About Bellman Principle and Solution Properties for Navier–Stokes Equations in the 3D Cauchy Problem

Abstract

Without belittling the achievements of many mathematicians in the studying of the Navier-Stokes equations, the real ways opened J. Leray and O.A. Ladyzhenskaya. The main goal of this work is to compare the smoothness property of a weak solution in the Cauchy problem after some moment if it is known solution regularity until this moment with the optimality property in the Bellman principle. Naturally, all these are connected with the existence problem of blow up solution in the Cauchy problem for Navier-Stokes equations in space attracting a lot of attention up to now. The smoothness control and controlling parameters can be varied. It is important to control the dissipation of kinetic energy to the fix moment or rate of change of kinetic energy square or the summability of velocity gradient to the fixed point in time and so on. There are possible other control parameters due to a weak solution.