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In this paper we propose a class of local definitions of weak lower scalarcurvature bounds that is well defined for $C^0$ metrics. We show the following:that our definitions are stable under greater-than-second-order perturbation ofthe 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 lowerscalar curvature bounds are preserved under evolution by the Ricci flow from$C^0$ initial data.

Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow