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 <a href="http://projecteuclid.org/euclid.jdg/1090349447">constructed</a> 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.