Remarks on the initial value problem for a stabilized Navier-Stokes equation

A new family of simple, straightforward modifications of the Navier-Stokes equations is presented that can be made arbitrarily close to the original equations. These modifications possess the following properties: (1) the initial value problem for these 'stabilized' equations is appropriately stated and can be treated by the methods of Hopf, Prodi, Serrin, and Walter; (2) the proof of existence proposed by Hopf for the initial value problem in the Navier-Stokes equation leads to a constructive solution to the initial value problem in the stabilized equations. The Navier-Stokes equation is modified by averaging in the convective term. The classical and Hopf forms of the initial value problem are also briefly reviewed.