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

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 $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$ even for smooth solutions close to constants. Analogous estimates forL ^p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.