Published Paper

Inserted: 7 jan 2004
Last Updated: 3 may 2011

Journal: Duke Math. J.
Volume: 127
Number: 2
Pages: 313-339
Year: 2004

Abstract:

We consider the Cauchy problem for the system $\partial_t u_i + \div_z (g(
u
) u_i) = 0$ $i\in \{1, \ldots, k\}$, in $m$ space dimensions and with $g\in C^3$. When $k\geq 2$ and $m=2$ we show a wide choice of $g$'s for which the BV norm of admissible solutions can blow up, even when the initial data have arbitrarily small oscillation, arbitrarily small total variation, and are bounded away from the origin. When $m\geq 3$ we show that this occurs whenever $g$ is not constant, i.e. unless the system reduces to $k$ decoupled transport equations with constant coefficients.

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