Appendix
Lemma A.1
For every \(t\ge 0,\) \(\Lambda _M(t),\) defined by (26), is demicontinuous, coercive, bounded and \(\delta -\)monotone.
Proof
Relations Eqs. (5) and (9) imply that there exists a constant \(\mathcal {C}>0\) such that
$$\begin{aligned}{} & {} \left( \varvec{\Gamma }_1^{-1}(t)\varvec{A}\varvec{\Gamma }_1(t)\varvec{y},\varvec{y}\right) \\{} & {} \quad =\left( \varvec{A}\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)\varvec{y}\right) =\nu _0\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\ge \frac{\nu _0}{C}\Vert \varvec{y}\Vert ^2=:\mathcal {C}\Vert \varvec{y}\Vert ^2,\ \forall \varvec{y}\in \varvec{V}, t\in [0,T].\end{aligned}$$
Next, we show that
$$\begin{aligned}{} & {} \left| (\varvec{\Gamma }_1^{-1}(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{y})-\varvec{\Gamma }_1^{-1}(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{z}),\varvec{y}-\varvec{z})\right| \nonumber \\{} & {} \quad \le \frac{1}{2}\mathcal {C}\Vert \varvec{y}-\varvec{z}\Vert ^2+C_M|\varvec{y}-\varvec{z}|^2,\end{aligned}$$
(41)
for all \(\varvec{y},\varvec{z}\in \varvec{V}.\) To this end, let \(\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ,\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert \le M\). Then, making use of Eq. (5), we have
$$\begin{aligned}\begin{aligned}&(\Gamma ^{-1}_1(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{y})-\Gamma ^{-1}_1(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{z}),\varvec{y}-\varvec{z})\\&\quad =(\varvec{B}(\varvec{\Gamma }_1(t)\varvec{y})-\varvec{B}(\varvec{\Gamma }_1(t)\varvec{z}),\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\\&\quad =b(\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))-b(\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\\&\quad =b(\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}),\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}{-}\varvec{z})){+}b(\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)(\varvec{y}{-}\varvec{z}),\varvec{\Gamma }_1(t)(\varvec{y}{-}\varvec{z}))\\&\quad =b(\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}),\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})),\end{aligned}\end{aligned}$$
in virtue of Eq. (12). Hence, taking into account relation Eq. (13), it yields that
$$\begin{aligned}\begin{aligned}&|(\Gamma ^{-1}_1(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{y})-\Gamma ^{-1}_1(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{z}),\varvec{y}-\varvec{z})| \\&\quad \le C|\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})|\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert \Vert \varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})\Vert \\&\text {(invoking relations (10), and (7) for }\chi (x)=|x|^2)\\&\quad \le \frac{1}{2}\mathcal {C}\Vert \varvec{y}-\varvec{z}\Vert ^2+C_M|\varvec{y}-\varvec{z}|^2. \end{aligned}\end{aligned}$$
Now consider the case where \(\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert>M, \Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert >M.\) We have
$$\begin{aligned}&(\varvec{\Gamma }_1^{-1}(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{y})-\varvec{\Gamma }_1^{-1}(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{z}),\varvec{y}-\varvec{z})\\&\quad =\frac{M^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2}\left[ b(\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))-b(\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\right] \\&\qquad +\left( \frac{M^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2}-\frac{M^2}{\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert ^2}\right) b(\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\\&\quad =\frac{M^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2}b(\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}),\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\\&\qquad +M^2\left( \frac{\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert ^2-\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert ^2}\right) b(\varvec{\Gamma }_1(t)\varvec{z},\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}))\\&\quad \le C_M\Vert \varvec{y}-\varvec{z}\Vert |\varvec{y}-\varvec{z}|\\&\qquad +\frac{CM^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert ^2} \left| \Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert ^2-\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\right| \\&\quad \Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert \Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert |\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})|^\frac{1}{2}\Vert \varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z}\Vert ^\frac{1}{2}\\&\quad \le C_M\Vert \varvec{y}-\varvec{z}\Vert |\varvec{y}-\varvec{z}|\\&\quad +\frac{CM^2}{\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert \Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert } (\Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert +\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert )\Vert \varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})\Vert ^\frac{3}{2}|\varvec{\Gamma }_1(t)(\varvec{y}-\varvec{z})|^\frac{1}{2}\\&\quad \le \frac{1}{2}\mathcal {C}\Vert \varvec{y}-\varvec{z}\Vert ^2_1+C_M|\varvec{y}-\varvec{z}|^2_2,\end{aligned}$$
where we have used Young’s inequality and relations Eqs. (10), (13) and (7).
Similarly one can also treat the last case \(\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert >M,\ \Vert \varvec{\Gamma }_1(t)\varvec{z}\Vert \le M\) (for details see pp. 255 in [20]).
