Proof. First, by applying the Fadeo-Galerkin approximation, we can establish the existence of approximation solution $u_{\hbar,n}^{\nu }$ (for more details, see [Reference Brzeźniak, Liu and Zhu11, Reference Chueshov and Millet21, Reference Duan and Millet28]). In order to obtain the existence of solution to (2.5), we need to take $n$ tend to infinity. And then we should give a priori estimates of $u_{\hbar,n}^{\nu }$ uniformly in $n\geq 1$ and in $\nu \in (0,\nu _0]$ for some $\nu _0>0$ under free boundary conditions as [Reference Chueshov and Millet21]. Next, we will give the energy estimates of $u_{\hbar,n}^{\nu }$. To simplify symbols, we replace the approximation solution $u_{\hbar,n}^{\nu }$ by $u_\hbar ^{\nu }$.

Let $\nu >0$, $\hbar =(f,g)\in {\tilde {\mathcal {U}}^{M}}$. Define $\tau _{N}=\inf \{t\geq 0,|u_\hbar ^{\nu }(t)|_{H}\geq N\}\wedge T$ for every $N>0$. By using Itô's formula to the function $|u_\hbar ^{\nu }(t\wedge \tau _{N})|_H^{2p}$, we obtain from (2.5)

(3.3)\begin{equation} |u_\hbar^{\nu}(t\wedge\tau_{N})|_{H}^{2p}+2p\nu\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\|u_\hbar^{\nu}(s)\|^{2} {\rm d}s\leq|u_\hbar^{\nu}(0)|_{H}^{2p}+\sum_{i=1}^{8}I_i(t), \end{equation}

where

\begin{align*} I_1(t)&=2p\sqrt{\nu}\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}(\Upsilon_\nu(s,u_\hbar^{\nu}(s)),u_\hbar^{\nu}(s))\,{\rm d}W(s),\\ I_2(t)&=\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}} \left(|u_\hbar^{\nu}(s)+\nu G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|u_\hbar^{\nu}(s)|_H^{2p}\right)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}s),\\ I_{3}(t)&=2p\nu \int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\int_{\partial D}k(r)|u_\hbar^{\nu}(r)|_{H}^{2}drds,\\ I_{4}(t)&=2p\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\langle B(u_\hbar^{\nu}(s),u_\hbar^{\nu}(s)), u_\hbar^{\nu}(s)\rangle {\rm d}s,\\ I_{5}(t)&=2p\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\big(\tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))f(s),u_\hbar^{\nu}(s) \big)\,{\rm d}s,\\ I_{6}(t)&={\nu}p(2p-1)\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}|\Upsilon_\nu(s,u_\hbar^{\nu}(s))|^{2}_{L_Q}{\rm d}s, \\ I_{7}(t)&=\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\bigg(|u_\hbar^{\nu}(s)+\nu G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|u_\hbar^{\nu}(s)|_H^{2p}\\ &\quad-\nu p|u_\hbar^{\nu}(s)|^{2p-2}(u_\hbar^{\nu}(s),G(s,u_\hbar^{\nu}(s),y))_H\bigg)\vartheta({\rm d}y)\,{\rm d}s,\\ I_8(t)&=2p\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}(\tilde{G}(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),u_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Next, we estimate the terms $I_1-I_8$ one by one. For the term $I_1$, the Burkholder–Davis–Gundy inequality, condition 2.5, Cauchy–Schwarz's and Young's inequalities imply

\begin{align*} {\mathbb E}\left(\sup_{s\in[0,t\wedge\tau_N]}I_1(s)\right)&\leq C\sqrt{\nu}p{\mathbb E}\left( \int_{0}^{t\wedge\tau_N}\ |u_\hbar^{\nu}(s)|_{H}^{4p-4}(\Upsilon_\nu(s,u_\hbar^{\nu}(s)),u_\hbar^{\nu}(s))^{2}{\rm d}s\right)^{1/2}\\ &\leq C\sqrt{\nu}p{\mathbb E}\left(\int_{0}^{t\wedge\tau_N}|u_\hbar^{\nu}(s)|_{H}^{4p-4}|\Upsilon_\nu(s,u_\hbar^{\nu}(s))|_H^{2}|u_\hbar^{\nu}(s)|_H^{2}{\rm d}s\right)^{1/2}\\ &\leq C\sqrt{\nu}p{\mathbb E}\left(\int_{0}^{t\wedge\tau_N}|u_\hbar^{\nu}(s)|_{H}^{4p-2}K(s)(1+|u_\hbar^{\nu}(s)|_H^{2})\,{\rm d}s\right)^{1/2}\\ &\leq C\sqrt{\nu}p{\mathbb E}\left(\int_{0}^{t\wedge\tau_N}K(s)(1+|u_\hbar^{\nu}(s)|_H^{4p})\,{\rm d}s\right)^{1/2}\\ &\leq \frac{1}{4}{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_{N}]}| u_\hbar^{\nu}(s)|_H^{2p}\right)\\ &\quad +C\nu p^{2}{\mathbb E}\left(\int_{0}^{t\wedge\tau_N}\ K(s)(1+|u_\hbar^{\nu}(s)|_H^{2p})\,{\rm d}s\right)+C. \end{align*}

From the Taylor formula, it follows that for every $p\ge 2$ there exists a positive constant $C_p>0$ such that for all $x, h\in H$

(3.4)\begin{equation} \left||x+h|_H^{p}-|x|_H^{p}-p|x|_H^{(p-2)}(x,h)_H\right|\le C_p\left(|x|_H^{(p-2)}+|h|_H^{(p-2)}\right)|h|_H^{2}. \end{equation}

Moreover, we arrive at

\[ \left||x+h|_H^{p}-|x|_H^{p}\right|\le\frac{p}{2}|x|_H^{p}+\left(C_p+\frac{p}{2}\right)|x|_H^{(p-2)}|h|^{2}_H+C_p|h|^{p}_H. \]

By using the Schwarz inequality, we obtain for all $x, h\in H$

(3.5)\begin{equation} \begin{aligned} \left(|x+h|_H^{p}-|x|_H^{p}\right)^{2} & \le 2\left[p^{2}|x|_H^{2p-2}|h|_H^{2}+c_p^{2}\left(|x|_H^{p-2}+|h|_H^{p-2}\right)^{2}|h|_H^{4}\right]\\ & \le 2p^{2}|x|^{2p-2}_H|h|_H^{2}+4c_p^{2}|x|_H^{2p-4}|h|_H^{4}+4C_p^{2}|h|_H^{2p}. \end{aligned} \end{equation}

For the term $I_2(t)$, by using the Burkholder–Davis–Gundy inequality, (3.5), condition 2.5, Hölder's and Young's inequalities, we deduce that

\begin{align*} &{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_{N}]}I_2(s)\right)\\ &\quad \le {\mathbb{E}}\left[\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}\left(|u_\hbar^{\nu}(s)+\nu G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|u_\hbar^{\nu}(s)|_H^{2p}\right)^{2}\vartheta({\rm d}y)\,{\rm d}s\right]^{1\over 2}\\ &\quad \le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}|u_\hbar^{\nu}(s)|_H^{4p-2}|G(s,u_\hbar^{\nu}(s),y)|_H^{2}\right.\\ &\quad +\left. |u_\hbar^{\nu}(s)|_H^{4p-4}|G(s,u_\hbar^{\nu}(s),y)|_H^{4}+|G(s,u_\hbar^{\nu}(s),y)|_H^{4p}\vphantom{\int_{0}^{t\wedge\tau_{N}}}\right)^{1\over 2}\\ &\quad\le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}K(s)|u_\hbar^{\nu}(s)|_H^{4p}{\rm d}s\right)^{1\over 2}+C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}K(s)\,{\rm d}s\right)^{1\over 2}\\ &\quad\le \frac{1}{4}{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_{N}]}| u_\hbar^{\nu}(s)|_H^{2p}\right)+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}K(s)|u_\hbar^{\nu}(s)|_H^{2p}{\rm d}s+C(\nu). \end{align*}

Using (2.1) and Young's inequality, for any $\varepsilon >0$, we bound the term $I_3(t)$ by

\begin{align*} {\mathbb{E}}I_{3}(t)&\leq 2\nu p\varepsilon{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}|{u_\hbar^{\nu}}(s)|_{H}^{2p-2}\|{u_\hbar^{\nu}}(s)\|^{2} {\rm d}s+2\nu pC(\varepsilon){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}|{u_\hbar^{\nu}}(s)|_{H}^{2p}{\rm d}s. \end{align*}

For the term $I_4(t)$, using (1.2) implies that ${\mathbb {E}}I_4(t)=0$ for any $t\in [0,T]$. For the term $I_5(t)$, since $(f,g)\in {\tilde {\mathcal {U}}^{M}}$, the growth condition 2.6, Cauchy–Schwarz's and Hölder's inequalities yield that

\begin{align*} {\mathbb{E}}I_{5}(t)&\leq 2p{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}| u_\hbar^{\nu}(s)|_{H}^{2p-2} |\tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))|_{L(H_0,H)}|u_\hbar^{\nu}(s)|_H|f(s)|_0{\rm d}s\\ &\leq 2p{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_{H}^{2p-2} \sqrt{K(t)}(1+|u_\hbar^{\nu}(s)|_H^{2})|f(s)|_0{\rm d}s\\ &\leq 2p{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} (1+|u_\hbar^{\nu}(s)|_H^{2p})(K(t)+|f(s)|_0^{2})\,{\rm d}s\\ &\leq 2p{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} (K(s)+|f(s)|_0^{2})| u_\hbar^{\nu}(s)|_H^{2p}{\rm d}s+2p{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} (K(s)+|f(s)|_0^{2})\,{\rm d}s. \end{align*}

For the term $I_6$, using the growth condition 2.5, we get

\begin{align*} {\mathbb{E}}I_6(t)&\leq \nu p(2p-1){\mathbb{E}}\int_0^{t\wedge\tau_N}K(s)\,{\rm d}s\\ &\quad +\nu p(2p-1){\mathbb{E}} \int_0^{t\wedge\tau_N}K(s)|u_\hbar^{\nu}(s)|_{H}^{2p}{\rm d}s, \text{for } \nu\in(0,\nu_0]. \end{align*}

For the term $I_7(t)$, by using (3.4) and condition 2.5, we have

\begin{align*} {\mathbb{E}}I_7(t)&\le C{\mathbb{E}}\left[\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\left(|u_\hbar^{\nu}(s)|_H^{2p-2}+|\nu G(s,u_\hbar^{\nu}(s),y)|_H^{2p-2}\right)\right.\\ &\quad \left. \,\times |\nu G(s,u_\hbar^{\nu}(s),y)|_H^{2}\vartheta({\rm d}y)\,{\rm d}s\right]\\ &\le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}\ |u_\hbar^{\nu}(s)|_H^{2p-2}\int_{\mathbb{Y}}|G(s,u_\hbar^{\nu}(s),y)|_H^{2}\vartheta({\rm d}y)\,{\rm d}s\right.\\ &\quad \left. +\,\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}} |G(s,u_\hbar^{\nu}(s),y)|_H^{2p}\vartheta({\rm d}y)\,{\rm d}s\right)\\ &\le C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}K(s)|u_\hbar^{\nu}(s)|_H^{2p}{\rm d}s+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}K(s)\,{\rm d}s. \end{align*}

For the term $I_8(t)$, by using Hölder's and Young's inequalities, one has

\begin{align*} &{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}(\tilde{G}(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),u_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le {\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}|\tilde{G}(s,u_\hbar^{\nu}(s),y)|_H|g(s,y)-1||u_\hbar^{\nu}(s)|_H\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\frac{|\tilde{G}(s,u_\hbar^{\nu}(s),y)|_H}{1+|u_\hbar^{\nu}(s)|_H}|g(s,y)-1|(1+|u_\hbar^{\nu}(s)|_H^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}\frac{|\tilde G(s,u_\hbar^{\nu}(s),y)|_H}{1+|u_\hbar^{\nu}(s)|_H}|g(s,y)-1|(|u_\hbar^{\nu}(s)|_{H}^{2p-2}+|u_\hbar^{\nu}(s)|_H^{2p})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ |u_\hbar^{\nu}(s)|_H^{2p}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Thus, for any $t\in [0,T]$, $\varepsilon \in (0,1)$, (3.3) and the estimates of $I_1-I_8$ imply that

\begin{align*} &{\mathbb{E}}|u_\hbar^{\nu}(t\wedge\tau_N)|_{H}^{2p}+2\nu p\big(1-\varepsilon\big){\mathbb{E}}\int_0^{t\wedge\tau_N}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\|u_\hbar^{\nu}(s)\|^{2}{\rm d}s\\ &\quad \leq C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} (K(s)+|f(s)|_0^{2})|u_\hbar^{\nu}(s)|_H^{2p}{\rm d}s+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} (K(s)+|f(s)|_0^{2})\,{\rm d}s\\ &\qquad+ C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s+C+{\mathbb E}|\varsigma|_{H}^{2p}\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}|u_\hbar^{\nu}(s)|_H^{2p}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s.\end{align*}

Applying Gronwall's inequality, condition 2.7 and lemma 2.9, we obtain

\[ {\mathbb{E}}|u_\hbar^{\nu}(t\wedge\tau_N)|_{H}^{2p}+2\nu p\big(1-\varepsilon\big) {\mathbb{E}}\int_0^{t\wedge\tau_N}|u_\hbar^{\nu}(s)|_{H}^{2p-2}\|u_\hbar^{\nu}(s)\|^{2}{\rm d}s\le C\left(1+{\mathbb E}|\varsigma|_{H}^{2p}\right). \]

Due to the estimates of the right-hand side of (3.3) does not depend on $N$, taking $N\to \infty$, we can infer that $\tau _N\to T$ $\mathbb {P}$-a.s. Hence, there exists a constant $C:=C({\mathbb E} |\varsigma |_H^{2p},\nu _0,M,T,p)$ such that

\[ \sup_{n} {\mathbb E} \left(\sup_{0\leq t\leq T}|u_{\hbar,n}^{\nu}(t)|_{H}^{2p}+\nu \int_0^{T}\big[\|u_{\hbar,n}^{\nu}(t)\|_{\mathcal{H}}^{4}+\|u_{\hbar,n}^{\nu}(s)\|^{2} \big]{\rm d}s \right) \leq C, \]

for any $n$, $\nu \in (0,\nu _0]$ and $(f,g)\in {\tilde {\mathcal {U}}^{M}}$. Letting $n\to \infty$ and using the classical argument as [Reference Brzeźniak, Liu and Zhu11], we obtain the existence of solution to (2.5). Taking $n\to \infty$ and combining the above estimates and the weakly lower semi-continuity of the norm, we can get (3.1) and (3.2).

