Sobolev spaces with respect to divergence and their properties

Question

Let $n \in \mathbb{N}$, $\Omega$ a non-empty bounded open set of $\mathbb{R}^n$ with Lipschitz boundary and $p \in [1,\infty]$. Define $$V_p:=\bigg\{\overrightarrow{q}\in L^p(\Omega;\mathbb{R}^n) \mid \exists \ f\in L^p(\Omega;\mathbb{R}), \forall \varphi \in C^1_c(\Omega;\mathbb{R}^n), \int_\Omega\overrightarrow{q}\cdot\nabla\varphi\operatorname{d}x = -\int_\Omega f\varphi \operatorname{d}x\bigg\}.$$ This space clearly has the interpretation of being the space of $L^p$ vector fields over $\Omega$ that admits a weak-divergence that lies also in $L^p$ (so, in the sequel, the corresponding $f$ w.r.t. to $\overrightarrow{q}$ in the definition will be denoted by $\operatorname{div}\overrightarrow{q}$).

I can't find in standard textbooks about Sobolev spaces a treatment of those spaces, so I'm wondering where I can find them in the literature. In particular, I have several questions that I want to answer, such as:

1. Are those spaces Banach spaces w.r.t. $\overrightarrow{q}\mapsto \| \operatorname{div}\overrightarrow{q} \|_p + \|\overrightarrow{q} \|_p$?
2. Do they admit traces on the boundary (or maybe just something that plays the role of the normal component of the trace on the boundary?) and do they have some integrability properties?
3. Does the divergence theorem hold? I.e., if $\overrightarrow{\nu}$ is the outward normal of $\Omega$ on $\partial\Omega$, does it hold true that $$\int_{\Omega} \operatorname{div}\overrightarrow{q} \operatorname{d}x = \int_{\partial\Omega} \overrightarrow{q}\cdot\overrightarrow{\nu} \operatorname{d}\mathcal{H}^{n-1},$$ where $\overrightarrow{q}$ over $\partial \Omega$ is the trace of $\overrightarrow{q}$ on $\partial \Omega$ and $\mathcal{H}^{n-1}$ is the $n-1$-dimensional Hausdorff measure?

References are very welcome.

Answer 1

Results about this space are likely in Chapter III in Galdi's "An introduction to the mathematical theory of the Navier-Stokes equations: steady state problems" but I haven't read it properly (and its a big book), at least for $p\in[1,\infty)$ and smoother boundary. They are indeed Banach.

I'm more familliar with the $p=2$ (Hilbert) case. At least here, I can tell you that they only necessarily have normal traces. The Stokes formula takes the form $$ u\in V_2, w\in H^1\implies (u,\nabla w) + (\nabla\cdot u,w) = \langle \gamma(u),\gamma_0(w)\rangle$$ where $\gamma_0: H^1(\Omega)\to H^{1/2}(\partial \Omega)$ is the trace map, and $\gamma:V_2\to H^{-1/2}(\partial \Omega)$ (the dual of $H^{1/2}(\partial \Omega)$) is such that $\gamma(u)=u\cdot n$ (the outward normal) for smooth functions $u$.

This can be found in pages 3--4 of Constantin and Foias's "Navier-Stokes equations". There's some discussion of selected parts of Galdi in the first chapter of Tsai's "Lectures on the Navier-Stokes Equations".