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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 timesand 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 thegraph and all higher-order derivatives of the corresponding map $f_t$.

Evolution of area-decreasing maps between two-dimensional Euclidean spaces