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| In a three dimensional bounded eventually multiply-connected domain of class $\mathcal{C}^{\,1,1}$, we prove existence and uniqueness of vector potentials associated with a divergence-free function and satisfying some boundary conditions in $L^{p}$ theory. We also present some results concerning various Sobolev's inequalities. Next, we consider the stationary Stokes equations with nonstandard boundary conditions of the form $\textbf{\textit{u}}\cdot\textbf{\textit{n}}=g$ and $\mathbf{curl}\,\textbf{\textit{u}}\times\textbf{\textit{n}}=\textbf{\textit{h}}\times\textbf{\textit{n}}$ or $\textbf{\textit{u}}\times\textbf{\textit{n}}=\textbf{\textit{g}}\times\textbf{\textit{n}}$ and $\pi=\pi_{0}$ on the boundary $\Gamma$. We prove the existence and uniqueness of weak, strong and very weak solutions correspending to each boundary condition in $L^{p}$ theory. To prove the solvability, we study the well-posdeness of some elliptic systems. For this end, it is necessary to establish $Inf-Sup$ conditions which play an essential role in our proofs. Furthermore, two Helmoholtz decompositions which consist of two kinds of boundary conditions such as $\textbf{\textit{u}}\cdot\textbf{\textit{n}}$ and $\textbf{\textit{u}}\times\textbf{\textit{n}}$ on $\Gamma$ are given. Finally, we give an application to the Navier-Stokes equations where the proof of solutions is obtained by applying a fixed point theorem over the Oseen equations. |
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