Global Existence of Martingale Solutions and Large Time Behavior for a 3D Stochastic Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Shear Dependent Viscosity

Abstract

In this paper, we consider a stochastic version of a nonlinear system which consists of the incompressible Navier–Stokes equations with shear dependent viscosity controlled by a power $p>2$, coupled with a convective nonlocal Cahn–Hilliard-equations. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluid having the same density. We prove the existence of a weak martingale solutions when $p\in [11/5,12/5)$, and their exponential decay when the time goes to infinity.

1 Introduction

We consider a mathematical model of two isothermal, incompressible, immiscible fluids evolving in three dimensional bounded domain $\mathcal {M}\subset \mathbb {R}^3$. This system of equations is well-known as a diffuse interface model (see, e.g., [1, 25, 26]) for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture. In a simplified setting where the density of the mixture is supposed to be one as well as the viscosity and the mobility, the model reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+(u.\nabla )u-{\text {div}}\varvec{T}(\varphi ,Du)+\nabla \pi =\mu \nabla \varphi +g_0(t),\\ {\text {div}}u=0,\\ \partial _t\varphi +(u.\nabla )\varphi =\Delta \mu ,\\ \mu =-\Delta \varphi +F'(\varphi ), \end{array}\right. } \end{aligned}$$

(1.1)

in $(0,T)\times \mathcal {M}$, where $T>0$ is a given final time, $\pi$ is the pressure, $g_0$ is a given volume force applied to the binary mixture fluid, u and $\varphi$ are unknown variables which represent the (volume) averaged velocity and the (relative) concentration difference of one of the fluids, respectively. The chemical potential $\mu$ is the variation of the free energy functional (cf. [9])

$$\begin{aligned} \mathcal {F}(\varphi )=\int _\mathcal {M}\left( \frac{1}{2}|\nabla \varphi |^2+F(\varphi )\right) dx. \end{aligned}$$

(1.2)

Here F is a double well potential (e.g., $F(r)=(r^2-1)^2, r\in \mathbb {R}$), which accounts for the presence of two components. The potential can be defined either on the whole real line (smooth potential) or on a bounded interval (singular potential). The latter case (in a logarithmic form) is the most appropriate choice from the modeling viewpoint (see [9]), while the former can be considered as an approximation. In the context of statistical mechanics, the square gradient term in (1.2) arises from attractive long-ranged interactions between the molecules of the fluid (see, e.g., [1] and references therein). The stress tensor $\varvec{T}$, up to the pressure term, dependents on the symmetric gradient $Du:=(\nabla u+\nabla ^{\text {tr}}u)/2$ of the flow velocity field u and, possibly, on $\varphi$, through a suitable constitutive law. In fact, when we are in presence of Newtonian mixture, the stress tensor is defined as

$$\begin{aligned} \varvec{T}(\varphi ,Du)=\nu (\varphi )Du, \end{aligned}$$

(1.3)

where $\nu$ is a given strictly positive function depending only on $\varphi$; and in this case, system (1.1) is what we call the Cahn–Hilliard–Navier–Stokes system (CH-NSs) or the H-model (cf. [2, 21, 35, 53]). The CH-NS model describes the chemical interactions between the two phases at the interface, which is achieved using a Cahn–Hilliard approach, and also the hydrodynamic properties of the mixture which is obtained using Navier–Stokes equations with surface tension terms acting at the interface (cf. [21]). Now, when the mixture has non-Newtonian features, then the stress tensor $\varvec{T}$ depends on some power of |Du|. For instance, it can be given as follows

$$\begin{aligned} \varvec{T}(\varphi ,Du)=(\nu _1(\varphi )+\nu _2(\varphi )|Du|^{p-2})Du \end{aligned}$$

(1.4)

where $\nu _1$ and $\nu _2$ are strictly positive functions and $p>2$. Systems like (1.1)–(1.3), also known as CH-NSs, have been analyzed by many authors and used in several different contexts (see, for instance, [2, 21, 22, 53], cf. also [14, 28] for numerical issues).

We note that system (1.1) has been deduced phenomenologically, i.e., as the (conserved) gradient flow associated with the Fréchet derivative of the free energy functional $\mathcal {F}$ defines in (1.2). However in [23, 24], starting from a microscopic model, another form of the free energy functional has been proposed and rigorously justified as a macroscopic limit of microscopic phase segregation models with particle conserving dynamics (cf. also [8]). In this case the gradient term is replaced by a nonlocal spatial operator, namely,

$$\begin{aligned} \begin{array}{ll} \mathcal {E}(\varphi )=\frac{\textstyle 1}{\textstyle 4}\int _\mathcal {M}\int _\mathcal {M}J(x-y)(\varphi (x)-\varphi (y))^2dxdy+\int _\mathcal {M}F(\varphi (x))dx, \end{array} \end{aligned}$$

(1.5)

where $J:\mathbb {R}^3\rightarrow \mathbb {R}$ is a sufficiently smooth interaction kernel such that $J(x)=J(-x)$. Taking the first variation of $\mathcal {E}$ the chemical potential becomes

$$\begin{aligned}\mu =a\varphi -J*\varphi +F'(\varphi ),\end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} a(x)=\int _\mathcal {M}J(x-y)dy~\text {and}~(J*\varphi )(x)=\int _\mathcal {M}J(x-y)\varphi (y)dy, \end{array} \end{aligned}$$

(1.6)

and consequently, we have the following nonlocal evolution system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+(u.\nabla )u-{\text {div}}\varvec{T}(\varphi ,Du)+\nabla \pi =\mu \nabla \varphi +g_0(t),\\ {\text {div}}u=0,\\ \partial _t\varphi +(u.\nabla )\varphi =\Delta \mu ,\\ \mu =a\varphi -J*\varphi +F'(\varphi ), \end{array}\right. } \end{aligned}$$

(1.7)

in $(0,T)\times \mathcal {M}$. As mention by Van der Waals in [46], we can observe (formally) that the nonlocal interaction term can be locally approximated by the square gradient, provided that the interaction J is sufficiently concentrated around 0; i.e., the functional $\mathcal {F}$ can be viewed as a local approximation of $\mathcal {E}$. Hence system (1.7) seems well justified and more general than the classical one, though the related literature is far less abundant. In the case (1.3), the solvability of system (1.7) has been analyzed first in [10] and then in [16,17,18,19] under various assumptions and generalizations. In [15], assuming that the stress tensor $\varvec{T}$ only depends on Du with a $(p-1)$-power growth, the authors proved the existence of a weak solution when $p\ge 11/5$ and they extend some previous results on time regularity and uniqueness when $p>11/5$. The aim of this paper is to study a stochastic version of the system (1.7), in the case that $\varvec{T}$ only depends on Du with a $(p-1)$-power growth.

However, in order to consider a more realistic model for our problem, it is sensible to consider some king of noise in the equation (1.7). This may reflect, for instance, some environmental effects on the phenomena, some external random forces, etc. 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. While this belief was based on empirical and experimental data, Rozovskii and Mikulevicius were able to derive the models rigorously in their recent work [36], thereby confirming the importance of this approach in hydrodynamic turbulence. More precisely, it is mentioned in [29] that some rigorous information on questions in turbulence might be obtained from stochastic version of the equations of fluid dynamics. To the best of our knowledge, the study of the stochastic version of the system (1.7) has not been analyzed

Considering the fact that the majority of work studies in SPDEs assumed that the fluids are Newtonian (since it is well-known that the incompressible Navier–Stokes equation governs the motions of single-phase fluids such as air or water), and that there are some conducting materials appearing in many practical and theoretical situations that cannot be characterized by Newtonian fluids (see for instance the introduction of Biskamp’s book [5] for some examples of these non-Newtonian conducting fluids), we will analyzed in this paper a stochastic version of problem (1.7) within a reasonably simple (but meaningful) non-Newtonian setting. Namely, for a final time $T>0$ and sufficiently smooth bounded domain $\mathcal {M}\subset \mathbb {R}^3$, assuming matched densities equal to unity and that the stress tensor $\varvec{T}$ only depends on Du with a $(p-1)$-power growth, we have to deal with the system of stochastic partial differential equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu+(u.\nabla )u-{\text {div}}\varvec{T}(Du)+\nabla \pi =\mu \nabla \varphi +g_0(t)+g_1(u,\varphi )+g_2(t,u,\varphi )\dot{W}_t,\\ {\text {div}}u=0,\\ \partial _t\varphi +(u.\nabla )\varphi =\Delta \mu ,\\ \mu =a\varphi -J*\varphi +F'(\varphi ),\\ u=0,~\frac{\partial \mu }{\partial \eta }=0~~\text {on}~(0,T)\times \partial \mathcal {M},\\ u(0)=u_0,~\varphi (0)=\varphi _0~\text {in}~\mathcal {M}, \end{array}\right. } \end{aligned}$$

(1.8)

where $\partial \mathcal {M}$ is the boundary of $\mathcal {M}$, $\eta$ is the outward normal to $\partial \mathcal {M}$, $u=(u_1,u_2,u_3)$, $\varphi$ and $\pi$ are unknown random fields defined on $[0,T]\times \mathcal {M}$, representing, respectively, the fluid velocity, the order (phase) parameter and the pressure, at each point of $[0,T]\times \mathcal {M}$. The external volume forces $g_0(t),~ g_1(u,\varphi )$, are given. The term $g_2(t,u,\varphi )\dot{W}_t$ is an external force depending on u and $\varphi$, where $\dot{W}_t$ denotes the time derivative of a cylindrical Wiener process. The quantities $u_0$ and $\varphi _0$ are given non-random initial velocity and phase field, respectively. These equations are of the nonlocal type because of the presence of the term J, which is the spatial-dependent internal kernel and $J*\varphi$ denotes the spatial convolution over $\mathcal {M}$. The purpose of the present manuscript is to prove some results related to problem (1.8), which are the stochastic analog of some of those obtained in [15] for the deterministic case. Our main results are the following:

• 1. We prove the existence of martingale solution for the stochastic system (1.8). We consider a sufficiently general forcing consisting of a regular part and a stochastic part both depending nonlinearly on the velocity of the fluids and the (order) phase parameter $\varphi$ (i.e. the relative concentration of one fluid or the difference of the two concentration). These forces terms are supposed to be non-Lipschitz. The method for the proof is based in the Galerkin, compactness, and monotonicity methods.

• 2. Having the existence of a martingale weak solutions in hand, we now move to the study of its asymptotic behavior as the time t is large. Then, we study the decay of the martingale weak solutions as times goes to infinity. More precisely, we prove that under some conditions on the forcing terms $g_i$, $i=0,1,2$, the couple $(u,\varphi )$ converges to zero exponentially in the mean square.

We note that for the proof of item (1) we drew our inspiration from the paper [15, 45] and for the proof of item (2) we mainly follow the idea in [6, 7].

The layout of the manuscript is as follows. In Section 2, we present the mathematical setting of our model, the stochastic framework and we gather all the necessary tools and the hypotheses. In Section 3 we introduce the notion of weak solutions and we state our first result for the existence of weak probabilistic solution. In Section 4, we derive the proof of our first main result by means of Galerkin methods and probabilistic and analytic compactness results. In Section 5, we prove our second main result concerning the exponential asymptotic behavior of these weak solutions.

2 Functional Setup and Preliminary

Here, we introduce some necessary notations and most of the hypotheses relevant for our analysis.

2.1 The Deterministic Framework

We introduce some notations and background following the mathematical theory of hydrodynamic equations such as Navier–Stokes equations. We denote by $\mathcal {D}(\mathcal {M})$ the space of functions $u\in \mathcal {C}^\infty (\mathcal {M})$ with compact support. Let $p\in (1,\infty )$, we introduce the following spaces

$$\begin{aligned} \begin{aligned} \mathcal {V}&=\{u\in \mathcal {D}(\mathcal {M})^3: {\text {div}}u=0\},\\ G_{{\text {div}}}&=\text {the closure of}~\mathcal {V}~\text {in}~(L^2(\mathcal {M}))^3,\\ V_{{\text {div}},p}&=\text {the closure of}~\mathcal {V}~\text {in}~(W^{1,p}(\mathcal {M}))^3. \end{aligned} \end{aligned}$$

We denote by $|.|$ the $(L^2(\mathcal {M}))^3$-norm, and by $(.,.)$ the $(L^2(\mathcal {M}))^3$-inner product.

The space $G_{{\text {div}}}$ is equipped with the scalar product and norm induced by $(L^2(\mathcal {M}))^3$ and thanks to Poincaré’s inequality we can endow the space $V_{{\text {div}},p}$ with the norm $\Vert u\Vert _{1,p}$ defined by

$$\begin{aligned}\Vert u\Vert ^p_{1,p}=\int _\mathcal {M}|\nabla u|^pdx.\end{aligned}$$

Note that this norm is equivalent to the usual $(W^{1,p}(\mathcal {M}))^3$-norm on $V_{{\text {div}},p}$.

We equip the space $V_{{\text {div}}}:=V_{{\text {div}},2}$ with the norm $\Vert .\Vert$ generated by the scalar product

$$\begin{aligned}((u,v))=\int _\mathcal {M}\nabla u.\nabla v dx. \end{aligned}$$

Owing to Poincaré’s inequality, $\Vert .\Vert$ and the usual $(H^1(\mathcal {M}))^3$-norm are equivalent on $V_{{\text {div}}}$.

For other Hilbert spaces X, the scalar product will be denoted by $(.,.)_X$. The notations $\left\langle .,.\right\rangle _Y$ and $\Vert .\Vert _Y$ will stand for the duality pairing between a Banach space Y and its duality $Y'$, and for the norm of Y, respectively.

With the view to implement the approximation scheme (see Section 4 below), we introduce the auxiliary Hilbert space $\mathbf{W} _s$ defined by (see [33])

$$\begin{aligned}{} \mathbf{W} _s=\text {the closure of}~\mathcal {V}~\text {in}~(H^s(\mathcal {M}))^3,\end{aligned}$$

where $s>\frac{5}{2}$ is fixed and we have the following Gelfand chain

$$\begin{aligned} \mathbf{W} _s\hookrightarrow V_{{\text {div}},p}\hookrightarrow G_{{\text {div}}}\cong G'_{{\text {div}}}\hookrightarrow V'_{{\text {div}},p}\hookrightarrow \mathbf{W} '_{s}, \end{aligned}$$

(2.1)

where each space is densely and compactly embedded into the next one.

Note that it is enough to take $s\ge \frac{5}{2}-\frac{3}{p}$ so as to (2.1) holds.

We set $H:=L^2(\mathcal {M}),~U:=H^1(\mathcal {M})$ and we also introduce the spaces (see [15] for more details)

$$\begin{aligned} \begin{array}{ll} H^{s_1}_{(0)}(\mathcal {M}):=\{\phi \in H^{s_1}(\mathcal {M}):\left\langle \phi ,1\right\rangle _{H^{s_1}}=0\},\\ H_{(0)}^{-1}(\mathcal {M}):=H_{(0)}^1(\mathcal {M})'=\{\phi \in U':\left\langle \phi ,1\right\rangle _U=0\}, \end{array} \end{aligned}$$

with $s_1\in \mathbb {R}$.

We set

$$\begin{aligned}\Vert \psi \Vert _{H^{s_1}}^2=\sum _{k\in \mathbb {N}}\iota _k^{s_1}c_k^2,~c_k=\int _\mathcal {M}\psi (x)\psi _k(x)dx,\end{aligned}$$

where $\{(\iota _k,\psi _k)\}_{k\in \mathbb {N}}$ are the eigenvalues and the eigenfunctions of the weak Laplace operator $A_N$ with homogeneous Neumann boundary condition, that is, for $f\in U'$ and $\phi \in U$ we have

$$\begin{aligned} \begin{array}{ll} A_N\phi =f\iff \int _{\mathcal {M}}\nabla \phi .\nabla \psi dx=\left\langle f,\psi \right\rangle _U,~\text {for all}~\psi \in U. \end{array} \end{aligned}$$

(2.2)

We recall that $A_N$ maps U onto $H_{(0)}^{-1}(\mathcal {M})$ and the restriction of $A_N$ to $H_{(0)}^1(\mathcal {M})$ is an isometry between $H_{(0)}^1(\mathcal {M})$ and the space $H_{(0)}^{-1}(\mathcal {M})$. Further, we denote by $A_N^{-1}:H_{(0)}^{-1}(\mathcal {M})\rightarrow H_{(0)}^{1}(\mathcal {M})$ the inverse map defined by

$$\begin{aligned} A_NA_N^{-1}f=f,~\forall f\in H_{(0)}^{-1}(\mathcal {M})~\text {and}~A_N^{-1}A_Nf=f,~\forall f\in H_{(0)}^1(\mathcal {M}). \end{aligned}$$

We know that, for every $f\in H_{(0)}^{-1}(\mathcal {M})$, $A_N^{-1}f$ is the unique solution with zero mean value of the Neumann problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi =f,\quad \text {in}~\mathcal {M},\\ \frac{\partial \phi }{\partial \eta }=0,\quad \text {on}~\partial \mathcal {M}. \end{array}\right. } \end{aligned}$$

In addition, we have

$$\begin{aligned} \begin{array}{ll} \left\langle A_N\phi ,A_N^{-1}f\right\rangle _U=\left\langle \phi ,f\right\rangle _U,~\text {for all}~\phi \in U, f\in H_{(0)}^{-1}(\mathcal {M}),\\ \left\langle f,A_N^{-1}g\right\rangle _U=\left\langle g,A_N^{-1}f\right\rangle _U=\int _{\mathcal {M}}\nabla (A_N^{-1}f).\nabla (A_N^{-1}g)dx,~\text {for all}~f,g\in H_{(0)}^{-1}(\mathcal {M}). \end{array} \end{aligned}$$

(2.3)

Note that $A_N$ can be also viewed as an unbounded linear operator on H with domain $D(A_N)=\{\phi \in H^2(\mathcal {M}):\frac{\partial \phi }{\partial \eta }=0~\text {on}~\partial \mathcal {M}\}$ and there a positive constant $c>0$ such that

$$\begin{aligned} \Vert A_N^{-1}\phi \Vert _U\le c\Vert \phi \Vert _{U'},~\Vert A_N^{-1}\phi \Vert _{H^2}\le c|\phi |. \end{aligned}$$

(2.4)

Remark 2.1

The natural no-flux condition $\frac{\partial \mu }{\partial \eta }=0$ implies the conservation of the following quantity

$$\begin{aligned} \begin{array}{ll} \left\langle \varphi (t)\right\rangle =\frac{1}{|\mathcal {M}|}\int _\mathcal {M}\varphi (t,x)dx, \end{array} \end{aligned}$$

where $|\mathcal {M}|$ stands for the Lebesgue measure of $\mathcal {M}$. More precisely, we have

$$\begin{aligned}\left\langle \varphi (t)\right\rangle =\left\langle \varphi (0)\right\rangle ,\ \ \forall t\ge 0.\end{aligned}$$

Thus, up to a shift of the order parameter field, we can always assume that the mean of $\varphi$ is zero at the initial time and, therefore it will remain zero for all positive times. Hereafter, we assume that

$$\begin{aligned}\left\langle \varphi (t)\right\rangle =\left\langle \varphi (0)\right\rangle =0\ \ \forall t\ge 0.\end{aligned}$$

Remark 2.2

We have just added a stochastic force in the equation for velocity u, not in the equation for the relative concentration $\varphi$ since it will involve tedious calculations and will increase significantly the size of the paper. In fact, we need to apply Itô’s formula to the functional $\mathcal {E}_{tot}$ defines in (3.3) below, which will require tedious calculations and probably more assumptions.

Let us set

$$\begin{aligned}&\mathbb {H}=G_{{\text {div}}}\times H,\\&\mathbb {W}_s=\mathbf{W} _s\times U,\ \ \text {for fix}\ \ s>5/2. \end{aligned}$$

The space $\mathbb {H}$ is a complete metric space with respect to the norm

$$\begin{aligned} \Vert (u,\varphi )\Vert _\mathbb {H}^2=|u|^2+|\varphi |^2. \end{aligned}$$

(2.5)

The space $\mathbb {W}_s$ will be equipped with the usual scalar product and norm of the cartesian space $H^s(\mathcal {M})\times H^1(\mathcal {M})$, respectively denoted by $((.,.))_s=((.,.))_{H^s}+((.,.))_{H^1}$ and $\Vert .\Vert _s^2=\Vert .\Vert _{H^s}^2+\Vert .\Vert _U^2$.

We define the Banach space $\mathbb {V}$ by

$$\begin{aligned}\mathbb {V}=V_{{\text {div}},p}\times U,\end{aligned}$$

with norm

$$\begin{aligned} \Vert (u,\varphi )\Vert _\mathbb {V}=\Vert u\Vert _{1,p}+\Vert \varphi \Vert _U. \end{aligned}$$

(2.6)

Remark 2.3

The norm defines in (2.6) is equivalent to any norm of the form

$$\begin{aligned}{}[[(u,\varphi )]]=C_1\Vert u\Vert _{1,p}+C_2\Vert \varphi \Vert _U, \end{aligned}$$

(2.7)

where $C_1$ and $C_2$ are positive constants depending only on p and $|\mathcal {M}|$.

Hereafter we set

$$\begin{aligned}\Vert (u,\varphi )\Vert _\mathbb {V}^{p,2}=\Vert u\Vert _{1,p}^p+\Vert \varphi \Vert _U^2~\text {and}~L^{p,2}(0,T;\mathbb {V})=L^p(0,T;V_{{\text {div}},p})\times L^2(0,T;U).\end{aligned}$$

The space $L^{p,2}(0,T;\mathbb {V})$ is a Banach space with respect to the norm

$$\begin{aligned} \begin{array}{ll} \Vert (u,\varphi )\Vert _{L^{p,2}(0,T;\mathbb {V})}^2=\Vert u\Vert _{L^p(0,T;V_{{\text {div}},p})}^p+\Vert \varphi \Vert _{L^2(0,T;U)}^2=\int _0^T\Vert (u(s),\varphi (s))\Vert _\mathbb {V}^{p,2}ds. \end{array} \end{aligned}$$

(2.8)

2.2 Nonlinear Operators

For $u, v, w\in V_{{\text {div}}}$, we define the trilinear operator b(., ., .) as

$$\begin{aligned} b(u,v,w)=\int _{\mathcal {M}}(u(x).\nabla )v(x).w(x)dx=\sum _{i,j=1}^3\int _{\mathcal {M}}u_i(x)\partial _{x_i}v_j(x)w_j(x)dx. \end{aligned}$$

We recall that

$$\begin{aligned} {\left\{ \begin{array}{ll} b(u,v,w)=-b(u,w,v),~\forall ~u,v,w\in V_{{\text {div}}},\\ b(u,v,v)=0,~\forall ~u,v\in V_{{\text {div}}}. \end{array}\right. } \end{aligned}$$

(2.9)

For more properties concerning the nonlinear operator b, we refer the readers to [51].

In order to introduce the weak formulation of problem (1.8), we introduce the following bilinear and trilinear forms as in [15].

$$\begin{aligned} \begin{array}{ll} \left\langle N(u),v\right\rangle _{V_{{\text {div}},p}}=\displaystyle \int _{\mathcal {M}}\varvec{T}(Du). Dvdx,\\ \left\langle B_0(u,u),v\right\rangle _{V_{{\text {div}},p}}=\displaystyle \int _\mathcal {M}[(u.\nabla )v].udx,\\ \left\langle R_1(\varphi ),v\right\rangle _{V_{{\text {div}},p}}=-\frac{\textstyle 1}{\textstyle 2}\displaystyle \int _\mathcal {M}\varphi ^2\nabla a.vdx+\displaystyle \int _\mathcal {M}(\nabla J*\varphi )\varphi .vdx,\\ \left\langle B_1(u,\varphi ),\psi \right\rangle _U=\displaystyle \int _\mathcal {M}\varphi u.\nabla \psi dx, \end{array} \end{aligned}$$

(2.10)

which are well defined for all $u, v\in V_{{\text {div}},p}$ and for all $\varphi \in L^{2\kappa +2}(\mathcal {M})$ and $\psi \in U=H^1(\mathcal {M})$, where p and $\kappa$ are chosen as in Theorem 3.1 below. Here $\varvec{T}$ designates the extra stress tensor of the non-Newtonian fluid, and it only depends on Du with a $(p-1)$-power growth (cf. (1.4) as in the introduction).

We note that $b(u,u,v):=-\left\langle B_0(u,u),v\right\rangle$, for all $u,v\in V_{{\text {div}},p}$. For simplicity we will write $B_0(u):=B_0(u,u)$.

2.3 Stochastic Setting and Assumptions

Let $\left( \Omega ,\mathcal {F},\mathbb {P}\right)$ be a complete probability space and $\mathbb {F}=\{\mathcal {F}_{t}\}_{t\in [0,T]}$ an increasing and right continuous family of sub $\sigma$-algebras of $\mathcal {F}$, such that $\mathcal {F}_{0}$ contains all the $\mathbb {P}$-null sets of $\mathcal {F}$. Given $K_1$ and $K_2$ two separable Banach spaces, we denote by $L(K_1)$ the set of bounded linear map in $K_1$, by $\mathcal {L}(K_1,K_2)$ the space of continuous linear mapping from $K_1$ into $K_2$. By $L_2(K_1,K_2)$, we mean the subspace of $\mathcal {L}(K_1,K_2)$ consisting of Hilbert–Schmidt operators when $K_1$ and $K_2$ are separable. It is known that $L_2(K_1,K_2)$ is a Hilbert space, and its norm is denoted by $\Vert .\Vert _{L_2(K_1,K_2)}$.

Let $\{\beta _{t}^{j}, t\ge 0, j=1,2,\ldots \}$ be a given sequence of mutually independent standard real $\mathcal {F}_{t}$-Wiener processes defined on $\left( \Omega ,\mathcal {F},\mathbb {P}\right)$, and suppose given K, a separable Hilbert space, and $\{e_{j}, j=1,2,\ldots \}$, an orthonormal basis of K. We denote by $\{W_t,t\ge 0\}$, the cylindrical Wiener process with values in K defined by

$$\begin{aligned} W_t=\sum _{j=1}^{\infty }\beta _{t}^{j}e_{j}. \end{aligned}$$

(2.11)

Remark 2.4

Let $\tilde{H}$ be a Hilbert space and $\mathbb {M}^2(\Omega \times [0,T];L_2(K,\tilde{H}))$ the space of all equivalence classes of $\mathbb {F}$-progressively measurable processes $\psi :\Omega \times [0,T]\rightarrow L_2(K,\tilde{H})$ satisfying

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\int _{0}^{T}\Vert \psi (s)\Vert _{L_{2(K,\tilde{H})}}^{2} ds<\infty . \end{array} \end{aligned}$$

• (i) For any $s\in [0,T]$ and $\psi \in \mathbb {M}^2(\Omega \times [0,T];L_2(K,\tilde{H}))$, we have

  $$\begin{aligned} \psi (s)\circ \mathfrak {J}^{-1}\in L_2(Q^{1/2}(K_1),\tilde{H}), \end{aligned}$$

  where $\mathfrak {J}$ is any one-to-one Hilbert–Schmidt operator from K into another Hilbert space $(K_1, (.,.)_{K_1})$ and $Q=\mathfrak {J}\mathfrak {J}^*\in L(K_1)$, $\mathfrak {J}^*$ the adjoint of $\mathfrak {J}$; since

  $$\begin{aligned} \Vert \psi (s)\Vert _{L_2(K,\tilde{H})}^2&=\sum _{i\in \mathbb {N}}(\psi (s)e_i,\psi (s)e_i)_{\tilde{H}}\\&=\sum _{i\in \mathbb {K}}\left( \psi (s)\circ \mathfrak {J}^{-1}(\mathfrak {J}e_i),\psi (s)\circ \mathfrak {J}^{-1}(\mathfrak {J}e_i)\right) _{\tilde{H}}=\Vert \psi (s)\circ \mathfrak {J}^{-1}\Vert _{L_2(Q^{1/2}(K_1),\tilde{H})}^2. \end{aligned}$$

  Hence

  $$\begin{aligned} \Vert \psi (s)\circ \mathfrak {J}^{-1}\Vert _{L_2(Q^{1/2}(K_1),\tilde{H})}=\Vert \psi (s)\Vert _{L_2(K,\tilde{H})}. \end{aligned}$$

• (ii) It follows from the theory of stochastic integration on infinite dimensional Hilbert space, cf. [37, Chapter 5, Section 26] and [11, Chapter 4], that the process $\chi$ defined by

  $$\begin{aligned} \begin{array}{ll} \chi (t)=\int _0^t\psi (s)dW_s:=\int _0^t\psi (s)\circ \mathfrak {J}^{-1}d\bar{W}(s),~t\in [0,T] \end{array} \end{aligned}$$

  (2.12)

  is a $\tilde{H}$-valued martingale; where

  $$\begin{aligned} \bar{W}(t):=\sum _{j=1}^\infty \beta _t^j\mathfrak {J}e_j,\ \ t\in [0,T]. \end{aligned}$$

  (2.13)

  Moreover, the following Itô isometry holds

  $$\begin{aligned} \begin{array}{ll} \mathbb {E}\Vert \int _0^t\psi (s)\circ \mathfrak {J}^{-1}d\bar{W}(s)\Vert _{\tilde{H}}^2=\mathbb {E}\int _0^t\Vert \psi (s)\Vert _{L_2(K,\tilde{H})}^2ds,~\forall t\in [0,T], \end{array} \end{aligned}$$

  (2.14)

  and the Burkholder–Davis–Gundy inequality

  $$\begin{aligned} \begin{array}{ll} \mathbb {E}\left( \displaystyle \sup _{s\in [0,T]}\left\| \displaystyle \int _0^s\psi (\tau )\circ \mathfrak {J}^{-1}d\bar{W}(\tau )\right\| _{\tilde{H}}^q\right) &{}\le c_q\mathbb {E}\left( \displaystyle \int _0^t\Vert \psi (s)\Vert _{L_2(K,\tilde{H})}^2 ds\right) ^{q/2},\\ &{}\forall t\in [0,T],~\forall q\in (1,\infty ). \end{array} \end{aligned}$$

  (2.15)

• (iii) $\bar{W}(t)$, $t\in [0,T]$ is a Q-Wiener process on $K_1$, see [34, Proposition 2.5.2]; and it is also a $K_1$-valued continuous, square integrable $\mathcal {F}_t$-martingale, see [34, Proposition 2.2.10].

