  We prove the local existence of unique smooth solutions of the Donaldson
geometric flow on the space of symplectic forms on a closed smooth
four-manifold, representing a fixed cohomology class. It is a semiflow on the
Besov space $B^{1,p}_2(M, \Lambda^2)$ for $p > 4$. The Donaldson geometric flow
was introduced by Simon Donaldson in [2]. For a detailed exposition see [6].
