Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity

A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier-Stokes equations in the plane when initial velocity field is merely bouncled not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in­time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when q > 2