• (iv) The series defines in (2.13) even converges in $L^2(\Omega ,\mathcal {F},\mathbb {P};\mathcal {C}([0,T];K_1))$, and thus always has a $\mathbb {P}$-a.s. continuous version.

Remark 2.5

for any $g\in \tilde{H}$, one has

$$\begin{aligned} \left( \int _{0}^{t}\psi (s)dW(s),g\right) _{\tilde{H}}=\left( \int _0^t\psi (s)\circ \mathfrak {J}^{-1}d\bar{W}(s),g\right) _{\tilde{H}}=\sum _{j=1}^{\infty }\int _0^t\left( \psi (s)e_{j},g\right) _{\tilde{H}}d\beta _{s}^{j},\ \ t\in [0,T], \end{aligned}$$

where each stochastic integral in the series is understood as an Itô’s stochastic integral with respect to the corresponding real valued Wiener process $\beta _s^j$. The above series converges in $L^2(\Omega ,\mathcal {F}_t,\mathbb {P};\mathcal {C}([0,t];\tilde{H}))$, for each $0<t\le T$; see [11] for details. In particular, we note that if $\psi \in \mathbb {M}^2(\Omega \times [0,T];L_2(K,\tilde{H}))$ and $g\in L^2(\Omega ,\mathcal {F},\mathbb {P}; L^\infty (0,T;\tilde{H}))$ is $\mathcal {F}_t$-progressively measurable, then the series

$$\begin{aligned}\sum _{j=1}^{\infty }\int _0^t(\psi (s)e_{j},g(s))d\beta _s^j,~ t\in [0,T],\end{aligned}$$

converges in $L^2(\Omega ,\mathcal {F}_t,\mathbb {P};\mathcal {C}([0,t];\mathbb {R}))$, and defined a real-valued continuous $\mathcal {F}_t$-martingale.

We will use the notation

$$\begin{aligned} \displaystyle \int _{0}^{t}(\psi (s)dW(s),g):=\displaystyle \int _0^t\left( \psi (s)\circ \mathfrak {J}^{-1}d\bar{W}(s),g\right) :=\sum _{j=1}^{\infty }\int _{0}^{t}(\psi (s)e_{j},g)_{\tilde{H}}d\beta _{s}^{j},~ t\in [0,T]. \end{aligned}$$

Hereafter, we shall fix one such $\mathfrak {J}$ and $(K_1,(.,.)_{K_1})$ as in Remark 2.4 and for the process $W_t$, $t\in [0,T]$, given by (2.11) we define $\bar{W}(t)$, $t\in [0,T]$ as in (2.13) for the fixed $\mathfrak {J}$.

The stochastic integral of $g_2(s,u(s),\varphi (s))$ (which is the unique $G_{{\text {div}}}$-valued $\mathcal {F}_t$-martingale) with respect to the K-cylindrical Wiener process $W_t$, $t\in [0,T]$ is given by

$$\begin{aligned} \begin{array}{ll} \int _0^tg_2(s,u(s),\varphi (s))dW_s&{}=\int _0^tg_2(s,u(s),\varphi (s))\circ \mathfrak {J}^{-1}d\bar{W}(s)\\ &{}:=\int _0^t\bar{g}_2(s,u(s),\varphi (s))d\bar{W}(s)\ \ t\in [0,T]. \end{array} \end{aligned}$$

(2.16)

Using the notations above, we rewrite problem (1.8) as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\textstyle du}{\textstyle dt}+N(u)-B_0(u)-R_1(\varphi )=g_0(t)+g_1(u,\varphi )+g_2(t,u,\varphi )\dot{W}_t~\text {in}~V'_{{\text {div}},p},\\ \frac{\textstyle d\varphi }{\textstyle dt}-B_1(u,\varphi )=\Delta \mu ~\text {in}~U'=(H^1(\mathcal {M}))',\\ \mu =a\varphi -J*\varphi +F'(\varphi ),\\ (u,\varphi )(0)=(u_0,\varphi _0), \end{array}\right. } \end{aligned}$$

(2.17)

or equivalently

$$\begin{aligned} {\left\{ \begin{array}{ll} u(t)+\displaystyle \int _0^t(N(u(s))-B_0(u(s))-R_1(\varphi (s)))ds=u_0+\displaystyle \int _0^tg_0(s)ds\\ \quad +\displaystyle \int _0^tg_1(u(s),\varphi (s))ds+\displaystyle \int _0^t\bar{g}_2(s,u(s),\varphi (s))d\bar{W}(s),\\ \varphi (t)-\displaystyle \int _0^tB_1(u(s),\varphi (s))ds=\varphi _0+\displaystyle \int _0^t\Delta \mu (s)ds,\\ \mu =a\varphi -J*\varphi +F'(\varphi ), \end{array}\right. } \end{aligned}$$

(2.18)

$\mathbb {P}$-a.s, and for all $t\in [0,T]$, where

$$\begin{aligned} \begin{array}{ll} g_0\in L^2(0,\infty ;G_{{\text {div}}}),~g_1: G_{{\text {div}}}\times H\rightarrow V'_{{\text {div}},p},~g_2:[0,\infty )\times G_{{\text {div}}}\times H\rightarrow L_2(K,G_{{\text {div}}}). \end{array} \end{aligned}$$

(2.19)

Remark 2.6

The pressure is excluded from (2.17) as usual; in fact, one (formally) has

$$\begin{aligned} \begin{array}{ll} \mu \nabla \varphi =\nabla \left( F(\varphi )+\frac{a}{2}\varphi ^2-(J*\varphi )\varphi \right) -\frac{\textstyle \nabla a}{\textstyle 2}\varphi ^2+(\nabla J*\varphi )\varphi . \end{array} \end{aligned}$$

This explains that $\mu \nabla \varphi =R_1(\varphi )$ in $V'_{{\text {div}},p}$. Here $J\in W^{1,1}_{{\text {loc}}}(\mathbb {R}^3)$. More assumptions on the kernel J will be given below (see (H5)).

We also introduce additional notations frequently used throughout the work. The mathematical expectation with respect to the probability measure $\mathbb {P}$ is denoted by $\mathbb {E}$. For a probability space $(\Omega ,\mathcal {F},\mathbb {P})$ and a Banach space X, we denote by $L^\gamma (\Omega ,\mathcal {F},\mathbb {P};L^{q}(0,T;X))$ $(1\le \gamma ,q<\infty )$ the space of random functions $u:[0,T]\times \mathcal {M}\times \Omega \rightarrow L^q(0,T;X)$ such that u is measurable w.r.t. $(t,\omega )$ and for all t, u is measurable w.r.t. $\mathcal {F}_t$, with $(\mathcal {F}_t)_{0\le t\le T}$ be a filtration of nondecreasing and right continuous family of sub $\sigma$-algebra of $\mathcal {F}$ with $\mathcal {F}_0$ containing all the $\mathbb {P}$-null sets. We furthermore endow this space with the norm

$$\begin{aligned} \Vert u\Vert _{L^\gamma (\Omega ,\mathcal {F},\mathbb {P};L^{q}(0,T;X))}=[\mathbb {E}\Vert u\Vert _{L^{q}(0,T;X)}^\gamma ]^{1/\gamma }. \end{aligned}$$

If $q=\infty$, we write

$$\begin{aligned} \Vert u\Vert _{L^\gamma (\Omega ,\mathcal {F},\mathbb {P};L^{\infty }(0,T;X))}=[\mathbb {E}\text {ess}\displaystyle \sup _{t\in [0,T]}\Vert u(t)\Vert _X^\gamma ]^{1/\gamma }. \end{aligned}$$

Let us introduce the hypotheses on $g_1(u,\varphi ), \varvec{T}, J$ and F that are relevant for the major part of the paper.

$(H_1)$:

We assume that $g_1:G_{{\text {div}}}\times H\rightarrow V'_{{\text {div}},p}$ is nonlinear mapping such that

(a):

$(u,\varphi )\mapsto g_1(u,\varphi )$ is continuous; there exists a positive constant C such that

$$\begin{aligned} \Vert g_1(u,\varphi )\Vert _{V'_{{\text {div}},p}}\le C(1+|u|+|\varphi |),\ \ \forall (u,\varphi )\in G_{{\text {div}}}\times H. \end{aligned}$$

$(H_2)$:

We suppose that $g_2:[0,T]\times G_{{\text {div}}}\times H\rightarrow L_2(K,G_{{\text {div}}})$ is nonlinear mapping such that it is continuous in both variables. We require that, for any $t\in [0,T]$ and $(u,\varphi )\in G_{{\text {div}}}\times H$, $g_2(t,u,\varphi )$ satisfy

$$\begin{aligned} \Vert g_2(t,u,\varphi )\Vert _{L_2(K, G_{{\text {div}}})}=\Vert \bar{g}_2(t,u,\varphi )\Vert _{L_2(Q^{1/2}(K),G_{{\text {div}}})}\le C(1+|u|+|\varphi |), \end{aligned}$$

with $\bar{g}_2$ defined as in (2.16).

As in ([15]), our assumption on the stress tensor $\varvec{T}$, the potential F and the kernel J are the following:

$(H_3)$:

$\varvec{T}(.)$ continuously depends on a symmetric tensor $e\in \mathbb {R}^{3\times 3}$ and satisfies the following conditions

$$\begin{aligned} \left( \varvec{T}(E)-\varvec{T}(S)\right) .(E-S)\ge {\left\{ \begin{array}{ll} c_1(1+|E|+|S|)^{p-2}|E-S|^2\\ c_2|E-S|^2+c_2|E-S|^p \end{array}\right. } \end{aligned}$$

(2.20)

$$\begin{aligned} |\varvec{T}(E)-\varvec{T}(S)|\le c_3(1+|E|+|S|)^{p-2}|E-S|,~\varvec{T}(0)=0, \end{aligned}$$

for all $E, S\in \mathbb {R}^{3\times 3}$, for some $c_i>0$, $i=1,2,3$ and some $p>2$. Here $|.|$ stands for a Euclidean norm of a tensor and $``.''$ at the left hand side of (2.20) denotes the scalar product of two tensors.

$(H_4)$:

$F\in \mathcal {C}^2(\mathbb {R})$ has a polynomially controlled growth

$$\begin{aligned} |F'(s)|^r\le c_4(1+|F(s)|),~r\in (1,2], \end{aligned}$$

(2.21)

for some $c_4>0$ and satisfies the coercivity condition:

$$\begin{aligned} F''(s)+a(x)\ge c_5\max \{1,|s|^{2\kappa }\}, \end{aligned}$$

(2.22)

for all $s\in \mathbb {R}$, almost any $x\in \mathcal {M}$, some $c_5>0$ and some $\kappa \ge 0$.

$(H_5)$:

$J\in W_{{\text {loc}}}^{1,1}(\mathbb {R}^3)$, $J(x)=J(-x)$ and $a(x)=\int _\mathcal {M}J(x-y)dy\ge 0$ a.e., in $\mathcal {M}$. Moreover, we set

$$\begin{aligned} a^*:=\displaystyle \sup _{x\in \mathcal {M}}\int _\mathcal {M}|J(x-y)|dy<\infty ,~ b^*:=\displaystyle \sup _{x\in \mathcal {M}}\int _\mathcal {M}|\nabla J(x-y)|dy<\infty . \end{aligned}$$

Remark 2.7

Assumption $J\in W_{{\text {loc}}}^{1,1}(\mathbb {R}^3)$ can be weakened. Indeed, it can be replaced by $J\in W^{1,1}(B_\delta )$, where $B_\delta :=\{z\in \mathbb {R}^3:|z|<\delta \}$ with $\delta :=\text {diam}(\mathcal {M})=\sup _{x,y\in \mathcal {M}}d_*(x,y)$, where $d_*(.,.)$ is the Euclidean metric on $\mathbb {R}^3$.

Remark 2.8

The hypothesis (2.22) is physically justified and relevant (see [20] for more details). From the mathematical viewpoint assumption (2.22) is satisfied, in particular, if $a\equiv 0$ and F is strictly convex.

Remark 2.9

(2.22) implies the existence of positive constants $c_6>0$ and $c_7>0$ such that

$$\begin{aligned} F(s)\ge c_6|s|^{2\kappa +2}-c_7,~\text {for all}~s\in \mathbb {R}. \end{aligned}$$

In order to formulate problem (1.8) in the framework of the proof of existence theorem in [27, Chapter IV, Section 2, pp. 167–177], which do not require the Lipschitz condition on the coefficients, we need some preliminaries which we state below.

Lemma 2.1

(Korn’s inequalities) Let $1<m<\infty$ and let $\mathcal {M}\subset \mathbb {R}^3$ be of class $\mathcal {C}^1$. Then, there exist two positive constants $\kappa _ m^i=\kappa _ m^i(\mathcal {M})$, $i=1,2$ such that

$$\begin{aligned} \kappa _m^1\Vert v\Vert _{1,m}\le \left( \int _{\mathcal {M}}|Dv|^mdx\right) ^{1/m}\le \kappa _m^2\Vert v\Vert _{1,m},~\forall v\in V_{{\text {div}},m}. \end{aligned}$$

Proof

The proof of this lemma can be found in [39, Chapter 5, Theorem 1.10]. $\square$

Let X be a Banach space and $X'$ be its topological dual. Let $\mathbb {T}$ be a function from X to $X'$ with domain $\mathbb {D}=\mathbb {D}(\mathbb {T})\subseteq X$.

Definition 2.1

[44, Definition 2.3] The function $\mathbb {T}$ is said to be

• (a) demicontinuous if for a sequence $v_n\in \mathbb {D}$, $v\in \mathbb {D}$ and $v_n\rightarrow v$ in X implies that $\mathbb {T}(v_n){\mathop {\rightharpoonup }\limits ^{\text {weakly}}}\mathbb {T}(v)$ in $X'$,

• (b) hemicontinuous if $v\in \mathbb {D}$, $u\in X$ and $v+t_nu\in \mathbb {D}$ for a sequence of positive real numbers $t_n$ such that $t_n\rightarrow 0$ implies $\mathbb {T}(v+t_nu){\mathop {\rightharpoonup }\limits ^{\text {weakly}}}\mathbb {T}(v)$ in $X'$,

• (c) locally bounded if for a sequence $v_n\in \mathbb {D}$, $v\in \mathbb {D}$ and $v_n\rightarrow v$ in X imply that $\mathbb {T}(v_n)$ is bounded in $X'$.

From the above definition, it is clear that a demicontinuous function is hemicontinuous and locally bounded.

We now introduce the following result concerning the operator $N:V_{{\text {div}},p}\rightarrow V'_{{\text {div}},p}$.

Proposition 2.1

We assume that T satisfies $(H_3)$ with $p\ge 2$. Then,

• (a) the operator N is demicontinuous,

• (b) the operator N is monotone; that is, $\left\langle N(u)-N(v),u-v\right\rangle _{V_{{\text {div}},p}}\ge 0,\ \ \forall u,v\in V_{{\text {div}},p}$.

• (c) There exists a positive constant C such that

  $$\begin{aligned} \Vert N(u)\Vert _{V'_{{\text {div}},p}}^{p'}\le C(1+\Vert u\Vert _{1,p}^p),\ \ \forall u\in V_{{\text {div}},p},~\text {with}~p'~\text {the conjugate index to}~p. \end{aligned}$$

  (2.23)

Proof

Let $p>2$. The item (b) of proposition 2.1 follows easily from (2.10)$_1$ and (2.20)$_1$.

Proof of item (a). Let $(v_n)_{n\ge 1}$ be a sequence of points of $V_{{\text {div}},p}$ and $v\in V_{{\text {div}},p}$ be such that $v_n\rightarrow v$ in $V_{{\text {div}},p}$. Let $u\in V_{{\text {div}},p}$. We have

$$\begin{aligned} \begin{aligned}&\left\langle N(v_n)-N(v),u\right\rangle _{V_{{\text {div}},p}}(\le )\int _\mathcal {M}|\varvec{T}(Dv_n)-\varvec{T}(Dv)||Du|dx\\&\quad (\le )~ c_3\int _\mathcal {M}(1+|Dv_n|+|Dv|)^{p-2}|D(v_n-v)||Du|dx\\&\quad \le c_{c_3,p}\left( \int _\mathcal {M}|Du|^pdx\right) ^{\frac{1}{p}}\left( \int _\mathcal {M}(1+|Dv_n|^p+|Dv|^p) dx\right) ^{\frac{p-2}{p}}\left( \int _\mathcal {M}|D(v_n-v)|^p dx\right) ^{\frac{1}{p}}, \end{aligned} \end{aligned}$$

(2.24)

where we have used Hölder’s inequality and (2.20)$_1$. Here $c_{c_3,p}$ is a positive constant depending on p and $c_3$. It then follows from (2.24) and Lemma (2.1) that

$$\begin{aligned} \left\langle N(v_n)-N(v),u\right\rangle _{V_{{\text {div}},p}}\le c_{c_3,p}(\kappa _p^2)^2\Vert u\Vert _{1,p}[1+(\kappa _p^2)^p(\Vert v_n\Vert _{1,p}^p+\Vert v\Vert _{1,p}^p)]\Vert v_n-v\Vert _{1,p} \end{aligned}$$

(2.25)

for all $u, v\in V_{{\text {div}},p}$. Therefore from (2.25), the fact that $v_n\rightarrow v$ in $V_{{\text {div}},p}$; i.e., $\Vert v_n-v\Vert _{1,p}\rightarrow 0$ as $n\rightarrow \infty$, $v\in V_{{\text {div}},p}$ and every convergent sequence is bounded, we deduce that

$$\begin{aligned} \left\langle N(v_n)-N(v),u\right\rangle _{V_{{\text {div}},p}}\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$ for every $u\in V_{{\text {div}},p}$. This proves that the operator N is demicontinuous; and hence hemicontinuous and locally bounded.

By definition we have

$$\begin{aligned} \Vert N(u)\Vert _{V'_{{\text {div}},p}}=\sup _{\Vert v\Vert _{1,p}=1}|\left\langle N(u),v\right\rangle _{V_{{\text {div}},p}}|. \end{aligned}$$

Hence, thanks to (2.10)$_1$, using Hölder’s and Korn’s inequalities we have

$$\begin{aligned} \begin{aligned} \Vert N(u)\Vert _{V'_{{\text {div}},p}}\le C\left[ \int _\mathcal {M}|\varvec{T}(Du)|^{p'}ds\right] ^\frac{1}{p'}\le C\left[ \int _\mathcal {M}(1+|Du|^p)ds\right] ^{1/p'}, \end{aligned} \end{aligned}$$

(2.26)

where we have also used (2.20)$_2$.

Finally, thanks to (2.26) in conjunction with Korn’s inequality, we obtain (2.23). $\square$

3 Statement of the Main Result

We introduce the concept of solution of the problem (2.17) or (2.18) that is interest to us.

Definition 3.1

By a solution of the problem (2.17) or (2.18), we mean a system $((\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}),\tilde{W}_t,\tilde{u},\tilde{\varphi })$, where

• (1) $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$ is a complete probability space; $\tilde{\mathbb {F}}:=\{\tilde{\mathcal {F}}_t:t\ge 0\}$ is a filtration on the probability space $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$, i.e., a nondecreasing family $\{\tilde{\mathcal {F}}_t:t\ge 0\}$ of sub $\sigma$-fields of $\tilde{\mathcal {F}}$: $\tilde{\mathcal {F}}_s\subset \tilde{\mathcal {F}}_t\subset \tilde{\mathcal {F}}$ for $0\le s<t<\infty$;

• (2) $\tilde{W}_t$ is a $\tilde{\mathcal {F}}_t$-cylindrical Wiener process on $G_{{\text {div}}}$;

• (3) for almost every $t\in [0,T]$, $\tilde{u}(t)$ and $\tilde{\varphi }(t)$ are $\tilde{\mathcal {F}}_t$ measurable;

• (4) for almost every t, $\tilde{u}(t)\in L^{q}(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p}))\cap L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;G_{{\text {div}}}))$, $\tilde{\varphi }(t)\in L^{q}(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))\cap L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L^{2\kappa +2}(\mathcal {M})))$, $2\le q<\infty$;

• (5) $\tilde{\mathbb {P}}$-a.s the following integral equations of Itô type hold:

  $$\begin{aligned} \begin{aligned}&(\tilde{u}(t)-u_0,v)+\displaystyle \int _0^t\left\langle (N(\tilde{u}(s))-B_0(\tilde{u}(s))-R_1(\tilde{\varphi }(s))),v\right\rangle _{V_{{\text {div}},p}} ds\\&\quad =\displaystyle \int _0^t\left\langle g_0(s),v\right\rangle _{V_{{\text {div}}},p} +\displaystyle \int _0^t\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),v\right\rangle _{V_{{\text {div}},p}}ds+\displaystyle \int _0^t(g_2(s,\tilde{u}(s),\tilde{\varphi }(s)),v)d\tilde{W}_s,\\&(\tilde{\varphi }(t)-\varphi _0,\psi )-\displaystyle \int _0^t\left\langle B_1(\tilde{u}(s),\tilde{\varphi }(s)),\psi \right\rangle _U ds=\displaystyle \int _0^t\left\langle \Delta \tilde{\mu }(s),\psi \right\rangle _U ds,\\&\tilde{\mu }=a\tilde{\varphi }-J*\tilde{\varphi }+F'(\tilde{\varphi }), \end{aligned} \end{aligned}$$

  (3.1)

  for any $t\in [0,T]$ and $(v,\psi )\in \mathbf{W} _s\times U:=\mathbf{W} _s\times H^1(\mathcal {M})$, $s\ge 5/2> 5/2-3/p$.

Now we can state our first result in the following theorem.

Theorem 3.1

Let $p\ge 11/5$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} \kappa \ge \frac{\textstyle 2(3-p)}{\textstyle 5p-6},\quad \text {if}~p<3,\\ 0<\kappa \le 2,\quad \text {if}~p=3,\\ \kappa>0\quad \text {if}~p>3. \end{array}\right. } \end{aligned}$$

(3.2)

Assume that $u_0\in G_{{\text {div}}}$, $\varphi _0\in H$ with $F(\varphi _0)\in L^1(\mathcal {M})$ and $g_0\in L^{p'}(0,T;V'_{{\text {div}},p})$. Suppose also that all the assumptions, namely, $(H_1)$ and $(H_5)$ are satisfied. Then problem (2.17) or (2.18) has a solution in the sense of the above definition. Moreover, we note that a solution $(\tilde{u},\tilde{\varphi })$ in the sense of definition 3.1 belongs to $L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};\mathcal {C}(0,T;\mathbb {H}))$, $q\in [2,\infty )$.

Proof

The proof will be carried out in Sects. 4.1–4.4. $\square$

In the rest of the paper, we set

$$\begin{aligned} \mathcal {E}_{tot}(u,\varphi ):=|u|^2+2\mathcal {E}(\varphi ),~\text {where}~\mathcal {E}~\text {is given by}~(1.5). \end{aligned}$$

(3.3)

4 Auxiliary Results

In this section, we introduce the Galerkin approximation scheme to reduce the original system to a system of finite-dimensional ordinary stochastic differential equations (SDEs). We derive crucial a priori estimates from the Galerkin approximation which will serve as a toolkit for the proof of Theorem 3.1. More precisely, the priori estimates will be used to prove the tightness of the family of laws of the sequence of solutions of the system of SDEs on appropriate topological spaces.

4.1 The Approximate Solution

We first assume that $\varphi _0\in D(\mathcal {B})=\{\varphi \in H^2(\mathcal {M}):\partial _\eta \varphi =0~\text {on}~\partial \mathcal {M}\}\subset H$ instead of $\varphi _0\in H$, where $\mathcal {B}=-\Delta +I$. The general case $\varphi _0\in H$ with $F(\varphi _0)\in L^1(\mathcal {M})$ can be dealt in the same fashion as in [10], by means of a density argument and by relying on the form of the potential F as a quadratic perturbation of a convex function.

As a Galerkin base in $V_{{\text {div}},p}$ we employ the family $\{w_j, j=1,2,\ldots \}$, where each $w_j$ solves

$$\begin{aligned} (w_j,v)_{W_s}=\lambda _j(w_j,v),\quad \forall v\in W_s. \end{aligned}$$

In U we choose as Galerkin base the family $\{\psi _j, j=1,2,\ldots \}$, where $\psi _j$ are the eigenfunctions of the operator $\mathcal {B}$. We set $\text {W}_s^n=\text {span}\{w_1,w_2,\ldots ,w_n\}$ and $H_n=\text {span}\{\psi _1,\psi _2,\ldots ,\psi _n\}$. Let $\mathcal {P}_n^1$ be the operator from $\mathbf{W} '_s$ to $\mathbf{W} _s^n$ defined by $\mathcal {P}_n^1u^*:=\sum _{j=1}^n\left\langle u^*,w_j\right\rangle _{W_s}w_j,~u^*\in \mathbf{W} '_s$. We will consider the restriction of the operator $\mathcal {P}_n^1$ to the space $G_{{\text {div}}}$ (still) denoted by $\mathcal {P}_n^1$. More precisely, we have $G_{{\text {div}}}\cong G'_{{\text {div}}}\hookrightarrow V'_{{\text {div}},p}\hookrightarrow \mathbf{W} '_{s}$ $(s>5/2)$, i.e. every element $u\in G_{{\text {div}}}$ induces a functional $u^*\in W'_s$ by the formula

$$\begin{aligned}\left\langle u^*,v\right\rangle _\mathbf{W _s}=(u,v),~v\in \mathbf{W} _s.\end{aligned}$$

Thus the restriction of $\mathcal {P}_n^1$ to $G_{{\text {div}}}$ is given by $\mathcal {P}_n^1u=\sum _{j=1}^n(u,w_j)w_j,~u\in G_{{\text {div}}}$. Hence in particular, $\mathcal {P}_n^1$ is the (., .)-orthogonal projection from $G_{{\text {div}}}$ onto $\mathbf{W} _s^n$. Similarly, we define by $\mathcal {P}_n^2$ the orthogonal projection from H onto $H_n$. Let $(\Omega ,\mathcal {F},\mathbb {P},W_t)$ ($W_t$ is a cylindrical Wiener processes evolving on K). We equip the probability space $(\Omega ,\mathcal {F},\mathbb {P})$ with the natural filtration of $W_t$ which is denoted by $\mathcal {F}_t$. We then look for the three functions of the form

$$\begin{aligned}u_n(t)=\sum _{k=1}^na_k^{(n)}(t)w_k,~\varphi _n(t)=\sum _{k=1}^nb_k^{(n)}(t)\psi _k,~\mu _n(t)=\sum _{k=1}^nc_k^{(n)}(t)\psi _k,\end{aligned}$$

which solves the following approximating problem

$$\begin{aligned} {\left\{ \begin{array}{ll} d(u_n,w_j)+[ (\varvec{T}(Du_n),Dw_j)+b(u_n,u_n,w_j)]dt=-(\varphi _n\nabla \mu _n,w_j)dt +(\mathcal {P}_n^1g_0,w_j)dt\\ \quad +(g_1(u_n,\varphi _n),w_j)dt+\sum _{i=1}^n( g_2(t,u_n,\varphi _n)e_i,w_j)d\beta _t^i,\\ d(\varphi _n,\psi _j)+(\nabla \mu _n,\nabla \psi _j)dt=(u_n\varphi _n,\nabla \psi _j)dt,\\ \mu _n=\mathcal {P}_n^2(a\varphi _n-J*\varphi _n+F'(\varphi _n)),\\ u_n(0)=\mathcal {P}_n^1u_0:=u_{0n},~\varphi _n(0)=\mathcal {P}_n^2\varphi _0:=\varphi _{0n}, \end{array}\right. } \end{aligned}$$

(4.1)

where $u_{0n}$, $\varphi _{0n}$ and $\mathcal {P}_n^1g_0$ are such that $u_{0n}\rightarrow u_0$ in $G_{{\text {div}}}$, $\varphi _{0n}\rightarrow \varphi _0$ in $H^2(\mathcal {M})$ and $\mathcal {P}_n^1g_0\rightarrow g_0$ in $L^{p'}(0,T;V'_{{\text {div}},p})$ as $n\rightarrow \infty$ respectively.

