6)

T

∫ 0

We also have the pathwise uniqueness and the existence of a strong solution in the two dimensions. 3.1. Compactness and tightness criterion Let (U , d1 ) be a complete separable metric space. Let D([0, T ]; U ) be the space of all U -valued càdlàg functions defined on [0, T ], i.e. the functions which are right continuous and have left limits at every t ∈ [0, T ]. This space is endowed with the Skorokhod topology. A sequence (vm ) ⊂ D([0, T ]; U ) converges to v ∈ D([0, T ]; U ) if and only if there exists a sequence (λm ) of homeomorphisms of [0, T ] such that λm tends to the identity uniformly on [0, T ] and vm ◦ λm tends to v uniformly on [0, T ]. The topology is metrizable by the following metric dT dT (v, u) := inf

λ∈σT

[

⏐ ⏐

sup d1 (v (t), u ◦ λ(t)) + sup |t + λ(t)| + sup ⏐⏐log

t ∈[0,T ]

t ∈[0,T ]

s̸ =t

⏐] λ(t) − λ(s) ⏐⏐ ⏐ , t −s

where σT is the set of increasing homeomorphisms of [0, T ]. Moreover, (D([0, T ]; U ), dT ) is a complete metric space. We recall the notion of a modulus of the function. It plays analogous role in the space D([0, T ]; U ) as the modulus of continuity in the space of continuous functions C ([0, T ]; U ). Definition 3.2 (see [27]). Let v ∈ (D[0, T ]; U ) and let δ > 0 be given. A modulus of v is defined by W[0,T ],U (v, δ ) := inf max

sup

Πδ tj ∈ω¯ t ≤s
d1 (v (t), v (s)),

(3.7)

where Πδ is the set of all increasing sequences ω ¯ = {0 = t0 < t1 < · · · , tm = T } with the following property tj+1 − tj ≥ δ, j = 0, 1, . . . , m − 1. We have the following criterion analogous to the Arzelà–Ascoli Theorem for the relative compactness of a subset of the space D([0, T ]; U ). If no confusion seems likely, we will denote the modulus by W[0,T ] (v, δ ). Theorem 3.2 (Proof see [27]). A set K ⊂ D([0, T ]; U ) has precompact if and only if it satisfies the following two conditions: (a) there exists a dense subset K1 ⊂ [0, T ] such that for every t ∈ K1 the set {v (t), v ∈ K } has compact closure in U .

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

29

(b) limδ→0 supv∈K W[0,T ] (v, δ ) = 0. We introduce the following functional spaces, analogous to those considered for example in [19,27]:

D([0, T ]; Vdi′ v ) := The space of càdlàg function v : [0, T ] → Vdi′ v with the topology T1 induced by the Skorokhod metric dT , C ([0, T ]; V ′ ) := the space of continuous functions φ : [0, T ] → V ′ with the topology T1′ ,

L2w (0, T ; Vdiv ) := the space L2 (0, T ; Vdiv ) with the weak topology T2 , L2w (0, T ; V ) := the space L2 (0, T ; V ) with the weak topology T2′ , L2 (0, T ; Gdiv ) := the space of measurable functions v : [0, T ] → Gdiv with the topology T3 , L2 (0, T ; V ) := the space of measurable functions φ : [0, T ] → V with the topology T3′ . We consider the space D([0, T ]; Gw div ):= the space of weakly càdlàg functions v : [0, T ] → Gdiv with the weakest topology T4 such that for all h ∈ Gdiv the mappings D([0, T ]; Gw div ) ∋ v ↦ → (v (.), h) ∈ D([0, T ]; R) are continuous. In particular, vn → v in D([0, T ]; Gw div ) if and only if for all h ∈ Gdiv : (vn (.), h) → (v (.), h) in the space D([0, T ]; R). Similarly, we define C ([0, T ]; Hw ) with the topology T4′ . The following theorems are due to [18,23], where we can see the details of the proof. Theorem 3.3 (Compactness Criterion for u). Let q ∈ (1, ∞) and let Z˜1q = Lqw (0, T ; Vdiv ) ∩ Lq (0, T ; Gdiv ) ∩ D([0, T ]; Vdi′ v ) ∩ D([0, T ]; Gw div )

(3.8)

and let T˜1 be the supremum of the corresponding topologies. Then a set K¯¯ 1 ⊂ Z˜1q is T˜1 -relatively compact if the following three conditions hold (a) supu∈K¯¯ supt ∈[0,T ] |u(t)| < ∞, 1

(b) supu∈K¯¯

1

∫T 0

∥u(s)∥qVdiv ds < ∞, i.e., K¯¯ 1 is bounded in Lq (0, T ; Vdiv ),

(c) limδ→0 supu∈K¯¯ W[0,T ],V ′ (u; δ ) = 0. div

1

Theorem 3.4 (Compactness Criterion for ϕ ). Let q ∈ (1, ∞) and let Z˜2q = Lqw (0, T ; V ) ∩ Lq (0, T ; H) ∩ C ([0, T ]; V ′ ) ∩ C ([0, T ]; Hw )

(3.9)

and let T˜2 be the supremum of the corresponding topologies. Then a set K¯¯ 2 ⊂ Z˜2q is T˜2 -relatively compact if the following three conditions hold (a) supϕ∈K¯¯ supt ∈[0,T ] |ϕ (t)| < ∞, 2

(b) supϕ∈K¯¯

2

∫T 0

∥ϕ (s)∥qV ds < ∞, i.e., K¯¯ 2 is bounded in Lq (0, T ; V ),

(c) limδ→0 supϕ∈K¯¯ sups,t ∈[0,T ],|t −s|≤δ d1 (ϕ (t), ϕ (s)) = 0. 2

We recall that the space Z˜1q and Z˜2q are not Polish spaces. 3.2. The Aldous condition Let (Ω , F , P) be a probability space with filtration F := {Ft }t ≥0 satisfying the usual hypotheses, see [27]. Let (U , d1 ) be a complete, separable metric space and (yn )n∈N be a sequence of F-adapted and U -valued processes. Definition 3.3. Let (yn )n∈N be a sequence of U -valued random variables. The sequence of laws of these processes forms a tight sequence if and only if

{ } [T] ∀ϵ > 0 ∀ζ > 0 ∃δ > 0 such that: sup P W[0,T ],U (yn , δ ) > ζ ≤ ϵ, n∈N

where W[0,T ],U is defined in (3.7). Definition 3.4. A sequence (yn )n∈N satisfies the Aldous condition in the space U if and only if

[A] ∀ϵ > 0 ∀ζ > 0 ∃δ > 0 such that for every sequence (τn )n∈N of F-stopping times with τn ≤ T one has sup sup P {|yn (τn + θ ) − yn (τn )|U ≥ ζ } ≤ ϵ. n∈N 0≤θ ≤δ

Also we have the following lemma where the proof can be found in [28, Theorem 2. 2. 2]. Lemma 3.1.

Condition [A] implies Condition [T].

30

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

The following lemma gives us a certain condition which guarantees that the sequence (yn )n∈N satisfies condition [A]. For the details proof of this lemma see [18, Appendix A, Lemma 6.3]. Lemma 3.2. Let (X , |.|X ) be a separable Banach space and let (yn )n∈N be a sequence of X -valued random variables. Assume that for every (τn )n∈N of F-stoppings times with τn ≤ T and for every n ∈ N and θ ≥ 0 the following condition holds

E |yn (τn + θ ) − yn (τn )|αX ≤ C θ β

(3.10)

for some α, β > 0 and some constant C > 0. Then the sequence (yn )n∈N satisfies the Aldous condition in the space X . In the view of Theorem 3.3, to show that the law of un is tight, we need the following result Let q ∈ (1, ∞) and let (un )n∈N be a sequence of càdlàg F-adapted, Vdi′ v -valued processes such that

Corollary 3.1.

¯ 1 such that (a) there exists a positive constant K ¯ 1, sup E[ sup |un (s)|] ≤ K s∈[0,T ]

n∈N

¯ 2 such that (b) there exists a positive constant K T

[∫ sup E n∈N

0

] ∥un (s)∥qVdiv ds ≤ K¯ 2 ,

(c) (un )n∈N satisfies the Aldous condition in Vdi′ v . Let P1n be the law of un on Z˜1q . Then for every ϵ > 0 there exists a compact subset K¯ ϵ1 of Z˜1q such that

P1n (K¯ ϵ1 ) ≥ 1 − ϵ. In the view of Theorem 3.4, to show that the law of ϕn is tight, we need the following result Let q ∈ (1, ∞) and let (ϕn )n∈N be a sequence of continuous F-adapted, V ′ -valued processes such that

Corollary 3.2.

¯ 3 such that (a) there exists a positive constant K ¯ 3, sup E[ sup |ϕn (s)|] ≤ K n∈N

s∈[0,T ]

¯ 4 such that (b) there exists a positive constant K T

[∫ sup E n∈N

] ∥ϕn (s)∥qV ds ≤ K¯ 4 ,

0

(c) (ϕn )n∈N satisfies the Aldous condition in V ′ . Let P2n be the law of ϕn on Z˜2q . Then for every ϵ > 0 there exists a compact subset K¯ ϵ2 of Z˜2q such that

P2n (K¯ ϵ2 ) ≥ 1 − ϵ. 3.3. Skorokhod embedding theorem We have the following Jakubowski’s version of the Skorokhod theorem due to [29]. Theorem 3.5. Let (B, τ ) be a topological space such that there exists a sequence (fk ) of continuous functions fk : B → R that separates points of B. Let (Xn ) be a sequence of B-valued random variables. Suppose that for every ϵ > 0 there exists a compact subset K˜ ϵ ⊂ B such that inf P

n∈N

({

Xn ∈ K˜ ϵ

})

> 1 − ϵ.

Then there exist a subsequence (Xnk )k∈N , a sequence (Ynk )k∈N of B-valued random variable and a B-valued random variable Y defined on some probability space (Ω , F , P) such that Law(Xnk ) = Law(Ynk ), k = 1, 2, . . .

and for all ω ∈ Ω τ

Ynk (ω) − → Y (ω) as k → ∞. We recall the following version of Skorokhod theorem due to [30,31] and [18]. Theorem 3.6. Let B1 , B2 be two separable Banach spaces and let πi : B1 × B2 → Bi , i = 1, 2, be the projection into Bi , i.e., B1 × B2 ∋ X = (X1 , X2 ) → πi (X ) ∈ Bi . Let (Ω , F , P) be a probability space and let Xn : Ω → B1 ×B2 , n ∈ N, be a family of random variables such that the sequence {Law (Xn ), n ∈ N} is weakly convergent on B1 × B2 . Finally let us assume that there exists a random variable X : Ω → B1 such that Law (π1 ◦ Xn ) = Law (X ), for all n ∈ N. Then there exist a probability space (Ω ′ , F ′ , P′ ), a family of B1 × B2 -valued random variables {Xn′ , n ∈ N} on (Ω ′ , F ′ , P′ ) and a random variable X ′ : Ω ′ → B1 × B2 such that

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

31

(a) Law (Xn′ ) = Law (Xn ), for all n ∈ N; (b) Xn′ → X ′ in B1 × B2 a.s.; (c) π1 ◦ Xn′ (ω) = π1 ◦ X ′ (ω) for all ω ∈ Ω ′ . We will use the following version of the Skorokhod embedding theorem where the proof can be found in [18, Appendix B, Corollary 7.3] or in [32, Corollary 5.3]. Theorem 3.7. Let M1 be a separable complete metric space and M2 be a topological space such there exists a sequence {fk }k∈K of continuous functions fk : M2 → R separating points of M2 . Let M = M1 × M2 with the Tychonoff topology induced by the projections

πi : M1 × M2 → Mi , i = 1, 2. Let (Ω , F , P) be a probability space and let Xn : Ω → M1 × M2 , n ∈ N, be a family of random variables such that the sequence {L(Xn ), n ∈ N} is tight on M1 × M2 . Finally let us assume that there exists a random variable X : Ω → M1 such that L(π1 ◦ Xn ) = L(X ) for all n ∈ N. Then there exist a subsequence (Xnk )k∈N , a probability space (Ω ′ , F ′ , P′ ), a family of M1 × M2 -valued random variables {Xk , k ∈ N} on (Ω ′ , F ′ , P′ ) and a random variable X ′ : Ω ′ → M1 × M2 such that (a) L(Xk ) = L(Xnk ) for all k ∈ N; (b) Xk → X ′ in M1 × M2 a.s., as k → ∞; (c) π1 ◦ Xk (ω) = π1 ◦ X ′ (ω) for all ω ∈ Ω ′ . 4. Proof of the main result The proof of the main result is split into seven steps. In Step 1, we first assume ϕ0 ∈ D(B) and introduce our Galerkin approximation. Step 2 deals with some a priori estimates of the scheme. Step 3 concerns the study of the tightness of the laws of approximating sequences un and ϕn . In Step 4, we construct new processes based on the application of Skorokhod theorem. In Step 5 and Step 6, we pass to the limit and prove the existence of a weak martingale solution for the problem (1.1). Finally in Step 7, we consider a general initial condition u0 ∈ Gdiv and ϕ0 ∈ H. Using a limiting argument and the previous steps, we prove the existence of a weak martingale solution of problem (1.1). 4.1. Step 1. The Galerkin approximation We first assume that ϕ0 ∈ D(B) ⊂ H instead of ϕ0 ∈ H. Since the injection of Vdiv × V ⊂ Gdiv × H is compact, let {(wj , ψj ), j = 1, 2, . . .} ⊂ Vdiv × V be an orthonormal basis on Gdiv × H, where {wj , j = 1, 2, . . .}, {ψj , j = 1, 2, . . .} are eigenvectors of the Stokes operator A and the Neumann operator B = A + I : D(B) → H : v ↦ → Av + v , respectively. We set Vdin v = span{w1 , w2 , . . . , wn } and Hn = span{ψ1 , ψ2 , . . . , ψn }. Let P˜ n be the operator from Vdi′ v to Vdin v defined by P˜ n u∗ :=

n ∑ ⟨

u∗ , wj wj , u∗ ∈ Vdi′ v .

⟩

j=1

We will consider the restriction of the operator P˜ n to the space Gdiv (still) denoted by P˜ n . More precisely, we have Gdiv ↪→ Vdi′ v , i.e. every element u ∈ Gdiv induces a functional u∗ ∈ Vdi′ v by the formula

⟨

u∗ , v = (u, v ), v ∈ Vdiv .

⟩

Thus the restriction of P˜ n to Gdiv is given by P˜ n u =

n ∑

(u, wj )wj , u ∈ Gdiv .

j=1

Hence in particular, P˜ n is the (., .)- orthogonal projection from Gdiv onto Vdin v . Similarly, we define by Pn the orthogonal projection from H onto Hn . We consider the probability space (Ω , F , P) with filtration F := {Ft }t ≥0 satisfying the usual hypotheses, see [27]. We then look for the three functions of the form un (t) =

n ∑

ank (t)wk , ϕn (t) =

k=1

n ∑

bnk (t)ψk , µn (t) =

k=1

n ∑

ckn (t)ψk ,

k=1

which solves the following approximating problem (un (t), v )

∫ t ⟨Aun (s), v⟩ ds + b(un (s), un (s), v )ds 0 0 ∫ t∫ ∫ t = (u0n , v ) − (v.∇µn )ϕn dxds + ⟨P˜ n h(s, un (s)), v⟩ds 0 O 0 ∫ t∫ + (P˜ n G(s, un (s− ); y), v )η˜ (ds, dy), ∫

t

+ν

0

Y

(4.1)

32

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

(ϕn , ψ ) + (∇ρ (., ϕn ), ∇ψ ) = ′

∫

∫

(un .∇ψ )ϕn dx + O

(∇ J ∗ ϕn ).∇ψ dx,

(4.2)

O

µn := Pn ρ (., ϕn ) − Pn (J ∗ ϕn ),

(4.3)

ϕn (0) = Pn ϕ0 , un (0) = P˜ n u0 = u0n ,

(4.4)

for every ψ ∈ Hn and every v ∈ The proof of the following result can be found in [23, Lemma 2.4]. Vdin v .

Lemma 4.1. For every (u, ϕ ) ∈ Vdiv × H (a) limn→∞ ∥P˜ n v − v∥Vdiv = 0, (b) limn→∞ ∥P˜ n v − v∥J˜m = 0, where m > 0, (c) limn→∞ |Pn ϕ − ϕ| = 0. 4.2. Step 2. A priori estimates In this section, using the Itô formula, see [25, Lemma 4.1] or [33, Theorem 5.1], and the Burkhölder–Davis–Gundy inequality, see [34], we will prove the following lemma about a priori estimates of the solutions un and ϕn of (4.1)–(4.4). In fact, these estimates hold provided the noise terms satisfy only condition (H3 ) in assumption 2.2. Lemma 4.2.

For every p ≥ 2, we have the following estimates:

( sup E

sup [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2

s∈[0,T ]

n∈N

2ν p/2 sup E

T

[∫

n∈N

)

˜

≤ K˜ 1 eK2 T ,

]p/2 [∫ ∥un (s)∥2 ds + 2 sup E n∈N

0

T

]p/2 |∇µn (s)|2 ds

0

≤ K˜ 1 (1 + K˜ 2 TeK˜ 2 T ) ≤ K˜ 1 e2K˜ 2 T , ˜ 1 , K˜ 2 are given in (4.38) and Etot is given in (2.15). where the constants K Proof. For all n ∈ N and all k > 0 let us define

τnk := inf{t ≥ 0 : |un (t)| ≥ k} ∧ T .

(4.5)

Since the process {un (t)}t ∈[0,T ] is F-adapted and right-continuous, τ is a stopping time. Moreover, since the process (un ) is càdlàg on [0, T ], the trajectories t ↦→ un (t) are bounded on [0, T ], P-a.s. Hence τnk ↑ T , P-a.s., as k ↑ ∞. Using the Itô formula to the formula φ (x) := 21 |x|2 , x ∈ Gdiv , we obtain for all t ∈ [0, T ] ∫ t ∧τnk ∫ t ∧τnk ⟨B0 (un (s), un (s)), un (s)⟩ ds ⟨Aun (s), un (s)⟩ ds + φ (un (t ∧ τnk )) − φ (u0n ) = −ν k n

0

0

∫

t ∧τnk

∫

t ∧τnk

∫ {

(un (s).∇µn (s))ϕn (s)ds +

− O

0

∫ + 0

⟨

⟩

P˜ n h(s, un (s)), un (s) ds

0

} φ (un (s− ) + P˜ n G(s, un (s− ); y)) − φ (un (s− )) η˜ (ds, dy)

(4.6)

Y t ∧τnk

∫

t ∧τnk

∫

+

∫ {

0

} φ (un (s) + P˜ n G(s, un (s); y)) − φ (un (s)) − (φ ′ (un (s)), P˜ n G(s, un (s); y)) ν1 (dy)ds.

Y

By (4.6), the fact that ⟨Aun (s), un (s)⟩ = ∥un (s)∥2 and ⟨B0 (un (s), un (s)), un (s)⟩ = 0, we obtain 2

|un (t ∧ τnk )| + 2ν

t ∧τnk

∫

∥un (s)∥2 ds = |u0n |2 − 2

0 t ∧τnk

∫ +2

⟨

t ∧τnk

∫

∫

(un (s).∇µn (s))ϕn (s)dxds O

0

⟩

P˜ n h(s, un (s)), un (s) ds

0 t ∧τnk

∫ + 0

0

2

2

|un (s− ) + P˜ n G(s, un (s− ); y)| − |un (s− )|

}

η˜ (ds, dy)

(4.7)

Y t ∧τnk

∫ +

∫ { ∫ {

} 2 |un (s) + P˜ n G(s, un (s); y)| − |un (s)|2 − 2(un (s), P˜ n G(s, un (s); y)) ν1 (dy)ds.

Y

By using µn as a test function in (4.2), we obtain (ϕn′ , µn ) + (∇ρ (., ϕn ), ∇µn ) =

∫

(un .∇µn )ϕn dx + O

∫

(∇ J ∗ ϕn ).∇µn dx. O

(4.8)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

33

We now have (ϕn′ , µn )

= (ϕn′ , aϕn + F ′ (ϕn ) − J ∗ ϕn ) ( ) ∫ 1 d 1 √ 2 | aϕn | + F (ϕn )dx − (ϕn , J ∗ ϕn ) = dt 2 2 O ( ∫ ∫ ) ∫ d 1 d = J(x − y)(ϕn (x) − ϕn (y))2 dxdy + F (ϕn )dx = E (ϕn (.)). dt

4

O

O

O

(4.9)

dt

Furthermore, observe that (∇ρ (., ϕn ), ∇µn ) = (∇µn + ∇ Pn (J ∗ ϕn ), ∇µn )

= |∇µn |2 + (∇ (Pn (J ∗ ϕn )), ∇µn ) = (∇ρn , ∇µn ),

(4.10)

where ρn := Pn ρ (., ϕn ) = µn + Pn (J ∗ ϕn ). Inserting (4.9) and (4.10) in (4.8), integrating the result equality over [0, t ∧ τnk ], we derive 2E (ϕn (t ∧ τnk )) + 2

t ∧τnk

∫ 0

= 2E (ϕn (0)) + 2

t ∧τnk

∫

t ∧τnk

∫

((∇ Pn (J ∗ ϕn )), ∇µn ) ds t ∧τnk ∫ (un .∇µn )ϕn dxds + 2 (∇ J ∗ ϕn (x, s)).∇µn (x, s)dxds.

|∇µn (s)|2 ds + 2

0

∫

∫

O

0

0

(4.11)

O

Now adding Eqs. (4.11) and (4.7), recalling (2.15), we obtain

∫ t ∧τnk |∇µn (s)|2 ds = Etot (u0n , ϕ0n ) ∥un (s)∥2 ds + 2 0 0 ∫ t ∧τnk ∫ t ∧τnk ⟨ ⟩ −2 P˜ n h(s, un (s)), un (s) ds ((∇ Pn (J ∗ ϕn (s))), ∇µn (s)) ds + 2 0 ∫0 t ∧τnk ∫ (∇ J ∗ ϕn (x, s)).∇µn (x, s)dxds + In1 (t ∧ τnk ) + In2 (t ∧ τnk ), +2 ∫

Etot (un (t ∧ τnk ), ϕn (t ∧ τnk )) + 2ν

t ∧τnk

(4.12)

O

0

where In1 (t

∧τ

k n)

t ∧τnk

∫ =

∫ {

0

} 2 2 |un (s− ) + P˜ n G(s, un (s− ); y)| − |un (s− )| η˜ (ds, dy)

(4.13)

Y

and In2 (t

∧τ

k n)

t ∧τnk

∫

∫

2 {|un (s) + P˜ n G(s, un (s); y)| − |un (s)|2

= 0

Y

−2(un (s), P˜ n G(s, un (s); y))}ν1 (dy)ds. By (2.13) and (4.5), the process In1 (t) :=

∫ t∫ { 0

(In1 (t

∧τ

k n ))t ∈[0,T ] ,

(4.14)

where

} 2 2 |un (s− ) + P˜ n G(s, un (s− ); y)| − |un (s− )| η˜ (ds, dy),

Y

t ∈ [0, T ], is an integrable martingale. Hence

EIn1 (t ∧ τnk ) = 0 for all t ∈ [0, T ].

(4.15)

From the Taylor formula, it follows that for each q ≥ 2 there exists a positive constant Cq > 0 such that for all x, h ∈ Gdiv the following inequality holds

⏐ ⏐ ⏐|x + h|q − |x|q − q|x|q−2 (x, h)⏐ ≤ Cq (|x|q−2 + |h|q−2 )|h|2 .

(4.16)

Using (4.16) and (2.13), we obtain the following estimate

|In2 (t ∧ τnk )| ≤

t ∧τnk

∫ 0

t ∧τnk

∫ ≤ C2 0

∫ ⏐ ⏐ 2 ⏐ ⏐ ⏐{|un (s) + P˜ n G(s, un (s); y)| − |un (s)|2 − 2(un (s), P˜n G(s, un (s); y))}⏐ ν1 (dy)ds Y

∫ ∫ ⏐ ⏐2 ⏐ ⏐˜ ⏐Pn G(s, un (s); y)⏐ ds ≤ c˜2 Y

t ∧τnk

(1 + |un (s)|2 )ds.

