*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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