We first note that since the operator N is hemicontinuous and monotone (see Proposition 2.1), we infer from [33, Chapitre II, page 171] that N is continuous from $V_{{\text {div}},p}$ into $V'_{{\text {div}},p}$ and locally bounded. System (4.1) is then a system of stochastic differential equations in a finite dimensional Banach spaces with continuous and locally bounded coefficients. From the existence theorem in [49, Chapter 3, Section 3, p. 59] (see also [27, Chapter 4, Section 2, pp. 167-177]), which do not require the Lipschitz condition on the coefficients, there exists on a short interval $[0,T_n)$, $T_n\le T$ a sequence of continuous functions $(u_n,\varphi _n)$ solving (4.1). It will follows from a priori estimates that $(u_n,\varphi _n)$ exists on [0, T].

4.2 A Priori Estimates

In this subsection, we derive some basic energy estimates for the sequence of the approximate solutions $u_n$, $\varphi _n$ and for the sequence $\mu _n$, $F(\varphi _n)$ and $F'(\varphi _n)$.

First, we prove the following lemma.

Lemma 4.1

the sequence $(u_n,\varphi _n,\mu _n, F(\varphi _n), F'(\varphi _n), n=1,2,\ldots )$ satisfies

$$\begin{aligned} \begin{aligned}&\mathbb {E}\sup _{s\in [0,T]}|u_n(s)|^q<C,\ \ \mathbb {E}\sup _{s\in [0,T]}\Vert \varphi _n(s)\Vert _{L^{2\kappa +2}(\mathcal {M})}^q<C,\\&\mathbb {E}\sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}^{q/2}< C,\ \ \mathbb {E}\sup _{s\in [0,T]}\Vert F'(\varphi _n(s))\Vert _{L^r(\mathcal {M})}^q< C, \end{aligned} \end{aligned}$$

(4.2)

and

$$\begin{aligned} \begin{aligned}&\mathbb {E}\left( \int _0^T\Vert u_n(s)\Vert _{1,p}^pds\right) ^\frac{q}{p}<C, \mathbb {E}\left( \int _0^T\Vert \varphi _n(s)\Vert _U^2ds\right) ^\frac{q}{2}<C,\mathbb {E}\left( \int _0^T\Vert \mu _n(s)\Vert _U^2ds\right) ^\frac{q}{2}<C, \end{aligned} \end{aligned}$$

(4.3)

for any $q\in [2,\infty )$, $r\in (1,2]$ and $p\ge 11/5$, with $U:=H^1(\mathcal {M})$. Here C is a positive constant depending on the parameters $T,q,p,\mathcal {M},\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)},\kappa ,r,c_2,c_4,c_5,c_6$ and the initial data $u_0,\varphi _0$ and $|F(\varphi _0)|_{L^1(\mathcal {M})}$. We recall that the constant $c_3$ is given by $(H_3)$, the constants $c_4$ and $c_5$ are given by $(H_4)$, the constant $c_6$ is defined as in Remark 2.9, and the constants $\kappa _2^1$ and $\kappa _p^1$ are given by Korn’s inequality (see Lemma 2.1).

Proof

Let $\tau _n^R, R,n\in \mathbb {N}$, be stopping time defined by

$$\begin{aligned} \begin{array}{ll} \tau _n^R=\inf \{t\in [0,T];|u_n(t)|^2+\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}}^2+\int _0^t(\Vert u_n(s)\Vert _{1,p}^p+\Vert \varphi _n(s)\Vert _U^2)ds\ge R^2\}\wedge T. \end{array} \end{aligned}$$

(4.4)

Let $t\in [0,\tau _n^R\wedge T]$. By applying Itô’s formula to the process $|u_n(t)|^2$, taking $\mu _n$ as test function in (4.1)$_2$, recalling that $b(u_n,u_n,u_n)=0$ (see (2.9)) and by summing the ensuing identities, we obtain

$$\begin{aligned} \begin{aligned}&[\mathcal {E}_{tot}(u_n(t),\varphi _n(t))+\bar{\kappa }_1]+2\displaystyle \int _0^t(\varvec{T}(Du_n),Du_n)ds+2\displaystyle \int _0^t|\nabla \mu _n|^2ds\\&\quad =\bar{\kappa }_1+\mathcal {E}_{tot}(u_{0n},\varphi _{0n}) +2\displaystyle \int _0^t\left\langle \mathcal {P}_n^1g_0,u_n\right\rangle _{V_{{\text {div}},p}}ds+2\displaystyle \int _0^t\left\langle g_1(u_n,\varphi _n),u_n\right\rangle _{V_{{\text {div}},p}}ds\\&\qquad +\sum _{j,k=1}^n\displaystyle \int _0^t[(g_2(s,u_n,\varphi _n)e_j,w_k)]^2ds+2\sum _{j=1}^n\displaystyle \int _0^t(g_2(s,u_n,\varphi _n)e_j,u_n)d\beta _s^j, \end{aligned} \end{aligned}$$

(4.5)

with

$$\begin{aligned} \begin{array}{ll} \bar{\kappa }_1=\frac{\textstyle \kappa |\mathcal {M}|}{\textstyle c_6^{1/\kappa }}\left( \frac{\textstyle |J|_{L^1(\mathbb {R}^3)}}{\textstyle \kappa +1}\right) ^{(\kappa +1)/\kappa }+2c_7|\mathcal {M}|. \end{array} \end{aligned}$$

(4.6)

Note that the constant $\bar{\kappa }_1$ is such that $[\mathcal {E}_{tot}(u_n(t),\varphi _n(t))+\bar{\kappa }_1]\ge 0$. In fact, we have

$$\begin{aligned} \begin{array}{ll} 2\mathcal {E}(\varphi _n(t))&{}=2|\sqrt{a}\varphi _n(t)|^2+2\int _\mathcal {M}F(\varphi _n(t,x))dx-(\varphi _n(t),J*\varphi _n(t))\\ &{}\ge \int _\mathcal {M}(a(x)-|J|_{L^1(\mathbb {R}^3)})(\varphi _n(t,x))^2dx+2c_6\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}-2c_7|\mathcal {M}|\\ &{}\ge c_6\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}-\bar{\kappa }_1, \end{array} \end{aligned}$$

(4.7)

where we have used Hölder’s inequality, Young’s inequality for convolutions, Young’s inequality and Remark 2.9. It then follows from (4.7) that

$$\begin{aligned} \begin{array}{ll} \mathcal {E}_{tot}(u_n(t),\varphi _n(t))+\bar{\kappa }_1&{}=|u_n(t)|^2+2\mathcal {E}(\varphi _n(t))+\bar{\kappa }_1\\ &{}\ge |u_n(t)|^2+c_6\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}\ge 0. \end{array} \end{aligned}$$

(4.8)

Now since

$$\begin{aligned} \sum _{j,k=1}^n[(g_2(s,u_n(s),\varphi _n(s))e_j,w_k)]^2\le \Vert g_2(s,u_n(s),\varphi _n(s))\Vert _{L_2(K,G_{{\text {div}}})}^2, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \mathcal {E}_{tot}(u_{0n},\varphi _{0n})&\le |u_{0n}|^2+2|J|_{L^1(\mathbb {R}^3)}|\varphi _{0n}|^2+2\int _{\mathcal {M}}F(\varphi _{0n}(x))dx\\&\le |u_0|^2+2|J|_{L^1(\mathbb {R}^3)}|\varphi _0|^2+2\int _{\mathcal {M}}F(\varphi _0(x))dx\\&\equiv \mathcal {K}(u_0,\varphi _0), \end{aligned} \end{aligned}$$

(4.9)

(where in (4.9), we have used the fact that, since $\varphi _0\in D(\mathcal {B})$ a.s., then we have $\varphi _{0n}\rightarrow \varphi _0$ in $H^2(\mathcal {M})$ a.s. and hence also in $L^\infty (\mathcal {M})$ a.s.), it follows from (4.5) that

$$\begin{aligned} \begin{aligned}&[\mathcal {E}_{tot}(u_n(t),\varphi _n(t))+\bar{\kappa }_1]+2\mathcal {Z}_p\displaystyle \int _0^t[\Vert u_n(s)\Vert ^2+\Vert u_n(s)\Vert _{1,p}^p]ds+2\displaystyle \int _0^t|\nabla \mu _n|^2ds\\&\quad \le \bar{\kappa }_1+\mathcal {K}(u_0,\varphi _0)+2\displaystyle \int _0^t[\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}+\Vert g_1(u_n(s),\varphi _n(s))\Vert _{V'_{{\text {div}},p}}]\Vert u_n(s)\Vert _{1,p}ds\\&\qquad +\displaystyle \int _0^t\Vert g_2(s,u_n(s),\varphi _n(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds+2\displaystyle \int _0^t\sum _{j=1}^n(g_2(s,u_n(s),\varphi _n(s))e_j,u_n)d\beta _s^j, \end{aligned} \end{aligned}$$

(4.10)

with $\mathcal {Z}_p=c_2\min \left( (\kappa _2^1)^2,(\kappa _p^1)^p\right)$. Note that in (4.10) we have also used the assumption $(H_3)$ (see (2.20)) in conjunction with Lemma 2.1.

Hereafter, we set (for the sake of simplicity):

$$\begin{aligned} \begin{aligned} \chi _n(t)&:=\mathcal {E}_{tot}(u_n(t),\varphi _n(t))+\bar{\kappa }_1,\ \ \vartheta _{p,n}:=[\Vert u_n(t)\Vert ^2+\Vert u_n(t)\Vert _{1,p}^p],\\ \varpi _n(t)&:=\int _0^t\sum _{j=1}^n(g_2(s,u_n(s),\varphi _n(s))e_j,u_n(s))d\beta _s^j,\ \ t\in [0,T]. \end{aligned} \end{aligned}$$

(4.11)

Setting $a_1=(\frac{p\mathcal {Z}_p}{2})^\frac{1}{p}\Vert u_n(s)\Vert _{1,p}, b=2(\frac{1}{p})^\frac{1}{p}(\frac{1}{\mathcal {Z}_p})^\frac{1}{p}\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}\times 2^\frac{1}{p}$, using Young’s inequality, we see that

$$\begin{aligned} \begin{array}{ll} 2\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}\Vert u_n(s)\Vert _{1,p}=a_1b&\le \frac{1}{p}a_1^p+\frac{1}{p'}b^{p'}=\frac{\mathcal {Z}_p}{2}\Vert u_n(s)\Vert _{1,p}^p+\frac{(2)^{\frac{p'(p+1)}{p}}}{p'\mathcal {Z}_p^{\frac{p'}{p}}p^{\frac{p'}{p}}}\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}. \end{array} \end{aligned}$$

(4.12)

Here $p'=\frac{p}{p-1}$. One can easily see that

$$\begin{aligned} \begin{array}{ll} 2\Vert g_1(u_n(s),\varphi _n(s))\Vert _{V'_{{\text {div}},p}}\Vert u_n(s)\Vert _{1,p}\le \frac{\mathcal {Z}_p}{2}\Vert u_n(s)\Vert _{1,p}^2+\frac{2}{\mathcal {Z}_p}\Vert g_1(u_n(s),\varphi _n(s))\Vert _{V'_{{\text {div}},p}}^2. \end{array} \end{aligned}$$

(4.13)

Now setting $r_1=\frac{p}{2}, r_2=\frac{p}{p-2}, a_1=(\frac{p}{2})^\frac{2}{p}\Vert u_n(s)\Vert _{1,p}^2, b=(\frac{2}{p})^\frac{2}{p}$, using Young’s inequality, we have

$$\begin{aligned} \begin{array}{ll} \Vert u_n(s)\Vert _{1,p}^2=a_1b\le \frac{1}{r_1}a_1^{r_1}+\frac{1}{r_2}b^{r_2}=\Vert u_n(s)\Vert _{1,p}^p+\frac{(p-2)\times 2^\frac{2}{p-2}}{p[p]^\frac{2}{p-2}}. \end{array} \end{aligned}$$

(4.14)

It then follows from estimates (4.12)–(4.14) that

$$\begin{aligned} \begin{array}{ll} &{} 2[\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}+\Vert g_1(u_n(s),\varphi _n(s))\Vert _{V'_{{\text {div}},p}}]\Vert u_n(s)\Vert _{1,p}\\ &{}\quad \le \mathcal {Z}_p\Vert u_n\Vert _{1,p}^p+\frac{\mathcal {Z}_p(p-2)}{2p}\left( \frac{2}{p}\right) ^{\frac{2}{p-2}}+\frac{2}{\mathcal {Z}_p}\Vert g_1(u_n,\varphi _n)\Vert _{V'_{{\text {div}},p}}^2+\frac{(2)^{\frac{p'(p+1)}{p}}}{p'\mathcal {Z}_p^{\frac{p'}{p}}p^{\frac{p'}{p}}}\Vert g_0\Vert _{V'_{{\text {div}},p}}^{p'}. \end{array} \end{aligned}$$

(4.15)

By Young inequality’s, we infer that

$$\begin{aligned} |\varphi _n|^2\le \frac{c_6}{\kappa +1}\Vert \varphi _n\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}+\frac{\kappa |\mathcal {M}|}{(\kappa +1)c_6^{1/\kappa }}. \end{aligned}$$

(4.16)

Now, owing to the assumption on $g_1$ and $g_2$, we can derive from the estimates (4.8), (4.15)–(4.16) and (4.10) that

$$\begin{aligned} \begin{array}{ll} &{}\chi _n(t)+2\mathcal {Z}_p\displaystyle \int _0^t\vartheta _{p,n}(s)ds+2\int _0^t|\nabla \mu _n(s)|^2ds\\ &{}\quad \le \bar{\kappa }_1+\mathcal {K}(u_0,\varphi _0)+C_1\displaystyle \int _0^t\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds+C_2\displaystyle \int _0^t(1+\chi _n(s))ds+2\varpi _n(t), \end{array} \end{aligned}$$

(4.17)

with $C_1:=C_1(\kappa _2^1,\kappa _p^1,p)$ and $C_2:=C_2(\kappa _2^1,\kappa _p^1,c_2,c_6,\kappa ,p,|\mathcal {M}|)$. $\chi _n(t), \vartheta _{p,n}(t)$ and $\varpi _n(t)$ are defined as in (4.11).

From (4.17), we infer that

$$\begin{aligned} \begin{aligned}&\mathbb {E}\displaystyle \sup _{s\in [0 ,t\wedge \tau _n^R]}[\chi _n(s)]+2\mathbb {E}\displaystyle \int _0^{t\wedge \tau _n^R}[\mathcal {Z}_p\vartheta _{p,n}(s)+|\nabla \mu _n(s)|^2]ds\\&\quad \le \bar{\kappa }_1+\mathcal {K}(u_0,\varphi _0)+C_1\displaystyle \int _0^{t\wedge \tau _n^R}\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds+C_2\mathbb {E}\displaystyle \int _0^{t\wedge \tau _n^R}(1+\chi _n(s))ds+2\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}|\varpi _n(s)|. \end{aligned} \end{aligned}$$

(4.18)

By Burkholder–Davis–Gundy’s inequality and Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned} 2\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}|\varpi _n(s)|&= 2\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}\left| \int _0^s\sum _{j=1}^n(g_2(\tau ,u_n(\tau ),\varphi _n(\tau ))e_j,u_n(\tau ))d\beta _\tau ^j\right| \\&\le C\mathbb {E}\left( \int _0^{t\wedge \tau _n^R}\sum _{j=1}^n(g_2(s,u_n(s),\varphi _n(s))e_j,u_n)^2ds\right) ^{1/2}\\&\le C\mathbb {E}\left( \int _0^{t\wedge \tau _n^R}\Vert g_2(s,u_n(s),\varphi _n(s))\Vert _{L_2(K,G_{{\text {div}}})}^2|u_n(s)|^2ds\right) ^{1/2}\\&\le C\left[ \mathbb {E}\sup _{s\in [0,t\wedge \tau _n^R]}|u_n(s)|^2\right] ^{1/2}\left[ \mathbb {E}\int _0^{t\wedge \tau _n^R}\Vert g_2(s,u_n,\varphi _n)\Vert _{L_2(K,G_{{\text {div}}})}^2ds\right] ^{1/2}. \end{aligned} \end{aligned}$$

(4.19)

Thanks to (4.19), using Young’s inequality and the assumption on $g_2$ (see $(H_2)$), we obtain

$$\begin{aligned} \begin{aligned} 2\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}|\varpi _n(s)|&\le \frac{1}{2}\mathbb {E}\sup _{s\in [0,t\wedge \tau _n^R]}|u_n(s)|^2+C\mathbb {E}\int _0^{t\wedge \tau _n^R}\Vert g_2(s,u_n,\varphi _n)\Vert _{L_2(K,G_{{\text {div}}})}^2ds\\&\le \frac{1}{2}\mathbb {E}\sup _{s\in [0,t\wedge \tau _n^R]}|u_n(s)|^2+C\mathbb {E}\int _0^{t\wedge \tau _n^R}(1+|u_n(s)|^2+|\varphi _n(s)|^2)ds. \end{aligned} \end{aligned}$$

Hence, from this previous inequality and (4.16), there exists a positive constant $C_3$ depending on $c_6,\kappa$ and $\mathcal {M}$ such that

$$\begin{aligned} \begin{aligned} 2\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}|\varpi _n(s)|&\le \frac{1}{2}\mathbb {E}\sup _{s\in [0,t\wedge \tau _n^R]}|u_n(s)|^2+ C_3\mathbb {E}\int _0^{t\wedge \tau _n^R}(1+[|u_n|^2+c_6\Vert \varphi _n\Vert _{L^{2\kappa +2}}^{2\kappa +2}])ds\\&\le \frac{1}{2}\mathbb {E}\sup _{s\in [0,t\wedge \tau _n^R]}[\chi _n(s)]+C_3\mathbb {E}\int _0^{t\wedge \tau _n^R}(1+[\chi _n(s)])ds, \end{aligned} \end{aligned}$$

(4.20)

where we have also used (4.8).

Thanks to (4.18) and (4.20), we obtain

$$\begin{aligned} \mathbb {E}\sup _{s\in [0 ,t\wedge \tau _n^R]}[\chi _n(s)]+2\mathbb {E}\int _0^{t\wedge \tau _n^R}[2\mathcal {Z}_p\vartheta _{p,n}(s)+2|\nabla \mu _n|^2]ds\le \mathcal {K}_1+\tilde{C}_2\mathbb {E}\int _0^{t\wedge \tau _n^R}\sup _{0\le s\le \tau }[\chi _n(s)]d\tau , \end{aligned}$$

(4.21)

with

$$\begin{aligned} \begin{array}{ll} \tilde{C}_2:=\tilde{C}_2(\kappa _2^1,\kappa _p^1,c_2,\kappa ,c_6,p,|\mathcal {M}|),\\ \mathcal {K}_1:=2\bar{\kappa }_1+2\mathcal {K}(u_0,\varphi _0)+C_1(\kappa _2^1,\kappa _p^1,p)\int _0^{T}\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds+\tilde{C}_2T, \end{array} \end{aligned}$$

(4.22)

$\chi _n$ and $\vartheta _{p,n}$ are defined in (4.11). Now, we define

$$\begin{aligned} \mathcal {Z}(t):=\mathbb {E}\int _0^{t\wedge \tau _n^R}\sup _{0\le s\le \tau }[\chi _n(s)]d\tau \end{aligned}$$

(4.23)

and then (4.21), implies

$$\begin{aligned} \mathcal {Z}'(t)\le \mathcal {K}_1+\tilde{C}_2\mathcal {Z}(t). \end{aligned}$$

(4.24)

This gives

$$\begin{aligned} \mathcal {Z}(t)\le \frac{\mathcal {K}_1}{\tilde{C}_2}(e^{\tilde{C}_2t}-1). \end{aligned}$$

(4.25)

Owing to (4.21), (4.23) and (4.25), we derive that

$$\begin{aligned} \mathbb {E}\sup _{s\in [0 ,t\wedge \tau _n^R]}[\chi _n(s)]+2\mathbb {E}\int _0^{t\wedge \tau _n^R}[2\mathcal {Z}_p\vartheta _{p,n}(s)+2|\nabla \mu _n(s)|^2]ds\le \mathcal {K}_1+\mathcal {K}_1e^{\tilde{C}_2t}. \end{aligned}$$

(4.26)

Arguing similarly as in [13, Inequality (4.57)], using (4.16) we get

$$\begin{aligned} \begin{array}{ll} |\nabla \varphi _n|^2\le \frac{8}{c_5^2}|\varphi _n|^2+\frac{4}{c_5^2}|\nabla \mu _n|^2\le \frac{8c_6}{c_5^2(\kappa +1)}\Vert \varphi _n\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}+\frac{8\kappa |\mathcal {M}|}{c_5^2(\kappa +1)c_6^{1/\kappa }}+\frac{4}{c_5^2}|\nabla \mu _n|^2. \end{array} \end{aligned}$$

(4.27)

Thanks to (4.8), (4.16), (4.27) and (4.26), we infer that

$$\begin{aligned} \begin{aligned}&\mathbb {E}\int _0^{t\wedge \tau _n^R}\Vert \varphi _n\Vert _U^2ds=\mathbb {E}\int _0^{t\wedge \tau _n^R}(|\varphi _n|^2ds+|\nabla \varphi _n|^2)ds\\&\quad \le [\tilde{\kappa }_1+\tilde{\kappa }_3]t+[\tilde{\kappa }_2+\tilde{\kappa }_4]t\mathbb {E}\displaystyle \sup _{s\in [0,t\wedge \tau _n^R]}\Vert \varphi _n(s)\Vert _{L^{2\kappa +2}}^{2\kappa +2}+\tilde{\kappa }_5\mathbb {E}\int _0^{t\wedge \tau _n^R}|\nabla \mu _n|^2ds\\&\quad \le [\tilde{\kappa }_1+\tilde{\kappa }_3]t+[\tilde{\kappa }_2+\tilde{\kappa }_4]\tilde{\kappa }_6t(1+e^{\tilde{C}_2t}) +\tilde{\kappa }_7(1+e^{\tilde{C}_2t})<C, \end{aligned} \end{aligned}$$

(4.28)

where $\tilde{\kappa }_1=\frac{\kappa |\mathcal {M}|}{(\kappa +1)c_6^{1/\kappa }},\tilde{\kappa }_2=\frac{c_6}{\kappa +1},\tilde{\kappa }_3=\frac{8\kappa |\mathcal {M}|}{c_5^2(\kappa +1)c_6^{1/\kappa }},\tilde{\kappa }_4=\frac{8c_6}{c_5^2(\kappa +1)},\tilde{\kappa }_5=\frac{4}{c_5^2}, \tilde{\kappa }_6=\frac{\mathcal {K}_1}{c_6}$ and $\tilde{\kappa }_7=\frac{\mathcal {K}_1}{c_5^2}$. Now, we will prove that

$$\begin{aligned}\tau _n^R\nearrow T~ \mathbb {P}-\text {almost surely as}~R\rightarrow \infty .\end{aligned}$$

Indeed, since $(u_n,\varphi _n)(.\wedge \tau _n^R):[0,T]\rightarrow \mathbf{W} _s^n\times H_n$ is continuous, we have

$$\begin{aligned} \begin{array}{ll} R^2\mathbb {P}(\tau _n^R<t)&{}\le \mathbb {E}[1_{\tau _n^R<t}(\rho _n(\tau _n^R)+\int _0^{\tau _n^R}\varrho _n(s)ds)]\\ &{}\le \mathbb {E}[1_{\tau _n^R<t}(\rho _n(\tau _n^R)+\int _0^{\tau _n^R}\varrho _n(s)ds)]+\mathbb {E}[1_{\tau _n^R\ge t}(\rho _n(\tau _n^R)+\int _0^{\tau _n^R}\varrho _n(s)ds)]\\ &{}=\mathbb {E}[\rho _n(\tau _n^R\wedge t)+\int _0^{\tau _n^R\wedge t}\varrho _n(s)ds], \end{array} \end{aligned}$$

(4.29)

with $t\wedge \tau _n^R=\tau _n^R$, since $\tau _n^R<t$, and for any $n\in \mathbb {N}$ and $t\in [0,T]$. Here $\rho _n(.)=|u_n(.)|^2+\Vert \varphi _n(.)\Vert _{L^{2\kappa +2}(\mathcal {M})}^2$ and $\varrho _n(.)=\Vert u_n\Vert _{1,p}^p(.)+\Vert \varphi _n\Vert _U^2(.)$.

From (4.29) and the inequalities (4.8), (4.26) and (4.28), we infer that

$$\begin{aligned} \mathbb {P}(\tau _n^R<t)\le \frac{C}{R^2}. \end{aligned}$$

(4.30)

Since the constant C in (4.30) does not depend on n and R, it then follows that

$$\begin{aligned} \lim _{R\rightarrow \infty }\mathbb {P}(\tau _n^R<t)=0~\text {for all}~t\in [0,T]~\text {and}~n\in \mathbb {N}, \end{aligned}$$

which implies that there exists a subsequence $\tau _n^{R_k}$, such that $\tau _n^{R_k}\rightarrow T$ a.s., which along with the fact that $(\tau _n^R, n)_{R\in \mathbb {N}}$ is increasing, yields that $\tau _n^R\nearrow T$ a.s. for any $n\in \mathbb {N}$. Therefore $T_n=T$.