(4.17)

0

Thus by Fubini’s theorem

E|In2 (t ∧ τnk )|

≤ c˜2 (t ∧ τnk ) + c˜2 ≤ c˜2 (t ∧ τ

k n)

t ∧τnk

∫

∫0 t ∧τnk + c˜2 0

with the constant c6 given by (H7 ).

E|un (s)|2 ds (4.18)

E(Etot (un (s), ϕn (s)) + 2c6 |O|)ds,

34

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Using the Cauchy–Schwarz inequality, Young inequality and (H1 ) we obtain

⏐⟨ ⟩⏐ ⏐ ˜ ⏐ ⏐ Pn h(s, un (s)), un (s) ⏐

≤ |P˜ n h(s, un (s))|V ′ ∥un (s)∥ div ν 2 ≤ ∥un (s)∥ + cν lh + cν lh |un (s)|2 2 ν ≤ ∥un (s)∥2 + cν lh + cν lh (Etot (un (s), ϕn (s)) + 2c6 |O|),

(4.19)

2

where cν is a constant depending only on ν . Also using Young’s inequality, we have

|((∇ Pn (J ∗ ϕn (s))), ∇µn (s))|

≤ |∇ Pn (J ∗ ϕn (s))||∇µn (s)| ⏐ ⏐ ≤ ⏐B1/2 Pn (J ∗ ϕn (s))⏐ |∇µn (s)| ≤ (|∇ J ∗ ϕn (s)| + |J ∗ ϕn (s)|)|∇µn (s)|

(4.20)

≤ |J |W 1,1 (R2 ,R) |ϕn (s)||∇µn (s)| ≤

1

4

|∇µn (s)|2 + cJ |ϕn (s)|2 ,

where cJ is a constant depending only on J. By (4.12), (4.15) and (4.18)–(4.20), we have for all t ∈ [0, T ]

E(Etot (un (t ∧ τnk ), ϕn (t ∧ τnk )) + 2c6 |O|) + ν E

t ∧τnk

∫

∥un (s)∥2 ds + E

t ∧τnk

∫

|∇µn (s)|2 ds

0

0

≤ (Etot (u0n , ϕ0n ) + 2c6 |O|) + (cν lh + c˜2 )t ∧ τnk ∫ t ∧τnk ( c ) + cν lh + c˜2 + αJ E(Etot (un (s), ϕn (s)) + 2c6 |O|)ds 0 ∫ ≤ |u0n |2 + 2|J |L1 (Rd ) |ϕ0n |2 + 2 F (ϕ0n (x))dx + 2c6 |O| + (cν lh + c˜2 )t ∧ τnk O ∫ t ∧τnk ( cJ ) ˜ E(Etot (un (s), ϕn (s)) + 2c6 |O|)ds + cν lh + c2 + α 0 ∫ t ∧τnk ( c ) ≤ K + cν lh + c˜2 + αJ E(Etot (un (s), ϕn (s)) + 2c6 |O|)ds,

(4.21)

0

where α = 2c5 − |J |L1 (Rd ) > 0, with the constant c5 given by (H7 ), while K is given by K = |u0 |2 + 2|J |L1 (Rd ) |ϕ0 |2 + 2

∫

F (ϕ0 (x))dx + 2c6 |O| + (cν lh + c˜2 )T .

(4.22)

O

Also we note that in (4.21) we have used the fact that, since ϕ0 is supposed to belong to D(B), then we have ϕ0n → ϕ0 in H 2 (O, Rd ) and hence also in L∞ (O, Rd ) (for d = 2, 3). In particular,

(

E(Etot (un (t ∧ τnk ), ϕn (t ∧ τnk ))) ≤ K + cν lh + c˜2 +

cJ )

t ∧τnk

∫

α

E(Etot (un (s), ϕn (s)) + 2c6 |O|)ds.

0

By the Gronwall Lemma we infer that for all t ∈ [0, T ] sup sup E[Etot (un (t ∧ τnk ), ϕn (t ∧ τnk ))] ≤ Ke

(

cJ

)

cν lh +˜c2 + α T

n∈N t ∈[0,T ]

.

Thus, in particular,

[∫

t ∧τnk

sup E n∈N

] (Etot (un (s), ϕn (s)) + 2c6 |O|)ds

cJ

(

≤ T Ke

)

cν lh +˜c2 + α T

.

(4.23)

0

Now passing to the limit as k → ∞ and using the Fatou Lemma we infer that

[∫ sup E n∈N

T

]

(Etot (un (s), ϕn (s)) + 2c6 |O|)ds ≤ T Ke

(

cJ

)

cν lh +˜c2 + α T

.

0

From (4.21), (4.24) and the fact that 1 + xeax ≤ e(a+1)x for all x ≥ 0, we infer that

[ ∫ ] ∫ t ∧τnk t ∧τnk 2 2 sup E ν ∥un (s)∥ ds + |∇µn (s)| ds n∈N 0 0 [ ] ( ( ( ) cJ ) cJ ) cJ T 2 c l +˜c + T cν l +˜c + ≤ K 1 + cν lh + c˜2 + Te h 2 α ≤ Ke ν h 2 α . α

(4.24)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

35

Passing to the limit as k → ∞ and using the Fatou Lemma we deduce that T

[ ∫

sup E ν n∈N

∥un (s)∥2 ds +

T

∫

0

] ( cJ ) T 2 c l +˜c + |∇µn (s)|2 ds ≤ Ke ν h 2 α .

0

We recall that for all t ∈ [0, T ], we have

∫ t ∫ t +2ν ∥un (s)∥2 ds + 2 |∇µn (s)|2 ds = Etot (u0n , ϕ0n ) 0 0 ∫ t⟨ ∫ t ⟩ P˜ n h(s, un (s)), un (s) ds −2 ((∇ Pn (J ∗ ϕn (s))), ∇µn (s)) ds + 2 0 ∫0 t ∫ (∇ J ∗ ϕn (x, s)).∇µn (x, s)dxds + In1 (t) + In2 (t), +2

Etot (un (t), ϕn (t))

O

0

In1 (t)

In2 (t)

where and are defined as in (4.13) and (4.14) respectively. From this last inequality, we deduce that t

∫

t

∫

|∇µn (s)|2 ds ≤ Etot (u0n , ϕ0n ) ∫ t ⏐⟨ ∫ t ⟩⏐ ⏐ ˜ ⏐ |((∇ Pn (J ∗ ϕn (s))), ∇µn (s))| ds + 2 +2c6 |O| + 2 ⏐ Pn h(s, un (s)), un (s) ⏐ ds 0 0 ∫ t∫ |(∇ J ∗ ϕn (x, s)).∇µn (x, s)| dxds + In1 (t) + In2 (t). +2

Etot (un (t), ϕn (t)) + 2c6 |O| + 2ν

∥un (s)∥2 ds + 2

0

0

(4.25)

O

0

By (4.25) and the estimates (4.19)–(4.20), we obtain

∫ t ∥un (s)∥2 ds + |∇µn (s)|2 ds 0 0 ∫ t ( c ) [Etot (un (s), ϕn (s)) + 2c6 |O|] ds ≤ [Etot (u0n , ϕ0n ) + 2c6 |O|] + cν lh t + cν lh + αJ

[Etot (un (t), ϕn (t)) + 2c6 |O|] + ν

∫

t

0

+

In1 (t)

+

(4.26)

In2 (t)

∫

≤ |u0 |2 + 2|J |L1 (Rd ) |ϕ0 |2 + 2 F (ϕ0 (x))dx + 2c6 |O| + cν lh t + In1 (t) + In2 (t) O ∫ ( cJ ) t [Etot (un (s), ϕn (s)) + 2c6 |O|] ds, + cν lh + α 0 where we have used also the fact that, since ϕ0 is supposed to belong to D(B), then we have ϕ0n → ϕ0 in H 2 (O, Rd ) and hence also in L∞ (O, Rd ) (for d = 2, 3). p Now raising both sides to the power 2 ≥ 1, taking supremum over s ∈ [0, t ∧ τnk ] and taking mathematical expectation we have [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2

sup

E

s∈[0,t ∧τnk ]

+ ν p/2 E

[∫

t ∧τnk

]p/2

[∫

∥un (s)∥2 ds

+E

≤ c K˜ + cE

]p/2 |∇µn (s)|2 ds

0

0 p/2

t ∧τnk

(

+ c cν lh +

sup s∈[0,t ∧τnk ]

|In1 (s)|

)p/2

cJ

α

[∫

t ∧τnk

E

]p/2

(4.27)

[Etot (un (s), ϕn (s)) + 2c6 |O|] ds

0

p/2

+ c E|In2 (t ∧ τnk )|

p/2

,

with

˜ = |u0 |2 + 2|J |L1 (Rd ) |ϕ0 |2 + 2 K

∫

F (ϕ0 (x))dx + 2c6 |O| + cν lh T .

(4.28)

O

Next we observe by the Hölder inequality that

(

c cν lh +

(

cJ

α

)p/2

≤ c cν lh +

[∫

t ∧τnk

E

]p/2 [Etot (un (s), ϕn (s)) + 2c6 |O|] ds

0 cJ

α

)p/2

T

p−2 2

t ∧τnk

∫ E

(4.29) p/2

[Etot (un (s), ϕn (s)) + 2c6 |O|]

0

with the constant α = 2c5 − |J |L1 (Rd ) > 0, c5 and c6 given by (H7 ).

ds,

36

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

By the Burkhölder–Davis–Gundy inequality, we have cE

sup s∈[0,t ∧τnk ]

|In1 (s)|

p/2

⏐∫ s ∫ { ⏐p/2 } ⏐ ⏐ 2 − − − 2 ⏐ ⏐ ˜ | u (s ) + P G(s , u (s ) ; y) | − | u (s ) | η ˜ (ds , dy) n n n n ⏐ ⏐ 0 Y s∈[0,t ∧τnk ] [∫ ] p/4 }2 t ∧τnk ∫ { 2 2 ≤ kp E |un (s− ) + P˜ n G(s, un (s− ); y)| − |un (s− )| ν1 (dy)ds .

= cE

sup

0

(4.30)

Y

From (4.16) and the Cauchy–Schwarz inequality we have the following inequalities for all x, h ∈ Gdiv

≤ (q|x|q−2 (x, h) + Cq (|x|q−2 + |h|q−2 )|h|2 )2

(|x + h|q − |x|q )2

≤ (q|x|q−1 |h| + Cq (|x|q−2 + |h|q−2 )|h|2 )2

(4.31)

≤ 2q2 |x|2q−2 |h|2 + 2Cq2 (|x|q−2 + |h|q−2 )2 |h|4 ≤ 2q2 |x|2q−2 |h|2 + 4Cq2 |x|2q−4 |h|4 + 4Cq2 |h|2q . Using (4.31) with q = 2, (2.13) and the Young inequality, we obtain

∫ {

}2 2 2 |un (s− ) + P˜ n G(s, un (s− ); y)| − |un (s− )| ν1 (dy) Y ∫ ∫ 2 4 2 |P˜ n G(s, un (s− ); y)| ν1 (dy) + 8C22 |P˜ n G(s, un (s− ); y)| ν1 (dy) ≤ 8|un (s− )| Y

≤

(4.32)

Y

− 2

− 4

˜ + 8C˜ 2 |un (s )| + 8C˜ 2 |un (s )| +

8C22 C4

− 4

˜ |un (s )|

8C22 C4

4 4 ≤ (8C˜ 4 C22 + 4C˜ 2 ) + (12C˜ 2 + 8C22 C˜ 4 )|un (s− )| ≤ χ1 + χ2 |un (s− )| ,

for all s ∈ [0, T ] and some positive constants χ1 and χ2 . Thus, from this last inequality and some well-known inequalities, we get

[∫

t ∧τnk

∫ {

0

≤2

]p/4

}2 − 2

2

ν1 (dy)ds

|un (s ) + P˜ n G(s, un (s ); y)| − |un (s )| −

−

Y p−2 2

(χ 1 T )

p/4

+2

p−2 2

χ

p/4 2

[∫

(4.33)

]p/4

t ∧τnk

.

4

|un (s)| ds 0

From (4.33), the Hölder, Young inequalities and the Fubini Theorem, we have cE

sup

p/2 In1 (s)

≤2

p−2 2

p−2 2

p/4

kp (χ1 T )

+2 ⎡( p−2 p−2 p/4 ≤ 2 2 kp (χ1 T )p/4 + 2 2 kp χ2 E ⎣ sup |

s∈[0,t ∧τnk ]

|

kp χ

p/4 2 E

[∫

≤2

≤2

p−2 2

p−2 2

kp (χ1 T )p/4 + 2 kp (χ1 T )

p/4

+

p−2 2

1

p/4

kp χ2

E

sup s∈[0,t ∧τnk ]

[

|un (s)| ds

|un (s)|2

)p/4 (∫

E

2

sup

|un (s)|

)p/4 ⎤ ⎦ |un (s)|2 ds

0 t ∧τnk

)p/2 ⎤1/2 ⎦ |un (s)|2 ds

0

+

s∈[0,t ∧τnk ]

t ∧τnk

]1/2 ⎡ (∫ ⎣E |un (s)|p

] p

4

0

s∈[0,t ∧τnk ]

[

]p/4

t ∧τnk

2p−3 k2p

p/2 2 (t

χ

p−2 k 2 n)

∧τ

∫

t ∧τnk

E|un (s)|p ds.

0

Hence cE

sup

p/2 In1 (s)

≤2

∫

t ∧τnk

|

s∈[0,t ∧τnk ]

p/2

+2p−3 k2p χ2 T

|

p−2 2

0

≤2

p−2 2

p/4

kp (χ1 T ) p/2

+2p−3 k2p χ2 T

p−2 2

+

∫

1

p−2 2

kp (χ1 T )

p/4

+

1 2

[ E

] sup

p

|un (s)|

s∈[0,t ∧τnk ]

E [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2 ds

(

2 t ∧τnk

) E

sup

[Etot (un (s), ϕn (s)) + 2c6 |O|]

s∈[0,t ∧τnk ]

E [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2 ds,

0

with Etot defined in (2.15) and c6 given in (H7 ).

p/2

(4.34)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

37

Using (4.17) and the Hölder inequality, we have

|In2 (t ∧ τnk )|

)p/2 + |un (s)|2 )ds [ (∫ k )p/2 ] t ∧τ p/2 ≤ 2p−1 c˜2 (t ∧ τnk )p/2 + 0 n |un (s)|2 ds ] [ p−2 ∫ t ∧τ k p/2 ≤ 2p−1 c˜2 (t ∧ τnk )p/2 + (t ∧ τnk ) 2 0 n |un (s)|p ds .

p/2

p/2

≤ c˜2

t ∧τnk (1 0

(∫

(4.35)

From (4.35) and the Fubini Theorem, we obtain p/2

] [ p−2 ∫ t ∧τ k p/2 ≤ 2p−1 c˜2 c E (t ∧ τnk )p/2 + (t ∧ τnk ) 2 0 n |un (s)|p ds ∫ t ∧τnk p−2 p−1 p/2 p/2 p−1 p/2 2 ≤ 2 c˜2 cT + 2 c˜2 cT E [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2 ds.

c E|In2 (t ∧ τnk )|

(4.36)

0

Now from (4.27), (4.29), (4.34) and (4.36) we obtain after arranging all the terms

E sups∈[0,t ∧τ k ] [Etot (un (s), ϕn (s)) + 2c6 |O|]

p/2

n

[∫

]p/2

t ∧τnk

2

+2E

|∇µn (s)| ds

≤ K˜ 1 + K˜ 2

0

+ 2ν

p/2

[∫

]p/2

t ∧τnk

2

∥un (s)∥ ds

E 0

t ∧τnk

∫

(4.37) p/2

E [Etot (un (s), ϕn (s)) + 2c6 |O|]

ds,

0

˜ 1 and K˜ 2 given by: with the constants K p/2

p

˜ 1 = c K˜ p/2 + 2 2 kp (χ1 T )p/4 + 2p c˜2 cT p/2 , K [ ( ] p−2 c )p/2 p−2 p/2 p−2 p/2 ˜ 2 = c cν lh + J K T 2 + 2p−2 k2p χ2 T 2 + 2p c˜2 cT 2 .

(4.38)

α

In particular, [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2

sup

E

s∈[0,t ∧τnk ]

≤ K˜ 1 + K˜ 2

t ∧τnk

∫

E sup [Etot (un (r), ϕn (r)) + 2c6 |O|]p/2 ds. r ∈[0,s]

0

Using the Gronwall Lemma we infer that for all t ∈ [0, T ]

˜ 1 eK2 (t ∧τn ) ≤ K˜ 1 eK2 T . [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2 ≤ K k

˜

sup

E

˜

(4.39)

s∈[0,t ∧τnk ]

Since the constant on the Right Hand Side of (4.39) is independent of n ∈ N and k > 0, passing to the limit as k → ∞, we obtain

( sup E n∈N

sup [Etot (un (s), ϕn (s)) + 2c6 |O|]p/2

s∈[0,T ]

)

˜

≤ K˜ 1 eK2 T .

(4.40)

From (4.37), (4.40), letting k → ∞ and using also the fact that 1 + xeax ≤ e(a+1)x for all x ≥ 0, we obtain 2ν p/2 sup E n∈N

T

[∫

]p/2 ∥un (s)∥2 ds

T

[∫ +2 sup E n∈N

0

]p/2 |∇µn (s)|2 ds

(4.41)

0

˜ ˜ ≤ K˜ 1 (1 + K˜ 2 TeK2 T ) ≤ K˜ 1 e2K2 T .

■

Estimates for F (ϕn ).

˜˜ > 0 depending only on l , ν , T , u , ϕ and q such that For every n ∈ N and all q ≥ 2, there exists a positive constant K 1 h 0 0

Lemma 4.3.

q/2 ˜˜ . sup E sup |F (ϕn (t))|L1 (O,Rd ) ≤ K 1

(4.42)

t ∈[0,T ]

n∈N

Proof. We first recall that by mean of (H4 ) and (H6 ), it is easy to see that for all t ∈ [0, T ] 1

∫ ∫

2

O

J(x − y)(ϕn (x, t) − ϕn (y, t))2 dxdy + 2 O

∫

F (ϕn (x, t))dx O

∫

√ 2 = | aϕn (t)| + 2 F (ϕn (x, t))dx − (ϕn (t), J ∗ ϕn (t)) ∫ O ∫ ≥ a(x)|ϕn (x, t)|2 dx + 2c5 |ϕn (x, t)|2 dx − 2c6 |O| − |J |L1 (Rd ) |ϕn (t)|2 O

O

≥ (2c5 − |J |L1 (Rd ) )|ϕn (t)|2 − 2c6 |O| = α|ϕn (t)|2 − 2c6 |O|, where

α = 2c5 − |J |L1 (Rd ) > 0, for all d = 2, 3.

(4.43)

38

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68 p/2

(a) We first assume that F (ϕn (x, t)) ≥ 0 and we prove that E supt ∈[0,T ] |F (ϕn (t))|L1 (O,Rd ) < ∞. By (4.26), we have

∫

√ 2 +| aϕn (t)| + 2

|F (ϕn (x, t))|dx − (ϕn (t), J ∗ ϕn (t)) + 2c6 |O| ∫ √ 2 = |un (t)|2 + | aϕn (t)| + 2 F (ϕn (x, t))dx − (ϕn (t), J ∗ ϕn (t)) + 2c6 |O| O ∫ t ( c ) [Etot (un (s), ϕn (s)) + 2c6 |O|] ds, ≤ K˜ + In1 (t) + In2 (t) + cν lh + αJ

|un (t)|2

O

0

˜ given by (4.28). with K In particular, we have for all t ∈ [0, T ] 2|F (ϕn (t))|L1 (O,Rd ) = 2

∫

|F (ϕn (x, t))|dx O

∫ ( cJ ) t [Etot (un (s), ϕn (s)) + 2c6 |O|] ds ≤ (ϕn (t), J ∗ ϕn (t)) + K˜ + In1 (t) + In2 (t) + cν lh + α 0 ∫ ( cJ ) t [Etot (un (s), ϕn (s)) + 2c6 |O|] ds + In1 (t) + In2 (t). ≤ K˜ + |J |L1 (Rd ) |ϕn (t)|2 + cν lh + α 0 Now raising both sides to the power

p 2

(4.44)

≥ 1, taking supremum over s ∈ [0, t ∧ τnk ] and taking mathematical expectation we have

p/2

2p/2 E sups∈[0,t ∧τ k ] |F (ϕn (s))|L1 (O,Rd ) n

[∫

p

(

p 2

≤ c K˜ + c cν lh +

cJ ) 2

α

t ∧τnk

E

] 2p [Etot (un (s), ϕn (s)) + 2c6 |O|] ds

0

p

+c E

sup s∈[0,t ∧τnk ]

p

p/2

|In1 (s)| 2 + c E|In2 (t ∧ τnk )| 2 + c |J |L1 (Rd ) E

p ( p cJ ) 2 ≤ c K˜ 2 + c cν lh + E α

p/2

+c E|In2 (t ∧ τnk )|

[∫

t ∧τnk

sup

|ϕn (s)|p

s∈[0,t ∧τnk ]

(4.45)

] 2p

[Etot (un (s), ϕn (s)) + 2c6 |O|] ds

p

+ cE

s∈[0,t ∧τnk ]

0

p/2

+ c |J |L1 (Rd ) E

sup

sup

|In1 (s)| 2

[Etot (un (s), ϕn (s)) + 2c6 |O|]p/2 .

s∈[0,t ∧τnk ]

By (4.29), (4.34), (4.36), (4.39) and (4.45), we infer that for all t ∈ [0, T ]: sup

E

s∈[0,t ∧τnk ]

p/2 |F (ϕn (s))|L1 (O,Rd ) ≤ K˜ 3

˜ 3 independent of t ∈ [0, T ], k > 0 and n ∈ N, i.e., for some constant K sup E n∈N

sup s∈[0,t ∧τnk ]

p/2

|F (ϕn (s))|L1 (O,Rd ) ≤ K˜ 3 .

Now as the constant on the Right Hand Side is independent of n ∈ N and k > 0, passing to the limit as k → ∞, we obtain p/2 ˜ 3. sup E sup |F (ϕn (s))|L1 (O,Rd ) ≤ K n∈N

s∈[0,T ]

p/2

(b) We assume that F (ϕn (x, t)) ≤ 0 and we prove that E supt ∈[0,T ] |F (ϕn (t))|L1 (O,Rd ) < ∞. In this case, we first observe that for all t ∈ [0, T ], we have: ∫ ∫ ∫ - O F (ϕn (x, t))dx = O −F (ϕn (x, t))dx = O |F (ϕn (x, t))|dx = |F (ϕn (t))|L1 (O,Rd ) . Now from (4.43), we deduce that 2|F (ϕn (t))|L1 (O,Rd )

√ 2 ≤ | aϕn (t)| − (ϕn (t), J ∗ ϕn (t)) + α|ϕn (t)|2 + 2c6 |O| ≤ (|a|L∞ + |J |L1 (Rd ) + α )|ϕn (t)|2 + 2c6 |O|.

Now raising both sides to the power we infer that for all t ∈ [0, T ]:

E

sup s∈[0,t ∧τnk ]

p 2

(4.46)

≥ 1, taking supremum over s ∈ [0, t ∧ τnk ], the mathematical expectation and using also (4.39),

p/2 |F (ϕn (s))|L1 (O,Rd ) ≤ K˜ 4

˜ 4 independent of t ∈ [0, T ], k > 0 and n ∈ N. for some constant K

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

39

Now letting k → ∞ and using also the fact that t ∧ τnk → T as k → ∞, we infer that p/2

˜ 4. sup E sup |F (ϕn (s))|L1 (O,Rd ) ≤ K n∈N

s∈[0,T ]

Hence from (a) and (b), we conclude the proof of Lemma 4.3.

■

Estimates for µn and ρ (., ϕn ).

˜ , K˜˜ 2 and K˜˜ 3 depending on T , p such that For every p ≥ 2 they exist some positive constants K

Corollary 4.1.

¯, sup E sup |ρ (., ϕn (t))|rLr (O,Rd ) ≤ K n∈N

t ∈[0,T ]

sup E sup |ρ (., ϕn (t))| n∈N

t ∈[0,T ]

≤ K˜˜ 2 and sup E

p Lr (O ,Rd )

n∈N

[∫

]p/2

T

∥µn (s)∥

≤ K˜˜ 3 .

