On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows

In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron are studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids, etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of mathematical physics (the main result). Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to solving of the problem of some substance transportation in the space by means of the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters. The results have applications to the description of atmospheric and ocean currents, nanosciences.