# Stokes operator without Dirichlet boundary condition

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

As a forward information, the space $H^1_0(\Omega)$ in $\mathcal{V}_{\sigma}$ should be replace by $H_N = \{u\in H^1(\Omega): \frac{\partial u}{\partial n}=0\}$, as it is done

J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), no.~2, 376--406. MR1705387

for other parabolic equations. Here are two references on the Navier-Stokes with Neumann boundary conditions:

Miyakawa, Tetsuro. The $L^{p}$ approach to the Navier-Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10 (1980), no. 3, 517--537. MR594132

and

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 2013, 6 (5), pp.1355-1369. MR3039703

Best regards,