Well-posedness for the Cahn-Hilliard-Navier-Stokes Equations Perturbed by Gradient-Type Noise, in Two Dimensions

Abstract

In this work, we study the problem of existence and uniqueness of solutions of the stochastic Cahn-Hilliard-Navier-Stokes system with gradient-type noise. We show that such kind of noise is related to the problem of modelling turbulence. We apply a rescaling argument to transform the stochastic system into a random deterministic one. We split the latter into two parts: the Navier-Stokes part and the Cahn-Hilliard part, respectively. The rescale operators possess good properties which allow to show that the rescaled Navier-Stokes equations have a unique solution, by appealing to $\delta -$monotone operators theory. While, well-posedness of the Cahn-Hilliard part is proved via a fixed point argument. Then, again a fixed point argument is used to prove global in time existence of a unique solution to the initial system. All the results are under the requirement that the initial data is in a certain small neighbourhood of the origin.

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. $\square$