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BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. (English) Zbl 0619.35073
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.

35L65 Hyperbolic conservation laws
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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