Now, since the constant $\mathcal {K}_1+\mathcal {K}_1e^{\tilde{C}_2t}$ in (4.26) does not depend on n and R, and since $\tau _n^R\nearrow T$ $\mathbb {P}$-almost surely as $R\rightarrow \infty$, we can conclude by passing to the limit in (4.26) that

$$\begin{aligned} \mathbb {E}\sup _{s\in [0 ,T]}[\chi _n(s)]+2\mathbb {E}\int _0^T[2\mathcal {Z}_p\vartheta _{p,n}(s)+2|\nabla \mu _n(s)|^2]ds\le \mathcal {K}_1+\mathcal {K}_1e^{\tilde{C}_2T}\equiv C. \end{aligned}$$

(4.31)

We infer from (4.17) that

$$\begin{aligned} \begin{array}{ll} \displaystyle \sup _{s\in [0,T]}[\chi _n(s)]+\displaystyle \int _0^T[2\mathcal {Z}_p\vartheta _{p,n}(s)+2|\nabla \mu _n|^2]ds&{}\le \bar{\kappa }_1+\mathcal {K}(u_0,\varphi _0)+C_1\displaystyle \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\\ &{}\quad +\,C_2T\ +C_2\displaystyle \int _0^T[\chi _n(s)]ds+2\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|. \end{array} \end{aligned}$$

(4.32)

Now squaring both sides of the above inequality to the power q/2, $q>2$, we obtain thanks to the Minkowski inequality and after taking the expected values

$$\begin{aligned} \begin{aligned}&\mathbb {E}\displaystyle \sup _{s\in [0,T]}[\chi _n(s)]^{\frac{q}{2}}+\left( \displaystyle \int _0^T[2\mathcal {Z}_p\vartheta _{p,n}(s)+2|\nabla \mu _n(s)|^2]ds\right) ^\frac{q}{2}\\&\quad \le C_5(q)\left[ \bar{\kappa }_1^\frac{q}{2}+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+C_1^\frac{q}{2}\left( \displaystyle \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}+(C_2T)^\frac{q}{2}\right] \\&\qquad +C_5(q)C_2^\frac{q}{2}T^\frac{q-2}{2}\mathbb {E}\displaystyle \int _0^T[\chi _n(s)]^{q/2}ds+C_5(q)\mathbb {E}\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|^{q/2}, \end{aligned} \end{aligned}$$

(4.33)

with

$$\begin{aligned} \tilde{\mathcal {K}}_q(u_0,\varphi _0)=|u_0|^q+2^\frac{q}{2}|J|_{L^1(\mathbb {R}^3)}^\frac{q}{2}|\varphi _0|^q+2^\frac{q}{2}|F(\varphi _0)|_{L^1(\mathcal {M})}^\frac{q}{2}. \end{aligned}$$

(4.34)

Using again the Burkholder–Davis–Gundy inequality as we did in the proof of Eq. (4.20) and the assumption on $g_2$, we can check that

$$\begin{aligned} \begin{array}{ll} C_5(q)\mathbb {E}\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|^{q/2}&{}\le \tilde{C}_5(q)\mathbb {E}\left( \displaystyle \int _0^T\sum _{j=1}^n(g_2(s,u_n,\varphi _n)e_j,u_n)^2ds\right) ^{q/4}\\ &{}\le \frac{1}{2}\mathbb {E}\displaystyle \sup _{s\in [0,T]}|u_n(s)|^q+\bar{C}_5T+\bar{C}_5\mathbb {E}\displaystyle \int _0^T\left( |u_n|^2+c_6|\varphi _n|_{L^{2\kappa +2}}^{2\kappa +2}\right) ^{q/2}ds, \end{array} \end{aligned}$$

(4.35)

with $\bar{C}_5:=\bar{C}_5(q,\kappa ,c_6,|\mathcal {M}|)$ and where we have also used (4.16).

From (4.35) and (4.8), we infer that

$$\begin{aligned} \begin{array}{ll} C_5(q)\mathbb {E}\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|^{q/2}\le \frac{1}{2}\mathbb {E}\sup _{s\in [0,T]}[\chi _n(s)]^{q/2}+\bar{C}_5T+\bar{C}_5\mathbb {E}\int _0^T[\chi _n(s)]^{q/2}ds. \end{array} \end{aligned}$$

(4.36)

Inserting (4.36) in (4.33) and multiplying the resulting inequality by 2, we obtain

$$\begin{aligned} \begin{aligned}&\mathbb {E}\displaystyle \sup _{s\in [0,T]}[\chi _n(s)]^{\frac{q}{2}}+2\mathbb {E}\left( \displaystyle \int _0^T[2\mathcal {Z}_p\vartheta _{p,n}(s)+ 2|\nabla \mu _n(s)|^2]ds\right) ^\frac{q}{2}\\&\quad \le 2C_5(q)\left[ \bar{\kappa }_1^\frac{q}{2}+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+C_1^\frac{q}{2}\left( \displaystyle \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}+(C_2T)^\frac{q}{2}\right] \\&\qquad +2\bar{C}_5T+C_6\mathbb {E}\displaystyle \int _0^T\sup _{\tau \in [0,s]}[\chi _n(\tau )]^{\frac{q}{2}}ds, \end{aligned} \end{aligned}$$

(4.37)

with $C_6=2C_5(q)C_2^\frac{q}{2}T^\frac{q-2}{2}+2\bar{C}_5(q,\kappa ,c_6,|\mathcal {M}|)$. Now dropping the integral term in the left-hand side of (4.37) and applying the deterministic Gronwall lemma, we arrive at

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{s\in [0,T]}[\chi _n(s)]^{q/2}\le C_7\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}\right] , \end{aligned}$$

(4.38)

with $C_7:=C_7(q,T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$ and $\tilde{\mathcal {K}}_q(u_0,\varphi _0)$ is given by (4.34).

Therefore the first two estimates in (4.2) follow from (4.38), (4.31), (4.8) and the fact that the domain $\mathcal {M}$ is bounded.

By using (4.37) and (4.38), it is straightforward to check that

$$\begin{aligned} \begin{array}{ll} &{}\mathbb {E}\left( \int _0^T\Vert u_n(s)\Vert _{1,p}^pds\right) ^\frac{q}{2}+\mathbb {E}\left( \int _0^T\Vert u_n(s)\Vert ^2ds\right) ^\frac{q}{2}+\mathbb {E}\left( \int _0^T|\nabla \mu _n(s)|^2ds\right) ^\frac{q}{2}\\ &{}\quad \le \tilde{C}_7\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}\right] , \end{array} \end{aligned}$$

(4.39)

with $\tilde{C}_7:=\tilde{C}_7(q,T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$.

So as to proved (4.3)$_1$, we make the following observation: For any $a_1>0$, $q>2$ and $p\ge 11/5$, we have using the Young inequality

$$\begin{aligned} a_1^{q/p}\le \frac{2}{p} a_1^{q/2}+\frac{\textstyle p-2}{\textstyle 2}. \end{aligned}$$

Now, applying this previous inequality with $a_1=\left( \int _0^T\Vert u_n(s)\Vert _{1,p}^pds\right) ^{q/p}$ in conjunction with (4.39), we infer that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\left( \int _0^T\Vert u_n(s)\Vert _{1,p}^pds\right) ^{q/p}&{}\le \frac{2}{p}\mathbb {E}\left( \int _0^T\Vert u_n(s)\Vert _{1,p}^pds\right) ^{q/2}+\frac{p-2}{2}\\ &{}\le \tilde{C}_7\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}\right] +\frac{p-2}{2}. \end{array} \end{aligned}$$

This proves (4.3)$_1$.

We will now prove that $\mathbb {E}\displaystyle \sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}^{q/2}<C$. The proof will be done in two cases:

First case: One can have $\underline{F(\varphi _n)>0}$.

From the first line of (4.7) in conjunction with (4.32), we obtain

$$\begin{aligned} \begin{array}{ll} 2\displaystyle \sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}&{}\le \displaystyle \sup _{s\in [0,T]}|(\varphi _n(s),J*\varphi _n(s))|+\bar{\kappa }_1+\mathcal {K}(u_0,\varphi _0)\\ &{}\quad +\,C_1\int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds+C_2T+C_2\int _0^T[\chi _n(s)]ds+2\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|. \end{array} \end{aligned}$$

(4.40)

Using young’s inequality for convolutions, we obtain

$$\begin{aligned} \sup _{s\in [0,T]}|(\varphi _n(s),J*\varphi _n(s))|\le |J|_{L^1(\mathbb {R}^3)}\sup _{s\in [0,T]}|\varphi _n(s)|^2. \end{aligned}$$

(4.41)

Making similar reasoning as in (4.20), we obtain

$$\begin{aligned} 2\mathbb {E}\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|\le \frac{1}{2}\mathbb {E}\displaystyle \sup _{s\in [0,T]}[\chi _n(s)]+C_3(c_6,\kappa ,|\mathcal {M}|)\int _0^{T}(1+[\chi _n(s)])ds. \end{aligned}$$

(4.42)

Taking the expected values in (4.40), using (4.41)–(4.42) and dividing the resulting inequality by 2, we obtain

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}&{}\le 2|J|_{L^1(\mathbb {R}^3)}\mathbb {E}\sup _{s\in [0,T]}|\varphi _n(s)|^2+2\bar{\kappa }_1+2\mathcal {K}(u_0,\varphi _0)\\ &{}\quad +\,2C_1\int _0^T\Vert g_0\Vert _{V'_{{\text {div}},p}}^{p'}ds+C_{2,3}T+(1+C_{2,3}T)\mathbb {E}\sup _{s\in [0,T]}[\chi _n(s)], \end{array} \end{aligned}$$

(4.43)

where $C_1:=C_1(\kappa _2^1,\kappa _p^1,p)$, $C_2:=C_2(\kappa _2^1,\kappa _p^1,c_2,c_6,\kappa ,p,|\mathcal {M}|)$, $C_3:=C_3(c_6,\kappa ,|\mathcal {M}|)$ and $C_{2,3}=2C_2+2C_3$.

Now, from (4.43), (4.16), (4.8) and (4.31), we obtain

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}\le \bar{C}_7\left[ 1+\mathcal {K}(u_0,\varphi _0)+\int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right] , \end{array} \end{aligned}$$

(4.44)

with $\bar{C}_7:=\bar{C}_7(T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$ and $\mathcal {K}(u_0,\varphi _0)$ is given by the last inequality of (4.9). Inserting (4.41) in (4.40), raising both sides to the power $q/2>1$, and taking the expectation in the resulting inequality, we obtain

$$\begin{aligned} \begin{array}{ll} \displaystyle \sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}^\frac{q}{2}&{}\le C(q)[|J|_{L^1}^\frac{q}{2}\displaystyle \sup _{s\in [0,T]}|\varphi _n(s)|^q+\bar{\kappa }_1^\frac{q}{2}+\tilde{\mathcal {K}}_q(u_0,\varphi _0)]\\ &{}\quad +\,C(q)C_1^\frac{q}{2}\left( \displaystyle \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}+C(q)(C_2T)^\frac{q}{2}\\ &{}\quad +\,C(q)C_2^\frac{q}{2}\left( \displaystyle \int _0^T[\chi _n(s)]ds\right) ^\frac{q}{2}+C(q)\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|^\frac{q}{2}. \end{array} \end{aligned}$$

(4.45)

Arguing similarly as in (4.36), we can check that

$$\begin{aligned} \begin{array}{ll} C(q)\mathbb {E}\displaystyle \sup _{s\in [0,T]}|\varpi _n(s)|^{q/2}\le \frac{1}{2}\mathbb {E}\displaystyle \sup _{s\in [0,T]}[\chi _n(s)]^{q/2}+\bar{C}_qT+\bar{C}_q\mathbb {E}\displaystyle \int _0^T[\chi _n(s)]^{q/2}ds, \end{array} \end{aligned}$$

with $\bar{C}_q:=\bar{C}_q(q,\kappa ,c_6,|\mathcal {M}|)$. Inserting now this previous inequality to (4.45) (after taking the expectation), using the inequality (4.38) and the fact that $\mathcal {M}$ is a bounded domain, we arrive at

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\displaystyle \sup _{s\in [0,T]}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}^{q/2}\le \iota _q\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^{q/2}\right] , \end{array} \end{aligned}$$

(4.46)

where $\iota _q:=\tilde{C}_q(q,T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$ and $\tilde{\mathcal {K}}_q(u_0,\varphi _0)$ is given by (4.34).

Second case: or $\underline{F(\varphi _n)<0}$.

From (4.7), we infer that

$$\begin{aligned} \begin{aligned} 2\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}&\le 2|\sqrt{a}\varphi _n(t)|^2+|J|_{L^1(\mathbb {R}^3)}|\varphi _n(t)|^2+\bar{\kappa }_1\\&\le 3|J|_{L^1(\mathbb {R}^3)}|\varphi _n(t)|^2+\bar{\kappa }_1\le \tilde{\kappa }_8\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}(\mathcal {M})}^{2\kappa +2}+\tilde{\kappa }_9, \end{aligned} \end{aligned}$$

(4.47)

where we have also used Young’s inequality for convolutions, the inequality (4.16), and the fact that $|a|_{L^\infty (\mathcal {M})}\le |J|_{L^1(\mathbb {R}^3)}$. Here $\tilde{\kappa }_8=\frac{3c_6|J|_{L^1(\mathbb {R}^3)}}{\kappa +1}$ and $\tilde{\kappa }_9=\frac{3\kappa |\mathcal {M}||J|_{L^1(\mathbb {R}^3)}}{(\kappa +1)c_6^{1/\kappa }}+\bar{\kappa }_1$.

Since (4.47) holds for every $t\in [0,T]$, we also infer that

$$\begin{aligned} 2\displaystyle \sup _{t\in [0,T]}\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}\le \tilde{\kappa }_8\displaystyle \sup _{t\in [0,T]}\Vert \varphi _n(t)\Vert _{L^{2\kappa +2}}^{2\kappa +2}+\tilde{\kappa }_9\le \tilde{\kappa }_8\displaystyle \sup _{t\in [0,T]}[\chi _n(s)]+\tilde{\kappa }_9. \end{aligned}$$

(4.48)

We note that in (4.48), we have also used (4.8).

Taking now the mathematical expectation in (4.48), making used of (4.31), we infer that

$$\begin{aligned} 2\mathbb {E}\displaystyle \sup _{t\in [0,T]}\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}\le \tilde{\kappa }_8[\mathcal {K}_1+\mathcal {K}_1e^{\tilde{C}_2T}]+\tilde{\kappa }_9, \end{aligned}$$

(4.49)

with $\mathcal {K}_1$, $\tilde{C}_2$ given by (4.22); and $\tilde{\kappa }_8, \tilde{\kappa }_9$ given by (4.47).

Also from (4.48), it is straightforward to check that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\displaystyle \sup _{t\in [0,T]}\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}^{q/2}\le C(q)(\tilde{\kappa }_9)^{q/2}+C(q)(\tilde{\kappa }_8)^{q/2}\mathbb {E}\displaystyle \sup _{t\in [0,T]}[\chi _n(t)]^{q/2}. \end{array} \end{aligned}$$

From this previous inequality and (4.38), we get

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\displaystyle \sup _{t\in [0,T]}\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}^{q/2}\le \iota _7\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}\right] , \end{array} \end{aligned}$$

(4.50)

with $\iota _7:=\iota _7(q,T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$ and $\tilde{\mathcal {K}}_q(u_0,\varphi _0)$ is defined as in (4.34). This completes the proof of estimate (4.2)$_3$, i.e.,

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{t\in [0,T]}\Vert F(\varphi _n(t))\Vert _{L^1(\mathcal {M})}^{q/2}<C,~\text {for both cases}. \end{aligned}$$

(4.51)

In view to prove estimate (4.2)$_4$, we begin by making the following observation: from assumption $(H_4)$ and the Young inequality, it is straightforward to check that

$$\begin{aligned} \begin{array}{ll} |F'(\varphi _n(s))|&\le \frac{\textstyle 1}{\textstyle r}|F'(\varphi _n(s))|^r+\frac{\textstyle r-1}{\textstyle r}\le \left( \frac{\textstyle c_4}{\textstyle r}+1\right) |F(\varphi _n(s))|+\frac{\textstyle r-1}{\textstyle r},~\forall s\in [0,T]. \end{array} \end{aligned}$$

(4.52)

Hence, (4.2)$_4$ follows from (4.52), (4.51) and (4.49).

We now give the proof of estimates (4.3)$_2$ and (4.3)$_3$.

From (4.28) and the fact that $\tau _n^R\nearrow T~ \mathbb {P}$-almost surely as $R\rightarrow \infty$, we obtain

$$\begin{aligned} \begin{aligned} \mathbb {E}\int _0^{T}\Vert \varphi _n(s)\Vert _U^2ds\le C, \end{aligned} \end{aligned}$$

(4.53)

Thanks to (4.16), (4.27), (4.38) and (4.39), we infer that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\left( \int _0^T\Vert \varphi _n(s)\Vert _U^2ds\right) ^{\frac{q}{2}}&{}\le C(q)\left[ \mathbb {E}\left( \int _0^T|\varphi _n(s)|^2ds\right) ^{q/2}+\mathbb {E}\left( \int _0^T|\nabla \varphi _n(s)|^2ds\right) ^{\frac{q}{2}}\right] \\ &{}\le \bar{\iota }_7\left[ 1+\tilde{\mathcal {K}}_q(u_0,\varphi _0)+\left( \int _0^T\Vert g_0(s)\Vert _{V'_{{\text {div}},p}}^{p'}ds\right) ^\frac{q}{2}\right] , \end{array} \end{aligned}$$

(4.54)

with $\bar{\iota }_7:=\bar{\iota }_7(q,T,\kappa ,c_6,|\mathcal {M}|,\kappa _2^1,\kappa _p^1,|J|_{L^1(\mathbb {R}^3)})$. So, by (4.53) and (4.54), we obtain (4.3)$_2$.

Arguing similarly as in [12, Inequality (3.65)], we check that

$$\begin{aligned} \begin{array}{ll} |\mu _n(s)|^2&{}\le C\left[ |\nabla \mu _n(s)|^2+4|\mathcal {M}|^{-1}|J|_{L^1(\mathbb {R}^3)}^2|\varphi _n(s)|^2\right] \\ &{}\le C\left[ c_4^2r^{-2}|\mathcal {M}|^{-2}\Vert F(\varphi _n(s))\Vert _{L^1(\mathcal {M})}^2+\left( \frac{\textstyle c_4+r-1}{\textstyle r}\right) ^2\right] . \end{array} \end{aligned}$$

(4.55)

Now, owing to (4.55), (4.49), (4.16) and (4.31), it follows that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\int _0^T\Vert \mu _n(s)\Vert _U^2ds=\mathbb {E}\int _0^T|\mu _n(s)|^2ds+\mathbb {E}\int _0^T|\nabla \mu _n(s)|^2ds<C. \end{array} \end{aligned}$$

(4.56)

Also, from (4.55), (4.51), (4.39) and (4.38), we get

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\left( \int _0^T\Vert \mu _n(s)\Vert _U^2ds\right) ^{q/2}\le \mathbb {E}\left( \int _0^T|\mu _n(s)|^2ds\right) ^{q/2}+\mathbb {E}\left( \int _0^T|\nabla \mu _n(s)|^2ds\right) ^{q/2}<C, \end{array} \end{aligned}$$

(4.57)

where $U=H^1(\mathcal {M})$.

The estimate (4.3)$_3$ follows from (4.56) and (4.57). This completes the proof of Lemma 4.1. $\square$

In the next propositions, we prove two uniform estimates for $u_n$ and $\varphi _n$ which are very crucial for our purpose.

Proposition 4.1

In addition to assumptions of Theorem 3.1, Let $s\in \mathbb {R}$ such that $s>5/2$. We assume that $t\mapsto u_n(t)$ is extended to zero outside the interval [0, T]. Then, there exists a positive constant C such that

$$\begin{aligned} \mathbb {E}\sup _{0<|\theta |\le \delta <1}\Vert u_n(t+\theta )-u_n(t)\Vert _\mathbf{W '_s}^{p'}\le C\delta ^\frac{p'}{p},~\forall t\in [0,T],~n\in \mathbb {N}^*~\text {and}~p'=p/(p-1). \end{aligned}$$

Proposition 4.2

Let the assumptions of Theorem 3.1 be satisfied. We assume that $t\mapsto \varphi _n(t)$ is extended to zero outside the interval [0, T]. Then, there exists a positive constant C such that

$$\begin{aligned} \mathbb {E}\sup _{0<|\theta |\le \delta <1}\Vert \varphi _n(t+\theta )-\varphi _n(t)\Vert _{V'_s}^{2}\le C\delta ^\frac{p'}{p},~\forall t\in [0,T],~n\in \mathbb {N}^*~\text {and}~p'=p/(p-1). \end{aligned}$$

Proof of Proposition 4.1

We rewrite the equation for $u_n$ as

$$\begin{aligned} \begin{array}{ll} &{} d(u_n,v)+[ (\varvec{T}(Du_n),Dv)+b(u_n,u_n,v)]dt=-(\varphi _n\nabla \mu _n,v)dt +(\mathcal {P}_n^1g_0,v)dt\\ &{}\quad +\,(g_1(u_n,\varphi _n),v)dt+\sum _{i=1}^n( g_2(t,u_n,\varphi _n)e_i,v)d\beta _t^i,\ \ \forall v\in \mathbf{W} _s^n. \end{array} \end{aligned}$$

(4.58)

Let us take $v\in \mathbf{W} _s$, and decompose it as $v=v_I+v_{II}$, where $v_I\in \mathbf{W} _s^n$ and $v_{II}\in (\mathbf{W} _s^n)^\bot$, and notice that $v_I$ and $v_{II}$ are orthogonal also in $\mathbf{W} _s$. Then, from (4.58) we can write

$$\begin{aligned} \begin{aligned} d(u_n,v)=d\left\langle u_n,v\right\rangle _\mathbf{W _s}&=d\left\langle u_n,v_I\right\rangle _\mathbf{W _s}\\&=-[ (\varvec{T}(Du_n),Dv_I)+b(u_n,u_n,v_I)]dt\\&\quad -\,(\varphi _n\nabla \mu _n,v_I)dt +(\mathcal {P}_n^1g_0,v_I)dt+(g_1(u_n,\varphi _n),v_I)dt\\&\quad +\,\sum _{i=1}^n( g_2(t,u_n,\varphi _n)e_i,v_I)d\beta _t^i. \end{aligned} \end{aligned}$$

(4.59)

Let us set $\tilde{\tilde{u}}_n(t)=u_n(t+\theta )-u_n(t)$, for any $\theta \in (0,\delta )$ and $\delta \in (0,1)$. Hence from (4.59), we have

$$\begin{aligned} \begin{aligned} \left\langle \tilde{\tilde{u}}_n(t),v\right\rangle _\mathbf{W _s}&=-\displaystyle \int _t^{t+\theta }[ (\varvec{T}(Du_n),Dv_I)+b(u_n,u_n,v_I)]d\tau -\displaystyle \int _t^{t+\theta }(\varphi _n\nabla \mu _n,v_I)d\tau \\&\quad +\,\displaystyle \int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau +\displaystyle \int _t^{t+\theta }(g_1(u_n,\varphi _n),v_I)d\tau +\displaystyle \int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i, \end{aligned} \end{aligned}$$

(4.60)

where, for the sake of simplicity, we have set

$$\begin{aligned} \displaystyle \int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i:=\sum _{j=1}^n\displaystyle \int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i. \end{aligned}$$

We set

$$\begin{aligned} \begin{array}{ll} y_t(\theta )&{}=|\int _t^{t+\theta }[ (\varvec{T}(Du_n),Dv_I)+b(u_n,u_n,v_I)]d\tau \\ &{}\quad -\,\int _t^{t+\theta }(\varphi _n\nabla \mu _n,v_I)d\tau +\int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau \\ &{}\quad +\,\int _t^{t+\theta }(g_1(u_n,\varphi _n),v_I)d\tau +\int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i|. \end{array} \end{aligned}$$

It follows from this that

$$\begin{aligned} \begin{aligned} y_t(\theta )&\le |\int _t^{t+\theta } (\varvec{T}(Du_n),Dv_I)d\tau |+|\int _t^{t+\theta }b(u_n,u_n,v_I)d\tau |\\&\quad +\,|\int _t^{t+\theta }(\varphi _n\nabla \mu _n,v_I)d\tau | +|\int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau |\\&\quad +\,|\int _t^{t+\theta }(g_1(u_n,\varphi _n),v_I)d\tau |+|\int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i|. \end{aligned} \end{aligned}$$

(4.61)

We have

$$\begin{aligned} \begin{array}{ll} |\int _t^{t+\theta } (\varvec{T}(Du_n),Dv_I)d\tau |&{}=|\int _t^{t+\theta }\left\langle N(u_n(\tau )),v_I\right\rangle _{V_{{\text {div}},p}}d\tau |\\ &{}\le \int _t^{t+\theta }|\left\langle N(u_n(\tau )),v_I\right\rangle _{V_{{\text {div}},p}}|d\tau \\ &{}\le \Vert v_I\Vert _{V_{{\text {div}},p}}\int _t^{t+\theta }\Vert N(u_n(\tau ))\Vert _{V'_{{\text {div}},p}}d\tau \\ &{}\le C\Vert v_I\Vert _\mathbf{W _s}\int _t^{t+\theta }\Vert N(u_n(\tau ))\Vert _{V'_{{\text {div}},p}}d\tau , \end{array} \end{aligned}$$

where we have used the Cauchy–Schwarz inequality and the fact $\mathbf{W} _s\hookrightarrow V_{{\text {div}},p}$ continuously.