Next, we begin to prove the uniqueness. When condition 2.5 with $K(\cdot )=C$ holds, we know that equations (2.4) has a unique strong solution $u^{\nu }\in {\mathcal {D}}([0,T];H)\cap L^{2}(0,T;V)$ by [Reference Brzeźniak, Liu and Zhu11] for any $T>0$. Set $\phi _\nu =(\psi _\nu, \varphi _\nu )\in \tilde {\mathcal {U}}^{M}$ and $\vartheta _\nu ={1\over \varphi _\nu }$. We conclude from lemma 2.3 in [Reference Budhiraja, Dupuis and Maroulas17] that $\mathcal {E}_t^{\nu }(\vartheta _\nu ):=\exp$ $\{\int _{[0,t]\times \mathbb {Y}\times [0,\nu ^{-1}]}\log (\vartheta _\nu (s,x))\bar {N}(dsdxdr)+\int _{[0,t]\times \mathbb {Y}\times [0,\nu ^{-1}]}(-\vartheta _\nu (s,x)+1)\bar {\vartheta }_T (dsdxdr)\}$, $\tilde {\mathcal {E}}_t^{\nu }(\psi _\nu ):=\exp \{{1\over \nu }\int _0^{t}\psi _\nu (s)d\beta (s)-{1\over 2\nu }\int _0^{t}\|\psi _\nu (s)\|^{2}{\rm d}s\}$ are $\{\bar {\mathcal {F}}_t^{\bar {\mathbb {V}}}\}$-marting ales and $\mathbb {Q}_t^{\nu }(G)=\int _G \bar {\mathcal {E}}_t^{\nu }(\psi _\nu,\vartheta _\nu )d\bar {{\mathbb {P}}}^{\bar {\mathbb {V}}}$ defines a probability measure on $\bar {\mathbb {V}}$, where $G\in \mathcal {B}(\bar {\mathbb {V}})$ and $\mathbb {Q}_t^{\nu }(G)=\tilde {\mathcal {E}}_t^{\nu }(\psi _\nu )\mathcal {E}_t^{\nu }(\vartheta _\nu )$. Then, we get the uniqueness from the fact that (2.4) has a unique stochastic analytically strong solution. Therefore, we complete the proof of proposition 3.1.

Proof. Denote $\xi _\hbar ^{\nu }=\text { curl }u_\hbar ^{\nu }$. We know that $u_\hbar ^{\nu }$ is a solution of the following elliptic problem (for details, see [Reference Bessaih6] and the references therein):

(3.7)\begin{equation} \begin{cases} -\Delta u_\hbar^{\nu}=\nabla^{\bot}\xi_\hbar^{\nu} & \text{ in } D,\\ u_\hbar^{\nu}\cdot\textbf{n}=\xi_\hbar^{\nu}=0 & \text{ on } \partial D, \end{cases} \end{equation}

where $\nabla ^{\bot }=(\partial _{2},-\partial _{1})$. By (3.7), we have

\[ -(\Delta u_\hbar^{\nu},\Delta u_\hbar^{\nu})=(\nabla^{\bot}\xi_\hbar^{\nu},\quad \Delta u_\hbar^{\nu})={-}(\nabla^{\bot}\xi_\hbar^{\nu},\ \nabla^{\bot}\xi_\hbar^{\nu}). \]

Then, by direct calculations, we can get

\[ |\Delta u_\hbar^{\nu}|_H^{2}=|\nabla^{\bot}\xi_\hbar^{\nu}|^{2}_{H} =\|\partial_{2}\xi_\hbar^{\nu}\|^{2}_{L^{2}(D)}+\|\partial_{1}\xi_\hbar^{\nu}\|^{2}_{L^{2}(D)} =|\nabla\xi_\hbar^{\nu}|_H^{2}. \]

In order to prove (3.6), by using (A.3), we only need to prove that there exists a constant $C(M,T):=C$ such that for any $\nu \in (0,\nu _0]$ and $\hbar =(f,g)\in {\tilde {\mathcal {U}}^{M}}$,

(3.8)\begin{equation} {\mathbb E}\left(\sup_{0\leq t\leq T}|\xi_\hbar^{\nu}(t)|_{H}^{2p}+\nu\int_0^{T} |\nabla \xi_\hbar^{\nu}(s)|_{H}^{2}{\rm d}s\right)\leq C(1+{\mathbb E}|\text{ curl }\varsigma|_{H}^{2p}). \end{equation}

Let $\bar {\tau }_N=\inf \{t\geq 0: |\xi _\hbar ^{\nu }(t)|_{H}\geq N\}\wedge T$ for fixed $N>0$. Applying the curl operator to (2.5), we obtain $\xi _\hbar ^{\nu }(0)=\text { curl }\varsigma$ and

(3.9)\begin{align} d\xi_\hbar^{\nu}(t)&=\nu A\xi_\hbar^{\nu}(t)\,{\rm d}t+\text{ curl} B(u_\hbar^{\nu}(t),u_\hbar^{\nu}(t))\,{\rm d}t\nonumber\\ &\quad+\sqrt{\nu}\text{ curl }\Upsilon_\nu (s,u_\hbar^{\nu}(t)) {\rm d}W(t)+\text{ curl }\tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(t))f(t)\,{\rm d}t\nonumber\\ &\quad+\nu\int_{\mathbb{ Y}}\text{curl}G(t,u_\hbar^{\nu}(t-),y)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}t)\notag\\ &\quad +\int_{\mathbb{Y}}\text{curl }\tilde G(t,u_\hbar^{\nu}(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t. \end{align}

We note that for $u\in D(A)$, $(\text { curl }B(u_\hbar ^{\nu },u_\hbar ^{\nu }),\xi _\hbar ^{\nu })=0$ by using (A.7) when $q=2$. Employing Itô's formula to the function $|\xi _\hbar ^{\nu }(s\wedge \bar {\tau }_N)|_{H}^{2p},p\in [2,\infty )$, for $t\in [0,T]$, we obtain from (3.9)

\[ |\xi_\hbar^{\nu}(s\wedge \bar{\tau}_N)|_{H}^{2p}+2p\nu\int_0^{t\wedge \bar{\tau}_N}|\nabla \xi_\hbar^{\nu}(s)|_{H}^{2} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2} {\rm d}s=|\text{ curl}\varsigma|_{H}^{2p}+\sum_{i=1}^{7}J_i(t), \]

where

\begin{align*} J_1(t)&=2p\sqrt{\nu}\int_0^{t\wedge\bar{\tau}_N} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}\big( \text{ curl } \Upsilon_\nu(s,u_\hbar^{\nu}(s)), \xi_\hbar^{\nu}(s)\big)\,{\rm d}W(s),\\ J_2(t)&=\int_{0}^{t\wedge\bar{\tau}_{N}}\ \int_{\mathbb{Y}} \left(|\xi_\hbar^{\nu}(s)+\nu\text{curl}G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|\xi_\hbar^{\nu}(s)|_H^{2p}\right)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}s),\\ J_3(t)&=2p \int_0^{t\wedge\bar{\tau}_N} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}\big(\text{ curl } \tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))f(s), \xi_\hbar^{\nu}(s)\big)\,{\rm d}s,\\ J_4(t)&=\nu p(2p-1)\int_0^{t\wedge\bar{\tau}_N} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}| \text{ curl }\Upsilon_\nu(s,u_\hbar^{\nu}(s))|^{2}_{L_Q}{\rm d}s,\\ J_5(t)&=\int_{0}^{t\wedge\bar{\tau}_{N}}\ \int_{\mathbb{Y}}\bigg(|\xi_\hbar^{\nu}(s)+\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|\xi_\hbar^{\nu}(s)|_H^{2p}\\ &\quad-\nu p|\xi_\hbar^{\nu}(s)|_H^{2p-2}(\xi_\hbar^{\nu}(s),\text{curl }G(s,u_\hbar^{\nu}(s),y))_H\bigg)\vartheta({\rm d}y)\,{\rm d}s,\\ J_6(t)&=2p\int_{0}^{t\wedge\bar{\tau}_{N}} \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}(\text{curl }\tilde G(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),\xi_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

For the term $J_1(t)$, it follows from the Burkholder–Davis–Gundy inequality, condition 2.11, (A.3) ($q=2$), Cauchy–Schwarz's, Hölder's and Young's inequalities that

\begin{align*} &{\mathbb E}\left(\sup_{s\in[0,t\wedge{\bar\tau}_{N}]}J_1(s)\right)\\ &\quad \leq 2p\sqrt{\nu}{\mathbb E}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}|\xi_\hbar^{\nu}(s)|_{H}^{4p-2}|\text{ curl }\Upsilon_\nu(u_\hbar^{\nu}(s)|^{2}_{L_Q}{\rm d}s\right)^{\frac{1}{2}}\\ &\quad \leq 2p\sqrt{\nu}{\mathbb E}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}|\xi_\hbar^{\nu}(s)|_{H}^{4p-2}K(s)(1+\|u_\hbar^{\nu}(s)\|^{2}_V)\,{\rm d}s\right)^{\frac{1}{2}}\\ &\quad \leq 2p\sqrt{\nu}{\mathbb E}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}|\xi_\hbar^{\nu}(s)|_{H}^{4p-2}K(s)(1+|u_\hbar^{\nu}(s)|^{2}_H+C|\xi_\hbar^{\nu}(s)|^{2}_Hds\right)^{\frac{1}{2}}\\ &\quad \leq \frac{1}{4}{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_{N}]}|\xi_\hbar^{\nu}(s)|_H^{2p}\right)+2p\sqrt{\nu}C{\mathbb E}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}|\xi_\hbar^{\nu}(s)|_{H}^{2p}K(s)\,{\rm d}s\right)\\ &\qquad+2p\sqrt{\nu}C{\mathbb{E}}\left(1+\sup_{s\in[0,T]}|u_\hbar^{\nu}(s)|^{2p}_H\right)\left(\int_{0}^{t\wedge{\bar \tau}_{N}}K(s)\,{\rm d}s\right)^{1\over 2}. \end{align*}

For the term $J_2(t)$, by applying the Burkholder–Davis–Gundy inequality, condition 2.11, Hölder's and Young's inequalities, we obtain

\begin{align*} &{\mathbb{E}}\left(\sup_{s\in[0,t\wedge{\bar\tau}_{N}]}J_2(s)\right)\\ &\quad\le{\mathbb{E}}\left[\int_{0}^{t\wedge{\bar \tau}_{N}}\ \int_{\mathbb{Y}}\left(|\xi_\hbar^{\nu}(s)+\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)|_{H}^{2p}-|\xi_\hbar^{\nu}(s)|_H^{2p}\right)^{2}\vartheta({\rm d}y)\,{\rm d}s\right]^{1\over 2}\\ &\quad\le C(\nu){\mathbb{E}}\Bigg(\int_{0}^{t\wedge{\bar \tau}_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_H^{4p-2}|\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{2}\\ &\qquad +|\xi_\hbar^{\nu}(s)|_H^{4p-4}|\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{4} +|\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{4p}\vartheta({\rm d}y)\,{\rm d}s\vphantom{\int_{0}^{t\wedge{\bar \tau}_{N}}}\Bigg)^{1\over 2}\\ &\quad\le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}\ K(s)|\xi_\hbar^{\nu}(s)|_H^{4p}{\rm d}s\right)^{1\over 2}+C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}K(s)\,{\rm d}s\right)^{1\over 2}\\ &\quad\le \frac{1}{4}{\mathbb{E}}\Big(\sup_{s\in[0,t\wedge{\bar \tau}_{N}]}|\xi_\hbar^{\nu}(s)|_H^{2p}\Big)+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ K(s)|\xi_\hbar^{\nu}(s)|_H^{2p}{\rm d}s\\ &\qquad +C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}\ K(s)\,{\rm d}s\right)^{1\over 2}. \end{align*}

For the term $J_3(t)$, we use Cauchy-Schwarz's inequality, condition 2.12, (A.3) with $q=2$, Hölder's and Young's inequalities to get

\begin{align*} {\mathbb{E}}J_3(t)&\leq 2p{\mathbb{E}}\int_{0}^{t\wedge{\bar\tau}_{N}}\ |\xi_\hbar^{\nu}(s)|_{H}^{2p-1}|\text{ curl } \tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))|_{L(H_{0},H)} |f(s)|_{0}{\rm d}s\\ &\leq 2p{\mathbb{E}}\int_{0}^{t\wedge{\bar\tau}_{N}}\ \Big(\sqrt{K(s)}+C\sqrt{K(s)}|\xi^{\nu}(s)|_{H}^{2p}+\sqrt{K(s)} |u_\hbar^{\nu}(s)|_{H}|\xi_\hbar^{\nu}(s)|_{H}^{2p-1}\Big)\\ &\quad \times|f(s)|_{0}{\rm d}s\\ &\leq 2p{\mathbb{E}}\int_{0}^{t\wedge{\bar\tau}_{N}}\ \Big(1+|\xi_\hbar^{\nu}(s)|_{H}^{2p}+|u_\hbar^{\nu}(s)|_{H}|\xi_\hbar^{\nu}(s)|_{H}^{2p-1}\Big) \Big(K(s)+|f(s)|^{2}_{0}\Big)\,{\rm d}s\\ &\leq 2pC{\mathbb{E}}\int_{0}^{t\wedge{\bar\tau}_{N}}\ \Big(K(s)+|f(s)|^{2}_{0}\Big)\Bigg(1+\sup_{s\in[0,T]}|u_\hbar^{\nu}(s)|_{H}^{2p}+|\xi_\hbar^{\nu}(s)|_{H}^{2p}\Bigg)\,{\rm d}s. \end{align*}

For the term $J_4(t)$, condition 2.11, (A.3) with $q=2$, Hölder's and Young's inequalities yield that for $\nu \in (0,\nu _0]$

\begin{align*} {\mathbb{E}}J_4(t)&\leq \nu p(2p-1){\mathbb{E}}\int_{0}^{t\wedge{\bar \tau}_{N}}\ |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}K(s)\Big(1+|u_\hbar^{\nu}(s)|_{H}^{2}+C |\xi_\hbar^{\nu}(s)|_{H}^{2}\big)\,{\rm d}s\\ &\leq\nu(2p-1){\mathbb{E}}\ \int_{0}^{t\wedge{\bar\tau}_{N}}\ K(s)\Big(1+\sup_{s\in[0,T]}|u_\hbar^{\nu}(s)|_{H}^{2p}+|\xi_\hbar^{\nu}(s)|_{H}^{2p}\Big)\,{\rm d}s. \end{align*}

For the term $J_5(t)$, it follows from condition 2.11 and Hölder's inequality that

\begin{align*} {\mathbb{E}}J_5(t)& \le C{\mathbb{E}}\left[\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}\left(|\xi_\hbar^{\nu}(s)|_H^{2p-2}+|\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{2p-2}\right)\right.\\ &\quad \times\left. |\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{2}\vartheta({\rm d}y)\,{\rm d}s\right]\\ &\le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_H^{2p-2}|\text{curl}G(s,u_\hbar^{\nu}(s),y)|_H^{2}\right.\\ &\quad +\left. |\text{curl }G(s,u_\hbar^{\nu}(s),y)|_H^{2p}\vartheta({\rm d}y)\,{\rm d}s\vphantom{\int_{0}^{t\wedge\bar\tau_{N}}}\right)\\ &\le C(\nu){\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}K(s)|\xi_\hbar^{\nu}(s)|_H^{2p}{\rm d}s+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ K(s)\,{\rm d}s. \end{align*}

