Let $\Omega\subset \mathbb{R}^N$ wehre $N=2,3$ be a Lipschitz domain. Define $$H=\{u\in L^2(\Omega)^N:\ \operatorname{div}u=0\ \mbox{and}\ \ \gamma(u)=0 \}$$
$$V=\{u\in H_0^1(\Omega):\ \operatorname{div}u=0\}$$
where $\gamma$ is defined as in this answer. It is knows that $V\subset H\subset V^\star$ is a Hilbert triple.
The Stokes operator $A:D(A)\to H$ is defined by $$\langle Au,v\rangle_{V^\star,V}=\int\nabla u\cdot\nabla v$$
where $D(A)=\{u\in V:\ Au\in H\}$. It is well know that the Stokes operator (see here chapter IV) satisfies all hypothesis of Spectral Theorem for unbounded operators, hence, it possess a sequence of positive (because $A$ is positive) eigenvalues $\lambda_k$ which tends to $\infty$. Therefore it is possible to define $D(A^s)$ for all $s\geq 0$, in fact $$D(A^s)=\{u\in H:\ \sum\lambda_k^{2s}(u,w_k)^2_H<\infty\}$$
where $w_k$ are the eigenfunctions and $A^s u=\sum \lambda_k^s(u,w_k)w_k$. Define $V_{s}= D(A^{s/2})$ and we take on $V_s$ the norm $\|u\|=\|A^s u\|_2$. If $\Omega\in C^{1,1}$, then the authors of the last citation show in Proposition IV.5.10 that $V_s= V\cap H^s(\Omega)^N$ for $s\in [1,2]$.
On the other hand, in this book page 161, the authors define $V^s$ (for $s\in [1,2]$) as the closure of $\{u\in C_0^\infty (\Omega):\ \operatorname{div}(u)=0 \}$ in $H_0^1(\Omega)\cap H^s(\Omega)$ with the norm $\sqrt{\|\cdot\|^2_{H_0^1}+\|\cdot\|^2_{H^s}}$.
Now, I have two questions:
I - Is $V^s=V_s$?
II - If $I$ is false, can I have at least $V_s\subset H_0^1(\Omega)\cap H^s(\Omega)$ continuously embedded?
Thank you
Q-I: For a bounded domain with Lipschitz boundary, coincidence $V^s=V_s$ is an established fact. For an unbounded domain with an unbounded boundary, the question is rather complicated, no matter how smooth is its boundary — only its geometry is important. This was discovered in 1976 by John Heywood. Generally, a codimension of the subspace $V^s\subset V_s$ is not necessarily bounded. For further details, see Ch.III of G.P. Galdi's book An Introdiction to the Mathematical Theory of the Navier-Stokes Equations (2nd ed., 2011).
Q-II: Whether or not I be false, the apriori embeddings $V^s\subset V_s\subset H_0^1(\Omega)\cap H^s(\Omega)$ always stay continuous.