*Published Paper*

**Inserted:** 28 feb 2005

**Last Updated:** 4 may 2011

**Journal:** Indiana Univ. Math. J.

**Volume:** 55

**Number:** 1

**Pages:** 1-13

**Year:** 2006

**Abstract:**

\documentclass{article}
\begin{document}
We show that there is no topological vector space $X\subset L^\infty\cap
L^1_{\rm loc} (\mathbf{R}^n \times \mathbf{R}^n)$ which embeds compactly
in $L^1_{\rm loc}$, contains $BV_{\rm loc}\cap L^\infty$ and enjoys the
following closure property: If $f\in X^n (\mathbf{R} \times
\mathbf{R}^n)$ has bounded divergence and $u_0\in X (\mathbf{R}^n)$, then
there exists $u\in X (\mathbf{R} \times \mathbf{R}^n)$ which solves
$$
\left\{ \begin{array}{l}
\partial_{t} u + {\rm div}\, (u f)\;=\; 0
u (0, \cdot) \;=\; u_{0}
\end{array}\right.
$$
in the sense of distributions. $X (\mathbf{R}^n)$ is defined as the set
of functions $u_0\in L^\infty (\mathbf{R}^n)$ such that $U(t,x):=
u_0 (x)$ belongs to $X (\mathbf{R}\times \mathbf{R}^n)$.
Our proof relies on an example of N. Depauw showing an ill--posed
transport equation whose vector field is ``almost $BV$''.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
\end{document} *

**Keywords:**
Transport equation, Hyperbolic systems of conservation laws