For the term $J_6(t)$, condition 2.12, Hölder's and Young's inequalities imply that

\begin{align*} &{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}(\text{curl}\tilde G(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),\xi_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}|\text{curl}\tilde G(s,u_\hbar^{\nu}(s),y)|_H|g(s,y)-1||\xi_\hbar^{\nu}(s)|_H\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}\frac{|\text{curl}\tilde G(s,u_\hbar^{\nu}(s),y)|_H}{1+|\xi_\hbar^{\nu}(s)|_{H}}\\ &\qquad \times |g(s,y) -1|(1+|\xi_\hbar^{\nu}(s)|_H^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}\frac{| \tilde G(s,\xi_\hbar^{\nu}(s),y)|_H}{1+|\xi_\hbar^{\nu}(s)|_{H}}|g(s,y)-1|(1+|\xi_\hbar^{\nu}(s)|_H^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|_{H}^{2p-2}|\tilde G(s,y)|_{0,H}|g(s,y)-1|(1+|\xi_\hbar^{\nu}(s)|_H^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}|\xi_\hbar^{\nu}(s)|_H^{2p}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Then, combining the estimates of $J_1-J_6$, we obtain

\begin{align*} &{\mathbb E}\Big(\sup_{s\in[0,t]}|\xi_\hbar^{\nu}(s\wedge \bar{\tau}_N)|_{H}^{2p}\Big)+2p\nu{\mathbb E}\int_0^{t\wedge \bar{\tau}_N}\ |\nabla \xi_\hbar^{\nu}(s)|_{H}^{2} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}{\rm d}s\\ &\quad\le {\mathbb E}\Big(|\text{ curl} \varsigma|_{H}^{2p}\Big)+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\bar\tau_{N}}\ \Big(K(s)+|f(s)|^{2}_{0}\Big)|\xi_\hbar^{\nu}(s)|^{2p}\\ &\qquad +C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge{\bar \tau}_{N}}\ K(s)\,{\rm d}s\right)^{1\over 2}\\ &\qquad +C(\nu){\mathbb E}\int_{0}^{t\wedge{\bar\tau}_{N}}\ \Big(K(s)+|f(s)|^{2}_{0}\Big)\Bigg(1+\sup_{s\in[0,T]}|u_\hbar^{\nu}(s)|_{H}^{2p}\Bigg)\,{\rm d}s\\ &\qquad+C{\mathbb E}\int_{0}^{t\wedge\bar\tau_{N}}|\xi_\hbar^{\nu}(s)|_H^{2p}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C{\mathbb E}\int_{0}^{t\wedge\bar\tau_{N}}\ \int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C(\nu){\mathbb E}\Big(1+\sup_{s\in[0,T]}|u_\hbar^{\nu}(s)|^{2p}_H\Big)\Bigg(\int_{0}^{t\wedge{\bar \tau}_{N}}\ K(s)\,{\rm d}s\Bigg)^{1\over 2}. \end{align*}

By using Gronwall's inequality, lemma 2.9 and proposition 3.1, we can infer that

(3.10)\begin{align} &{\mathbb E}\Bigg(\sup_{s\in[0,t\wedge \bar{\tau}_N)]}|\xi_\hbar^{\nu}(s|_{H}^{2p}\Bigg)+2p\nu{\mathbb E}\int_0^{t\wedge \bar{\tau}_N}|\nabla \xi_\hbar^{\nu}(s)|_{H}^{2} |\xi_\hbar^{\nu}(s)|_{H}^{2p-2}{\rm d}s\\ &\quad \le C(1+{\mathbb E}|\text{ curl }\varsigma|_{H}^{2p}). \end{align}

From the fact that estimate (3.10) is uniform in $N$, we can deduce that $\bar {\tau }_N\to T$ as $N\to \infty$. This together with the monotone convergence theorem yields that (3.8). Then, the proof of proposition 3.2 is completed.

Proof. We consider the following Navier–Stokes equations:

(4.3)\begin{equation} \begin{aligned} {\rm d}u_\hbar^{0\nu}(t) & ={-}\big[\nu Au_\hbar^{0\nu}(t)+ B(u_\hbar^{0\nu}(t),u_\hbar^{0\nu}(t))\big] {\rm d}t+\tilde{\Upsilon}_0(t, u_\hbar^{0\nu}(t))f(t)\,{\rm d}t\\ & \quad+\int_{\mathbb{Y}}\tilde G(t,u_\hbar^{0\nu}(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t \end{aligned} \end{equation}

with $\nu >0$, the initial data $u_\hbar ^{0\nu }(0)=\varsigma$, $\nabla \cdot u_\hbar ^{0\nu }=0$ and (1.3). When $\varsigma \in H$, $(\tilde {\Upsilon }_0,\tilde {G})$ satisfies conditions 2.6, 2.7 and $\hbar =(f,g)\in {{\bar S}^{M}}$ with $M>0$, similar to the proof of proposition 3.1, we can conclude that (4.3) has a unique weak solution $u_\hbar ^{0\nu }\in {\mathcal {C}}([0,T]; H)\cap L^{2}(0,T;V)$. Furthermore, when $\varsigma \in V$ and $(\tilde {\Upsilon }_0,\tilde G)$ satisfies conditions 2.7 and 2.12, similar to the proof of proposition 3.2, we can infer that $u_\hbar ^{0\nu }\in {\mathcal {C}}([0,T];V)$.

By taking $\nu \to 0$, we need to get some uniform estimates of $u_\hbar ^{0\nu }$ about $\nu >0$ for proving the existence of solutions to (4.1). Similar to the proof of proposition 3.1, multiplying (4.3) by $2u_\hbar ^{0\nu }$ and integrating over $[0,t]\times D$ and applying (A.5), Hölder's and Young's inequalities, condition 2.6, we obtain for every $\nu >0$

\begin{align*} &|u_\hbar^{0\nu}(t)|_{H}^{2}+2\nu\int_{0}^{t}\|u_\hbar^{0\nu}(s)\|^{2}{\rm d}s\\ &\quad\leq|\varsigma|_{H}^{2}+2\int_{0}^{t}\ \big(\tilde{\Upsilon}_0(s,u_\hbar^{0\nu}(s))f(s),u_\hbar^{0\nu}(s)\big)\,{\rm d}s\\ &\qquad +2\int_{0}^{t}\int_{\mathbb{Y}}(\tilde G(t,u_\hbar^{0\nu}(s),y)(g(s,y)-1),u_\hbar^{0\nu}(s))\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\leq|\varsigma|_{H}^{2}+2\int_0^{t}K(s)\,{\rm d}s+2\ \int_0^{t}\Big(K(s)+|f(s)|^{2}_0\Big)|u_\hbar^{0\nu}(s)|_{H}^{2}{\rm d}s\\ &\qquad+C\int_{0}^{t}\ \int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C\int_{0}^{t} |u_\hbar^{0\nu}(s)|_H^{2}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Then, using condition 2.7, remark 2.8, lemma 2.9 and Gronwall's inequality, we deduce that there exists a constant $C$ such that

(4.4)\begin{equation} \sup_{\nu>0}\sup_{0\leq t\leq T}|u_\hbar^{0\nu}(t)|_{H}^{2}\leq C(M,T)(1+|\varsigma|_{H}^{2}). \end{equation}

Denote $\xi _\hbar ^{0\nu }(t):=\text { curl }u_\hbar ^{0\nu }(t)$. Similar to the proof of proposition 3.2, employing the curl operator to (4.3) and applying (A.6), we get

(4.5)\begin{equation} \begin{aligned} d\xi_\hbar^{0\nu}(t) & ={-}\nu A\xi_\hbar^{0\nu}(t)- B(u_\hbar^{0\nu}(t),\xi_\hbar^{0\nu}(t))\,{\rm d}t+\text{ curl } \tilde{\Upsilon}_0(t, u_\hbar^{0\nu}(t))f(t)\,{\rm d}t\\ & \quad+\int_{\mathbb{Y}}\text{ curl }\tilde G(t,u_\hbar^{0\nu}(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t \end{aligned} \end{equation}

with the initial data $\xi _\hbar ^{0\nu }(0)=\text { curl }\varsigma$.

Multiplying (4.5) by $2\xi _\hbar ^{0\nu }$ and integrating over $[0,T]\times D$ and using condition 2.12, (A.7) ($q=2$), (A.3), Hölder's and Young's inequalities, one deduces that

\begin{align*} &|\xi_\hbar^{0\nu}(t)|_{H}^{2}+2\nu \int_{0}^{t} \|\xi_\hbar^{0\nu}(s)\|^{2}{\rm d}s\\ &\quad\leq |\text{ curl }\varsigma|_{H}^{2}+2 \int_{0}^{t}|\text{ curl }\tilde{\Upsilon}_0(s, u_\hbar^{0\nu}(s))|_{L(H_{0},H)} |f(s)|_{0}|\xi_\hbar^{0\nu}(s)|_{H}{\rm d}s\\ &\qquad+2 \int_{0}^{t}\ \int_{\mathbb{Y}}|\text{ curl }\tilde G(s,u_\hbar^{0\nu}(s),y)|_H|g(s,y)-1||\xi_\hbar^{0\nu}(s)|_{H}\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\leq\|\varsigma\|^{2}+\Big(1+\sup_{s\in [0,T]}|u_\hbar^{0\nu}(s)|_{H}^{2}\Big)\int_0^{t} \Big(K(s)+|f(s)|_{0}^{2}\Big)\,{\rm d}s\\ &\qquad +2\ \int_0^{t} \left(K(s)+|f(s)|_{0}^{2}\right)|\xi_\hbar^{0\nu}(s)|_{H}^{2}{\rm d}s\\ &\qquad+C\int_{0}^{t}\ \int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C\int_{0}^{t}|\xi_\hbar^{0\nu}(s)|_H^{2}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

It follows from condition 2.7, remark 2.8, lemma 2.9, (4.4) and Gronwall's inequality that there exists a constant $C$ such that for every $\hbar =(f,g)\in {{\bar S}^{M}}$, it holds that

(4.6)\begin{equation} \sup_{\nu>0}\sup_{0\leq t\leq T}|\xi_\hbar^{0\nu}(t)|_{H}^{2}\leq C(M,T)(1+\|\varsigma\|^{2}). \end{equation}

This together with (4.4) and (A.3) yields that

(4.7)\begin{equation} \sup_{\nu >0}\sup_{0\leq t\leq T}\|u_\hbar^{0\nu}\|\leq C(M,T)(1+\|\varsigma\|). \end{equation}

Moreover, for every $\nu >0$, we can infer that $u_\hbar ^{0\nu }\in {\mathcal {C}}([0,T]; H)\cap L^{\infty }(0,T; V)$ and it holds that

\begin{align*} u_\hbar^{0\nu}(t)&=\varsigma-\nu\int_{0}^{t} Au_\hbar^{0\nu}(s)-\int_{0}^{t}B(u_\hbar^{0\nu}(s),u_\hbar^{0\nu}(s))\,{\rm d}s+ \int_{0}^{t}\tilde{\Upsilon}_0(s,u_\hbar^{0\nu}(s))f(s)\,{\rm d}s\\ &\quad+\int_{0}^{t}\int_{\mathbb{Y}}\tilde G(s,u_\hbar^{0\nu}(s),y)(g(s,y)-1)\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

By applying (4.4), (4.7), conditions 2.6, 2.7 and 2.12 on $(\tilde {\Upsilon }_0,\tilde G)$ and (A.4) when $q=2$ and $r=1$, remark 2.8, lemma 2.9, one deduces that for any $\nu \in (0,1]$ and $\hbar =(f,g)\in {{\bar S}^{M}}$:

\[ \|u_\hbar^{0\nu}\|_{W^{1,2}(0,T;V')}\leq C(1+\|\varsigma\|). \]

Then, there exists a function ${v\in W^{1,2}(0,T;V')\cap L^{\infty }(0,T; V)}$ such that

\begin{align*} u_\hbar^{0\nu}&\to v\text{ weakly in }L^{2}\big(0,T; V\big)\cap W^{1,2}\big(0,T; V'\big),\\ u_\hbar^{0\nu}&\to v\text{ strongly in }L^{2}\big(0,T; H\big),\\ u_\hbar^{0\nu}&\to v\text{ weak star in }L^{\infty}\big(0,T; V\big), \end{align*}

as $\nu \to 0$. Taking $\nu \rightarrow 0$ in (4.3), we conclude that $u_\hbar ^{0}:=v$ is a solution of (4.1) as [Reference Temam52, theorem 3.1] (see also [Reference Zhai and Zhang60]). Moreover, since (4.7) is uniformly bounded in $\nu >0$, taking $\nu \to 0$ in (4.7), we can get (4.2).

Condition 4.2 Let $q\in [2,\infty )$; $\Upsilon _\nu \in {\mathcal {C}}([0,T]\times H^{2,q}; R(H_0,H^{1,q}))$ for $\nu >0$ and $G\in {\mathcal {C}}([0,T]\times H^{2,q}\times L^{q}(\mathbb {Y}); H^{1,q})$, there exists $K(t)\in L^{1}([0,T],\mathbb {R}^{+})$ such that for every $u\in H\cap H^{2,q}$ and $\nu >0$, if $\xi =\text {curl }u$, it holds

\begin{align*} \|\textrm{curl }\Upsilon_\nu(t,u)\|_{R(H_0,L^{q})}^{2}&\leq K(t)(1+\|u\|_q^{2}+\|\xi\|_q^{2}),\\ \int_{\mathbb{Y}}\|\textrm{curl }G(t,u(t),y)\|^{q}_{L^{q}}\vartheta({\rm d}y)&\le \int_{\mathbb{Y}}\|G(t,\textrm{curl }u(t),y)\|^{q}_{L^{q}}\vartheta({\rm d}y)\\ &\le K(t)(1+\|\textrm{curl } u\|^{q}_q). \end{align*}

Condition 4.3 Let $q\in [2,\infty )$; $\tilde {\Upsilon }_\nu \in {\mathcal {C}}([0,T]\times H^{1,q}; L(H_0, H^{1,q}))$ for $\nu \geq 0$, $\tilde G\in {\mathcal {C}}([0,T]\times H^{1,q}\times L^{1}(\mathbb {Y}); H^{1,q})$, and there exists $K(t)\in L^{1}([0,T],\mathbb {R}^{+})$ such that for every $u\in H^{1,q}$ and $\nu >0$ (resp. $u\in H^{2,q}$ for $\nu =0$) if $\xi =\text { curl }u$, it holds

\begin{align*} \|\text{ curl }\tilde{\Upsilon}_\nu(t,u)\|_{L(H_{0},L^{q})}&\leq \sqrt{K(t)}(1+\|u\|_{q}+\|\xi\|_{q}),\\ \int_{\mathbb{Y}}\|\textrm{curl }\tilde G(t,u(t),y)\|_{q}\vartheta({\rm d}y)&\le \int_{\mathbb{Y}}\|\tilde G(t,\textrm{curl }u(t),y)\|_{q}\vartheta({\rm d}y). \end{align*}

Proof. First, we verify the estimates (4.8) and (4.9). Thanks to (A.3), we only need to check the $L^{q}(D)$ upper bounds of $\text { curl }u_\hbar ^{0}(t)$. Now, we establish the a priori estimates of the Galerkin approximation solution $u_{\hbar,n}^{0}$, and assume that $u_{\hbar,n}^{0}\in H^{2,q}$. To simplify the notation, we drop the index $n$.