Now, from this last inequality and by Hölder’s inequality, we infer that

$$\begin{aligned} \begin{array}{ll} |\int _t^{t+\theta } (\varvec{T}(Du_n),Dv_I)d\tau |\le C\Vert v_I\Vert _\mathbf{W _s}\theta ^\frac{1}{p}\left( \int _t^{t+\theta }\Vert N(u_n(\tau ))\Vert _{V'_{{\text {div}},p}}^{p'}d\tau \right) ^\frac{1}{p'}. \end{array} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{array}{ll} \left| \int _t^{t+\theta } (\varvec{T}(Du_n),Dv_I)d\tau \right| ^{p'}&{}\le C \Vert v_I\Vert _\mathbf{W _s}^{p'} \theta ^\frac{p'}{p}\int _t^{t+\theta }\Vert N(u_n(s))\Vert _{V'_{{\text {div}},p}}^{p'}ds\\ &{}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^\frac{p'}{p}\int _t^{t+\theta }\Vert N(u_n(s))\Vert _{V'_{{\text {div}},p}}^{p'}ds:=C\Vert v\Vert _\mathbf{W _s}^{p'}y_t^1(\theta ,p). \end{array}. \end{aligned}$$

(4.62)

Note that (see [15, Inequalities (4.14)])

$$\begin{aligned} |b(u_n,u_n,v_I)|\le C|u_n|^2\Vert v_I\Vert _\mathbf{W _s}\le C|u_n|^2\Vert v\Vert _\mathbf{W _s}, \end{aligned}$$

(4.63)

and the following holds

$$\begin{aligned} \begin{array}{ll} |\int _t^{t+\theta }b(u_n,u_n,v_I)d\tau |\le \int _t^{t+\theta }|b(u_n,u_n,v_I)|d\tau &{}\le C\Vert v\Vert _\mathbf{W _s}\int _t^{t+\theta }|u_n(\tau )|^2d\tau \\ &{}\le C\Vert v\Vert _\mathbf{W _s}\theta ^\frac{1}{p}\left( \int _t^{t+\theta }|u_n(\tau )|^{2p'}d\tau \right) ^{\frac{1}{p'}}. \end{array} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \left| \int _t^{t+\theta }b(u_n,u_n,v_I)d\tau \right| ^{p'}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^\frac{p'}{p}\int _t^{t+\theta }|u_n(\tau )|^{2p'}d\tau :=C\Vert v\Vert _\mathbf{W _s}^{p'}y_t^2(\theta ,p). \end{aligned}$$

(4.64)

As in [15, Inequalities (4.15)], we also have

$$\begin{aligned} |(\varphi _n\nabla \mu _n,v_I)|\le C|\varphi _n||\nabla \mu _n|\Vert v_I\Vert _\mathbf{W _s}\le C|\varphi _n||\nabla \mu _n|\Vert v\Vert _\mathbf{W _s}. \end{aligned}$$

(4.65)

By (4.65) and the Hölder inequality, we have

$$\begin{aligned} \begin{array}{ll} |\int _t^{t+\theta }(\varphi _n\nabla \mu _n,v_I)d\tau |\le \int _t^{t+\theta }|(\varphi _n\nabla \mu _n,v_I)|d\tau &{}\le C\Vert v\Vert _\mathbf{W _s}\int _t^{t+\theta }|\varphi _n||\nabla \mu _n|d\tau \\ &{}\le C\Vert v\Vert _\mathbf{W _s}\theta ^\frac{1}{p}\left( \int _t^{t+\theta }|\varphi _n|^{p'}|\nabla \mu _n|^{p'}d\tau \right) ^\frac{1}{p'}. \end{array} \end{aligned}$$

Thus,

$$\begin{aligned} \begin{array}{ll} \left| \int _t^{t+\theta }(\varphi _n\nabla \mu _n,v_I)d\tau \right| ^{p'}&{}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^{\frac{p'}{p}}\int _t^{t+\theta }|\varphi _n|^{p'}|\nabla \mu _n|^{p'}d\tau \\ &{}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^{\frac{p'}{p}}\left( \int _t^{t+\theta }|\varphi _n|^\frac{2p'}{2-p'}d\tau \right) ^\frac{2-p'}{2}\left( \int _t^{t+\theta }|\nabla \mu _n|^2d\tau \right) ^\frac{p'}{2}\\ &{}:=C\Vert v\Vert _\mathbf{W _s}^{p'}y_t^3(\theta ,p). \end{array} \end{aligned}$$

(4.66)

We have by Cauchy–Schwarz’s inequality

$$\begin{aligned} \begin{array}{ll} |\int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau |&{}\le \int _t^{t+\theta }|(\mathcal {P}_n^1g_0,v_I)|d\tau \\ &{}\le \Vert v_I\Vert _\mathbf{W _s}\int _t^{t+\theta }\Vert \mathcal {P}_n^1g_0(\tau )\Vert _\mathbf{W '_s}d\tau \\ &{}\le \Vert v_I\Vert _\mathbf{W _s}\int _t^{t+\theta }\Vert g_0(\tau )\Vert _\mathbf{W '_s}d\tau \\ &{}\le C\Vert v\Vert _\mathbf{W _s}\int _t^{t+\theta }\Vert g_0(\tau )\Vert _{V'_{{\text {div}},p}}d\tau , \end{array} \end{aligned}$$

where we have also used the fact that $\Vert \mathcal {P}_n^1\Vert _{\mathcal {L}(\mathbf{W} '_s,\mathbf{W} (_s))}\le 1$ and $V'_{{\text {div}},p}\hookrightarrow \mathbf{W} '_s$ continuously. Hence, by Hölder’s inequality, we obtain

$$\begin{aligned} \begin{array}{ll} \left| \int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau \right| \le C\Vert v\Vert _\mathbf{W _s}\theta ^\frac{1}{p}\left( \int _t^{t+\theta }\Vert g_0(\tau )\Vert _{V'_{{\text {div}},p}}^{p'}d\tau \right) ^\frac{1}{p'} \end{array} \end{aligned}$$

From this previous inequality, we have

$$\begin{aligned} \begin{array}{ll} \left| \int _t^{t+\theta }(\mathcal {P}_n^1g_0,v_I)d\tau \right| ^{p'}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^\frac{p'}{p}\int _t^{t+\theta }\Vert g_0(\tau )\Vert _{V'_{{\text {div}},p}}^{p'}d\tau :=C\Vert v\Vert _\mathbf{W _s}^{p'}y_t^4(\theta ,p) \end{array}. \end{aligned}$$

(4.67)

Making similar reasoning as in (4.67), we obtain

$$\begin{aligned} \begin{array}{ll} \left| \int _t^{t+\theta }(g_1(u_n(\tau ),\varphi _n(\tau )),v_I)d\tau \right| ^{p'}&\le C\Vert v\Vert _\mathbf{W _s}^{p'}\theta ^\frac{p'}{p}\int _t^{t+\theta }\Vert g_1(u_n,\varphi _n)\Vert _{V'_{{\text {div}},p}}^{p'}d\tau :=C\Vert v\Vert _\mathbf{W _s}^{p'}y_t^5(\theta ,p). \end{array} \end{aligned}$$

(4.68)

Raising both sides of the inequality (4.61) to the power $p'$ ($p'$ is the conjugate index to p), and thanks to previous inequalities, we obtain

$$\begin{aligned} \begin{array}{ll} y_t(\theta )^{p'}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\sum _{i=1}^5y_t^i(\theta ,p)+\left| \int _t^{t+\theta }(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i\right| ^{p'}. \end{array} \end{aligned}$$

(4.69)

Thanks to (2.23), we have

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^1(\theta ,p)\le C\delta ^\frac{p'}{p} \left( \delta +\mathbb {E}\int _0^T\Vert u_n(\tau )\Vert _{1,p}^pd\tau \right) . \end{array} \end{aligned}$$

Owing to Lemma 4.1 (see inequality (4.3)$_1$), we derive from this last inequality that

$$\begin{aligned} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^1(\theta ,p)\le C\delta ^\frac{p'}{p}. \end{aligned}$$

(4.70)

Also, thanks to Lemma 4.1, we obtain

$$\begin{aligned} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^2(\theta ,p)\le C\delta ^\frac{p'}{p}\mathbb {E}\sup _{\tau \in [0,T]}|u_n(\tau )|^{2p'}\le C\delta ^\frac{p'}{p}. \end{aligned}$$

(4.71)

By Hölder’s inequality, we have

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}y_t^3(\theta ,p)&{}\le \delta ^{\frac{p'}{p}}\mathbb {E}\left( \int _t^{t+\delta }|\varphi _n|^\frac{2p'}{2-p'}d\tau \right) ^\frac{2-p'}{2}\left( \int _t^{t+\delta }|\nabla \mu _n|^2d\tau \right) ^\frac{p'}{2}\\ &{}\le \delta ^{\frac{p'}{p}}\left[ \mathbb {E}\int _t^{t+\delta }|\varphi _n|^\frac{2p'}{2-p'}d\tau \right] ^\frac{2-p'}{2}\left[ \mathbb {E}\int _t^{t+\delta }|\nabla \mu _n|^2d\tau \right] ^\frac{p'}{2}\\ &{}\le C \delta ^{\frac{p'}{p}}\left[ \mathbb {E}\displaystyle \sup _{\tau \in [0,T]}|\varphi _n(\tau )|^\frac{2p'}{2-p'}\right] ^\frac{2-p'}{2}\left[ \mathbb {E}\int _0^T|\nabla \mu _n|^2d\tau \right] ^\frac{p'}{2}. \end{array} \end{aligned}$$

Therefore, thanks to (4.16) in conjunction with Lemma 4.1, we have

$$\begin{aligned} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^3(\theta ,p)\le C \delta ^{\frac{p'}{p}}. \end{aligned}$$

(4.72)

By the assumption on $g_0$, it follows that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^4(\theta ,p)\le C \delta ^{\frac{p'}{p}}\mathbb {E}\int _0^T\Vert g_0(\tau )\Vert _{V'_{{\text {div}},p}}^{p'}d\tau \le C \delta ^{\frac{p'}{p}}. \end{array} \end{aligned}$$

(4.73)

Using the assumptions on $g_1$ in conjunction with Lemma 4.1, we obtain

$$\begin{aligned} \mathbb {E}\sup _{\theta \in (0,\delta )}y_t^5(\theta ,p) \le C \delta ^{\frac{p'}{p}}. \end{aligned}$$

(4.74)

By the Burkholder–Davis–Gundy lemma, we have

$$\begin{aligned} \begin{aligned}&\mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\left| \displaystyle \int _t^{t+\theta }\sum _{i=1}^n(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i\right| ^{p'}\\&\quad \le C\mathbb {E}\left( \displaystyle \int _t^{t+\delta }\sum _{i=1}^n (\mathcal {P}_n^1 g_2(\tau ,u_n(\tau ),\varphi _n(\tau ))e_i,v_I)^2d\tau \right) ^\frac{p'}{2}\\&\quad \le C |v_I|^{p'}\mathbb {E}\left( \displaystyle \int _t^{t+\delta }\sum _{i=1}^n |\mathcal {P}_n^1g_2(\tau ,u_{\tau },\varphi _n(\tau ))|^2d\tau \right) ^\frac{p'}{2}\\&\quad \le C\Vert v\Vert _\mathbf{W _s}^{p'}\mathbb {E}\left( \displaystyle \int _t^{t+\delta }\Vert g_2(\tau ,u_n(\tau ),\varphi _n(\tau ))\Vert _{L_2(K,G_{{\text {div}}})}^2d\tau \right) ^\frac{p'}{2}. \end{aligned} \end{aligned}$$

From this last estimate and the assumption on $g_2$, we derive that

$$\begin{aligned} \begin{aligned}&\mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\left| \int _t^{t+\theta }\sum _{i=1}^n(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i\right| ^{p'}\\&\quad \le C\Vert v\Vert _\mathbf{W _s}^{p'}\mathbb {E}\left( \delta +\int _t^{t+\delta }(|u_n(\tau )|^2+|\varphi _n(\tau )|^2)d\tau \right) ^\frac{p'}{2}\\&\quad \le C\Vert v\Vert _\mathbf{W _s}^{p'}\left( \delta +\delta \mathbb {E}\displaystyle \sup _{\tau \in [t,t+\delta ]}|u_n(\tau )|^2+\delta \mathbb {E}\displaystyle \sup _{\tau \in [t,t+\delta ]}|\varphi _n(\tau )|^2\right) ^\frac{p'}{2}. \end{aligned} \end{aligned}$$

(4.75)

Thanks to (4.16), (4.75), and Lemma 4.1, we see that

$$\begin{aligned} \mathbb {E}\sup _{\theta \in (0,\delta )}\left| \int _t^{t+\theta }\sum _{i=1}^n(\mathcal {P}_n^1 g_2(\tau ,u_n,\varphi _n)e_i,v_I)d\beta _\tau ^i\right| ^{p'}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\delta ^\frac{p'}{2}\le C\Vert v\Vert _\mathbf{W _s}^{p'}\delta ^\frac{p'}{p}. \end{aligned}$$

(4.76)

Now, from (4.69) and the estimates (4.70)–(4.76), we have

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\displaystyle \sup _{v\in \mathbf{W} _s,\Vert v\Vert _\mathbf{W _s}=1}y_t(\theta )^{p'}\le C\delta ^\frac{p'}{p}. \end{aligned}$$

(4.77)

Hence

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\Vert \tilde{\tilde{u}}_n(t)\Vert _\mathbf{W '_s}^{p'}=\mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\displaystyle \sup _{v\in \mathbf{W} _s,\Vert v\Vert _\mathbf{W _s}=1}|\left\langle \tilde{\tilde{u}}_n(t),v\right\rangle _\mathbf{W _s}|^{p'}\le C\delta ^\frac{p'}{p}, \end{aligned}$$

(4.78)

with $\tilde{\tilde{u}}_n(t)=u_n(t+\theta )-u_n(t)$, and for any positive integer n, $t\in [0,T]$ and $\delta \in (0,1)$. The Proposition 4.1 follows readily from this last inequality and noting that a similar argument can be carried out to find a similar estimate for negative values of $\theta$.

Proof of Proposition 4.2

The second equations for the Galerkin approximation is written as

$$\begin{aligned} \begin{array}{ll} d(\varphi _n,\psi )+(\nabla \mu _n,\nabla \psi )dt=(u_n\varphi _n,\nabla \psi )dt,\ \ \forall \psi \in H_n. \end{array} \end{aligned}$$

(4.79)

Let $\psi \in U:=H^1(\mathcal {M})$ and decompose it as $\psi =\psi _I+\psi _{II}$, where $\psi _I\in H_n$ and $\psi _{II}\in H_n^\bot$. Recall that $\psi _I$ and $\psi _{II}$ are orthogonal also in U. Then, from (4.79), we deduce

$$\begin{aligned} d(\varphi _n,\psi )=d\left\langle \varphi _n,\psi \right\rangle _U=d\left\langle \varphi _n,\psi _I\right\rangle _U=-(\nabla \mu _n,\nabla \psi _I)dt+(u_n\varphi _n,\nabla \psi _I)dt. \end{aligned}$$

(4.80)

It follows from (4.80) that

$$\begin{aligned} \begin{array}{ll} \left\langle \varphi _n(t+\theta )-\varphi _n(t),\psi \right\rangle _U&{}=-\int _t^{t+\theta }(\nabla \mu _n(\tau ),\nabla \psi _I)d\tau +\int _t^{t+\theta }(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)d\tau \\ &{}:=X_t(\theta ), \end{array} \end{aligned}$$

(4.81)

for any $0<\theta<\delta <1$.

We have

$$\begin{aligned} \begin{array}{ll} (X_t(\theta ))^2\le C\left| \int _t^{t+\theta }(\nabla \mu _n(\tau ),\nabla \psi _I)d\tau \right| ^2+C\left| \int _t^{t+\theta }(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)d\tau \right| ^2. \end{array} \end{aligned}$$

(4.82)

Note that

$$\begin{aligned} \begin{array}{ll} C\left| \int _t^{t+\theta }(\nabla \mu _n(\tau ),\nabla \psi _I)d\tau \right| ^2&{}\le C\left( \int _t^{t+\theta }|(\nabla \mu _n(\tau ),\nabla \psi _I)|d\tau \right) ^2\\ &{}\le C|\nabla \psi _I|^2\theta \int _t^{t+\theta }|\nabla \mu _n(\tau )|^2d\tau \\ &{}\le C\Vert \psi \Vert _U^2\theta \int _t^{t+\theta }|\nabla \mu _n(\tau )|^2d\tau :=X_t^1(\theta ). \end{array} \end{aligned}$$

(4.83)

Owing to Lemma 4.1, we derive that

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}X_t^1(\theta )\le C\Vert \psi \Vert _U^2\delta \le C\Vert \psi \Vert _U^2\delta ^{\frac{p'}{p}}. \end{aligned}$$

(4.84)

As far as the second term in the right hand side of (4.82) is concerned we notice that when $p<3$ and $\kappa \ge \frac{2(3-p)}{5p-6}$, due to the embedding $W^{1,p}(\mathcal {M})\hookrightarrow L^{\frac{3p}{3-p}}(\mathcal {M})\hookrightarrow L^{\frac{2\kappa +2}{\kappa }}(\mathcal {M})$, we can check that (see [15, Inequality (4.18)])

$$\begin{aligned} \begin{array}{ll} |(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)|\le C\Vert u_n(\tau )\Vert _{V_{{\text {div}},p}}\Vert \varphi _n(\tau )\Vert _{L^{2\kappa +2}}\Vert \psi \Vert _U. \end{array} \end{aligned}$$

(4.85)

When $p=3$ and $\kappa \in (0,2]$ or $p>3$ and $\kappa >0$, due to the embedding $W^{1,p}(\mathcal {M})\hookrightarrow L^{\frac{2\kappa +2}{\kappa }}(\mathcal {M})$, we have also

$$\begin{aligned} |(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)|\le C\Vert u_n(\tau )\Vert _{1,p}\Vert \varphi _n(\tau )\Vert _{L^{2\kappa +2}}\Vert \psi \Vert _U. \end{aligned}$$

(4.86)

Now from (4.85), we estimate the second term in the right hand side of (4.82) as follows

$$\begin{aligned} \begin{array}{ll} C\left| \int _t^{t+\theta }(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)d\tau \right| ^2 &{}\le C\theta \int _t^{t+\theta }|(u_n(\tau )\varphi _n(\tau ),\nabla \psi _I)|^2 d\tau \\ &{}\le C\theta \Vert \psi \Vert _U^2\int _t^{t+\theta }\Vert u_n(\tau )\Vert _{1,p}^2\Vert \varphi _n(\tau )\Vert _{L^{2\kappa +2}}^2d\tau \\ &{}\le C\theta \Vert \psi \Vert _U^2\left( \int _t^{t+\theta }\Vert u_n\Vert _{1,p}^pd\tau \right) ^\frac{2}{p}\left( \int _t^{t+\theta }\Vert \varphi _n\Vert _{L^{2\kappa +2}}^\frac{2p}{p-2}d\tau \right) ^\frac{p-2}{p}=X_t^2(\theta ), \end{array} \end{aligned}$$

where we have also used the Hölder inequality.

Hence, from this previous inequality, we infer that

$$\begin{aligned} \begin{array}{ll} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}X_t^2(\theta ) &{}\le C\delta \Vert \psi \Vert _U^2\left[ \mathbb {E}\int _t^{t+\delta }\Vert u_n\Vert _{1,p}^pd\tau \right] ^\frac{2}{p}\left[ \mathbb {E}\int _t^{t+\delta }\Vert \varphi _n\Vert _{L^{2\kappa +2}}^\frac{2p}{p-2}d\tau \right] ^\frac{p-2}{p}\\ &{}\le C\delta ^\frac{2(p-1)}{p}\Vert \psi \Vert _U^2\mathbb {E}\displaystyle \sup _{\tau \in [t,t+\delta ]}\Vert \varphi _n(\tau )\Vert _{L^{2\kappa +2}}^2\left[ \mathbb {E}\int _t^{t+\delta }\Vert u_n\Vert _{1,p}^pd\tau \right] ^\frac{2}{p}. \end{array} \end{aligned}$$

(4.87)

Owing to Lemma 4.1, we derive from (4.87) that

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}X_t^2(\theta )\le C\delta ^\frac{2(p-1)}{p}\Vert \psi \Vert _U^2\le C\delta ^\frac{p'}{p}\Vert \psi \Vert _U^2,\ \ \text {with}\ \ p'=\frac{p}{p-1}. \end{aligned}$$

(4.88)

Collecting (4.84), (4.88), from (4.82) we then get

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\displaystyle \sup _{\psi \in \mathbf{W} _s,\Vert \psi \Vert _V=1}(X_t(\theta ))^2\le C\delta ^\frac{p'}{p}. \end{aligned}$$

(4.89)

Therefore

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\Vert \varphi _n(t+\theta )-\varphi _n(t)\Vert _{U'}^2=\mathbb {E}\displaystyle \sup _{\theta \in (0,\delta )}\displaystyle \sup _{\psi \in V,\Vert \psi \Vert _U=1}|\left\langle \varphi _n(t+\theta )-\varphi _n(t),\psi \right\rangle _U|^2\le C\delta ^\frac{p'}{p}, \end{aligned}$$

(4.90)

for any positive integer n, $t\in [0,T]$ and $\delta \in (0,1)$.

Finally, collecting all the estimates and making a similar reasoning with $\theta <0$, we thus deduce

$$\begin{aligned} \mathbb {E}\displaystyle \sup _{0<|\theta |\le \delta <1}\Vert \varphi _n(t+\theta )-\varphi _n(t)\Vert _{U'}^2\le C\delta ^\frac{p'}{p}, \end{aligned}$$

for any positive integer n, $t\in [0,T]$. This completes the proof of Proposition 4.2. $\square$

4.3 Tightness and Compactness Results

In this subsection, we study the tightness property of the Galerkin solutions and derive several weak convergence results. The estimates from the previous Propositions (Propositions 4.1–4.2) play an important role in this part of the paper.

Throughout this subsection, we fix $s\in \mathbb {R}$ such that $s>5/2$. Let us consider the spaces

$$\begin{aligned} \begin{array}{ll} &{}\mathfrak {X}_1= L^2(0,T;\mathbb {H})\cap \mathcal {C}(0,T;\mathbb {W}'_s),\\ &{}\mathfrak {X}_2=\mathcal {C}(0,T;K) \end{array} \end{aligned}$$

and we denote by $\mathfrak {B}(\mathfrak {X}_1)$ and $\mathfrak {B}(\mathfrak {X}_2)$ their borel $\sigma$-algebras, respectively.

Now, before proving that the family of laws $\{\mathcal {L}(u_n,\varphi _n): n\in \mathbb {N}\}$ is tight on the Polish space $\mathfrak {X}_1$, we recall the following result which will be needed in the sequel. The proof of this result can be found in the book of Métivier [38, Chapter VI, Lemma 2 and Lemma 3].

Lemma 4.2

Let $\mathbb {B}$, $\mathbb {B}_0$ and $\mathbb {B}_1$ be three reflexive Banach spaces satisfying the compact embedding $\mathbb {B}_0\hookrightarrow \mathbb {B}\hookrightarrow \mathbb {B}_1$. Let $q\in (1,\infty )$ and $\mathfrak {Q}$ be a subset of $L^q(0,T;\mathbb {B})$, which is included in a compact set of $L^q(0,T;\mathbb {B}_1)$ and

$$\begin{aligned} \sup _{v\in \mathfrak {Q}}\int _0^T\Vert u(s)\Vert ^q_{\mathbb {B}_0}ds<\infty . \end{aligned}$$

Then, $\mathfrak {Q}$ is relatively compact in $L^q(0,T;\mathbb {B})$.

We shall prove the following important result.

Lemma 4.3

The family of laws $\{\mathcal {L}(u_n,\varphi _n): n\in \mathbb {N}\}$ is tight in $\mathfrak {X}_1$.

Proof

We firstly prove that $\{\mathcal {L}(u_n,\varphi _n): n\in \mathbb {N}\}$ is tight in $\mathcal {C}(0,T;\mathbb {W}'_s)$. For this aim, we first observe that for a fixed number $R>0$ we have

$$\begin{aligned} \mathbb {P}(\Vert (u_n(t),\varphi _n(t))\Vert _\mathbb {H}>R)\le \frac{1}{R^2}\mathbb {E}\sup _{t\in [0,T]}\Vert (u_n(t),\varphi _n(t))\Vert _\mathbb {H}^2, \end{aligned}$$

from which along with (4.2)$_1$, (4.2)$_2$ and the fact that the domain $\mathcal {M}$ is bounded we infer that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\mathbb {P}(\Vert (u_n(t),\varphi _n(t))\Vert _\mathbb {H}>R)\le \frac{C}{R^2}, \end{aligned}$$

(4.91)

for any $t\in [0,T]$.

Since, by the compact embedding $\mathbb {H}\subset \mathbb {W}'_s$, balls in $\mathbb {H}$ are compact for the strong topology in $\mathbb {W}'_s$, then this implies that the family $\{(u_n(t),\varphi _n(t)): n\in \mathbb {N}\}$ is relatively compact in $\mathbb {W}'_s$ for any $t\in [0,T]$. Therefore, by Propositions 4.1–4.2 and [47, Lemma 1, page 71] we derive that the laws of the family $\{(u_n,\varphi _n): n\in \mathbb {N}\}$ are tight in $\mathcal {C}(0,T;\mathbb {W}'_s)$. This means that for any $\epsilon >0$ there exists a compact subset $K_\epsilon$ of $\mathcal {C}(0,T;\mathbb {W}'_s)$ such that

$$\begin{aligned} \mathbb {P}((u_n,\varphi _n)\in K_\epsilon )\ge 1-\frac{\epsilon }{2},\quad n\in \mathbb {N}. \end{aligned}$$

(4.92)

We also observe that for a fixed number $R>0$, we have

$$\begin{aligned} \begin{array}{ll} \mathbb {P}\left( \Vert (u_n,\varphi _n)\Vert _{L^{p,2}(0,T;\mathbb {V})}>R\right) \le \frac{1}{R^2}\mathbb {E}\Vert (u_n,\varphi _n)\Vert _{L^{p,2}(0,T;\mathbb {V})}^2=\frac{1}{R^2}\left( \mathbb {E}\int _0^T[\Vert u_n\Vert _{1,p}^p+\Vert \varphi _n\Vert _U^2]ds\right) , \end{array} \end{aligned}$$

from which along with (4.3)$_1$, (4.3)$_2$ we derive that

$$\begin{aligned} \mathbb {P}\left( \Vert (u_n,\varphi _n)\Vert _{L^{p,2}(0,T;\mathbb {V})}>R\right) \le \frac{C}{R^2}. \end{aligned}$$

(4.93)

Now, taking $R=\sqrt{\frac{2C}{\epsilon }}:=\varepsilon _1$, where C is the constant appearing in (4.93), we infer that

$$\begin{aligned} \begin{array}{ll} \mathbb {P}(\Vert (u_n,\varphi _n)\Vert _{L^{p,2}(0,T;\mathbb {V})}\le \varepsilon _1)&{}=1-\mathbb {P}(\Vert (u_n,\varphi _n)\Vert _{L^{p,2}(0,T;\mathbb {V})}>\varepsilon _1)\\ &{}\ge 1-\frac{C}{\varepsilon _1^2}=1-\frac{\epsilon }{2},\ \ n\in \mathbb {N}. \end{array} \end{aligned}$$

(4.94)

Now, let

$$\begin{aligned}\mathfrak {Q}_\epsilon =\{(u,\varphi )\in L^{p,2}(0,T;\mathbb {V}):\Vert (u,\varphi )\Vert _{L^{p,2}(0,T;\mathbb {V})}\le \varepsilon _1\}\cap K_\epsilon .\end{aligned}$$

Since $L^{p,2}(0,T;\mathbb {V})\cap \mathcal {C}(0,T;\mathbb {W}'_s)$ is compactly embedded in $L^2(0,T;\mathbb {H})\cap L^2(0,T;\mathbb {W}'_s)$, then $\mathfrak {Q}_\epsilon$ satisfies the conditions of Lemma 4.2. Hence $\mathfrak {Q}_\epsilon$ is relatively compact in $L^2(0,T;\mathbb {H})$. Moreover, $\mathbb {P}((u_n,\varphi _n)\in \mathfrak {Q}_\epsilon )\ge 1-\epsilon$, $n\in \mathbb {N}$. This proves that the family of laws $\{\mathcal {L}(u_n,\varphi _n): n\in \mathbb {N}\}$ is tight in $L^2(0,T;\mathbb {H})$ and we can easily conclude the proof of the lemma. $\square$

Hereafter, the law of the cylindrical Brownian motion $W=(W_t)_{t\in [0,T]}$ is denoted by $\Pi$ and we mention that it is possible to find a set $\bar{\Omega }\in \mathcal {F}$ of measure zero such that $W(\omega )\in \mathcal {C}(0,T;K)$ for any $\omega \in \Omega /\bar{\Omega }$. For any $n\in \mathbb {N}$, we construct a family of probability laws on $\mathfrak {X}_2=\mathcal {C}(0,T;K)$ by setting

$$\begin{aligned} \Pi _n(.)=\mathbb {P}(W\in .)\in P_r(\mathfrak {X}_2)=\Pi ,\ \ \forall n\ge 1 \end{aligned}$$

(4.95)

and where $P_r(\mathfrak {X}_2)$ denotes the set of all probability measures on $(\mathfrak {X}_2,\mathfrak {B}(\mathfrak {X}_2))$.

We now prove the following important result.

Theorem 4.1

The family of laws of $((u_n,\varphi _n); W)$ is tight on the Polish space $\mathfrak {X}_1\times \mathfrak {X}_2$.

Proof

We have already proved that the family of laws $\{\mathcal {L}(u_n,\varphi _n):n\in \mathbb {N}\}$ is tight in $\mathfrak {X}_1$ (cf. Lemma 4.3). Now we shall proved that the family $\{\Pi _n: n=1,2,\ldots \}$ is tight in $P_r(\mathfrak {X}_2)$. For this, we endow the space $\mathfrak {X}_2$ with the uniform convergence, and then $\mathfrak {X}_2$ is now a Polish space. Hence, it follows from [3, Theorem 6.8] that $P_r(\mathfrak {X}_2)$ endowed with the Prohorov’s metric is a separable and complete metric space. By construction, the family of probability laws $\{\Pi _n:n=1,2,\ldots \}$ is reduced to one element which is the law of W and belongs to $P_r(\mathfrak {X}_2)$. Thus, by [43, Chapter II, Theorem 3.2] we infer that the family $\{\Pi _n:n=1,2,\ldots \}$ is tight on $P_r(\mathfrak {X}_2)$. Finally from the fact that $\{\mathcal {L}(u_n,\varphi _n):n\in \mathbb {N}\}$ is tight in $\mathfrak {X}_1$, the family $\{\Pi _n: n=1,2,\ldots \}$ is tight in $P_r(\mathfrak {X}_2)$ in conjunction with [31, Corollary 1.3], we infer that the family of laws of the joint processes $((u_n,\varphi _n),W)$ is tight in $\mathfrak {X}_1\times \mathfrak {X}_2$. $\square$

Proposition 4.3

Let $\mathfrak {X}=\mathfrak {X}_1\times \mathcal {C}(0,T;K)$. There exist a Borel probability measure $\mu _1$ on $\mathfrak {X}$ and a subsequence of $((u_n,\varphi _n), W)$ such that their laws weakly converge to $\mu _1$.

Proof

Thanks to the theorem 4.1, the laws of $((u_n,\varphi _n), W)$ form a tight family on $\mathfrak {X}$. Since $\mathfrak {X}$ is a Polish space, we get the result from the application of Prohorov’s theorem (cf. [3, Theorem I. 5. 1, page 59]). $\square$

The following result relates the above convergence in law to almost sure convergence.

