Let $\Omega$ be a domain, then the following stokes operator is quite well known:

$\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $

$f \rightarrow u$ such that $ - \Delta u = f $

where $\mathcal{H}$ is the closure in $L^2$ of $\{ \phi , \phi \in D(\Omega)^n div \phi = 0 \}$ and $ \mathcal{V}_{\sigma}$ is the closure in $L^2$ of $\{ v \in H^1_0(\Omega)^n, \nabla \cdot v=0 \}$

I am concerned with what happens when we take of the vanishing at boundary condition, namely when we are interested with the Laplacian in the space $H^1_{\sigma}(\Omega)$, the closure in $L^2$ of $\{ v \in H^1(\Omega)^n, \nabla \cdot v=0 \}$

In that setting we have boundary terms that appears: simply considering smooth functions then the identity:

$\int_{\Omega} \Delta \Phi \cdot \phi + \int_{\Omega} \nabla \Phi : \nabla \phi = \int_{\partial \Omega} \phi \cdot \frac{\partial \Phi}{\partial n}$

suggests that we cannot define the Laplacian for any $u \in H^1_{\sigma}(\Omega)$ we need the term $\frac{\partial \Phi}{\partial n}$ to make sense for $u$ so maybe we could ask something like $\nabla u_i \in H^{-\frac{1}{2}}(\Omega)$ and then define $-\Delta u$ in the dual of $H^1_{\sigma}(\Omega)\cap \{u \in H^1_{\sigma}(\Omega), s.t. \nabla u_i \in H^{-\frac{1}{2}}(\Omega) \}$ as $\phi \rightarrow -\sum \langle\phi_i;\nabla u_i\rangle + \int_{\Omega} \nabla u : \nabla \phi$

But there remain an important problem, this operator does not seem to be self adjoint. I am interested in having some spectral theorem that would allow me to construct solutions for the time dependent stokes problem without the boundary condition $u|_{\partial \Omega} =0$ using some Galerkin method:

$\partial_t u - \Delta u = f + \nabla p$

$div\ u = 0$

$u \cdot n = 0 $ in $\partial \Omega$

$u_{t=0} = u_0$

Would you know some literature reference for this problem that I could read? To be more specific I am interested with Navier boundary conditions in fact.

Thanks