2 V ds

(4.47)

0

Here the constant r belongs to [1, 2]. Proof. Applying Lemmas 4.2, 4.3 with p = 2, q = 2 and (H6 ) we easily derive (4.47)1 . Using (H6 ) and the Hölder inequality, we have for all t ∈ [0, T ] p Lr (O ,Rd )

|ρ (., ϕn (t))|

[∫

r

]p/r

|a(x)ϕn (x, t) + F (ϕn (x, t))| dx (∫ )p/r ∫ p(r −1) r r ′ r |a(x)ϕn (x, t)| dx + |F (ϕn (x, t))| dx ≤2 O O ( )p/r p(r −1) 2−r ≤2 r |a|rL∞ |O| r |ϕn (t)|r + c3 |F (ϕn (t))|L1 (O,Rd ) + c4 |O| ′

=

O

≤2

p(r −1) r

×3

p−r r

[|a|pL∞ |ϕn (t)|p |O|

p(2−r) 2r

p

p

p

+ c3r |F (ϕn (t))|Lr1 + (c4 |O|) r ].

Now taking the supremum over [0, T ] and the mathematical expectation, we obtain p

E sup |ρ (., ϕn (t))|Lr (O,Rd ) t ∈[0,T ]

≤ K˜ 5 E sup

(

t ∈[0,T ]

≤

˜5 K α p/2

˜5 = 2 where K

p ) |ϕn (t)|p + |F (ϕn (t))|Lr1 + K˜ 5

(4.48) p r L1 (O ,Rd )

E sup [Etot (un (t), ϕn (t)) + 2c6 |O|]p/2 + K˜ 5 E sup |F (ϕn (t))| t ∈[0,T ]

p(r −1) r

×3

t ∈[0,T ]

p−r r

(

p

max |a|L∞ |O|

p(2−r) 2r

+ K˜ 5 ,

) p p , c3r , (c4 |O|) r .

By (4.40), Lemma 4.3 and (4.48) we derive (4.47)2 . Now, we prove (4.47)3 . We first note that from (H6 ) and the Young inequality, it is easy to see that for all s ∈ [0, T ]

|F ′ (ϕn (s))| ≤

1 r

r

|F ′ (ϕn (s))| +

r −1 r

≤

c3 r

|F (ϕn (s))| +

c4 r

+

r −1 r

.

(4.49)

By (4.49), the Poincaré–Wirtinger inequality (see for instance [35]) and the Young inequality, we infer that

⏐∫ ⏐) ( ⏐ 1 ⏐ ⏐ µn (x, s)dx⏐ = K˜ 6 (|∇µn (s)| + 1 |(µn (s), 1)|) |µn (s)| ≤ K˜ 6 |∇µn (s)| + ⏐ ⏐ |O| ⏐) ⏐∫ |O| O ( ⏐ 1 ⏐ ′ ⏐ ≤ K˜ 6 |∇µn (s)| + |a(x)ϕn (x, s) + F (ϕn (x, s)) − J ∗ ϕn (x, s)|dx⏐⏐ ⏐ |O | O ( ) ∫ ∫ ∫ 1 1 | a| ∞ ≤ K˜ 6 |∇µn (s)| + L |ϕn (x, s)|dx + |F ′ (ϕn (x, s))|dx + |J ∗ ϕn (x, s)|dx |O| O |O| O |O| O ) ( 1 c3 c4 + r − 1 ˜ ≤ K6 |∇µn (s)| + (|a|L∞ + |J |L1 (Rd ) )|ϕn (s)| + |F (ϕn (s))|L1 + . 1 r |O| r |O| 2

(4.50)

From (4.50), we obtain

|µn (s)|2

[ ] ≤ 4K˜ 6 |∇µn (s)|2 + |O1 | (|a|L∞ + |J |L1 (Rd ) )2 |ϕn (s)|2 [ ( )2 ] c32 c4 +r −1 2 + 4K˜ 6 r 2 |O | F ( ϕ (s)) | + . n 1 d r L (O ,R ) |2

(4.51)

40

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Using (4.51), we infer that:

[∫

] 2p

T

∥µn (s)∥

T

[∫

2 V ds

∫

2

] 2p

T 2

|µn (s)| ds + |∇µn (s)| ds 0 ) 2p ) 2p (∫ T (∫ T p−2 p−2 ≤2 2 |µn (s)|2 ds +2 2 |∇µn (s)|2 ds 0 0 ) 2p ( (∫ T )p p p−2 5p−6 c4 + r − 1 2 2 |∇µn (s)| ds + ≤ (2 2 K˜ 6 + 2 2 ) =

0

0

p

+ 22p−2 K˜ 62 p

+

22p−2 c3 r p |O |p

1

|O|

T

p/2

(4.52)

r

0 p

p 2

(|a|L∞ + |J |L1 (Rd ) )p T 2 sup |ϕn (s)|p s∈[0,T ]

p 2

˜ 6 sup |F (ϕn (s))| K s∈[0,T ]

p L1 (O ,Rd )

.

By (4.40), Lemma 4.3 and (4.52), we infer that

] 2p

T

[∫

∥µn (s)∥

sup E n∈N

2 V ds

≤ K˜ 6 .

(4.53)

0

This concludes the proof of Corollary 4.1.

■

Remark 4.1. Since a stochastic integral with respect to the time homogeneous compensated Poisson random measure is defined for all progressively measurable processes, each un satisfies a version of (4.1) with s− replaced by s. 4.3. Step 3. Tightness In this section, we will apply Corollaries 3.1 and 3.2 with q = 2 and we will deduce the tightness of the laws of approximating sequences un and ϕn . So, let us consider the space Z˜2 = Z˜12 × Z˜22 ,

where Z˜12 = L2w (0, T ; Vdiv ) ∩ L2 (0, T ; Gdiv ) ∩ D([0, T ]; Vdi′ v ) ∩ D([0, T ]; Gw div )

(4.54)

and Z˜22 = L2w (0, T ; V ) ∩ L2 (0, T ; H) ∩ C ([0, T ]; V ′ ) ∩ C ([0, T ]; Hw ).

For each n ∈ N, the solutions (un , ϕn ) of the Galerkin equation define measures L(un , ϕn ) on (Z˜2 , T ), where T is the supremum of T˜1 and T˜2 . Using Corollaries 3.1 and 3.2 we will prove that the set of measures {L(un , ϕn ), n ∈ N} is tight on (Z˜2 , T ). Now, we show that all the conditions of Corollaries 3.1 and 3.2 are satisfied. By (2.15)2 , (4.43), Lemma 4.2 and the fact that (a2 + b2 )p/2 ≥ ap + bp for all a, b > 0 and p ≥ 2 we infer that

[ ] ˜ |un (s)|p + |ϕn (s)|p ≤ min(1, α −p/2 )K˜ 1 eK2 T = K˜ p1,T ,

sup E sup n∈N

s∈[0,T ]

(4.55)

˜ p,T independent to t ∈ [0, T ], k > 0 and n ∈ N. In particular from (4.55), for some constant K T

[∫

|un (s)|p ds +

sup E n∈N

0

T

∫ 0

] |ϕn (s)|p ds ≤ T K˜ p1,T = K˜ p2,T .

(4.56)

By (4.3) and (H5 ), we have (µn , −∆ϕn ) = (∇µn , ∇ϕn )

= (∇ϕn , (F ′′ (ϕn ) + a)∇ϕn + ϕn ∇ a − ∇ (J ∗ ϕn )) ≥ c0 |∇ϕn |2 + (∇ϕn , ϕn ∇ a − ∇ (J ∗ ϕn )) ≥ c0 |∇ϕn |2 − 2|∇ J |L1 |ϕn ||∇ϕn | ≥

c0 2

|∇ϕn |2 −

2 c0

|∇ J |2L1 |ϕn |2 .

Hence c0 2

|∇ϕn |2 −

2 c0

|∇ J |2L1 |ϕn |2 ≤ (∇µn , ∇ϕn ) ≤

c0 4

|∇ϕn |2 +

1 c0

|∇µn |2 .

From this last inequality, we obtain

|∇ϕn |2 ≤

8 c02

|ϕn |2 +

4 c02

|∇µn |2 .

(4.57)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

41

From Lemma 4.2 we observe that for p ≥ 2 T

[∫ supn∈N E

0

[∫ ]p/2 + supn∈N E ∥un (s)∥2 ds

T

]p/2 |∇µn (s)|2 ds

0

1

˜ 1 e2K2 T min(1, ν −p/2 )K 2 = K˜ p3,T . ≤

˜

(4.58)

Using (4.56), (4.58) with p = 2 and (4.57), we infer that T

[∫ sup E n∈N

] |∇ϕn (s)|2 ds ≤ K˜ T1 .

(4.59)

0

Again putting p = 2 in (4.56) and using also (4.59), we get T

∫

∥ϕn (s)∥2V ds = sup E

sup E n∈N

n∈N

0

T

[∫

]

˜ T2 . (|ϕn (s)|2 + |∇ϕn (s)|2 )ds ≤ K

(4.60)

0

From (4.56) and (4.58), putting p = 2, we obtain T

∫ sup E

∥un (s)∥

n∈N

0

2 Vdiv ds

T

[∫ = sup E n∈N

]

˜ T3 . (|un (s)| + ∥un (s)∥ )ds ≤ K 2

2

(4.61)

0

Now, since L2 (Ω , P; L∞ (0, T ; Gdiv )) ⊂ L1 (Ω , P; L∞ (0, T ; Gdiv )), from (4.55) with p = 2 we infer that,

˜ T4 . sup E[ sup |un (s)|] ≤ K

(4.62)

s∈[0,T ]

n∈N

Using similar argument as of (4.62), we obtain

˜ T5 . sup E[ sup |ϕn (s)|] ≤ K

(4.63)

s∈[0,T ]

n∈N

Finally from (4.62), (4.61), (4.63) and (4.60), we conclude the first two points (a) and (b) of Corollaries 3.1 and 3.2 are satisfied respectively. Now we prove the following tightness Lemma. The set of measures {L(un , ϕn ), n ∈ N} is tight on (Z˜2 , T ).

Lemma 4.4.

Proof. We have already shown that the first two conditions of Corollaries 3.1 and 3.2 for un and ϕn are satisfied respectively. Now we prove that the sequences (un )n∈N and (ϕn )n∈N satisfy the Aldous condition in the space Vdi′ v and V ′ respectively for tightness.

We will use Lemma 3.2. Let (τn )n∈N be a sequence of stopping times such that 0 ≤ τn ≤ T . By (4.1) and Remark 4.1, we get

∫ t ∫ t ∫ t = u0n − ν Aun (s)ds − P˜ n B0 (un (s), un (s))ds − P˜ n (ϕn (s)∇µn (s))ds 0 0 ∫ t ∫ t 0∫ + P˜ n h(s, un (s))ds + P˜ n G(s, un (s− ); y)η˜ (ds, dy)

un (t)

0

0

(4.64)

Y

=: I1n + I2n (t) + I3n (t) + I4n (t) + I5n (t) + I6n (t), for all t ∈ [0, T ]. Let θ > 0. We will check that each term Iin , i = 1, . . . , 6, satisfies condition (3.10) in Lemma 3.2. It is clear that I1n satisfies condition (3.10) with α = 1 and β = 1.

Now, since A : Vdiv → Vdi′ v and |Av|V ′ ≤ ∥v∥, by the Fubini theorem, the Hölder inequality and (4.61), we have the following div estimates

⏐∫ ⏐ τn +θ

E|I2n (τn + θ ) − I2n (τn )|V ′ = ν E ⏐ div

= νE ≤ νE

∫ (∫

τn +θ

sup

v∈Vdiv ,∥v∥Vdiv ≤1 τn +θ

∫

≤ν E

τn +θ τn

) 21

∥un (s)∥2 ds

1 2

)

′ Vdi v

Aun (s).v (x)ds dx

|Aun (s)|V ′ ds ≤ ν E div

τn

( ∫

τn

O

τn

⏐ ⏐

Aun (s)ds⏐

∫

τn +θ

τn

1

∥un (s)∥ds ≤ νθ 2 E

) 21

T

( ∫

θ ≤ν E

0

∥un (s)∥2Vdiv ds

Thus I2n satisfies condition (3.10) with α = 1 and β =

(∫

1 . 2

τn +θ

τn

)1/2 ∥un (s)∥2 ds

√ 1 θ 2 ≤ ν K˜ T3 . θ 1/2 .

42

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

We consider the term I3n . By (4.55), (4.61), (2.9), using the Fubini theorem and the Hölder inequality, we infer that in the case d = 2

τ + θ) −

I3n ( n

E|

I3n (

⏐∫ τn +θ ( ) ⏐⏐ ⏐ ⏐ ˜ = E⏐ Pn B0 (un (s), un (s)) ds⏐⏐ ′ τn Vdi v ∫ τn +θ ⏐ ⏐ ⏐˜ ⏐ ≤E ⏐Pn B0 (un (s), un (s))⏐ ′ ds Vdiv ∫τn τn +θ |un (s)|∥un (s)∥ds ≤ cE

θ )|V ′

div

τn

1 2

≤ cθ

[

2

] 12 [ ∫

E sup |un (s)|

] 21

T

∥un (s)∥

E

s∈[0,T ]

0

2 Vdiv ds

√ √ 1 ≤ c K˜ 21,T . K˜ T3 .θ 2 .

This means that satisfies condition (3.10) with α = 1 and β = 12 in 2D. But in the case d = 3, from (4.55), (4.61), (2.10), using the Fubini theorem and the Hölder inequality, we obtain I3n

⏐∫ ⏐ = E ⏐⏐

E|I3n (τn + θ ) − I3n (θ )|V ′

div

∫ ≤E

τn +θ τn τn +θ

∫τn

≤ cE

(

) ⏐⏐

P˜ n B0 (un (s), un (s)) ds⏐⏐

⏐ ⏐ ⏐˜ ⏐ ⏐Pn B0 (un (s), un (s))⏐

′ Vdi v

τn +θ

3

1

] 14 [ ∫ [ 1 E ≤ c θ 4 E sup |un (s)|2 s∈[0,T ]

≤ cθ

ds

|un (s)| 2 ∥un (s)∥ 2 ds

τn

1 4

′ Vdi v

[

2

τn +θ τn

] 14 [ ∫

E sup |un (s)|

] 43

T

∥un (s)∥

E

s∈[0,T ]

] 43 ∥un (s)∥2 ds

0

2 Vdiv ds

1

3

1

≤ c(K˜ 21,T ) 4 . (K˜ T3 ) 4 .θ 4 .

Hence satisfies condition (3.10) with α = 1 and β = in 3D . Let us move to the term I4n . For every γ ∈ (0, 23 ], using the embedding H 1 (O, Rd ) ↪→ L2(2−γ )/γ (O, Rd ) and the Gagliardo–Nirenberg inequality in dimension d = 2, respectively, we have 1 4

I3n

|P˜ n (ϕn ∇µn )|V ′

div

≤ c |ϕn |

2−γ L 1−γ

|∇µn |

2(1−γ )/2−γ

≤ c |ϕn |L2

(4.65) γ /2−γ

∥ϕn ∥V

|∇µn |.

By Fubini’s theorem and Hölder’s inequality, we have

⏐∫ ⏐ = E ⏐⏐

2−γ

E|I4n (τn + θ ) − I4n (θ )|V ′

div

⏐2−γ ⏐

τn +θ

P˜ n (ϕn (s)∇µn (s))ds⏐⏐

τn

V′

div )2−γ ⏐ ⏐ ⏐˜ ⏐ ≤E ⏐Pn (ϕn (s)∇µn (s))⏐ ′ ds Vdiv τn ∫ ⏐2−γ τn +θ ⏐ ⏐˜ ⏐ ≤ θ 1−γ E ⏐Pn (ϕn (s)∇µn (s))⏐ ′ ds.

(∫

τn +θ

(4.66)

Vdiv

τn

Now from (4.65), (4.66) and the Hölder inequality, we infer that 2−γ

E|I4n (τn + θ ) − I4n (θ )|V ′

≤ c θ 1−γ c E

∫

τn +θ

γ

|ϕn (s)|2(1−γ ) ∥ϕn (s)∥V |∇µn (s)|2−γ ds ( ) ∫ τn +θ γ ≤ c θ 1−γ E sup |ϕn (s)|2(1−γ ) ∥ϕn (s)∥V |∇µn (s)|2−γ ds s∈[0,T ] τn ⎡ ⎤ (∫ τn +θ ) γ2 (∫ τn +θ ) 2−γ 2 ⎦ ≤ c θ 1−γ E ⎣ sup |ϕn (s)|2(1−γ ) ∥ϕn (s)∥2V ds |∇µn (s)|2 ds div

s∈[0,T ]

τn

τn

[

≤ c θ 1−γ E sup |ϕn (s)|2 s∈[0,T ]

τn

]1−γ [ ∫

T

∥ϕn (s)∥2V ds

E

]γ /2

⎛

T

[∫

|∇µn (s)|2 ds

⎝E

0

] 2−γ γ

⎞ γ2 ⎠ .

0

By (4.55), (4.58), (4.60) and (4.67) we get 2−γ

E|I4n (τn + θ ) − I4n (θ )|V ′

div

γ

≤ c [K˜ 21,T ]1−γ . [K˜ T2 ]. [K˜ 32(2−γ ) ] 2 . θ 1−γ . γ

,T

Thus in the case d = 2, condition (3.10) holds with α = 2 − γ and β = 1 − γ for all γ ∈ (0, 32 ].

(4.67)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

43

For the case d = 3, by using the Sobolev embedding and the Gagliardo–Nirenberg inequality, one has

|P˜ n (ϕn ∇µn )|V ′

≤ c |ϕn |L3 (O,R3 ) |∇µn | 1/2 ≤ c |ϕn |1/2 ∥ϕn ∥V |∇µn |.

div

(4.68)

By (4.68) and similar reasoning as in (4.67), we infer that 4

E|I4n (τn + θ ) − I4n (θ )|V3 ′

div

≤ cθ

1 3

[ E sup |ϕn (s)|

2

]1/3 [ ∫

∥ϕn (s)∥

E

s∈[0,T ]

]1/3 [ (∫

T

2 V ds

T 2

)2 ]1/3

|∇µn (s)| ds

E

.

(4.69)

0

0

By (4.69), (4.55), (4.58), (4.60) and (4.67) we get 4/3 E|I4n (τn + θ ) − I4n (θ )|V ′ ≤ c [K˜ 21,T ]1/3 . [K˜ T2 ]. [K˜ 43,T ]1/3 . θ 1/3 . div

Hence in the case d = 3, condition (3.10) holds with α = 4/3 and β = 1/3. Let us consider the term I5n . By Fubini’s theorem, the Hölder inequality, (H1 ) and (4.56) we get

⏐∫ ⏐

E|I5n (τn + θ ) − I5n (θ )|V ′ = E ⏐⏐ div

∫ ≤E ≤θ

τn +θ

τn +θ

[ ∫ E

τn

ds ≤ θ 2 E

(∫

τn +θ τn

div

′ Vdi v

) 21 ⏐ ⏐2 ⏐˜ ⏐ ⏐Pn h(s, un (s))⏐ ′ ds

] 21 [ ∫ ⏐ ⏐2 1 1 ⏐˜ ⏐ ⏐Pn h(s, un (s))⏐ ′ ds ≤ clh2 θ 2 E Vdiv

τn

⏐ ⏐

P˜ n h(s, un (s))ds⏐⏐

1

V′

τn 1 2

⏐ ⏐ ⏐˜ ⏐ ⏐Pn h(s, un (s))⏐

τn +θ

Vdiv

τn +θ

] 21

(1 + |un (s)| )ds

τn

1 2

2

] [ ∫ T 1 1 1/2 (1 + |un (s)|2 )ds ≤ clh [T + K˜ 22,T ]1/2 . θ 1/2 . ≤ clh2 θ 2 E 0

Thus satisfies condition (3.10) holds with α = 1 and β = 1/2. Let us consider the term I6n . By the Itô isometry, condition (2.13), Eq. (4.55), continuity of the embedding Gdiv ↪→ Vdi′ v , we have I5n

⏐∫ ⏐

2

E|I6n (τn + θ ) − I6n (θ )|V ′ = E ⏐⏐ div

⏐∫ ⏐ ≤ c E ⏐⏐ ≤ c C˜ 2 E

τn +θ τ∫n

∫

τn +θ

τn

Y

τn +θ

⏐2 ⏐

∫

τn

Y

P˜ n G(s, un (s); y)η˜ (ds, dy)⏐⏐

⏐2 ∫ ⏐ ˜Pn G(s, un (s); y)η˜ (ds, dy)⏐ = c E ⏐

τn +θ τn

V′

div ∫ ⏐ ⏐2 ⏐˜ ⏐ ⏐Pn G(s, un (s); y)⏐ ν1 (dy)ds

Y

˜ 21,T ). θ. (1 + |un (s)| )ds ≤ c C˜ 2 (1 + E sup |un (s)|2 ). θ ≤ c C˜ 2 (1 + K 2

s∈[0,T ]

satisfies condition (3.10) holds with α = 2 and β = 1. Hence By Lemma 3.2 the sequence (un )n∈N satisfies the Aldous condition in the space Vdi′ v . From (4.2), we have I6n

ϕn (t)

= ϕ0n −

t

∫

un (s). ∇ϕn (s)ds −

t

∫

0

Aµn (s)ds 0

(4.70)

=: kn1 + kn2 (t) + kn3 (t), for all t ∈ [0, T ]. It is clear that kn1 satisfies condition (3.10) with α = 1 and β = 1. Let us consider the term kn2 . By Fubini’s theorem, the Hölder inequality, the Ladyzhenskaya inequality and the continuous embedding V = H 1 (O, Rd ) ↪→ L4 (O, Rd ), we have

E ⏐kn2 (τn + θ ) − kn2 (τn )⏐V ′

⏐

⏐

⏐∫ τn +θ ⏐ ⏐ ⏐ = E ⏐⏐ un (s).∇ϕn (s)ds⏐⏐ V′ ∫ ττnn +θ ≤E |un (s).∇ϕn (s)|V ′ ds ∫τn τn +θ ≤ cE |un (s)|L4 (O,Rd ) |∇ϕn (s)|ds τn ∫ τn +θ 1 d d ≤ C 4 cE |un (s)|1− 4 ∥un (s)∥ 4 |∇ϕn (s)|ds, τn

for d = 2, 3, where C = 2, 4 for d = 2, 3 respectively. We will prove for d = 2 and 3 separately.

(4.71)

44

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

In the case d = 2, from (4.71), (4.55), (4.59), (4.61) and Hölder’s inequality, we get

E ⏐kn2 (τn + θ ) − kn2 (τn )⏐V ′

⏐

⏐

τn +θ

∫

1

≤ 2 4 cE

1

1

|un (s)| 2 ∥un (s)∥ 2 |∇ϕn (s)|ds

τn

[ ] 14 [ ∫ 1 1 ≤ 2 4 θ 4 c E sup |un (s)|2 E s∈[0,T ]

1 4

[

1 4

≤ 2 θ c E sup |un (s)|

2

1

τn

] 14 [ ∫

|∇ϕn (s)|2 ds

] 21 [ ∫ E

] 21 [ ∫

T 2

|∇ϕn (s)| ds

E

s∈[0,T ]

1

τn +θ

1

τn

] 14 ∥un (s)∥2 ds ] 14

T 2

∥un (s)∥ ds

E

0 1

τn +θ

0

1

≤ 2 4 c [K˜ 21,T ] 4 . [K˜ T1 ] 2 . [K˜ T3 ] 4 . θ 4 . In the case d = 3, using (4.55), (4.59), (4.61) and Hölder’s inequality, we infer that

E ⏐kn2 (τn + θ ) − kn2 (τn )⏐V ′

⏐

⏐

1 2

τn +θ

∫

1

≤ 2 2 cE 1 8

1

3

|un (s)| 4 ∥un (s)∥ 4 |∇ϕn (s)|ds

τn

[

≤ 2 θ c E sup |un (s)|

2

] 18 [ ∫ E

s∈[0,T ]

1 2

1 8

[

≤ 2 θ c E sup |un (s)|

2

1

] 21 [ ∫

2

|∇ϕn (s)| ds

τn

] 18 [ ∫

E

1 2

T 2

1

2

] 38

∥un (s)∥ ds

τn

] 38

2

∥un (s)∥ ds

E

0 1

τn +θ

T

] [ ∫

|∇ϕn (s)| ds

E

s∈[0,T ]

1

τn +θ

0

3

≤ 2 2 c [K˜ 21,T ] 8 . [K˜ T1 ] 2 . [K˜ T3 ] 4 . θ 8 . Hence kn2 satisfies condition (3.10) with α = 1 and β = 1/4 in dimension 2 and α = 1 and β = 3/8 in three dimensional. Finally we consider the term kn3 . By Fubini’s theorem, the Hölder inequality, the fact that |Aψ|V ′ ≤ ∥ψ∥V (for all ψ ∈ HE2 (O)) and Eq. (4.47)2 , we have

⏐∫ ⏐

E ⏐kn3 (τn + θ ) − kn3 (τn )⏐V ′ = E ⏐⏐

⏐

⏐

∫ ≤E

τn +θ τn

[ ∫ 1 ≤θ2 E

1 2

|µn (s)|V ′ ds ≤ θ E T

∥µn (s)∥2V ds

] 12

τn +θ τn

(∫

⏐ ⏐

∫

Aµn (s)ds⏐⏐

≤E

V′

τn +θ τn

)

2 V ds

∥µn (s)∥

1 2

τn +θ τn

≤θ

1 2

|Aµn (s)|V ′ ds

[ ∫ E

] 12

τn +θ τn

∥µn (s)∥

2 V ds

≤ [K˜˜ 3 ]1/2 .θ 1/2 .