Proposition 4.4

There exist a complete probability space $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$ and a sequence of $\mathfrak {X}$-valued random variables, denoted by $\{(\tilde{u}_n,\tilde{\varphi }_n,\tilde{W}_n): n\in \mathbb {N}\}$, defined on $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$ such that their laws are equal to the laws of $\{(u_n,\varphi _n,W): n\in \mathbb {N}\}$ on $\mathfrak {X}$. Also, there exists an $\mathfrak {X}$-random variable $\{(\tilde{u},\tilde{\varphi }),\tilde{W}\}$ defined on $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$ such that

$$\begin{aligned} \begin{array}{ll} &{}\mathcal {L}(\tilde{u},\tilde{\varphi },\tilde{W})=\mu _1,\\ &{} \tilde{W}_n\rightarrow \tilde{W}~\text {in}~\mathcal {C}(0,T;K)~\tilde{\mathbb {P}}\text {-a.s.},\\ &{}(\tilde{u}_n,\tilde{\varphi }_n)\rightarrow (\tilde{u},\tilde{\varphi })~\text {in}~L^2(0,T;\mathbb {H})~\tilde{\mathbb {P}}\text {-a.s.},\\ &{}(\tilde{u}_n,\tilde{\varphi }_n)\rightarrow (\tilde{u},\tilde{\varphi })~\text {in}~\mathcal {C}(0,T;\mathbb {W}_s'). \end{array} \end{aligned}$$

(4.96)

Proof

The proof of Proposition 4.4 is a consequence of Proposition 4.3 and Skorokhod’s Theorem [50]. $\square$

Proposition 4.5

Let $Q=\mathfrak {I}\mathfrak {I}^*$ where $\mathfrak {I}$ is the canonical injection, which is Hilbert–Schmidt, from K into $K_1$. Then, the stochastic process $\tilde{W}=(\tilde{W}_t)_{t\in [0,T]}$ is a $K_1$-valued Q-Wiener process on $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$. Furthermore, if $0\le s<t\le T$, then the increments $\tilde{W}_t-\tilde{W}_s$ are independent of the $\sigma$-algebra $\tilde{\mathcal {F}}_s$ generated by $\tilde{u}(\tau )$, $\tilde{\varphi }(\tau )$, $\tilde{W}_\tau$ for $\tau \in [0,s]$.

Proof

The proof is a verbatim reproduction of similar result in [45, Proposition 3.11]. $\square$

Now, since $(\tilde{u}_n,\tilde{\varphi }_n,\tilde{W}_n)$ and $(u_n,\varphi _n,W)$ have the same law (cf. Proposition 4.4), it follows from Lemma 4.1 that $(\tilde{u}_n,\tilde{\varphi }_n)$ satisfies the estimates

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}|\tilde{u}_n(s)|^q<C,\\ &{}\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}\Vert \tilde{\varphi }_n(s)\Vert _{L^{2\kappa +2}(\mathcal {M})}^q<C,\\ &{}\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}\Vert F(\tilde{\varphi }_n(s))\Vert _{L^1(\mathcal {M})}^{q/2}< C,\\ &{}\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}\Vert F'(\tilde{\varphi }_n(s))\Vert _{L^r(\mathcal {M})}^q< C,\\ &{}\tilde{\mathbb {E}}\left( \displaystyle \int _0^T\Vert \tilde{u}_n(s)\Vert _{1,p}^pds\right) ^{q/p}<C,\\ &{}\tilde{\mathbb {E}}\left( \displaystyle \int _0^T\Vert \tilde{\varphi }_n(s)\Vert _U^2ds\right) ^{q/2}<C,\\ &{}\tilde{\mathbb {E}}\left( \displaystyle \int _0^T\Vert \tilde{\mu }_n(s)\Vert _U^2ds\right) ^{q/2}<C, \end{array} \end{aligned}$$

(4.97)

for any $n\in \mathbb {N}$, $q\in [2,\infty )$ and $p\ge 11/5$. Here $\tilde{\mu }_n=a\tilde{\varphi }_n-J*\tilde{\varphi }_n+F'(\tilde{\varphi }_n)$ and $U=H^1(\mathcal {M})$.

We will now prove the following important lemma:

Lemma 4.4

We can extract a subsequence $\{(\tilde{u}_{n_k},\tilde{\varphi }_{n_k}): k\in \mathbb {N}\}$ from $\{(\tilde{u}_n,\tilde{\varphi }_n): n\in \mathbb {N}\}$ such that

$$\begin{aligned} \begin{array}{ll} &{}(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\rightarrow (\tilde{u},\tilde{\varphi })\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;\mathbb {H}),\\ &{} J*\tilde{\varphi }_{n_k}\rightarrow J*\tilde{\varphi }\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U)),\\ &{}\tilde{\mu }_{n_k}\rightharpoonup \tilde{\mu }=a\tilde{\varphi }-J*\tilde{\varphi }+F'(\tilde{\varphi })\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U)). \end{array} \end{aligned}$$

(4.98)

Also the processes $\tilde{u}$, $\tilde{\varphi }$ and $\tilde{\mu }$ satisfy the estimates (4.97).

Proof

Thanks to the estimate (4.97)$_5$ and the Eberlein–Smulian theorem (see [52, Chapter 21, Proposition 21.23-(h)]), we infer that there exists a subsequence $\tilde{u}_{n_k}$ of $\tilde{u}_n$ satisfying

$$\begin{aligned} \begin{array}{ll} &{}\tilde{u}_{n_k}\rightharpoonup \tilde{u}\ \ \text {in}\ \ L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p})),\\ &{} \tilde{u}_{n_k}(T)\rightharpoonup \eta _1\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};G_{{\text {div}}}),\ \ \text {for any}\ \ q\in [2,\infty ). \end{array} \end{aligned}$$

(4.99)

We claim that $\eta _1=\tilde{u}(T)$ in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};G_{{\text {div}}})$. The prove of this will be given later.

Thanks to (4.99)$_1$, reasoning similarly as in [13, Equation (4.88)], we check that

$$\begin{aligned}\tilde{u}_{n_k}\rightarrow \tilde{u}\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}}; L^2(0,T;G_{{\text {div}}})).\end{aligned}$$

Thus, modulo the extraction of a subsequence (still) denoted $(\tilde{u}_{n_k})_{k\ge 1}$ we have

$$\begin{aligned} \tilde{u}_{n_k}\rightarrow \tilde{u}\ \ d\tilde{\mathbb {P}}\otimes dt\text {-a.e. in}\ \ G_{{\text {div}}}. \end{aligned}$$

(4.100)

Thanks to (4.96)$_2$ and (4.97)$_2$, [13, Equation (4.89)] we obtain

$$\begin{aligned}\tilde{\varphi }_{n_k}\rightarrow \tilde{\varphi }\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}}; L^2(0,T;H)),\end{aligned}$$

and thus, modulo the extraction of a subsequence (still) denoted $(\tilde{\varphi }_{n_k})_{k\ge 1}$ one has

$$\begin{aligned} \tilde{\varphi }_{n_k}\rightarrow \tilde{\varphi }\ \ d\tilde{\mathbb {P}}\otimes dt\text {-a.e. in}\ \ H. \end{aligned}$$

(4.101)

Naturally the convergent (4.98)$_1$ follows from the previous convergence.

Later, we will show that

$$\begin{aligned} \tilde{\varphi }_{n_k}\rightharpoonup \eta _2=\tilde{\varphi }(T)\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};H). \end{aligned}$$

(4.102)

Due to (4.98)$_1$ and the fact that the map $J*: H\rightarrow U$ is linear and bounded, we easily derive the convergence (4.98)$_2$.

Now, we will show that $\tilde{\mu }_{n_k}\rightharpoonup \tilde{\mu }=a\tilde{\varphi }-J*\tilde{\varphi }+F'(\tilde{\varphi })\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))$.

First of all, from the estimate (4.97)$_7$ and the Eberlein–Smulian theorem (see [52, Chapter 21, Proposition 21.23-(h)]), we can extract a subsequence of $\tilde{\mu }_n$ denoted by $\tilde{\mu }_{n_k}$ such that for any $q\in [2,\infty )$

$$\begin{aligned} \tilde{\mu }_{n_k}\rightharpoonup \tilde{\mu }\ \ \text {in}\ \ L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U)). \end{aligned}$$

(4.103)

From the estimates (4.97)$_{2,4}$ and the fact that the domain $\mathcal {M}$ is bounded, we infer that $\tilde{\rho }(.,\tilde{\varphi }_n)=a(.)\tilde{\varphi }_n+F'(\tilde{\varphi }_n)$ is bounded in $L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L^r(\mathcal {M})))$, for any $q\in [2,\infty )$. Therefore from the Banach–Alaoglu theorem, we conclude there exits a subsequence of $\rho (.,\tilde{\varphi }_n)$ denoted by $\rho (.,\tilde{\varphi }_{n_k})$ such that for any $q\in [2,\infty )$

$$\begin{aligned} \rho (.,\tilde{\varphi }_{n_k}){\mathop {\rightharpoonup }\limits ^{*}}\rho \ \ \text {in}\ \ L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L^r(\mathcal {M}))). \end{aligned}$$

(4.104)

From the pointwise convergence (4.101) we have $\rho (.,\tilde{\varphi }_{n_k})\rightarrow a(.)\tilde{\varphi }+F'(\tilde{\varphi })$ d$\tilde{\mathbb {P}}$-almost everywhere in $(0,T)\times \mathcal {M}$ and therefore from (4.104) we have $\rho =a\tilde{\varphi }+F'(\tilde{\varphi })$, i.e., $\rho (x,\varphi )=a(x)\tilde{\varphi }+F'(\tilde{\varphi })$; $x\in \mathcal {M}$.

Now, we introduce the following set

$$\begin{aligned}\varvec{D}=\{\varvec{\Phi }=\phi (\omega )\chi (t)v:\phi \in L^\infty (\tilde{\Omega },\tilde{\mathbb {P}}), \chi \in \mathcal {D}(0,T)~\text {and}~v\in H_n\},\end{aligned}$$

where $H_n$ is defined in the Sect. 4.1. This set is dense in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;H))$.

For any $\varvec{\Phi }=\phi (\omega )\chi (t)v\in \varvec{D}$ and every $n_k\ge n$ (n is fixed), we have

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\left( \phi (\omega )\int _0^T\left( \tilde{\mu }_{n_k}(t),v\right) \chi (t)dt\right) =\tilde{\mathbb {E}}\left( \phi (\omega )\int _0^T(\rho (.,\tilde{\varphi }_{n_k}(t))-J*\tilde{\varphi }_{n_k}(t),v)\chi (t)dt\right) . \end{array} \end{aligned}$$

By passing to the limit as $n_k\rightarrow \infty$ in this identity and using the convergence (4.103), (4.104) and (4.98)$_2$, on account of the density of $\varvec{D}$ in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;H))$ we get $\tilde{\mu }=a\tilde{\varphi }+F'(\tilde{\varphi })-J*\tilde{\varphi }$ i.e. (4.98)$_3$. In particular we obtain $\rho (.,\tilde{\varphi })=a(.)\tilde{\varphi }+F'(\tilde{\varphi })\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))$.

Finally, making similar reasoning as in [13, Proposition 4.2, Inequality (4.86)] we can check that the stochastic processes $\tilde{u}, \tilde{\varphi }$ and $\tilde{\mu }$ satisfies the same estimates as in (4.97). This ends the proof of Lemma 4.4. $\square$

Proposition 4.6

Let $p\in [11/5,12/5)$ and $T>0$. There exits five processes $\varvec{N}, \varvec{B}_0, \varvec{R}_1\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p})$, $\varvec{B}_1\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U')$ and $\varvec{g}_1\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;V'_{{\text {div}},p})$ such that

$$\begin{aligned} \begin{array}{ll} &{}\mathcal {P}_{n_k}^1 N(\tilde{u}_{n_k})\rightharpoonup \varvec{N}\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}),\\ &{}\mathcal {P}_{n_k}^1 B_0(\tilde{u}_{n_k})\rightharpoonup \varvec{B}_0\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}),\\ &{}\mathcal {P}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}\rightharpoonup \varvec{R}_1\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}),\\ &{}\mathcal {P}_{n_k}^2\tilde{u}_{n_k}.\nabla \tilde{\varphi }_{n_k}\rightharpoonup \varvec{B}_1\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U'),\\ &{}\mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\rightharpoonup \varvec{g}_1\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;V'_{{\text {div}},p})). \end{array} \end{aligned}$$

(4.105)

Proof

Using the assumption $(H_1)$ for $g_1$, the estimates (4.97)$_{1,2}$, and applying also the Banach–Alaoglu theorem we easily derive (4.105)$_5$.

Owing to (2.23), (4.97)$_5$ and application of Banach–Alaoglu’s theorem, we get (4.105)$_1$.

By mean of Hölder’s and Gagliardo–Nirenberg’s inequalities, we have for all $v\in V_{{\text {div}},p}$

$$\begin{aligned} \begin{array}{ll} |b(\tilde{u}_{n_k},\tilde{u}_{n_k},v)|&{}\le c \Vert \tilde{u}_{n_k}\Vert _{L^{3p/(4p-6)}}\Vert \tilde{u}_{n_k}\Vert _{1,p}\Vert v\Vert _{L^{3p/(3-p)}}\\ &{}\le c|\tilde{u}_{n_k}|^{(10p-18)/(5p-6)}\Vert \tilde{u}_{n_k}\Vert _{W^{1,p}}^{(12-5p)/(5p-6)}\Vert \tilde{u}_{n_k}\Vert _{1,p}\Vert v\Vert _{L^{3p/(3-p)}}\\ &{}\le c|\tilde{u}_{n_k}|^{(10p-18)/(5p-6)}\Vert \tilde{u}_{n_k}\Vert _{W^{1,p}}^{(12-5p)/(5p-6)}\Vert \tilde{u}_{n_k}\Vert _{1,p}\Vert v\Vert _{W^{1,p}}\\ &{}\le c|\tilde{u}_{n_k}|^{(10p-18)/(5p-6)}\Vert \tilde{u}_{n_k}\Vert _{1,p}^{6/(5p-6)}\Vert v\Vert _{1,p}, \end{array} \end{aligned}$$

where we have also used the fact that $W^{1,p}\hookrightarrow L^\frac{\textstyle 3p}{\textstyle 3-p}$ and that $V_{{\text {div}},p}$-norm is equivalent to the $W^{1,p}$-norm. Hence,

$$\begin{aligned} \Vert B_0(\tilde{u}_{n_k})\Vert _{V'_{{\text {div}},p}} =\displaystyle \sup _{v\in V_{{\text {div}},p},\Vert v\Vert _{1,p}\le 1}\left| b(\tilde{u}_{n_k},\tilde{u}_{n_k},v)\right| \le c|\tilde{u}_{n_k}|^\frac{10p-18}{5p-6}\Vert \tilde{u}_{n_k}\Vert _{1,p}^\frac{6}{5p-6}. \end{aligned}$$

(4.106)

Thanks to (4.106), using Hölder’s inequality, we see that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\left[ \int _0^T\Vert \mathcal {P}_{n_k}^1B_0(\tilde{u}_{n_k})\Vert _{V'_{{\text {div}},p}}^{p'}dt\right] ^{2/p'}&{}\le c\tilde{\mathbb {E}}\left[ \left( \int _0^T|\tilde{u}_{n_k}|^\frac{2(5p-9)}{5p-11} dt\right) ^\frac{2(5p-11)}{5p-6}\left( \int _0^T\Vert \tilde{u}_{n_k}\Vert _{1,p}^pdt\right) ^{\frac{12}{p(5p-6)}}\right] \\ &{}\le c \left[ \tilde{\mathbb {E}}\displaystyle \sup _{t\in [0,T]}|\tilde{u}_{n_k}(t)|^\frac{4(5p-9)}{5p-8}\right] ^\frac{5p-8}{5p-6}\left[ \tilde{\mathbb {E}}\left( \int _0^T\Vert \tilde{u}_{n_k}\Vert _{1,p}^pdt\right) ^\frac{6}{p}\right] ^\frac{2}{5p-6}. \end{array} \end{aligned}$$

Therefore, from this previous inequality and the estimates (4.97)$_1$–(4.97)$_5$, we infer that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\left[ \int _0^T\Vert \mathcal {P}_{n_k}^1B_0(\tilde{u}_{n_k}(t))\Vert _{V'_{{\text {div}},p}}^{p'}dt\right] ^{2/p'}<C. \end{array} \end{aligned}$$

(4.107)

Thanks to (4.107) and the Banach–Alaoglu theorem, we derive that (4.105)$_2$ holds.

By using the Sobolev embedding $W^{1,p}\hookrightarrow L^\frac{3p}{3-p}$, the Hölder inequality and the fact that $\kappa \ge \frac{2(3-p)}{5p-6}$, we have

$$\begin{aligned} |(\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k},v)|&\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{6p/(5p-6)}}|\nabla \tilde{\mu }_{n_k}|\Vert v\Vert _{L^{3p/(3-p)}}\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{2\kappa +2}}|\nabla \tilde{\mu }_{n_k}|\Vert v\Vert _{1,p}, \end{aligned}$$

for all $v\in V_{{\text {div}},p}$. Hence

$$\begin{aligned} \Vert \tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}\Vert _{V'_{{\text {div}},p}}\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{2\kappa +2}}|\nabla \tilde{\mu }_{n_k}|. \end{aligned}$$

(4.108)

From (4.108), (4.97)$_2$–(4.97)$_7$ and the Hölder inequality, we obtain

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}[\int _0^T\Vert \tilde{\varphi }_{n_k}(t)\nabla \tilde{\mu }_{n_k}(t)\Vert _{V'_{{\text {div}},p}}^{p'}dt]^\frac{2}{p'}&{}\le c\tilde{\mathbb {E}}\left[ \left( \int _0^T\Vert \tilde{\varphi }_{n_k}(t)\Vert _{L^{2\kappa +2}}^\frac{2p}{p-2}dt\right) ^\frac{p-2}{p}\int _0^T|\nabla \tilde{\mu }_{n_k}(t)|^2dt\right] \\ &{}\le c[\tilde{\mathbb {E}}\displaystyle \sup _{t\in [0,T]}\Vert \tilde{\varphi }_{n_k}(t)\Vert _{L^{2\kappa +2}}^4]^\frac{1}{2}\left[ \tilde{\mathbb {E}}\left( \int _0^T|\nabla \tilde{\mu }_{n_k}|^2dt\right) ^2\right] ^\frac{1}{2}<C. \end{array} \end{aligned}$$

(4.109)

Thus from (4.109) and the Banach–Alaoglu theorem, we derive (4.105)$_3$.

Now, we complete our proof by proving (4.105)$_4$. For this we first give the following estimate:

$$\begin{aligned} \begin{array}{ll} |(\tilde{u}_{n_k}.\nabla \tilde{\varphi }_{n_k},\psi )|=|(\tilde{u}_{n_k},\tilde{\varphi }_{n_k}\nabla \psi )|&{}\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{2\kappa +2}}\Vert \tilde{u}_{n_k}\Vert _{L^{2(\kappa +1)/\kappa }}\Vert \psi \Vert _U\\ &{}\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{2\kappa +2}}\Vert \tilde{u}_{n_k}\Vert _{L^{3p/(3-p)}}\Vert \psi \Vert _U\\ &{}\le c\Vert \tilde{\varphi }_{n_k}\Vert _{L^{2\kappa +2}}\Vert \tilde{u}_{n_k}\Vert _{1,p}\Vert \psi \Vert _U, \end{array} \end{aligned}$$

(4.110)

where we have used the fact that, since $\kappa \ge \frac{2(3-p)}{5p-6}$, then the following Sobolev embedding hold: $W^{1,p}\hookrightarrow L^{3p/(3-p)}\hookrightarrow L^{(2\kappa +2)/\kappa }$.

Using now (4.110), (4.97)$_{2,5}$ and the Hölder inequality, we get

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\int _0^T\Vert \mathcal {P}_{n_k}^2\tilde{u}_{n_k}(t).\nabla \tilde{\varphi }_{n_k}(t)\Vert _{U'}^2dt \le c[\tilde{\mathbb {E}}\displaystyle \sup _{t\in [0,T]}\Vert \tilde{\varphi }_{n_k}(t)\Vert _{L^{2\kappa +2}}^4]^\frac{1}{2}\left[ \tilde{\mathbb {E}}\left( \int _0^T\Vert \tilde{u}_{n_k}\Vert _{1,p}^pdt\right) ^\frac{4}{p}\right] ^\frac{1}{2}<C. \end{array} \end{aligned}$$

(4.111)

Finally (4.105)$_4$ follows from (4.111) and an application of Banach–Alaoglu’s theorem. The proof of Proposition 4.6 is now complete. $\square$

Hereafter, $\left\langle .,.\right\rangle$ denotes the dual pairing between $V_{{\text {div}},p}$ and $V'_{{\text {div}},p}$ relative to $G_{{\text {div}}}$, and that between $U=H^1(\mathcal {M})$ and $U'$ relative to H.

4.4 Passage to the Limit and the End of Proof of Theorem 3.1

Here we prove several convergence which will enable us to conclude that the limiting objects that we found in Proposition 4.4 are in fact a weak martingale solution to our problem.

Lemma 4.4 will be used to prove the following convergence results.

Proposition 4.7

$$\begin{aligned} \begin{aligned}&\int _0^.\mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s))ds\rightarrow \int _0^.\varvec{g}_1(s)ds\\&\quad = \displaystyle \int _0^.g_1(\tilde{u}(s),\tilde{\varphi }(s))ds\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;V'_{{\text {div}}}));\\&\mathcal {P}_{n_k}^1 B_0(\tilde{u}_{n_k})\rightharpoonup \varvec{B}_0=B_0(\tilde{u})\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}));\\&\mathcal {P}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}\rightharpoonup \varvec{R}_1=\tilde{\varphi }\nabla \tilde{\mu }\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}));\\&\mathcal {P}_{n_k}^2\tilde{u}_{n_k}.\nabla \tilde{\varphi }_{n_k}\rightharpoonup \varvec{B}_1=\tilde{u}.\nabla \tilde{\varphi }\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U'). \end{aligned} \end{aligned}$$

(4.112)

Proof

We have already proved in Proposition 4.6 that $\mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})$ belongs to a bounded set of $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}))$; $\mathcal {P}_{n_k}^1 K_0(\tilde{u}_{n_k})$; $\mathcal {P}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}$ belong to a bounded set of $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}))$ and $\mathcal {P}_{n_k}^2\tilde{u}_{n_k}.\nabla \tilde{\varphi }_{n_k}$ belongs to a bounded set of $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U')$.

Hereafter we denote by (for the sake of simplicity) $\tilde{\tilde{u}}_{n_k}(.):=\tilde{u}_{n_k}(.)-\tilde{u}(.)$ and $\tilde{\tilde{\varphi }}_{n_k}(.):=\tilde{\varphi }_{n_k}(.)-\tilde{\varphi }(.)$.

In order to prove (4.112)$_{2,3}$, we introduce the following set

$$\begin{aligned}\varvec{\mathbb {D}}=\{\varvec{\Phi }=\phi (\omega )\chi (t)w_j:\phi \in L^\infty (\tilde{\Omega },\tilde{\mathbb {P}}), \chi \in \mathcal {D}(0,T)~\text {and}~j=1,2,\ldots \},\end{aligned}$$

where $\{w_j:j=1,2,\ldots \}$ is defined in Sect. 4.1. Since this set is dense in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p}))$, it then follows from [52, Proposition 21.23] that the claims (4.112)$_{2,3}$ are achieved if we prove that

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle B_0(\tilde{u}_{n_k}(s))-B_0(\tilde{u}(s)),w_j\right\rangle \chi (s)ds)\rightarrow 0;\\ &{}\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle \tilde{\varphi }_{n_k}(s)\nabla \tilde{\mu }_{n_k}(s)-\tilde{\varphi }(s)\nabla \tilde{\mu }(s),w_j\right\rangle \chi (s)ds)\rightarrow 0 \end{array} \end{aligned}$$

for any $\varvec{\Phi }=\phi (\omega )\chi (t)w_j\in \varvec{\mathbb {D}}$. For this purpose we first note that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle B_0(\tilde{u}_{n_k})-B_0(\tilde{u}),w_j\right\rangle \chi (s)ds)&{}=-\tilde{\mathbb {E}}(\phi (\omega )\int _0^T b(\tilde{\tilde{u}}_{n_k},\tilde{u}_{n_k},w_j)\chi (s)ds)\\ &{}\ \ +\tilde{\mathbb {E}}(\phi (\omega )\int _0^T b(\tilde{u},\tilde{u}-\tilde{u}_{n_k},w_j)\chi (s)ds)\\ &{}=I_1+I_2 \end{array} \end{aligned}$$

(4.113)

and

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle \tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}-\tilde{\varphi }\nabla \tilde{\mu },w_j\right\rangle \chi (s)ds)&{}=\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{\varphi }_{n_k}-\tilde{\varphi }]\nabla (\tilde{\mu }_{n_k}-\tilde{\mu }),w_j\right) \chi (s)ds)\\ &{}\quad -\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{\mu }_{n_k}-\tilde{\mu }]\nabla \tilde{\varphi },w_j\right) \chi (s)ds)\\ &{}\quad +\,\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{\varphi }_{n_k}-\tilde{\varphi }]\nabla \tilde{\mu },w_j\right) \chi (s)ds)\\ &{}=I_3+I_4+I_5. \end{array} \end{aligned}$$

(4.114)

The mapping $b(\tilde{u},.,w_j)$ from $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;V_{{\text {div}},p}))$ into $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;\mathbb {R}))$ is linear and continuous. Therefore, by invoking (4.99), $b(\tilde{u},\tilde{u}-\tilde{u}_{n_k},w_j)\rightarrow 0$ weakly in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;\mathbb {R}))$; and then $I_2\rightarrow 0$ as $n_k\rightarrow \infty$. Next, from the properties of the operator b and Hölder’s inequality, we see that

$$\begin{aligned} \begin{array}{ll} |\tilde{\mathbb {E}}(\phi (\omega )\int _0^T b(\tilde{u}_{n_k}-\tilde{u},\tilde{u}_{n_k},w_j)\chi (s)ds)| \le c\Vert \varvec{\Phi }\Vert _{L^\infty }\left[ \tilde{\mathbb {E}}\left( \int _0^T\Vert \tilde{u}_{n_k}\Vert _{1,p}^pds\right) ^\frac{2}{p}\right] ^\frac{1}{2}\left[ \int _0^T|\tilde{u}_{n_k}-\tilde{u}|^2ds\right] ^{\frac{1}{2}}, \end{array} \end{aligned}$$

with $\Vert \varvec{\Phi }\Vert _{L^\infty }=\Vert \varvec{\Phi }\Vert _{L^\infty (\tilde{\Omega }\times [0,T]\times \mathcal {M})}$.

Thanks to (4.97)$_5$ in conjunction with (4.98)$_1$ we see that the right-hand side of the above inequality converges to 0 as $n_k\rightarrow \infty$. Hence, $I_1$ converges to 0 and then (4.112)$_2$ holds.

By Lemma 4.4 and Hölder’s inequality, we have

$$\begin{aligned} \begin{array}{ll} |\tilde{\mathbb {E}}(\phi (\omega )\int _0^T([\tilde{\tilde{\varphi }}_{n_k}(s)]\nabla \tilde{\mu },w_j)\chi (s)ds)|&\le c\Vert \varvec{\Phi }\Vert _{L^\infty }[\tilde{\mathbb {E}}\int _0^T|\nabla \tilde{\mu }|^2ds]^\frac{1}{2}[\tilde{\mathbb {E}}\int _0^T|\tilde{\tilde{\varphi }}_{n_k}|^2ds]^\frac{1}{2}\rightarrow 0, \end{array} \end{aligned}$$

(4.115)

and therefore $I_5\rightarrow 0$ as $n_k\rightarrow \infty$. We recall that $\tilde{\tilde{\varphi }}_{n_k}=\tilde{\varphi }_{n_k}-\tilde{\varphi }$.

