The Navier-Stokes equation for an incompressible fluid in ${\bf R}^2$ with a measure as the initial vorticity

Global (in time) solutions of the Navier-Stokes equation, and the associated vorticity equation, for an incompressible fluid in ~2 are constructed, with a measure w as the initial vorticity.Regularity of the velocity and the vorticity fields as well as mono tonicity (in time) of the LP -norms of the vorticity are proved.Estimates for the singularity at t = 0 and the decay rate at t = oo of their LP -norms are deduced, and found to be almost identical with those for the solutions of the linear (heat) equation.Uniqueness is proved under a mild restriction on the atomic part wa of w, with no restriction on the size of the continuous part We.For example, it suffices that the £P -norm of the atoms (regarded as a sequence) for some p E [4/3, 2) do not exceed a certain numerical value 1Jp, which is explicitly given.