  We consider the mean curvature flow of the graph of a smooth map
$f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If
$f$ satisfies an area-decreasing property, the solution exists for all times
and the evolving submanifold stays the graph of an area-decreasing map $f_t$.
Further, we prove uniform decay estimates for the mean curvature vector of the
graph and all higher-order derivatives of the corresponding map $f_t$.