Denote $\xi _\hbar ^{0}(t):=\text { curl }u_\hbar ^{0}(t)$. Applying the curl operator to (4.1) and using (A.6), we have

(4.10)\begin{equation} \begin{aligned} d\xi_\hbar^{0}(t) & ={-}B\big(u_\hbar^{0}(t),\xi_\hbar^{0}(t)\big)\,{\rm d}t+\text{ curl }\tilde{\Upsilon}_0(t,u_\hbar^{0}(t))f(t)\,{\rm d}t\\ & \quad+\int_{\mathbb{Y}}\text{ curl } \tilde G(t,u_\hbar^{0}(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t,\ \xi_\hbar^{0}(0)=\text{ curl }\varsigma. \end{aligned} \end{equation}

Multiplying (4.10) by $q|\xi _\hbar ^{0}(t)|^{q-2}\xi _\hbar ^{0}(t)$ and integrating over $[0,t]\times D$, we obtain

\begin{align*} &\|\xi_\hbar^{0}(t)\|_{q}^{q}+q\int_0^{t}\ \int_{D} (u_\hbar^{0}(s)\cdot\nabla)\xi_\hbar^{0}(s)|\xi_\hbar^{0}(s)|^{q-2}\xi_\hbar^{0}(s)dxds\\ &\quad=\|\text{ curl }\varsigma\|_q^{q}+q\int_0^{t}\int_{D}\text{ curl }\tilde{\Upsilon}_0(s,u_\hbar^{0}(s))f(s)|\xi_\hbar^{0}(s)|^{q-2}\xi_\hbar^{0}(s) {\rm d}x {\rm d}s\\ &\qquad+q\int_0^{t} \int_{D}\int_{\mathbb{Y}}\text{ curl }\tilde G(t,u_\hbar^{0}(t),y)(g(t,y)-1)|\xi_\hbar^{0}(t)|^{q-2}\xi_\hbar^{0}(t)\vartheta({\rm d}y)dxds. \end{align*}

We note that for every $s$, $\int _{D}(u_\hbar ^{0}(s)\cdot \nabla )\xi _\hbar ^{0}(s)|\xi _\hbar ^{0}(s)|^{q-2}\xi _\hbar ^{0}(s) {\rm d}x=0$ by using the fact that $\xi _\hbar ^{0}(t)=\text { curl }u_\hbar ^{0}(t)$, and (A.7). Then, by using Hölder's and Young's inequalities and condition 4.3, one has

\begin{align*} \|\xi_\hbar^{0}(t)\|_{q}^{q}&\leq \|\text{ curl }\varsigma\|_q^{q}+q\int_0^{t} \|\text{ curl }\tilde{\Upsilon}_\nu(s,u_\hbar^{0}(s))\|_{L(H_{0},L^{q})} |f(s)|_0\|\xi_\hbar^{0}(s)\|_q^{q-1}{\rm d}s\\ &\quad +q\int_0^{t}\ \int_{D} \int_{\mathbb{Y}}\text{ curl }\tilde G(t,u_\hbar^{0}(t),y)(g(t,y)-1)|\xi_\hbar^{0}(t)|^{q-2}\xi_\hbar^{0}(t)\vartheta({\rm d}y)dxds\\ &\leq \|\text{ curl }\varsigma\|_q^{q}+q\Big(1+\sup_{0\leq s\leq T}\|u_\hbar^{0}(s)\|^{q}_q\Big)\int_0^{t} \left(K(s)+|f(s)|_{0}^{2}\right)\,{\rm d}s\\ &\quad+q\int_0^{t}\left(K(s)+|f(s)|_{0}^{2}\right)\|\xi_\hbar^{0}(s)\|_q^{q}ds\\ &\quad+C\int_{0}^{t}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\quad+C\int_{0}^{t}\|\xi_\hbar^{0}(s)\|_q^{q}\int_{\mathbb{Y}}\|\tilde G(s,y)\|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

From the fact that $H^{1,2}\hookrightarrow L^{q}(D)$, (4.2), lemma 2.9 and Gronwall's inequality, we conclude that for any $n\geq 1$ and $\hbar =(f,g)\in {\bar S}^{M}$

(4.11)\begin{equation} \sup_{0\leq t\leq T}\|\xi_{\hbar,n}^{0}(t)\|_{q}^{q}\leq C\Big(\| \text{ curl }\varsigma\|_q^{q}+1+\|\varsigma\|^{q}\Big). \end{equation}

Note that the upper bounds of inequality 4.11 do not depend on $n$. We can take $n\to \infty$ in (4.11) and use classical arguments to get (4.9). For more details, see e.g. [Reference Bessaih and Millet8].

Next, we prove the uniqueness of the solution $u_\hbar ^{0}$. By using (4.9) for some $q>2$ and (A.2), we can infer that any solution $u_\hbar ^{0}$ to (4.1) belongs to $L^{\infty }((0,T)\times D)$.

Let $u_\hbar ^{0}$ and $v_\hbar ^{0}$ be two solutions of (4.1) with the same initial data. Denote $\varpi :=u_\hbar ^{0}-v_\hbar ^{0}$. Then $\varpi$ is a weak solution of the following equations:

\begin{align*} d\varpi(s)&={-}\left[B(u_\hbar^{0}(s),u_\hbar^{0}(s))-B(v_\hbar^{0}(s),v_\hbar^{0}(s))\right]{\rm d}s \\ &\quad +\left[\tilde{\Upsilon}_0(s,u_\hbar^{0}(s))-\tilde{\Upsilon}_0(s,v_\hbar^{0}(s))\right]f(s)\,{\rm d}s\\ &\quad+\int_{\mathbb{Y}}\left[\tilde G(s,u_\hbar^{0}(s),y)-\tilde G(s,v_\hbar^{0}(s),y)\right](g(s,y)-1)\vartheta({\rm d}y)\,{\rm d}s,\quad \varpi(0)=0. \end{align*}

Multiplying the above equations by $\varpi (t)$ and integrating on $D$, and using condition 2.6 on $(\tilde {\Upsilon }_0,\tilde G)$, Schwarz's and Hölder's inequalities and (4.9), we obtain for any $q\in (1,\infty )$

\begin{align*} &\frac{1}{2}\frac{d}{{\rm d}t}|\varpi(t)|_H^{2}\\ &\quad={-}(B(\varpi(t),u_\hbar^{0}(t)), \varpi(t)) +\big(\left[\tilde{\Upsilon}_0(t,u_\hbar^{0}(t))-\tilde{\Upsilon}_0(t,v_\hbar^{0}(t))\right]f(t),\varpi(t)\big)\\ &\qquad+\int_{\mathbb{Y}}(\left[\tilde G(t,u_\hbar^{0}(t),y)-\tilde G(t,v_\hbar^{0}(t),y)\right](g(t,y)-1),\varpi(t))\vartheta({\rm d}y)\\ &\quad\leq\int_{D}|\varpi(t)|^{2}(x)|\nabla u_\hbar^{0}(t)|(x)\,{\rm d}x \\ &\qquad +|\left(\tilde{\Upsilon}_0(t,u_\hbar^{0}(t))-\tilde{\Upsilon}_0(t,v_\hbar^{0}(t))\right)|_{L(H_0,H)} |f(t)|_0|\varpi(t)|_H\\ &\qquad +C|\varpi(t)|_H^{2}\int_{\mathbb{Y}}\|\tilde G(t,y)\|_{1,H}|g(t,y)-1|\vartheta({\rm d}y)\\ &\quad\leq\|\nabla u_\hbar^{0}(t)\|_q\|\varpi(t)\|_{L^{\infty}(D)}^{\frac{2}{q}} |\varpi(t)|_H^{\frac{2(q-1)}{q}}+\sqrt{K(t)}|u_\hbar^{0}(t)-v_\hbar^{0}(t)|_H|f(t)|_{0}|\varpi(t)|_H\\ &\qquad+C|\varpi(t)|_H^{2}\int_{\mathbb{Y}}\|\tilde G(t,y)\|_{1,H}|g(t,y)-1|\vartheta({\rm d}y). \end{align*}

Applying the fact that $\|\text { curl }\varsigma \|_q\leq C \|\text { curl }\varsigma \|_\infty, q\in [2,\infty ), C\geq 1$, $\varpi (0)=0$ and (4.9), we obtain for $t\in [0,T]$

\begin{align*} \mathcal{W}'(t)&\leq 2qC(M)(1+\|\varsigma\|+\|\text{ curl }\varsigma\|_{L^{\infty}(D)})\bar{\mathcal{W}}^{2\over q}\mathcal{W}(t)^{1-{1\over q}}+2(K(t)+|f(t)|_0^{2})\mathcal{W}(t)\\ &\quad +\mathcal{W}(t)\int_{\mathbb{Y}}\|\tilde G(t,y)\|_{1,H}|g(t,y)-1|\vartheta({\rm d}y), \end{align*}

where $\bar {\mathcal {W}}:=\sup _{0\leq t\leq T}\|\varpi (t)\|_{L^{\infty }(D)}$ and $\mathcal {W}(t):=|\varpi (t)|_{H}^{2}$. Then

\begin{align*} \mathcal{W}(t)^{1/q}&\leq 2 C q\, \bar{C}_4(M) [1+\|\varsigma\|+\|\text{ curl } \varsigma \|_{L^{\infty}(D)}] \bar{\mathcal{W}}^{\frac{2}{q}} t\\ &\quad +\int_{0}^{t} 2 (K(s)+|f(s)|_0^{2}) \mathcal{W}(s)^{1/q} {\rm d}s\\ &\quad+C\int_{0}^{t}\mathcal{W}(s)^{1/q}\int_{\mathbb{Y}}\|G(s,y)\|_{1,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Hence, by using Gronwall's inequality, for $q\in [2,\infty )$ and any $T^{*}\in [0,T]$, we deduce that

\begin{align*} &\sup_{0\leq t\leq T^{*}}|\varpi(t)|_{H}^{2}\\ &\quad \leq C\Bigg(2C(M)[1+ \|\varsigma\|+\|\text{ curl }\varsigma\|_{L^{\infty}(D)}]\bar{\mathcal{W}}^{2}T^{*}\\ &\qquad\times \exp\Bigg\{\int_0^{T^{*}}\ \Big[(K(t)+|f(t)|_0^{2})+\int_{\mathbb{Y}} \|\tilde G(t,y)\|_{1,H}|g(t,y)-1|\vartheta({\rm d}y)\Big]{\rm d}t\Bigg\}\Bigg)^{q}. \end{align*}

Choosing $T_1^{*}>0$ small enough and taking $q\to \infty$, for every $t\in [0,T_1^{*}]$, we can infer that $|\varpi (t)|_{H}^{2}=0$. Repeating this process by using the value at time $T_1^{*}$ as the initial data and employing (A.2), (4.9) and (4.2), one concludes that there is a $T^{*}>0$ such that for every integer $k=0,1,\cdots$ and any $t\in [T_1^{*}+kT^{*},T_1^{*}+(k+1)T^{*}]\cap [0,T]$, it holds that $|\varpi (t)|_{H}^{2}=0$. This yields the uniqueness. We complete the proof of theorem 4.4.

Proof. Now, we study the energy estimation of the approximation solution $u^{\nu }_{h,n}$ in $H^{1,q}$. To simplify the notation, we drop the index $n$. It follows from (A.1) and proposition 3.2 that for $0<\nu \leq {\nu }_0$ and $\hbar =(f,g)\in {{\tilde {\mathcal {U}}}^{M}}$, ${\mathbb E}(\sup _{0\leq t\leq T}\|u_\hbar ^{\nu }(t)\|_q^{q})\leq C(q,M) (1+{\mathbb E}\|\varsigma \|^{q})$. Denote $\xi _\hbar ^{\nu }:=\text { curl }u_\hbar ^{\nu }$. By using (A.3), we have $\|\nabla u_\hbar ^{\nu }\|_q\le \|\text {curl } u_\hbar ^{\nu }\|_q$. Then, for getting (5.1), we only need to prove

(5.2)\begin{equation} \sup_{0<\nu\leq{\nu}_0}\sup_{\hbar\in{{\tilde{\mathcal{U}}}^{M}}}{\mathbb E}\Bigg( \sup_{0\leq t\leq T}\|\xi_\hbar^{\nu}(t)\|_{q}^{q} \Bigg)\leq C(M,q)\big(1+{\mathbb E}\|\text{ curl }\varsigma\|_{q}^{q}\big). \end{equation}

Denote $\langle \cdot,\cdot \rangle$ by the duality between $L^{q}(D)$ and $L^{q^{*}}(D)$, where ${1\over q}+{1\over q^{*}}=1$. For fixed $N>0$, set $\tau _N=\inf \{t\geq 0:\|\xi _\hbar ^{\nu }(t)\|_q\geq N\}\wedge T$. By using Itô's formula, we obtain

\[ \|\xi_\hbar^{\nu}(t\wedge \tau_N)\|_q^{q}\leq \|\text{curl}\varsigma\|_q^{q}+\sum_{i=1}^{8} K_i(t), \]

where

\begin{align*} K_1(t)&=q\sqrt{\nu}\int_0^{t\wedge\tau_N}\langle|\xi_\hbar^{\nu}(s)|^{q-2} \xi_\hbar^{\nu}(s),\text{ curl }\Upsilon_\nu(s,u_\hbar^{\nu}(s))\rangle {\rm d}W(s),\\ K_2(t)&=\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\left(\|\xi_\hbar^{\nu}(s)+\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_{q}^{q}-\|\xi_\hbar^{\nu}(s)\|_q^{q}\right)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}s),\\ K_3(t)&={-}q\nu\int_0^{t\wedge \tau_N}\langle |\xi_\hbar^{\nu}(s)|^{q-2}\xi_\hbar^{\nu}(s), A\xi_\hbar^{\nu}(s)\rangle {\rm d}s,\\ K_4(t)&={-}q\int_0^{t\wedge \tau_N}\langle|\xi_\hbar^{\nu}(s)|^{q-2} \xi_\hbar^{\nu}(s),\text{ curl }B(u_\hbar^{\nu}(s), u_\hbar^{\nu}(s))\rangle {\rm d}s,\\ K_5(t)&=q\int_0^{t\wedge \tau_N}\langle|\xi_\hbar^{\nu}(s)|^{q-2} \xi_\hbar^{\nu}(s), \textrm{curl }\tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))f(s) \rangle {\rm d}s,\\ K_6(t)&=\frac{q}{2}(q-1)\nu\int_0^{t\wedge\tau_N} \|\mbox{curl } \Upsilon_\nu(s,u_\hbar^{\nu}(s))\|_{R(H_0,L^{q})}^{2}\|\xi_\hbar^{\nu}(s)\|^{q-2}_qds,\\ K_7(t)&=\nu\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\bigg(\|\xi_\hbar^{\nu}(s)+\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_{q}^{q}-\|\xi_\hbar^{\nu}(s)\|_q^{q}\\ &\quad-\nu q|\xi_\hbar^{\nu}(s)|^{q-2}\langle\xi_\hbar^{\nu}(s),\text{curl }G(s,u_\hbar^{\nu}(s),y)\rangle\bigg)\vartheta({\rm d}y)\,{\rm d}s,\\ K_8(t)&=q\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}|\xi_\hbar^{\nu}(s)|^{q-2}(\text{curl }\tilde G(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),\xi_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