Hence, we have by Eq. (10) that
$$\begin{aligned}\begin{aligned}(\Lambda _M(t)\varvec{y},\varvec{y})&=(\varvec{\Gamma }_1^{-1}(t)\varvec{A}\varvec{\Gamma }_1(t)\varvec{y},\varvec{y})+(\Gamma ^{-1}_1(t)\varvec{B}_M(\varvec{\Gamma }_1(t)\varvec{y}),\varvec{y})\\&\quad =\nu _0\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2+b(\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)\varvec{y},\varvec{\Gamma }_1(t)\varvec{y})\\&\quad =\nu _0\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\le \nu _0C\Vert \varvec{y}\Vert ^2,\ \forall \varvec{y}\in \varvec{V}.\end{aligned}\end{aligned}$$
Thus, \(\Lambda _M(t)\) is demi continuous and bounded. Also, we have by Eq. (9) that
$$\begin{aligned}(\Lambda _M(t)\varvec{y},\varvec{y})=\nu _0\Vert \varvec{\Gamma }_1(t)\varvec{y}\Vert ^2\ge \mathcal {C}\Vert \varvec{y}\Vert ^2,\ \forall \varvec{y}\in \varvec{V}.\end{aligned}$$
Hence, \(\Lambda _M\) is coercive. Moreover, in virtue of Eq. (41), we see that \(\Lambda _M(t)\) is \(\delta -\)monotone, that is,
$$\begin{aligned}(\Lambda _M(t)\varvec{y}-\Lambda _M(t)\varvec{z},\varvec{y}-\varvec{z})+\delta |\varvec{y}-\varvec{z}|^2\ge 0,\ \forall \varvec{y},\varvec{z}\in \varvec{V},\ \forall t\in [0,T],\end{aligned}$$
where \(\delta \) is such that \(\delta >C_M.\) \(\square \)
Lemma A.2
The solution \(\varvec{Y}_M\) to Eq. (29) satisfies, for M large enough,
$$\begin{aligned}\Vert \varvec{Y}_M(t)\Vert \le M\ \ \mathbb {P}-\text {a.s.},\ \forall t\in [0,Nh].\end{aligned}$$
Proof
We aim to apply the Itô’s formula to the \(\varvec{V}-\)norm in Eq. (29). To do this, we consider the Galerkin approximation of Eq. (29). More precisely, based on the eigenfunctions set of the Stokes operator, \(\left\{ \varvec{e}_j\right\} _j\), we define \(\varvec{H}_n=\text {span}\left\{ \varvec{e}_1,\varvec{e}_2,...,\varvec{e}_n\right\} \) and denote by \(\varvec{Y}_M^n\in \varvec{H}_n\) the solution to
$$\begin{aligned}{} & {} d{\varvec{Y}^n_M}(t)+[\varvec{A}^n{\varvec{Y}^n_M}(t)+\varvec{B}^n_M{\varvec{Y}^n_M}(t)]dt=\varvec{\Gamma }_1(t)F^n(\varphi )(t)dt\nonumber \\{} & {} \quad +\varvec{P}\varvec{L}^n_1{\varvec{Y}^n_M}(t)\circ dW_1(t),\ t>0,\end{aligned}$$
(42)
where \(\varvec{A}^n:\varvec{H}_n\rightarrow \varvec{H}_n,\ \varvec{B}^n:\varvec{V}\cap \varvec{H}_n \rightarrow \varvec{H}_n,\ F^n:\varvec{H}_n\rightarrow \varvec{H}_n,\ \varvec{L}^n_1:\varvec{V}\cap \varvec{H}_n\rightarrow \varvec{H}_n\) are defined respectively by
$$\begin{aligned}\begin{aligned}&\varvec{A}^n\varvec{u}=\sum _{i=1}^n(\varvec{A}\varvec{u},\varvec{e}_i)\varvec{e}_i,\ \varvec{B}^n(\varvec{u})=\sum _{i=1}^n(\varvec{B}(\varvec{u}),\varvec{e}_i)\varvec{e}_i,\\&\quad F^n(\varphi )=\sum _{i=1}^n(F(\varphi ),\varvec{e}_i)\varvec{e}_i,\ \varvec{P}\varvec{L}_1^n\varvec{u}=\sum _{i=1}^n(\varvec{P}\varvec{L}_1\varvec{u},\varvec{e}_i)\varvec{e}_i.\end{aligned}\end{aligned}$$
Itô’s formula in Eq. (42), yields
