
Existence and Smoothness of Solution of NavierStokes Equation on R^{3}DOI: 10.4236/ijmnta.2015.42008, PP. 117126 Keywords: NavierStokes Equation, Millennium Problem, Nonlinear Dynamics, Fluid, Physics Abstract: NavierStokes equation has for a long time been considered as one of the greatest unsolved problems in three and more dimensions. This paper proposes a solution to the aforementioned equation on R^{3}. It introduces results from the previous literature and it proves the existence and uniqueness of smooth solution. Firstly, the concept of turbulent solution is defined. It is proved that turbulent solutions become strong solutions after some time in NavierStokes set of equations. However, in order to define the turbulent solution, the decay or blowup time of solution must be examined. Differential inequality is defined and it is proved that solution of NavierStokes equation exists in a finite time although it exhibits blowup solutions. The equation is introduced that establishes the distance between the strong solutions of NavierStokes equation and heat equation. As it is demonstrated, as the time goes to infinity, the distance decreases to zero and the solution of heat equation is identical to the solution of NS equation. As the solution of heat equation is defined in the heatsphere, after its analysis, it is proved that as the time goes to infinity, solution converges to the stationary state. The solution has a finite τ time and it exists when τ → ∞ that implies that it exists and it is periodic. The aforementioned statement proves the existence and smoothness of solution of NavierStokes equation on R^{3} and represents a major breakthrough in fluid dynamics and turbulence analysis.