0

Hence kn3 satisfies condition (3.10) with α = 1 and β = 1/2. By Lemma 3.2 the sequence (ϕn )n∈N satisfies the Aldous condition in the space V ′ .

■

4.4. Step 4. Application of skorokhod theorem and construction of new processes This section is devoted to the construction of new probability space and processes. The estimates from the previous subsection will play an important role. Also with the use of the Skorokhod Theorem for nonmetric spaces we will deduce the existence of new probability space and new processes. ¯ := N ∪ {∞}. Let (S, T ) be a measurable space. We will denote by Let us denote N := {0, 1, 2, . . . , }, N

¯ valued measures on (S, T ). MN¯ (S) := the set of all N On the set MN¯ (S) we consider the σ -field MN¯ (S) defined as the smallest σ -field such that for all B ∈ S: the map

¯ is measurable. iB : MN¯ (S) ∋ ν1 → ν1 (B) ∈ N By Lemma 4.4 the set of measures {L(un , ϕn ), n ∈ N} is tight on the space (Z˜2 = Z˜12 × Z˜22 , T ), with Z˜12 and Z˜22 given by (4.54). Let η˜ n := η˜ , n ∈ N. Then the set of measures {L(η˜ n ), n ∈ N} is tight on the space MN¯ ([0, T ] × Y ). Hence the set {L(un , ϕn , η˜ n ), n ∈ N} is tight on Z˜2 × MN¯ ([0, T ] × Y ). By Remark B.2 in [19] (applying with H = Gdiv , V = Vdiv , ZT = Z˜2 ) and Theorem 3.7, there exist a subsequence (η˜ k )k∈N , a probability ¯ , F¯ , P¯ ) and, on this space, Z˜2 × MN¯ ([0, T ] × Y )-valued random variables (u¯ , ϕ, ¯ η¯ ), (u¯ k , ϕ¯ k , η¯ k ), k ∈ N such that space (Ω (1) L((u¯ k , ϕ¯ k , η¯ k )) = L((u¯ nk , ϕ¯ nk , η¯ nk )) for all k ∈ N; ¯ , F¯ , P¯ ) as k → ∞; (2) (u¯ k , ϕ¯ k , η¯ k ) → (u¯ , ϕ, ¯ η¯ ) in Z˜2 × MN¯ ([0, T ] × Y ) with probability 1 on (Ω ¯ (3) η¯ k (ω) = η¯ (ω) for all ω ∈ Ω . To simplify our notation, we will denote these sequences again by ((un , ϕn , η˜ ))n∈N and ((u¯ n , ϕ¯ n , η¯ n ))n∈N . Using the definition of the space Z˜2 = Z˜12 × Z˜22 , see (4.54), in particular, we have

¯ u¯ n → u¯ in L2w (0, T ; Vdiv ) ∩ L2 (0, T ; Gdiv ) ∩ D([0, T ]; Vdi′ v ) ∩ D([0, T ]; Gw div ) P − a.s. and ϕ¯ n → ϕ¯ in L2w (0, T ; V ) ∩ L2 (0, T ; H) ∩ C ([0, T ]; V ′ ) ∩ C ([0, T ]; Hw ) P¯ − a.s.

(4.72)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

45

4.5. Step 5. Properties of the new processes and the limiting processes Due to the Kuratowski Theorem, the following result holds:

˜ ˜ Lemma 4.5. The Borel subsets of D([0, T ], Gw div ) are Borel subsets of Z12 and the Borel subsets of C ([0, T ], Hw ) are Borel subsets of Z22 . Hence we have we obtain the following proposition. Proposition 4.1. u¯ n and ϕ¯ n take values in Vdin v and Hn respectively. The laws of un and u¯ n are equal on D([0, T ], Gw div ) and the laws of ϕn and ϕ¯ n are equal on C ([0, T ], Hw ). Since the random variables un and u¯ n are identically distributed, we have the following inequalities. From Lemma 4.2 and for every p ≥ 2, we infer that

¯ sup sup E n∈N

s∈[0,T ]

[ ] |¯un (s)|p ≤ Cp1,T and sup E¯

T

[∫

n∈N

0

]p/2 ≤ Cp2,T . ∥¯un (s)∥2 ds

(4.73)

¯ , P¯ ; From (4.73) and the Banach–Alaoglu theorem, we conclude there exist a subsequence of (u¯ n ) convergent weakly star in Lp (Ω ¯ , P¯ ; L2 (0, T ; Vdiv )). So for p = 2 and (4.72)1 , we infer that L∞ (0, T ; Gdiv )) and weakly in Lp (Ω ¯ sup sup E n∈N

s∈[0,T ]

[ ] |¯u(s)|2 < ∞ and sup E¯

T

[∫

n∈N

] ∥¯u(s)∥2 ds < ∞.

(4.74)

0

From Lemma 4.2, Eqs. (4.55), (4.57) and (4.58), we have T

[∫

∥ϕn (s)∥2V ds

sup E n∈N

]p/2

0

≤ Cp3,T .

(4.75)

Since the random variables ϕn and ϕ¯ n are identically distributed, from Lemma 4.2 applying with p = 2 and (4.60), we have

¯ sup sup E n∈N

s∈[0,T ]

[ ] |ϕ¯ n (s)|2 < Cp4,T and sup E¯

T

[∫

n∈N

0

] ∥ϕ¯ n (s)∥2V ds < Cp5,T .

(4.76)

¯ , P¯ ; From (4.76)1 and Banach–Alaoglu theorem we conclude there exists a subsequence of (ϕ¯ n ) convergent weakly star in L2 (Ω L∞ (0, T ; H)). So from (4.72)2 , we infer that ¯ sup sup E n∈N

s∈[0,T ]

[ ] |ϕ¯ (s)|2 < ∞.

(4.77)

Similarly from (4.76)2 , (4.72)2 and Banach–Alaoglu theorem we conclude there exists a subsequence of (ϕ¯ n ) convergent weakly in ¯ , P¯ ; L2 (0, T ; V )), i.e., L2 ( Ω

¯ sup E n∈N

T

[∫

] ∥ϕ¯ (s)∥2V ds < ∞.

(4.78)

0

Also from Lemma 4.2 and (4.75), we have for p ≥ 2 sup E sup n∈N

s∈[0,T ]

[ ] |ϕ¯ n (s)|p ≤ Cp6,T and sup E

T

[∫

n∈N

∥ϕ¯ n (s)∥2V ds 0

]p/2

≤ Cp3,T .

(4.79)

From (4.47)1 and (4.47)2 in Corollary 4.1 and Banach–Alaoglu theorem we conclude there exist subsequences of (ρ¯ (., ϕ¯ n )) and (µ ¯ n) ¯ , P¯ ; L2 (0, T ; Lr (O, Rd ))) and weakly in µ ¯ , P¯ ; L2 (0, T ; V )). In particular, we have convergent weakly star in ρ¯ in Lp (Ω ¯ in Lp (Ω p

sup E sup |ρ¯ (., ϕ¯ n (t))|Lr (O,Rd ) ≤ Cp7,T and sup E n∈N

t ∈[0,T ]

n∈N

T

[∫

∥µ ¯ n (s)∥2V ds 0

]p/2

≤ Cp8,T .

(4.80)

The following proposition is the essence for the proof of our existence result. Proposition 4.2.

One has:

¯ , P¯ ; L2 (0, T ; Gdiv )), u¯ n → u¯ in L2 (Ω

(4.81)

¯ , P¯ ; L2 (0, T ; H)) ϕ¯ n → ϕ¯ in L2 (Ω

(4.82)

One also has

¯ , P¯ ; L2 (0, T ; V )), J ∗ ϕ¯ n → J ∗ ϕ¯ in L2 (Ω

(4.83)

ρ¯ (., ϕ¯ n ) → ρ¯ := ρ¯ (., ϕ¯ ) = aϕ¯ + F ′ (ϕ¯ ) ∈ L2 (0, T ; V ) P¯ − a.s.,

(4.84)

and hence

¯ , P¯ ; L2 (0, T ; V )). µ ¯ = ρ¯ (., ϕ¯ ) − J ∗ ϕ¯ = aϕ¯ − J ∗ ϕ¯ + F ′ (ϕ¯ ) ∈ L2 (Ω

(4.85)

46

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

For every p ≥ 2, r ∈ (1, 2] and d = 2, 3, we have

] 2p ∥µ ¯ (s)∥2V ds ≤ C¯ p , 0 )p/2 ≤ C¯ p . ∥¯u(t)∥2Vdiv dt

¯ sup |¯u(s)|p ≤ C¯ p , E¯ sup |ϕ¯ (s)|p ≤ C¯ p , E¯ E s∈[0,T ]

s∈[0,T ]

p Lr (O ,Rd )

E sup |ρ¯ (., ϕ¯ (t))| t ∈[0,T ]

≤ C¯ p and E¯

T

(∫ 0

[∫

T

(4.86)

Proof. Due to Lemma 4.2 for p = 4, one has:

¯ sup E

T

[∫

2

]2

|¯un (s)| ds

n∈N

< ∞ and sup E¯ n∈N

0

T

[∫

|ϕ¯ n (s)| ds 2

]2

< ∞.

(4.87)

0

Now let us consider the positive nondecreasing function ψ (x) = x4 , defined on R+ . The function ψ obviously satisfies lim

ψ (x)

= ∞.

x Thanks to (4.87), we have: x→∞

¯ ψ (|¯un |L2 (0,T ;G ) ) < ∞, E div and

¯ ψ (|ϕ¯ n |L2 (0,T ;H) ) < ∞, E which along with the uniform integrability criteria in [36, Chapter 3, Exercise 6] implies that the families {|¯un |L2 (0,T ;Gdiv ) : n ∈ N} and

{|ϕ¯ n |L2 (0,T ;H) : n ∈ N} are uniform integrable with respect to the probability measure.

Thus, we can deduce from Vitali’s Theorem (see, for instance, [36, Chapter 3, Proposition 3.2]), (4.72)1 and (4.72)2 that

¯ , P¯ ; L2 (0, T ; Gdiv )) u¯ n → u¯ in L2 (Ω

(4.88)

and

¯ , P¯ ; L2 (0, T ; H)). ϕ¯ n → ϕ¯ in L2 (Ω

(4.89)

¯ , P¯ ; L (0, T ; H)). The proof of (4.83) follows easily from the fact that ϕ¯ n → ϕ¯ in L (Ω ¯ , P¯ ; L∞ (0, T ; Lr (O, Rd ))) and ϕ¯ n → ϕ¯ Since ρ¯ (., ϕ¯ n ) tends to ρ¯ (., ϕ¯ ) in L2 (Ω in r ¯ , P¯ ; L1 (0, T ; L r −1 (O, Rd ))) and z1 ∈ L2 (Ω ¯ , P¯ ; L2 (0, T ; H)), L2 ( Ω 2

¯ lim E

n→∞

¯ lim E

¯ (ρ (., ϕ¯ n (t)), z(t))dt = E

∫0 T

n→∞

= E¯

T

∫

2

¯ , P¯ ; L2 (0, T ; H)), we have for all z ∈ L2 (Ω

T

∫

(ρ¯ (t), z(t))dt , 0

¯ (ρ¯ (., ϕ¯ n (t)), z1 (t))dt = lim E n→∞

0

T

∫

(aϕ¯ n (t) + F ′ (ϕ¯ n (t)), z1 (t))dt

(4.90)

0

T

∫

(aϕ¯ (t) + F ′ (ϕ¯ (t)), z1 (t))dt . 0 r

¯ , P¯ ; L2 (0, T ; L r −1 (O, Rd ))) in (4.90), we deduce that Taking in particular z = z1 ∈ L2 (Ω ρ¯ = aϕ¯ + F ′ (ϕ¯ ) P¯ − a.s.

(4.91)

¯ × [0, T ]) be the set of ψ ∈ L∞ (Ω ¯ , P¯ ; L∞ (0, T ; Hn )) with Now let DHn (Ω ¯ × [0, T ]; R). ψ = z φ, z ∈ Hn , φ ∈ L∞ (Ω ¯ × [0, T ]) Since µ ¯ n = Pn ρ¯ (., ϕ¯ n ) − Pn (J ∗ ϕ¯ n ), we have for every ψ = z φ ∈ DHn (Ω ¯ E

T

∫

(µ ¯ n (t), z)φ (t)dt = E¯

T

∫

0

(ρ¯ (., ϕ¯ n (t)) − J ∗ ϕ¯ n (t), z)φ (t)dt . 0

¯ , P¯ ; L2 (0, T ; V )) weakly and (4.83) Now letting n → ∞ in the previous equality, using the fact that µ ¯n → µ ¯ in L2 (Ω ¯ E

T

∫

(µ ¯ (t), z)φ (t)dt = E¯ 0

∫

T

(ρ¯ (., ϕ¯ (t)) − J ∗ ϕ¯ (t), z ) φ (t)dt .

(4.92)

0

¯ × [0, T ]) is dense in L2 (Ω ¯ , P¯ ; L2 (0, T ; Lr /(r −1,) (O, Rd ))), we obtain from (4.92) and (4.91) that Therefore, as DHn (Ω µ ¯ = ρ¯ (., ϕ¯ ) − J ∗ ϕ¯ = aϕ¯ + F ′ (ϕ¯ ) − J ∗ ϕ, ¯ i.e. (4.85). ¯ , P¯ ; L2 (0, T ; V )) ⊂ L2 (Ω ¯ , P¯ ; L2 (0, T ; Lr (O, Rd ))). In particular we obtain as elements of the space L2 (Ω ¯ , i.e. (4.84). ρ¯ (., ϕ¯ ) ∈ L2 (0, T ; V ) and a.e. in Ω Also from (4.83)2 , (4.83), and the fact that Pn ∈ L(V , V ), Pn ∈ L(H , H), we have

¯ − a.s., Pn (J ∗ ϕn ) → J ∗ ϕ¯ in L2 (0, T ; V ) P

(4.93)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

47

and

¯ − a.s. Pn ρ¯ (., ϕ¯ n ) → ρ¯ (., ϕ¯ ) in L2 (0, T ; H) P

(4.94)

µ ¯ n = Pn ρ¯ (., ϕ¯ n ) − Pn (J ∗ ϕ¯ n ) → µ ¯ in L2 (0, T ; H) P¯ − a.s.

(4.95)

Thus

Proof of (4.86). ¯ , P¯ ; L∞ (0, T ; Gdiv )). Since the dual space of Using (4.73)1 we infer that the sequence (u¯ n )n∈N is uniformly bounded in Lp (Ω p 1 p ¯ ¯ ∞ p − 1 ¯ ¯ (Ω , P; L (0, T ; Gdiv )), by Banach–Alaoglu theorem, it follows that there exists a subsequence of u¯ n , again L (Ω , P; L (0, T ; Gdiv )) is L ¯ , P¯ ; L∞ (0, T ; Gdiv )) such that u¯ n tends to v weakly star in Lp (Ω ¯ , P¯ ; L∞ (0, T ; Gdiv )). More denoted by the same and there exists v ∈ Lp (Ω p ¯ , P¯ ; L1 (0, T ; Gdiv )), we have precisely, for all φ ∈ L p−1 (Ω

¯ E

T

∫

¯ (u¯ n (t), φ (t))dt → E

T

∫

0

(v (t), φ (t))dt as n → ∞.

(4.96)

0

¯ , P; L2 (0, T ; Vdiv )) and the injection of Vdiv in Gdiv is compact, we infer that Also since u¯ n tends to u¯ in L2 (Ω ¯ E

T

∫

¯ (u¯ n (t), φ (t))dt → E

T

∫

0

¯ , P¯ ; L2 (0, T ; Gdiv )). (u¯ (t), φ (t))dt , for all φ ∈ L2 (Ω

(4.97)

0 p

¯ , P¯ ; L2 (0, T ; Gdiv )) ⊂ L p−1 (Ω ¯ , P¯ ; L1 (0, T ; Gdiv )) in (4.96) and (4.97), we deduce that Now taking in particular φ ∈ L2 (Ω ¯ E

T

∫

¯ (v (t), φ (t))dt = E

T

∫

0

(u¯ (t), φ (t))dt . 0

¯ , P¯ ; L∞ (0, T ; Gdiv )). So (4.86)1 is proved. Hence we have u¯ = v and u¯ ∈ Lp (Ω ¯ , P¯ ; L2 (0, T ; Vdiv )). Since the dual space of Lp (Ω ¯ , P¯ ; L2 (0, T ; Vdiv )) Again from (4.73), the sequence (u¯ n )n∈N is uniformly bounded in Lp (Ω p ¯ , P¯ ; L2 (0, T ; Vdi′ v )), by the Banach–Alaoglu theorem, we deduce that there exists a subsequence of u¯ n and there exists v ∈ is L p−1 (Ω p

¯ , P¯ ; L2 (0, T ; Vdiv )) such that u¯ n → v weakly in Lp (Ω ¯ , P¯ ; L2 (0, T ; Vdiv )); i.e., for all φ ∈ L p−1 (Ω ¯ , P¯ ; L2 (0, T ; Vdi′ v )), we have Lp ( Ω ¯ lim E

T

∫

n→∞

⟨¯un (t), φ (t)⟩ dt = E¯

T

∫

⟨v (t), φ (t)⟩ dt .

(4.98)

0

0

¯ , P¯ ; L2 (0, T ; Vdiv )), we get for all φ ∈ L2 (Ω ¯ , P¯ ; L2 (0, T ; Vdi′ v )) As u¯ n → u¯ in L2 (Ω ¯ lim E

T

∫

n→∞

⟨¯un (t), φ (t)⟩ dt = E¯ 0

T

∫

⟨¯u(t), φ (t)⟩ dt .

(4.99)

0 p

¯ , P¯ ; L2 (0, T ; Vdi′ v )) ⊂ L p−1 (Ω ¯ , P¯ ; L2 (0, T ; Vdi′ v )) in (4.98)–(4.99), we obtain Now taking in particular φ ∈ L2 (Ω ¯ E

T

∫

⟨v (t), φ (t)⟩ dt = E¯

T

∫

0

⟨¯u(t), φ (t)⟩ dt . 0

¯ , P¯ ; L2 (0, T ; Vdiv )); i.e. Thus u¯ = v ∈ Lp (Ω ¯ E

T

(∫

∥¯u(t)∥2Vdiv dt

0

)p/2

≤ C¯ p .

(4.100)

Reasoning similarly as in the proof of (4.86)1 and (4.100), we can check (4.86)2 , (4.86)3 and (4.86)4 . Finally (4.86) is proved and then the proof of Proposition 4.2 is now complete. ■ Let us fix v ∈ Vdiv . Let us denote

∫ t ⟨Au¯ n (s), v⟩ ds Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) := (u¯ n (0), v ) − ν 0 ∫ t⟨ ∫ ∫ ∫ t⟨ ⟩ ⟩ t − P˜ n (B0 (u¯ n ), u¯ n ), v ds − (v.∇ µ ¯ n )ϕ¯ n dxds + P˜ n h(s, u¯ n (s)), v ds 0 ∫0 t ∫ ( ) 0 O ˜ Pn G(s, u¯ n (s); y), v η¯ n (ds, dy), for all t ∈ [0, T ] + 0

(4.101)

Y

and fixing ψ ∈ V , denote

∧n (u¯ n , ϕ¯ n , ψ )(t) :

∫ t ∫ t∫ = (ϕ¯ n (0), ψ ) − (u¯ n .∇ψ )ϕ¯ n dxds (∇ ρ¯ (., ϕ¯ n (s)), ∇ψ) ds + 0 0 O ∫ t∫ + (∇ J ∗ ϕ¯ n ).∇ψ dxds, t ∈ [0, T ]. 0

(4.102)

O

For v ∈ Vdiv , t ∈ [0, T ] we denote by

∫ t ∫ t ⟨Au¯ (s), v⟩ ds − ⟨B0 (u¯ (s), u¯ (s), v )⟩ ds Υ (u¯ , ϕ, ¯ η, ¯ v )(t) := (u¯ (0), v ) − ν 0 ∫ t∫ ∫ t ∫ t ∫0 ⟨h(s, u¯ (s)), v⟩ ds + − (v.∇ µ ¯ )ϕ¯ dxds + (G(s, u¯ (s); y), v )η¯ (ds, dy), 0

O

0

0

Y

(4.103)

48

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

and for ψ ∈ V ,

∧(u¯ , ϕ, ¯ ψ )(t) :

∫ t∫ ∫ t (u¯ .∇ψ )ϕ¯ dxds = (ϕ¯ (0), ψ ) − (∇ ρ¯ (., ϕ¯ (s)), ∇ψ) ds + 0 O 0 ∫ t∫ + (∇ J ∗ ϕ¯ ).∇ψ dxds. 0

(4.104)

O

We will show that (see Proposition 4.3) lim |Υn (u¯ n , ϕ¯ n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|L2−γ (Ω¯ ,P¯ ;L2−γ (0,T ;R)) = 0, γ ∈ (0,

n→∞

2 3

] in the case d = 2

and

(4.105)

lim |Υn (u¯ n , ϕ¯ n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|

4

4

¯ ,P¯ ;L 3 (0,T ;R)) L 3 (Ω

n→∞

= 0, in the case d = 3.

We will also show that (see Proposition 4.4) lim |∧n (u¯ n , ϕ¯ n , ψ ) − ∧(u¯ , ϕ, ¯ ψ )|L2 (Ω¯ ,P¯ ;L2 (0,T ;R)) = 0.

n→∞

(4.106)

From (4.105)1 and Fubini’s Theorem, we infer that

|Υn (u¯ n , ϕ¯ n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|L2−γ (Ω¯ ,P¯ ;L2−γ (0,T ;R)) ∫ T |Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) − Υ (u¯ , ϕ, ¯ η, ¯ v )(t)|2−γ dt = E¯ 0 ) ∫ (∫ T 2−γ |Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) − Υ (u¯ , ϕ, ¯ η, ¯ v )(t)| dt dP¯ (ω¯ ) = ¯ 0 Ω ) ∫ T (∫ |Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) − Υ (u¯ , ϕ, ¯ η, ¯ v )(t)|2−γ dP¯ (ω¯ ) dt = ¯ 0 Ω ∫ T ¯ |Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) − Υ (u¯ , ϕ, = E ¯ η, ¯ v )(t)|2−γ dt .

