Navier-Stokes Initial Value Problem for Boundary-Free Incompressible Fluid Flow

Abstract

Convergence proofs are reported for a general local iteration solution to the Navier-Stokes initial value problem and estimates of the accuracy of the nth iterative approximation are derived. Without appeal to methods of functional analysis, it is shown that a Kiselev-Ladyzhenskaya weak solution is, in fact, a classical solution. Any one of three alternative conditions on the initial velocity field is found to be sufficient to guarantee the existence of a global solution. Breakdown phenomenon which may prevent a local solution from being continued for all t ≥ 0 to a global solution is analyzed. The mathematical theory suggests that breakdown is precluded for a suitably smooth initial velocity field, irrespective of the over-all initial velocity field magnitude.