For the term $K_1(t)$, by applying the Burkholder–Davies–Gundy inequality, condition 4.2, Hölder's and Young's inequalities, we deduce that

\begin{align*} &{\mathbb E}\Bigg(\sup_{s\in[0,t\wedge\tau_N]}K_1(s)\Bigg)\\ &\quad\leq\sqrt{\nu}C q{\mathbb E}\Bigg(\int_0^{t\wedge\tau_N} \|\text{ curl }\Upsilon_\nu(s,u_\hbar^{\nu}(s))\|_{R(H_0,L_q)}^{2} \|\xi_\hbar^{\nu}(s)\|_q^{2(q-1)}{\rm d}s\Bigg)^{\frac{1}{2}} \\ &\quad\leq\sqrt{\nu}C q {\mathbb E}\Bigg(\sup_{0\leq s\leq t} \|\xi_\hbar^{\nu}(s\wedge \tau_N)\|^{\frac{q}{2}}_q \Bigg[\int_0^{t\wedge \tau_N}\|\xi^{\nu}(s)\|_q^{q-2}K(s)\\ &\quad \times \Big(1+ \|u_\hbar^{\nu}(s)\|_q^{2}+\|\xi_\hbar^{\nu}(s)\|_q^{2}\Big)\,{\rm d}s\Bigg]^{\frac{1}{2}}\Bigg)\\ &\quad\leq\frac{1}{2}{\mathbb E}\Bigg(\sup_{0\leq s\leq t}\|\xi_\hbar^{\nu}(s\wedge \tau_N)\|^{q}_q\Bigg)+\nu C q^{2}{\mathbb E}\int_0^{t\wedge\tau_N}\ K(s)\|\xi_\hbar^{\nu}(s)\|_q^{q}ds\\ &\qquad+\nu C q^{2}\Bigg(1+{\mathbb E}\big(\sup_{0\leq t\leq T}\|u_\hbar^{\nu}(t)\|_q^{q}\big)\Bigg)\int_0^{t\wedge\tau_N}\ K(s)\,{\rm d}s. \end{align*}

For the term $K_2(t)$, we use the Burkholder–Davis–Gundy inequality, condition 4.2, Hölder's and Young's inequalities again to obtain

\begin{align*} &{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_N]}K_2(s)\right)\\ &\quad\le {\mathbb{E}}\left[\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\left(\|\xi_\hbar^{\nu}(s)+\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_{q}^{q}-\|\xi_\hbar^{\nu}(s)\|_q^{q}\right)^{2}\vartheta({\rm d}y)\,{\rm d}s\right]^{1\over 2}\\ &\quad\le C(\nu){\mathbb{E}}\bigg(\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_q^{2q-2}\|\text{curl}G(s,u_\hbar^{\nu}(s),y)\|_q^{2}\\ &\qquad +\|\xi_\hbar^{\nu}(s)\|_q^{2q-4}\|\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{4} +\|\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{2q}\vartheta({\rm d}y)\,{\rm d}s\bigg)^{1\over 2}\\ &\quad \le C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}\ K(s)\|\xi_\hbar^{\nu}(s)\|_q^{2q}{\rm d}s\right)^{1\over 2}+C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}\ K(s)\,{\rm d}s\right)^{1\over 2}\\ &\quad\le \frac{1}{2}{\mathbb{E}}\left(\sup_{s\in[0,t\wedge\tau_{N}]}\|\xi_\hbar^{\nu}(s)\|_q^{q}\right)+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ K(s)\|\xi_\hbar^{\nu}(s)\|_q^{q}{\rm d}s\\ &\qquad +C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}\ K(s)\,{\rm d}s\right)^{1\over 2}. \end{align*}

By using the facts that $\xi _\hbar ^{\nu }=0$ on $\partial D$ and $A=-\Delta$, one deduces that

\begin{align*} {\mathbb{E}}K_3(t)&={-}q\nu{\mathbb{E}}\int_0^{t\wedge\tau_N}\ \int_D\langle \nabla\big(|\xi_\hbar^{\nu}(s)|^{q-2}\xi_\hbar^{\nu}(s)\big),\nabla \xi_\hbar^{\nu}(s)\rangle dxds \\ &={-}q(q-1)\nu{\mathbb{E}}\int_0^{t\wedge\tau_N}\ \int_D |\xi_\hbar^{\nu}(s)|^{q-2}|\nabla\xi_\hbar^{\nu}(s)|^{2}dxds. \end{align*}

It follows from $\xi _\hbar ^{\nu }=\text { curl }u_\hbar ^{\nu }$ and (A.7) that ${\mathbb {E}}K_4(t)=0$. For the term $K_5(t)$, we employ Hölder's and Young's inequalities and condition 4.3 to bound

\begin{align*} {\mathbb{E}}K_5(t)&\leq q{\mathbb{E}}\int_0^{t\wedge\tau_N}\||\xi_\hbar^{\nu}(s)|^{q-1}\|_{q^{*}} \|\text{ curl }\tilde{\Upsilon}_\nu(s,u_\hbar^{\nu}(s))f(s)\|_qds\\ &\leq {\mathbb{E}}\left[\Big(1+\sup_{s\in[0,T]}\| u_\hbar^{\nu}(s)\|_q^{q}\Big)\int_0^{t\wedge\tau_N} \Big(K(s)+|f(s)|^{2}_0\Big)\,{\rm d}s\right.\\ &\qquad \left.+\,q\int_0^{t\wedge\tau_N} \|\xi_\hbar^{\nu}(s)\|_q^{q} \Big(K(s)+|f(s)|^{2}_0\Big)\,{\rm d}s\right]. \end{align*}

Condition 4.2, Hölder's and Young's inequalities imply that for any $\nu \in (0,\nu _{0}]$

\begin{align*} {\mathbb{E}}K_6(t)&\leq\frac{q(q-1)}{2}\nu{\mathbb{E}}\int_0^{t\wedge\tau_N} \|\xi_\hbar^{\nu}(s)\|_q^{q-2}K(s)\Big(1+\|u_\hbar^{\nu}(s)\|_q^{2}+\|\xi_\hbar^{\nu}(s)\|_q^{2}\Big)\,{\rm d}s\\ &\leq\frac{q(q-1)}{2}\nu{\mathbb{E}}\Big(1+{2\over q}\sup_{s\in[0,T]}\| u_\hbar^{\nu}(s)\|_q^{q}\Big)\int_0^{t\wedge \tau_N} K(s)\,{\rm d}s\\ &\quad+\nu(q-1)^{2}{\mathbb{E}}\int_0^{t\wedge \tau_N}\ K(s)\|\xi_\hbar^{\nu}(s)\|_q^{q} {\rm d}s. \end{align*}

For the term $K_7(t)$, by using condition 4.2, Hölder's and Young's inequalities, we have

\begin{align*} {\mathbb{E}}K_7(t)&\le C{\mathbb{E}}\left[\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\left(\|\xi_\hbar^{\nu}(s)\|_q^{q-2}+\|\nu\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{q-2}\right)\|\right.\\ &\quad \left.\times \nu\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{2}\vartheta({\rm d}y)\,{\rm d}s\vphantom{\int_{0}^{t\wedge\tau_{N}}}\right]\\ &\le C(\nu){\mathbb{E}} \left(\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}(\|\xi_\hbar^{\nu}(s)\|_q^{q-2}\|\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{2}\right.\\ &\quad +\left.\|\text{curl }G(s,u_\hbar^{\nu}(s),y)\|_q^{q})\vartheta({\rm d}y)\,{\rm d}s\vphantom{\int_{0}^{t\wedge\tau_{N}}}\right)\\ &\le C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ K(s)\|\xi_\hbar^{\nu}(s)\|_q^{q}{\rm d}s+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ K(s)\,{\rm d}s. \end{align*}

For the term $K_8(t)$, applying condition 4.3, Hölder's and Young's inequalities, one has

\begin{align*} &{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_q^{q-2}(\text{curl }\tilde G(s,u_\hbar^{\nu}(s),y)(g(s,y)-1),\xi_\hbar^{\nu}(s))\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_q^{q-2}\|\text{curl }\tilde G(s,u_\hbar^{\nu}(s),y)\|_q|g(s,y)-1|\|\xi_\hbar^{\nu}(s)\|_q\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\ \int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_q^{q-2}\frac{\|\text{curl }\tilde G(s,u_\hbar^{\nu}(s),y)\|_q}{1+\|\xi_\hbar^{\nu}(s)\|_{q}}\\ &\qquad \times |g(s,y)-1|(1+\|\xi_\hbar^{\nu}(s)\|_q^{2})\vartheta({\rm d}y)\,{\rm d}s \\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}} \int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_q^{q-2}\frac{\| \tilde G(s,\xi_\hbar^{\nu}(s),y)\|_q}{1+\|\xi_\hbar^{\nu}(s)\|_{q}}|g(s,y)-1|(1+\|\xi_\hbar^{\nu}(s)\|_q^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}\|\xi_\hbar^{\nu}(s)\|_{q}^{q-2}|\tilde G(s,y)|_{0,q}|g(s,y)-1|(1+\|\xi_\hbar^{\nu}(s)\|_q^{2})\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}|\tilde G(s,y)|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\|\xi_\hbar^{\nu}(s)\|_q^{q}\int_{\mathbb{Y}}|\tilde G(s,y)|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Then, we obtain

\begin{align*} &{\mathbb E}\|\xi_\hbar^{\nu}(t\wedge \tau_N)\|_q^{q}\\ &\quad \leq {\mathbb E}\|\mbox{curl }\varsigma\|_q^{q}+C(\nu,q){\mathbb E}\left[\Big(1+\sup_{s\in[0,T]}\|u_\hbar^{\nu}(s)\|_q^{q}\Big)\int_0^{t\wedge\tau_N} \Big(K(s)+|f(s)|^{2}_0\Big)\,{\rm d}s\right]\\ &\qquad+C(\nu,q){\mathbb E}\int_0^{t\wedge\tau_N} \|\xi_\hbar^{\nu}(s)\|_q^{q} \Big(K(s)+|f(s)|^{2}_0\Big)\,{\rm d}s+C(\nu){\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}K(s)\,{\rm d}s\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\int_{\mathbb{Y}}|\tilde G(s,y)|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s+C(\nu){\mathbb{E}}\left(\int_{0}^{t\wedge\tau_{N}}K(s)\,{\rm d}s\right)^{1\over 2}\\ &\qquad+C{\mathbb{E}}\int_{0}^{t\wedge\tau_{N}}\|\xi^{\nu}(s)\|_q^{q}\int_{\mathbb{Y}}|\tilde G(s,y)|_{0,q}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Using Grownall's inequality and lemma 2.9, we have

\[ \sup_{0<\nu\leq{\nu}_0}\sup_{\hbar\in{{\tilde{\mathcal{U}}}^{M}}}{\mathbb E}\Big( \sup_{0\leq t\leq T}\|\xi_\hbar^{\nu}(t)\|_{q}^{q} \Big)\leq C(M,q)\big(1+{\mathbb E}\|\text{ curl }\varsigma\|_{q}^{q}\big). \]

Then, one deduces that (5.2) holds for the approximation sequence $\xi _{\hbar,n}^{\nu }$. We notice that the constant $C(M,q)$ does not depend on $n$. Hence, letting $n\to \infty$, we can obtain (5.1) by using the weak convergence discussion. Therefore, we complete the proof.

Condition 6.1 For every $q\in [2,+\infty )$, we assume that $(\Upsilon _0,G)$ satisfies conditions 2.5, 2.7, 2.11 and 4.2 and $\Upsilon _0$ satisfies

\[ \sup_{0\leq t\leq T}\big|\Upsilon_\nu(t,u)-\Upsilon_0(t,u)\big|_{L(H_0,H)}\leq C_\nu\big(1+|u|_H\big), \text{ for } u\in H \text{ and }\nu>0, \]

where $C_\nu \in [0,+\infty )$ and it converges to $0$ as $\nu \to 0$.

Proof. We divide the following four steps to prove the proposition.

Step 1: We note that $u_{\hbar _\nu }^{\nu }=\varsigma +\sum _{i=1}^{6} H_i$, where

\begin{align*} H_1&={-}\nu\int_{0}^{t}Au_{\hbar_\nu}^{\nu}(s)\,{\rm d}s,\quad H_2={-}\int_{0}^{t}B(u_{\hbar_\nu}^{\nu}(s),u_{\hbar_\nu}^{\nu}(s))\,{\rm d}s, \\ H_3&=\sqrt{\nu}\int_{0}^{t}\Upsilon_{\nu}(s,u_{\hbar_\nu}^{\nu}(s))\,{\rm d}W(s), \quad H_4=\int_{0}^{t}{\Upsilon}_{\nu}(s,u_{\hbar_\nu}^{\nu}(s))f_\nu(s)\,{\rm d}s,\\ H_5&=\nu\int_{0}^{t}\ \int_{\mathbb{ Y}}G(t,u_{\hbar_\nu}^{\nu}(t-),y)\tilde{N}^{\nu^{{-}1}g_\nu}({\rm d}y,{\rm d}t),\\ H_6&=\int_{0}^{t}\ \int_{\mathbb{Y}}G(t,u_{\hbar_\nu}^{\nu}(t),y)(g_\nu(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t. \end{align*}

For $\nu \in (0,\nu _0]$, by using Minkowski's and Cauchy-Schwarz's inequalities, we obtain

\begin{align*} \|H_{1}\|^{2}_{W^{1,2}(0,T; H)}&=\nu \int_{0}^{T}\Big|\int_{0}^{t}Au_{\hbar_\nu}^{\nu}(s)\,{\rm d}s\Big|^{2}_{H}{\rm d}t +\nu\int_{0}^{T}|Au_{\hbar_\nu}^{\nu}(t)|^{2}_{H}{\rm d}t \\ &\leq C(T,p)\nu\int_{0}^{T}|Au_{\hbar_\nu}^{\nu}(s)|_H^{2}{\rm d}s. \end{align*}

Hence, for $\nu \in (0,\nu _0]$, by virtue of (3.6), one deduces

\[ {\mathbb E}\big(\|H_{1}\|^{2}_{W^{1,2}(0,T; H)}\big)\leq C(M,T,\nu_0) \left(1+{\mathbb E}\|\varsigma\|^{4}\right). \]

Similarly, it follows from (3.6) that for all $p\in [2,\infty )$ and $\nu \in (0,\nu _0]$

\begin{align*} {\mathbb E}\|H_{1}\|^{p}_{W^{1,p}(0,T;V')}&\leq\nu C(T){\mathbb E}\int_0^{T}\|A u_{\hbar_\nu}^{\nu}(s)\|_{V'}^{p} {\rm d}s\nonumber\leq\nu C(T){\mathbb E}\int_0^{T}\| u_{\hbar_\nu}^{\nu}(s)\|^{p}ds\\ &\leq C(T,p,\nu_0)(1+{\mathbb E}\|\varsigma\|^{p}). \end{align*}

Minkowski's and Hölder's inequalities and (A.8) imply that for $p,q\in [4,\infty )$ and $\nu \in (0,\nu _0]$

\[ \|H_{2}\|^{p}_{W^{1,p}(0,T;H)}\leq C(T,p,\nu_0)\int_0^{T}\|u_{\hbar_\nu}^{\nu}(t)\|_{H^{1,q}}^{p}\|u_{\hbar_\nu}^{\nu}(t)\|^{p}{\rm d}t. \]

Then, for $\nu \in (0,\nu _0]$, by using Hölder's inequality, (3.6) and (5.1), we have

\[ {\mathbb E}\big(\|H_{2}\|_{W^{1,p}(0,T;H)}^{p}\big)\leq C(T,M,p,q) \left(1+{\mathbb E}\|\varsigma\|^{pq/(q-p)}\right)^{1-p/q}\left(1+{\mathbb E}\|\varsigma\|_{H^{1,q}}^{q}\right)^{p/q}. \]

