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\begin{document}
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\title[CKN]{Navier-Stokes Seminar: Caffarelli-Kohn-Nirenberg Theory}
\author{}
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\begin{center}
\huge{\bf Navier-Stokes Seminar: Caffarelli-Kohn-Nirenberg Theory}\\ 
\vspace{1cm}
\huge{{(insert name here)}}\\
Universit\"at Ulm, Summer 2019\\
\vspace{1cm}
%\huge{Version as of 13/02/2019} \\
\end{center}
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%\maketitle
%\frontmatter
%\mainmatter
\chapter*{Preface}
These are lecture notes geberated by the seminar course on the Caffarelli-Kohn-Nirenberg Theory for the Navier-Stokes equations at the Universit\"at Ulm in the summer term of 2019. We mainly follow the \cite{CKNpaper} in a modern fashion. This work is aimed at enthusiastic Masters and PhD students. %Except for the first and the last chapter, the notes follow the excellent recent textbook~\cite{RRS}. Students are encouraged to consult further literature, such as the classical books~\cite{CF, Tem, lions}. 

%Except for the very end of the course, I chose to work exclusively on the three-dimensional torus such as to simplify the presentation. However all mentioned results from the first four chapters have a straightforward extension to the whole space $\R^3$, or to (sufficiently regular) bounded domains, which certainly represent the physically most relevant case.

I would like to thank everyone taking the seminar for typing parts of these notes.

Corrections and suggestions should be sent to \verb+jack.skipper@uni-ulm.de+.

%\item Anschließend an die vorige Bemerkung ergibt sich aus Tippfehlern im Skript für Sie kein Anspruch, denselben Fehler Ihrerseits machen zu dürfen, z.B.~in einer Klausur.%\footnote{Quod licet Iovi, non licet bovi.} 

  

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\chapter{Talk 1: Introduction}

The Navier-Stokes equations are 
\begin{align}
\partial_t u(x,t)+ (u\cdot \nabla )u(x,t)+\nabla p(x,t )&=\nu \Delta u(x,t)\\
 \diverg u(x,t)&=0.
\end{align}
Here, $(x,t)\in \Omega\times[0,T]$, where $\Omega\subset \R^d$ some domain, and we have the unknown velocity field
\begin{equation}
u\colon\quad \Omega \times [0,T] \to \R^d;
 \end{equation} 
the unknown pressure field
\begin{equation}
p\colon \quad\Omega \times [0,T] \to \R;
\end{equation} 
and the given constant viscosity $\nu>0$. It can be written in components, for $ i= 1, \dots, d$: 
\begin{align}
\partial_t u_i + \sum^d_{j=1}u_j\partial_j u_i +\partial_i p &= \nu \sum^d_{j=1} \partial_{x_j}^2 u_i\\
\sum_{j=1}^d \partial_j u_j&=0.
\end{align}

The Navier-Stokes Equations (NSE) describe the time evolution of the velocity and pressure of a viscous incompressible fluid (e.g.\ water) without external forces. 
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\section{topic}


\begin{thebibliography}{10}

%Caffarelli, L., Kohn, R., & Nirenberg, L. (1982). Partial regularity of suitable weak solutions of the Navier‐Stokes equations. Communications on pure and applied mathematics, 35(6), 771-831.

\bibitem{CKNpaper}
L.~Caffarelli, R.~Kohn and L.~Nirenberg. Partial regularity of suitable weak solutions of the Navier‐Stokes equations. \emph{Communications on pure and applied mathematics}, 1982.

%
%\bibitem{CF}
%P.~Constantin and C.~Foias. Navier-Stokes equations. \emph{University of Chicago Press}, 1988.
%
%\bibitem{kato}
%T.~Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on nonlinear partial differential equations. \emph{Springer, New York, NY}, 1984.
%
%\bibitem{lions}
%P.-L.~Lions. Mathematical topics in fluid mechanics. Vol.\ 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, 3. \emph{Oxford Science Publications. The Clarendon Press, Oxford University Press, New York}, 1996. 
%
%\bibitem{RRS}
%J.~C.~Robinson, J.~L.~Rodrigo, and W.~Sadowski. The Three-Dimensional Navier–Stokes equations: Classical theory. Cambridge Studies in Advanced Mathematics, Vol.~157. \emph{Cambridge University Press}, 2016.
%
%\bibitem{Tem}
%R.~Temam. Navier-Stokes equations. Theory and numerical analysis. Revised edition. Studies in Mathematics and its Applications, 2. \emph{North-Holland Publishing Co., Amsterdam-New York}, 1979.

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%
\end{thebibliography}
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