  In this paper we propose a class of local definitions of weak lower scalar
curvature bounds that is well defined for $C^0$ metrics. We show the following:
that our definitions are stable under greater-than-second-order perturbation of
the metric, that there exists a reasonable notion of a Ricci flow starting from
$C^0$ initial data which is smooth for positive times, and that the weak lower
scalar curvature bounds are preserved under evolution by the Ricci flow from
$C^0$ initial data.
