A remark on the Navier-Stokes flow with bounded initial data having a special structure

Abstract

The Navier-Stokes equations with bounded initial data admit unique local-in-time smooth mild solutions. It is shown that the solution can be extended globally-in-time, if the initial velocity has a special structure. Thanks to the structure, the annihilation of the pressure occurs, and then the mild solution is a solution to the viscous Burgers equations. By the maximum principle, it is derived an a priori bound for velocity, uniformly in time and space.