BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one (Q1089518)

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form \[ \| \nabla u(\bar t)\|_{T.V.}\leq F(\| \nabla u(0)\|_{T.V.}), \] \(F\in C({\mathbb{R}})\), \(F(0)=0\), F Lipschitzean at 0, even for smooth solutions close to constants. Analogous estimates for \(L^ p\) norms \[ \| u(\bar t)-\bar u\|_{L^ p}\leq F(\| u(0)- \bar u\|_{L^ p}),\quad p\neq 2 \] with F as above are also false. In one dimension such estimates are the backbone of the existing theory.