By the Burkholder–Davis–Gundy inequality and Hölder's inequality, it holds that

\begin{align*} {\mathbb E}\int_{0}^{T}|H_{3}(t)|^{p}_{H}{\rm d}t &\leq C_p\nu^{p/2}\int_{0}^{T}{\mathbb E}\left(\int_{0}^{T}|\Upsilon_\nu(s,u_{\hbar_\nu}^{\nu}(s))|^{2}_{L_{Q}}{\rm d}s\right)^{p/2}{\rm d}t\\ &\leq C_p\nu^{p/2}\Big(1+{\mathbb E}\sup_{s\in[0,T]}|u_{\hbar_\nu}^{\nu}(s)|_H\Big)\int_0^{T}\Big(\int_0^{T} K(t)\,{\rm d}t\Big)^{p\over 2}{\rm d}t\\ &\leq C(T,p,\nu_0)(1+{\mathbb E}|\varsigma|_H^{p}). \end{align*}

Let $p\in [4,\infty )$, $\alpha \in (0,\frac {1}{2})$. By virtue of the Burkholder–Davis–Gundy inequality and Hölder's inequality, one deduces that

\begin{align*} &{\mathbb E}\int_{0}^{T}\int_{0}^{T} \frac{|H_{3}(t)-H_{3}(s)|_{H}^{p}}{|t-s|^{1+\alpha p}}\,{\rm d}t\,{\rm d}s \\ &\quad =\nu^{p/2}\int_{0}^{T}\int_{0}^{T}\frac{{\mathbb E}|\int_{s\wedge t}^{s\vee t}\Upsilon_\nu(r,u_{\hbar_\nu}^{\nu}(r))\,{\rm d}W(r)|^{p}_{H}}{|t-s|^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad \leq C(T,p,\nu_0)(1+{\mathbb E}|\varsigma|_H^{p})\int_{0}^{T}\int_{0}^{T}{|t-s|^{-(1+\alpha p)}}\,{\rm d}s\,{\rm d}t\\ &\quad \leq C(T,p,\nu_0)(1+{\mathbb E}|\varsigma|_H^{p}). \end{align*}

Then, we obtain

\[ {\mathbb E}\big(\|H_3\|_{W^{\alpha,p}(0,T;H)}^{p}\big)\leq C(p,T){\nu}_0^{p/2}\left( 1+{\mathbb E}|\varsigma|_{H}^{p}\right). \]

Furthermore, we use Hölder's inequality and condition 2.5 to infer that for $\nu \in (0,{\nu }_0]$ and $p\in [4,\infty )$

\[ \int_0^{T}|H_4(t)|_H^{p} {\rm d}t\leq C\Big(1+\sup_{s\in[0,T]}|u_{\hbar_\nu}^{\nu}(s)|_{H}^{p}\Big)\int_0^{T}\Big(\int_0^{T}K(t)\,{\rm d}t\Big)^{p}dt. \]

Let $\alpha \in (0,\frac {1}{2})$. By using Minkowski's and Hölder's inequalities, condition 2.5 and Fubini's theorem, we deduce that for $\nu \in (0,{\nu }_0]$

\begin{align*} &\int_0^{T}\int_0^{T}\frac{ |H_4(t)-H_4(s)|_H^{p}}{(t-s)^{1+\alpha p}}{\rm d}s {\rm d}t\\ &\quad \leq 2\int_0^{T} \int_0^{T}(t-s)^{{-}1-\alpha p}\Big|\int_s^{t}|{\Upsilon}_\nu(r,u_{\hbar_\nu}^{\nu}(r))|_{L_Q}|f_\nu(r)|_0{\rm d}r\Big|^{p}\,{\rm d}s\,{\rm d}t\\ &\quad \leq C\Big(1+\sup_{r\in[0,T]}|u_{\hbar_\nu}^{\nu}(r)|_{H}^{p}\Big) \int_0^{T}\int_0^{T}(t-s)^{{-}1-\alpha p}\,{\rm d}s\,{\rm d}t. \end{align*}

Using the two above estimates and (3.1), for $\alpha \in (0,\frac {1}{2})$, $p\in [4,\infty )$ and $\nu \in (0,{\nu }_0]$, we get

\[ {\mathbb E}\big(\|H_4\|_{W^{\alpha,p}(0,T;H)}^{p}\big)\leq C(p,\alpha,T,M)\left(1+{\mathbb E} |\varsigma|_{H}^{p}\right). \]

For the term $H_5$, by the Burkholder–Davis–Gundy inequality and Hölder's inequality, and proposition 3.1, we obtain

\begin{align*} {\mathbb E}\int_{0}^{T}|H_{5}(t)|^{p}_{H}{\rm d}t&\leq C_p \nu^{p/2}\int_{0}^{T}{\mathbb E}\left(\int_{0}^{T}\ \int_{\mathbb{ Y}}|G(t,u_{\hbar_\nu}^{\nu}(t),y)|_H^{2}\vartheta({\rm d}y)\,{\rm d}t\right)^{p/2}{\rm d}t \\ &\leq C_p\nu^{p/2}\int_{0}^{T}{\mathbb E}\left(\int_{0}^{T}|u_{\hbar_\nu}^{\nu}|_H^{2}{\rm d}t\right)^{p/2}{\rm d}t\\ &\quad +C(p,T)\nu^{p/2}\Bigg(\int_0^{T}K(t)\,{\rm d}t\Bigg)^{p\over 2}\\ &\leq C(p,T)\nu^{p/2}\left(1+\sup_{0\leq t\leq T}{\mathbb E}|u_{\hbar_\nu}^{\nu}(t)|_{H}^{p}\right)+C(p,T)\nu^{p/2}\\ &\leq C(p,T){\nu}_0^{p/2}\left(1+{\mathbb E}|\varsigma|_{H}^{p}\right). \end{align*}

Let $p\in [4,\infty )$, $\alpha \in (0,\frac {1}{2})$. The Burkholder–Davis–Gundy and Hölder's inequalities, conditions 2.5 and proposition 3.1 imply that

\begin{align*} &{\mathbb E}\int_{0}^{T}\ \int_{0}^{T} \frac{|H_{5}(t)-H_{5}(s)|_{H}^{p}}{|t-s|^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad=\nu^{p/2}\int_{0}^{T}\int_{0}^{T}\frac{{\mathbb E}|\int_{s\wedge t}^{s\vee t}\int_{\mathbb{Y}}G(r,u_{\hbar_\nu}^{\nu}(r),y)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}r)|^{p}_{H}}{|t-s|^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C_p\nu^{p/2} \int_{0}^{T}\int_{0}^{T}{\mathbb E}\left| \int_{s\wedge t}^{s \vee t}|G(r,u_{\hbar_\nu}^{\nu}(r-),y)|^{2}_Hdr\right|^{p/2}{|t-s|^{-(1+\alpha p)}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C_p\nu^{p/2}\left(1+\sup_{0\leq t\leq T}{\mathbb E}|u_{\hbar_\nu}^{\nu}(t)|_{H}^{p} \right)\int_{0}^{T}\int_{0}^{T}|t-s|^{-(1+\alpha p)}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C(p,T){\nu}_0^{p/2}\left(1+{\mathbb E}|\varsigma|_{H}^{p}\right). \end{align*}

For the term $H_6$, by using conditions 2.5, 2.7, lemma 2.9 and proposition 3.1, we obtain

\begin{align*} &{\mathbb E}\int_{0}^{T} \int_{0}^{T} \frac{|H_{6}(t)-H_{6}(s)|_{H}^{p}}{|t-s|^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad=\int_{0}^{T}\int_{0}^{T}\frac{{\mathbb E}|\int_{s\wedge t}^{s\vee t}\int_{\mathbb{Y}}G(r,u_{\hbar_\nu}^{\nu}(r),y)(g_\nu(r,y)-1)\vartheta({\rm d}y)\,{\rm d}r|^{p}_{H}}{|t-s|^{1+\alpha p}}\,{\rm d}s\,{\rm d}t \\ &\quad\leq C_p\int_{0}^{T}\int_{0}^{T}{\mathbb E}\left|\int_{s\wedge t}^{s \vee t}\int_{\mathbb{Y}}|G(r,u_{\hbar_\nu}^{\nu}(r-),y)|_H|g_\nu(r,y)-1|\vartheta({\rm d}y)\,{\rm d}r\right|^{p}\\ &\qquad \times {|t-s|^{-(1+\alpha p)}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C_p\int_{0}^{T}\int_{0}^{T}{\mathbb E}\left|\int_{s\wedge t}^{s \vee t}\int_{\mathbb{Y}}|G(r,y)|_{0,H}|g_\nu(r,y)-1|(1+|u_{\hbar_\nu}^{\nu}|_H)\vartheta({\rm d}y)\,{\rm d}r\right|^{p}\\ &\qquad \times {|t-s|^{-(1+\alpha p)}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C_p\left(1+\sup_{0\leq t\leq T}{\mathbb E}|u_{\hbar_\nu}^{\nu}(t)|_{H}^{p} \right)\int_{0}^{T}\int_{0}^{T}|t-s|^{-(1+\alpha p)}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C_p\left(1+{\mathbb E}|\varsigma|_{H}^{p}\right). \end{align*}

Combining the estimates of $H_1-H_6$, we can infer that for $p\in [4,\infty )$, $\alpha \in (0,1/2)$, there is a constant $C(p,M,T)>0$ such that for any $\nu \in (0,{\nu }_{0}]$, it holds that

(6.2)\begin{equation} {\mathbb E}\big(\|u_{\hbar_\nu}^{\nu}\|_{W^{\alpha,2}(0,T; H)}^{2}\big)+{\mathbb E}\big( \|u_{\hbar_\nu}^{\nu}\|_{W^{\alpha,p}(0,T; V')}^{p}\big)\leq C(p,M,T). \end{equation}

Step 2: Using lemmas 4.3 and 4.4 in [Reference Zhai and Zhang60], we deduce $u_{\hbar _\nu }^{\nu }$ is tight in ${\mathcal {D}}([0,T];D(A^{-\beta }),\beta >1/2$. It follows from (3.6) and (6.2) that the process $\{u_{\hbar _\nu }^{\nu }\}_{\nu \in (0,{\nu }_0]}$ is bounded in probability in

\[ W^{\alpha,2}(0,T;H)\cap L^{2}(0,T;V)\cap W^{\alpha,p}(0,T;V'). \]

Using theorem 2.1 in [Reference Flandoli and Gatarek29] (see also [Reference Brzeźniak and Peszat12] and the references therein), we can infer that the space $W^{\alpha,2}(0,T; H)\cap L^{2}(0,T;V)$ ($W^{\alpha,2}(0,T; V^{\prime })\cap L^{2}(0,T;V$)) is compactly embedded in $L^{2}([0,T]; \mathcal {H})$ ($L^{2}([0,T]; H$)).

Therefore, for $\nu \in (0,{\nu }_0]$, using the Prokhorov theorem, we know that the distribution $\mathcal {L}((f_\nu,g_\nu ), u_{\hbar _\nu }^{\nu })$ of the process $((f_\nu,g_\nu ), u_{\hbar _\nu }^{\nu })$ is tight in

\[ {\mathcal{Z}}:=\tilde{\mathcal{U}}^{M}\times L^{2}([0,T];\mathcal{H})(\textrm{or }L^{2}([0,T]; H))\cap{\mathcal{D}}([0,T];D(A^{-\beta})). \]

Let $\{\nu _n\}_{n\geq 0}$ be a sequence in $(0,{\nu }_0]$ such that $\nu _n\to 0$. Thus, we can choose a subsequence, still denoted by $((f_{\nu _n},g_{\nu _n}), u_{\hbar _{\nu _n}}^{\nu _n})$, that converges in distribution to $((f,g),u)$ in ${\mathcal {Z}}$ as $n\to \infty$.

Step 3: Applying the Jakubowski–Skorohod Theorem in [Reference Jakubowski33], we infer that there exists a stochastic basis $(\tilde {\Omega }, \tilde {\mathcal {F}}, \{\tilde {\mathcal {F}}_{t}\}_{t\ge 0}, \tilde {{{\mathbb {P}}}})$ and on this basis, ${\mathcal {Z}}$-valued random variables $((\tilde {f},\tilde {g}), \tilde {u})$ and $\{((\tilde {f}_{\nu _n},\tilde {g}_{\nu _n}), \tilde {u}_{\hbar _{\nu _n}}^{\nu _n})\}_{n\geq 0}$ such that $((\tilde {f}_{\nu _n},\tilde {g}_{\nu _n}), \tilde {u}_{\hbar _{\nu _n}}^{\nu _n})$ and $((f_{\nu _n},g_{\nu _n}), u_{\hbar _{\nu _n}}^{\nu _n})$ have the same distribution on ${\mathcal {Z}}$, and as $n\to \infty$, $((\tilde {f}_{\nu _n},\tilde {g}_{\nu _n}), \tilde {u}_{\hbar _{\nu _n}}^{\nu _n})\to ((\tilde {f},\tilde {g}),\tilde {u})$ in ${\mathcal {Z}}$ $\tilde {\mathbb {P}}$-a.s. For simplicity, we drop the tilde and the index $n$. Denote $\varsigma :=u_{\hbar _\nu }^{\nu }(0)$. Notice that here the corresponding solution is the stochastic weak solution.

