Oscillatory solutions to transport equations

We show that there is no topological vector space $X\subset L^\infty\cap L^1_{\loc} (\rn{}\times \rn{n})$ which embeds compactly in $L^1_{\loc}$, contains $BV_{\loc}\cap L^\infty$ and enjoys the following closure property: If $f\in X^n (\rn{}\times \rn{n})$ has bounded divergence and $u_0\in X (\rn{n})$, then there exists $u\in X (\rn{}\times \rn{n})$ 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. $X (\rn{n})$ is defined as the set of functions $u_0\in L^\infty (\rn{n})$ such that $\tilde{u} (t,x):= u_0 (x)$ belongs to $X (\rn{}\times \rn{n})$. Our proof relies on an example of N.~Depauw showing an ill--posed transport equation whose vector field is ``almost $BV$''.