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.