Oscillatory solutions to transport equations (original) (raw)

Let n ≥ 3. We show that there is no topological vector space X ⊂ L ∞ ∩ L 1 loc (R × R n) which embeds compactly in L 1 loc , contains BV loc ∩ 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 × 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 thatũ(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 ".