(4.107)

0

From (4.105)2 and Fubini’s theorem, we infer that

|Υn (u¯ n , ϕ¯ n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|L4/3 (Ω¯ ,P¯ ;L4/3 (0,T ;R)) ∫ T ¯ |Υn (u¯ n , ϕ¯ n , η¯ n , v )(t) − Υ (u¯ , ϕ, E ¯ η, ¯ v )(t)|4/3 dt . = 0

From (4.106) and Fubini’s theorem, we infer that

|∧n (u¯ n , ϕ¯ n , ψ ) − ∧(u¯ , ϕ, ¯ ψ )|L2 (Ω¯ ,P¯ ;L2 (0,T ;R)) ∫ T ¯ |∧n (u¯ n , ϕ¯ n , ψ ) − ∧(u¯ , ϕ, = E ¯ ψ )|2 dt . 0

Hence for proving (4.105) we will show that each term of right hand side of (4.101) tends to the corresponding terms in (4.103) in ¯ , P¯ ; L2−γ (0, T ; R)), L 43 (Ω ¯ , P¯ ; L 43 (0, T ; R)), if d = 2 or d = 3, respectively. Similarly for proving (4.106), we will show that each L2−γ (Ω ¯ , P¯ ; L2 (0, T ; R)). To achieve this we need to prove term of right hand side of (4.102) tends to the corresponding terms in (4.104) in L2 (Ω the following propositions. Proposition 4.3.

|(u¯ n (t) − u¯ (t), v )|2 dt = 0, ∫ T0 ¯ |(u¯ n (0) − u¯ (0), v )|2 dt = 0, limn→∞ 0 E ⏐2 ∫ T ⏐⏐∫ t ¯ ⏐ ⟨Au¯ n (s) − Au¯ (s), v⟩ ds⏐⏐ dt = 0, limn→∞ 0 E 0 ⟩ ⏐2 ∫ T ⏐⏐∫ t ⟨ ⏐ ¯ limn→∞ 0 E ⏐ 0 P˜ n (B0 (u¯ n (s))), (u¯ n (s)) − B0 (u¯ (s), u¯ (s)), v ds⏐ dt = 0, ⏐ ⏐ ⟩ ⟨ ∫ T ⏐∫ t ⏐2 ¯⏐ limn→∞ 0 E P˜ n h(s, u¯ n (s)) − h(s, u¯ (s)), v ds⏐ dt = 0, 0 ⏐2−γ ∫ T ⏐⏐∫ t ∫ ⏐ ¯⏐ [(v.∇ µ limn→∞ 0 E ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐ dt = 0, 0 O 4 ⏐ ⏐ ∫ T ⏐∫ t ∫ ⏐3 ¯⏐ [(v.∇ µ limn→∞ 0 E ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐ dt = 0, 0 O ⏐2 ∫ T ⏐⏐∫ t ∫ ⏐ ¯⏐ limn→∞ 0 E (P˜ n G(s, u¯ n ; y) − G(s, u¯ ; y), v )η¯ (ds, dy)⏐ dt = 0. 0 Y

¯ (1) limn→∞ E (2) (3) (4) (5) (6) (7) (8)

For v ∈ Vdiv

∫T

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

49

Proof. Proof of (1) We have

|(u¯ n (.) − u¯ (.), v )|2L2 (Ω¯ ,P¯ ;L2 (0,T ;R))

= E¯

T

∫

|(u¯ n (t) − u¯ (t), v )|2 dt

∫0 T

|⟨¯un (t) − u¯ (t), v⟩ |2 dt 0 ∫ T |¯un (t) − u¯ (t)|2V ′ dt . ≤ ∥v∥2Vdiv E¯

= E¯

(4.108)

div

0

¯ -a.s. and the embedding Gdiv ↪→ Vdi′ v is continuous, by By (4.54)1 , u¯ n → u¯ in D([0, T ]; Vdi′ v ) and from (4.73)1 , sups∈[0,T ] |¯un (s)|2 < ∞, P the dominated convergence theorem we infer that u¯ n → u¯ in L2 (0, T ; Vdi′ v ). Hence T

∫

|(u¯ n (t) − u¯ (t), v )|2 dt → 0.

(4.109)

0

Moreover, by (4.73)1 , (4.86)1 and the Hölder inequality, for every q/2 ≥ 1 and every n ∈ N, we have

⏐q/2 ⏐ |¯un (t) − u¯ (t)|2 dt ⏐⏐

T

⏐∫ ⏐

¯⏐ E ⏐

0

≤ c E¯ ≤ c˜ E¯

T

∫

( ) |¯un (t)|q + |¯u(t)|q dt

[0

q

sup |¯un (t)| + sup |¯u(t)|

t ∈[0,T ]

t ∈[0,T ]

q

]

(4.110)

< ∞,

for some constant c˜ > 0. Hence from (4.109), (4.110) and Vitali’s theorem we have

¯ lim E

T

∫

n→∞

|(u¯ n (t) − u¯ (t), v )|2 dt = 0, i.e. (1) holds. 0

¯ ¯ ¯ ¯ ¯ Proof of (2). From (4.54)1 , u¯ n → u¯ in D([0, T ]; Gw div ) P-a.s. and u is right continuous as t = 0. Thus we get (un (0), v ) → (u(0), v ) P-a.s. Now from (4.73)1 and Vitali’s theorem, we infer that ¯ lim E

T

∫

n→∞

|(u¯ n (0) − u¯ (0), v )|2 dt = 0, i.e. (2) holds. 0

¯ -a.s., so using also the fact that Au = ((u, .)), u ∈ Vdiv we obtain for all v ∈ Vdiv , Proof of (3). By (4.54)1 u¯ n → u¯ in L2w (0, T ; Vdiv ) P t ∈ [0, T ] t

∫

⟨Au¯ n (s), v⟩ ds

lim

n→∞

0

∫ t ((u¯ n (s), v ))ds = lim n→∞ 0 ∫ t ∫ t ⟨Au¯ (s), v⟩ ds. ((u¯ (s), v ))ds = = 0

(4.111)

0

Using (4.73)2 , the fact that Au = ((u, .)), u ∈ Vdiv and Hölder’s inequality we get for all n ∈ N, t ∈ [0, T ] and q/2 > 1,

⏐∫ t ⏐ 2q ⏐∫ t ⏐ 2q (∫ t ) 2q ⏐ ⏐ ⏐ ⏐ ¯E ⏐ ⟨Au¯ n (s), v⟩ ds⏐ = E¯ ⏐ ((u¯ n (s), v ))ds⏐ ≤ E¯ ∥¯un (s)∥∥v∥Vdiv ds ⏐ ⏐ ⏐ ⏐ 0 0 0 q (∫ ) (∫ ) 4q t t q q 2 ≤ ∥v∥V2div E¯ ∥¯un (s)∥ds ≤ c ∥v∥V2div E¯ ∥¯un (s)∥2 ds 0

0

[ (∫ [ (∫ ) 2q ] 21 t q q 2 2 2 ≤ c ∥v∥Vdiv E¯ ≤ c˜ ∥v∥Vdiv E¯ ∥¯un (s)∥ ds 0

1 ) ]2 q 2

T

∥¯un (s)∥2 ds

(4.112)

< ∞,

0

for some constant c > 0 and c˜ > 0. Hence from (4.111), (4.112) and Vitali’s theorem we infer that for all t ∈ [0, T ]

⏐∫ ⏐

¯⏐ lim E ⏐ n→∞

0

t

⏐2 ⏐ ⟨Au¯ n (s) − Au¯ (s), v⟩ ds⏐⏐ = 0.

¯ From the fact that supn∈N E T

∫ lim

n→∞

0

[∫

T 0

] ∥¯un (s)∥2 ds ≤ C22,T and the dominated convergence theorem, we conclude that for all t ∈ [0, T ]

⏐∫ t ⏐2 ⏐ ⏐ ¯E ⏐ ⟨Au¯ n (s) − Au¯ (s), v⟩ ds⏐ dt = 0, i.e. (3) holds. ⏐ ⏐ 0

Proof of (4). Now we consider the nonlinear terms.

50

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Using the property of b, we have for all t ∈ [0, T ] and n ∈ N

∫ t⟨

⟩

P˜ n B0 (u¯ n , u¯ n ) − B0 (u¯ , u¯ ), v ds

0

= =

∫ t⟨

⟩

∫

∫0 t ⟨

⟩

∫0 t

B0 (u¯ n , u¯ n ), (P˜ n − I)v ds +

B0 (u¯ n , u¯ n ), (P˜ n − I)v ds −

∫ 0t −

t

⟨B0 (u¯ n , u¯ n ) − B0 (u¯ , u¯ ), v⟩ ds (4.113)

⟨B0 (u¯ n − u¯ , v ), u¯ n ⟩ ds

0

⟨B0 (u¯ , v ), u¯ n − u¯ ⟩ ds = I1n (t) + I2n (t) + I3n (t).

0

Using the fact that the embedding Vdiv ↪→ Gdiv is continuous and the properties of b we have for all t ∈ [0, T ]

⏐∫ t ⟨ ∫ t ⟩ ⏐⏐ ⏐ |I1n (t)| = ⏐⏐ B0 (u¯ n , u¯ n ), (P˜ n − I)v ds⏐⏐ ≤ c ∥(P˜ n − I)v∥Vdiv |¯un (s)|∥¯un (s)∥ds 0 0 ∫ t ∫ T ≤ c˜ ∥(P˜ n − I)v∥Vdiv ∥¯un (s)∥2 ds ≤ c˜ ∥(P˜ n − I)v∥Vdiv ∥¯un (s)∥2 ds. 0

0

¯ -a.s., see Eq. (4.72)1 , we infer that Hence by Lemma 4.1, using the fact that u¯ n → u¯ weakly in Lw (0, T ; Vdiv ) P 2

¯ − a.s., as n → ∞. I1n (t) → 0 P Let us move to the term I2n (t). By the properties of b, the fact the embedding Vdiv ↪→ Gdiv is continuous and the Hölder inequality, we obtain for all t ∈ [0, T ], n ∈ N and d = 2

⏐∫ t ⏐ ⏐ ⏐ |I2n (t)| = ⏐⏐ ⟨B0 (u¯ n − u¯ , v ), u¯ n ⟩ ds⏐⏐ 0 ∫ t 1 1 1 1 |¯un (s) − u¯ (s)| 2 ∥¯un (s) − u¯ (s)∥ 2 ∥v∥|¯un (s)| 2 |¯un (s)| 2 ds ≤c 0 (∫ t ) 14 (∫ t ) 21 (∫ t ) 14 2 2 2 |¯un (s) − u¯ (s)| ds ≤ c ∥v∥Vdiv ∥¯un (s)∥ ds ∥¯un (s) − u¯ (s)∥ ds 0

0

1 2 L2 (0,T ;G

≤ c˜ ∥v∥Vdiv |¯un − u¯ |

T

(∫

2

0

) 12 (∫

T

∥¯un (s)∥ ds

div )

∫

2

T 2

∥¯un (s)∥ ds +

0

(4.114)

) 41

∥¯u(s)∥ ds

0

.

0

Thus from (4.72)1 , (4.73)2 and (4.74)2 , we infer that

¯ − a.s., as n → ∞. I2n (t) → 0 P Again by the properties of b, the fact the embedding Vdiv ↪→ Gdiv is continuous and the Hölder inequality, we obtain for all t ∈ [0, T ], n ∈ N and d = 3

⏐∫ t ⏐ ⏐ ⏐ ⏐ |I2n (t)| = ⏐ ⟨B0 (u¯ n − u¯ , v ), u¯ n ⟩ ds⏐⏐ 0 ∫ t 3 1 1 3 ≤c |¯un (s) − u¯ (s)| 4 ∥¯un (s) − u¯ (s)∥ 4 ∥v∥|¯un (s)| 4 |¯un (s)| 4 ds 0 (∫ t ) 18 (∫ t ) 21 (∫ t ) 38 ≤ c ∥v∥Vdiv |¯un (s) − u¯ (s)|2 ds ∥¯un (s)∥2 ds ∥¯un (s) − u¯ (s)∥2 ds 0

0

1 4 L2 (0,T ;G

≤ c˜ ∥v∥Vdiv |¯un − u¯ |

T

(∫ div )

2

0

) 12 (∫

T

∥¯un (s)∥2 ds +

∥¯un (s)∥ ds 0

∫

0

T

∥¯u(s)∥2 ds

(4.115)

) 83

.

0

Hence by (4.72)1 , (4.73)2 and (4.74)2 , we infer that

¯ − a.s., as n → ∞. I2n (t) → 0 P Let us consider the term I3n (t). Proceeding similarly as in (4.114), we see that in the case d = 2

|I3n (t)|

⏐∫ t ⏐ ⏐ ⏐ = ⏐⏐ ⟨B0 (u¯ , v ), u¯ n − u¯ ⟩ ds⏐⏐ 0

1 2 L2 (0,T ;G

≤ c˜ ∥v∥Vdiv |¯un − u¯ |

T

(∫ div )

2

T 2

∥¯u(s)∥ ds 0

for some positive constant c˜ > 0. Hence from (4.72)1 , (4.73)2 and (4.74)2 , we infer that

¯ − a.s., as n → ∞. I3n (t) → 0 P

) 21 (∫

T

∫

2

∥¯un (s)∥ ds + 0

) 41

∥¯u(s)∥ ds 0

,

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

51

Again arguing as in (4.115), we obtain in the case d = 3

⏐∫ t ⏐ ⏐ ⏐ |I3n (t)| = ⏐⏐ ⟨B0 (u¯ , v ), u¯ n − u¯ ⟩ ds⏐⏐ 0

≤ c˜ ∥v∥Vdiv |¯un − u¯ |

1 4 L2 (0,T ;G

T

(∫

2

) 12 (∫

T 2

T

∥¯un (s)∥ ds +

∥¯u(s)∥ ds

div )

∫

0

0

2

) 83

∥¯u(s)∥ ds

,

0

for some positive constant c˜ > 0. Hence from (4.72)1 , (4.73)2 and (4.74)2 , we infer that

¯ − a.s., as n → ∞. I3n (t) → 0 P ¯ -a.s. Now collecting all the previous convergence, using (4.113), we obtain P

∫ t ⟩ ˜Pn B0 (u¯ n (s), u¯ n (s)), v ds = ⟨B0 (u¯ (s), u¯ (s)), v⟩ ds,

∫ t⟨

lim

n→∞

0

(4.116)

0

for all v ∈ Vdiv , t ∈ [0, T ] and d = 2, 3. By (2.9), (2.10), (4.73)1 , (4.73)2 , the fact that P˜ n ∈ L(Vdi′ v , Vdiv ), we obtain for all t ∈ [0, T ], q/2 > 1 and n ∈ N

⏐∫ t ⟨ (∫ t ⏐ ) 2q ⏐ ⟩ ⏐⏐ 2q ⏐ ⏐˜ ⏐ ˜Pn B0 (u¯ n (s), u¯ n (s)), v ds⏐ ≤ c E¯ ¯E ⏐ ⏐Pn B0 (u¯ n (s), u¯ n (s))⏐ ′ ∥v∥Vdiv ds ⏐ ⏐ Vdiv 0 0 q ) ) 2q (∫ (∫ T t q q 2 2 2 2 2 ¯ ¯ ≤ c˜ ∥v∥Vdiv E ≤ c˜ ∥v∥Vdiv E ∥¯un (s)∥Vdiv ds ∥¯un (s)∥Vdiv ds < ∞. 0

(4.117)

0

Hence from (4.116), (4.117) and using Vitali’s theorem we get for all t ∈ [0, T ], v ∈ Vdiv

⏐∫ t ⟨ ⏐

¯⏐ lim E ⏐ n→∞

0

⟩ ⏐⏐2

P˜ n B0 (u¯ n (s), u¯ n (s)) − B0 (u¯ (s), u¯ (s)), v ds⏐⏐ = 0, d = 2, 3.

Hence from (4.118) and the dominated convergence theorem, we infer that T

∫ n→∞

0

⟩ ⏐⏐2

⏐∫ t ⟨ ⏐

¯⏐ E ⏐

lim

0

P˜ n B0 (u¯ n (s), u¯ n (s)) − B0 (u¯ (s), u¯ (s)), v ds⏐⏐ dt = 0, i.e. (4) holds.

Proof of (5). For every t ∈ [0, T ], n ∈ N and v ∈ Vdiv , we have

∫ t⟨

⟩

P˜ n h(s, u¯ n (s)) − h(s, u¯ (s)), v ds

0

=

∫ t⟨

⟩

h(s, u¯ n (s)), (P˜ n − I)v ds +

0

t

∫

⟨h(s, u¯ n (s)) − h(s, u¯ (s)), v⟩ ds 0

= I˜1n (t) + I˜2n (t). Now we prove that I˜1n (t) → 0, I˜2n (t) → 0 P-a.s., respectively. Let us consider the term I˜1n (t). By (H1 ) and the Hölder inequality, we have

|I˜1n (t)|

≤ ∥(P˜ n − I)v∥Vdiv

∫

t

|h(s, u¯ n (s))|V ′ ds div (∫ t ) 12 1 ≤ ∥(P˜ n − I)v∥Vdiv t 2 |h(s, u¯ n (s))|2V ′ ds div 0 ( ) 12 ∫ t 1 2 ˜ ≤ ∥(Pn − I)v∥Vdiv t 2 tlh + lh |¯un (s)| ds 0 ( ) 21 ∫ t 1 2 ˜ ∥¯un (s)∥ ds . ≤ ∥(Pn − I)v∥Vdiv t 2 tlh + lh c 0

0

Hence by Lemma 4.1 and Eq. (4.73)2 , we infer that I˜1n (t) → 0 P − a.s., (C1).

¯ -a.s., using (H1 ), we infer that From (4.72)1 , u¯ n → u¯ in D([0, T ]; Vdi′ v ) P lim I˜2n (t) = lim

n→∞

n→∞

∫ 0

t

⟨h(s, u¯ n (s)) − h(s, u¯ (s)), v⟩ ds = 0 P¯ − a.s., (C2).

(4.118)

52

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

By (C1) and (C2), we have for all t ∈ [0, T ], v ∈ Vdiv

∫ t⟨

lim

⟩

P˜ n h(s, u¯ n (s)), v ds =

n→∞

t

∫

0

⟨h(s, u¯ (s)), v⟩ ds P¯ − a.s.

(4.119)

0

By (4.73)2 , (H1 ), the fact that |u| ≤ ∥u∥, for all u ∈ Vdiv , using the Hölder inequality, we obtain for all t ∈ [0, T ], q/2 > 1 and n ∈ N

⏐∫ t ⟨ ) 2q (∫ t ⟩ ⏐⏐ 2q q ⏐ 2 ⏐ ⏐ ˜ ˜ ¯ ¯ |Pn h(s, u¯ n (s))|V ′ ds E Pn h(s, u¯ n (s)), v ds⏐ ≤ ∥v∥V E ⏐ div div 0 0 [ (∫ q (∫ ) ) 2q ] 12 t t q q 4 2 2 |P˜ n h(s, u¯ n (s))|V ′ ds |P˜ n h(s, u¯ n (s))|V ′ ds ≤ c ∥v∥V2div E¯ ≤ c ∥v∥V2div E¯ div

0

div

0

(4.120)

[ (∫ [ ) 2q ] 12 ) 2q ] 12 (∫ t t q q q q 2 2 2 2 2 lh (1 + |¯un (s)| )ds ∥¯un (s)∥ ds ≤ c ∥v∥Vdiv E¯ ≤ c˜ ∥v∥Vdiv (tlh ) 2 + lh E¯ 0

0

1

[

q

q

q

≤ c˜ ∥v∥V2div (Tlh ) 2 + lh2 E¯

) 2q ] 2

T

(∫

∥¯un (s)∥2 ds

< ∞.

0

Hence from (4.119), (4.120) and using Vitali’s theorem we get for all t ∈ [0, T ], v ∈ Vdiv

⏐∫ t ⟨ ⟩ ⏐⏐2 ⏐ ⏐ ˜ ¯ ¯ ¯ lim E ⏐ Pn h(s, un (s)) − h(s, u(s)), v ds⏐⏐ = 0. n→∞

(4.121)

0

So from (4.121) and the dominated convergence theorem, we infer that T

∫ lim

n→∞

0

⏐∫ t ⟨ ⟩ ⏐⏐2 ⏐ ˜Pn h(s, u¯ n (s)) − h(s, u¯ (s)), v ds⏐ dt = 0, i.e. (5) holds. ¯E ⏐ ⏐ ⏐

(4.122)

0

Proof of (6), (7). Now let us consider (6). Let m > 2d + 1. Let us first assume that v ∈ J˜m . By integration by parts, the fact that v = 0 on ∂ O and div v = 0, we have for all t ∈ [0, T ], n ∈ N

∫ t∫

∫ t∫

[v.∇ µ] ¯ ϕ¯ dxds [v.∇ µ ¯ n ]ϕ¯ n dxds − 0 O ∫ t∫ ∫ O t ∫ [v.∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds [v.∇ (µ ¯n −µ ¯ )]ϕ¯ n dxds + = 0 ∫ O∫ 0 ∫ O∫ t t [v.∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds. [v.∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + =− 0

O

0

O

0

Now by (2.5) and the Hölder inequality, we have

⏐ ⏐∫ t ∫ ∫ t∫ ⏐ ⏐ ⏐ [v.∇ µ] ¯ ϕ¯ dxds⏐⏐ [v.∇ µ ¯ n ]ϕ¯ n dxds − ⏐ 0 O 0 O ⏐∫ t ∫ ⏐ ⏐∫ t ∫ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≤⏐ [v.∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds⏐ + ⏐ [v.∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ O

0

O

0

t

∫

|∇ ϕ¯ n ∥µ ¯ n − µ| ¯ ds + |v|L∞

≤ |v|L∞

t

∫

0

|∇ µ∥ ¯ ϕ¯ n − ϕ| ¯ ds

(4.123)

0

∫ ≤ c ∥v∥J˜m

t

|∇ ϕ¯ n ∥µ ¯ n − µ| ¯ ds + c ∥v∥J˜m

⎡0 (∫ ≤ c ∥v∥J˜m ⎣

T

∥ϕ¯ n ∥2V ds

) 21 (∫

0

t

∫

|∇ µ∥ ¯ ϕ¯ n − ϕ| ¯ ds 0

) 21

T

|µ ¯ n − µ| ¯ 2 ds

T

(∫

∥µ∥ ¯ 2V ds

+

0

) 21 (∫

0

T

|ϕ¯ n − ϕ| ¯ 2 ds

) 21

⎤ ⎦.

0

By (4.72)2 and (4.95), ϕ¯ n → ϕ¯ , µ ¯n → µ ¯ in L2 (0, T ; H) P¯ -a.s., respectively, using also (4.76)2 , (4.86)3 and (4.123), we infer that for all ˜ v ∈ Jm , t ∈ [0, T ]

∫ t∫

[v.∇ µ ¯ n ]ϕ¯ n dxds =

lim

n→∞

0

O

∫ t∫ 0

[v.∇ µ] ¯ ϕ¯ dxds. O

Let now v ∈ Vdiv and let ϵ > 0. Since J˜m is dense in Vdiv , there exists vϵ ∈ J˜m such that ∥v − vϵ ∥Vdiv < ϵ .

