<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt]{article}

\usepackage{amssymb, amsthm, amsmath, amsfonts}
\usepackage{cancel, enumerate, url}

%--- Margins
\setlength{\oddsidemargin}{-.2in} 
\setlength{\evensidemargin}{-.2in}
\setlength{\textwidth}{6.9in} 
\setlength{\topmargin}{-.5in} 
\setlength{\textheight}{9in}

\setlength{\parskip}{1ex}
		
													
\begin{document}

\begin{center}

%--- Title
{\huge Existence and regularity results for an important PDE system}
%--- Title

\bigskip

%--- Authors
Piotr Biler,  Piotr.Biler@math.uni.wroc.pl

Michal  Kowalczyk,  kowalczy@dim.uchile.cl

Piotr Rybka,  rybka@mimuw.edu.pl
%--- Authors

\bigskip

\textbf{\large Abstract}

\end{center}

%--- Abstract
We consider a three-dimensional version of the fancy system from \cite{bib1}.
For every smooth and divergence-free initial condition $v_{0} (x) := v(x,0)$
satisfying the additional assumption:
\begin{center} for every multi-index $\alpha$ and any $K&gt;0$ there exists a constant $C=C(\alpha, K)$ such that $| \partial ^{\alpha} v_{0}(x) | \leq \frac{1}{(1+|x|)^{K}}$ for all $x \in \mathbb{R}^{3}$\end{center}
we prove that there exists a smooth and globally defined solution $v(x,t)$ of the system of two equations:
\begin{center} $\frac{\partial v}{\partial t} + (v \cdot \nabla)v=- \nabla p + \nu \Delta v \qquad \textnormal{and} \qquad \nabla\cdot v = 0$. \end{center}
In addition we show that there is a constant $E&gt;0$ such that
$\int_{\mathbb{R}^{3}} |v(x,t)|^{2} &lt;E$ for all $t\geq 0$.
%--- Abstract


%--- References
\begin{thebibliography}{99}

\bibitem{bib1}
\newblock  Au. Thor,
\newblock \emph{The Most Important Equations of the Universe,}
\newblock A Noble Publishing Co, Pleasant City, 1969.
\end{thebibliography}
%--- References







\end{document}
</pre></body></html>