  An effort has been made to solve the Cauchy problem of the Navier-Stokes
equations in the whole space by two methods. It is proved that the sum of the
three vorticity components is a time-invariant in fluid motion. It has been
proved that, given smooth, localized initial data with finite energy and
enstrophy, the vorticity equation admits a global, unique and smooth solution.
Second, the vorticity equation has been converted into a non-linear integral
equation by means of similarity reduction. The solution of the integral
equation has been constructed in a series expansion. The series is shown to
converge for initial data of finite size. The complete vorticity field is
characterized, as an instantaneous description, by a multitude of vorticity
constituents. The flow field is composed of vortical elements of broad
spatio-temporal scales. Inference of the solutions leads itself to a
satisfactory account for the observed dynamic characteristics of transition
process, and of turbulent motion. In the limit of vanishing viscosity, the
equations of motion cannot develop flow-field singularities in finite time. In
the Maxwell-Boltzmann kinetic theory, the density function of the Maxwellian
molecules possesses a phase-space distribution resembling the continuum
turbulence. Qualitatively, the apparent macroscopic randomness of turbulence
can be attributed to a ramification of molecular fluctuations.