Applying (3.1), (3.6) and (5.1), for $\nu \in (0,\nu _0]$, $\alpha \in (0,1/2)$ and $q\in [2,\infty )$, we get

\begin{align*} &\tilde{\mathbb E}\Bigg(\sup_{0\leq t\leq T}|u_{\hbar_\nu}^{\nu}(t)|^{2}_{H}\Bigg) \leq C,\quad \tilde{\mathbb E}\left(\int_{0}^{T}\|u_{\hbar_\nu}^{\nu}(t)\|^{2} {\rm d}t\right) \leq C,\\ & \tilde{\mathbb E} \Big(\sup_{0\leq t\leq T}\|u_{\hbar_\nu}^{\nu}(t)\|^{q}_{H^{1,q}(D)}\Big)\leq C. \end{align*}

Then, there is a subsequence which converges weakly to $u$ in $L^{2}(\tilde {\Omega }\times (0,T); V)\cap L^{q}(\tilde {\Omega }\times (0,T); H^{1,q})$ as $n\to \infty$. Hence,

\[ u\in L^{2}(0,T; V)\cap L^{\infty}\big(0,T; H\cap H^{1,q}\big)\ \tilde{{{\mathbb{P}}}}-\textrm{a.s.} \]

Step 4: We want to check that $u$ is a weak solution of the following equations:

\begin{align*} {\rm d}u(t)+B(u(t),u(t))\,{\rm d}t & ={\Upsilon}_{0}(t, u(t))f(t)\,{\rm d}t\\ &\quad +\int_{\mathbb{Y}}G(t,u(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t, \quad u(0)= {\varsigma} . \end{align*}

Taking $\varphi \in D(A^{\beta })$ with $\beta >1/2$, then we have

\[ (u_{\hbar_\nu}^{\nu}(t)-\varsigma ,\varphi)+\int_0^{t}\ \big(B(u_{\hbar_\nu}^{\nu}(s),u_{\hbar_\nu}^{\nu}(s))-\Upsilon_{0}(s,u_{\hbar_\nu}^{\nu}(s))f(s), \varphi\big) {\rm d}s=\sum_{i=1}^{8} L_i, \]

where

\begin{align*} L_1&={-}\nu\int_{0}^{t}\left(Au_{\hbar_\nu}^{\nu}(s),\varphi\right)\,{\rm d}s , \quad L_2= \sqrt{\nu}\int_{0}^{t}\left(\Upsilon(s,u_{\hbar_\nu}^{\nu}(s)),\varphi\right)\,{\rm d}W(s),\\ L_3&=\nu\int_{0}^{t}\int_{\mathbb{ Y}}(G(s,u_{\hbar_\nu}^{\nu}(s-),y)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}s),\varphi),\\ L_4& ={-}\int_0^{t} \big[\left\langle B(u_{\hbar_\nu}^{\nu}(s)-u(s),u_{\hbar_\nu}^{\nu}(s)),\varphi\right\rangle +\left\langle B(u(s), u_{\hbar_\nu}^{\nu}(s)-u(s)),\varphi\right\rangle\big]{\rm d}s,\\ L_5&=\int_0^{t} \left(\big[\Upsilon_\nu(s,u_{\hbar_\nu}^{\nu}(s)-\Upsilon_0(s,u_{\hbar_\nu}^{\nu}(s))\big]f_{\nu}(s),\varphi\right)\,{\rm d}s,\\ L_6&=\int_0^{t} \left(\big[\Upsilon_{0}(s,{u}_{\hbar_\nu}^{\nu}(s))-\Upsilon_{0}(s,u(s))\big]f_{\nu}(s),\varphi\right)\,{\rm d}s,\\ L_7&=\int_0^{t}\left(\Upsilon_{0}(s,u(s))\left[f_{\nu}(s)-f(s)\right],\varphi\right)\,{\rm d}s,\\ L_8&=\int_{0}^{t}\int_{\mathbb{Y}}\left[(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g_\nu(s,y)-1),\varphi)\right.\\ &\quad \left.-\,(G(s,u(s),y)(g(s,y)-1),\varphi)\right]\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

For the term $L_1$, the fact that $D(A^{\beta })\subset V, \beta >1/2$, Cauchy–Schwarz's inequality and (3.6) imply that for $t\in [0,T]$ and $\nu \in (0,{\nu }_0]$

\begin{align*} \tilde{\mathbb E}|L_{1}|&\leq \nu \tilde{\mathbb E}\int_{0}^{t}\|u_{\hbar_\nu}^{\nu}(s)\|\|\varphi\|{\rm d}s \leq\nu\sqrt{t}\|\varphi\|\left(\tilde{\mathbb E}\int_{0}^{t}\|u_{\hbar_\nu}^{\nu}(s)\|^{2}{\rm d}s\right)^{1/2}\\ &\leq{\nu}C(T,M)\|\varphi\|\left(1+\tilde{\mathbb E}\|\varsigma\|^{4}\right)^{1/2}\to 0 \text{ as }\nu\to 0. \end{align*}

For the term $L_2$, by using the Burkholder–Davis–Gundy inequality, Cauchy– Schwarz's inequality, condition 2.5 and (3.1), we have

\begin{align*} \tilde{\mathbb E}|L_{2}|&\leq\sqrt{\nu}\tilde{\mathbb E}\left(\int_{0}^{t}|\Upsilon(s,u_{\hbar_\nu}^{\nu}(s))|^{2}_{L_{Q}}\|\varphi\|^{2}\right)^{1/2}\\ &\leq\sqrt{\nu}\|\varphi\|C(T,M)\left(1+\tilde{\mathbb E} |\varsigma|_{H}^{4}\right)^{1/2}\to 0 \text{ as } \nu\to 0. \end{align*}

For the term $L_3$, we use the Burkholder–Davis–Gundy inequality, Hölder's inequality and condition 2.5 to deduce

\begin{align*} \tilde{\mathbb E}|L_{3}|&\le \nu\tilde{\mathbb E}\left(\sup_{t\in[0,T]}\int_{0}^{t}\int_{\mathbb{ Y}}(G(s,u_{\hbar_\nu}^{\nu}(s-),y)\tilde{N}^{\nu^{{-}1}}({\rm d}y,{\rm d}s),\varphi)\right)\\ &\le \nu\tilde{\mathbb E}\left(\int_{0}^{T}\int_{\mathbb{ Y}}(G(s,u_{\hbar_\nu}^{\nu}(s-),y),\varphi)^{2}\vartheta({\rm d}y)\,{\rm d}t\right)^{1\over 2}\\ &\le \nu\|\varphi\|C(T,M)\left(1+\tilde{\mathbb E} |\varsigma|_{H}^{4}\right)^{1/2}\to 0 \text{ as } \nu\to 0. \end{align*}

For the term $L_4$, using (2.2), Hölder's inequality and (3.6), the fact that $u_{\hbar _\nu }^{\nu }-u\to 0$ in $L^{2}(0,T;\mathcal {H})$ $\tilde {{{\mathbb {P}}}}$-a.s. as $\nu \to 0$, (3.6), (4.2), the Vitali convergence theorem, we obtain

\begin{align*} \tilde{\mathbb E}|L_4|&\leq C\tilde{\mathbb E}\int_{0}^{t}\|u_{\hbar_\nu}^{\nu}(s)-u(s)\|_{\mathcal{H}}\left(\|u_{\hbar_\nu}^{\nu}(s)\|_{\mathcal{H}} +\|u(s)\|_{\mathcal{H}}\right)\|\varphi\|{\rm d}s\nonumber\\ &\leq C\|\varphi\|\left(\tilde{\mathbb E}\int_{0}^{t} \|u_{\hbar_\nu}^{\nu}(s)-u(s)\|^{2}_{\mathcal{H}} {\rm d}s \right)^{\!1/2} \left(\tilde{\mathbb E}\int_{0}^{t}\big[\|u_{\hbar_\nu}^{\nu}(s)\|^{2}_{V} +\|u(s)\|^{2}_{V}\big]{\rm d}s\right)^{\!1/2}\\ &\leq C(T,M)\|\varphi\| \left(1+\tilde{\mathbb E}\|\varsigma\|^{4}\right)^{1/2}\left(\tilde{\mathbb E} \int_{0}^{t}\|u_{\hbar_\nu}^{\nu}(s)-u(s)\|^{2}_{\mathcal{H}}{\rm d}s \right)^{1/2}\to 0. \end{align*}

For the term $L_5$, condition 6.1, Cauchy–Schwarz's inequality and (3.1) yield that

\begin{align*} \tilde{\mathbb E}|L_5|&\leq\tilde{\mathbb E}\int_{0}^{t}\big| {\Upsilon}_{\nu}(s,u_{\hbar_\nu}^{\nu}(s))-\Upsilon_0(s,u_{\hbar_\nu}^{\nu}(s))\big|_{L(H_0,H)} |f_\nu(s)|_0|\varphi|_{H}{\rm d}s\\ &\leq |\varphi|_{H}\sqrt{MT}\tilde{\mathbb E}\Big(\int_{0}^{t}\big| {\Upsilon}_{\nu}(s, u_{\hbar_\nu}^{\nu}(s))-\Upsilon_0(,u_{\hbar_\nu}^{\nu}(s))\big|^{2}_{L(H_0,H)}{\rm d}s\Big)^{1/2}\\ &\leq C_\nu\,|\varphi|_{H}\sqrt{MT}\tilde{\mathbb E}\left( \int_{0}^{t}\big[1+|u_{\hbar_\nu}^{\nu}(s)|_H^{2}\big]{\rm d}s\right)^{1/2}\\ &\leq C_\nu|\varphi|_{H}C(T,M)\left(1+\tilde{\mathbb E}|\varsigma|^{2}_{H}\right)^{1/2}\to 0 \text{ as } \nu\to 0. \end{align*}

For $\nu \in (0,\nu _0]$, by condition 2.5 and Hölder's inequality, we obtain

\begin{align*} \tilde{\mathbb E}|L_6|&\leq\tilde{\mathbb E}\int_{0}^{t}\left| {\Upsilon}_{0}(s,u_{\hbar_\nu}^{\nu})-\Upsilon_{0}(s,u(s))\right|_{L_Q(H_0,H)}|f_{\nu}(s)|_0 |\varphi|_{H}{\rm d}s\\ &\leq C|\varphi|_{H}\tilde{\mathbb E}\left(\int_{0}^{t}\sqrt{K(s)}|u_{\hbar_\nu}^{\nu}(s)-u(s)|_H|f_{\nu}(s)|_0 {\rm d}s\right). \end{align*}

For any $\varepsilon >0$, let $A_{\nu,\varepsilon }=\{t\in [0,T]: |u_{\hbar _\nu }^{\nu }(s)-u(s)|_H>\varepsilon \}$, then we have

\begin{align*} \lim_{\nu\to 0}\lambda_T(A_{\nu,\varepsilon})\le \lim_{\nu\to 0}\frac{\tilde{\mathbb E}\int_0^{T}|u_{\hbar_\nu}^{\nu}(t)-u(t)|_H^{2}{\rm d}t}{\varepsilon^{2}}=0. \end{align*}

Set $\tilde {M}=\sup _{\nu \in (0,\nu _0]}\tilde {\mathbb E}(\sup _{t\in [0,T]}|u_{\hbar _\nu }^{\nu }(t)|_H)\vee \tilde {\mathbb E}(\sup _{t\in [0,T]}|u(t)|_H)<\infty$. Then, one gets

\begin{align*} \tilde{\mathbb E}|L_{6}|&\le 2\tilde{M}|\varphi|_{H}\sqrt{\int_{A_{\nu,\varepsilon}}K(t)\,{\rm d}t}\sqrt{\int_0^{T}|f(t)|_0^{2}{\rm d}t}+\varepsilon|\varphi|_{H}\int_{A^{c}_{\nu,\varepsilon}}\sqrt{K(t)}|f(t)|_0{\rm d}t\\ &\le 2\tilde{M}|\varphi|_{H}\sqrt{\int_{A_{\nu,\varepsilon}}K(t)\,{\rm d}t}\sqrt{\int_0^{T}|f(t)|_0^{2}{\rm d}t}\\ &\quad +\varepsilon|\varphi|_{H}\left(\int_0^{T}K(t)\,{\rm d}t+\int_0^{T}|f(t)|_0^{2}{\rm d}t\right)\\ &\to 0 \text{ as } \nu\to 0. \end{align*}

For the term $L_7$, we have

\[ \tilde{\mathbb E}|L_7|=\tilde{\mathbb E}\left|\int_0^{t}\Big(f_{\nu}(s)-f(s),\Upsilon_{0}^{*}(s,u(s))\varphi\Big)\,{\rm d}s \right|. \]

It follows from condition 2.5 that

\[ \int_0^{T}|\Upsilon_{0}^{*}(s,u(s))\varphi|_0^{2}{\rm d}s\leq |\varphi|_{H}^{2}\int_0^{T}K(s)\big(1+|u(s)|_{H}^{2}\big)\,{\rm d}s\le C. \]

We note that $f_{\nu }-f\to 0$ in $L^{2}(0,T;H_0)$ for the weak topology $\tilde {{{\mathbb {P}}}}$-a.s. as $\nu \to 0$. Hence, $\int _0^{t}(f_{\nu }(s)-f(s),\Upsilon _{0}^{*}(s,u(s))\varphi ) {\rm d}s\to 0$ $\tilde {{{\mathbb {P}}}}$-a.s. as $\nu \to 0$. On the other hand, (4.2) implies that $\tilde {\mathbb E}(\int _0^{t}(f_{\nu }(s)-f(s),\Upsilon _{0}^{*}(s,u(s))\varphi ) {\rm d}s)^{2}\le C$. Applying the Vitali convergence theorem and (6), we can infer that $\tilde {\mathbb E}|L_7|\to 0$ as $\nu \to 0$.

Finally, by condition 2.6, lemma 2.9 and the fact that $u_{\hbar _\nu }^{\nu }\to u$ in $L^{2}(0,T;H)$ $\tilde {{{\mathbb {P}}}}$-a.s. as $\nu \to 0$, we get for fixed $\tilde {\omega }\in \tilde {\Omega }$

\begin{align*} &\int_0^{t}\int_{\mathbb{Y}}[(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g(s,y)-1),\varphi)\\ &\quad -(G(s,u(s),y)(g(s,y)-1),\varphi)]\vartheta({\rm d}y)\,{\rm d}s\to 0.\end{align*}

In fact, we have

\begin{align*} &\int_0^{t}\ \int_{\mathbb{Y}}|(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g(s,y)-1),\varphi)-(G(s,u(s),y)(g(s,y)-1),\varphi)|\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C\int_0^{t}\int_{\mathbb{Y}}\|G(s,y)\|_{1,H}|u_{\hbar_\nu}^{\nu}(s)-u(s)|_{H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C\tilde{M}\int_{A_{\nu,\varepsilon}}\int_{\mathbb{Y}}\|G(s,y)\|_{1,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\qquad +\varepsilon\int_{A^{c}_{\nu,\varepsilon}}\int_{\mathbb{Y}}\|G(s,y)\|_{1,H}|g(s,y)-1|\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C\tilde{M}\tilde{\eta}+C\varepsilon\to 0, \end{align*}

since $\tilde {\eta }$ and $\varepsilon$ are arbitrary. Moreover, condition 2.5, lemmas 2.9 and 2.10 yields that as $\nu \to 0$, it holds

\begin{align*} &\int_0^{t}\int_{\mathbb{Y}}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g_\nu(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s\\ &\quad \to\int_0^{t}\int_{\mathbb{Y}}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

For the completeness, we give the details of the proof. First, from lemma 2.10, we know that for every $\varepsilon >0$, there exists a compact set $\mathbb {K}\subset \mathbb {Y}$ so that

\begin{align*} &\int_0^{t}\int_{\mathbb{K}^{c}}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g_\cdot(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s\\ &\quad\le C\sup\int_{[0,T]\times \mathbb{K}^{c}}\|G(s,u_{\hbar_\nu}^{\nu}(s),y)\|_{0,H}|g_\cdot(s,y)-1)|\vartheta({\rm d}y)\,{\rm d}s\le\varepsilon. \end{align*}

Here $g_\cdot (s,y)=g_\nu (s,y)$ or $g(s,y)$. Then, to prove the above convergence, we need to prove that for every compact $\mathbb {K}\subset \mathbb {Y}$, it holds that

\begin{align*} &\int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g_\nu(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s\\ &\quad \to\int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Here $\mathbb {K}_T:=[0,T]\times \mathbb {K}$. If $G(s,\cdot,y)$ is bounded, then $\int _{\mathbb {Y}_T}(G(s,\cdot,y),\varphi )\vartheta ({\rm d}y){\rm d}s<\infty$, and we just need to prove that

\[ \int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)g_\nu(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s\to\int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)g(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s. \]

For this, we introduce $\theta (\cdot )={\vartheta (\cdot \cap [0,T]\times \mathbb {K})\over \vartheta ([0,T]\times \mathbb {K})}$ as appendix in [Reference Budhiraja, Chen and Dupuis13]. Hence, we know that $\theta$ is a probability measure on $[0,T]\times \mathbb {K}$. By Hölder's inequality and lemma 2.9, we have

\begin{align*} &\int_{\mathbb{K}_T}|(G(s,u_{\hbar_\nu}^{\nu}(s),y)-G(s,u(s),y),\varphi)|\theta({\rm d}y)\,{\rm d}s\\ &\quad\le C\frac{\vartheta(\mathbb{K})}{\vartheta([0,T]\times \mathbb{K})} \left(\int_{\mathbb{K}_T}\|G(s,u(s),y),\varphi)\|^{2}_{1,H}\vartheta({\rm d}y)\,{\rm d}s\right)^{1\over 2}\\ &\qquad \times \left(\int_0^{T}\|u_{\hbar_\nu}^{\nu}(s)-u(s)\|^{2}_{H}{\rm d}s\right)^{1\over 2}\to 0. \end{align*}