$$\begin{aligned} \begin{aligned}&d\ln (e+\Vert \varvec{Y}^n_{M}(t)\Vert ^2)\\&\quad =\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}d\Vert \varvec{Y}^n_M(t)\Vert ^2-\frac{1}{\nu _0}\frac{1}{(e+\Vert \varvec{Y}^n_M(t)\Vert ^2)^2}\left[ ( \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(t),\varvec{A}\varvec{Y}^n_M(t))\right] ^2dt\\&\quad \le C\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}\left[ -( \varvec{A}^n\varvec{Y}^n_M(t),\varvec{A}\varvec{Y}^n_M(t))-( \varvec{B}^n_M(\varvec{Y}^n_M(t)),\varvec{A}\varvec{Y}^n_M(t))\right. \ \\&\qquad \left. \ +(\varvec{\Gamma }_1(t)F^n(\varphi (t)),\varvec{A}\varvec{Y}^n_M(t))) + ((\varvec{P}\varvec{L}^n_1)^2\varvec{Y}^n_M(t), \varvec{A}\varvec{Y}^n_M(t)) \right. \\&\qquad \left. +( \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(t),\varvec{A}\varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(t))\right] dt \\&\qquad +C\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}( \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(t)dW_1(t),\varvec{A}\varvec{Y}^n_M(t))\end{aligned}\end{aligned}$$
(43)
Making use of relation Eq. (11) for the fourth term and the fifth term in the RHS of Eq. (43), we get
$$\begin{aligned} \begin{aligned}&d\ln (e+\Vert \varvec{Y}^n_{M}(t)\Vert ^2)\\&\quad \le C\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}\left[ -|\varvec{A}\varvec{Y}^n_M(t)|^2+|b^n(\varvec{Y}^n_M(t),\varvec{Y}^n_M(t),\varvec{A}\varvec{Y}^n_M(t))|\right. \\&\qquad \left. +|(\varvec{\Gamma }_1(t)F^n(\varphi (t)), \varvec{A}\varvec{Y}^n_M(t))|\right] dt\\&\qquad +\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}\left[ C_\varepsilon \Vert \varvec{Y}^n_M(t)\Vert ^2+\varepsilon |\varvec{A}\varvec{Y}^n_M(t)|^2\right] dt\\&\qquad +C\sum _{|\alpha |=1}\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}(D^\alpha \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(t)dW_1(t),D^\alpha \varvec{Y}^n_M(t)).\end{aligned}\end{aligned}$$
(44)
It follows, by integration over time, that
$$\begin{aligned} \begin{aligned}&\ln (e+\Vert \varvec{Y}^n_M(t)\Vert ^2)\le \ln (e+\Vert \varvec{Y}^n_0\Vert ^2)\\&\qquad +C\int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}\left[ (-1+\varepsilon )|\varvec{A}\varvec{Y}^n_M(s)|^2+|(\varvec{\Gamma }_1(s)F^n(\varphi (s)), \varvec{A}\varvec{Y}^n_M(s))|\right] ds\\&\qquad +C\int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|b^n(\varvec{Y}^n_M(s),\varvec{Y}^n_M(s),\varvec{A}\varvec{Y}^n_M(s))|ds\\&\qquad +C\sum _{|\alpha |=1}\int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}(D^\alpha \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(s),D^\alpha \varvec{Y}^n_M(s))dW_1(s).\end{aligned}\end{aligned}$$
(45)
In the following, we will focus on bounding the RHS of Eq. (45). To this end, taking advantage of Eq. (14), then using Eq. (31) and Young’s inequality, we deduce that
$$\begin{aligned}&\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}|b^n(\varvec{Y}^n_M(t),\varvec{Y}^n_M(t),\varvec{A}\varvec{Y}^n_M(t))|\\&\quad \le C\frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}|\varvec{Y}^n_M(t)|^\frac{1}{2}\Vert \varvec{Y}^n_M(t)\Vert |\varvec{A}\varvec{Y}^n_M(t)|^\frac{3}{2}\\&\quad \le C\frac{1}{\root 4 \of {e+\Vert \varvec{Y}^n_M(t)\Vert ^2}}\Vert \varvec{Y}^n_M(t)\Vert \frac{1}{\root 4 \of {(e+\Vert \varvec{Y}^n_M(t)\Vert ^2)^3}}|\varvec{A}\varvec{Y}^n_M(t)|^\frac{3}{2}\\&\quad \le C\Vert \varvec{Y}^n_M(t)\Vert ^\frac{1}{2}\frac{1}{\root 4 \of {(e+\Vert \varvec{Y}^n_M(t)\Vert ^2)^3}}|\varvec{A}\varvec{Y}^n_M(t)|^\frac{3}{2}\\&\quad \le C\Vert \varvec{Y}^n_M(t)\Vert ^2+\varepsilon \frac{1}{e+\Vert \varvec{Y}^n_M(t)\Vert ^2}|\varvec{A}\varvec{Y}^n_M(t)|^2.\end{aligned}$$