(4.124)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

53

We have

∫ t∫

∫ t∫ [v.∇ µ ¯ n ]ϕ¯ n dxds − [v.∇ µ] ¯ ϕ¯ dxds 0 0 O ∫O t ∫ ∫ t∫ =− [v.∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [v.∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds 0 O 0 O ∫ t∫ ∫ t∫ =− [(v − vϵ ).∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [(v − vϵ ).∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds 0 O ∫ t ∫0 O ∫ t∫ + [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds, O

0

O

0

which implies that

⏐∫ t ∫ ⏐ ∫ t∫ ⏐ ⏐ ⏐ [v.∇ µ ¯ n ]ϕ¯ n dxds − [v.∇ µ] ¯ ϕ¯ dxds⏐⏐ ⏐ 0 0 O ⏐∫ O ⏐ ⏐∫ t ∫ ⏐ ∫ ⏐ t ⏐ ⏐ ⏐ ≤ ⏐⏐ [(v − vϵ ).∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds⏐⏐ + ⏐⏐ [(v − vϵ ).∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ 0 O ⏐∫ 0 t ∫ O ⏐ ∫ t∫ ⏐ ⏐ ⏐ +⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ 0 0 O O ∫ t ∫ t ¯ )|J˜m′ ds ¯n −µ ¯ )∇ ϕ¯ n )|J˜m′ ds + ∥v − vϵ ∥J˜m |P˜ n ((ϕ¯ n − ϕ¯ )∇ µ ∥v − vϵ ∥J˜m |P˜ n ((µ ≤ 0 0 ⏐ ⏐∫ t ∫ ∫ t∫ ⏐ ⏐ [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + + ⏐⏐ 0 O 0 O ∫ t ∫ t ¯ µ| ¯ ds ∥v − vϵ ∥J˜m |ϕ¯ n − ϕ∥∇ ¯ n − µ∥∇ ¯ ϕ¯ n |ds + c ∥v − vϵ ∥J˜m |µ ≤c ⏐∫ 0t ∫ ⏐ ∫ t∫0 ⏐ ⏐ + ⏐⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ 0 O 0 O ∫ t ∫ t |ϕ¯ n − ϕ∥∇ ¯ µ| ¯ ds |µ ¯ n − µ∥∇ ¯ ϕ¯ n |ds + c ϵ ≤ cϵ 0 ⏐ ⏐∫ t 0∫ ∫ t∫ ⏐ ⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ . + ⏐⏐ O

0

O

0

Using this previous inequality and the Hölder inequality, we infer that for all t ∈ [0, T ]

⏐ ⏐∫ t ∫ ∫ t∫ ⏐ ⏐ ⏐ [v.∇ µ] ¯ ϕ¯ dxds⏐⏐ [v.∇ µ ¯ n ]ϕ¯ n dxds − ⏐ 0 O 0 O ) 12 ) 12 (∫ T ) 12 (∫ T ) 21 (∫ T (∫ T 2 2 2 2 |∇ ϕ¯ n | ds |ϕ¯ n − ϕ| ¯ ds |∇ µ| ¯ ds + cϵ |µ ¯ n − µ| ¯ ds ≤ cϵ 0 0 0 ⏐∫ t ∫ 0 ⏐ ∫ t∫ ⏐ ⏐ + ⏐⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ O T

0

(∫

) 21 (∫

O

0

) 21

T

(∫

T

≤ cϵ |µ ¯ n − µ| ¯ ds ∥ϕ¯ ∥ + cϵ |ϕ¯ n − ϕ| ¯ ds 0 0 ⏐∫ t ∫ 0 ⏐ ∫ t∫ ⏐ ⏐ + ⏐⏐ [vϵ .∇ ϕ¯ n ](µ ¯n −µ ¯ )dxds + [vϵ .∇ µ] ¯ (ϕ¯ n − ϕ¯ )dxds⏐⏐ . 2

2 n V ds

O

0

2

) 21 (∫

T

∥µ∥ ¯

) 12

2 V ds

0

O

0

Passing to the upper limit as n → ∞ in the above inequality, by (4.124), (4.95), (4.76)2 , (4.72)2 and (4.86)3 we obtain

[∫ t ∫

[v.∇ µ ¯ n ]ϕ¯ n dxds −

lim sup

n→∞

0

∫ t∫

O

0

] [v.∇ µ] ¯ ϕ¯ dxds ≤ c˜ ϵ.

O

Since ϵ > 0 was chosen in an arbitrary way, we deduce that for all v ∈ Vdiv , t ∈ [0, T ]

∫ t∫

[v.∇ µ ¯ n ]ϕ¯ n dxds =

lim

n→∞

Let m >

d 2

0

O

∫ t∫ 0

+ 1. We assume that v ∈ J˜m .

[v.∇ µ] ¯ ϕ¯ dxds, P¯ − a.s. O

(4.125)

54

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

By (2.5), (4.79)1 , (4.80)2 , using the Hölder inequality, we infer that for all t ∈ [0, T ], n ∈ N and q/2 > 1

⏐∫ t ∫ ⏐ 2q ⏐ ⏐ ¯E ⏐ (v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐ ⏐

≤ |v|L∞ E¯

O

0

t

(∫

q 2

|∇ µ ¯ n ||ϕ¯ n |ds

) 2q

0

[ (∫ ) 2q ] 12 [ (∫ t ) 2q ] 12 t q 2 2 2 ¯ |∇ µ ¯ n | ds ≤ c ∥v∥J˜ E¯ E |ϕ¯ n | ds m

0

0

1

[ (∫ q 2 ≤ c˜ ∥v∥J˜ E¯ m

T

∥µ ¯ ∥

2 n V ds

) 2q ] 2 [

¯ sup |ϕ¯ n (s)|q E

] 12

s∈[0,T ]

0

< ∞.

Hence since J˜m is dense is Vdiv , we deduce that

⏐ 2q ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

O

0

< ∞, for all v ∈ Vdiv t ∈ [0, T ] and n ∈ N.

(v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐

(4.126)

Now reasoning similarly as in (4.67), we have in the case d = 2 that for all γ ∈ (0, 23 ], n ∈ N and t ∈ [0, T ]

⏐∫ t ∫ ⏐2−γ ⏐ ⏐ ⏐ ¯ E (v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐ ⏐ 0 O (∫ t )2−γ 2−γ ≤ ∥v∥Vdiv E¯ |P˜ n (ϕ¯ n (s)∇ µ ¯ n (s))|V ′ ds div (∫0 T )2−γ 2−γ ˜ ¯ |Pn (ϕ¯ n (s)∇ µ ¯ n (s))|V ′ ds ≤ ∥v∥Vdiv E div 0

[

≤ c E sup |ϕ¯ n (s)|2

]1−γ [ ∫

∥ϕ¯ n (s)∥2V ds

E

s∈[0,T ]

]γ /2

T

⎛

∥µ ¯ n (s)∥2V ds

⎝E

0

] 2−γ γ

T

[∫

⎞ γ2 ⎠ .

0

Hence by (4.76)1 , (4.76)2 , (4.80)2 and the previous inequality, we infer that for all t ∈ [0, T ], n ∈ N

⏐2−γ ⏐∫ t ∫ ⏐ ⏐ ⏐ ¯ < ∞ in the case d = 2. (v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐ E ⏐

(4.127)

O

0

Making similar reasoning as in (4.69), we have in the case d = 3 that for all n ∈ N and t ∈ [0, T ]

⏐ 34 ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

O

0

(v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐

4 3

≤ ∥v∥Vdiv E¯ 4 3

≤ ∥v∥Vdiv E¯

) 43

t

(∫

|P˜ n (ϕ¯ n (s)∇ µ ¯ n (s))|V ′ ds div

0 T

(∫

|P˜ n (ϕ¯ n (s)∇ µ ¯ n (s))|V ′ ds

4 3

div

0

[

≤ c ∥v∥Vdiv E sup |ϕ¯ n (s)| s∈[0,T ]

) 34

2

]1/3 [ ∫

T

∥ϕ¯ n (s)∥

E

]1/3 [ (∫

2 V ds

0

T

∥µ ¯ n (s)∥

E

)2 ]1/3

2 V ds

.

0

Hence by (4.76)1 , (4.76)2 , (4.80)2 and the previous inequality, we infer that for all t ∈ [0, T ], n ∈ N

⏐∫ t ∫ ⏐ 34 ⏐ ⏐ ⏐ ¯ E (v.∇ µ ¯ n )ϕ¯ n dxds⏐⏐ < ∞ in the case d = 3. ⏐ 0

(4.128)

O

Hence from (4.125), (4.126), (4.127) and using Vitali’s theorem we get for all t ∈ [0, T ], v ∈ Vdiv and γ ∈ (0, 23 ]

⏐∫ t ∫ ⏐2−γ ⏐ ⏐ ⏐ ¯ [(v.∇ µ lim E ⏐ ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐⏐ = 0, in the case d = 2, n→∞ 0

(4.129)

O

and from (4.125), (4.126), (4.128) and using Vitali’s theorem, we get for all t ∈ [0, T ] and v ∈ Vdiv

⏐∫ t ∫ ⏐ 34 ⏐ ⏐ ⏐ ¯ [(v.∇ µ lim E ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐⏐ = 0, in the case d = 3. ⏐ n→∞ 0

O

(4.130)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

55

Then from (4.129), (4.130) and the dominated convergence theorem, we obtain T

∫ lim

n→∞

0

⏐∫ t ∫ ⏐2−γ ⏐ ⏐ ¯E ⏐ [(v.∇ µ ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐⏐ dt = 0, in the case d = 2, ⏐ O

0

and T

∫ lim

n→∞

0

⏐∫ t ∫ ⏐ 34 ⏐ ⏐ ⏐ ¯ [(v.∇ µ E ¯ n )ϕ¯ n − (v.∇ µ ¯ )ϕ¯ ] dxds⏐⏐ dt = 0, in the case d = 3. ⏐ O

0

Hence the points (6) and (7) in Proposition 4.3 are then proved. Proof of (8). Let us consider the term (8). Assume first that v ∈ Gdiv . From (2.12), we have for all t ∈ [0, T ]

∫ t∫

|(G(s, u¯ n ; y) − G(s, u¯ ; y), v )|2 ν1 (dy)ds ∫ ∫ |G(s, u¯ n ; y) − G(s, u¯ ; y)|2 |v|2 ν1 (dy)ds ≤ 0 Y∫ t |¯un (s) − u¯ (s)|2 ds ≤ lg |v|2 0 ∫ T |¯un (s) − u¯ (s)|2 ds. ≤ lg |v|2 0

Y t

0

¯ -a.s. Hence for all t ∈ [0, T ] and the previous inequality, we infer that By (4.72)2 , u¯ n → u¯ in L2 (0, T ; Gdiv ), P

∫ t∫

|(G(s, u¯ n (s); y) − G(s, u¯ (s); y), v )|2 ν1 (dy)ds = 0.

lim

n→∞

0

(4.131)

Y

By (2.13), (4.73)1 and (4.86)1 , for every n ∈ N, t ∈ [0, T ] and every q/2 ≥ 0,

⏐ 2q ⏐∫ t ∫ ⏐ ⏐ 2 ¯E ⏐ |(G(s, u¯ n (s); y) − G(s, u¯ (s); y), v )| ν1 (dy)ds⏐⏐ ⏐ 0 Y ⏐ 2q ⏐∫ t ∫ ⏐ ⏐ } { q¯ ⏐ ≤ c |v| E ⏐ |G(s, u¯ n (s); y)|2 + |G(s, u¯ (s); y)|2 ν1 (dy)ds⏐⏐ ⏐ 2q ⏐∫0 t Y ⏐ ⏐ 2 2 q¯ ⏐ ≤ c˜ |v| E ⏐ (2 + |¯un (s)| + |¯u(s)| )ds⏐⏐ [ 0 ] q q q q ˜ ¯ ¯ 2 ≤ c˜ |v| 2 + E sup |¯un (s)| + E sup |¯u(s)| < ∞, s∈[0,T ]

(4.132)

s∈[0,T ]

for some positive constant c˜˜ > 0 depending only on T and q. Hence from (4.131), (4.132) and using also Vitali’s theorem, we infer that for all t ∈ [0, T ]

¯ lim E

∫ t∫

n→∞

0

|(G(s, u¯ n ; y) − G(s, u¯ ; y), v )|2 ν1 (dy)ds = 0, for all v ∈ Gdiv . Y

Now since the restriction of P˜ n to the space Gdiv is (., .)-projection onto Vdin v , see Section 4.1, we also obtain

¯ lim E

n→∞

∫ t∫ ⏐ ⏐2 ⏐ ˜ ⏐ ⏐(Pn G(s, u¯ n ; y) − G(s, u¯ ; y), v )⏐ ν1 (dy)ds = 0, for all v ∈ Gdiv . 0

Y

So by the properties of the integral with respect to the compensated Poisson random measure and the fact that η¯ n = η¯ , we obtain

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ lim E ⏐ n→∞

0

Y

(P˜ n G(s, u¯ n ; y) − G(s, u¯ ; y), v )η¯ (dy, ds)⏐⏐ = 0, for all v ∈ Gdiv .

Again since Vdiv is dense in Gdiv and from the previous limit, we infer that for all t ∈ [0, T ]

⏐∫ t ∫ ⏐2 ⏐ ⏐ ⏐ ˜ ¯ lim E ⏐ (Pn G(s, u¯ n ; y) − G(s, u¯ ; y), v )η¯ (dy, ds)⏐⏐ = 0, for all v ∈ Vdiv . n→∞ 0

(4.133)

Y

Moreover, reasoning similarly as in (4.132), with q = 2, we can prove that

⏐∫ t ∫ ⏐

¯⏐ E ⏐

0

= E¯

⏐2 ⏐

(P˜ n G(s, u¯ n ; y) − G(s, u¯ ; y), v )η¯ (dy, ds)⏐⏐

∫ t Y∫ ⏐ ⏐2 ⏐ ˜ ⏐ ⏐(Pn G(s, u¯ n ; y) − G(s, u¯ ; y), v )⏐ ν1 (dy)ds < ∞. 0

Y

(4.134)

56

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

From (4.133), (4.134) and the dominated convergence theorem we obtain for all v ∈ Vdiv ⊂ Gdiv T

∫ n→∞

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

lim

0

0

Y

(P˜ n G(s, u¯ n ; y) − G(s, u¯ ; y), v )η¯ (dy, ds)⏐⏐ dt = 0, i.e. (8) holds.

The proof of Proposition 4.3 is now complete.

■

For ψ ∈ V ∫T ¯ |(ϕ¯ n (t) − ϕ¯ (t), ψ )|2 dt = 0, lim→∞ E ∫ T0 ¯ |(ϕ¯ n (0) − ϕ¯ (0), ψ )|2 dt = 0, limn→∞ 0 E ⏐2 ∫ T ⏐⏐∫ t ¯ ⏐ (∇ ρ¯ (., ϕ¯ n (τ )) − ∇ρ (., ϕ¯ (τ )), ∇ψ) dτ ⏐⏐ dt = 0, limn→∞ 0 E 0 ⏐2 ∫ T ⏐⏐∫ t ∫ ⏐ ¯ limn→∞ 0 E ⏐ 0 O [(u¯ n .∇ψ )ϕ¯ n − (u¯ .∇ψ )ϕ¯ ] dxdτ ⏐ dt = 0, ⏐2 ∫ T ⏐⏐∫ t ∫ ⏐ ¯⏐ [(∇ J ∗ ϕ¯ n ).∇ψ − (∇ J ∗ ϕ¯ ).∇ψ ] dxdτ ⏐ dt = 0. limn→∞ 0 E 0 O

Proposition 4.4. (1) (2) (3) (4) (5)

Proof. Proof of (1). We have

|(ϕ¯ n (.) − ϕ¯ (.), ψ )|2L2 (Ω¯ ,P¯ ;L2 (0,T ;R))

= E¯

T

∫

|(ϕ¯ n (t) − ϕ¯ (t), ψ )|2 dt

0

∫

T

|⟨ϕ¯ n (t) − ϕ¯ (t), ψ⟩|2 ds 0 ∫ T 2 ¯ ≤ ∥ψ∥V E |ϕ¯ n (t) − ϕ¯ (t)|2V ′ dt . = E¯

(4.135)

0

¯ -a.s. and the embedding H ↪→ V ′ is continuous, by the By (4.54)2 , ϕ¯ n → ϕ¯ in C ([0, T ]; V ) and from (4.76)1 , sups∈[0,T ] |ϕ¯ n (s)|2 < ∞, P dominated convergence theorem we infer that ϕ¯ n → ϕ¯ in L2 (0, T ; V ′ ). Hence ′

T

∫

|(ϕ¯ n (t) − ϕ¯ (t), ψ )|2 dt = 0.

lim

n→∞

(4.136)

0

Moreover, by (4.79)1 , (4.86)2 and Hölder’s inequality, for every q/2 ≥ 1 and every n ∈ N, we have

⏐q/2 ⏐ |ϕ¯ n (t) − ϕ¯ (t)| dt ⏐⏐

T

⏐∫ ⏐ ¯⏐ E ⏐

2

0

≤ c E¯ ≤ c˜ E¯

T

∫

(

[0

) |ϕ¯ n (t)|q + |ϕ¯ (t)|q dt

sup |ϕ¯ n (t)|q + sup |ϕ¯ (t)|q

t ∈[0,T ]

]

t ∈[0,T ]

(4.137)

< ∞,

for some constant c˜ > 0. Hence from (4.135)–(4.137) and Vitali’s theorem we have

¯ lim E

T

∫

n→∞

|(ϕ¯ n (t) − ϕ¯ (t), ψ )|2 dt = 0, i.e. (1) holds. 0

¯ -a.s. and ϕ¯ is continuous as t = 0. Thus we get (ϕ¯ n (0), ψ ) → (ϕ¯ (0), ψ ) P¯ -a.s. Proof of (2). From (4.54)2 , ϕ¯ n → ϕ¯ in C ([0, T ]; Hw ) P Now from (4.79)1 and Vitali’s theorem, we infer that ¯ lim E

T

∫

n→∞

|(ϕ¯ n (0) − ϕ¯ (0), ψ )|2 dt = 0, i.e. (2) holds. 0 ′

Proof of (3). We first assume that ψ ∈ Vs , where s > 2 is such that ∆ψ ∈ H s−2 (O, Rd ) ↪→ Lr (O, Rd ), with d = 2, 3, r given by (H6 ) (4−d)r +2d . and r ′ = r −r 1 the conjugate index of r. Since H s−2 (O, Rd ) ↪→ L2d/(d+4−2s) (O, Rd ), see [22], we see that is enough to take s ≥ 2r More precisely we assume that ψ ∈ Vs , with s ≥

(4−d)r +2d , 2r

d = 2, 3. ∗

¯ -a.s., we infer that for all Now by integration by parts, using the fact that ρ¯ (., ϕ¯ n ) ⇀ ρ¯ := ρ¯ (., ϕ¯ ) in L∞ (0, T ; Lr (O; Rd )), P t ∈ [0, T ] t

∫

(∇ ρ¯ (., ϕ¯ n (s)), ∇ψ )ds

lim

n→∞

0

∫ t = lim (ρ¯ (., ϕ¯ n (s)), −∆ψ )ds n→∞ 0 ∫ t = (ρ¯ (s), −∆ψ )ds ∫0 t ∫ t = (∇ ρ¯ (s), ∇ψ )ds = (∇ ρ¯ (., ϕ¯ (s)), ∇ψ )ds. 0

0

Hence, since Vs ⊂ V is dense in V we can conclude that the previous equality also holds for all ψ ∈ V , i.e., t

∫

(∇ ρ¯ (., ϕ¯ n (s)), ∇ψ )ds =

lim

n→∞

0

∫ 0

t

(∇ ρ¯ (., ϕ¯ (s)), ∇ψ )ds, P- a.s.

(4.138)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

57

By integration by parts, (4.80)1 and Hölder’s inequality we get for all n ∈ N, t ∈ [0, T ] and q ≥ 2

⏐q ⏐

t

⏐∫ ⏐

⏐∫ ⏐

⏐q ⏐

t

¯⏐ (∇ ρ¯ (., ϕ¯ n (τ )), ∇ψ )dτ ⏐⏐ = E ⏐

¯⏐ E ⏐

(ρ¯ (., ϕ¯ n (τ )), −∆ψ )dτ ⏐⏐

0⏐∫ 0 (∫ ⏐q )q t ⏐ t ⏐ ⏐ ⏐ ¯ ¯ ′ |ρ¯ (., ϕ¯ n (τ ))|Lr |−∆ψ|Lr dτ = E ⏐ ⟨ρ¯ (., ϕ¯ n (τ )), −∆ψ⟩ dτ ⏐ ≤ E 0 0 )q (∫ T )q (∫ t q q ¯ ¯ |ρ¯ (., ϕ¯ n (τ ))|Lr dτ |ρ¯ (., ϕ¯ n (τ ))|Lr ds ≤ |−∆ψ| r ′ E ≤ |−∆ψ| r ′ E L

L

0

(4.139)

0

≤ cT |ψ| ¯ sup |ρ¯ (., ϕ¯ n (τ ))|qLr (O,Rd ) < ∞, q Vs E

τ ∈[0,T ]

for some constant cT depending on T . Also since the space Vs ⊂ V is dense in V , we infer that (4.139) also holds for all ψ ∈ V . From (4.3), (4.76)2 and (4.80)2 , we have

¯ E

T

∫

∥ρ¯ (., ϕ¯ (τ ))∥

τ

2 Vd

∫

T

∥µ ¯ n (τ ) + Pn (J ∗ ϕ¯ n (τ ))∥2V dτ ∫ T T 2 ¯ ¯ ∥µ ¯ n (τ ) ∥ V d τ + 2 E ∥Pn (J ∗ ϕ¯ n (τ ))∥2V dτ ≤ 2E 0 ∫ T ∫0 T ∥µ ¯ n (τ )∥2V dτ + 2|J |2L1 (Rd ) E¯ ∥ϕ¯ n (τ )∥2V dτ < ∞. ≤ 2E¯ = E¯

∫0

0

(4.140)

0

0

Now from (4.140), we have for all t ∈ [0, T ], n ∈ N, ψ ∈ V

⏐2 ⏐∫ t ⏐ ⏐ ¯E ⏐ (∇ ρ¯ (., ϕ¯ n (τ )), ∇ψ )dτ ⏐ ⏐ ⏐ 0

)2

t

(∫

|∇ ρ¯ (., ϕ¯ n (τ ))||∇ψ|dτ ∫ t |∇ ρ¯ (., ϕ¯ n (τ ))|2 dτ ≤ t |∇ψ|2 E¯ 0 ∫ T 2 ¯ ∥ρ¯ (., ϕ¯ n (τ ))∥2V dτ < ∞. ≤ T ∥ψ∥V E ≤ E¯

0

(4.141)

0

By (4.138), (4.139) and using the Vitali theorem, we infer that for all t ∈ [0, T ], ψ ∈ V

⏐2 ⏐∫ t ⏐ ⏐ ⏐ ¯ (∇ ρ¯ (., ϕ¯ n (τ )) − ∇ ρ¯ (., ϕ¯ (τ )), ∇ψ )dτ ⏐⏐ = 0. lim E ⏐ n→∞

(4.142)

0

Hence from (4.141), (4.142) and the dominated convergence theorem, we infer that for all ψ ∈ V T

∫ lim

n→∞

0

⏐2 ⏐∫ t ⏐ ⏐ ¯E ⏐ (∇ ρ¯ (., ϕ¯ n (τ )) − ∇ ρ¯ (., ϕ¯ (τ )), ∇ψ )dτ ⏐ dt = 0, i.e. (3) holds. ⏐ ⏐ 0

Proof of (4). Let s > 2d + 1, d = 2, 3. We first assume that ψ ∈ Vs . By Hölder’s inequality, using the fact that |v| ≤ c ∥v∥ for all v ∈ Vdiv we have

⏐∫ t ∫ ⏐ ⏐ ⏐ ⏐ [(u¯ n .∇ψ )ϕ¯ n − (u¯ .∇ψ )ϕ¯ ] dxdτ ⏐⏐ ⏐ 0

O

⏐∫ t ∫ ⏐ ⏐ ⏐ ⏐ =⏐ [(u¯ n .∇ψ )(ϕ¯ n − ϕ¯ ) + ((u¯ n − u¯ ).∇ψ )ϕ] ¯ dxdτ ⏐⏐ 0 O ⏐∫ t ∫ ⏐ ⏐∫ t ∫ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ≤⏐ [(u¯ n .∇ψ )(ϕ¯ n − ϕ¯ )]dxdτ ⏐ + ⏐ [((u¯ n − u¯ ).∇ψ )ϕ] ¯ dxdτ ⏐⏐ O

0

|∇ψ|L∞ |¯un (τ )||ϕ¯ n (τ ) − ϕ¯ (τ )|dτ +

≤

O

0

t

∫ 0

|∇ψ|L∞ |¯un (τ ) − u¯ (τ )|dτ 0

T

(∫ ≤ c |ψ|Vs

∥¯un (τ )∥2 dτ

) 12 (∫

+ c |ψ|Vs

T

|ϕ¯ n (τ ) − ϕ¯ (τ )|2 dτ

T

|¯un (τ ) − u¯ (τ )| dτ 2

) 21 (∫

0

≤ c |ψ|Vs

∥¯un (τ )∥ dτ 2

) 21 (∫

T

|¯un (τ ) − u¯ (τ )| dτ 2

0

|ϕ¯ (τ )| dτ 2

) 12

T

|ϕ¯ n (τ ) − ϕ¯ (τ )| dτ 2

) 12

0

0

(∫

T 0

T

(∫

) 12

0

0

(∫

+ c˜ |ψ|Vs

t

∫

) 21 (

sup |ϕ¯ (τ )|

2

τ ∈[0,T ]

) 12

.

