w=0 ~~ {\rm on}~S;
    \quad
    w=\psi ~~{\rm on} ~ {{\Omega}},
    \end{array}\right.$$ for $\psi\in [L^2({{\Omega}})]^2 \times{\widehat L_2({{\Omega}})}$ ($N_0\psi$ solves with $g\equiv 0$). It follows from (\[stokes-bnd1\]) and (\[stokes-bnd2\]) that $$N_0 :\, [H^s_*({{\Omega}})]^2\times \widehat{H}^s_*({{\Omega}})\mapsto [H^{1/2+s}({{\cal O}})]^3\cap X
    ~~continuously$$ for every $-1/2\le s\le3/2$.

We just combine proof of Proposition 2.1 [@Chu_2010] and Proposition 2.2 [@ChuRyz2011]. The argument in these references rely on the consideration in [@lad-NSbook; @temam-NS] and also in [@GSS2005].

Well-Posedness Theorem {#sec:WP}
======================

To define weak (variational) solutions to - we need the following class ${{\cal L}}_T$ of test functions $\phi$ on ${{\cal O}}$: $${{\cal L}}_T=\left\{\phi \left|\begin{array}{l}
\phi\in L_2(0,T; \left[H^1({{\cal O}})\right]^3),\; \phi_t\in L_2(0,T;  [L_2({{\cal O}})]^3),  \\
{\rm div}\phi=0,\; \phi|_S=0,\; \phi|_{{\Omega}}=b=(b^1;b^2;d),  \\
d\in  L_2(0,T; {\widehat H^2_{0}({{\Omega}})}),\; b^j \in L_2(0,T; H^1_0({{\Omega}})), \; j=1,2,\; \\
d_t\in  L_2(0,T; \widehat{H}^1_0({{\Omega}})), \; b^j_t \in L_2(0,T; L_2({{\Omega}})), \; j=1,2.
\end{array}\right.\right\}.$$ We also denote ${{\cal L}}_T^0=\{\phi\in {{\cal L}}_T\, :\, \phi(T)=0\}$.

\[de:solution\]

A pair of vector functions $(v(t);u(t))$ with $v=(v^1;v^2;v^3)$ and $u=(u^1;u^2;w)$ is said to be a weak solution to the problem in – on a time interval $[0,T]$ if

-   $v\in L_\infty(0,T;X)\bigcap L_2(0,T; V)$;

-   $u \in L_\infty(0,T; H^1_0({{\Omega}})\times H^1_0({{\Omega}})\times H^2_0({{\Omega}}))$;

-   $u_t \in L_2(0,T; \big[H^{1/2}_*({{\Omega}})\big]^3)$ and the compatibility condition $v(t)|_{{\Omega}}=(u^1_t;u^2_t;w_t)(t)$ holds for almost all $t\in [0,T]$;

-   $({{\varrho}}u^1_t; 