Hence, integrating over time and using Eq. (31), we get
$$\begin{aligned} \begin{aligned}&\int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|b^n(\varvec{Y}^n_M(s),\varvec{Y}^n_M(s),\varvec{A}\varvec{Y}^n_M(s))|ds\\&\quad \le C\int _0^t\Vert \varvec{Y}^n_M(s)\Vert ^2ds+\varepsilon \int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|\varvec{A}\varvec{Y}^n_M(s)|^2ds\\&\quad \le C(h)R^2+\varepsilon \int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|\varvec{A}\varvec{Y}^n_M(s)|_2^2ds.\end{aligned}\end{aligned}$$
(46)
Next, in virtue of Eq. (28) and Young’s inequality, we have
$$\begin{aligned}{} & {} \int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|(\varvec{\Gamma }_1(s)F(\varphi (s)),\varvec{A}\varvec{Y}^n_M(s))|ds\le C(h,R)\nonumber \\{} & {} \quad +\varepsilon \int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert ^2}|\varvec{A}\varvec{Y}^n_M(s)|_2^2ds.\end{aligned}$$
(47)
Furthermore, we let
$$\begin{aligned} Z(t)=\sum _{|\alpha |=1}\int _0^t\frac{1}{e+\Vert \varvec{Y}^n_M(s)\Vert _1^2}(D^\alpha \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(s),D^\alpha \varvec{Y}^n_M(s))dW_1(s).\end{aligned}$$
By the Burkholder-Davis-Gundy inequality we have
$$\begin{aligned}\mathbb {E}\left[ \sup _{s\in [0,t]}\left| \int _0^sdZ_r\right| \right] \le C\mathbb {E}\left[ \left<\int _0^\cdot dZ_r\right>_t^{1/2}\right] .\end{aligned}$$
Clearly seen,
$$\begin{aligned}\begin{aligned}|(D^\alpha \varvec{P}\varvec{L}^n_1\varvec{Y}^n_M(s), D^\alpha \varvec{Y}^n_M(s))|\le C\Vert \varvec{Y}^n_M(s)\Vert ^2.\end{aligned}\end{aligned}$$
Therefore, the quadratic variation
$$\begin{aligned}\left<\int _0^\cdot dZ_s\right>_t\le C\int _0^t\frac{\Vert \varvec{Y}^n_M(s)\Vert ^4}{(e+\Vert \varvec{Y}^n_M(s)\Vert ^2)^2}ds\le Ct.\end{aligned}$$
In consequence, the Burkholder-Davis-Gundy inequality gives
$$\begin{aligned} \mathbb {E}\left[ \sup _{s\in [0,t]}\left| \int _0^sdZ_r\right| \right] \le C\sqrt{t}.\end{aligned}$$
(48)
Now, we return to relation Eq. (45). We take the supreme over time, and apply the expectation, then invoke the relations Eqs. (46)–(48). It follows that, if \(\varepsilon \) is small enough, we have
$$\begin{aligned} \begin{aligned}\mathbb {E}\sup _{0\le r\le t}\ln (e+\Vert \varvec{Y}^n_M(r)\Vert ^2)\le C(h,R),\ \forall t\in [0,Nh].\end{aligned}\end{aligned}$$
(49)
Thus, in light of Eq. (49), we get
$$\begin{aligned}\mathbb {P}\left( \Vert \varvec{Y}_M^n(t)\Vert >M\right) \le \frac{\mathbb {E}[\ln (\Vert \varvec{Y}_M^n(t)\Vert ^2+e)]}{M^2+e}\le \frac{C(h,R)}{M^2+e}\rightarrow 0 \text { for }M\rightarrow \infty .\end{aligned}$$
It follows that, for M large enough, we have
$$\begin{aligned} \Vert \varvec{Y}^n_M(t)\Vert \le M\ \mathbb {P}-\text {a.s.}, \forall t\in [0,Nh].\end{aligned}$$
(50)
To conclude with the existence part, we prove the convergence of the Galerkin scheme. To this aim, let us once again apply Itô’s formula to \(\varvec{Y}_M-\varvec{Y}^n_M\), where \(\varvec{Y}_M\) solves Eq. (29) and \(\varvec{Y}_M^n\) solves Eq. (42), respectively. We get