58

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Hence by (4.72)1 , (4.72)2 , u¯ n → u¯ in L2 (0, T ; Gdiv ), ϕ¯ n → ϕ¯ in L2 (0, T ; H), respectively, (4.74)2 , (4.77) and the previous inequality, we infer that for all t ∈ [0, T ], ψ ∈ Vs

∫ t∫

[(u¯ n (τ ).∇ψ )ϕ¯ n (τ ) − (u¯ (τ ).∇ψ )ϕ¯ (τ )] dxdτ = 0.

lim

n→∞

(4.143)

O

0

Since Vs is dense in V , we infer that this previous equality also holds for all ψ ∈ V . By (4.73)1 , (4.79)1 and using the Hölder inequality, we have for all t ∈ [0, T ], n ∈ N, q/2 ≥ 2 and ψ ∈ Vs

⏐ 2q ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

O

0

(u¯ n (τ ).∇ψ )ϕ¯ n (τ )dxdτ ⏐⏐ t

(∫

q 2

) 2q

|¯un (τ )||ϕ¯ n (τ )|dτ ≤ |∇ψ|L∞ E¯ [(∫0 ) q (∫ T

|¯un (τ )| dτ

≤ c ∥ψ∥Vs E¯

0

[ ≤ c˜ ∥ψ∥Vs

T

4

|ϕ¯ n (τ )| dτ

2

2

(4.144)

) 4q ]

0

sup |¯un (τ )|q

] 12 [

sup |ϕ¯ n (τ )|q

] 12

< ∞.

τ ∈[0,T ]

τ ∈[0,T ]

Again as Vs is dense subset of V , we infer that (4.144) holds for all ψ ∈ V . Now by (4.143), (4.144) and Vitali’s theorem, we infer that for all t ∈ [0, T ], ψ ∈ V

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ lim E ⏐ n→∞

O

0

[(u¯ n (τ ).∇ψ )ϕ¯ n (τ ) − (u¯ (τ ).∇ψ )ϕ¯ (τ )] dxdτ ⏐⏐ = 0.

(4.145)

By (4.73)2 , (4.75), the fact that V ↪→ L4 (O, Rd ) continuously, |v| ≤ c ∥v∥, ∀v ∈ Vdiv and using the Hölder inequality, we infer that for all t ∈ [0, T ], n ∈ N and ψ ∈ V

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

O

0

(u¯ n (τ ).∇ψ )ϕ¯ n (τ )dxdτ ⏐⏐

≤ |∇ψ|2 E¯

t

(∫

|¯un (τ )|L4 |ϕ¯ n (τ )|L4 dτ

)2

0

≤ c |∇ψ|2 E¯

t

[(∫

|¯un (τ )|2L4 dτ

0

≤ c˜ |∇ψ|2 E¯

t

[(∫

∥¯un (τ )∥2 dτ

|ϕ¯ n (τ )|2L4 dτ

0

)]

t

) (∫

0

[ (∫ 2 ≤ c˜ ∥ψ∥ E¯

t

) (∫

∥ϕ¯ n (τ )∥2V dτ

(4.146)

)]

0 T

)2 ]1/2 [ (∫ 2 ¯ ∥¯un (τ )∥ dτ E

T

∥ϕ¯ n (τ )∥

τ

2 Vd

)2 ]1/2

< ∞.

0

0

Now from (4.145), (4.146) and the dominated convergence theorem, we infer that for all ψ ∈ V T

∫ lim

n→∞

0

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

0

O

[(u¯ n (τ ).∇ψ )ϕ¯ n (τ ) − (u¯ (τ ).∇ψ )ϕ¯ (τ )] dxdτ ⏐⏐ dt = 0, i.e. (4) holds.

(4.147)

Proof of (5). Let ψ ∈ V . Using the Hölder inequality, we have for all t ∈ [0, T ], n ∈ N

⏐∫ t ∫ ⏐ ∫ t ⏐ ⏐ ⏐ ⏐ ≤ |∇ψ||∇ J |L1 [ ∇ J ∗ ( ϕ ¯ − ϕ ¯ )] .∇ψ dxd τ |ϕ¯ n (τ ) − ϕ¯ (τ )|dτ n ⏐ ⏐ 0 O 0 (∫ T ) 21 (∫ T ) 12 1 1 2 2 ≤ T 2 |∇ψ||∇ J |L1 |ϕ¯ n (τ ) − ϕ¯ (τ )| dτ ≤ T 2 ∥ψ∥V |∇ J |L1 |ϕ¯ n (τ ) − ϕ¯ (τ )| dτ . 0

0

¯ -a.s. and using the previous equality we infer that for all t ∈ [0, T ], ψ ∈ V Hence by (4.72)2 , ϕ¯ n → ϕ¯ in L2 (0, T ; H), P

∫ t∫

(∇ J ∗ ϕ¯ n (τ )).∇ψ dxdτ =

lim

n→∞

O

0

∫ t∫

(∇ J ∗ ϕ¯ (τ )).∇ψ dxdτ .

(4.148)

O

0

By (4.79)2 and the Hölder inequality, we get for all t ∈ [0, T ], n ∈ N, ψ ∈ V

⏐∫ t ∫ ⏐ 2q ⏐ ⏐ ¯E ⏐ (∇ J ∗ ϕ¯ n (τ )).∇ψ dxdτ ⏐⏐ ⏐ 0

O

q 2 L1 (Rd )

q 2

(∫

) 2q

t

|∇ ϕ¯ n (τ )|dτ 0 (∫ ) 4q t q q q ≤ t 4 |∇ψ| 2 |J |L21 (Rd ) E¯ |∇ ϕ¯ n (τ )|2 dτ

≤ |∇ψ| |J |

¯ E

0

q

q

q

≤ T 4 ∥ψ∥V2 |J |L21 (Rd )

[ (∫ ¯ E

T

∥ϕ¯ n (τ )∥2V dτ 0

(4.149)

1 ) ]2 q 2

< ∞.

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

59

From (4.148), (4.149) and Vitali’s theorem, we infer that for all t ∈ [0, T ] and ψ ∈ V

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ lim E ⏐ n→∞

O

0

[(∇ J ∗ ϕ¯ n (τ )).∇ψ − (∇ J ∗ ϕ¯ (τ )).∇ψ ] dxdτ ⏐⏐ = 0.

(4.150)

Now by (4.79)2 , (4.150) and the dominated convergence theorem, we obtain T

∫ lim

n→∞

0

⏐2 ⏐

⏐∫ t ∫ ⏐

¯⏐ E ⏐

0

O

[(∇ J ∗ ϕ¯ n (τ )).∇ψ − (∇ J ∗ ϕ¯ (τ )).∇ψ ] dxdτ ⏐⏐ dt = 0,

which proves (5). Hence the proof of Proposition 4.4 is now completed proved.

■

4.6. Step 6. Convergence of the new processes to the corresponding limiting processes From Proposition 4.3, we deduce that lim |(¯un (.) − u¯ (.), v) |L2−γ (Ω¯ ,P¯ ;L2−γ (0,T ;R)) = 0, in the case d = 2,

(4.151)

lim |(¯un (.) − u¯ (.), v) |L4/3 (Ω¯ ,P¯ ;L4/3 (0,T ;R)) = 0, in the case d = 3,

(4.152)

lim |Υn (u¯ n , ϕ¯ n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|L2−γ (Ω¯ ,P¯ ;L2−γ (0,T ;R)) = 0, when d = 2,

(4.153)

lim |Υn (u¯ n , ϕ¯n , η¯ n , v ) − Υ (u¯ , ϕ, ¯ η, ¯ v )|L4/3 (Ω¯ ,P¯ ;L4/3 (0,T ;R)) = 0, when d = 3.

(4.154)

n→∞

n→∞

n→∞

and n→∞

From Proposition 4.4, we also have lim |(ϕ¯ n (.) − ϕ¯ (.), ψ) |L2 (Ω¯ ,P¯ ;L2 (0,T ;R)) = 0

(4.155)

lim |∧n (u¯ n , ϕ¯ n , ψ ) − ∧(u¯ , ϕ, ¯ ψ )|L2 (Ω¯ ,P¯ ;L2 (0,T ;R)) = 0.

(4.156)

n→∞

and n→∞

¯ -a.s. Since (un , ϕn ) is a solution of the Galerkin approximation equations (4.1)–(4.4), for all t ∈ [0, T ], we have P (un (t), v) = Υn (un , ϕn , η˜ n , v )(t) and

(ϕn (t), ψ) = ∧n (un , ϕn , ψ )(t). In particular, T

∫

E |(un (t), v) − Υn (un , ϕn , η˜ n , v )(t)|2−γ dt = 0, when d = 2, 0 T

∫

E |(un (t), v) − Υn (un , ϕn , η˜ n , v )(t)|4/3 dt = 0, when d = 3,

0

and T

∫

E |(ϕn (t), ψ) − ∧n (un , ϕn , ψ )(t)|2 dt = 0. 0

Now since the law of (un , ϕn , η˜ n ) is equal to the law of (u¯ n , ϕ¯ n , η¯ n ), we derive that T

∫

E |(¯un (t), v) − Υn (u¯ n , ϕ¯ n , η¯ n , v )(t)|2−γ dt = 0, when d = 2,

(4.157)

E |(¯un (t), v) − Υn (u¯ n , ϕ¯ n , η¯ n , v )(t)|4/3 dt = 0, when d = 3,

(4.158)

E |(ϕ¯ n (t), ψ) − ∧n (u¯ n , ϕ¯ n , ψ )(t)|2 dt = 0.

(4.159)

0 T

∫ 0

and T

∫ 0

By (4.151), (4.152), (4.153), (4.154), (4.157), (4.158) and (4.155), (4.156) and (4.159), respectively we deduce that T

∫

E |(¯u(t), v) − Υ (u¯ , ϕ, ¯ η, ¯ v )(t)|2−γ dt = 0, when d = 2, 0 T

∫ 0

E |(¯u(t), v) − Υ (u¯ , ϕ, ¯ η¯ n , v )(t)|4/3 dt = 0, when d = 3,

60

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

and T

∫

E |(ϕ¯ (t), ψ) − ∧(u¯ , ϕ, ¯ ψ )(t)|2 dt = 0. 0

¯ ¯ -almost all ω¯ ∈ Ω Thus for a.e. t ∈ [0, T ] and P ¯ η, ¯ v )(t) = 0 (¯u(t), v) − Υ (u¯ , ϕ, and

¯ ψ )(t) = 0, (ϕ¯ (t), ψ) − ∧(u¯ , ϕ, ¯ ¯ -almost all ω¯ ∈ Ω More precisely for a.e. t ∈ [0, T ] and P (¯u(t), v) + ν

t

∫

⟨Au¯ (s), v⟩ ds 0

∫ t∫ ⟨B0 (u¯ (s), u¯ (s)), v⟩ ds = (u¯ (0), v ) − (v.∇ µ ¯ )ϕ¯ dxds 0 O ∫ t∫ ∫0 t ⟨h(s, u¯ (s)), v⟩ ds + + (G(s, u¯ (s); y), v) η¯ (dy, ds), t

∫

+

0

0

(4.160)

Y

and

(ϕ¯ (t), ψ) +

∫ t∫ = (ϕ¯ (0), ψ ) + (u¯ (x, s).∇ψ (x))ϕ¯ (x, s)dxds 0 O ∫ t∫ (∇ J ∗ ϕ¯ (x, s)).∇ψ (x)dxds. +

t

∫

(∇ ρ¯ (., ϕ¯ (s)), ∇ψ) ds 0

0

(4.161)

O

¯ ¯ ∈ C ([0, T ]; Hw ), i.e., ϕ¯ is As (u¯ , ϕ¯ ) is Z˜2 = Z˜12 × Z˜22 -valued random variable, in particular u¯ ∈ D([0, T ]; Gw div ), i.e., u is weakly càdlàg ϕ

weakly continuous, we infer that the functions on the left-hand sides of (4.160) and (4.161) are càdlàg with respect to t and weakly continuous with respect to t, respectively. Hence the equalities (4.160) and (4.161) hold for all t ∈ [0, T ] and all v ∈ Vdiv and ψ ∈ V , respectively. Recalling (4.84), i.e., ρ¯ (., ϕ¯ ) = aϕ¯ + F ′ (ϕ¯ ), P-a.s. and (4.85), i.e., µ ¯ = ρ¯ (., ϕ¯ ) − J ∗ ϕ¯ P-a.s., we infer that the system ¯ , F¯ , F¯ , P¯ , u¯ , ϕ, (Ω ¯ η¯ ) in the sense of Definition 3.1. 4.7. Step 7: General initial condition Now we assume that u0 ∈ Gdiv and ϕ0 ∈ H such that F (ϕ0 ) ∈ L1 (O, Rd ). As in [37], for every m ∈ N we define ϕ0m ∈ D(B) as

( )−1 1 ϕ0m := I + B ϕ0 . m

Since B is maximal and monotone, we then deduce from the maximal monotone operators theory that ϕ0m → ϕ0 in H as m → ∞. Let (Ωm , Fm , Fm , Pm , η˜ m , um , ϕm ) be a weak martingale solution corresponding to the initial data u0 and ϕ0m . Using (2.14), we can write

∫

F (ϕ0m )dx = O

∫ (

a∗

Q (ϕ0m ) −

2

O

) |ϕ0m |2 dx.

(4.162)

1 We now multiply the equation ϕ0m − ϕ0 = − m Bϕ0m by Q ′ (ϕ0m ). We obtain

∫

Q ′ (ϕ0m )(ϕ0m − ϕ0 )dx

=−

O

=−

1

∫

m O ∫ 1 m

Q ′ (ϕ0m )Bϕ0m dx Q ′′ (ϕ0m )|∇ϕ0m |2 dx −

O

1 m

(4.163)

∫

Q ′ (ϕ0m )ϕ0m dx ≤ 0, O

′

since Q is monotone nondecreasing and we can suppose, without loss of generality, that Q ′ (0) = 0. Therefore, due to the convexity of Q we have

∫

Q (ϕ0m )dx ≤ O

∫

Q (ϕ0 )dx + O

∫ O

Q (ϕ0m )(ϕ0m − ϕ0 )dx ≤ ′

∫

Q (ϕ0 )dx.

(4.164)

O

Hence, thanks to (4.162), (4.164), written for each ϕ0m , we can control the nonlinear term O F (ϕ0m )dx that figures on the right hand side of (4.21), (4.26). Now we can argue as we did in Step 2 to get the estimates for the sequences um , ϕm and µm and ρ (., ϕm ). Similarly ¯ , F¯ , P¯ ), and random using a Skorokhod Theorem for nonmetric spaces as in Steps 3–4, we deduce the existence of a probability space (Ω ¯ , F¯ , P¯ ) such that the convergence in Steps 4–5 of the proof holds. Finally, proceeding similarly as variables (u¯ m , ϕ¯ m , η¯ m ), (u¯ , ϕ, ¯ η¯ ) on (Ω ¯ , F¯ , F¯ , P¯ , η, in Steps 6 of the proof, we see that (Ω ¯ u¯ , ϕ¯ ) is a weak martingale solution in the sense of Definition 3.1 corresponding to the initial data u0 and ϕ0 . The first part of the proof of Theorem 3.1 is now completed.

∫

5. Existence and uniqueness of strong solution in dimension 2 In this section we establish the pathwise uniqueness of solutions of the problem (2.16) and use Yamada–Watanabe’s classical famous result to derive the existence of a strong probabilistic solution for our problem. In the following lemma we will show that almost trajectories of the solution (u¯ , ϕ¯ ) are almost everywhere equal to Vdiv × V -valued function defined on [0, T ]. Before given the Lemma, we add the following assumptions:

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

61

(H8 ) We assume that there exists a positive constant Lh > 0 such that

|h(t , u) − h(t , v )|V ′ ≤ Lh |u − v|. div

(H9 ) There exist c1 , c2 > 0 and κ > 0 such that F ′′ (s) + a(x) ≥ c1 |s|2κ − c2 , for all s ∈ R, a.e., x ∈ O. Recall that (H9 ) implies the existence of constants c7 > 0 and c8 > 0 such that F (s) ≥ c7 |s|2+2κ − c8 , for all s ∈ R.

(5.1)

We also establish the following proposition that will be used in the proof of our lemma. Proposition 5.1. There exists a positive constant K⋆ , such that

( sup E

)

sup [Etot (u¯ n (s), ϕ¯ n (s)) + c9 ]

s∈[0,T ]

n∈N

≤ K⋆ ,

[

where Etot is defined by (2.15) and c9 = 2c8 + κ c7

(5.2)

(

2c5 (1+κ )c7

] ) 1+κ κ

|O|.

Moreover, from (5.2), we infer that 2 ¯ sup |ϕ¯ n (s)|22κ+ E ≤ L κ+2 (O ,Rd ) s∈[0,T ]

K⋆

c7

.

(5.3)

Proof. By (H7 ), (5.1), using Hölder’s and Young’s inequalities, we infer that for all t ∈ [0, T ], n ∈ N

∫ √ 2 = 2| aϕ¯ n (t)| + 2 F (ϕ¯ n (x, t))dx − (ϕ¯ n (t), J ∗ ϕ¯ n (t)) O ∫ 2κ ≥ (a(x) − |J |L1 (Rd ) )|ϕ¯n (x, t)|2 dx + 2c7 |ϕ¯ n (t)|2L2+κ+ 2 (O ,Rd ) − 2c8 |O | O ∫ ∫ 2κ |ϕ¯n (x, t)|2 dx ≥ (a(x) + 2c5 − |J |L1 )|ϕ¯n (x, t)|2 dx + 2c7 |ϕ¯n (t)|2L2+κ+ 2 − 2c8 |O | − 2c5 O O ∫ 2κ |ϕ¯n (x, t)|2 dx ≥ 2c7 |ϕ¯n (t)|2L2+κ+ 2 (O ,Rd ) − 2c8 |O | − 2c5

2E (ϕ¯ n (t))

O

≥ c7 |ϕ¯n (t)|

2+2κ L2κ+2 (O ,Rd )

− c9 ,

where the constant c9 is given by

[ c9 =

2c8 + κ c7

(

] ) 1+κ κ

2c5

|O|.

(1 + κ )c7

Hence for all t ∈ [0, T ] and n ∈ N, we have 2κ 2E (ϕ¯ n (t)) + c9 ≥ c7 |ϕ¯n (t)|2L2+κ+ 2 (O ,Rd ) .

Now by (5.4) and similar reasoning as in the proof of Lemma 4.2, see Section 4.2, with p = 2, we derive (5.2).

(5.4) ■

Remark 5.1. Note that from (5.3), we can check that

¯ , P¯ ; L2 (0, T ; Vdi′ v )). P˜ n ϕ¯ n ∇ µ ¯ n ∈ L2 ( Ω ¯ , P¯ ; From (5.3) and Banach–Alaoglu theorem we conclude there exists a subsequence of (ϕ¯ n ) convergent weakly star in L2κ+2 (Ω L (0, T ; L2κ+2 (O; Rd ))). So from (4.72)1 and similar reasoning as in the proof of (4.86)1 , we obtain ∞

2 ¯ sup |ϕ¯ (s)|22κ+ E ≤ c10 . L κ+2 (O ,Rd )

(5.5)

s∈[0,T ]

for some positive constant c10 > 0. Now we state our lemma.

¯ , F¯ , F¯ , P¯ , u¯ , ϕ, Lemma 5.1. We assume that the assumptions (H1 )–(H7 ) and (H9 ) are satisfied. Let (Ω ¯ η¯ ) be a martingale solution of (2.16) ¯ , the trajectories u¯ (ω., ¯ -almost all ω¯ ∈ Ω in the sense of Definition 3.1. Let (u0 , ϕ0 ) ∈ Gdiv × H. Hence P ¯ ) are almost everywhere equal to a càdlàg Gdiv -valued function and ϕ¯ (ω., ¯ ) is almost everywhere equal to a continuous H-valued function defined on [0, T ]. Proof. From (4.160) for all t ∈ [0, T ] u¯ (t)

∫ t ∫ t ∫ t = u¯ 0 − ν Au¯ (s)ds − B0 (u¯ (s), u¯ (s))ds − ϕ¯ (s)∇ µ ¯ (s)ds 0 0 ∫ t ∫ t ∫0 + h(s, u¯ (s))ds + G(s, u¯ (s); y)η¯ (dy, ds). 0

0

Y

(5.6)

62

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

From (4.161), the fact that div u¯ = 0, t

∫

ϕ¯ (t) = ϕ¯ 0 −

u¯ (s).∇ ϕ¯ (s)ds −

∂µ ¯ ∂ξ

= 0, we obtain for all t ∈ [0, T ] t

∫

Aµ ¯ (s)ds.

(5.7)

0

0

Now we prove that each term on the right-hand sides of (5.6) and (5.7) is Vdi′ v , V ′ -valued random variables defined on [0, T ], respectively. By (4.74)2 , we have

¯ E

T

∫

2 ′ dt Vdi v

|Au¯ (t)| 0

≤ E¯

T

∫

∥¯u(t)∥2 dt < ∞.

(5.8)

0

By (2.9), (4.86)1 , (4.100) and the Hölder inequality, we get

¯ E

T

∫

∫ T ≤ c E¯ |¯u(t)|2 ∥¯u(t)∥2 dt 0 ]1/2 [ (∫ [ 4 ¯ E ≤ c E¯ sup |¯u(t)|

|B0 (u¯ (t), u¯ (t))|2V ′ dt div

0

t ∈[0,T ]

T 2

∥¯u(t)∥ dt

)2 ]1/2

(5.9)

< ∞.

0

Let v ∈ Vdiv . Using the Hölder inequality, the fact that the embedding H 1 (O, R2 ) ↪→ L all v1 ∈ Vdiv

⏐ d=2 ∫ ⏐ d=2 ∫ ⏐ ⏐ ⏐∑ ⏐ ∑ ⏐ ∂ µ¯ (x) ⏐ ∂µ ¯ (x) ⏐ ⏐ ⏐ |ϕ¯ (x)|dx ⏐ |⟨ϕ∇ ¯ µ, ¯ v⟩ | = ⏐ ϕ¯ (x)dx⏐ ≤ vi (x) |vi (x)| ⏐ ⏐ ⏐ ∂ xi ∂ xi ⏐ i=1 O i=1 O ) 12++22κκ ⏐ 2+2κ ) 2+12κ (∫ ⏐ d=2 (∫ ∑ ⏐ ∂ µ¯ (x) ⏐ 1+2κ 2+2κ 2+2κ ⏐ ⏐ |vi (x)| 1+2κ dx ≤ |ϕ¯ (x)| dx ⏐ ∂ xi ⏐ O O i=1 ⎡( ⎤ 1+2κ ⏐ ) 2(12++22κκ ) (∫ ) 1+κ2κ 2+2κ ∫ ⏐ d=2 ∑ ⏐ ∂ µ¯ (x) ⏐2 2+2κ ⏐ ⏐ ⎣ ⎦ ≤ |ϕ| ¯ L2+2κ (O,R2 ) |vi (x)| κ dx ⏐ ∂ xi ⏐ dx O O i=1 ⏐ ⏐ d=2 ⏐ d=2 ⏐ ∑ ∑ ⏐ ∂ µ¯ (x) ⏐ ⏐ ∂ µ¯ (x) ⏐ ⏐ |vi | 2+2κ ⏐ ⏐ ⏐ ≤ c | ϕ| ¯ = |ϕ| ¯ L2+2κ (O,R2 ) L2+2κ (O ,R2 ) ⏐ ∂ xi ⏐ ∥vi ∥ ⏐ ∂ xi ⏐ L κ (O,R2 )

2+2κ

κ

(O, R2 ) is continuous and |v1 | ≤ c ∥v1 ∥, for

(5.10)

i=1

i=1

≤ c |ϕ| ¯ L2+2κ (O,R2 ) |∇ µ|∥v∥ ¯ ≤ c |ϕ| ¯ L2+2κ (O,R2 ) ∥µ∥ ¯ V ∥v∥Vdiv . Hence

|ϕ∇ ¯ µ| ¯ V′ = div

sup

v∈Vdiv ,∥v∥Vdiv ≤1

|⟨ϕ∇ ¯ µ, ¯ v⟩ | ≤ c |ϕ| ¯ L2+2κ (O,R2 ) ∥µ∥ ¯ V.

(5.11)

By (5.5), (5.11), (4.86)3 , using the Hölder inequality, we have T

∫

|ϕ¯ (t)∇ µ ¯ (t)|2V ′ dt div 0 ∫ T 2 |ϕ¯ (t)|L2+2κ (O,R2 ) ∥µ ¯ (t)∥2V dt ≤ c E¯ 0 ⎡ 1 [ ] 1+κ (∫ ⎣ ¯ ≤ c E¯ sup |ϕ¯ (t)|22++22κκ E 2 ¯ E

t ∈[0,T ]

L

(O ,R )

T

∥µ ¯ (t)∥2V dt

) 1+κ κ

(5.12)

κ ⎤ 1+κ

⎦

< ∞.