For fixed $\varvec{\Phi }\in \varvec{\mathbb {D}}$ and $\tilde{\varphi }\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))$, the mapping

$$\begin{aligned} \Gamma \mapsto \tilde{\mathbb {E}}\left( \phi (\omega )\int _0^T\left( \Gamma (s)\nabla \tilde{\varphi }(s),w_j\right) \chi (s)ds\right) \end{aligned}$$

is a continuous linear functional on $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))$. Then, by invoking (4.103), we infer that $I_4\rightarrow 0$ as $n_k$ goes to infinity.

Arguing similarly as in (4.115), we have

$$\begin{aligned} \begin{array}{ll} &{}|\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{\tilde{\varphi }}_{n_k}(s)]\nabla (\tilde{\mu }_{n_k}(s)-\tilde{\mu }(s)),w_j\right) \chi (s)ds)|\\ &{}\quad \le c\Vert \varvec{\Phi }\Vert _{L^\infty }\left[ \tilde{\mathbb {E}}\int _0^T\left[ |\nabla \tilde{\mu }(s)|^2+|\nabla \tilde{\mu }_{n_k}(s)|^2\right] ds\right] ^\frac{1}{2}\left[ \tilde{\mathbb {E}}\int _0^T|\tilde{\tilde{\varphi }}_{n_k}(s)|^2ds\right] ^\frac{1}{2}. \end{array} \end{aligned}$$

(4.116)

Using now (4.116), (4.97)$_7$ in conjunction with Lemma 4.4, it follows that $I_3$ converges to 0 as $n_k$ goes to infinity; and then we get the convergence (4.112)$_3$.

Let us moves to the proof of (4.112)$_4$. For this let us introduce the following set

$$\begin{aligned}\tilde{\mathbb {D}}=\{\tilde{\Phi }=\phi (\omega )\chi (t)\psi _j:\phi \in L^\infty (\tilde{\Omega },\tilde{\mathbb {P}}), \chi \in \mathcal {D}(0,T)~\text {and}~j=1,2,\ldots \},\end{aligned}$$

where $\left\{ \psi _j:j=1,2,\ldots \right\}$ is defined in Sect. 4.1. Since this set is dense in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U))$, it then follows from [52, Proposition 21.23] that the claims (4.112)$_4$ are achieved if we prove that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle \tilde{u}_{n_k}(s).\nabla \tilde{\varphi }_{n_k}(s)-\tilde{u}(s).\nabla \tilde{\varphi }(s),\psi _j\right\rangle \chi (s)ds)\rightarrow 0, \end{array} \end{aligned}$$

for any $\tilde{\Phi }=\phi (\omega )\chi (t)\psi _j\in \tilde{\mathbb {D}}$. For this purpose, we begin by rewriting the last identity as follows

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left\langle \tilde{u}_{n_k}(s).\nabla \tilde{\varphi }_{n_k}(s)-\tilde{u}(s).\nabla \tilde{\varphi }(s),\psi _j\right\rangle \chi (s)ds)\\ &{}\quad =-\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{u}_{n_k}-\tilde{u}]\tilde{\varphi }_{n_k},\nabla \psi _j\right) \chi ds)-\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( \tilde{u}[\tilde{\varphi }_{n_k}-\tilde{\varphi }],\nabla \psi _j\right) \chi ds)\\ &{}\quad =I_6+I_7. \end{array} \end{aligned}$$

By the Hölder inequality and Lemma 4.4, we have

$$\begin{aligned} \begin{array}{ll} |\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( \tilde{u}[\tilde{\tilde{\varphi }}_{n_k}],\nabla \psi _j\right) \chi ds)|&\le c\Vert \tilde{\Phi }\Vert _{L^\infty }[\tilde{\mathbb {E}}\int _0^T|\tilde{u}(s)|^2ds]^\frac{1}{2}[\tilde{\mathbb {E}}\int _0^T|\tilde{\tilde{\varphi }}_{n_k}(s)|^2ds]^{1/2}\rightarrow 0, \end{array} \end{aligned}$$

as $n_k\rightarrow \infty$. Hence, $I_7\rightarrow 0$ as $n_k\rightarrow \infty$.

By Hölder’s inequality, we obtain

$$\begin{aligned} \begin{array}{ll} |\tilde{\mathbb {E}}(\phi (\omega )\int _0^T\left( [\tilde{\tilde{u}}_{n_k}(s)]\tilde{\varphi }_{n_k}(s),\nabla \psi _j\right) \chi (s) ds)|&\le c \Vert \tilde{\Phi }\Vert _{L^\infty }[\tilde{\mathbb {E}}\int _0^T|\tilde{\varphi }_{n_k}|^2ds]^\frac{1}{2}[\tilde{\mathbb {E}}\int _0^T|\tilde{\tilde{u}}_{n_k}(s)|^2ds]^\frac{1}{2}, \end{array} \end{aligned}$$

from which in conjunction with (4.97)$_6$ and Lemma 4.4, we infer that $I_6\rightarrow 0$ as $n_k$ goes to infinity. Hence (4.112)$_4$ holds.

Thanks to (4.100)–(4.101), the continuity of $\mathcal {P}_{n_k}^1 g_1(u_{n_k},\varphi _{n_k})$ and the applicability of dominated convergence theorem, we infer that (4.112)$_1$ holds. The proof of Proposition 4.7 is now complete. $\square$

Remark 4.1

Almost surely the paths of the process $(\tilde{u},\tilde{\varphi })$ are $\mathbb {H}=G_{{\text {div}}}\times H$-valued weakly continuous. Indeed, we note that from (4.96)$_4$ and the fact that the processes $\tilde{u}$ and $\tilde{\varphi }$ satisfy the estimates (4.97)$_1$ and (4.97)$_2$, respectively, see Lemma 4.4; it follows that almost surely $(\tilde{u},\tilde{\varphi })\in \mathcal {C}(0,T;\mathbb {W}'_s)\cap L^2(0,T;\mathbb {H})$. Hence, we infer from [48, Theorem 2.1] that $\tilde{\mathbb {P}}$-a.s. $(\tilde{u},\tilde{\varphi })\in \mathcal {C}(0,T;\mathbb {H}_w)$, where $\mathcal {C}(0,T;\mathbb {H}_w)$ denotes the space of weakly continuous functions $\mathbf{u} :[0,T]\rightarrow \mathbb {H}$. By closely follows the proof of [48, Theorem 2.1], we derive from [48, Eq. (2.1), p. 544] that $(\tilde{u}(t),\tilde{\varphi }(t))\in \mathbb {H}$ for all $t\in [0,T]$. We can used the same argument to prove that $\tilde{\mathbb {P}}$-a.s. $(\tilde{u}_{n_k}(t),\tilde{\varphi }_{n_k}(t))\in \mathbb {H}$ for all $t\in [0,T]$.

Now, to simplify notation let us define the processes $\mathfrak {M}_{n_k}(t)$ and $\mathfrak {R}_{n_k}(t)$, $t\in (0,T]$ by

$$\begin{aligned} \begin{aligned} \mathfrak {M}_{n_k}(t)&:=\tilde{u}_{n_k}(t)-\tilde{u}_{0n_k}+\displaystyle \int _0^t\mathcal {P}_{n_k}^1N(u_{n_k})ds-\displaystyle \int _0^t\mathcal {P}_{n_k}^1B_0(\tilde{u}_{n_k})ds\\&\quad +\,\displaystyle \int _0^t\mathcal {P}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k}ds-\displaystyle \int _0^t\mathcal {P}_{n_k}^1g_0(s)ds-\displaystyle \int _0^t\mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})ds \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \mathfrak {R}_{n_k}(t):=\tilde{\varphi }_{n_k}(t)-\tilde{\varphi }_{0n_k}-\displaystyle \int _0^t\mathcal {P}_{n_k}^2\Delta \tilde{\mu }_{n_k}(s)ds+\displaystyle \int _0^t\mathcal {P}_{n_k}^2\tilde{u}_{n_k}(s).\nabla \tilde{\varphi }_{n_k}(s)ds=0. \end{aligned}$$

Proposition 4.8

Let $11/5\le p< 12/5$, $\mathfrak {M}(t)$ and $\mathfrak {R}(t)$, $t\in (0,T]$ be two processes define by

$$\begin{aligned} \begin{array}{ll} &{}\mathfrak {M}(t)=\tilde{u}(t)-u_0+\int _0^t(\varvec{N}(s)-B_0(\tilde{u}(s))+\tilde{\varphi }(s)\nabla \tilde{\mu }(s)-g_0(s)-g_1(\tilde{u}(s),\tilde{\varphi }(s)))ds,\\ &{}\mathfrak {R}(t)=\tilde{\varphi }(t)-\varphi _0-\int _0^t\Delta \tilde{\mu }(s)ds+\int _0^t\tilde{u}(s).\nabla \tilde{\varphi }(s)ds. \end{array} \end{aligned}$$

Then, for any $t\in (0,T]$, we have

$$\begin{aligned} \begin{array}{ll} &{}\mathfrak {M}_{n_k}(t)\rightharpoonup \mathfrak {M}(t)\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p})),\\ &{}\mathfrak {R}_{n_k}(t)\rightharpoonup \mathfrak {R}(t)\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U')), \end{array} \end{aligned}$$

as $n_k\rightarrow \infty$.

Proof

Thanks to Remark 4.1, the convergence (4.105)$_1$ and Proposition 4.7, we see that

$$\begin{aligned} \mathfrak {M}_{n_k}(t)\rightharpoonup \mathfrak {M}(t)\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}))\end{aligned}$$

as $n_k\rightarrow \infty$.

Also, thanks to Remark 4.1, (4.98)$_3$ in Lemma 4.4 and (4.112)$_4$ in Proposition 4.7, we see that

$$\begin{aligned} \mathfrak {R}_{n_k}(t)\rightharpoonup \mathfrak {R}(t)\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U'))\end{aligned}$$

as $n_k\rightarrow \infty$. $\square$

Let $\mathcal {N}$ be the set of null sets of $\tilde{\mathcal {F}}$ and for any $t\ge 0$ and $k\in \mathbb {N}$, let

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathcal {F}}^{n_k}_t:=\sigma (\sigma ( (\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s),\tilde{W}_{n_k}(s));s\le t)\cup \mathcal {N}),\\ &{}\tilde{\mathcal {F}}_t:=\sigma (\sigma ( (\tilde{u}(s),\tilde{\varphi }(s),\tilde{W}(s));s\le t)\cup \mathcal {N}), \end{array} \end{aligned}$$

be the completion of the natural filtration generated by $(\tilde{u}_{n_k},\tilde{\varphi }_{n_k},\tilde{W}_{n_k})$ and $(\tilde{u},\tilde{\varphi },\tilde{W})$, respectively.

We infer from Proposition 4.4 that the law of $(u_n,\varphi _n,W)$ are equal to those of $(\tilde{u}_n,\tilde{\varphi }_n,\tilde{W}_n)$ on $\mathfrak {X}=\mathfrak {X}_1\times \mathcal {C}(0,T;K)$, with $\mathfrak {X}_1=L^2(0,T;\mathbb {H})\cap \mathcal {C}(0,T;\mathbb {W}'_s)$. Hence, it is easy to check that $\tilde{W}_n$ is a sequence of $K_1$-valued Wiener process adapted to the filtration $\tilde{\mathbb {F}}^{n_k}:=\{\tilde{\mathcal {F}}_t^{n_k}: t\in [0,T]\}$. Also from Proposition 4.5, we see that $\tilde{W}$ is a $K_1$-valued Wiener process adapted to the filtration $\tilde{\mathbb {F}}:=\{\tilde{\mathcal {F}}_t:t\in [0,T]\}$. The $\mathbb {W}'_s$-valued stochastic processes $(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})$ and $(\tilde{u},\tilde{\varphi })$ are adapted with respect to $\tilde{\mathbb {F}}^{n_k}$ and $\tilde{\mathbb {F}}$ as well. Hence, since their sample paths are continuous in $\mathbb {W}'_s$, we infer that there are also predictable in $\mathbb {W}'_s$.

We now give the following important result.

Proposition 4.9

For each $t\in (0,T]$ we have

$$\begin{aligned} \begin{array}{ll} \mathcal {M}_{n_k}(t):=\int _0^t\mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s))d\tilde{W}_{n_k}\rightarrow \int _0^tg_2(s,\tilde{u}(s),\tilde{\varphi }(s))d\tilde{W} \end{array} \end{aligned}$$

(4.117)

in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;G_{{\text {div}}}))$ and the following identity holds $\tilde{\mathbb {P}}$-a.s

$$\begin{aligned} \begin{array}{ll} \mathfrak {M}(t)=\int _0^tg_2(s,\tilde{u}(s),\tilde{\varphi }(s))d\tilde{W}(s). \end{array} \end{aligned}$$

(4.118)

Proof

The proof of (4.117) and (4.118) is similar to that of Lemma 3.13 and Proposition 3.16 of [45]. $\square$

The process $(\tilde{u},\tilde{\varphi })$ satisfies the following property

Proposition 4.10

For any $q\in [2,\infty )$, we have $(\tilde{u},\tilde{\varphi })\in L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};\mathcal {C}(0,T;\mathbb {H}))$.

Proof

In fact, from (4.118) in Proposition 4.9 and the last convergence in Proposition 4.8, we have for each $t\in (0,T]$

$$\begin{aligned} \begin{array}{ll} &{}\tilde{u}(t)=u_0+\int _0^tG(s)ds+\int _0^tS(s)d\tilde{W}(s),\\ &{}\tilde{\varphi }(t)=\varphi _0+\int _0^t\Delta \tilde{\mu }(s)ds-\int _0^t\tilde{u}(s).\nabla \tilde{\varphi }(s)ds, \end{array} \end{aligned}$$

(4.119)

where

$$\begin{aligned} \begin{array}{ll} G(.):=-\varvec{N}(.)+B_0(\tilde{u}(.))-\tilde{\varphi }(.)\nabla \tilde{\mu }(.)+g_0(.)+g_1(\tilde{u}(.),\tilde{\varphi }(.)),~ S(.):=g_2(.,\tilde{u}(.),\tilde{\varphi }(.)). \end{array} \end{aligned}$$

From the properties of $g_0$ and thanks to (4.105)$_1$ and (4.112)$_{1,2,3}$, we have $G(.)\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}))$. By $\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}|\tilde{u}(s)|^q<C,\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}\Vert \tilde{\varphi }(s)\Vert ^q_{L^{2\kappa +2}}<C$ (see Lemma 4.4) and assumption $(H_2)$, we obtain $S(.)\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L_2(K,G_{{\text {div}}})))$. Thus, we infer from [30, Chapter I, Theorem 3.2] that there exists $\tilde{\Omega }_1\in \tilde{\mathbb {F}}$ such that $\tilde{\mathbb {P}}(\tilde{\Omega }_1)=1$ and for each $\omega \in \tilde{\Omega }_1$ the function $\tilde{u}(\omega ,.)$ takes values in $G_{{\text {div}}}$, and it is continuous in $G_{{\text {div}}}$ with respect to t.

Since $\tilde{\mu }$ also satisfies the estimate (4.97)$_7$ (cf. Lemma 4.4), we can easily show that $\Delta \tilde{\mu }\in L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U'))$ and thanks to (4.112)$_4$ in Proposition 4.7 and (4.119)$_2$, we derive that $\tilde{\varphi }_t\in L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^2(0,T;U'))$. Also since $\tilde{\varphi }$ satisfies the estimate (4.97)$_6$ (cf. Lemma 4.4), we infer from [51, Lemma 1.2, page 261] that for $\tilde{\mathbb {P}}$-almost all $\omega \in \tilde{\Omega }$ the trajectory $\tilde{\varphi }(\omega ,.)$ is equal almost everywhere to a continuous H-valued functions defined in [0, T]. Now, since $\tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}|\tilde{u}(s)|^q<C,~ \tilde{\mathbb {E}}\displaystyle \sup _{s\in [0,T]}\Vert \tilde{\varphi }(s)\Vert ^q_{L^{2\kappa +2}}<C$ we infer that $(\tilde{u},\tilde{\varphi })\in L^q(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};\mathcal {C}(0,T;\mathbb {H}))$. $\square$

To complete the proof of Theorem 3.1, we need to prove some additionally results.

Proposition 4.11

We have the following identity

$$\begin{aligned}\varvec{N}(.)=N(\tilde{u})\ \ \text {in}\ \ L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p})).\end{aligned}$$

Before proving this proposition, we first state and prove the following important result.

Proposition 4.12

The following energy identity holds

$$\begin{aligned} \begin{aligned}&\tilde{\mathbb {E}}\mathcal {E}_{tot}(\tilde{u}(T),\tilde{\varphi }(T))+2\tilde{\mathbb {E}}\int _0^T[\left\langle \varvec{N}(s),\tilde{u}(s)\right\rangle +|\nabla \tilde{\mu }(s)|^2]ds\\&\quad =\mathcal {E}_{tot}(u_0,\varphi _0)+\tilde{\mathbb {E}}\int _0^T\Vert g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2+2[\left\langle g_0(s)+ g_1(\tilde{u},\tilde{\varphi }),\tilde{u}\right\rangle ]ds. \end{aligned} \end{aligned}$$

(4.120)

Proof

Since the processes G(.) and S(.) define in (4.119) belong to $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^{p'}(0,T;V'_{{\text {div}},p}))$ and $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^\infty (0,T;L_2(K,G_{{\text {div}}})))$, respectively, we can apply the Itô formula to the process $|\tilde{u}|^2$ (see, for instance, [42, Theorem 4.2.5]) and derive that

$$\begin{aligned} 2\tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s),\tilde{u}(s)\right\rangle ds&=|\tilde{u}_0|^2-\tilde{\mathbb {E}}|\tilde{u}(T)|^2-2\tilde{\mathbb {E}}\int _0^T\left\langle \tilde{\varphi }(s)\nabla \tilde{\mu }(s),\tilde{u}(s)\right\rangle ds\nonumber \\&\quad +\,\tilde{\mathbb {E}}\int _0^T2[\left\langle g_0(s)+ g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ]+\Vert g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds, \end{aligned}$$

(4.121)

where we have also used (2.9)$_1$.

We note that (4.119)$_2$ can be rewritten in the following form

$$\begin{aligned} \left\langle \partial _t\tilde{\varphi },\psi _j\right\rangle +(\nabla \tilde{\mu },\nabla \psi _j)=(\tilde{u}\tilde{\varphi },\nabla \psi _j). \end{aligned}$$

(4.122)

Now taking $\tilde{\mu }$ as a test function in (4.122) and multiplying the resulting equality by 2, we get

$$\begin{aligned} 2\frac{\textstyle d}{\textstyle dt}\mathcal {E}(\tilde{\varphi }(.))+2|\nabla \tilde{\mu }|^2=2(\tilde{u}\tilde{\varphi },\nabla \tilde{\mu }). \end{aligned}$$

(4.123)

Integrating now (4.123) between 0 and T and adding the resulting equality to (4.121), we obtain (4.120).

$\square$

We now give the proof of Proposition 4.11.

Proof

For the proof, we will use the method of monotonicity (see, for instance, [40, Chapitre 3, Section 3, p. 103]).

From Propositions 4.1–4.2, we have for any $j=1,2,\ldots ,n_k$

$$\begin{aligned} \begin{array}{ll} (\tilde{u}_{n_k}(T),w_j)&{}=(\tilde{u}_{0n_k},w_j)-\int _0^T\left\langle \tilde{\mathcal {P}}_{n_k}^1[ N(\tilde{u}_{n_k})-B_0(\tilde{u}_{n_k})],w_j\right\rangle ds\\ &{}\quad -\,\int _0^T\left\langle \tilde{\mathcal {P}}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k},w_j\right\rangle ds+\int _0^T(\tilde{\mathcal {P}}_{n_k}^1g_0,w_j)ds\\ &{}\quad +\,\int _0^T\left\langle \tilde{\mathcal {P}}_{n_k}^1g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k}),w_j\right\rangle ds +\int _0^T(\tilde{\mathcal {P}}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k}),w_j)d\tilde{W}_{n_k} \end{array} \end{aligned}$$

(4.124)

and

$$\begin{aligned} \begin{array}{ll} (\tilde{\varphi }_{n_k}(T),\psi _j)=(\tilde{\varphi }_{0n_k},\psi _j)+\int _0^T(\mathcal {P}_{n_k}^2\Delta \tilde{\mu }_{n_k},\psi _j)ds+\int _0^T\left\langle \mathcal {P}_{n_k}^2\tilde{u}_{n_k}.\nabla \tilde{\varphi }_{n_k},\psi _j\right\rangle ds \end{array} \end{aligned}$$

(4.125)

where $(w_j,\psi _j)$ are introduce in Sect. 4.1.

Now from (4.124), (4.112)$_{1,2,3}$ (see Proposition 4.7), using (4.117), Remark 4.1 and (4.100), we derive that

$$\begin{aligned} \begin{array}{ll} (\tilde{u}(T),w_j)=(u_0,w_j)+\int _0^T\left\langle G(s),w_j\right\rangle ds+\int _0^T(S(s),w_j)d\tilde{W}(s), \end{array} \end{aligned}$$

where G(.) and S(.) are defined as in the proof of Proposition 4.10. Also from (4.99)$_2$ and (4.124), we infer that

$$\begin{aligned} \begin{array}{ll} (\eta _1,w_j)=(\tilde{u}_0,w_j)+\int _0^T\left\langle G(s),w_j\right\rangle ds+\int _0^T(S(s),w_j)d\tilde{W}(s), \end{array} \end{aligned}$$

for all $j\ge 1$. Hence, from this two previous equalities, we infer that $\eta _1=\tilde{u}(T)$ in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};G_{{\text {div}}})$.

Thanks to (4.125), using Remark 4.1, (4.112)$_4$ in Proposition 4.7 and (4.103), we get

$$\begin{aligned} \begin{array}{ll} (\tilde{\varphi }(T),\psi _j)=(\varphi _0,\psi _j)+\int _0^T(\Delta \tilde{\mu },\psi _j)ds+\int _0^T\left\langle \tilde{u}.\nabla \tilde{\varphi },\psi _j\right\rangle ds. \end{array} \end{aligned}$$

(4.126)

Owing to (4.125) and (4.102), we get

$$\begin{aligned} \begin{array}{ll} (\eta _2,\psi _j)=(\tilde{\varphi }_0,\psi _j)+\int _0^T(\Delta \tilde{\mu },\psi _j)ds+\int _0^T\left\langle \tilde{u}.\nabla \tilde{\varphi },\psi _j\right\rangle ds. \end{array} \end{aligned}$$

Thus, from this two previous equalities, we derive that $\eta _2=\tilde{\varphi }(T)$ in $L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};H)$.

Applying also the Itô formula to the process $|\tilde{u}_{n_k}|^2$, we derive that

$$\begin{aligned} \begin{aligned}&2\tilde{\mathbb {E}}\displaystyle \int _0^T\left\langle N(\tilde{u}_{n_k}),\tilde{u}_{n_k}\right\rangle ds=|\tilde{u}_{0n_k}|^2-\tilde{\mathbb {E}}|\tilde{u}_{n_k}(T)|^2-2\tilde{\mathbb {E}}\displaystyle \int _0^T\left\langle \mathcal {P}_{n_k}^1\tilde{\varphi }_{n_k}\nabla \tilde{\mu }_{n_k},\tilde{u}_{n_k}\right\rangle ds\\&\quad +\,\tilde{\mathbb {E}}\displaystyle \int _0^T2[\left\langle \mathcal {P}_{n_k}^1[g_0(s)+ g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})],\tilde{u}_{n_k}\right\rangle ]+\Vert \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\Vert _{L_2(K,G_{{\text {div}}})}^2ds. \end{aligned} \end{aligned}$$

(4.127)

Making similar reasoning as in the proof of (4.120), we can easily check that the processes $\tilde{u}_{n_k}$ and $\tilde{\varphi }_{n_k}$ satisfy

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathbb {E}}\mathcal {E}_{tot}(\tilde{u}_{n_k}(T),\tilde{\varphi }_{n_k}(T))+2\tilde{\mathbb {E}}\int _0^T[\left\langle N(\tilde{u}_{n_k}(s)),\tilde{u}_{n_k}(s)\right\rangle +|\nabla \tilde{\mu }_{n_k}(s)|^2]ds\\ &{}\quad =\mathcal {E}_{tot}(\tilde{u}_{0n_k},\tilde{\varphi }_{0n_k})+\tilde{\mathbb {E}}\int _0^T\Vert \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s))\Vert _{L_2(K,G_{{\text {div}}})}^2\\ &{}\qquad +2\tilde{\mathbb {E}}\int _0^T[\left\langle \mathcal {P}_{n_k}^1[g_0(s)+ g_1(\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s))],\tilde{u}_{n_k}(s)\right\rangle ]ds. \end{array} \end{aligned}$$

(4.128)

Take now an arbitrary $v\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p}))$ and set

$$\begin{aligned} \begin{aligned} \mathcal {Z}_{n_k}(T)&:=\tilde{\mathbb {E}}\mathcal {E}_{tot}(\tilde{u}_{n_k}(T),\tilde{\varphi }_{n_k}(T))+2\tilde{\mathbb {E}}\int _0^T\left\langle N(\tilde{u}_{n_k})-N(v),\tilde{u}_{n_k}-v\right\rangle ds\\&\quad +\,\tilde{\mathbb {E}}\int _0^T2|\nabla (\tilde{\mu }_{n_k}-\tilde{\mu })|^2+\Vert \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})-g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2ds. \end{aligned} \end{aligned}$$

(4.129)

Using (4.128), we see that $\mathcal {Z}_{n_k}(T)$ can be rewritten in the form

$$\begin{aligned} \mathcal {Z}_{n_k}(T)&=\mathcal {E}_{tot}(\tilde{u}_{0n_k},\tilde{\varphi }_{0n_k})-2\tilde{\mathbb {E}}\int _0^T\left\langle N(\tilde{u}_{n_k}),v\right\rangle ds-2\tilde{\mathbb {E}}\int _0^T\left\langle N(v),\tilde{u}_{n_k}-v\right\rangle ds\nonumber \\&\quad -\,4\tilde{\mathbb {E}}\int _0^T(\nabla \tilde{\mu }_{n_k},\nabla \tilde{\mu })ds+2\tilde{\mathbb {E}}\int _0^T|\nabla \tilde{\mu }|^2ds-\tilde{\mathbb {E}}\int _0^T\Vert g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2ds\nonumber \\&\quad +\,2\tilde{\mathbb {E}}\int _0^T\Vert \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\Vert _{L_2(K,G_{{\text {div}},p})}^2ds+2\tilde{\mathbb {E}}\int _0^T\left\langle \mathcal {P}_{n_k}^1g_0(s),\tilde{u}_{n_k}\right\rangle ds\nonumber \\&\quad +\,2\int _0^T\left( g_2(s,\tilde{u},\tilde{\varphi })-\mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k}),g_2(s,\tilde{u},\tilde{\varphi })\right) _{L_2(K,G_{{\text {div}}})}ds\\&\quad +\,2\tilde{\mathbb {E}}\int _0^T\left\langle \mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k},\tilde{\varphi }_{n_k})-g_1(\tilde{u},\tilde{\varphi }),\tilde{u}_{n_k}\right\rangle ds+2\tilde{\mathbb {E}}\int _0^T\left\langle g_1(\tilde{u},\tilde{\varphi }),\tilde{u}_{n_k}\right\rangle ds.\nonumber \end{aligned}$$

(4.130)

Now, from the fact that $\tilde{u}_{0n_k}\rightarrow u_0$ in $G_{div}$ and $\tilde{\varphi }_{0n_k}\rightarrow \varphi _0$ in $H^2(\mathcal {M})$ and hence also in $L^\infty (\mathcal {M})$, we derive that

$$\begin{aligned} \mathcal {E}_{tot}(\tilde{u}_{0n_k},\tilde{\varphi }_{0n_k})\rightarrow \mathcal {E}_{tot}(u_0,\varphi _0)\ \ \text {as}\ \ n_{k}\rightarrow \infty . \end{aligned}$$

From (4.105)$_1$ and (4.99)$_1$ we obtain

$$\begin{aligned} \begin{array}{ll} &{}\tilde{\mathbb {E}}\int _0^T\left\langle N(\tilde{u}_{n_k}(s)),v\right\rangle ds\rightarrow \tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s),v\right\rangle ds,\\ &{}\tilde{\mathbb {E}}\int _0^T\left\langle N(v),\tilde{u}_{n_k}(s)-v\right\rangle ds\rightarrow \tilde{\mathbb {E}}\int _0^T\left\langle N(v),\tilde{u}(s)-v\right\rangle ds,\\ &{}\tilde{\mathbb {E}}\int _0^T\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}_{n_k}(s)\right\rangle ds\rightarrow \tilde{\mathbb {E}}\int _0^T\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ds \end{array} \end{aligned}$$

as $n_k\rightarrow \infty$.