$$\begin{aligned}\begin{aligned}&|\varvec{Y}_M(t)-\varvec{Y}_M^n(t)|^2+2\nu _0\int _0^t\Vert \varvec{Y}_M(s)-\varvec{Y}_M^n(s)\Vert ^2ds\\&\quad \le |\varvec{Y}_0-\varvec{Y}_0^n|^2+\int _0^t(\varvec{B}_M\varvec{Y}_M(s)-\varvec{B}_M^n\varvec{Y}_M^n(s),\varvec{Y}_M(s)-\varvec{Y}_M^n(s))ds\\&\qquad +\int _0^t(\varvec{\Gamma }_1F(\varphi )-\varvec{\Gamma }_1F^n(\varphi ),\varvec{Y}_M(s)-\varvec{Y}_M^n(s))ds\\&\qquad +\int _0^t(\varvec{P}\varvec{L}_1\varvec{Y}_M(s)-\varvec{P}\varvec{L}^n_1\varvec{Y}_M^n(s),\varvec{Y}_M(s)-\varvec{Y}_M^n(s))dW_1(s)\end{aligned}\end{aligned}$$
Using similar ideas as in Eq. (41) for the nonlinear term \(\varvec{B}_M\) in the RHS, the above leads to
$$\begin{aligned}\begin{aligned}&|\varvec{Y}_M(t)-\varvec{Y}_M^n(t)|^2+C\int _0^t\Vert \varvec{Y}_M(s)-\varvec{Y}_M^n(s)\Vert ^2ds\\&\quad \le |\varvec{Y}_0-\varvec{Y}_0^n|_2^2+\int _0^t\Vert F(\varphi )-F^n(\varphi )\Vert _{\varvec{V}'}^2ds\\&\quad \ \ +\int _0^t(\varvec{P}\varvec{L}_1\varvec{Y}_M(s)-\varvec{P}\varvec{L}^n_1\varvec{Y}_M^n(s),\varvec{Y}_M(s)-\varvec{Y}_M^n(s))dW_1(s).\end{aligned}\end{aligned}$$
Consequently, taking the supreme and the expectation, we get
$$\begin{aligned}\begin{aligned}&\mathbb {E}\sup _{0\le t\le T}|\varvec{Y}_M(t)-\varvec{Y}_M^n(t)|^2+C\int _0^t\mathbb {E}\sup _{0\le s\le t}\Vert \varvec{Y}_M(s)-\varvec{Y}_M^n(s)\Vert ^2ds\\&\quad \le |\varvec{Y}_0-\varvec{Y}_0^n|^2+C\int _0^t\mathbb {E}\Vert F(\varphi )-F^n(\varphi )\Vert _{\varvec{V}'}^2ds.\end{aligned}\end{aligned}$$
Since \(\varvec{Y}_0^n\rightarrow \varvec{Y}_0\) in \(\varvec{H}\) and \(F^n(\varphi )\rightarrow F(\varphi )\) in \(L^2(0,T;\varvec{V}')\), the above implies that \(\varvec{Y}_M^n\rightarrow \varvec{Y}_M\) in \(C([0,T];\varvec{H})\cap L^2(0,T;\varvec{V})\). The conclusion follows immediately. \(\square \)
Lemma A.3
For all \(n\in \left\{ 0,1,...,N-1\right\} \), there exist positive constants, depending only on h (and independent on n), \(C_1,C_2(h)-C_5(h),\) such that
$$\begin{aligned} \begin{aligned}\Vert \mathcal {G}_n(\varphi )\Vert _{\mathcal {X}_n}\le C_1(n+1)\sum _{\alpha \in \mathbb {N}^2,|\alpha |=1,2}|D^\alpha \varphi _{nh}|_2+C_2(h)R^2+C_3(h)R^3,\ \forall \varphi \in B_{R,n}, \end{aligned}\nonumber \\\end{aligned}$$
(51)
and
$$\begin{aligned} \Vert \mathcal {G}_n\varphi _1-\mathcal {G}_n\varphi _2\Vert _{\mathcal {X}_n}\le (C_{4}(h)R+C_{5}(h)R^2)\Vert \varphi _1-\varphi _2\Vert _{\mathcal {X}_n},\ \forall \varphi _1,\varphi _2\in B_{R,n}.\end{aligned}$$
(52)
Proof
We estimate \(\mathcal {G}_n\varphi \) in the norm of \(\mathcal {X}_n\). To this end, let \(\alpha \in \mathbb {N}^2\) be such that \(|\alpha |=1\). By using Eq. (17) with \(p=q=2\) and relations Eq. (7), for \(\chi (x)=|x|^2\), and Eq. (8), we see that
