Engineering Physics Annotation << Back Existence and Smoothness of Solutions, for General, and two Special Cases of the Navier-Stokes Equation Mark W., Kuznetsov A.T. The article examines the existence and smoothness of a solution to the Navier-Stokes equation for three cases: 1. General case, where the driving force , present in the equation depends on the radius vector and time; 2. A special case where the driving force , present in the equation does not depend on the radius vector; 3. A special case, where the Navier-Stokes equation is supplemented with the continuity equation and the movement of an incompressible liquid or gas is considered. In the general case, two solutions are obtained, one of which has negative values, but this cannot happen, since the fl uid cannot move against the driving force. And the analysis of the second solution, when substituting small values of the argument, allows us to consider the second solution of the Navier-Stokes equation as a smooth function. The special case with a constant radius vector of the driving force function has two solutions, one of which is a smooth function, and the other describes the motion of the fl uid, at high speeds – turbulent fl uid motion. The transition from one solution of the equation (laminar fl uid fl ow) to another solution of the Navier-Stokes equation (turbulent fl uid fl ow) can be carried out based on the analysis of data from the Reynolds criterion. The second special case, in which the Navier-Stokes equation is supplemented with the continuity equation, is the case of an incompressible fl uid. In this case, there is no solution to the equation, because the fl uid fl ow speed in the pipe is zero. The considered cases of solving the Navier-Stokes equation show that there is no unique solution that has a simple transition to its particular cases. Keywords: Navier-Stokes equation, continuity equation, incompressible fl uid, driving force, Reynolds criterion. Pp. 21-29.