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We provide the first general result for the asymptotics of the areapreserving mean curvature flow in two dimensions showing that flat flowsolutions, starting from any bounded set of finite perimeter, converge withexponential rate to a finite union of equally sized disjoint disks. A similarresult is established also for the periodic two-phase Mullins-Sekerka flow.

The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions