| Contents |
| The aim of this work is to study the analyticity of the Stokes operator with Navier-type boundary conditions on $\boldsymbol{L}^{p}$-spaces in order to get strong, weak and very weak solutions to the following evolution Stokes problem:
\begin{equation}\label{lens}
\left\{
\begin{array}{cccc}
&\frac{\partial\boldsymbol{u}}{\partial t} - \Delta \boldsymbol{u
}\,+\,\nabla\pi=\boldsymbol{f},\quad
\mathrm{div}\,\boldsymbol{u}= 0 & \textrm{in} &\Omega\times (0,T), \\
&\boldsymbol{u}\cdot\boldsymbol{n}=0,\qquad
\boldsymbol{\mathrm{curl}}\,\boldsymbol{u}\times \boldsymbol{n} = \boldsymbol{0}, &\textrm{on}&\Gamma\times (0,T),\\
&\boldsymbol{u}(0)=\boldsymbol{u}_{0}& \textrm{in} & \Omega.
\end{array}
\right.
\end{equation}
%where the unkowns $\boldsymbol{u}$ and $\pi$ stand respectively for the velocity field and the pressure of a fluid occupying a domain $\Omega$.
In this work we prove that the Stokes operator with Navier-type boundary conditions generates a bounded analytic semi-group on the space
$$
\boldsymbol{L}^{p}_{\sigma,T}(\Omega) =\,\big\{\boldsymbol{v}\in\boldsymbol{L}^{p}(\Omega);\,\,\,\textrm{div}\,\boldsymbol{v} = 0\,\, \mathrm {in}\, \Omega \quad\mathrm{and}\quad \boldsymbol{v}\cdot\boldsymbol{n}\,= 0 \, \, \mathrm{on}\, \Gamma \big\}.
$$
The idea is to study the resolvent of the Stokes operator:
\begin{equation}\label{*}
\left\{
\begin{array}{r@{~}c@{~}l}
\lambda \boldsymbol{u} - \Delta \boldsymbol{u}\,+\,\nabla\pi = \boldsymbol{f}, &&\mathrm{div}\,\boldsymbol{u} = 0 \,\,\, \qquad\qquad \mathrm{in} \,\,\, \Omega, \\
\boldsymbol{u}\cdot \boldsymbol{n} = 0, && \boldsymbol{\mathrm{curl}}\,\boldsymbol{u}\times\boldsymbol{n}=\boldsymbol{0}\qquad
\mathrm{on}\,\,\, \Gamma,
\end{array}
\right.
\end{equation}
where $\lambda\in\mathbb{C}^{\ast}$ with $\mathrm{Re}\,\lambda\geq 0$. We prove the existence of weak, strong and very weak solutions to Problem (\ref{*}) satisfying the following resolvent estimate
\begin{equation}\label{**}
\Vert\boldsymbol{u}\Vert_{\boldsymbol{L}^{p}(\Omega)}\,\leq\,
\frac{C(\Omega,p)}{\vert\lambda\vert} \,\Vert\boldsymbol{f}\Vert_{\boldsymbol{L}^{p}(\Omega)}.
\end{equation} |
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