## Séminaire d'Analyse

**Very weak solutions and regularity for some elliptic equations**

Chérif Amrouche

jeudi 26 janvier 2012 11:00 - Tours - Salle 2290 (Bât E2)

**Résumé :**

\noindent{\bf Part I.} The concept of very weak solution introduced by Giga \cite{G} for the stationary Stokes equations (and also by Lions-Magenes \cite{LM} for the Laplace's equation, in a domain $\Omega$ of class $\mathcal{C}^{\infty}$) has been intensively studied in the last years for the stationary Navier-Stokes equations with Dirichlet boundary conditions. We give here a new and simpler proof of the existence of very weak solutions $ ({\textbf {\textit u}}, \pi)\in {\bf L}^p(\Omega)\times W^{-1, p}(\Omega),$ with $1< p < \infty$, for the stationary Navier-Stokes equations, based on density arguments and an adequate functional framework in order to define more rigourously the traces of non regular vector fields. We also obtain regularity results in fractional Sobolev spaces. All these results are obtained in the case of a bounded open set, connected of class ${\mathcal C}^{1,1}$ of $\mathbb{R}^3$ and can be extended to the Laplace's equation and to other dimensions. \medskip \noindent{\bf Part II.} We consider in this part the Stokes and Navier-Stokes problems in a bounded domain, eventually multiply connected, whose boundary consists of multi-connected components. We investigate the solvability in $L^p$-theory, with $1 < p <\infty$, under the non standard boundary conditions \begin{equation} {\textbf {\textit u}}\cdot\textbf{\textit{n}} = g,\quad\mathbf{curl}\,{\textbf {\textit u}}\times\textbf{\textit{n}}=\textbf{\textit{h}}\quad\mathrm{or}\quad {\textbf {\textit u}}\times\textbf{\textit{n}}=\textbf{\textit{g}},\quad \pi = \pi_*\quad\mathrm{on}\,\Gamma. \end{equation} The main ingredients for this solvability are given by the Inf-Sup conditions, some Sobolev's inequalities for vector fields and the theory of vector potentials satisfying \begin{equation} \boldsymbol{\psi}\cdot\textbf{\textit{n}}= 0 \quad\mathrm{or}\quad \boldsymbol{ \psi}\times\textbf{\textit{n}}=\bf{0}\quad\mathrm{on}\,\mathrm{\Gamma}. \end{equation} Those inequalities play a fundamental key and are obtained thanks to Calderon-Zygmund inequalities and integral representations. In the study of ellpitical problems, we consider both generalized solutions and strong solutions that very weak solutions. Finally, this work is an extension of \cite{ABDG} where the authors give the hilbertian theory for vector potentials. \begin{thebibliography}{9} \bibitem{ABDG} {\sc C. Amrouche, C. Bernardi, M. Dauge and V. Girault}, Vector Potentials In Three-dimensional Non-smooth Domains, {\sl M2AS}, Vol.~21, pp.~823--864, 1998. \bibitem{[1]} \textsc {C.~Amrouche and V.~Girault}, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, {\sl Czechoslovak Mathematical Journal}, 44, 119, 109--140, 1994. \bibitem{[4]} \textsc{C.~Amrouche and M.~A.~Rodr\'{\i}guez-Bellido}, Stokes, Oseen and Navier-Stokes equations with singular data, {\sl Arch. Ration. Mech. Anal.}, 199, 597--651, 2011 \bibitem{ASa} {\sc C. Amrouche, N.E.H. Seloula}, $L^p$-Theory for Vector Potentials and Sobolev's Inequalities for Vector Fields, Applicatin to the Stokes equations with pressure boundary conditions, to appear in {\sl M3AS} \bibitem{ASb} {\sc C. Amrouche, N.E.H. Seloula}, On the Stokes Equations with the Navier-type boundary conditions, {\sl Diff. Eq. Appl.} Volume 3, Number 4 (2011), 581-607 \bibitem{G} \textsc{Y.~Giga}, Analyticity of the semigroup generated by the Stokes operator in $L^p$-spaces, {\sl Math.~Z.~} 178, 287-329, 1981. \bibitem{LM} \textsc{J.~L.~Lions and E.~Magenes}, Probl\`emes aux limites non homog\`enes et applications, vol 1, Dunod, Paris, 1968. \end{thebibliography}

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