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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

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1 Answer 1

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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,

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