$$\begin{aligned} \begin{aligned}&\left| D^\alpha e^{(t-s)S}\Gamma _2^{-1}(s)[\varvec{\Gamma }_1(s)\varvec{y}(s)\cdot \nabla (\Gamma _2(s)\varphi (s))]\right| _2 \\&\quad \le c(t-s)^{-\frac{1}{4}}\left| \Gamma _2^{-1}[\varvec{\Gamma }_1(s)\varvec{y}(s)\cdot \nabla (\Gamma _2(s)\varphi (s))]\right| _2\\&\quad =c(t-s)^{-\frac{1}{4}}\left| \varvec{\Gamma }_1(s)\varvec{y}(s)\cdot \nabla (\Gamma _2(s)\varphi (s))\right| _2\\&\quad \le c(t-s)^{-\frac{1}{4}}|\varvec{\Gamma }_1(s)\varvec{y}(s)||\nabla \Gamma _2(s)\varphi (s)|_2\\&\quad \le c(t-s)^{-\frac{1}{4}}|\varvec{y}(s)||\nabla \varphi (s)|_2\\&\quad \le C(h)R^2(t-s)^{-\frac{1}{4}}(n+1)^{-1}(s-nh)^{-\frac{1}{8}}\Vert \varphi \Vert _{\mathcal {X}_n},\ \forall nh\le s\le t\le (n+1)h,\end{aligned}\end{aligned}$$
(53)
by invoking the hypothesis on the \(\varvec{H}\)-norm of \(\varvec{y}\) and relation Eq. (34). It yields that if \(\varphi \in B_{R,n}\) then
$$\begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Gamma _2^{-1}(s)[\varvec{\Gamma }_1(s)\varvec{y}(s)\cdot \nabla (\Gamma _2(s)\varphi (s))]\right| _2ds\nonumber \\&\quad \le C(h)R^3(n+1)^{-1}\int _{nh}^t (t-s)^{-\frac{1}{4}}(s-nh)^{-\frac{1}{8}}ds\nonumber \\&\quad =C(h)R^3(n+1)^{-1} \int _{0}^{t-nh} (t-nh-s)^{-\frac{1}{4}}s^{-\frac{1}{8}}ds\nonumber \\&\quad =C(h)R^3(n+1)^{-1}\int _{0}^{t-nh} (t-nh-s)^{-\frac{1}{4}}s^{\frac{3}{4}}s^{-\frac{3}{4}}s^{-\frac{1}{8}}ds\nonumber \\&\quad \le C(h)h^{\frac{3}{4}}R^3(n+1)^{-1}\int _{0}^{t-nh} (t-nh-s)^{-\frac{1}{4}}s^{-\frac{7}{8}}ds\nonumber \\&\quad =C(h)h^{\frac{3}{4}}B\left( \frac{1}{8},\frac{3}{4}\right) R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}}\nonumber \\&\quad =c_1(h)R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ t\in [nh,(n+1)h], \end{aligned}$$
(54)
where B(a, b) is the classical beta function.
We go on with
$$\begin{aligned} \begin{aligned}&\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^3\right| _2\\&\quad \le \sum _{\beta \in \mathbb {N}^2,|\beta |=1}\left| D^\alpha e^{(t-s)S}D^\beta D^\beta (\varphi (s)-M(\varphi (s)))^3\right| _2\\&\quad =\sum _{\beta \in \mathbb {N}^2,|\beta |=1}\left| D^\alpha D^\beta e^{(t-s)S} D^\beta (\varphi (s)-M(\varphi (s)))^3\right| _2\\&\quad =3\sum _{\beta \in \mathbb {N}^2,|\beta |=1}\left| D^{\alpha +\beta } e^{(t-s)S} (\varphi (s)-M(\varphi (s)))^2D^\beta (\varphi (s)-M(\varphi (s)))\right| _2.\end{aligned}\end{aligned}$$
(55)
Involving relation Eq. (17) for \(q=\frac{6}{5},p=2, |\alpha +\beta |=2\), then using the generalized Hölder inequality for \(\frac{5}{6}=\frac{1}{3}+\frac{1}{2}\), we deduce that
$$\begin{aligned} \begin{aligned}&\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^3\right| _2\\&\quad \le 3c (t-s)^{-\frac{2}{3}}\sum _{\beta \in \mathbb {N}^2,|\beta |=1}|(\varphi (s)-M(\varphi (s)))^2D^\beta \varphi (s)|_\frac{6}{5}\\&\quad \le 3c(t-s)^{-\frac{2}{3}}\sum _{\beta \in \mathbb {N}^2,|\beta |=1}|(\varphi (s)-M(\varphi (s)))^2|_3|D^\beta \varphi (s)|_2\\&\quad \le C(t-s)^{-\frac{2}{3}}\sum _{\beta \in \mathbb {N}^2,|\beta |=1}|\varphi (s)-M(\varphi (s))|_6^2|D^\beta \varphi (s)|_2\\&\quad \text { (using the Sobolev embedding }H^1\subset L^6)\\&\quad \le C(t-s)^{-\frac{2}{3}}\sum _{\beta \in \mathbb {N}^2,|\beta |=1}\Vert \varphi (s)-M(\varphi (s))\Vert ^2_1|D^\beta \varphi (s)|_2\\&\quad \le C(t-s)^{-\frac{2}{3}}|\nabla (\varphi (s)-M(\varphi (s)))|_2^2|\nabla \varphi (s)|_2. \end{aligned}\end{aligned}$$
(56)
