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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}}\\
\huge{Universit\"at Ulm, Summer 2019}\\
\vspace{1cm}
%\huge{Version as of 13/02/2019} \\
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%\newpage
%\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: Insert title}


\begin{centering}
\large{\textbf{By insert name }}\\
\end{centering}
\hspace{1cm}

%XXXXXXXXX


%XXXXXXXXXXX
For this introduction we will use the original paper of \cite{CKNpaper} and the excellent book \cite{JamesJoseBook}. 

The three-dimensional Navier-Stokes equations are 
\begin{align}
\begin{split}
\label{eq:NSE}
\partial_t u(x,t)+ (u\cdot \nabla )u(x,t)+\nabla p(x,t )-\Delta u(x,t) &= f(x,t)\\
 \diverg u(x,t)&=0.
\end{split}
\end{align}


Here, $(x,t)\in \Omega\times[0,T]$, where $\Omega\subset \R^3$ or $\T^3$ or $\R^3$ some domain, and we have the unknown velocity field
\begin{equation}
u\colon \Omega \times [0,T] \to \R^3;
 \end{equation} 
the unknown pressure field
\begin{equation}
p\colon \Omega \times [0,T] \to \R;
\end{equation} 
and the given force $f\colon \Omega\times[0,T]\to\R^3$ with $\diverg f=0$ in $\Omega\times[0,T]$. Together with initial data and boundary data, \eqref{eq:NSE} turns into an initial boundary value problem
\begin{align}\label{eq:InitialBoundary}
	u(x,0)&= u_0(x), && x\in \Omega,\\
	u(x,t)&=0, && x\in \partial\Omega \quad\mathrm{for}\quad 0<t<T.
\end{align}

With compatibility conditions for $u_0$ and $f$ we see that
\begin{equation}
-\Delta p=\partial_i\partial_j(u_iu_j) \quad \mathrm{for} \; a.e\; t.
\end{equation}
   
%\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. 
%%(diagram)

\section{Outline: The Navier-Stokes Equations}
\subsection{Weak and Strong}

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\bibitem[RRS16]{JamesJoseBook} %RRS
J.C.~Robinson, J.L.~Rodrigo and W.~Sadowski. The Three-Dimensional Navier–Stokes Equations: Classical Theory. \emph{Cambridge University Press.}, 2016.


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V.~Scheffer. An inviscid flow with compact support in space-time . \emph{J. Geom. Anal.} 1993.


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J.~Serrin. On the interior regularity of weak solutions for the Navier-Stokes equations. \emph{Arch. Ration. Mech. Anal.}, 1962.

\bibitem[Ser63]{Serrinweakstong} %Serrin
J.~Serrin. The initial value problem for the Navier-Stokes equations. \emph{In Nonlinear Problems (Proc. Sympos., Madison, Wis.)}, 1963.

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H~Sohr and W.~von~Wahl. On the regularity of the pressure of weak solutions of Navier-Stokes equations. \emph{Archiv der Mathematik}, 1986.

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M.~Struwe.  On partial regularity results for the Navier‐Stokes equations.  \emph{Communications on Pure and Applied Mathematics}, 1988.


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S.~Takahashi.  On interior regularity criteria for weak solutions of the Navier-Stokes equations.  \emph{Manuscripta Mathematica}, 1990.

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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.


\bibitem[Wie11]{Emileulerinfinesol}
E.~Wiedemann.  Existence of weak solutions for the incompressible Euler equations.  \emph{In Annales de l'Institut Henri Poincare (C) Non Linear Analysis}, 2011.





\end{thebibliography}
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