Solving the Navier-Stokes Equation for a Viscous Incompressible Fluid in an n-Dimensional Bounded Region and in the Entire Space ℝn

We consider initial-value problems for a system of motion equations of a viscous incompressible fluid in Lagrangian variables with n = 2 or n = 3 . We show that the fluid motion is independent of pressure. In the absence of external forces, the pressure is constant and the fluid is in free motion. This motion is purely turbulent and is described by quasi-linear equations of parabolic type. We prove existence and uniqueness of the classical solution of the initial-value problem in a bounded region and in the entire space. Necessary conditions of solvability are given. Steady-state equations of fluid motion are derived. Applied problems involving fluid flow in a pipe, onset of turbulence, and existence of Taylor vortices in a solid torus are solved.