  We prove sharp bounds on the enstrophy growth in viscous scalar conservation
laws. The upper bound is, up to a prefactor, the enstrophy created by the
steepest viscous shock admissible by the $L^\infty$ and total variation bounds
and viscosity. This answers a conjecture by D. Ayala and B. Protas
(\emph{Physica D}, 2011), based on numerical evidence, for the viscous Burgers
equation.