Then, we deduce that $(G(s,u_{\hbar _\nu }^{\nu }(s),y),\varphi )\to G(s,u(s),y),\varphi )$ $\theta$-a.s. Setting $\tilde {\vartheta }^{\nu }={\vartheta ^{g^{\nu }}(\cdot \cap \mathbb {K})\over \int _\mathbb {K} g_\nu \vartheta ({\rm d}y){\rm d}s }$ and $\tilde {\vartheta }={\vartheta ^{g}(\cdot \cap \mathbb {K})\over \int _\mathbb {K} g\vartheta ({\rm d}y){\rm d}s}$, we know that $\tilde {\vartheta }^{\nu }$ converges weakly to $\tilde {\vartheta }$. By lemma 2.8 in [Reference Boue and Dupuis9] and the fact that $\int _\mathbb {K} g_\nu \vartheta ({\rm d}y){\rm d}s\to \int _\mathbb {K} g\vartheta ({\rm d}y){\rm d}s$, we get

\[ \int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)g_\nu(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s\to\int_{\mathbb{K}_T}(G(s,u(s),y)g(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s, \]

and

\[ \int_{\mathbb{K}_T}(G(s,u_{\hbar_\nu}^{\nu}(s),y)g(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s\to\int_{\mathbb{K}_T}(G(s,u(s),y)g(s,y),\varphi)\vartheta({\rm d}y)\,{\rm d}s. \]

For general $G(s,\cdot,y)$, it is enough to get

\[ \tilde\Gamma=\sup\int_{\mathbb{K}_T}\|G(s,\cdot,y)\|_{0,H}1_{\{\|G(s,\cdot,y)\|_{0,H}>{\mathcal{M}}\}}g_\cdot(s,y)\vartheta({\rm d}y)\,{\rm d}s\to 0. \]

Since $ab\le e^{La}+{1\over L}l(b)$ for $a,b\in (0,\infty ), L\in [1,\infty )$, where $l(b)=b\ln b-b+1$ (see remark 3.3 in [Reference Budhiraja, Chen and Dupuis13]) and using the definition of $S^{M}$, we have

\begin{align*} |\Gamma|&\le \sup\int_{\mathbb{K}_T\cap\{\|G\|_{0,H}>{\mathcal{M}}\}}e^{L\|G(s,\cdot,y)\|_{0,H}}\vartheta({\rm d}y)\,{\rm d}s+{1\over L} \sup_{\nu}\int_{\mathbb{Y}_T}l(g_\cdot(s,y))\vartheta({\rm d}y)\,{\rm d}s\\ &\le \sup\int_{\mathbb{K}_T\cap\{\|G\|_{0,H}>{\mathcal{M}}\}}e^{L\|G(s,\cdot,y)\|_{0,H}}\vartheta({\rm d}y)\,{\rm d}s+{M\over L}. \end{align*}

For any $\varepsilon >0$, we can choose a sufficiently large $L$ such that ${M\over L}<\varepsilon$ and take ${\mathcal {M}}_0$ large enough so that $\int _{([0,T]\times \mathbb {K})\cap \{\|G\|_{0,H}>{\mathcal {M}}\}}e^{L\|G(s,\cdot,y)\|_{0,H}}\vartheta ({\rm d}y){\rm d}s\le \varepsilon$ for $\mathcal {M}>\mathcal {M}_0$ (see remark 2.8). Since $\varepsilon$ is arbitrary, we have $\tilde \Gamma \to 0$ as ${\mathcal {M}}\to \infty$. Hence, for fixed $\tilde {\omega }\in \tilde {\Omega }$, combining the above arguments, we arrive at

\begin{align*} &\int_0^{t}\int_{\mathbb{Y}}(G(s,u_{\hbar_\nu}^{\nu}(s),y)(g_\nu(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s\\ &\quad \to\int_0^{t}\int_{\mathbb{Y}}(G(s,u(s),y)(g(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

Therefore, as $\nu \to 0$, for any $t\in [0,T]$ and $\varphi \in D(A^{\beta })$ with $\beta >1/2$

\begin{align*} &\tilde{\mathbb E}\Bigg((u_{\hbar_\nu}^{\nu}(t),\varphi)-\int_0^{t} \big\langle-B(u(s),u(s))+\sigma_{0}(s,u(s))f(s), \varphi\big\rangle {\rm d}s\\ &\quad+\int_0^{t}\int_{\mathbb{Y}}(G(s,u(s),y)(g(s,y)-1),\varphi)\vartheta({\rm d}y)\,{\rm d}s\Bigg)\to 0. \end{align*}

On the other hand, taking $\varphi \in D(A^{\beta })$, we deduce that

\[ \sup_{t\in [0,T]}|(u_{\hbar_\nu}^{\nu}(t)-u(t),\varphi)|\to 0, \quad \tilde{{{\mathbb{P}}}} \text{ -a.s. as } \nu\to 0. \]

By (3.6), (4.2) and the Vitali convergence theorem, we can infer that

\[ \tilde{\mathbb E}\left(\sup_{t\in [0,T]}|(u_{\hbar_\nu}^{\nu}(t)-u(t), \varphi)|\right)\to 0 \text{ as }\nu\to 0 . \]

Then, $u$ is a solution of (4.1). By applying theorem 4.4, we obtain $u=u_\hbar ^{0}$. It follows from theorems 4.1 and 4.4 that $u_\hbar ^{0}$ belongs to $\mathcal {C}([0,T];H)\cap L^{\infty }(0,T; V\cap H^{1,q})$. Note that $\tilde {u}_{\hbar _{\nu _{n}}}^{\nu _{n}}$ and $u_{\hbar _{\nu _{n}}}^{\nu _{n}}$ has the same law. Hence, when $\nu _{n}\to 0$, we can chose a subsequence $\{\nu _{n_k}\}_{k\geq 0}$ such that $u_{\hbar _{\nu _{n_k}}}^{\nu _{n_k}}$ converges to $u_\hbar ^{0}$ in distribution in ${\mathcal {X}}$. Then, $u_{\hbar _\nu }^{\nu }$ converges in distribution to $u_\hbar ^{0}$. By applying the facts that $u_{\hbar _\nu }^{\nu }(s)={\mathcal {G}}^{\nu }_\xi (\sqrt {\nu }(W(s)+\frac {1}{\sqrt \nu }\int _0^{\cdot } f_\nu (s){\rm d}s),\nu N^{\nu ^{-1}g_\nu })$ and $u_h^{0}={\mathcal {G}}^{0}_\varsigma (\int _0^{\cdot } f(s){\rm d}s,\vartheta ^{g})$, we get the desired result. Therefore, we complete the proof.

Proof. Let $\{u_n\}_{n\geq 1}$ be a sequence corresponding to the solutions of (4.1) with control $\{(f_n,g_n)\}_{n\geq 1}$ in ${{\bar S}^{M}}$:

\begin{align*}& {\rm d}u_n(t)+B(u_n(t),u_n(t))\,{\rm d}t\\ &\quad =\tilde{\Upsilon}_{0}(t,u_n(t))f_n(t)\,{\rm d}t+\int_{\mathbb{Y}}\tilde G(t,u_n(t),y)(g_n(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t \end{align*}

with the initial data $u_n(0)=\varsigma$. First, for any $p,q>2$ and $\alpha <\frac {1}{2}$, we will prove that $u_n$ is bounded in $W^{1,2}(0,T;L^{q})\cap W^{\alpha,p}(0,T; L^{q})\cap L^{2}(0,T; H^{1,q})$. In fact, we can rewrite the above equality as $u_n(t)=\varsigma +W_1(t)+W_2(t)+W_3(t)$, where

\begin{align*} W_1(t)&={-}\int_0^{t}B(u_n(s),u_n(s))\,{\rm d}s, W_2(t)=\int_0^{t}\tilde{\Upsilon}_{0}(s,u_n(s))f_n(s)\,{\rm d}s,\\ W_3(t)&=\int_0^{t}\int_{\mathbb{Y}}\tilde G(s,u_n(s),y)(g_n(s,y)-1)\vartheta({\rm d}y)\,{\rm d}s. \end{align*}

For the term $W_1$, by using Hölder's inequality, (A.8), and (4.9), we have

\[ \|W_1\|_{W^{1,q}(0,T; L^{q})}^{q}\leq C(T)\sup_{t\in[0,T]} \|u_n(t)\|_{H^{1,q}}^{2q}\leq C(q,T,M)\left(1+\|\varsigma\|+\|\text{ curl }\varsigma\|_q\right)^{2q}. \]

For the term $W_2$, it follows from Minkowski's inequality, (A.1), (A.3), condition 2.6 and (4.2) that

\begin{align*} \nonumber\|W_2\|_{L^{q}}^{2}&\leq C(T,q)\Bigg(\int_0^{T}\| \tilde{\Upsilon}_{0}(t,u_n(t))f_n(t)\|_qdt\Bigg)^{2} \\ &\leq C(T,q)\Bigg(\int_0^{T}\|\tilde{\Upsilon}_{0}(t,u_n(t))f_n(t)\|{\rm d}t\Bigg)^{2}\\ &\leq C(T,q)\Bigg(\int_0^{T}|\text{ curl } \tilde{\Upsilon}_{0}(t,u_n(t))|_{L(H_0,H)}|f_n(t)|_0{\rm d}t\Bigg)^{2}\\ &\leq C(T,q)\left(1+\sup_{t\in [0,T]}\|u_n(t)\|^{2}\right)\int_0^{T}K(t)\,{\rm d}t\int_0^{T}|f(t)|_0^{2}{\rm d}t\\ &\leq C(T,q,M)\left(1+\|\varsigma\|^{2}\right). \end{align*}

Applying Minkowski's and Hölder's inequalities, the Sobolev embedding theorem, (A.3), conditions 2.6 and 2.12, (4.2) and (A.1), we obtain

\begin{align*} &\int_0^{T}\|W_2(t)\|_q^{p} {\rm d}t\\ &\quad \leq\int_0^{T}\Big| \int_0^{t}\|\tilde{\Upsilon}_{0}(s, u_n(s))f_n(s)\|_qds\Big|^{p}dt\\ &\quad \leq C(q)\int_0^{T}\Big|\int_0^{t}\|\tilde{\Upsilon}_{0}(s,u_n(s))f_n(s)\|{\rm d}s\Big|^{p}dt \\ &\quad \leq C(q)\sup_{t\in [0,T]}| \tilde{\Upsilon}_{0}(t,u_n(t))|_{L(H_0,V)}^{p}\int_0^{T}\Big|\int_0^{t}\Big(K(s)+|f_n(s)|_0^{2}\Big)\,{\rm d}s\Big|^{p}dt\\ &\quad \leq C(p,q,T,M) \left(1+\sup_{t\in [0,T]}\|u(t)\|^{p} \right)\\ &\quad \leq C(p,q,T,M)\left(1+\|\varsigma\|^{p}\right). \end{align*}

Moreover, for $\alpha \in (0,\frac {1}{2})$, we use conditions 2.6 and 2.12 to get

\begin{align*} &\int_0^{T}\int_0^{T} \frac{\|W_2(t)-W_2(s)\|_q^{p}}{(t-s)^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq 2 C(q)\int_0^{T}\int_0^{T}(t-s)^{{-}1-\alpha p} \Big|\int_s^{t}\|\tilde{\Upsilon}_{0}(r,u_n(r))f_n(r)\|{\rm d}r\Big|^{p}\,{\rm d}s\,{\rm d}t\\ &\quad\leq 2 C(q)C\left(1+\sup_{r\in[0,T]} \|u_n(r)\|^{p} \right) \int_0^{T}\int_0^{T}(t-s)^{{-}1-\alpha p}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C(q,T,M)\left(1+\|\varsigma\|^{p}\right). \end{align*}

For the term $W_3$, by using condition 2.6, 2.7 and lemma 2.9, one deduces that

\begin{align*} &\|W_3(t)\|^{2}_{L^{2}(0,T;L^{q})}\\ &\quad\leq C(T,q)\int_0^{T}\left\|\int_0^{T}\int_{\mathbb{Y}}\tilde G(t,u_n(t),y)(g_n(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t\right\|_q^{2}{\rm d}t\\ &\quad\leq C\Big(1+\sup_{t\in[0,T]}\|u_n(t)\|^{2}\Big)\int_0^{T}\left(\int_0^{T}\int_{\mathbb{Y}}\|\tilde G(t,y)\|_{0,q}(g_n(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t\right)^{2}{\rm d}t\\ &\quad\leq C(T,q,M)\left(1+\|\varsigma\|^{2}\right). \end{align*}

Then, for $\alpha \in (0,\frac {1}{2})$, we have

\begin{align*} &\int_0^{T}\int_0^{T} \frac{\|W_3(t)-W_3(s)\|_q^{p}}{(t-s)^{1+\alpha p}}\,{\rm d}s\,{\rm d}t\\ &\quad\leq 2 C(q)\int_0^{T}\int_0^{t}(t-s)^{{-}1-\alpha p} \\ &\qquad \times \Big|\int_s^{t}\int_{\mathbb{Y}}\|\tilde G(r,u_n(r),y)(g_n(r,y)-1)\|\vartheta({\rm d}y)\,{\rm d}r\Big|^{p}\,{\rm d}s\,{\rm d}t\\ &\quad\leq 2 C(q)C\left(1+\sup_{r\in[0,T]}\|u_n(r)\|^{p}\right) \int_0^{T}\int_0^{t}(t-s)^{{-}1-\alpha p}\,{\rm d}s\,{\rm d}t\\ &\quad\leq C(q,T)\left(1+\|\varsigma\|^{p}\right). \end{align*}

From the above estimates of $W_1-W_3$, we can infer that $\{u_n\}_{n\ge 1}$ is relatively compact in ${\mathcal {C}}([0,T];D(A^{-\beta }))\cap L^{2}(0,T;{\mathcal {H}})$ with $\beta >1/2$ by using Aubin-Lions Lemma (see also [Reference Bessaih and Millet8, Reference Lions38]). Then, there exists a subsequence, still denoted $\{u_n\}$, which converges to some element $u$ in ${\mathcal {C}}([0,T];D(A^{-\beta }))\cap L^{2}(0,T; {\mathcal {H}})$. Finally, it remains to check that $u$ is a solution to the following equations:

\begin{align*} {\rm d}u(t)+B(u(t),u(t))\,{\rm d}t&=\tilde{\Upsilon}_{0}(t,u(t))f(t)\,{\rm d}t\\ &\quad +\int_{\mathbb{Y}}\tilde G(t,u(t),y)(g(t,y)-1)\vartheta({\rm d}y)\,{\rm d}t,\quad u(0)=\varsigma. \end{align*}

Similar to the proof of step 4 in proposition 6.3, we can prove the desired result.

Proof of theorem 6.2. According to theorem 2.4, using propositions 6.3 and 6.4, we can get the desired result of theorem 6.2. Then, we complete the proof.