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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.

##### MSC:
 35L65 Hyperbolic conservation laws 35B35 Stability in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs
##### Keywords:
systems of conservation laws; smooth solutions; estimates
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##### References:
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