For the last inequality, we have used that, in virtue of the Poincaré inequality Eq. (15), the \(H^1\)-norm of \(f-M(f)\) is equivalent with the \(|\nabla \cdot |_2\)- norm of \(f-M(f)\). Therefore, if \(\varphi \in B_{R,n}\), we have
$$\begin{aligned} \begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^3\right| _2ds\\&\quad \le C\int _{nh}^t(t-s)^{-\frac{2}{3}}|\nabla \varphi (s)|_2^3ds\\&\quad =C\int _{nh}^t(t-s)^{-\frac{3}{4}}(t-s)^\frac{1}{12}|\nabla \varphi (s)|_2^3ds\\&\quad \le C(n+1)^{-3}R^3h^\frac{1}{12}\int _{nh}^t(t-s)^{-\frac{3}{4}}(s-nh)^{-\frac{3}{8}}ds\\&\quad \le C(h)B\left( \frac{5}{8},\frac{1}{4}\right) R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}}\\&\quad =c_2(h)R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ nh\le t\le (n+1)h.\end{aligned}\end{aligned}$$
(57)
Obviously, with similar arguments as above, we also have
$$\begin{aligned} \begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^2\right| _2ds\\&\quad \le C(h)(n+1)^{-2}R^2\int _{nh}^t(t-s)^{-\frac{3}{4}}(s-nh)^{-\frac{2}{8}}ds\\&\quad \le C(h)(n+1)^{-1}R^2\int _{nh}^t(t-s)^{-\frac{3}{4}}(s-nh)^\frac{1}{8}(s-nh)^{-\frac{3}{8}}ds\\&\quad \le C(h)h^\frac{1}{8}R^2B\left( \frac{5}{8},\frac{1}{4}\right) (n+1)^{-1}(t-nh)^{-\frac{1}{8}}\\&\quad =c_3(h)R^2(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ nh\le t\le (n+1)h.\end{aligned}\end{aligned}$$
(58)
We move on to the second order derivatives. Let \(\alpha \in \mathbb {N}^2\) be such that \(|\alpha |=2\). We have by Eq. (17) for \(p=q=2,\ |\alpha |=2,\) that
$$\begin{aligned} \begin{aligned}&\left| D^\alpha e^{(t-s)S}\Gamma _2^{-1}(s)[\varvec{\Gamma }_1(s)y(s)\cdot \nabla (\Gamma _2(s)\varphi (s))]\right| _2\\&\quad \le c(t-s)^{-\frac{1}{2}}|y(s)||\nabla \varphi (s)|_2\\&\quad \le C(h)R^2(t-s)^{-\frac{1}{2}}(n+1)^{-1}(s-nh)^{-\frac{1}{8}}\Vert \varphi \Vert _{\mathcal {X}_n}\end{aligned}\end{aligned}$$
(59)
Thus, for \(\varphi \in B_{R,n}\), we have
$$\begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Gamma _2^{-1}(s)[\varvec{\Gamma }_1(s)y(s)\cdot \nabla (\Gamma _2(s)\varphi (s))]\right| _2\nonumber \\&\quad \le C(h)R^3(n+1)^{-1}\int _{nh}^t(t-s)^{-\frac{1}{2}}(s-nh)^{-\frac{1}{8}}ds\nonumber \\&\quad =C(h)R^3(n+1)^{-1}\int _{nh}^t(t-s)^{-\frac{1}{2}}(s-nh)^{\frac{1}{2}}(s-nh)^{-\frac{1}{2}}(s-nh)^{-\frac{1}{8}}ds\nonumber \\&\quad \le C(h)h^\frac{1}{2}R^3(n+1)^{-1}\int _{nh}^t(t-nh-s)^{-\frac{1}{2}}s^{-\frac{5}{8}}ds\nonumber \\&\quad =Ch^\frac{1}{2}R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}}B\left( \frac{3}{8},\frac{1}{2}\right) \nonumber \\&\quad =c_4(h)R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ nh\le t\le (n+1)h. \end{aligned}$$
(60)
For the cubic term we use Eq. (17) with \(q=\frac{6}{5},p=2, |\alpha |=2\), then, as before, the generalized Hölder inequality for \(\frac{5}{6}=\frac{1}{3}+\frac{1}{2}\). We deduce that
