Of course, the classical flow need not be well defined when starting from a minimal surface. This is part of the reason why it was an amazing result when Huisken--Ilmanen constructed a "weak inverse mean curvature flow" which
(1) Can start at a minimal surface (technically, for certain things to work nicely, it should be outer-minimizing)
(2) Exists for all time in an asymptotically flat manifold.
AND
(3) Still satisfies Geroch monotonicity, i.e. the Hawking mass is monotone along the flow.
That one could find a "flow" which satisfies (1) and (2) while still keeping (3) is incredible.
Their paper is very readable, although quite long, so I'd recommend that you take a look at it, rather than I try to explain the ideas here.