Thanks to (4.98)$_3$, we have

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\int _0^T(\nabla \tilde{\mu }_{n_k}(s),\nabla \tilde{\mu }(s)ds\rightarrow \tilde{\mathbb {E}}\int _0^T|\nabla \tilde{\mu }(s)|^2ds\ \ \text {as}\ \ n_{k}\rightarrow \infty . \end{array} \end{aligned}$$

It follows from (4.99)$_1$ and (4.112)$_1$ that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\int _0^T\left\langle \mathcal {P}_{n_k}^1g_1(\tilde{u}_{n_k}(s),\tilde{\varphi }_{n_k}(s))-g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}_{n_k}(s)\right\rangle ds\rightarrow 0\ \ \text {as}\ \ n_{k}\rightarrow \infty . \end{array} \end{aligned}$$

Note that $\left\langle \mathcal {P}_{n_k}^1g_0(s),\tilde{u}_{n_k}\right\rangle =\left\langle \mathcal {P}_{n_k}^1g_0(s),\tilde{u}_{n_k}-\tilde{u}\right\rangle +\left\langle \mathcal {P}_{n_k}^1g_0(s),\tilde{u}\right\rangle$. From this observation and making use of the convergence (4.98)$_1$ and $\mathcal {P}_{n_k}^1g_0\rightarrow g_0$ in $L^{p'}(0,T;V'_{{\text {div}},p})$, we infer that

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\int _0^T\left\langle \mathcal {P}_{n_k}^1g_0(s),\tilde{u}_{n_k}(s)\right\rangle ds\rightarrow \tilde{\mathbb {E}}\int _0^T\left\langle g_0(s),\tilde{u}(s)\right\rangle ds\ \ \text {as}\ \ n_{k}\rightarrow \infty . \end{array} \end{aligned}$$

Now, thanks to the continuity of $\mathcal {P}_{n_k}^1g_2$ and (4.96)$_4$, we can arguing as in the proof of (4.100) to derive that

$$\begin{aligned} \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\rightarrow g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\ \ \text {in}\ \ L_2(K,G_{{\text {div}}})\ \ d\tilde{\mathbb {P}}\otimes dt\text {-a.e}.. \end{aligned}$$

(4.131)

From (4.131), (4.97)$_{1,2}$ and $(H_2)$, we can apply the Vitali convergence theorem to derive that

$$\begin{aligned} \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\rightarrow g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\ \ \text {in}\ \ L^4(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^4(0,T;L_2(K,G_{{\text {div}}}))). \end{aligned}$$

(4.132)

It follows from (4.132) that

$$\begin{aligned} \begin{array}{ll} &{} \tilde{\mathbb {E}}\int _0^T[2\Vert \mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k})\Vert _{L_2(K,G_{{\text {div}},p})}^2-\Vert g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2]ds\\ &{}\rightarrow \tilde{\mathbb {E}}\displaystyle \int _0^T\Vert g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2ds,\\ &{}\int _0^T\left( g_2(s,\tilde{u},\tilde{\varphi })-\mathcal {P}_{n_k}^1g_2(s,\tilde{u}_{n_k},\tilde{\varphi }_{n_k}),g_2(s,\tilde{u},\tilde{\varphi })\right) _{L_2(K,G_{{\text {div}}})}ds\rightarrow 0, \end{array} \end{aligned}$$

as $n_k\rightarrow \infty$.

Letting $n_k\rightarrow \infty$ in (4.130) and using all the previous convergences, we see that

$$\begin{aligned} \begin{array}{ll} \mathcal {Z}_{n_k}(T)&{}\rightarrow \mathcal {E}_{tot}(u_0,\varphi _0)-2\tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s),v\right\rangle ds-2\tilde{\mathbb {E}}\int _0^T\left\langle N(v),\tilde{u}(s)-v\right\rangle ds\\ &{} -2\tilde{\mathbb {E}}\int _0^T|\nabla \tilde{\mu }(s)|^2ds+\tilde{\mathbb {E}}\int _0^T\Vert g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds\\ &{}+2\tilde{\mathbb {E}}\int _0^T\left\langle g_0(s),\tilde{u}(s)\right\rangle ds+2\tilde{\mathbb {E}}\int _0^T\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ds. \end{array} \end{aligned}$$

(4.133)

On the other hand, thanks to item (b) in Proposition 2.1, to the lower semicontinuity of the norms, using Fatou’s Lemma we have

$$\begin{aligned} \lim _{n_k\rightarrow \infty }\displaystyle \inf \mathcal {Z}_{n_k}(T)\ge \tilde{\mathbb {E}}\mathcal {E}_{tot}(\tilde{u}(T),\tilde{\varphi }(T)). \end{aligned}$$

(4.134)

Hence, we obtain

$$\begin{aligned} \begin{array}{ll} \tilde{\mathbb {E}}\mathcal {E}_{tot}(\tilde{u}(T),\tilde{\varphi }(T))&{}\le \mathcal {E}_{tot}(u_0,\varphi _0)-2\tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s),v\right\rangle ds-2\tilde{\mathbb {E}}\int _0^T\left\langle N(v),\tilde{u}(s)-v\right\rangle ds\\ &{}\quad -\,2\tilde{\mathbb {E}}\int _0^T|\nabla \tilde{\mu }(s)|^2ds+\tilde{\mathbb {E}}\int _0^T\Vert g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds\\ &{}\quad +2\tilde{\mathbb {E}}\int _0^T\left\langle g_0(s),\tilde{u}(s)\right\rangle ds+2\tilde{\mathbb {E}}\int _0^T\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ds, \end{array} \end{aligned}$$

which, combined with (4.120) in Proposition 4.12, yields the variational inequality

$$\begin{aligned} \begin{array}{ll} 2\tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s)-N(v(s)),v(s)-\tilde{u}(s)\right\rangle ds\le 0 \end{array} \end{aligned}$$

(4.135)

for any $v\in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p}))$. Let $\zeta \in L^2(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}};L^p(0,T;V_{{\text {div}},p}))$ and $\epsilon >0$. By taking $v=\tilde{u}\pm \epsilon \zeta$, we derive from (4.135) that

$$\begin{aligned} \begin{array}{ll} 2\tilde{\mathbb {E}}\int _0^T\left\langle \varvec{N}(s)-N(\tilde{u}(s)\pm \epsilon \zeta (s)),\pm \epsilon \zeta (s)\right\rangle ds\le 0, \end{array} \end{aligned}$$

(4.136)

from which along the hemicontinuity of N we conclude the proof of Proposition 4.11. $\square$

We can now give the proof of Theorem 3.1, which concerns the existence of a weak martingale solution.

Proof

Endowing the complete probability space $(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})$ with the filtration $\tilde{\mathbb {F}}=\{\tilde{\mathcal {F}}_t:t\in [0,T]\}$, where the $\sigma$-algebra $\tilde{\mathcal {F}}_t$ is defined by

$$\begin{aligned}\tilde{\mathcal {F}}_t:=\sigma (\sigma ( (\tilde{u}(s),\tilde{\varphi }(s),\tilde{W}(s));s\le t)\cup \mathcal {N}),\end{aligned}$$

and combining Propositions 4.5, 4.9, 4.10 and 4.11 , we derive that the system $\{(\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {F}},\tilde{\mathbb {P}}),(\tilde{u},\tilde{\varphi },\tilde{W})\}$ is a martingale solution to (2.17) or (2.18) which satisfy all the items of Definition 3.1. This ends the proof of the existence theorem. $\square$

5 Exponential Decay of the Weak Solution

In this section, we will prove that any weak solution $(\tilde{u},\tilde{\varphi })$ to (2.17) or (2.18) converges to zero exponentially in the mean square. So in the rest of this section, we will assume the existence of such solution.

We first note from (2.20) that

$$\begin{aligned} \begin{array}{ll} \left\langle N(\tilde{u}),\tilde{u}\right\rangle _{V_{{\text {div}},p}}&=\int _{\mathcal {M}}\varvec{T}(D\tilde{u}). D\tilde{u}dx\ge c_1\int _\mathcal {M}\left( |D\tilde{u}|^2+|D\tilde{u}|^p\right) dx. \end{array} \end{aligned}$$

Owing to Korn’s inequalities, we infer from this previous inequality that

$$\begin{aligned} \left\langle N(\tilde{u}),\tilde{u}\right\rangle _{V_{{\text {div}},p}}\ge \tilde{c}_1(\Vert \tilde{u}\Vert _{1,2}^2+\Vert \tilde{u}\Vert _{1,p}^p)\ge \tilde{c}_1\tilde{c}_2|\tilde{u}|^2, \end{aligned}$$

where $\tilde{c}_2$ is the constant in Poincaré’s inequality. Setting $\tilde{c}_3=\tilde{c}_1\tilde{c}_2$, then

$$\begin{aligned} \left\langle N(\tilde{u}),\tilde{u}\right\rangle _{V_{{\text {div}},p}}\ge \tilde{c}_3|\tilde{u}|^2. \end{aligned}$$

(5.1)

We also remark that by setting $\left\langle \tilde{\mu }\right\rangle =\frac{1}{|\mathcal {M}|}\int _\mathcal {M}\tilde{\mu }dx$ and since the mean of $\tilde{\varphi }$ is zero (cf. Remark 2.1), we have

$$\begin{aligned} \left( \tilde{\mu },\tilde{\varphi }\right) =\left( \tilde{\mu }-\left\langle \tilde{\mu }\right\rangle ,\tilde{\varphi }\right) \le C_p|\nabla \tilde{\mu }||\tilde{\varphi }|, \end{aligned}$$

(5.2)

where $C_p$ is the Poincaré–Wirtinger constant.

To prove exponential stability, furthermore we assume that the constant $c_5$ is such that

$$\begin{aligned} \frac{c_5}{2}\ge |J|_{L^1(\mathbb {R}^3)}. \end{aligned}$$

(5.3)

Theorem 5.1

We assume that $g_1=0$, $F'(0)=0$ and there exists a constant $\zeta >0$ such that

$$\begin{aligned} \Vert g_2(t,\tilde{u}(t),\tilde{\varphi }(t))\Vert _{L_2(K,G_{{\text {div}}})}^2\le \gamma (t)+\left( \zeta +\delta (t)\right) \left( |\tilde{u}(t)|^2+|\tilde{\varphi }(t)|^2\right) , \end{aligned}$$

(5.4)

where $\gamma (t)$ and $\delta (t)$ are nonnegative integrable functions such that there exist real numbers $\theta >0$, $M_\gamma \ge 1$, $M_\delta \ge 1$ with

$$\begin{aligned} \gamma (t)\le M_\gamma e^{-\theta t},\ \ \delta (t)\le M_\delta e^{-\theta t},\ \ t\ge 0. \end{aligned}$$

(5.5)

We also suppose that there exist positive constants $c_{g_1}, M_\alpha$, $M_\beta$ and two integrable functions $\alpha (.)$ and $\beta (.)$ satisfying

$$\begin{aligned} 0<\alpha (t)\le M_\alpha e^{-\theta t},\ \ 0<\beta (t)\le M_\beta e^{-\theta t}, \end{aligned}$$

(5.6)

and

$$\begin{aligned} \left\langle g_1(\tilde{u}(t),\tilde{\varphi }(t)),\tilde{u}(t)\right\rangle _{V_{{\text {div}},p}}\le \alpha (t)+\left( c_{g_1}+\beta (t)\right) \left( |\tilde{u}(t)|^2+|\tilde{\varphi }(t)|^2\right) , \end{aligned}$$

(5.7)

for any $t\ge 0$ and $(\tilde{u},\tilde{\varphi })\in \mathbb {H}$.

Furthermore, we assume that $F''(\varphi _0)\in L^2(\mathcal {M})$ and we suppose that

$$\begin{aligned} \begin{array}{ll} 2\tilde{c}_3>2c_{g_1}+\zeta \ \ \text {and}\ \ \frac{2(c_5-|J|_{L^1(\mathbb {R}^3)})}{C_p^2}>\frac{2c_{g_1}+\zeta }{(c_5-|J|_{L^1(\mathbb {R}^3)})}. \end{array} \end{aligned}$$

(5.8)

Then any weak solution $(\tilde{u}(t),\tilde{\varphi }(t))$ to (2.17) or (2.18) converges to zero exponentially in the mean square. That is, there exist real numbers $b\in (0,\theta )$, $M_0=M_0(u_0,\varphi _0)>0$ such that

$$\begin{aligned}\tilde{\mathbb {E}}\Vert (\tilde{u}(t),\tilde{\varphi }(t))\Vert _\mathbb {H}^2\le M_0e^{-bt},\ \ t\ge 0.\end{aligned}$$

Proof

We recall that in Theorem 3.1, we have proved that the process $(\tilde{u},\tilde{\varphi },\tilde{W})$ is a weak martingale solution of problem (2.17) in the sense of Definition 3.1. Now from (3.1)$_1$ and Itô’s formula (see, for instance, [41, Theorem I. 3. 3. 2, page 147]) we obtain

$$\begin{aligned} \begin{aligned} |\tilde{u}(t)|^2&=|u_0|^2-2\int _0^t\left\langle N(\tilde{u}(s))-R_1(\tilde{\varphi }(s))-g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ds\\&\quad +\,\int _0^t\Vert g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds+2\int _0^t\left( g_2(s,\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right) d\tilde{W}(s), \end{aligned} \end{aligned}$$

(5.9)

where we have also used the fact that $\left\langle B_0(\tilde{u}(s)),\tilde{u}(s)\right\rangle =-b_0(\tilde{u}(s),\tilde{u}(s),\tilde{u}(s))=0$. Here $\left\langle .,.\right\rangle$ denotes the dual pairing between $V_{{\text {div}},p}$ and $V'_{{\text {div}},p}$ relative to $G_{{\text {div}}}$ .

Thanks to (3.1)$_2$, we have

$$\begin{aligned} (\tilde{\varphi }_t(t),\tilde{\mu }(t))=-|\nabla \tilde{\mu }(t)|^2+\left( B_1(\tilde{u}(t),\tilde{\varphi }(t)),\tilde{\mu }(t)\right) \ \ t\in [0,T]. \end{aligned}$$

Note that since $F'(0)=0$ and $\frac{d}{dt}(\int _\mathcal {M}F(0,x)dx)=0$, we have

$$\begin{aligned} \begin{aligned} (\tilde{\varphi }_t(t),\tilde{\mu }(t))&=(\tilde{\varphi }_t(t),a\tilde{\varphi }(t)-J*\tilde{\varphi }(t)+F'(\tilde{\varphi }(t)))\\&=\frac{d}{dt}\left\{ \frac{1}{2}|\sqrt{a}\tilde{\varphi }(t)|^2-\frac{1}{2}(J*\tilde{\varphi }(t),\tilde{\varphi }(t))+\int _\mathcal {M}[F(\tilde{\varphi }(t,x))-F(0)-F'(0)\tilde{\varphi }(t,x)]dx\right\} . \end{aligned} \end{aligned}$$

(5.10)

Using Taylor’s formula, we have

$$\begin{aligned} \begin{array}{ll} \int _\mathcal {M}[F(\tilde{\varphi }(t,x))-F(0)-F'(0)\tilde{\varphi }(t,x)]dx=\frac{\textstyle 1}{\textstyle 2}\int _\mathcal {M}F''(\xi \tilde{\varphi }(t,x))(\tilde{\varphi }(t,x))^2dx \end{array} \end{aligned}$$

for some $0<\xi <1$. Thus, from (5.10), we get

$$\begin{aligned} \begin{array}{ll} (\tilde{\varphi }_t(t),\tilde{\mu }(t))&{}=\frac{1}{2}\frac{\textstyle d}{\textstyle dt}\left\{ |\sqrt{a}\tilde{\varphi }(t)|^2-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))+\int _\mathcal {M}F''(\xi \tilde{\varphi }(t,x))(\tilde{\varphi }(t,x))^2dx\right\} \\ &{}=\frac{\textstyle 1}{\textstyle 2}\frac{\textstyle d}{\textstyle dt}\left\{ \int _{\mathcal {M}}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} . \end{array} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\left\{ \int _\mathcal {M}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} \\&\quad =-|\nabla \tilde{\mu }(t)|^2+\left( B_1(\tilde{u}(t),\tilde{\varphi }(t)),\tilde{\mu }(t)\right) . \end{aligned} \end{aligned}$$

(5.11)

Now integrating (5.11) between 0 and t, multiplying the resulting equality by 2 and adding it to (5.9), we obtain

$$\begin{aligned} \begin{aligned}&|\tilde{u}(t)|^2+\displaystyle \int _{\mathcal {M}}(a(x)+F''(\xi \tilde{\varphi }(t,x)))(\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))+2\displaystyle \int _0^t[\left\langle N(\tilde{u}),\tilde{u}\right\rangle +|\nabla \tilde{\mu }|^2]ds\\&\quad =|u_0|^2+\displaystyle \int _\mathcal {M}\left( a(x)+F''(\xi \varphi _0(x))(\varphi _0(x))^2dx-(J*\varphi _0,\varphi _0)\right) +2\displaystyle \int _0^t\left\langle g_1(\tilde{u},\tilde{\varphi }),\tilde{u}\right\rangle ds\\&\qquad +\displaystyle \int _0^t\Vert g_2(s,\tilde{u},\tilde{\varphi })\Vert _{L_2(K,G_{{\text {div}}})}^2ds+2\displaystyle \int _0^t\left( g_2(s,\tilde{u},\tilde{\varphi }),\tilde{u}\right) d\tilde{W}(s). \end{aligned} \end{aligned}$$

(5.12)

Since (5.8) is satisfied, we can choose a constant $b\in (0,\theta )$ such that

$$\begin{aligned} 2\tilde{c}_3>2c_{g_1}+\zeta +b\ \ \text {and}\ \ \frac{2(c_5-|J|_{L^1(\mathbb {R}^3)})}{C_p^2}\ge \frac{\zeta +2c_{g_1}}{c_5-|J|_{L^1(\mathbb {R}^3)}}+b. \end{aligned}$$

(5.13)

Hence, applying again the Itô formula to the real process

$e^{bt}\left[ |\tilde{u}(t)|^2+\left\{ \displaystyle \int _{\mathcal {M}}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} \right]$, using (5.12) and since the mathematical expectation of the stochastic integral vanishes, we obtain

$$\begin{aligned} \begin{aligned}&e^{bt}\tilde{\mathbb {E}}\left[ |\tilde{u}(t)|^2+\left\{ \int _{\mathcal {M}}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} \right] \\&\qquad +2\tilde{\mathbb {E}}\int _0^te^{bs}\left\langle N(\tilde{u}(s)),\tilde{u}(s)\right\rangle ds+2\tilde{\mathbb {E}}\int _0^te^{bs}|\nabla \tilde{\mu }(s)|^2ds\\&\quad =|u_0|^2+\left\{ \int _\mathcal {M}\left( a(x)+F''(\xi \varphi _0(x))(\varphi _0(x))^2dx-(J*\varphi _0,\varphi _0)\right) \right\} \\&\qquad +2\tilde{\mathbb {E}}\int _0^te^{bs}\left\langle g_1(\tilde{u}(s),\tilde{\varphi }(s)),\tilde{u}(s)\right\rangle ds+\tilde{\mathbb {E}}\int _0^te^{bs}\Vert g_2(s,\tilde{u}(s),\tilde{\varphi }(s))\Vert _{L_2(K,G_{{\text {div}}})}^2ds\\&\qquad +b\tilde{\mathbb {E}}\int _0^te^{bs}\left[ |\tilde{u}(s)|^2+\left\{ \int _{\mathcal {M}}\left( a+F''(\xi \tilde{\varphi }(s,x))\right) (\tilde{\varphi }(s,x))^2dx-(J*\tilde{\varphi }(s),\tilde{\varphi }(s))\right\} \right] ds. \end{aligned} \end{aligned}$$

(5.14)

Using Assumption $(H_4)$, (5.3) and Young’s inequality for convolutions, we obtain

$$\begin{aligned} \begin{array}{ll} \int _\mathcal {M}\left( a+F''(\xi \tilde{\varphi })\right) \tilde{\varphi }^2dx\ge c_5|\tilde{\varphi }|^2\ge |J|_{L^1(\mathbb {R}^3)}|\tilde{\varphi }|^2\ge (J*\tilde{\varphi },\tilde{\varphi }). \end{array} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{array}{ll} \int _\mathcal {M}\left( a+F''(\xi \tilde{\varphi })\right) \tilde{\varphi }^2dx-(J*\tilde{\varphi },\tilde{\varphi })\ge 0. \end{array} \end{aligned}$$

(5.15)

Using the Taylor series expansion, Assumption $(H_4)$ and the fact that $F'(0)=0$, we see that

$$\begin{aligned} \begin{array}{ll} (\tilde{\mu },\tilde{\varphi })=(a\tilde{\varphi }-J*\tilde{\varphi }+F'(\tilde{\varphi }),\tilde{\varphi })&{}=(a\tilde{\varphi }-J*\tilde{\varphi }+F'(\tilde{\varphi })-F'(0),\tilde{\varphi })\\ &{}=(a\tilde{\varphi }+F''(\xi \tilde{\varphi })\tilde{\varphi },\tilde{\varphi })-(J*\tilde{\varphi },\tilde{\varphi })\\ &{}\ge (c_5-|J|_{L^1(\mathbb {R}^3)})|\tilde{\varphi }|^2, \end{array} \end{aligned}$$

(5.16)

for some $0<\xi <1$ and where we have also used the Young inequality for convolutions. From the above relation in conjunction with (5.2), it follows that

$$\begin{aligned} |\tilde{\varphi }|\le \delta _1|\nabla \tilde{\mu }|, \end{aligned}$$

(5.17)

where $\delta _1=\frac{C_p}{[c_5-|J|_{L^1(\mathbb {R}^3)}]}$. Using Cauchy–Schwarz’s inequality, (5.2) and (5.17), one has

$$\begin{aligned} (a\tilde{\varphi }+F''(\xi \tilde{\varphi })\tilde{\varphi },\tilde{\varphi })-(J*\tilde{\varphi },\tilde{\varphi })=(\tilde{\mu },\tilde{\varphi })\le \delta _1C_p|\nabla \tilde{\mu }|^2. \end{aligned}$$

(5.18)

Thanks to Hölder’s inequality, Young’s inequality for convolutions and (2.22), we obtain

$$\begin{aligned} \begin{array}{ll} &{}\int _\mathcal {M}\left( a(x)+F''(\xi \varphi _0(x))(\varphi _0(x))^2dx-(J*\varphi _0,\varphi _0)\right) \\ &{}\quad \le \left( |a|_{L^\infty (\mathcal {M})}+|J|_{L^1(\mathbb {R}^3)}+\right) |\varphi _0|^2+|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}|\varphi _0|_{L^\infty (\mathcal {M})}|\varphi _0|\\ &{}\quad \le 2|J|_{L^1(\mathbb {R}^3)}|\varphi _0|^2+|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}|\varphi _0|_{L^\infty (\mathcal {M})}|\varphi _0|, \end{array} \end{aligned}$$

(5.19)

where we used the fact that $\varphi _0\in H^2(\mathcal {M})\hookrightarrow L^\infty (\mathcal {M})$ and that $|a|_{L^\infty (\mathcal {M})}\le |J|_{L^1(\mathbb {R}^3)}$.

Inserting now these estimates (5.18)–(5.19) in (5.14), using also (5.17), (5.7), (5.4) and (5.1), we get

$$\begin{aligned} \begin{aligned}&e^{bt}\tilde{\mathbb {E}}\left[ |\tilde{u}(t)|^2+\left\{ \int _{\mathcal {M}}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} \right] \\&\qquad +2\tilde{c}_3\tilde{\mathbb {E}}\int _0^te^{bs}|\tilde{u}(s)|^2ds+2\tilde{\mathbb {E}}\int _0^te^{bs}|\nabla \tilde{\mu }(s)|^2ds\\&\quad \le |u_0|^2+2|J|_{L^1(\mathbb {R}^3)}|\varphi _0|^2+|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}|\varphi _0|_{L^\infty (\mathcal {M})}|\varphi _0|\\&\qquad +\int _0^t\left( 2M_\alpha +M_\gamma \right) e^{(b-\theta )s}ds+(2c_{g_1}+\zeta +b)\tilde{\mathbb {E}}\int _0^te^{bs}|\tilde{u}(s)|^2ds\\&\qquad +[\tilde{c}_4+\tilde{c}_5b]\tilde{\mathbb {E}}\int _0^te^{bs}|\nabla \tilde{\mu }(s)|^2ds+\tilde{\mathbb {E}}\int _0^t\left( 2M_\beta +M_\delta \right) e^{(b-\theta )s}\Vert (\tilde{u}(s),\tilde{\varphi }(s))\Vert _\mathbb {H}^2ds, \end{aligned} \end{aligned}$$

(5.20)

where $\tilde{c}_4=(2c_{g_1}+\zeta )\delta _1^2$, $\tilde{c}_5=\delta _1C_p$ and $\delta _1$ is given by (5.17). Hence, from (5.13) and (5.20) we have

$$\begin{aligned} \begin{aligned}&e^{bt}\tilde{\mathbb {E}}\left[ |\tilde{u}(t)|^2+\left\{ \displaystyle \int _{\mathcal {M}}\left( a(x)+F''(\xi \tilde{\varphi }(t,x))\right) (\tilde{\varphi }(t,x))^2dx-(J*\tilde{\varphi }(t),\tilde{\varphi }(t))\right\} \right] \\&\quad \le |u_0|^2+2|J|_{L^1(\mathbb {R}^3)}|\varphi _0|^2+|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}|\varphi _0|_{L^\infty (\mathcal {M})}|\varphi _0|\\&\qquad +\displaystyle \int _0^t\left( 2M_\alpha +M_\gamma \right) e^{(b-\theta )s}ds+\tilde{\mathbb {E}}\displaystyle \int _0^t\left( 2M_\beta +M_\delta \right) e^{(b-\theta )s}\Vert (\tilde{u}(s),\tilde{\varphi }(s))\Vert _\mathbb {H}^2ds. \end{aligned} \end{aligned}$$

(5.21)

By using (5.16) and (5.21), we can easily see that

$$\begin{aligned} \begin{aligned} e^{bt}\tilde{\mathbb {E}}\Vert (\tilde{u}(t),\tilde{\varphi }(t))\Vert _\mathbb {H}^2&\le \frac{|u_0|^2}{\delta _2}+2|J|_{L^1(\mathbb {R}^3)}\frac{|\varphi _0|^2}{\delta _2}+\frac{|\varphi _0|}{\delta _2}|\varphi _0|_{L^\infty (\mathcal {M})}|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}\\&\quad +\,\frac{1}{\delta _2}\int _0^t\left( 2M_\alpha +M_\gamma \right) e^{(b-\theta )s}ds+\frac{1}{\delta _2}\tilde{\mathbb {E}}\int _0^t\left( 2M_\beta +M_\delta \right) e^{(b-\theta )s}\Vert (\tilde{u},\tilde{\varphi })\Vert _\mathbb {H}^2ds, \end{aligned} \end{aligned}$$

(5.22)

where $\delta _2=\min (1,(c_5-|J|_{L^1(\mathbb {R}^3)}))$. Now, by applying the deterministic Gronwall’s lemma, we can infer the existence of $M_0\equiv M_0(|u_0|^2+2|J|_{L^1(\mathbb {R}^3)}|\varphi _0|^2+|F''(\xi \varphi _0)|_{L^2(\mathcal {M})}|\varphi _0|_{L^\infty (\mathcal {M})}|\varphi _0|)$ such that

$$\begin{aligned} \tilde{\mathbb {E}}\Vert (\tilde{u}(t),\tilde{\varphi }(t))\Vert _\mathbb {H}^2\le M_0e^{-bt}\ \ \text {for all}\ \ t>0. \end{aligned}$$

This completes the proof of Theorem 5.1. $\square$