On Existence of General Solution of the Navier - Stokes Equations for 3D Non-Stationary Incompressible Flow

The paper deals with presenting the new form of general solution for the Navier - Stokes equations. The equations of motion for a 3D non-stationary incompressible flow are considered. The flow velocity field and the equation of momentum should be split to the sum of two components: the irrotational (curl-free) one, and solenoidal (divergence-free) ones. The obviously irrotational (curl-free) part of equation of momentum is used for obtaining of the components of pressure gradient. The irrotational (curl-free) vector field of flow velocity is given by the proper potential according to the continuity equation. Another part of equation of momentum could also be split to the sum of two equations: one with zero curl for the field of flow velocity (viscous-free) and proper equation with the viscous effects but variable curl. A solenoidal equation with the viscous effects is represented by the proper heat equation for each component of a flow velocity with variable curl. The non-viscous case is presented by the system of three linear partial differential equations with respect to the time-parameter, depending on the components of solution of the above heat equation for the component of flow velocity with the variable curl. So, the existence of general solution of the Navier Stokes equation is proved to be the question of existence of the proper solution of Riccati-type for such a system of linear partial differential equations. The final solution is proved to be the sum of the irrotational (curl-free) and solenoidal (variable curl) components.