0

By (H1 ) and (4.74)1 , we have

¯ E

T

∫

|h(t , u¯ (t))|2V ′ dt ≤ lh E¯ div

0

∫

T

¯ sup |¯u(t)|2 < ∞. (1 + |¯u(t)|2 )dt ≤ T (lh + 1)E t ∈[0,T ]

0

(5.13)

By (2.13), (4.74)1 , using Itô’s isometry, we obtain

⏐∫ ⏐2 ⏐ ⏐ ⏐ G(t , u¯ (t); y)η¯ (dy, dt)⏐ ⏐ ⏐ 0 Y ∫ T∫ ≤ E¯ |G(t , u¯ (t); y)|2 ν1 (dy)dt 0 Y ∫ T ¯ sup |¯u(t)|2 < ∞. ≤ C˜ 2 E¯ (1 + |¯u(t)|2 )dt ≤ T (C˜ 2 + 1)E ¯ E

∫

T

0

t ∈[0,T ]

Hence we have shown that G is Gdiv -valued. Now we show that the right-hand side of (5.7) is V ′ -valued.

(5.14)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

63

Making similar reasoning as in (5.10), we infer that

|¯u.∇ ϕ| ¯ V′

=

sup

|⟨¯u.∇ ϕ, ¯ ψ⟩ |

sup

|(u¯ , ϕ∇ψ ¯ )|

ψ∈V ,∥ψ∥V ≤1

=

ψ∈V ,∥ψ∥V ≤1

(5.15)

≤ c |ϕ| ¯ L2+2κ (O,R2 ) ∥¯u∥Vdiv . From (4.100), (5.5), (5.15) and the Hölder inequality we get T

∫

|¯u(t).∇ ϕ¯ (t)|2V ′ dt 0 ∫ T ≤ c E¯ |ϕ¯ (t)|2L2+2κ (O,R2 ) ∥¯u(t)∥2Vdiv dt 0 ⎡ 1 ] 1+κ [ (∫ ⎣ ¯ ≤ c E¯ sup |ϕ¯ (t)|22++22κκ E 2 ¯ E

L

t ∈[0,T ]

(O ,R )

T

∥¯u(t)∥2Vdiv dt

0

) 1+κ κ

(5.16)

κ ⎤ 1+κ

⎦

< ∞.

From (2.4) and (4.86)3 , we have

¯ E

T

∫

|Aµ ¯ (t)|

2 V ′ dt

0

≤ E¯

T

∫

∥µ ¯ (t)∥2V dt < ∞.

(5.17)

0

¯ , P¯ ; L2 (0, T ; V ′ )). Also from (4.78), we have ϕ¯ ∈ L2 (Ω ¯ , P¯ ; L2 (0, T ; V )). Hence From (5.7), (5.16) and (5.17), we infer that ϕ¯ t ∈ L2 (Ω ¯ ¯ from [38, Lemma 1.2, page 261] we infer that for P-almost all ω ¯ ∈ Ω the trajectory ϕ¯ (ω, ¯ .) is equal almost everywhere to a continuous H-valued function defined in [0, T ]. ¯ the trajectory u¯ (ω, From (4.74), (5.6), (5.8), (5.9), (5.12)–(5.14), we infer that for ω ¯ ∈Ω ¯ .) is almost everywhere equal to a càdlàg Gdiv -valued function defined on [0, T ]. Hence the proof of Lemma 5.1 is completed. ■ Lemma 5.2. We assume that (u1 , ϕ1 ) and (u2 , ϕ2 ) are two solutions of (2.16) defined on the same stochastic basis S := (Ω , F , F, P, η˜ ) with the same initial condition (u0 , ϕ0 ) ∈ Gdiv × H. Hence (u1 (t), ϕ1 (t)) = (u2 (t), ϕ2 (t)) P − a.s. for any t ∈ [0, T ].

(5.18)

Proof. We set u = u1 − u2 , ϕ = ϕ1 − ϕ2 , with (u(0), ϕ (0)) = (0, 0). We rewrite the Korteweg force by making explicit the dependence on ϕ as follows

) ( ( ) ϕ2 ϕ2 − ∇ a − (J ∗ ϕ )∇ϕ. µ∇ϕ = aϕ − J ∗ ϕ + F ′ (ϕ ) ∇ϕ = ∇ F (ϕ ) + a 2

2

Now we can write (2.16)1 as follows du(t)

+ν Au(t)dt + B0 (u(t), u(t))dt [ 2 ] ϕ (t) = − ∇ a − (J ∗ ϕ (t))∇ϕ (t) + h(t , u(t)) dt [ (2 )] ∫ ϕ 2 (t) + ∇ F (ϕ (t)) + a dt + G(t , u(t − ); y)η˜ (dt , dy). 2

(5.19)

Y

Substituting (u1 , ϕ1 ) and (u2 , ϕ2 ) into (5.19), (2.16)2 , (2.16)3 and taking the difference between these equations, we obtain:

⎧ ⎪ ⎪ ⎪ ⎪ du(t) + ν Au(t)dt + [B0 (u1 (t), u1 (t)) − B0 (u2 (t), u2 (t))]dt ⎪ ⎪ ⎪ )] [ ( ⎪ ⎪ ϕ22 (t) ϕ12 (t) ∇a ⎪ ⎪ ⎪= ∇ F (ϕ1 (t)) − F (ϕ2 (t)) + a −a dt − [ϕ (t)(ϕ1 (t) + ϕ2 (t)) ]dt ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −(J ∗ ϕ (t))∇ϕ1 (t)dt − (J ∗ ϕ2 (t))∇ϕ (t)dt ∫ ⎪ ⎪ ⎪ ⎪ +[ h(t , u1 (t)) − h(t , u2 (t))]dt + [G(s, u1 (s− ); y) − G(s, u2 (s− ); y)]η˜ (dt , dy), ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dϕ (t) + u(t).∇ϕ1 (t)dt + u2 (t).∇ϕ (t)dt = −Aµ(t)dt , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩µ(t) = aϕ (t) − J ∗ ϕ (t) + F ′ (ϕ (t)) − F ′ (ϕ (t)). 1

2

(5.20)

64

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Using the Itô formula to the function φ (x) := |x|2 , x ∈ Gdiv , and the fact that divu = 0, we obtain for all t ∈ [0, T ]

+2ν

|u(t)|2

t

∫

∥u(s)∥2 ds + 2

t

∫

0

⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩ ds 0

t

∫ 0

((J ∗ ϕ2 (s))∇ϕ (s), u(s))ds + 2

t

∫

0

⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ ds 0

∫ t∫

{ } φ (u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)) − φ (u(s− )) η˜ (ds, dy)

+ 0

Y

∫ t∫ 0

((J ∗ ϕ (s))∇ϕ1 (s), u(s))ds 0

t

∫ −2

t

∫

(ϕ (s)(ϕ1 (s) + ϕ2 (s))∇ a, u(s))ds − 2

=−

{ } φ (u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)) − φ (u(s− )) ν1 (dy)ds Y

∫ t∫

(φ ′ (u(s− )), G(s, u1 (s− ); y) − G(s, u2 (s− ); y))ν1 (dy)ds.

− 0

Y

That is for all t ∈ [0, T ]

|u(t)|2 + 2ν

t

∫

∥u(s)∥2 ds + 2

t

∫

⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩ ds 0

0 t

∫

(ϕ (s)(ϕ1 (s) + ϕ2 (s))∇ a, u(s))ds − 2

=− t

∫

((J ∗ ϕ (s))∇ϕ1 (s), u(s))ds 0

0

((J ∗ ϕ2 (s))∇ϕ (s), u(s))ds + 2

−2

t

∫

t

∫

⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ ds 0

0

∫ t {∫ 0

2

2

2

2

|u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| − |u(s− )|

+

}

η˜ (ds, dy)

Y

∫ t {∫

|u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| − |u(s− )|

+ 0

}

ν1 (dy)ds

Y

∫ t∫

(u(s− ), G(s, u1 (s− ); y) − G(s, u2 (s− ); y))ν1 (dy)ds.

−2 0

Y

This previous equality is equivalent to

|u(t)|

2

+2ν

t

∫

∫

2

∥u(s)∥ ds + 2

t

⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩ ds

0

0 t

∫

(ϕ (s)(ϕ1 (s) + ϕ2 (s))∇ a, u(s))ds − 2

=− t

((J ∗ ϕ2 (s))∇ϕ (s), u(s))ds + 2

−2 0

t

∫

⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ ds

(5.21)

0

∫ t {∫ 0

2

− 2

|u(s ) + G(s, u1 (s ); y) − G(s, u2 (s ); y)| − |u(s )| −

+

−

−

}

η˜ (dy, ds)

Y

∫ t∫

2

|G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| ν1 (dy)ds.

+ 0

((J ∗ ϕ (s))∇ϕ1 (s), u(s))ds 0

0

∫

t

∫

Y

If we apply the Lagrangian theorem to F ′ (F is regular enough to do so), we get F ′ (ϕ1 ) − F ′ (ϕ2 ) = F ′′ (ϕ2 + θϕ )ϕ, with 0 < θ < 1. 1 We also note that since ϕ (0) = ϕ1 (0) − ϕ2 (0) = 0, it is clear that (ϕ (t), 1) = 0. We multiply (5.20)2 by A− N ϕ and get

−1/2

|AN

ϕ (t)|2

t

∫

aϕ (s) + F ′′ (ϕ2 (s) + θϕ (s))ϕ (s), ϕ (s) ds

(

+2

)

0 t

∫

1 u.∇ϕ1 , A− N ϕ ds − 2

(

= −2

)

0 t

∫

(J ∗ ϕ (s), ϕ (s)) ds.

+2 0

t

∫

1 u2 (s).∇ϕ (s), A− N ϕ (s) ds

( 0

)

(5.22)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

65 −1/2

Adding the corresponding relations (5.21) and (5.22) together yields the following evolution equations for |u(t)|2 and |AN

ϕ (t)|2 , we

obtain for all t ∈ [0, T ] −1/2

|u(t)|2 + |AN

ϕ (t)|2 + 2

t

∫

(

aϕ (s) + F ′′ (ϕ2 (s) + θϕ (s))ϕ (s), ϕ (s) ds + 2ν

)

t

∫

0

∥u(s)∥2 ds 0

t

∫

⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩ ds

+2 0 t

∫

(J ∗ ϕ (s), ϕ (s)) ds − 2

=2

t

∫

(

1 u.∇ϕ1 , A− N ϕ ds − 2

)

0

0

(ϕ (s)(ϕ1 (s) + ϕ2 (s))∇ a, u(s))ds − 2

−

1 u2 (s).∇ϕ (s), A− N ϕ (s) ds

)

t

∫

(5.23)

((J ∗ ϕ (s))∇ϕ1 (s), u(s))ds 0

0 t

∫

((J ∗ ϕ2 (s))∇ϕ (s), u(s))ds + 2

−2

t

∫

0

⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ ds 0

∫ t {∫

2

2

|u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| − |u(s− )|

+

}

η˜ (dy, ds)

Y

0

∫ t∫

2

|G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| ν1 (dy)ds.

+ 0

( 0

t

∫

t

∫

Y

Now we are going to estimate each term of the left and right hand-sides of (5.23). From (H5 ), we deduce that t

∫

(

2

aϕ (s) + F ′′ (ϕ2 (s) + θϕ (s))ϕ (s), ϕ (s) ds ≥ 2c0

)

t

∫

|ϕ (s)|2 ds.

(5.24)

0

0

By (2.7), (2.11), using the Young inequality, we obtain 2 |⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩|

= 2 |⟨B0 (u(s), u1 (s)), u(s)⟩| (5.25)

≤ c ∥u(s)∥|u(s)|∥u1 (s)∥ ν ≤ ∥u(s)∥2 + cν ∥u1 (s)∥2 |u(s)|2 . 6

We have 2 |(J ∗ ϕ (s), ϕ (s))|

⏐( )⏐ ⏐ 1/2 ⏐ −1/2 ≤ 2 ⏐ AN (J ∗ ϕ (s)), AN ϕ (s) ⏐ ⏐( )⏐ ⏐ 1/2 ⏐ −1/2 = 2 ⏐ AN (J ∗ ϕ (s) − ⟨J ∗ ϕ (s)⟩), AN ϕ (s) ⏐ −1/2

≤ 2|∇ (J ∗ ϕ (s) − ⟨J ∗ ϕ (s)⟩)||AN −1/2

≤ 2|∇ J |L1 |ϕ (s)||AN ≤

c0 5

(5.26)

ϕ (s)|

ϕ (s)| −1/2

|ϕ (s)|2 + c |∇ J |2L1 |AN

ϕ (s)|2 ,

1/2

1/2

where we have used the fact that |AN v|2 = (AN v, v ) = |∇v|2 = ∥v∥2 , for all v ∈ HE2 (O) and hence |AN v|2 = ∥v∥2 , which also holds, 1/2 by density, for all v ∈ V0 = D(AN ). By integration by parts, using the fact that ∇.u = 0 and u|∂ O = 0, we obtain

−2 u(s).∇ϕ1 (s), AN ϕ (s)

(

−1

)

∫ =2 +2

j=1

=2

AN (ϕ )(x, s)ϕ1 (x, s)∇.u(s, x)dx −1

O 2 ∫ ∑

2 ∫ ∑ j=1

uj (s, x)ϕ1 (x, s)

∂ −1 A (ϕ )(s, x)dx ∂ xj N

uj (s, x)ϕ1 (s, x)

∂ −1 A ϕ (s, x)dx ∂ xj N

O

O

( ) 1 = 2 u(s).∇ A− N ϕ (s), ϕ1 (s) .

66

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Thus

( ) ( ) 1 −1 − 2 u(s).∇ϕ1 (s), A− N ϕ (s) = 2 u(s).∇ AN ϕ (s), ϕ1 (s) .

(5.27)

By (5.27), the fact that |v| ≤ c ∥v∥, for all v ∈ Vdiv , using the Hölder inequality, the Ladyzhenskaya inequality and the Young inequality, we derive that 1 ⏐ 2 ⏐ u(s).∇ A− N ϕ (s), ϕ1 (s)

⏐(

)⏐

1 ≤ 2|ϕ1 (s)|L4 |u(s)|L4 |∇ A− N ϕ (s)| 1 ≤ c |ϕ1 (s)|L4 (O,R2 ) ∥u(s)∥|∇ A− N ϕ (s)| ν −1/2 ≤ ∥u(s)∥2 + cν |ϕ1 (s)|2L4 |AN ϕ (s)|2 .

(5.28)

6

Similarly we have 1 ⏐ 2 ⏐ u2 (s).∇ϕ (s), A− N ϕ (s)

⏐(

)⏐

⏐ ( )⏐ 1 ⏐ = 2 ⏐− u2 (s).∇ A− N ϕ (s), ϕ (s) 1 ≤ 2|ϕ (s)||u2 (s)|L4 |∇ A− N ϕ (s)|L4

≤ ≤

c0

2

10 c0 10

1 |ϕ (s)|2 + cc0 |u2 (s)|2L4 |∇ A− N ϕ (s)|L4 (O ,R2 )

(5.29)

−1 1 |ϕ (s)|2 + cc0 |u2 (s)|2L4 |∇ A− N ϕ (s)||∇ AN ϕ (s)|H 1 (O ,R2 )

where we have also used the Gagliardo–Nirenberg inequality in dimension 2. From (5.29) and the Young inequality, we obtain 1 ⏐ 2 ⏐ u2 (s).∇ϕ (s), A− N ϕ (s)

⏐(

)⏐

≤ ≤ ≤ ≤ ≤

c0 10 c0 10 c0 10 c0 10 c0 5

−1 1 |ϕ (s)|2 + cc0 |u2 (s)|2L4 |∇ A− N ϕ (s)||AN ϕ (s)|H 2 1 −1 |ϕ (s)|2 + cc0 |u2 (s)|2L4 |∇ A− N ϕ (s)||(AN + I)AN ϕ (s)| 1 |ϕ (s)|2 + cc0 |u2 (s)|2L4 |∇ A− N ϕ (s)||ϕ (s)|

−1/2

|ϕ (s)|2 + cc0 |u2 (s)|2L4 |AN

(5.30)

ϕ (s)||ϕ (s)| −1/2

|ϕ (s)|2 + cc0 |u2 (s)|4L4 (O,R2 ) |AN

ϕ (s)|2 .

Again by the Hölder inequality, the Ladyzhenskaya inequality and the Young inequality, we infer that

|(ϕ (ϕ1 + ϕ2 )∇ a, u)|

≤ |∇ a|L∞ |ϕ||ϕ1 + ϕ2 |L4 (O,R2 ) |u|L4 (O,R2 ) ≤ c |∇ a|L∞ |ϕ||ϕ1 + ϕ2 |L4 (O,R2 ) |u|1/2 ∥u∥1/2 ≤ ≤

c0 5 c0 5

|ϕ|2 + cc0 |ϕ1 + ϕ2 |2L4 (O,R2 ) |u|∥u∥ |ϕ|2 +

ν 6

(5.31)

∥u∥2 + c(cc0 , ν )|ϕ1 + ϕ2 |4L4 (O,R2 ) |u|2 .

By integration by parts and the fact that u|∂ O , div u = ∇.u = 0, we obtain

−2 ((J ∗ ϕ )∇ϕ2 , u) = 2

2 ∫ ∑ j=1

ϕ2

(

O

) ∂ J ∗ ϕ uj dx ∂ xj

From this last equality, the Hölder inequality, the Ladyzhenskaya inequality and the Young inequality, we get

|−2 ((J ∗ ϕ )∇ϕ2 , u)|

≤ 2|ϕ2 |L4 |u|L4 |∇ J ∗ ϕ| ≤ c |ϕ2 |L4 |u|1/2 ∥u∥1/2 |∇ J ∗ ϕ| ≤ ≤

c0 5 c0 5

|ϕ|2 + cc0 |ϕ2 |2L4 |u||∇ J |2L1 ∥u∥ |ϕ|2 +

ν 6

∥u∥2 + c(ν, |∇ J |L1 )|ϕ2 |4L4 (O,R2 ) |u|2 .

(5.32)

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

67

Similarly, we have

|−2 ((J ∗ ϕ1 )∇ϕ, u)|

⏐ ⏐ ⏐ ⏐∑ ) ( ∫ ⏐ ⏐ 2 ∂ j J ∗ ϕ1 u dx⏐⏐ = 2 ⏐⏐ ϕ1 ∂ xj ⏐ ⏐ j=1 O ≤ 2|ϕ||u|L4 |∇ J ∗ ϕ1 |L4 (5.33)

≤ c |ϕ||u|1/2 ∥u∥1/2 |∇ J |L1 |ϕ1 |L4 ≤ ≤

c0 5 c0 5

|ϕ|2 + c(c0 , |∇ J |L1 )|∇ J |2L1 ∥u∥|ϕ1 |2L4 |u| |ϕ|2 +

ν 6

∥u∥2 + (ν, c0 , |∇ J |L1 )|ϕ1 |4L4 (O,R2 ) |u|2 .

By (2.12), we have

∫

2

2

|G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| ν1 (dy) ≤ lg |u1 (s− ) − u2 (s− )| .

(5.34)

Y

By (H8 ) and Young’s inequality, we have 2|⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ | ≤ Lh |u(s)|∥u(s)∥ ≤

ν 6

∥u(s)∥2 + cν L2h |u(s)|2 .

(5.35)

Now we set

= cν ∥u1 ∥2 + cν |ϕ1 |2L4 + |ϕ2 |4L4 + cc0 |u2 |4L4 + c(c0 , ν )|ϕ1 + ϕ2 |4L4

z¯ (t) :

+ c(ν, |∇ J |L1 )|ϕ2 |4L4 + c(ν, c0 , |∇ J |L1 |ϕ1 |4L4 ) which belongs to L1 (0, T ) P¯ − a.s., where cν > 0, cc0 > 0, c(c0 , |∇ J |L1 ) > 0 and c(ν, c0 , |∇ J |L1 ) > 0 are positive constants depending on ν , J and c0 . We define for almost every t ∈ [0, T ] z(t) := e−

∫t

¯ 0 z (s)ds

.

(

−1/2

Applying Itô’s formula to the real process z(t) |u(t)|2 + |AN

) ϕ (t)|2 and using (5.23), we infer that for all t ∈ [0, T ]

∫ t ) ( ) z(s) aϕ (s) + F ′′ (ϕ2 (s) + θϕ (s))ϕ (s), ϕ (s) ds ϕ (t)|2 + 2 0 ∫ t ∫ t 2 + 2ν z(s)∥u(s)∥ ds + 2 z(s) ⟨B0 (u1 (s), u1 (s)) − B0 (u2 (s), u2 (s)), u(s)⟩ ds 0 ∫ t ∫ 0t ( ) 1 z(s) u.∇ϕ1 , A− z(s) (J ∗ ϕ (s), ϕ (s)) ds − 2 =2 N ϕ ds 0 ∫ 0t ∫ t ( ) −1 −2 z(s) u2 (s).∇ϕ (s), AN ϕ (s) ds − z(s)(ϕ (s)(ϕ1 (s) + ϕ2 (s))∇ a, u(s))ds ∫0 t ∫0 t −2 z(s)((J ∗ ϕ (s))∇ϕ1 (s), u(s))ds − 2 z(s)((J ∗ ϕ2 (s))∇ϕ (s), u(s))ds 0 0 ∫ t +2 z(s) ⟨h(s, u1 (s)) − h(s, u2 (s)), u(s)⟩ ds {∫ } ∫ t0 2 2 + z(s) |u(s− ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| − |u(s− )| η˜ (dy, ds) 0 Y ∫ t ∫ 2 + z(s) |G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| ν1 (dy)ds 0 Y ∫ t ( ) −1/2 − z¯ (s)z(s) |u(s)|2 + |AN ϕ (s)|2 ds. −1/2

(

z(t) |u(t)|2 + |AN

(5.36)

0

Now (5.36) and (5.24)–(5.35) imply

∫ t ∫ t ( ) −1/2 2 2 2 z(t) |u(t)| + |AN ϕ (t)| + c0 z(s)|ϕ (s)| ds + ν z(s)∥u(s)∥2 ds 0 0 ∫ t ( ) −1/2 ≤ (lg + cν L2h + c |∇ J |2L1 ) z(s) |u(s)|2 + |AN ϕ (s)|2 ds 0 {∫ } ∫ t 2 2 − + z(s) |u(s ) + G(s, u1 (s− ); y) − G(s, u2 (s− ); y)| − |u(s− )| η˜ (dy, ds). 0

Y

(5.37)

68

G. Deugoué, A.N. Ngana and T.T. Medjo / Physica D 398 (2019) 23–68

Taking the mathematical expectation in (5.37), we obtain for all t ∈ [0, T ]

[

−1/2

(

E z(t) |u(t)|2 + |AN

ϕ (t)|2 ∫ t [

)] (

−1/2

ϕ (s)|2 0 ∫ t ( ) −1/2 ≤ (lg + cν L2h + c |∇ J |2L1 ) E |u(s)|2 + |AN ϕ (s)|2 ds,

≤ (lg + cν L2h + c |∇ J |2L1 )

E z(s) |u(s)|2 + |AN

)]

ds

(5.38)

0

since 0 < z(s) ≤ 1. Therefore by using Gronwall’s lemma, from (5.38) we infer that

[

(

−1/2

E z(t) |u(t)|2 + |AN

ϕ (t)|

2

)]

= 0.

Hence, we obtain (5.18). This completes the proof of lemma.

■

From an infinite dimensional version of Yamada–Watanabe famous result [39] due to Ondreajat [40], we can conclude that problem ¯ , F¯ , P¯ , η, (2.16) has a unique strong solution (u¯ , ϕ¯ ), in the sense that the system (Ω ¯ u¯ , ϕ¯ ) is a weak solution of (2.16) and (u¯ , ϕ¯ ) is F¯ -adapted. Acknowledgments The author would like to thank the anonymous referees whose comments helped to improve the contents of this article. References [1] C.G. Gal, M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier–Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 401–436. [2] C.G. Gal, M. Grasselli, Trajectory attractors for binary fluid mixture in 3D, Chinese Ann. Math. Ser. B 31 (2010) 655–678. [3] L. Zhao, H. Wu, H. Huang, Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci. 7 (2009) 939–962. [4] C.G. Gal, M. Grasselli, Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. 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Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

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