$$\begin{aligned} \begin{aligned}&\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^3\right| _2\\&\quad \le 6c(t-s)^{-\frac{2}{3}}\left\{ \left| ((\varphi (s)-M(\varphi (s)))^2\Delta \varphi (s)\right| _\frac{6}{5}\right. \ \\&\quad \left. \ +\left| (\varphi (s)-M(\varphi (s)))|\nabla \varphi (s)|^2|\right| _\frac{6}{5}\right\} \\&\quad \le C(t-s)^{-\frac{2}{3}}[|\varphi (s)-M(\varphi (s))|_6^2|\Delta \varphi (s)|_2+|\nabla \varphi (s)|^2_6|\varphi (s)-M(\varphi (s))|_2]\\&\quad \text {(making use of the embedding }H^1\subset L^6\hbox { and } \hbox {(15))}\\&\quad \le C(t-s)^{-\frac{2}{3}}[|\nabla \varphi (s)|_2^2|\Delta \varphi (s)|_2+\Vert \nabla \varphi (s)\Vert _1^2|\nabla \varphi (s)|_2]\\&\quad \le C(t-s)^{-\frac{2}{3}}(n+1)^{-3}(s-nh)^{-\frac{3}{8}}\Vert \varphi \Vert ^3_{\mathcal {X}_n}\\&\quad =C(t-s)^{-\frac{3}{4}}(t-s)^\frac{1}{12}(n+1)^{-3}(s-nh)^{-\frac{3}{8}}\Vert \varphi \Vert ^3_{\mathcal {X}_n}\\&\quad \le C(h)(t-s)^{-\frac{3}{4}}(n+1)^{-3}(s-nh)^{-\frac{3}{8}}\Vert \varphi \Vert ^3_{\mathcal {X}_n},\ nh\le s\le t\le (n+1)h.\end{aligned}\nonumber \\\end{aligned}$$
(61)
So, \(\varphi \in B_{R,n}\) implies that
$$\begin{aligned} \begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^3\right| _2ds\\&\quad \le C(h)B\left( \frac{5}{8},\frac{1}{4}\right) R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}}\\&\quad =c_5(h)R^3(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ nh\le t\le (n+1)h.\end{aligned}\end{aligned}$$
(62)
Similarly, with tricks as above, for the square term we have
$$\begin{aligned} \begin{aligned}&\int _{nh}^t\left| D^\alpha e^{(t-s)S}\Delta (\varphi (s)-M(\varphi (s)))^2\right| _2ds\\&\quad \le c_6(h)R^2(n+1)^{-1}(t-nh)^{-\frac{1}{8}},\ nh\le t\le (n+1)h.\end{aligned}\end{aligned}$$
(63)
Next, we deal with the term \(e^{(t-nh)S}\varphi _{nh}.\) Let \(\alpha \in \mathbb {N}^2\) with \(|\alpha |=1,2\) We have, by using Eq. (17) for \(q=\frac{4}{3}\) and \(p=2\), that
$$\begin{aligned}\left| D^\alpha e^{(t-nh)S}\varphi _{nh}\right| _2\le c (t-nh)^{-\frac{1}{8}}|D^\alpha \varphi _{nh}|_{\frac{4}{3}}\le c\left( 2,\frac{4}{3}\right) C_{2,\frac{4}{3}}|D^\alpha \varphi |_2.\end{aligned}$$
Here, \(C_{2,\frac{4}{3}}\) is the constant from the continuous embedding \(L^2\subset L^\frac{4}{3}.\) Therefore, for
$$\begin{aligned}C_1=c\left( 2,\frac{3}{2}\right) C_{2,\frac{3}{2}}\end{aligned}$$
we have
$$\begin{aligned} |D^\alpha e^{(t-nh) S}\varphi _{nh}|_2\le (n+1)C_1(n+1)^{-1}(t-nh)^{-\frac{1}{8}}|D^\alpha \varphi _{nh}|_{2},\ nh\le t\le (n+1)h.\nonumber \\\end{aligned}$$
(64)
Gathering together the relations Eqs. (54)–(64), we deduce that, there exist positive constants, depending only on h (and independent on n), \(C_1,C_2(h),C_3(h)\) such that
$$\begin{aligned} \begin{aligned}\Vert \mathcal {G}_n(\varphi )\Vert _{\mathcal {X}_n}\le C_1(n+1)\sum _{\alpha \in \mathbb {N}^2,|\alpha |=1,2}|D^\alpha \varphi _{nh}|_2+C_2(h)R^2+C_3(h)R^3,\ \forall \varphi \in B_{R,n}, \end{aligned}\nonumber \\\end{aligned}$$
(65)
\(n\in \left\{ 0,1,...,N-1\right\} .\)
In a similar manner as above, one may also prove that there exist constants \(C_{4}(h)>0,C_{5}(h)>0\), depending only on h, such that
$$\begin{aligned}{} & {} \Vert \mathcal {G}_n\varphi _1-\mathcal {G}_n\varphi _2\Vert _{\mathcal {X}_n}\le (C_{4}(h)R+C_{5}(h)R^2)\Vert \varphi _1-\varphi _2\Vert _{\mathcal {X}_n},\nonumber \\{} & {} \quad \forall \varphi _1,\varphi _2\in B_{R,n},n\in \left\{ 0,1,...,N-1\right\} .\end{aligned}$$
(66)
Thereby completing the proof. \(\square \)