It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in L_{\sigma }^{q}(\Omega ),$ $Q\in L_{loc}^{q}(\Omega ),$ $% \nabla Q\in \left( L^{q}(\Omega )\right) ^{n}$ and
\begin{eqnarray*} \ L_{\sigma }^{q}(\Omega ):=\overline{\left\{ v\in C_{0}^{\infty }(\Omega )^{n}:\nabla\cdot v=0\right\} }^{L^{q}(\Omega )^{n}}. \ \ \end{eqnarray*} We define the natural projection $P_{q}:L^{q}\rightarrow L_{\sigma }^{q}$. Then we define the Stokes operator $ A_{q}=-P_{q}\Delta $ and also the space $ \ D_{q}^{\alpha ,s}:=\left\{ v\in L_{\sigma }^{q}(\Omega ):\left\Vert v\right\Vert _{D_{q}^{\alpha ,s}}:=\left\Vert v\right\Vert _{q}+\left( \int_{0}^{\infty }\left\Vert t^{1-\alpha }A_{q}e^{-tA_{q}}v\right\Vert _{q}^{s}\frac{dt}{t}\right) ^{1/s}<\infty \right\} . \ $
My question is: Does anyone knows if I can assume $ \ \left\{ v\in C_{0}^{\infty }(\Omega )^{n}:\nabla \cdot v=0\right\} \subset D_{q}^{\alpha ,s}? \ $ If not, I would like to know if there is some set of smooth functions that is contained in $D_{q}^{\alpha ,s}.$
Additional information: Let me say a couple of words about the context of my question. I am working with the stokes equation: \begin{eqnarray*} \frac{\partial \mathbf{u}}{\partial t}-\Delta \mathbf{u}+\nabla P &=&f, \\ \nabla \cdot\mathbf{u}=0, \text{ in }\Omega \times (0,T) \\ \mathbf{u} &=&0\text{ on }\partial \Omega ,\mathbf{u}(x,0)=% \mathbf{u}_{0}, \end{eqnarray*} A theorem from Giga&Sohr (Th. 4.3, L^{p} estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, 1991, 72-94.) says that there are solutions satisfying: \begin{eqnarray*} \ \int_{0}^{T}\left\Vert \frac{\partial \mathbf{u}}{\partial t}(t)\right\Vert _{r}^{p}dt+\int_{0}^{T}\left\Vert \nabla ^{2}\mathbf{u}(t)\right\Vert _{r}^{p}dt+\int_{0}^{T}\left\Vert \nabla P\right\Vert _{r}^{p}dt\leq C\left( \int_{0}^{T}\left\Vert f(t)\right\Vert _{r}^{p}dt+\left\Vert \mathbf{u}% _{0}\right\Vert _{D_{r}^{1-1/p,p}}\right) . \ \end{eqnarray*} I am using this this theorem several times in a proof, but I am changing time and again the parameters $p$ and $r$ in $\left\Vert \mathbf{u}_{0}\right\Vert _{D_{r}^{1-1/p,p}}.$ Therefore I would like just to say that the initial data satisfies \begin{eqnarray*} \ \mathbf{u}_{0}\in \left\{ v\in C_{0}^{\infty }(\Omega )^{n}:\nabla\cdot v=0\right\} \end{eqnarray*} and hence forget the problem with the parameters.
