Oscillatory solutions to transport equations

Let n ≥ 3. We show that there is no topological vector space X C L ∞ n L 1 loc (R x R n ) that embeds compactly in L 1 loc , contains BV loc n L ∞ , and enjoys the following closure property: If f ∈ X n (R × R n ) has bounded divergence and u 0 ∈ X(R n ), then there exists u ∈ X(R x R n ) which solves ∂ t u + div (uf) = 0 u(0,·) = u 0 in the sense of distributions. X(R n ) is defined as the set of functions u 0 ∈ L ∞ (R n ) such that Ū(t,x):= u 0 (x) belongs to X (R × R n ). Our proof relies on an example of N. Depauw showing an ill-posed transport equation whose vector field is "almost BV".