Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>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 <jats:inline-formula>
<jats:tex-math><?CDATA $L^\infty$?>
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<mml:mi>L</mml:mi>
<mml:mi mathvariant="normal">∞</mml:mi>
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<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="nonad073fieqn1.gif" xlink:type="simple" />
</jats:inline-formula> and total variation bounds and viscosity. This answers a conjecture by Ayala and Protas (2011 <jats:italic>Physica</jats:italic> D <jats:bold>240</jats:bold> 1553–63), based on numerical evidence, for the viscous Burgers equation.</jats:p>