s\le 1/2;\\
& H^s_*(D)= H^s_0(D),~~~ -1/2<  s<\infty,~~ s-1/2\not\in
\{ 0,1,2,\ldots\}.\end{aligned}$$ In particular, $H^s_*(D)= H^s_0(D)= H^s(D)$ for $|s|<1/2$. Note that in the notations of [@LiMa_1968] the space $H^{m+1/2}_*(D)$ is the same as $H^{m+1/2}_{00}(D)$ for every $m= 0,1,2,\ldots$ , and for $s=m+{{\sigma}}$ with $0<{{\sigma}}<1$ we have $$\|u \|^*_{s,D}=\left\{ \| u\|^2_{s,D}
+\sum_{|{{\alpha}}|=m}\int_D \,\frac{|D^{{\alpha}}u(x)|^2}{d(x,{\partial}D)^{2{{\sigma}}}}\, dx
\right\}^{1/2},$$ where $d(x,{\partial}D)$ is the distance between $x$ and ${\partial}D$. The norm $\|\cdot \|^*_{s,D}$ is equivalent to $\|\cdot \|_{s,D}$ in the case when $s>-1/2$ and $s-1/2\not\in\{ 0,1,2,\ldots\}$.

Understanding adjoint spaces [[with respect to ]{}]{}duality between $C_0^\infty(D)$ and $[C_0^\infty(D)]'$ by Theorems 4.8.1 and 4.8.2 from [@Triebel78] we also have that $$\begin{aligned}
 [H^s_*(D)]'= H^{-s}(D),~ s\in{{\mathbb R}}, ~~~\mbox{and} ~~~
 [H^s(D)]' =  H_*^{-s} (D),~ s\in (-\infty,1/2).\end{aligned}$$ Below we also use the factor-spaces $H^s(D)/{{\mathbb R}}$ with the naturally induced norm.

To describe fluid velocity fields we introduce the following spaces.

Let $\mathscr{C}({{\cal O}})$ be the class of $C^\infty$ vector-valued solenoidal (i.e., divergence-free) functions on $\overline{{{\cal O}}}$ which vanish in a neighborhood of $S$. We denote by $X$ the closure of ${\mathscr{C}}({{\cal O}})$ [[with respect to ]{}]{}the $L_2$-norm and by $V$ the closure [[with respect to ]{}]{}the $H^1({{\cal O}})$-norm. One can see that $$\label{X-space}
X=\left\{ v=(v^1;v^2;v^3)\in [L_2({{\cal O}})]^3\, :\; {\rm div}\, v=0,
\gamma_n v\equiv (v,n)=0~\mbox{on}~ 