2
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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,

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.