| Contents |
| In a first part of this work, we will study the following Stokes problem in exterior domain of $\R^{3}$:
\begin{equation*}
(\mathcal{S})\quad
-\Delta\textbf{\textit{u}}+\nabla\,\pi=\textbf{\textit{f}}\quad \mathrm{and} \quad\mathrm{div}\,\textbf{\textit{u}}=h\quad\mbox{in}\,\,\Omega,
\quad\textbf{\textit{u}}=\textbf{\textit{g}}\quad\mbox{on}\,\,\Gamma,
\end{equation*}
where $\textbf{\textit{u}}$ denote the velocity and $\pi$ the pressure and both are unknown. We are interested in the existence and the uniqueness of very weak solutions. Here, we extend a result proved by Farwig et al $\cite{Farwig}$ and we prove the existence and the uniqueness of a second type of very weak solution.
In a second part, we will study the linearized Navier-Stokes equations in an exterior domain of $\R^3$ at the steady state, that is, the Oseen equations:
\begin{equation*}
(\mathcal{O})\quad
-\Delta\,\textbf{\textit{u}}+\mathrm{div}(\textbf{\textit{v}}\otimes\textbf{\textit{u}})+\nabla\,\pi=\textbf{\textit{f}}\quad\mathrm{and}\quad\mathrm{div}\,\textbf{\textit{u}}=h\quad\mathrm{in}\quad \Omega,\quad\textbf{\textit{u}}=\textbf{\textit{g}}\quad\mathrm{on}\quad\Gamma.
\end{equation*}
We are interested in the existence and the uniqueness of weak, strong and very weak solutions. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space $\R^3$ and the properties in bounded domains. |
|