(v(s),\psi)_{{\cal O}}+(M_{{\alpha}}w_t(s),\delta)_{{\Omega}}+{{\varrho}}({\bar{u}}_t(s),\bar{\beta})_{{\Omega}}\\
 =
-\int_0^s\big[
 \nu E(v,\psi) +({\Delta}w, {\Delta}\delta)_{{\Omega}}+ a({\bar{u}}, \bar{\beta}) \\ +{{\gamma}}( M_{{\alpha}}w_t, \delta)_{{{\Omega}}}
   + q(\widehat{u}, \beta) - q(\widetilde{u}, \beta)  \big] d\tau  \label{weak_sol-dif}
        \end{gathered}$$ for all $s\in [0,T]$ and $\psi\in \widetilde{V}$ with $\psi\big\vert_{{\Omega}}=\beta=(\beta^1;\beta^2;\delta)$ and $\bar{\beta}=(\beta^1;\beta^2)$, where $\widetilde{V}$ is defined by .

One can see that $u(t)$ lies in $H^1_0({{\Omega}})\times H^1_0({{\Omega}})\times H^2_0({{\Omega}})$ for each $t\ge 0$ and $$\psi= i[v](s)\equiv \int_0^sv({{\sigma}})d{{\sigma}}\in {\widetilde{V}}~~\mbox{with}~~ \psi\big\vert_{{\Omega}}=\beta=u(s)
~~\mbox{for each $s\in [0,T]$,}$$ where we have introduced the notation $$ i[h](s)= \int_0^sh({{\sigma}})d{{\sigma}},~~ s\in [0,T],$$ for any integrable function $h(t)$ with values in some Banach space.

Substituting this $\psi$ in for the fixed $s\in [0,T]$ after integration from $0$ to $t$ over $s$ we obtain that $$\xi(t)+{{\varrho}}\|{\bar{u}}(t)\|_{{\Omega}}^2   \le2 \left| \int_0^t ds \int_0^s d\tau \left[q(\widehat{u}(\tau), u(s))- q(\widetilde{u}(\tau), u(s))\right] \right|. \label{app_dif_est}$$ where $$\begin{aligned}
\xi(t) =& \|i[v](t) \|_{{\cal O}}^2+ \|M^{1/2}_{{\alpha}}w(t)\|_{{\Omega}}^2 +2\nu \int_0^t ds E\left(i[v](s),
i[v](s))\right) \notag  \\
& +
2{{\gamma}}\int_0^t||M^{1/2}_{{\alpha}}w(s)||_{{\Omega}}^2 ds
+\|{\Delta}i[w](t) \|_{{\Omega}}^2 + a\left(i[{\bar{u}}](t), i[{\bar{u}}](t)\right).\notag $$

Now we estimate the integral term in . We first note that the term $q(\widehat{u}(\tau), u(s))$ has the structure $$q(\widehat{u}(\tau), u(s))=\sum_{j=1}^4q_j(\widehat{u}(\tau), u(s)).$$ Here

-   $q_1(\widehat{u}, u(s))$ is a linear combination of terms of the form $(\varkappa_{0} \widehat{w},w(s))_{{\Omega}}$ and $(\varkappa_{li} D_l\widehat{u}^i,w(s))_{{\Omega}}$, where $D_l={\partial}_{x_l}$ and $\varkappa_{0}$, $\varkappa_{li}$ are smooth function, $l,i=1,2$;

-   $q_2(\widehat{u}, u(s))$ is a linear 