  We capture optimal decay for the Mullins-Sekerka evolution, a nonlocal,
parabolic free boundary problem from materials science. Our main result
establishes convergence of BV solutions to the planar profile in the physically
relevant case of ambient space dimension three. Far from assuming small or
well-prepared initial data, we allow for initial interfaces that do not have
graph structure and are not connected, hence explicitly including the regime of
Ostwald ripening. In terms only of initially finite (not small) excess mass and
excess surface energy, we establish that the surface becomes a Lipschitz graph
within a fixed timescale (quantitatively estimated) and remains trapped within
this setting. To obtain the graph structure, we leverage regularity results
from geometric measure theory. At the same time, we extend a duality method
previously employed for one-dimensional PDE problems to higher dimensional,
nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy,
dissipation, and graph height are obtained.
